curriculum standards for the state of qatar - international · pdf file ·...

Curriculum Standards for the State of Qatar

Mathematics: Grades K to 12

Developed for the Education Institute by CfBT

2 | Qatar mathematics standards | Introduction © Supreme Education Council 2004


Contents

Foreword 5

1 Introduction 7 The new curriculum standards 9

About the mathematics standards 12

2 Scope and sequence charts for mathematics 21 Grades K to 4 23

Grades 5 to 9 27

Grades 10 to 12 (foundation) 35

Grades 10 to 12 (advanced) 39

3 Mathematics standards 51 Kindergarten 53

Grade 1 59

Grade 2 69

Grade 3 79

Grade 4 91

Grade 5 105

Grade 6 119

Grade 7 133

Grade 8 149

Grade 9 165

Grade 10 (foundation) 181



Grade 10 (advanced) 233

Grade 11 (advanced) 257

Grade 12 (advanced: quantitative methods) 281

Grade 12 (advanced: mathematics for science) 305

4 Appendix 321 Sources used for international comparisons for mathematics 323


Foreword

Qatar’s Comprehensive Education Reform Initiative

These Curriculum Standards lie at the heart of ‘Education for a New Era’, Qatar’s education reform initiative. They draw on international expectations of what students should know, understand and be able to do at each stage of their schooling, as well as on the current best practices in Qatar’s public schools.

The standards focus on the content essential for preparing students to be engaged and productive citizens. Critical thinking, enquiry and reasoning are emphasised in all grades to ensure that students develop the ability to work creatively, think analytically and solve problems.

The standards have been developed for the Education Institute of the Supreme Education Council by an international team of curriculum experts, guided by the staff of the Institute. Working groups of local teachers and curriculum specialists have helped to ensure that the standards reflect Qatari values and culture, and are relevant to the needs and interests of Qatari students.

Principals and teachers should find these standards to be an excellent resource on which to base their planning, teaching and assessment. Quality instruction and high levels of scholastic achievement are crucial to the future success of our youth and our nation.

Sabah Esmail Al-Haidoos Acting Director, Education Institute


1 Introduction


The new curriculum standards

Background

The curriculum standards are goals for students’ learning. They set out what Qatari students should know, understand and be able to do by the end of each grade from Kindergarten to Grade 12. They are intended to help each Independent School to plan its curriculum, to guide writers of teaching and learning materials and to inform the design of tests and examinations.

The standards cover four subjects: Arabic, English, mathematics and science. Today’s students must have a high degree of competence in all these subjects, and must benefit from the best opportunities in higher education, if they are to compete successfully in the worldwide economy. At the same time, they need to develop a feeling of identity with their country and a deep understanding of Qatar’s traditions, achievements and culture.

The standards are based on the premise that all Qatari students are capable of learning successfully and of achieving high levels of performance. The standards are aligned to expectations in those countries that demand the most of their students, including those that achieve excellent results in international tests.

Students who master the knowledge, concepts and skills specified in the standards, and who perform well in the new national tests based on the standards, should also score well in international qualifying examinations for admission to first-class universities around the world.

Structure of the curriculum standards

The standards are presented in Section 3 of this document. They are preceded by scope and sequence charts. The charts give an overview of the standards and summarise the content for each grade. They are intended to help schools to see at a glance how students’ knowledge, understanding and skills should progress from one grade to the next.

The standards from Kindergarten to Grade 9 are for all students. Each set of standards is structured into strands as follows:

• Arabic word knowledge, listening and speaking, reading and writing

• English words, grammar, reading and writing, listening and speaking

• Mathematics reasoning and problem solving, number and algebra, geometry and measures, data handling

• Science scientific enquiry, physical processes, life science, materials, Earth and space

In each strand, the standards are grouped into topics. The strands and topics for a given subject do not necessarily involve equal amounts of teaching time, and are not necessarily given equal emphasis in the national tests. The approximate proportion of the overall teaching and assessment time that should be devoted to them is discussed in more detail on page 14 of this introduction.

The standards for Grades 10 to 12 have two different pathways. All students should continue to study all four subjects but not necessarily to the same depth or level. The standards for these grades are therefore at two levels, foundation and


advanced. Students, with advice from teachers and parents, will choose a course based on one or the other, but not both. This is to allow teachers to prepare students according to the students’ individual needs and aspirations.

The foundation standards include revision and consolidation of standards for earlier grades as well as some new material. Advanced standards include the foundation standards, so that advanced students are taught the foundation standards before moving on to more in-depth study (for example, more challenging problems, more demanding critiques of texts, more complex topics).

Key performance standards

The standards in Section 3 are numbered to make them easy to reference. The numbers in shaded rectangles, e.g. 1.2, identify the key performance standards. These are the standards that should be taught to all students and that all students should master. The national tests are based on these standards.

The remaining non-key standards represent extension or enrichment objectives for the more able, or consolidation objectives for those who learn more slowly. As such, they will not necessarily be taught to all students. Some of them are key standards in an earlier or higher grade.

The shaded panels at the start of each strand are summaries of the key standards for that strand. They should be useful to teachers when they make informal assessments of students’ progress and when they are reporting to parents.

Illustrations of the standards

The standards aim to provide enough detail to give teachers a clear understanding of:

• what students should learn by the end of each grade in each of the four subjects;

• the emphasis to be placed on higher order skills, such as critical thinking, enquiry, reasoning and problem solving.

The standards are illustrated with examples to show what is expected. The examples should help teachers to interpret the standards and to develop lesson plans, learning resources and assessment materials.

Notes in the margin are also intended to help teachers to interpret the standards. For example, a margin note might add further detail about what to include or not include in the teaching of a standard, or refer to a linked standard.

Spelling, units of measurement, numbers and equations

The mathematics and science standards are offered in English and Arabic versions. The Arabic standards are provided in Arabic, and the English standards in English.

The spelling conventions adopted in the English standards, and the English versions of the mathematics and science standards, are based on standard British English.

The units of measurement and abbreviations used are the Système Internationale (SI) units. They are therefore written in their internationally recognised form: for example, the word centimetre and its abbreviation cm are used in both the Arabic and English language versions of the standards. Thin spaces, not commas, are used to separate groups of three digits in numbers with more than four digits: for example, 48 746, not 48,746.


In both the Arabic and English versions of the standards, numbers and symbols, including chemical symbols, are written using Roman or Greek script. Mathematical and chemical equations and formulae are presented from left to right.

Schools will need to make their own decisions about spelling conventions and how numbers, symbols, equations and formulae are presented to students in lessons and learning resources, taking account of the language of instruction and the age of the students.

Using the standards in schools

The standards are intended to help schools to meet students’ learning needs but are not in themselves a ‘syllabus’. They can be used in schools in several different ways. For example:

• Principals and senior managers might use the standards to help them to plan, resource, monitor and evaluate the school curriculum, and to support the development of school policies for teaching and learning.

• Subject leaders and teams of teachers who are teaching the same subject can use the standards to develop schemes of work or programmes of study, classroom resources and assessment materials.

• Teams of teachers, from one or more schools, who are teaching the same grade can use the standards to develop integrated programmes for the grade.

• Individual teachers can use the standards to help them to plan lessons for a class, set learning objectives for students, assess and monitor students’ progress, and report to parents.

Decisions about how individual teachers might best teach the standards are left to schools. There are no prescribed teaching methods. Teachers should choose appropriate methods in line with their school’s policies. For example, they may use direct instruction and explicit teaching, or may guide students to learn through experimentation and discovery. Traditional methods might have a place but students will need a much wider range of active experiences if they are to solve problems, think creatively, enquire, criticise and evaluate.

Equally, there are no prescribed textbooks or other teaching and learning resources. Schools can select a variety of support materials from the very best that exist. It is unlikely that any single set of textbooks could address the standards adequately. Teachers should use their creativity in developing their own resources, or in finding and introducing published resources, choosing those that are culturally relevant and interesting to students.

The standards do not imply that each student in a grade is necessarily at the same level of achievement. Teachers should exercise flexibility and imagination when they are planning lessons based on the standards. They should move high-achieving students forward or give extra support to students who are experiencing difficulty with more basic content. They should make their own professional judgement about these matters based on their knowledge of the students in their classes. This is particularly important for students whose mother tongue is other than Arabic (whose academic potential may be underestimated), gifted and talented students, and students with special educational needs.

Similarly, there are no prescribed methods of assessing and recording students’ progress. The only requirement on Independent Schools is that every student participates in the national tests based on the standards. Each school can design and implement its own assessment policy to help teachers to plan and improve teaching and learning.


About the mathematics standards This section provides more detail about the mathematics standards. It covers:

• the aims of the mathematics standards;

• recommended teaching time for mathematics in the whole school curriculum;

• the strands of the mathematics standards:

– the recommended proportion of mathematics teaching time to be given to each strand;

– a broad description of the scope of each strand; • the place of information technology in the mathematics standards.

The aims of the mathematics standards

The overall aims or goals of the mathematics standards are that students should:

• become mathematical problem solvers capable of solving familiar and unfamiliar problems in mathematical, real-world and other subjects’ settings;

• develop proficiency in mental and written calculations, algebraic manipulation and other techniques, including visualisation and geometric imagery;

• use calculators and computers to support and develop their mathematical work;

• communicate mathematical ideas accurately and precisely through natural and mathematical language such as numbers, signs, symbols, diagrams, graphs and mathematical terms;

• select and use different types of reasoning, including different kinds of proof;

• make connections between different mathematical ideas and between mathematics and other subjects;

• appreciate the variety of ways that mathematical ideas are applied and used in modern society;

• appreciate the contribution of mathematicians to the history and development of mathematics.

Recommended teaching time for the mathematics standards

The mathematics standards assume that the approximate time needed to teach them in a school year of around 900 teaching hours (excluding special events, tests and examinations) is as follows:

• in Grades 1 to 6 16% to 20% of the overall teaching time of 900 hours about 145 to 180 hours per year

• in Grades 7 to 9 13% to 15% of the overall teaching time of 900 hours about 120 to 135 hours per year

• in Grades 10 to 12, students choose either a foundation or an advanced course: advanced course 15% to 16.5% of the overall teaching time of 900 hours about 135 to 150 hours per year foundation course around 10% to 12% of the overall teaching time of 900 hours about 90 to 110 hours per year


Some models for the number of mathematics lessons per week might be:

In a timetable of five 60-minute lessons per day (25 lessons per week, for 36 weeks)

Grades 1 to 6 5 mathematics lessons per week

Grades 7 to 9 3–4 lessons per week (e.g. 7 per fortnight)

Grades 10 to 12 (advanced courses) 4 lessons per week

Grades 10 to 12 (foundation course) 3 lessons per week

In a timetable of six 50-minute lessons per day (30 lessons per week, for 36 weeks)


Grades 7 to 9 4–5 lessons per week (e.g. 9 per fortnight)

Grades 10 to 12 (advanced courses) 5 lessons per week

Grades 10 to 12 (foundation course) 3–4 lessons per week (e.g. 7 per fortnight)

In a timetable of seven 50-minute lessons per day (35 lessons per week, for 31 weeks)


Grades 7 to 9 5 lessons per week

Grades 10 to 12 (advanced courses) 5–6 lessons per week (e.g. 11 per fortnight)


In a timetable of seven 45-minute lessons per day (35 lessons per week, for 34 weeks)


Grades 7 to 9 5 lessons per week

Grades 10 to 12 (advanced courses) 5–6 lessons per week (e.g. 11 per fortnight)


The strands of the mathematics standards

The mathematics standards are organised in strands as follows: • reasoning and problem solving; • number and algebra, plus calculus from Grade 10; • geometry and measures, which includes trigonometry from Grade 9; • data handling, which is separated into statistics and probability from Grade 10

onwards.

The advanced standards for mathematics for Grade 12 have two options: quantitative methods, to support the social sciences and economics, and mathematics for science. This allows schools to offer advanced students a choice of one or the other, depending on the students’ career needs and the other subjects they are studying.

The content strands (number and algebra, geometry and measures, and data handling) cover the knowledge, skills and understanding that students need to master. The reasoning and problem solving strand involves the higher order,


generic skills needed to solve mathematical problems. The reasoning and problem solving standards apply to each of the content strands. They should be taught and assessed at the same time as the content strands.

The identification of strands does not mean that each should be allocated an equal amount of teaching time or that each will have the same emphasis in the national tests. The approximate proportion of the overall teaching and assessment time that should be devoted to each of them is set out below.

Number and algebra

Geometry and measures

Data handling

Kindergarten 60% 35% 5%

Grades 1 to 6 60% 30% 10%

Number and algebra1

Number Algebra

Geometry and measures2

Data handling

Grade 7 30% 25% 27.5% 17.5%

Grade 8 25% 30% 27.5% 17.5%

Grade 9 15% 40% 27.5% 17.5% 1 Since the relative proportions of algebra change throughout Grades 7 to 9, these are shown separately. 2 Includes trigonometry in Grade 9.

Foundation Number and algebra

Geometry, measuresand trigonometry

Probability and statistics

Grade 10 55% 30% 15%

Grade 11 55% 30% 15%

Grade 12 50% 25% 25%

Advanced Number, algebra and calculus

Geometry, measuresand trigonometry


Grade 10 55% 30% 15%

Grade 11 55% 30% 15%

Grade 12 (quantitative)

40% – 60%

Grade 12 (for science)

75% 25% –

The reasoning and problem solving strand

The standards in the reasoning and problem solving strand should be integrated at all times with the standards from each of the other three strands so that students make the connections in their learning associated with success in mathematics.

The proportion of each content strand devoted to reasoning and problem solving should increase steadily from grade to grade. By Grades 10 to 12, problem solving and reasoning should be fostered in almost all lessons as an essential part of learning mathematics and developing understanding of its applications.

Mathematical reasoning includes explaining mathematical facts, solving problems and puzzles, understanding mathematical procedures and formulae, and justifying and giving reasons for results. The logical thinking involved in reasoning leads to mathematical proof, which lies at the heart of the subject.


Problem solving, applications of mathematics and the use of reasoning should feature throughout the teaching of each topic. A problem can serve as an introduction, to assess students’ prior knowledge or to set a context for the work. It can be used to provide motivation for acquiring a skill or it can be set as a class activity or as homework towards the end of a topic, so that students use and apply the mathematics that they have been taught.

Students need to be able to select the relevant mathematics required to solve a problem and to recognise that an idea that they meet in one strand of mathematics can be applied in another. The standards can be incorporated in:

• problems and applications that extend content beyond what has just been taught;

• familiar (routine) and unfamiliar (non-routine) problems in a range of numeric, algebraic and geometric contexts, some with a unique solution and some with several possible solutions;

• activities that develop short chains of deductive reasoning and concepts of proof in algebra and geometry;

• opportunities to sustain thinking through more complex problems.

Number and algebra

Number

The standards for number centre on the number system (systems for writing whole numbers and decimals, fractions, negative numbers, rational numbers, percentages, numbers in standard form, and so on), properties of numbers, calculations, and estimation and approximation.

The number standards for the earlier grades stress the importance of mental calculation methods. This does not mean that students are not taught written methods, but the balance between mental and written methods is in favour of mental methods. Alongside their oral work students learn to read, interpret and complete statements such as 5 + 8 = or 13 = + 5, and then to record the results of mental calculations using a horizontal format like 43 – 8 = 35.

The approach in the primary grades builds on the use of the number line, first with numbers marked and then a blank line on which numbers are inserted, to record steps in calculations such as 47 + 26 or 243 – 77.

The progression in the standards towards written methods is crucial, since standard

written methods are based on steps which are done mentally and which need to be secured first. For example, the calculation of 487 + 356, done by the traditional method in columns, requires the mental calculations 7 + 6 = 13, 8 + 5 + 1 = 14 and 4 + 3 + 1 = 8, while a division calculation such as 987 ÷ 23 can involve mental experiment with multiples of 23 before the correct multiple is chosen.

The written column methods that are introduced gradually are efficient procedures for calculating which can be generally applied. Countries that are most successful at


teaching number avoid the premature teaching of column methods in order not to jeopardise the development of mental calculation strategies. The bridge from recording informal part-written, part-mental methods to learning written column methods begins in earnest only when students can reliably add or subtract two-digit numbers mentally, usually when they are about eight years old.

When they have reached the stage of working out more complex calculations using written methods, students still need to practise and refine their mental calculation strategies. The standards include the development of estimation skills in all aspects of calculation, but particularly in multiplication and division.

The standards include explicit teaching of the skills needed to use a calculator efficiently. The use of a basic calculator is introduced in Grade 5 and of a scientific calculator in Grade 7.

The connections between fractions, decimals, percentages, ratio and proportion start to be established from Grade 6, particularly the equivalence between fractions, decimals and percentages. In Grades 7, 8 and 9 the standards extend to positive and negative numbers and, in particular, to fractions and their representations as terminating or recurring decimals.

After calculation, the application of proportional reasoning is the most important aspect of elementary arithmetic. Proportionality underlies key aspects of number, algebra, geometry and measures, and data handling. It is also central in applications of mathematics in subjects such as science, technology, geography and art. The study of proportion begins in Grade 4 but it is in Grades 7, 8 and 9 where secure foundations are established.

Problems involving proportion are often solved by informal methods, particularly when the numbers involved are easy to deal with mentally. The standards include methods that can be applied generally. For example, the unitary method is useful for solving problems involving proportion, and multiplicative methods involving fractions or decimals are useful for solving percentage problems.

In the higher grades, the standards for number include study of prime numbers, rational and irrational numbers, real numbers and complex numbers.

Algebra

The algebra standards that are introduced gradually from Grade 5 are generalised arithmetic. The origins are in the art of manipulating sums, products and powers of numbers. The same rules are seen to hold true for all numbers, of whatever type, so it becomes possible to generalise the rules with letters in place of numbers.

The algebra standards include work on equations, formulae and identities, and sequences, functions and graphs. Students’ developing skill in solving equations depends on their ability to add, subtract, multiply and divide directed numbers and to simplify expressions. They learn that expressions and equations can be manipulated in their own right according to given rules and conventions. The algebraic use of brackets – multiplying out brackets, and the inverse process of factorising – also develops from examples with numbers.

Many applications of algebra involve finding an appropriate formula to model a situation, for example, to generate the general term of a sequence. The standards include justifying formulae from physical patterns as well as from number sequences. This extends to ‘proving’ a solution, not just illustrating or verifying particular solutions.

The standards relating to functions and graphs should be taught and learned in tandem. In Grades 7 and 8, the main emphasis is on linear functions and their


graphs. A graphics calculator, or graph plotting software, has an important role in these standards since it helps students to learn from exploring problems.

In Grades 7 to 9, direct proportion is viewed as a linear relationship of the form y = mx. The graphical representation of this equation helps students to visualise ideas such as rate of change and gradient. The algebraic representation of a proportion (e.g. a : b = c : d or a/b = c/d) underpins a general method for solving problems involving proportional reasoning.

In the higher grades, algebra is the vital tool for solving equations and inequalities and using them as mathematical models of real situations. In Grades 10 to 12, a range of functions is introduced, with emphasis on their use in a wide variety of applications. Mathematical modelling becomes a key feature.

In the highest grades, the advanced standards introduce calculus to further the study of functions and of mathematical applications. Students solve problems by translating everyday language into the abstract language of algebra, and vice versa.


In the early grades, the geometry standards include knowing and using geometrical properties that students have discovered intuitively from practical work before they can prove them analytically. The aim from Grade 7 is for students to use and develop their knowledge of shapes and space to support short chains of geometrical reasoning. These chains of reasoning are essential steps towards the recognition of congruent triangles and the geometric proofs that are introduced in higher grades.

The geometry standards include using instruments accurately, drawing shapes and appreciating how they can link together, for example, in tessellations. In the earlier grades, constructions develop from drawing on grids. Later, these extend to constructions that involve protractors, and then to standard constructions which, in principle, are exact and which use only compasses and a straight edge. Geometrical reasoning can show students why a construction method works, such as the method to construct the perpendicular bisector of a line segment.

The standards related to transformations are best achieved through practical work which helps students to understand the topic more fully, and through visualisation problems such as: ‘When a triangle is rotated through 180° about the mid-point of one side, what shape is formed by the combination of the original and final triangles?’ Linking the geometry standards at appropriate points to the properties of Islamic patterns can help students to appreciate their culture more fully.

Proportionality arises in enlargement by different scale factors. Scaling has a wide range of applications, for example, in maps, plans and scale drawings. Similar figures have sides or dimensions that are in proportion. Recognition of the similarity of all circles leads to an understanding that the circumference is directly proportional to the diameter, while awareness of the similarity of triangles with the same angles leads to an understanding of trigonometry.

The measurement standards involve understanding concepts of length, mass and capacity, and estimating and making measurements using standard metric units. They progress to the recognition of the relative sizes of units such as millimetres, centimetres, metres and kilometres, and conversions between them. Compound measures, such as speed and density, are introduced in Grade 8. The measurement standards include appreciation of the imprecision of measurement and recognition of the accuracy to which measurements can be stated.

Standards linked to perimeter, area and volume extend to a range of shapes, including rectangles, triangles, parallelograms, circles, cuboids and prisms. The theorem of Pythagoras and trigonometric ratios are introduced in Grade 9.


The geometry standards in Grades 10 to 12 develop understanding of geometric proof, particularly in relation to circle geometry and circular functions. They extend the study of trigonometry and its applications, including geometric representations for the geometry associated with position and distance on the surface of the Earth. The foundation standards include a basic introduction to vector geometry as an alternative way of specifying two- and three-dimensional space. Vector geometry is developed further in the advanced standards of mathematics for science.

Data handling

The data handling standards are described by the cycle shown in the diagram.

The standards for data handling are best achieved in the context of real statistical

enquiries using situations of interest and relevance to students, including appropriate links to other subjects.

As students get older, the standards for data handling ensure that they engage with large sets of real data from a wider range of sources and contexts. The standards include the collection and use of primary data from questionnaires or results of an experiment, and secondary data from published sources, including reference books, ICT databases and the Internet. ICT is also used to process and represent the data.

Real data present problems that ‘textbook’ or contrived data can bypass, such as the accuracy of recording or how to deal with ambiguous data. The sizes of numbers can be problematic, either because they are large or, in the case of a pie chart, because they are not factors of 360. The time to process and represent real data is likely to be greater than with textbook examples but by using real data students gain useful skills that can be transferred to other investigations.

Quantitative probability is introduced from Grade 7. The data handling cycle shown above applies also to probability. Two aspects are developed in tandem: probability as the proportion of successes in an experiment, and probability derived from theoretical considerations. As students compare practical and theoretical results, they should sense that as the number of trials becomes very large the proportion of successes converges to the theoretical probability.

In the higher grades, the data handling strand includes definitions and calculations of various averages for central tendency and measures of spread. It extends to the analysis of data by classification, frequency distributions and other graphical displays, taking account of randomness and bias in sampling. The strand has important connections with number and algebra in the study of permutations and combinations, and Pascal’s triangle.

The probability of compound events is introduced in the highest grades. Theoretical probabilistic models are introduced in the foundation standards for Grades 10 to 12 and developed further in the advanced standards for quantitative mathematics. The latter also introduce correlation and regression, and hypothesis testing.


The place of information technology in mathematics

Information and communication technology (ICT) is a powerful tool in mathematics. Used appropriately, it helps students to develop better knowledge and skills and to make a successful transition to the world beyond school.

ICT does not replace the need for all students to master mental and written calculation skills and other mathematical techniques. The standards introduce these skills first before the use of technology is introduced and developed.

Possible opportunities for students to use ICT as a tool for doing mathematics are highlighted in margin notes next to the standards. Some of the standards refer directly to the students’ ability to use ICT. These opportunities allow them to extend their mathematical skills beyond what is possible with print resources. The focus is on using technology as a tool for learning mathematics, not as an end in itself. For example:

• Basic calculators, when used to carry out tedious calculations, allow students to focus on the strategies needed to solve a problem. They can also be a useful support when students are learning arithmetic to help them to grasp ideas such as place value. Scientific calculators are used for more complex calculations, including work with trigonometric ratios and some statistical work.

• Graphics calculators and function graph plotters can help students to learn about graphing – they are not just an efficient tool to use when students have mastered the basics. They allow students to see instantly the graphs of complex functions and to explore the impact of changes. They can also display statistical diagrams, and can be used to develop algebraic identities.

• Programming languages such as Logo allow students to explore angle, direction and distance, and simple transformations, particularly rotations.

• Dynamic geometry systems (DGS) are interactive tools for constructing geometric diagrams. They allow students to see figures in two- and three-dimensional space and to experiment with the effects of transformations.

• Spreadsheets and databases allow students to enter data, compile statistics and produce a range of graphs, charts and tables. The students can decide on the most appropriate way to display the data and can readily make and test hypotheses about the impact of a change in the data set.

• The Internet can be used as a source of relevant data. It also allows students to exchange ideas and test hypotheses with a far wider audience.

• Independent learning systems, which tutor students and provide instruction, and number games and ‘drill-and-practice’ software can help to reinforce students’ basic skills and mathematical techniques.

Language and communication in mathematics

Communication of mathematical ideas assumes competence in a number of language standards. The language required by the mathematics standards is generally introduced at the same or at a lower grade in the language standards. Mathematics teachers should work closely with their colleagues teaching Arabic and English to ensure continuity, particularly where the medium of instruction is English.

The following are brief descriptions of the strands of the English standards and how they relate to the mathematics standards.

• Word knowledge Knowledge of words and parts of words is especially useful for increasing students’ understanding of mathematical texts and word problems. Systematic vocabulary development is essential. It is important to teach not only


content-specific words, such as square, ratio or median, but also academic word families. Words taught in English lessons should be recycled, practised and applied in the mathematics classroom. This is especially important when students are working with word problems, are presenting information and are constructing surveys and questionnaires. The complete list of recommended words for Grades K to 9 included in an appendix of the English standards is a good resource for identifying non-content-specific words that could prove difficult for students.

• Listening and speaking The mathematics classroom should support students’ developing skills, such as asking for information, giving advice, agreeing and disagreeing. Appropriate modelling and rephrasing strategies should be encouraged in the classroom. Mathematics teachers should introduce and reinforce the listening strategies taught in English (for example, the speaking and listening strand for English Grade 8 gives a comprehensive list of listening strategies). The mathematics standards expect students to present and explain methods and solutions from the earliest grades. Strategies for speaking learned in English can be reinforced and built on. Active paired and group work to discuss ideas are as essential in mathematics as they are in language lessons.

• Reading and writing Good literacy skills are crucial in all subjects, including mathematics. It is essential that mathematics teachers promote reading and writing in mathematics lessons. Reading strategies are important for accessing information in mathematics textbooks and for understanding and identifying key information in mathematical problems. Mathematics teachers should:

– give students opportunities to apply reading for meaning with increasing attention to information-gathering strategies, inference and deduction;

– draw attention to the typical language and organisational features of mathematical resources such as textbooks;

– encourage students to use self-monitoring and self-correction strategies when reading expository texts.

Writing is an important means of communicating mathematical and scientific information. Mathematics teachers should include in their lessons both informal writing (journals, learning logs) and more formal report writing involving the complete writing process of planning, revising and editing.

Acknowledgements

The examples that illustrate the mathematics standards include some that are based on or are adapted from:

• released mathematics items used in the international tests TIMSS (1993) or TIMSS-R (1998), or example items for TIMSS 2003, all published by the International Association for the Evaluation of Educational Achievement, The Hague, Belgium;

• examples from the Mathematics for Education and Industry (MEI) syllabus of the General Certificate of Education Advanced Level examinations administered by Oxford Cambridge and RSA Examinations (OCR);

• examples drawn from the National Curriculum tests for England.

The generosity of the MEI Office, OCR and the Qualifications and Curriculum Authority for England for agreeing that these examples may be used is acknowledged gratefully.


2 Scope and sequence charts for mathematics

23 | Qatar mathematics standards | Scope and sequence chart | Kindergarten to Grade 4 © Supreme Education Council 2004

Mathematics scope and sequence chart: Kindergarten to Grade 4

Kindergarten (optional) Grade 1 Grade 2 Grade 3 Grade 4

REASONING AND PROBLEM SOLVING

To be applied in other strands

• Simple practical problems

• Representing problems and explaining solutions with pictures or objects

• Routine and non-routine problem solving in all strands

• Representing problems and explaining solutions with objects, numbers, symbols or simple diagrams

• Describing a relationship


• Representing problems and explaining solutions with words, numbers, symbols or diagrams

• Explaining methods and a simple line of reasoning



• Explaining methods and reasoning




• Checking results

NUMBER AND ALGEBRA

Whole numbers

• Counting to 10, then 20

• Conservation of number

• Zero; numerals 0 to 20

• Comparing numbers to 20; ordering on number line

• Number notation and place value to 100

• Ordinal numbers

• Comparing 2-digit numbers; ordering on number line

• Number notation and place value to 1000

• Comparing 3-digit numbers; ordering on number line

• Number notation and place value to 10 000

• Comparing 4-digit numbers; ordering on number line; use of >, =, <

• Number notation and place value in whole numbers

• Rounding to nearest 10 or 100

Whole-number calculations

• Addition/subtraction of small numbers to solve practical problems (no written recording)

• Concepts of addition and subtraction; +, – and = signs

• Relationship between addition and subtraction, and missing-number problems

• Addition/subtraction facts to 10, and pairs of numbers with a total of 20

• Addition/subtraction of 0

• Commutative law (+)

• Doubles of 1 to 10

• Special cases of mental addition /subtraction (e.g. near doubles)

• Addition/subtraction of 1-digit number/multiple of 10 (not crossing tens), horizontal recording

• Relationship between addition and subtraction, and missing-number problems

• Associative law (+ and –)

• Addition/subtraction facts to 20

• Adding three 1-digit numbers

• Doubles of 1 to 15, and corresponding halves

• Mental addition/subtraction of multiples of 1, 10 or 100 to/from a 2- or 3-digit number (including crossing tens)

• Written addition/subtraction of two-digit numbers (TU ± TU, TU + TU + TU)

• Concepts of multiplication and division; × and ÷ signs; commutative law for ×

• Adding three 1-digit numbers mentally

• Mental addition/subtraction of any pair of 2-digit numbers

• Written addition/subtraction of numbers with up to four digits (HTU ± HTU, ThHTU ± HTU, ThHTU ± ThHTU)

• Multiplication facts to 10 × 10

• Multiplication by 0 and 1

• Distributive law

• Mental multiplication/division (TU by U, simple cases, not crossing 100, no remainders)

• Multiplication/division by 10 and 100 (whole-number answers)

• Written multiplication/division (TU × U, TU ÷ U)

• Mental and written addition and subtraction of whole numbers

• Multiplication and division facts to 10 × 10

• Mental multiplication/division (TU by U, no remainders, crossing 100)

• Multiplication/division by multiples of 10 and 100, using factors (whole-number answers)

• Written methods for: – multiplication/division

(HTU × U, HTU ÷ U) – multiplication

(TU × TU, HTU × TU)

• Estimating answers to calculations to check accuracy



Whole-number calculations (continued)

• Simple word problems involving whole numbers, money or measures

• Relationship between multiplication and division

• Multiplication/division within 2, 5 and 10 times tables

• One-step word problems involving whole numbers, money or measures

• Remainders

• Relationship between multiplication and division, and missing-number problems

• One- and two-step word problems involving whole numbers, money or measures

• Missing-number problems involving inverse operations

• One- and two-step word problems involving whole numbers, money, measures, simple decimals or fractions

Number patterns

• Counting in 2s, 5s, 10s • Counting in 3s, 4s

• Counting in 10s, 100s

• Counting in 6s, 7s, 8s, 9s

• Counting in 1000s

• Odd and even numbers

• Factors, multiples, primes (less than 100)

• Number pairs related by rule

Money • Coin and bank-note recognition and equivalence

• Finding total and working out change (up to QR 100, riyals only)

• Paying exact number of riyals using smaller bank-notes

• Counting half and quarter riyal coins up to QR 2

• Finding total (up to QR 500); working out change

• Finding cost of number of items given unit cost (within 2, 5 and 10 times tables)

• Finding total (up to QR 10 000), working out change

• Finding cost of number of items given unit cost, or unit cost given total cost and number of items (whole numbers of riyals)

• Reading and writing money in decimal form

• Conversion of riyals in decimal notation to dirhams, and vice versa

Decimals • Decimal notation and place value to 2 places, including comparing and ordering

• Rounding to whole number or 1 decimal place

• Mental methods (simple cases)

• Written methods for: – addition and subtraction of

decimals with up to 2 places – multiplication of decimals with

up to 2 places by 1-digit number

• Multiplication and division of decimals with up to 2 places by 10 and 100

• Estimating answers to check reasonableness



Fractions • Halves and quarters • Adding half and quarter riyal coins up to QR 2 (see Money)

• Simple fractions of shapes and numbers

• Equivalence of simple fractions

• Comparing and ordering simple fractions

• Simple equivalent fractions

• Fraction and decimal equivalents for one half, one quarter, three quarters, one tenth and one hundredth

• Addition and subtraction of two fractions with same or related denominators

• Mixed numbers and improper fractions

• Product of proper fraction and whole number

GEOMETRY AND MEASURES

Geometry • Shape recognition: recognising common 2-D and 3-D shapes in the environment, e.g. circle, square, triangle; cube, cone

• Recognition of types of lines (straight, curved, etc.)

• Vocabulary of position and direction

• Using shapes to make models, pictures, patterns

• Shape recognition: circle, square, rectangle, triangle

• Extending/completing repeating patterns according to shape, size, position or colour

• Straight/curved lines, flat/curved surfaces

• Shape recognition: sphere, cube, cuboid, cone, cylinder, pyramid (square-based), pentagon, hexagon, octagon

• Completing geometric patterns according to one or two of shape, size, colour, orientation

• Concept of angle: whole, half and quarter turns

• Eight-point compass

• Identifying right angles in 2-D shapes

• Shape recognition: equilateral, isosceles, right-angled triangles

• Regular and irregular polygons

• Simple symmetrical patterns

• Perpendicular and parallel lines

• Angles greater or less than a right angle; ordering angles by estimating relative size

• Side and angle properties of squares, rectangles, parallelograms

• Line symmetry

Constructions • Using a ruler to measure and draw lines to nearest centimetre

• Using a ruler to measure and draw lines to nearest millimetre

• Constructing squares and rectangles

Length, mass/weight, capacity

• Direct comparison: – length – mass – capacity

• Measurement in non-standard units:

– length – mass – capacity

• Estimations and measurement in single unit:

– length (m or cm) – mass (kg or g) – capacity (l or ml)

• Knowing relationships between km, m, cm; kg and g; l and ml

• Using mixed units to: – record measurements of

m and cm kg and g l and ml

– convert between cm and m, and m and cm

• Using decimal notation to write: – measurements of m and cm,

or cm and mm – conversions of m to cm, cm to

m, cm to mm, mm to cm

• Reading scales with increasing accuracy

• Simple problems involving scale



Time • Using vocabulary such as day, week, morning, afternoon, evening; today, tonight, yesterday, tomorrow, birthday, Eid

• Days of week

• Telling time (o’clock)

• Reading time (o’clock, half past)

• Ordering familiar events

• Days of week and months of year

• Reading time to 5 minutes; notation 6:35

• Calculating a time interval less than 1 hour (multiple of 5 minutes, not crossing the hour)

• Reading time to minute

• Measuring and comparing time in minutes or seconds using a stopwatch

• Using simple timetables

• Calculating a time interval less than 1 hour (multiple of 5 minutes, crossing the hour) or more than 1 hour (whole number of hours)

• Converting weeks to days, and vice versa

• Months of year, using a calendar, Hijri calendar

• Calculating a time interval less than 1 hour (in minutes) or more than 1 hour (multiple of 15 minutes)

Area and perimeter

• Concept of perimeter

• Perimeter of squares and rectangles

• Perimeter of regular polygons (whole-number sides)

• Concept of area

• Comparing two areas using unit squares

• Calculating area of squares and rectangles using formula

• Perimeter of irregular polygons (whole-number sides)

• Rectangles with same area, different perimeter, or same perimeter, different area

• Simple problems involving the area and/or perimeter of squares and rectangles

DATA HANDLING

Data handling • Sorting common objects (single criterion)

• Collecting, representing and interpreting data in:

– simple pictograms (symbol represents 1 complete unit)

• Carroll diagram (single criterion)

• Collecting, representing and interpreting data in:

– pictograms (symbol represents 2, 5 or 10 complete units)

• Recording data systematically (e.g. in a tally chart)

• Representing and interpreting data in:

– bar charts with simple scales (e.g. intervals of 2, 4, 5 or 10)

• Recording and reading data in simple two-way tables

• Representing and interpreting data in:

– bar charts with scales (e.g. intervals of 2, 4, 5, 10, 20, 100)

– Carroll diagrams (two criteria)

27 | Qatar mathematics standards | Scope and sequence chart | Grades 5 to 9 © Supreme Education Council 2004

Mathematics scope and sequence chart: Grades 5 to 9

Grade 5 Grade 6 Grade 7 Grade 8 Grade 9


To be applied in other strands

• Routine/non-routine problems in number, algebra, geometry, measures, data handling

• Representing and interpreting problems


• Deciding when to use a calculator and interpreting display

• Giving examples to match a general statement


• Representing and interpreting problems; changing from one representation to another


• Checking results for reasonableness

• Searching for all possibilities

• Identifying simple patterns and explaining them in words


• Representing and interpreting problems; changing from one representation to another

• Explaining methods and reasoning, including reasoning to establish truth of a statement

• Presenting and explaining solution in context of problem

• Making generalised statements using words and symbols


• Representing and interpreting problems in different ways

• Choosing techniques

• Explaining methods and reasoning, including reasoning to establish truth of a statement

• Using step-by-step reasoning in geometry, or using symbols, and finding counter-examples

• Finding alternative solutions


• Representing and interpreting problems

• Breaking problems into parts

• Choosing techniques

• Explaining methods and reasoning, developing simple proofs and identifying exceptional cases

• Recognising degree of accuracy

• Finding alternative solutions

NUMBER

Whole numbers and decimals up to 3 places

• Place value to 3 decimal places; comparing, ordering, rounding

• Mental addition of several 1-digit numbers

• Multiplication/division of whole numbers and decimals by 10, 100, 1000, and their multiples

• Mental multiplication/division of whole numbers using factors

• Quotient as fraction or decimal

• Written methods for: – multiplication/division

(ThHTU × U, ThHTU ÷ U HTU × TU, HTU ÷ TU)

– addition/subtraction of decimals with up to 3 places

– multiplication/division of decimals with up to 2 places by 1-digit whole number

• Notation for recurring decimals

• Multiplying and dividing dividend and divisor by the same number

• Order of operations (not brackets)

• Mental multiplication/division of simple decimals and whole numbers

• Written methods for multiplication/division of:

– whole numbers with up to 4 digits by a 2-digit whole number

– decimals with up to 2 places by 2-digit whole number

• Ordering and symbols: <, ≤, >, ≥, =, ≠

• Rounding; ≈ symbol

• Order of operations; brackets

• Multiplication/division of whole numbers and decimals by 0.1, 0.01, 0.001

• Mental addition, subtraction, multiplication and division of simple decimals

• Written methods for: – multiplication/division of

decimals with up to 2 places by decimal with up to 2 places

• Understanding principles of commutative, associative and distributive laws and their use in mental and written calculations

• Mental, and written calculations in appropriate cases

• Significant figures

• Rounding, including measures, to a given degree of accuracy



Calculator • Using a basic calculator

• Calculator methods, including inverse operations

• Real-life word problems involving numbers or measures

• Estimating/checking answers

• Further use of basic calculator

• Calculator methods, including inverse operations


• Estimating, checking and rounding answers, including interpreting display in context of problem

• Using a scientific calculator

• Calculator methods, including inverse operations, combined operations, using the memory


• Estimating, checking and rounding answers

• Further use of scientific calculator, including function keys



• Further use of scientific calculator, including function keys



Properties of numbers and integers, powers and roots

• Squares of numbers 1 to 12, corresponding square roots

• Properties of numbers less than 100: multiple, factor, prime

• Prime numbers, factors, prime factors, multiples, HCF, LCM

• Divisibility by 2, 3, 4, 5, 6 and 10

• Representing and ordering directed numbers (words, models, number line)

• Four operations with directed numbers

• Positive integer powers of whole numbers; +ve and –ve square roots of squares to 144; √ sign

• Use of scientific calculator to calculate positive integer powers and square roots

• Positive integer powers of whole numbers and decimals; cube roots of perfect cubes to ±216; 3√ sign

• Use of scientific calculator to evaluate powers and roots

• Use of calculator or spreadsheet to find approximate values of square roots of whole numbers to 100

• Use of laws of indices for multiplication and division of small integer powers, including positive and negative integer powers of 10

• Standard form, including interpreting a calculator display

• Use of calculator or spreadsheet to find approximate values of roots

Fractions • Equivalent fractions

• Expressing a fraction in its simplest form

• Comparing and ordering related and unrelated fractions using diagrams (denominators up to 12)

• Finding fractions of quantities

• Adding and subtracting simple fractions

• Finding equivalent fractions, including simplest form

• Converting fractions to decimals and vice versa

• Comparing and ordering unrelated fractions by using division to convert to a decimal (with/without a calculator)

• Addition and subtraction of proper fractions, different denominators

• Multiplication of a proper fraction by a fraction

• Division of a proper fraction by a whole number

• Simple mental calculations

• Converting fractions to decimals and vice versa, recognising any terminating decimal as a fraction

• Comparing and ordering unrelated fractions by:

– converting to decimals (with/without calculator)

– forming common denominator

• Relating operations to models

• Calculating mentally with proper fractions in simple cases

• Addition and subtraction of mixed numbers, including combined operations

• Multiplication and division of one mixed number by another

• Using fraction key on calculator

• Four operations for fractions, including combined operations and brackets

• Calculating mentally with fractions in appropriate cases

• Using a scientific calculator to calculate with fractions



Percentages • Concept of percentage; simple equivalent fractions, decimals, percentages

• Calculating a given percentage of a quantity (simple cases)

• Converting fractions and decimals to percentages and vice versa

• Calculating a given percentage of a quantity

• Finding the whole, given a percentage part (simple cases)

• Expressing one quantity as a percentage of another

• Simple mental calculations

• Estimating and calculating a given percentage of a quantity

• Expressing one quantity as a percentage of another

• Finding the whole, given a percentage part

• Calculating percentage increases or decreases

• Calculating mentally with percentages in simple cases

• Using a scientific calculator to calculate with percentages

• Calculating mentally with percentages in appropriate cases

• Simple interest

Ratio and proportion

• Ratio of two quantities (simple cases); dividing a quantity in a given ratio

• Equivalent ratios, including lowest terms

• Direct proportion, including simple currency conversions

• Relating ratios to fractions

• Ratio of two or more quantities: equivalent ratios, including lowest terms

• Comparing ratios

• Direct proportion, unitary method

• Using a scientific calculator to calculate with ratios and proportions

• Ratio problems involving similar triangles (see Geometry)

• Scale drawings (see Geometry)

ALGEBRA

Sequences, functions and graphs

• Simple rules to generate sequences and ordered pairs, and to find missing terms

• Coordinates in first quadrant

• Term-to-term and position-to-term rules to generate sequences and find missing terms

• Graphs of simple linear functions (first quadrant)

• Numeric or geometric sequences (term-to-term or position-to-term rules); formulating nth term in words or symbols

• Coordinates in all four quadrants

• Graphs of y = mx + c, on paper and using ICT (all quadrants)

• Use of graphics calculator to generate sequences and to plot graphs

• Formulating nth term of sequence using symbols

• Definition of a function

• Representing functions; identifying intercepts and intervals where function increases or decreases

• Graphs of proportional/linear functions from practical situations, including conversion and distance–time graphs


• Gradients of lines given by y = mx + c, and of lines parallel and perpendicular to y = mx + c

• Simultaneous linear equations, graphical methods, on paper and using ICT

• Graphs of simple quadratic and cubic functions, on paper and using ICT; approximate solution of quadratic equations by graphical methods

• Graphs arising from practical situations, including non-linear functions




Expressions, equations and formulae

• Simple formulae using words, including unit conversions

• Using letters to represent numbers

• Algebraic expressions in one variable:

– simplification, excluding brackets

– evaluation by substituting integers

• Construction and evaluation of simple formulae using letters

• Algebraic expressions and formulae:

– simplification, collection of like terms, removal of brackets


– formulation of simple algebraic expressions or equations to model a situation

• Linear equations or inequalities (integer coefficients, unknown on one or both sides, brackets)

• Algebraic expressions and formulae:

– simplification of sum, difference, product and quotient of expressions


– formulation of linear expressions or equations to model a situation

– factorisation by removing common factors

– addition and subtraction of simple algebraic fractions with integer denominators

• Linear equations or inequalities (coefficients as decimals or fractions with integer denominators)

• Algebraic expressions: – evaluation by substituting

integers of polynomials such as 3x2 + 4 and 2x3

– expansion of (x ± b)(x ± d), where b and d are integers

– squaring a linear expression; factorisation of corresponding quadratic expression

– factorisation by removing common factors of: ax ± ay ax + bx + ay + by

– identities such as a2 – b2 = (a + b)(a – b) (a + b)2 = a2 + 2ab + b2

– addition, subtraction, multiplication and division of simple algebraic fractions

• Changing the subject of a simple formula

• Fractional linear equations (numerical and linear algebraic denominators, brackets), simple cases

• Simultaneous linear equations, two unknowns, by elimination and by substitution

• Quadratic equations of the form a2x2 – b2y2 = 0 x2 ± 2ax + a2 = 0 by recognising the factors

• Approximate solutions using ICT of equations such as x3 + x = 20



GEOMETRY

Geometry • Rotation symmetry

• Labelling conventions for angles

• Relating angles in degrees to whole, half and quarter turns

• Ordering angles

• Simple cases of finding unknown angles involving:

– angles in a straight line and at a point

– angle sum of triangle – angle properties of equilateral,

isosceles and right-angled triangles, and rectangles and squares

• Nets of cube and cuboid

• Simple reflections and rotations

• Labelling conventions for geometric figures

• Angle sum of quadrilateral

• Finding unknown angles in involving:


– vertically opposite angles – angle sum of triangle – angle and side properties of

equilateral, isosceles and right-angled triangles, rectangles, squares and parallelograms (excluding properties related to diagonals)

• Identification of properties of 2-D shapes, including angle, side, diagonal and symmetry properties of quadrilaterals

• Identification of properties of 3-D shapes; planes of symmetry; nets of cubes, cuboids, prisms, pyramids, cones

• Finding unknown angles involving:


– vertically opposite angles – corresponding, alternate and

supplementary angles – angle sum and exterior angle

of triangle – angle and side properties of

triangles – angle, side and diagonal

properties of squares, rectangles, parallelograms, rhombuses

– angle bisectors and perpendicular bisectors

• Recognising similar triangles

• Sum of interior angles and sum of exterior angles of a polygon; calculation of unknown angles of a polygon

• Reflection and rotation symmetry of 2-D shapes, on paper and using ICT; symmetrical properties of triangles, quadrilaterals and regular polygons

• Deducing properties in a given plane figure using known properties of angles and shapes

• Transformations: using coordinates in all four quadrants, on paper and using ICT, to identify and draw in relation to a simple 2-D shape:

– a reflection in lines parallel to or at 45° to the axes

– a rotation about the origin, or a vertex of the shape, or a mid-point of a side, through multiples of 90°

– a translation parallel to one of the axes

– an enlargement (whole-number scale factor)

– the combination of two simple transformations

• Visualising and describing 3-D shapes

• Congruent triangles and similar triangles:

– establishing congruence or similarity

– finding unknown sides and angles of similar or congruent figures

• Deducing properties in a given plane figure using known properties of angles and shapes

• Using Pythagoras’ theorem and its converse to solve 2-D problems (see Trigonometry)

• Given the coordinates of A and B, finding:

– the mid-point of line segment AB

– the length of line segment AB – the point that divides line

segment AB in a given ratio

• Transformations: – identifying line of reflection,

centre and angle of rotation, centre of enlargement and scale factor in simple cases

– enlargement of a simple 2-D shape (positive fractional scale factor)



Constructions • Coordinates in first quadrant (see Sequences, functions and graphs)

• Using ruler, set square and protractor to:

– measure and draw line segments and angles in degrees (acute)

– construct 2-D shapes on grids

• Using ruler, set square and protractor to:

– measure and draw angles (acute, obtuse, reflex)

– draw perpendicular and parallel lines

– construct rectangles and squares

– construct triangles, given SAS or ASA

• Using ruler, set square, protractor and compasses to:

– draw parallel and perpendicular lines

– draw circles and arcs, including Islamic patterns

– construct angle bisectors and perpendicular bisectors

– construct simple plane shapes from given data

• Using ICT to explore constructions

• Using ruler, protractor and compasses to construct geometrical figures, including scale drawings

• Using ICT to explore constructions

• Plans and elevations

MEASURES

Measurement • Conversions between: km, m, cm and mm, kg and g, l and ml, using decimal notation

• Converting units of time

• 24-hour clock; calculating time intervals in hours and minutes

• Reading timetables

• Using rulers, weighing scales, measuring cylinders, stopwatches

• Reading numbers from scales and dials

• Choosing units to estimate measurements

• Compound measures: rate and speed

• Appreciation of precision of measurement – upper and lower bounds

• Conversions of measures, including area and volume, and units of speed (e.g. km/h to m/s)

• Further problems involving speed

• Using calculator for time or speed calculations

• Compound measures: density

Mensuration • Perimeter and area of: – squares and rectangles – shapes formed from two

squares or rectangles, e.g. L-shape, T-shape

• Finding side of square given area or perimeter, and similar problems with rectangles

• Volume of cube, cuboid (cm3, m3)

• Perimeter and area of triangles and parallelograms

• Surface area of cuboid

• Finding edge of cube given volume or surface area, and similar problems with cuboids

• Equivalence of 1000 cm3 and 1 litre, or 1 cm3 and 1 ml, and volume of liquid in cuboid container

• Perimeter and area of triangles, rectangles, parallelograms, trapeziums and related shapes

• Parts of a circle: circumference and area of circle

• Volume and surface area of cubes and cuboids and related solids

• Recall equivalence of 1 litre and 1000 cm3

• Volume and surface area of right prisms and cylinders

• Area of plane shapes related to circles



Trigonometry • Sine, cosine, tangent (right-angled triangle)

• Pythagoras’ theorem (excluding proof)

DATA HANDLING

Statistics • Collecting and organising discrete data, grouping as appropriate

• Construction and interpretation of:

– frequency diagrams for grouped discrete data

– line graphs – Venn diagrams

• Collecting and tabulating data

• Using ICT to generate graphs, charts and tables, including pie charts

• Interpretation of graphs and charts, including pie charts

• Mean, median, mode and range to describe a simple distribution

• Collecting, classifying and tabulating data

• Construction (on paper and using ICT) and interpretation of:

– tables – pictograms – bar charts – line graphs – pie charts (no calculation of

angles)

• Different representations of same data set

• Mean, median, mode, range

• Sources of error in data collection

• Designing data collection sheet

• Collecting data from secondary sources


– frequency tables and diagrams (grouped continuous data, equal intervals)

– pie charts

• Comparison of two distributions

• Identifying questions that can be answered by statistical methods


– scatter diagrams – lines of best fit; basic

understanding of correlation – frequency diagrams, choosing

appropriate equal class intervals

• Calculating mean, median, range of small sets of discrete or continuous data

• Finding modal class and estimating mean, median and range for grouped data

Probability • Informal language of probability

• Probability scale

• Equally likely outcomes for single events

• Total probability of all mutually exclusive outcomes is 1

• Estimating probabilities from experimental data, understanding that different outcomes may result from repeated experiments

• Calculating theoretical probabilities

• Systematic listing of all possible outcomes

• Comparing experimental and theoretical probabilities in simple cases

• Relative frequency as an estimate of probability, including grouped data

• P(A) + P(B) (A and B mutually exclusive)

• Further comparison of experimental and theoretical probability in different contexts

35 | Qatar mathematics standards | Scope and sequence chart | Grades 10 to 12 foundation © Supreme Education Council 2004

Mathematics scope and sequence chart: Grades 10 to 12 foundation Grade 10 Grade 11 Grade 12


To be applied to all strands

• Routine and non-routine problem solving

• Modelling real-world applications

• Identifying and using connections between mathematical topics

• Breaking complex problems into smaller tasks; using problem solving strategies to set up and solve relevant equations and to perform appropriate calculations

• Developing and explaining short chains of reasoning, using correct notation and terms; generalising; generating mathematical proofs; identifying exceptional cases

• Solving problems systematically; conjecturing possibilities; synthesising, presenting, interpreting and criticising mathematical information; working to expected degrees of accuracy

• Recognising when to use ICT; using ICT efficiently





• Developing and explaining chains of reasoning, using correct notation and terms; generalising; generating mathematical proofs; identifying exceptional cases







• Developing and explaining longer chains of reasoning, using correct notation and terms; generalising; generating mathematical proofs; discussing exceptional cases



NUMBER AND ALGEBRA

General • Real-world numerical and algebraic applications

• Linking algebraic reasoning to geometrical ideas

• Contributions to mathematics by Islamic scholars

• Real-world numerical and algebraic applications


• Real-world numerical and algebraic applications


Number • Powers, nth roots; exact calculations with surds; standard form

• Calculations with any real numbers, including mental calculations; multiplicative nature of proportional reasoning; using, forming, simplifying and comparing ratios; percentage calculations, including percentage of a percentage, inverse percentage

• Compound interest problems; limiting value of compounding interest more and more frequently

Set theory • The number sets: (reals), (integers), (natural numbers), (rationals); irrational numbers

• Common set theory symbols: E (universal set), ∅ (null set), ∈ (is a member of), ∉ (is not a member of), ∀ (for all), brace notation; A ∪ B (union of sets); A ∩ B (intersection of sets); A′ (complement of set A); A ∪ A′ = E ; further Venn diagrams


Grade 10 Grade 11 Grade 12


• Algebraic generalisations for odd and even numbers; vocabulary related to primes, factors, multiples, divisors

• Sequences from term-to-term and position-to-term definitions; simple growth patterns; Pascal’s triangle; arithmetic sequences; sum of first n consecutive integers

• Function, domain and range; functional relationships between related variables; graphs of simple functional relationships from familiar contexts; recognising when a graph represents a function; function notation y = f(x)

• Translation of ‘y is proportional to x’ into equation y = kx representing a straight line through the origin, with gradient k; common examples of direct proportion

• Plotting straight line equations y = mx + c; m as gradient of line and c as intercept on y-axis; establishing Cartesian equations of lines from appropriate information; conditions for two straight lines to be parallel or perpendicular, including special cases

• Finding point of intersection of two lines: exactly using algebraic methods, approximately using graphical methods; interpreting solutions in physical contexts

• Quadratic functions of the form y = ax2 + c; their graphs, intercepts with the coordinate axes, axis of symmetry and coordinates of the maximum or minimum point; modelling with quadratic functions

• Geometric sequences and their sums; evaluating any recurring decimal as an exact fraction

• Further work on straight lines, including implicit form ax + by + c = 0

• Regions of linear inequality; solution of simple problems

• Further functional relationships between related variables and their graphs

• Tangent line at a point on the graph of a function, its gradient and its interpretation in physical applications

• Approximate solutions of ax2 + bx + c = 0 from graph of y = ax2 + bx + c

• Further quadratic functions y = ax2 + bx + c; their graphs, intercepts with coordinate axes, axis of symmetry and coordinates of maximum or minimum point; when such functions are increasing, decreasing or stationary; modelling situations with quadratic functions

• Translation of ‘y is proportional to x2’ into equation y = kx2 representing a parabola

• Translation of ‘y is inversely proportional to x’ into equation y = k/x, where x ≠ 0 and x- and y-axes are each asymptotes to the curve; examples of inverse proportionality

• Recurrence relations in physical applications

• Using physical contexts to plot and interpret graphs of linear, quadratic and cubic functions, reciprocal function y = k/x (x ≠ 0), sine and cosine functions, modulus function and a range of simple non-standard functions; using a graphics calculator to show approximate solutions to physical problems requiring location and interpretation of intersection points of two or more graphs

• Inverse functions of simple functions

• Addition, subtraction, multiplication of two functions; division in simple cases

• Composite functions and notation y = f(g(x)); deconstruction of composite functions into constituent functions

• Transformation of y = f(x) to y = f(x) + a, y = f(x – a), y = af(x), y = f(ax), and interpretations as translation in the y-direction, the x-direction, and stretch or compression in the y- and x-direction respectively

• Exponential growth and decay and associated graphs y = ax, where a > 0; using graphics calculator to plot graphs of exponential function, ex, and natural logarithm function, ln x; solution of equation y = ax and its use in problems; log function (base 10) on a calculator


• Working with symbols; distinguishing expressions, equations, formulae and identities; recognising that rules of algebra generalise the rules of arithmetic

• Brackets and correct order of precedence of operations when performing numerical or algebraic calculations

• Combining numeric or algebraic fractions; multiplying combinations of monomial and binomial expressions

• Simplifying numeric and algebraic fractions; rationalising denominators of fractions containing surds

• Solving any linear equation with one unknown, and a pair of simultaneous linear equations

• Rearranging formulae connecting at least two variables

• Further working with symbols in expressions, equations, formulae and identities

• Multiplication of combinations of monomial, binomial and trinomial expressions; linear factors of quadratic expressions; difference of two squares

• Substitution of an expression into another formula, including a linear into a quadratic expression

• Exact solutions of quadratic equations by factorisation, by completing the square, by using the quadratic formula

• Solution set of two simultaneous equations, one linear and one quadratic; physical problems modelled simultaneously by two such functions

• Further working with expressions, equations, formulae and identities

Using ICT • Using ICT to explore sequences, functions and graphs • Using ICT to explore sequences, functions and graphs • Using ICT to explore sequences, functions and graphs




General • Real-world geometrical applications • Real-world geometrical applications • Real-world geometrical applications

Using ICT • Using dynamic geometry systems (DGS) to explore pattern, similarity, congruence and constructions, and to conjecture geometric properties and theorems

• Using dynamic geometry systems (DGS) to explore pattern, similarity, congruence and constructions, and to conjecture geometric properties and theorems

• Using dynamic geometry systems (DGS) for further exploration of pattern, similarity, congruence and constructions, including plans and elevations

Geometry • Angles at a point, angles on a straight line, alternate and corresponding angles; formal arguments to establish congruency of two triangles; using congruency of two triangles to generate further knowledge

• Similarity of two triangles and other rectilinear shapes; preservation of shape and angles, but not of size, in a similarity transformation; ratio of lengths of sides and areas of similar figures; ratio of volume of a scale model to volume of the actual object

• Regular polygons and their interior and exterior angles

• Constructions using straight edge and compass

• Simple loci, including those arising in physical situations

• Points of intersection of straight line with circle

• Relevant vocabulary associated with a circle; proof of standard circle theorems

• Transformations of rectilinear figures using combinations of translations, rotations about centre of rotation, enlargements about centre of enlargement, and reflections about a line; positive, negative and fractional scale factors in enlargements; Islamic patterns

• Maps and scale drawings


Trigonometry • Solution of triangles using standard trigonometric ratios

• Proof of Pythagoras’ theorem; using Pythagoras’ theorem to find the distance between two points, to solve right-angled triangles and to set up the Cartesian equation of a circle of radius r, centred at the origin of an xy-coordinate system

• Sine rule and cosine rule; triangle problems in two and three dimensions; area of triangle using 1⁄2 ab sin C

• Using Pythagoras’ theorem to find Pythagorean triples; set up the Cartesian equation of a circle of radius r, centred at point (α, β); unit circle x2 + y2 = 1 and plots of graphs of circular functions sin θ and cos θ for any angle θ°, where 0° ≤ θ° ≤ 360°; the identity sin2 θ° + cos2 θ° ≡ 1 for any angle θ° and simple related identities; simple problems modelled by circular functions

Measures and mensuration

• Perimeters and areas of rectilinear and circular shapes, and volumes of rectilinear solids, cones, cylinders and spheres

• Bearings

• SI units

• Compound measures, including those that reinforce links with science and technology

• Radian measure for calculating sector areas and arc lengths

• Latitude, longitude and great circles and their use in solving problems relating to position, distance and displacement on the Earth’s surface

• Further compound measures, including those that reinforce links with science and technology

• Further compound measures, including those that reinforce links with other disciplines, including the social sciences

• Approximation methods to calculate the area of an irregular two-dimensional flat surface and the volume of a prism with a constant, but irregular-shaped, cross-section



Vectors • Vectors: position vector and translation as a vector displacement; knowing that the vector displacement depends only on the starting point and the finish point, and not on intermediate steps

• Addition and subtraction of two vectors in up to three dimensions and the corresponding vector diagrams

• Scalar product of two vectors; multiplication of a vector by a scalar; magnitude and direction of a vector; vector displacement and velocity; unit vectors and components

• Solution of physical problems using vectors

PROBABILITY AND STATISTICS


• Using statistical data collected from samples to make inferences about the population as a whole

• Distinguishing qualitative from quantitative data, and discrete from continuous data

• Measures of central tendency

• Simple histograms

• Scatter diagrams between two random variables associated with common contexts; elementary qualitative discussion of correlation, including positive and negative correlation; drawing a line of best fit by eye through the scatter points when there appears to be some correlation

• Representative samples; random and biased samples; location of sources of bias

• Planning surveys and questionnaires to collect meaningful primary data from samples to make estimates of, or test hypotheses about, quantities or attributes characteristic of the population as a whole

• Using secondary data from published sources, including the Internet


• Histograms, frequency and (relative) frequency distributions and associated distributions; using grouped continuous data

• Stem-and-leaf diagrams and box-and-whisker plots; making inferences and drawing conclusions from analysis of data in a range of situations

• Random variables

• Empirical probability (relative frequency) of a particular value; using simple mathematical models to calculate theoretical probability of particular outcome for a random variable; knowing that probability values lie between 0 and 1

• Risk as probability of occurrence of an adverse event; risk in everyday situations

• Sum of probabilities for all outcomes of mutually exclusive and exhaustive events is 1; when two events A and B are mutually exclusive, probability of A or B, P(A ∪ B), is P(A) + P(B); two events A and B are independent if the probability of A and B occurring together, P(A ∩ B), is P(A) × P(B)

• Tree diagrams for representing and calculating the probabilities of compound events when events are independent or when one is conditional on another

• Measures of spread

• Trends over time and moving averages

• Simulation using random numbers to model simple situations, including waiting times

Using ICT • Using a calculator with statistical functions for analysing large data sets

• Using ICT packages to produce statistical tables and graphs

• Using a calculator with statistical functions for analysing large data sets




39 | Qatar mathematics standards | Scope and sequence chart | Grades 10 to 12 advanced | Quantitative methods © Supreme Education Council 2004

Mathematics scope and sequence chart: Grades 10 to 12 advanced: Quantitative methods Grade 10 Grade 11 Grade 12














• Developing and explaining chains of reasoning, using correct notation and terms; generalising; generating mathematical proofs; discussing exceptional cases




• Modelling statistical situations



• Developing and explaining chains of reasoning, using correct notation and terms confidently; generalising; discussing exceptional cases and statistical outliers



NUMBER AND ALGEBRA




• Real-world numerical and algebraic applications • Real-world numerical and algebraic applications


• Calculations with any real numbers, including mental calculations; multiplicative nature of proportional reasoning; using, forming, simplifying and comparing ratios; percentage calculations, including percentage of a percentage, inverse percentage; compound interest

• Evaluating recurring decimal as an exact fraction (see also Geometric sequences)

• Limiting value of compounding interest more and more frequently

• Laws of indices and of logarithms in any base

• The number e and the series expansion for ex

• Using appropriate function keys on a scientific calculator to work with indices, logarithms and exponentials





• Solution sets of equations and inequalities


• Algebraic generalisations from odd and even numbers

• Sequences from term-to-term and position-to-term definitions; simple growth patterns; Pascal’s triangle; arithmetic sequences; sum of first n consecutive positive integers

• Geometric sequences and their sums; recurring decimal as an example of infinite geometric series


• Translating ‘y is proportional to x’ into equation y = kx, representing a straight line through the origin with gradient k; common examples of direct proportion; quadratic proportion

• Plotting straight line equations y = mx + c; m as gradient of line and c as intercept on y-axis; establishing Cartesian equations of lines from appropriate information; conditions for two straight lines to be parallel or perpendicular, including special cases; implicit form ax + by + d = 0



• Regions of linear inequality; simple quadratic inequalities

• Quadratic functions of the form y = ax2 + bx + c; their graphs, intercepts with the coordinate axes, axis of symmetry and coordinates of the maximum or minimum point; modelling with quadratic functions

• Finite and infinite convergent geometric sequences

• Sums of first n squares and cubes; further work on sequences, series; recurrence relations; arrangements; sigma notation

• Binomial theorem and binomial coefficients

• Combinations and permutations

• Odd and even functions; symmetry properties


• Approximate solutions of ax2 + bx + c = 0 from graph

• Translating ‘y is inversely proportional to x’ into equation y = k/x, where x ≠ 0 and x- and y-axes are each asymptotes to the curve; examples of inverse proportion

• Using physical contexts to plot and interpret graphs of linear, quadratic, cubic, reciprocal, sine and cosine functions, modulus function and simple non-standard functions; using a graphics calculator to find approximate solutions to physical problems


• Composite functions; notation y = f(g(x)); deconstruction of composite functions into constituent functions

• Transformation of y = f(x) to y = f(x) + a, y = f(x + a), y = af(x), y = f(ax), and interpretations as translation, stretch or compression

• Exponential growth and decay and associated graphs y = ax, where a > 0; using graphics calculator to plot graphs of exponential function, ex, and natural logarithm function, ln x; solution of equation y = ax and its use in problems

• Further algebraic manipulation, factorisation and simplification

• Partial fractions

• The remainder theorem and factor theorem

• Key features of functions: polynomial functions rational functions exponential and logarithm functions circular functions modulus function

• Symmetry of functions

• Inverse functions

• Transforming and combining functions






• Combining numeric or algebraic fractions

• Multiplication of combinations of monomial, binomial and trinomial expressions, including squares of linear binomial expressions; linear factors of quadratic expressions; factorisation of difference of two squares





• Substituting values in formulae and expressions; substituting an expression into another formula

• Solving equations and inequalities associated with all of the above functions

• The quadratic equation revisited: use of discriminant; number of real roots


Limits • Exploring limits in various contexts

• Defining the tangent and its slope at a point on a curve

Differential calculus

• First principles definition of a derivative and its interpretation as a rate of change

• Derivative of the standard functions given above

• Second order derivatives


• Use of the derivative to analyse behaviour of functions

• Derivatives of combinations of functions

• Use of the derivative to maximise or minimise a function

Integral calculus

• Integration as the ‘inverse’ of differentiation

• The indefinite integral

• The definite integral; its interpretation as an area function and as a measure or probability under a distribution curve

• Use of trapezium rule as approximation to definite integral

• Integrals of xn (including n = –1) and ekx

• Some techniques of integration

Using ICT • Using ICT to explore number and algebra • Using ICT to explore number and algebra • Using ICT to explore number and algebra




General • Real-world geometrical applications, including Islamic patterns

• Real-world geometrical applications




• Similarity of two triangles and other rectilinear shapes; preservation of shape and angles, but not of size; ratio of lengths of sides and areas of similar figures; ratio of volume of a scale model to volume of the actual object


• Proof that the perpendicular from the centre of a circle to a chord bisects the chord, and that two tangents from an external point to a circle are of equal length





• Transformations of rectilinear figures using combinations of translations, rotations about centre of rotation, enlargements about centre of enlargement, and reflections about a line; positive, negative and fractional scale factors in enlargements



• Proof of Pythagoras’ theorem; using Pythagoras’ theorem to find Pythagorean triples, distance between two points, set up the Cartesian equation of a circle of radius r, centred at point (α, β); unit circle x2 + y2 = 1 and graphs of circular functions sin θ and cos θ for any angle θ°, where 0° ≤ θ° ≤ 360°; simple problems modelled by circular functions


• Using Pythagoras’ theorem to show that sin2 θ° + cos2 θ° ≡ 1 for any angle θ°; simple related identities; problems modelled by circular functions

• Plots of graphs of circular functions sin θ and cos θ for any angle θ, using radian measure (see also Sequences, functions and graphs)




• SI units


• Radian measure; sector areas and arc lengths

• Bearings; latitude, longitude and great circles and their use in solving problems relating to position, distance and displacement on the Earth’s surface


• Further work on rates and other compound measures, including those that reinforce links with other disciplines, science, technology and the social sciences



















• Scatter diagrams between two random variables associated with common contexts; elementary qualitative discussion of correlation, including positive and negative correlation; drawing a line of best fit through the scatter points when there appears to be some correlation

• Empirical probability (relative frequency) of a particular value; using simple mathematical models to calculate theoretical probability of a particular outcome for a random variable; knowing that probability values lie between 0 and 1


• Sum of probabilities for all outcomes of mutually exclusive and exhaustive events is 1; when two events A and B are mutually exclusive, probability of A or B, P(A ∪ B), is P(A) + P(B); two events A and B are independent if the probability of A and B occurring together, P(A ∩ B), is P(A) × P(B); P(A ∩ B) is P(A) × P(B | A) when B is conditional on A




• Classifying data

• Statistical measurement and measurement scales

• Sampling from a population and associated vocabulary

• Random numbers and their uses in sampling, measurement and calculating probabilities

• Random numbers and their uses in simulations


• Collecting and organising data: surveys and questionnaires statistical charts and distributions measures of central tendency and of spread linear coding for mean and variance percentiles, quartiles and range

• Laws of probability for single and combined events; sample space; tree diagrams

• Expectation of a random variable

• Properties of probability distributions

• Discrete probability distributions, including the binomial and the Poisson distributions; the Poisson distribution as an approximation to the binomial distribution

• The normal distribution and its properties

• The normal distribution as an approximation to the binomial distribution and to the Poisson distribution

• Linear correlation; correlation coefficient; least squares regression; rank correlation coefficient

• Making inferences from data: level of significance confidence interval null and alternative hypotheses chi-squared test







45 | Qatar mathematics standards | Scope and sequence chart | Grades 10 to 12 advanced | Mathematics for science © Supreme Education Council 2004

Mathematics scope and sequence chart: Grades 10 to 12 advanced: Mathematics for science Grade 10 Grade 11 Grade 12














• Developing and explaining chains of reasoning, using correct notation and terms; generalising; generating mathematical proofs; discussing exceptional cases




• Modelling real-world applications; using error bounds



• Developing and explaining longer chains of reasoning, using logical implication and using correct notation and terms confidently; generalising; generating mathematical proofs and disproving by counter-example; discussing exceptional cases



NUMBER AND ALGEBRA




• Real-world numerical and algebraic applications • Real-world numerical and algebraic applications


• Calculations with any real numbers, including mental calculations; multiplicative nature of proportional reasoning; using, forming, simplifying and comparing ratios; percentage calculations, including percentage of a percentage, inverse percentage; compound interest

• Evaluating recurring decimal as an exact fraction (see also Geometric sequences)

• Limiting value of compounding interest more and more frequently

• Laws of indices and of logarithms in any base

• The number e

• Using appropriate function keys on a scientific calculator to work with indices, logarithms and exponentials

• Introduction to complex numbers and Argand diagram





• Solution sets of equations and inequalities


• Algebraic generalisations from odd and even numbers

• Sequences from term-to-term and position-to-term definitions; simple growth patterns; Pascal’s triangle; arithmetic sequences; sum of first n consecutive positive integers

• Geometric sequences and their sums; recurring decimal as an example of infinite geometric series


• Translating ‘y is proportional to x’ into equation y = kx, representing a straight line through the origin with gradient k; common examples of direct proportion; quadratic proportion

• Plotting straight line equations y = mx + c; m as gradient of line and c as intercept on y-axis; establishing Cartesian equations of lines from appropriate information; conditions for two straight lines to be parallel or perpendicular, including special cases; implicit form ax + by + d = 0



• Regions of linear inequality; simple quadratic inequalities

• Quadratic functions of the form y = ax2 + bx + c; their graphs, intercepts with the coordinate axes, axis of symmetry and coordinates of the maximum or minimum point; modelling with quadratic functions

• Finite and infinite convergent geometric sequences

• Sums of first n squares and cubes; further work on sequences, series; recurrence relations; arrangements; sigma notation

• Binomial theorem and binomial coefficients


• Odd and even functions; symmetry properties


• Approximate solutions of ax2 + bx + c = 0 from graph

• Translating ‘y is inversely proportional to x’ into equation y = k/x, where x ≠ 0 and x- and y-axes are each asymptotes to the curve; examples of inverse proportion

• Using physical contexts to plot and interpret graphs of linear, quadratic, cubic, reciprocal, sine and cosine functions, modulus function and simple non-standard functions; using a graphics calculator to find approximate solutions to physical problems


• Composite functions; notation y = f(g(x)); deconstruction of composite functions into constituent functions

• Transformation of y = f(x) to y = f(x) + a, y = f(x + a), y = af(x), y = f(ax), and interpretations as translation, stretch or compression

• Exponential growth and decay and associated graphs y = ax, where a > 0; using graphics calculator to plot graphs of exponential function, ex, and natural logarithm function, ln x; solution of equation y = ax and its use in problems

• Further algebraic manipulation, factorisation and simplification

• Partial fractions

• The remainder theorem and factor theorem

• Binomial series


• Key features of functions: polynomial functions rational functions exponential and logarithm functions circular functions modulus function

• Composite functions; inverse functions

• Symmetry of functions

• Further transformations of functions

• Periodic functions

• Simple examples of curves given by parametric equations






• Combining numeric or algebraic fractions

• Multiplication of combinations of monomial, binomial and trinomial expressions, including squares of linear binomial expressions; linear factors of quadratic expressions; factorisation of difference of two squares





• Substituting values in formulae and expressions; substituting an expression into another formula

• Solving equations and inequalities associated with all of the above functions

• The quadratic equation revisited: use of discriminant; number of real roots


Limits • Exploring limits in various contexts

• Defining the tangent and its slope at a point on a curve

Differential calculus

• First principles definition of a derivative and its interpretation as a rate of change


• Higher order derivatives


• Use of the derivative to analyse behaviour of functions

• Derivatives of combinations of functions and of composite functions

• Use of the derivative in optimisation, in mechanics and in other physical examples, including geometry

• Use of the derivative in numerical approximations and in approximations to functions and locating roots of equations



Integral calculus

• Integration as the ‘inverse’ of differentiation

• The indefinite integral

• The definite integral; its interpretation as an area function

• Use of trapezium rule as an approximation to a definite integral

• Limits of integration and discussion of simple convergent integrals as an integration limit tends to infinity

• Integrals of the standard functions

• Some techniques of integration

• Use of integrals in mechanics and in the solution of a range of physical problems, including geometry

• Solution of simple differential equations and their use in mathematical modelling of situations

Using ICT • Using ICT to explore number and algebra • Using ICT to explore number and algebra • Using ICT to explore number and algebra


General • Real-world geometrical applications, including Islamic patterns

• Real-world geometrical applications • Real-world geometrical applications



• Using a dynamic geometry system (DGS) for further exploration of geometry and motion


• Similarity of two triangles and other rectilinear shapes; preservation of shape and angles, but not of size; ratio of lengths of sides and areas of similar figures; ratio of volume of a scale model to volume of the actual object


• Proof that the perpendicular from the centre of a circle to a chord bisects the chord, and that two tangents from an external point to a circle are of equal length





• Transformations of rectilinear figures using combinations of translations, rotations about centre of rotation, enlargements about centre of enlargement, and reflections about a line; positive, negative and fractional scale factors in enlargements





• Proof of Pythagoras’ theorem; using Pythagoras’ theorem to find Pythagorean triples, distance between two points, set up the Cartesian equation of a circle of radius r, centred at point (α, β); unit circle x2 + y2 = 1 and graphs of circular functions sin θ and cos θ for any angle θ°, where 0° ≤ θ° ≤ 360°; simple problems modelled by circular functions


• Using Pythagoras’ theorem to show that sin2 θ° + cos2 θ° ≡ 1 for any angle θ°; simple related identities; problems modelled by circular functions

• Plots of graphs of circular functions sin θ and cos θ for any angle θ, using radian measure (see also Sequences, functions and graphs)

• Trigonometric identities and equations


• SI units


• Radian measure; sector areas and arc lengths

• Bearings; latitude, longitude and great circles and their use in solving problems relating to position, distance and displacement on the Earth’s surface


• Further work on rates and other compound measures, including those that reinforce links with other disciplines, science, technology and the social sciences

• Further compound measures, especially those generated as a derivative or integral of some function used in context

• Areas and volumes by integration





• Vectors: velocity and acceleration vectors and their use in analysing motion

• Speed as the magnitude of velocity; unit vectors and components

• Scalar product of two vectors and its use to determine magnitudes and the angle between two vectors

• Vector equation of a straight line
















• Scatter diagrams between two random variables associated with common contexts; elementary qualitative discussion of correlation, including positive and negative correlation; drawing a line of best fit through the scatter points when there appears to be some correlation

• Empirical probability (relative frequency) of a particular value; using simple mathematical models to calculate theoretical probability of a particular outcome for a random variable; knowing that probability values lie between 0 and 1


• Sum of probabilities for all outcomes of mutually exclusive and exhaustive events is 1; when two events A and B are mutually exclusive, probability of A or B, P(A ∪ B), is P(A) + P(B); two events A and B are independent if the probability of A and B occurring together, P(A ∩ B), is P(A) × P(B); P(A ∩ B) is P(A) × P(B | A) when B is conditional on A








3 Mathematics standards

53 | Qatar mathematics standards | Kindergarten © Supreme Education Council 2004

Mathematics standards

Summary of students’ performance by the end of Kindergarten

Reasoning and problem solving

Students solve simple practical problems involving numbers, shapes or objects. They represent a mathematical problem and its solution with pictures or objects. They explain and give simple reasons for their methods.

Number

Students count, order, add and subtract numbers when solving practical problems involving up to 10 objects. They read and write numerals up to 10. They recognise up to five objects without counting.


Students use shapes to make models, pictures or patterns, or other shapes. They describe properties of lines, shapes, positions and directions using everyday language. They identify and name a cube, cone, circle, square and triangle. They make direct comparisons of the length, weight or capacity of two or three common objects. They describe time using words such as today, tomorrow, morning, afternoon, and the names of the days of the week.

Data handling

Students identify similarities or differences between two common objects. They sort a set of common objects using a criterion that they have chosen.

Content and assessment weightings for Kindergarten

The mathematics standards for Grades K to 9 are grouped into four strands: reasoning and problem solving; number and algebra; geometry and measures; and data handling.

The reasoning and problem solving strand cuts across the other three strands and should be integrated with them in teaching and assessments. For Kindergarten, about 30% of the teaching and assessment of each of the other three strands should be devoted to reasoning and problem solving.

For Kindergarten, the weightings of the three content strands relative to each other are as follows:

Number and algebra


Data handling

60% 35% 5%

The standards are numbered for easy reference. Those in shaded rectangles, e.g. 1.2, are the performance standards for all students.

The standards for Kindergarten should be reviewed and consolidated in Grade 1 since not all students will have experienced Kindergarten education.

K




By the end of Kindergarten, students solve simple practical problems involving numbers, shapes or objects. They represent a mathematical problem and its solution with pictures or objects. They explain and give simple reasons for their methods.

Students should:

1 Solve practical mathematical problems

1.1 Find their own way of solving simple practical problems involving numbers, shapes or objects.

Pack this set of solid shapes into this box. Try to fit them all in.

Find out which of these three containers will hold the most water.

Here are five buttons. How many more buttons are needed to make up a set of eight buttons?

1.2 Represent a problem and its solution with pictures or objects.

1.3 Explain orally their way of solving a problem and their reasons.

Number

By the end of Kindergarten, students count, order, add and subtract numbers when solving practical problems involving up to 10 objects. They read and write numerals up to 10. They recognise up to five objects without counting.

Students should:

2 Understand, order and use whole numbers up to 10 in real or play situations

2.1 Recite the sequence one, two, three, … up to ten.

2.2 Count up to 10 objects and say how many there are.

Count these buttons.

K

Key standards

Key performance standards are shown in shaded rectangles, e.g. 1.2.

Cross-references

Standards are referred to using the notation RP for reasoning and problem solving, NA for number and algebra, GM for geometry and measures and DH for data handling, e.g. standard NA 2.4.

Examples of problems

The examples of problems in italics are intended to clarify the standards, not to represent the full range of possible problems.

Kindergarten students are not expected to read or write answers to examples. It is assumed that the teacher asks these and similar questions orally.

Reciting numbers to 10

Include:

• counting backwards;

• continuing a sequence such as 3, 4, 5, …


2.3 Given a spoken number to 10, represent it using real objects.

Put out nine toy cars.

2.4 Know that if a set of objects is rearranged, then the number of objects remains the same (conservation of number).

2.5 Visualise up to five objects without counting.

2.6 Recognise zero when counting.

2.7 Read and write numerals to 10, including 0.

2.8 Write a number to indicate the number of objects in a set.

How many cubes are there? Write the number.

2.9 Given a written number to 10, represent it using real objects or drawings.

Draw 7 squares.

2.10 Compare two sets of objects and identify which set has more or less objects.

2.11 Identify the number that is 1 more/less than a given number to 10.

Add one more flower to this set of 7 flowers. How many flowers are there now?

2.12 Order numbers to 10 and position them on a number line.

What number goes in the box?

3 Understand, order and use whole numbers up to 20 in real or play situations

3.1 Recite the sequence one, two, three, … up to twenty.

3.2 Count up to 20 objects and say how many there are.

3.3 Read and write numerals 11 to 20.

3.4 Write a number to indicate the number of objects in a set of 11 to 20 objects.

3.5 Given a spoken or written number to 20, represent it using real objects or drawings.

3.6 Compare three sets of objects, and identify which set has more than, the same numbers as or fewer objects than another.

3.7 Identify the number that is 1 more/less than a given number to 20.

3.8 Order numbers to 20 and position them on a number line.

4 Understand, carry out and describe simple additions and subtractions in real or play problem situations

4.1 Relate addition to combining two groups of objects, and to counting on.

4.2 Relate subtraction to taking away a number of objects and finding how many are left, and to counting back.

Representing numbers

Exclude reading and writing numbers.


Include reading and writing numbers.

Comparing

Include the vocabulary more than, less than, fewer than.

Reciting numbers to 20

Include:

• counting backwards;

• continuing a sequence such as 13, 14, 15, …

Comparing

Include the vocabulary more than, less than, fewer than, and finding how many more and how many less.


4.3 Use real objects to determine answers to addition and subtraction problems involving numbers up to 10.

There are four cups on the table. Put two more cups on the table, one for Saif and one for Mosa. How many cups are on the table now?

There were five cubes in this box. Hessa has taken out two of the cubes. How many cubes are left in the box?

4.4 Build up knowledge of doubles of numbers to 5 (e.g. 4 and 4 is 8).

4.5 Use knowledge of doubles of numbers to 5 to develop other facts (e.g. use knowledge that 4 + 4 = 8 to work out 4 + 5 = 9, 4 + 3 = 7).

4.6 Build up knowledge of pairs of numbers with a sum of 10, without counting, and begin to remember them (e.g. 8 + 2 = 10).


By the end of Kindergarten, students use shapes to make models, pictures or patterns, or other shapes. They describe properties of lines, shapes, positions and directions using everyday language. They identify and name a cube, cone, circle, square and triangle. They make direct comparisons of the length, weight or capacity of two or three common objects. They describe time using words such as today, tomorrow, morning, afternoon, and the names of the days of the week.

Students should:

5 Describe geometric features of common objects in the environment

5.1 Name and describe common geometric shapes, e.g. circle, square, triangle, cube, cone.

Which of these shapes is a square?

5.2 Recognise and describe the shape of lines, using words like straight, curved, wavy, zigzag.

5.3 Describe the position or direction of an object using everyday language, e.g. near, far, up, down, left, right, in front of, behind, next to, above, below, …

Here are pictures of a ball, a horse, and a boat. Put the picture of the ball above the picture of the horse. Put the picture of the boat to the left of the picture of the ball.

5.4 Use flat and solid shapes to create models, pictures and patterns, or other shapes.

Here are five squares of the same size. Use some or all of the squares to make a bigger square.

Adding and subtracting

Exclude any written recording of calculations in a number sentence or equation, e.g. 2 + 4 = 6.

Doubles

Encourage the use of fingers.

Shapes

Include both flat and solid shapes.

Creating shapes

Include describing the models, pictures and patterns that are made.


6 Understand that objects have properties that can be compared, such as length, weight or capacity

6.1 Make direct comparisons of two or three objects by noting which is: taller, tallest; shorter, shortest; longer, longest; heavier, heaviest; lighter, lightest; holds more, holds most; holds less, holds least.

Which pencil is longer?

7 Recognise that time passes and name parts of the day, week or year

7.1 Use terms such as day, week, morning, afternoon, evening; today, tonight, yesterday, tomorrow, birthday, Eid.

7.2 Name the days of the week in order.

7.3 Read time to the hour on a clock face.

7.4 Identify the time to the nearest hour of everyday events, such as ‘lunchtime is 1 o’clock’, ‘bedtime is 8 o’clock’, ‘prayers are at 12 o’clock’, and relate these times to the position of hands on a clock face.

Data handling

By the end of Kindergarten, students identify similarities or differences between two common objects. They sort a set of common objects using a criterion that they have chosen.

Students should:

8 Sort and classify common objects

8.1 Choose own criterion for sorting a set of common objects (e.g. plain, patterned; with holes, without holes; square, not square).

8.2 Identify how two or more common objects are the same or different.

Describe how these two buttons are the same.

How are they different?

Weight

Compare directly in hands, or using a lever balance.

Strictly speaking, mass, not weight, is compared. But in the lower grades, mass and weight are treated as the same, so ‘weight’ is used.

59 | Qatar mathematics standards | Grade 1 © Supreme Education Council 2004


Summary of students’ performance by the end of Grade 1


Students represent and interpret mathematical problems using objects, numbers, symbols or simple diagrams. They explain their solutions orally. They describe a simple relationship between two numbers, quantities, shapes or objects using appropriate mathematical terms.

Number and algebra

Students count up to 100 objects reliably, read and write numbers to 100, and use ordinal numbers. They use their knowledge of place value in two-digit numbers to order numbers and to calculate. They know by heart addition and subtraction facts to 10 and pairs of numbers with a total of 20, and know that adding or subtracting 0 leaves a number unchanged. They work out calculations such as 42 + 7, 26 – 4, 30 + 50 and 76 – 20, if necessary with the support of real objects or a number line, recording calculations in a number sentence using the symbols +, – and =. They solve missing-number problems such as 3 + = 9 or 8 – = 5. They choose, use and explain the appropriate operation and mental calculation strategies to solve simple routine and non-routine problems involving the addition and subtraction of numbers, money or non-standard measures.


Students use mathematical names for common 2-D shapes and describe simple properties such as the number of sides and corners. They extend or complete repeating patterns according to shape, size, position or colour. They measure length, weight and capacity using everyday non-standard units. They order a set of familiar events, read the time on a clock to the hour or half hour, and name in order the days of the week and the months of the year.

Data handling

Students sort objects according to a single criterion, and represent them on a diagram. They make and interpret a simple pictogram in which the symbol represents one unit. They solve simple addition and subtraction problems by using data from graphs.

Content and assessment weightings for Grade 1


The reasoning and problem solving strand cuts across the other three strands and should be integrated with them in teaching and assessments. For Grade 1, about 30% of the teaching and assessment of each of the other three strands should be devoted to reasoning and problem solving.

Grade 1


For Grades 1 to 6, the weightings of the three content strands relative to each other are as follows:

Number and algebra


Data handling

60% 30% 10%

The standards are numbered for easy reference. Those in shaded rectangles, e.g. 1.2, are the performance standards for all students. The national tests for mathematics will be based on these standards.

Grade 1 teachers should also teach the Kindergarten standards, integrating them with Grade 1 standards, since some students may not have attended Kindergarten.




By the end of Grade 1, students represent and interpret mathematical problems using objects, numbers, symbols or simple diagrams. They explain their solutions orally. They describe a simple relationship between two numbers, quantities, shapes or objects using appropriate mathematical terms.

Students should:

1 Solve practical mathematical problems

1.1 Represent a problem and its solution by using objects, numbers, symbols or simple diagrams.

There are 10 crayons in each box.

How many crayons are there altogether?

Draw a line on this square to make two triangles.

1.2 Explain orally in own words the method used to solve a simple problem.

1.3 Describe a simple relationship between two numbers, quantities, shapes or objects using appropriate mathematical terms.

Write the missing number in each box.

19 is 1 less than . 19 is 10 less than .

Describe this arrangement of shapes so that other students can draw it.

Using a structured set of shapes (e.g. four shapes, in three colours, two sizes, two thicknesses), create a pattern in which there is just one difference between each shape and the next. Describe the difference.

Grade 1

Key standards


Cross-references




Grade 1 students are not expected to read the examples. It is assumed that the teacher asks the majority of these and similar questions orally.


Number and algebra

By the end of Grade 1, students count up to 100 objects reliably, read and write numbers to 100, and use ordinal numbers. They use their knowledge of place value in two-digit numbers to order numbers and to calculate. They know by heart addition and subtraction facts to 10 and pairs of numbers with a total of 20, and know that adding or subtracting 0 leaves a number unchanged. They work out calculations such as 42 + 7, 26 – 4, 30 + 50 and 76 – 20, if necessary with the support of real objects or a number line, recording calculations in a number sentence using the symbols +, – and =. They solve missing-number problems such as 3 + = 9 or 8 – = 5. They choose, use and explain the appropriate operation and mental calculation strategies to solve simple routine and non-routine problems involving the addition and subtraction of numbers, money or non-standard measures.

Students should:

2 Understand place value in and order whole numbers to 100

2.1 Use ordinal numbers (first, second, third, …, tenth) to describe the position of an object in a row of objects or the order of a set of events.

2.2 Count to 100 and back to zero by reciting zero, one, two, three, …

Count from 48 to 63.

Count back from 95 to 67.

2.3 Count reliably and give a reasonable estimate of up to 100 objects.

2.4 Read and write numbers 0 to 100 in numerals and words.

2.5 Represent the place value of two-digit numbers (tens and ones) using real objects, models and expanded notation, e.g. 43 = 40 + 3.

What number is the arrow pointing to?

2.6 Compare and order two-digit numbers and position them on a number line.

This number square is torn.

What was the largest number on the square?

2.7 Know that a two-digit number lies between two multiples of 10, e.g. know that that 46 lies between 40 and 50.

2.8 Identify the number that is 10 more/less than a given two-digit number.

2.9 Identify whole numbers lying between two given two-digit whole numbers, e.g. the whole numbers lying between 38 and 43.

3 Continue simple number patterns

3.1 Count groups of objects, e.g. put 20 items into groups of 5 and count the number of groups.

Ordinal numbers

Exclude the term ordinal numbers.

Counting to 100

Include counting backwards.

Counting objects and representing numbers

Include counting by grouping, e.g. four groups of 10 and 3 make 43.

Represent numbers using models such as a number line, abacus or 100-square, or using place value cards.

Ordering

Include arranging numbers in ascending and descending order.

10 more and 10 less

Link to counting in tens as in standard NA 3.3.


3.2 Count to 100 and back to zero in 10s, 5s and 2s.

Write the numbers that are missing from these sequences.

3.3 Count on or back from a given two-digit number in 1s or 10s.

4 Add and subtract whole numbers and apply these skills to solving routine and non-routine problems

Addition and subtraction

4.1 Understand addition as putting together, and subtraction as taking away, comparing or finding the difference, by adding and subtracting using real objects or counting on or back on a number line or 100-square.

4.2 Understand the meaning of the symbols +, –, = and use them to record additions and subtractions in a number sentence.

4.3 Know that adding or subtracting 0 to or from a number leaves the number unchanged.

4.4 Know by heart:

• all addition and subtraction facts to 10, e.g. 8 – 5, 3 + 6;

• pairs of numbers with a total of 20.

4.5 Represent equivalent forms of a number to 20 using models such as objects, diagrams, a number line, number expressions.

e.g. represent 12 as 4 + 4 + 4, 10 + 2, 20 – 8, 3 + 3 + 3 + 3, 15 – 3

4.6 Know doubles of numbers 1 to 10 (e.g. 7 and 7 is 14), and corresponding halves; use this knowledge to develop other facts, e.g. 7 + 8 = 15.

At noon, there are 4 rokaas and 4 sajdas. How many rokaas and sajdas are there altogether?

4.7 Use and explain mental methods to add and subtract, without crossing the tens boundary, supported by a model such as a number line or 100-square:

• a multiple of 10 and ones e.g. 50 + 7, 40 – 8

• a two-digit number and ones e.g. 32 + 6, 47 – 2

• two multiples of 10 e.g. 30 + 50, 90 – 60

• a two-digit number and tens e.g. 53 + 20, 96 – 50

4.8 Use known facts to add and subtract mentally in special cases.

e.g. add/subtract 9 by adding/subtracting 10 and adjusting by 1.

Counting in tens

Link to standard NA 2.8.


Initially, use numbers in the range 0–20.

Include finding how much more and how much less one number is than another.

Number facts

Include:

• writing number stories for each number to 10;

• completing equations such as:

+ 2 = 7, 3 + = 9

Mental methods

Exclude formal written methods in columns. Use informal recording on a number line or 100-square, e.g. for 46 + 9.


4.9 Know that the commutative law applies to addition but not subtraction and use it to simplify mental additions by putting the larger number first.

Example of commutative law of addition 5 + 18 = 18 + 5

Problem solving involving whole numbers

4.10 Given a problem ‘story’ or ‘situation’ involving the addition or subtraction of numbers, money or measures, identify a relevant operation, write a related number sentence and do the required calculation, supported where necessary by real objects or a number line.

Fatma is 9 years old. Her sister is 13 years old. How many years older is Fatma’s sister? Circle which of these you could use to work out the answer.

20 – 13 13 + 9 13 – 9 20 – 9

Ahmad has finished reading page 4 of his book. There are 16 pages in his book. How many more pages has he left to read?

4.11 Given a number sentence involving addition or subtraction, create a problem ‘story’ that might lead to it.

4.12 Understand the use of symbols such as to stand for an unknown number; solve problems such as + 4 = 7, 10 – = 3, supported at first by real objects or a number line.

4.13 Solve non-routine problems, such as simple number puzzles.

Use all the numbers 1 to 5. Write one number in each circle. Each line must have a total of 8.

Amna has less than 20 cubes. She counted the cubes in twos. She had 1 left over. She counted the cubes in fives. She had 3 left over. How many cubes does Amna have?

5 Solve simple problems involving money

5.1 Identify the values of Qatari coins and bank-notes to QR 100.

5.2 Identify different combinations of notes with the same value, e.g. know that a QR 10 note is equivalent in value to two QR 5 notes or ten QR 1 notes, or that two half riyal coins or four quarter riyal coins are equivalent to QR 1.

Ali buys a book for QR 35. He pays exactly QR 35 with five bank-notes. What could the five notes be?

Commutative law

Exclude knowing the name of the law.

Problem solving

Include finding how much more and how much less and difference between.

Missing-number problems

Keep within the range 0–10 inclusive.

Money

Include the notation QR.

Use simulated bank-notes.


5.3 Find simple totals and work out change for amounts up to QR 100.

Jassim has QR 50. He wants to buy a football.

football QR 75

How much more money does Jassim need?

6 Recognise halves and quarters in everyday situations

6.1 Recognise that one whole is equivalent to two identical halves or four identical quarters. For example:

• find halves and quarters of paper shapes by paper folding;

• estimate the quantity of water in a cylindrical container using phrases like about half full, about one quarter full;

• make half turns and quarter turns in outdoor games.


By the end of Grade 1, students use mathematical names for common 2-D shapes and describe simple properties such as the number of sides and corners. They extend or complete repeating patterns according to shape, size, position or colour. They measure length, weight and capacity using everyday non-standard units. They order a set of familiar events, read the time on a clock to the hour or half hour, and name in order the days of the week and the months of the year.

Students should:

7 Name common 2-D shapes and describe their properties using everyday language

7.1 Identify and name the circle, square, triangle and rectangle; describe simple properties of shapes using everyday language.

Draw arrows to show which shapes belong in the set.

Identifying 2-D shapes

Include faces of 3-D objects.


Complete this shape so that it is a square.

7.2 Describe, extend or complete repeating patterns made from shapes, according to shape, size, position or colour.

Draw the next two shapes in this pattern.

Continue the pattern in the next two circles.

8 Use direct comparison and non-standard units to measure and compare objects

8.1 Compare the length, weight or capacity of two or more everyday objects by using direct comparison or non-standard units.

Five children used cubes to balance one of their shoes. This table shows the number of cubes they needed.

cubes

Adel 16

Taleb 13

Najib 18

Sadeq 20

Idris 15

Whose shoe was heaviest? Whose shoe was two cubes lighter than Najib’s shoe?

Aziza measured the height of these two dolls. She used blocks. How many blocks taller is the large doll?

Weight


Comparing lengths

Include the use of simple approximations, e.g. about 3 pencils long.


9 Relate time to familiar events

9.1 Read the time on a clock to the hour and half hour.

Which of these clocks shows half past 3?

Tamim went to the beach. He left home at half past 10. The journey took half an hour. What time did he get to the beach?

9.2 Relate events to clock time using vocabulary such as before 3 o’clock, after half past 6, between 4 o’clock and 5 o’clock.

9.3 Order a set of familiar events.

9.4 Use units of time such as week, month, year to describe the duration of an event or when it will take place.

9.5 Name the days of the week and the months of the year in order.

Data handling

By the end of Grade 1, students sort objects according to a single criterion, and represent them on a diagram. They make and interpret a simple pictogram in which the symbol represents one unit. They solve simple addition and subtraction problems by using data from graphs.

Students should:

10 Make and interpret simple pictograms

10.1 Collect a simple set of data, organise it and represent it in a pictogram in which the symbol represents one unit.

10.2 Interpret simple pictograms in which the symbol represents one unit.

Some children made this graph about four containers.

How many more cups of water does the teapot hold than the jug? How many bottles can the teapot fill?

Time

Exclude the notation 10:30.

Vocabulary of time

Include terms used in Kindergarten.

Pictograms

Exclude graphs where the symbol represents more than one unit.

Include both horizontal and vertical formats.


10.3 Solve simple addition and subtraction problems by using data from graphs.

Some children made this graph.

How many children liked red best? How many more children liked yellow than orange?

10.4 Sort a set of common objects according to a single criterion, and represent them on a diagram.

These shapes have been sorted. Put a cross ( ) on the shape that is in the wrong place.

Graphs

Graphs like these lay the foundation for later work on bar charts.

Sorting

Represent the objects using the objects themselves, pictures of objects or drawings.

The diagram shown is a Carroll diagram; exclude knowing the name of the diagram.





Students represent and interpret mathematical problems by using numbers, objects, signs and symbols, or simple diagrams. They explain in their own words or by using diagrams the method used to solve a simple problem. They explain a simple line of reasoning.

Number and algebra

Students understand and use place value in numbers up to 1000. They know by heart addition and subtraction facts to 20. They use these facts and their knowledge of the inverse relationship between addition and subtraction to calculate with larger numbers and to check results. They choose, use and explain mental methods to add and subtract multiples of 1, 10 or 100 to two- and three-digit numbers and written column methods to add and subtract numbers with two digits. They understand the operations of multiplication and division. They know by heart multiplication tables for 2, 5 and 10, and other facts to 5 × 5, and use them to multiply and divide, recording results in a number sentence. They use the four operations to solve routine and non-routine one-step problems. They find the unknown number in problems such as + 6 = 11 and 15 – = 8 and solve real-life word problems involving money or a standard unit of measurement.


Students identify common 2-D and 3-D shapes, straight and curved lines and flat and curved surfaces. They measure and compare length, weight and capacity using standard metric units, reading scales to the nearest division. They use a ruler to measure and draw lines to the nearest centimetre. They tell the time to the nearest 5 minutes and calculate a simple time difference.

Data handling

Students make and interpret pictograms where the symbol represents a group of 2, 5 or 10 units. They answer questions by using data from simple graphs and tables.




Grade 2



Number and algebra


Data handling

60% 30% 10%


Grade 2 teachers should review and consolidate Grade 1 standards where necessary.




By the end of Grade 2, students represent and interpret mathematical problems by using numbers, objects, signs and symbols, or simple diagrams. They explain in their own words or by using diagrams the method used to solve a simple problem. They explain a simple line of reasoning.

Students should:

1 Use mathematical reasoning to solve simple problems

1.1 Represent a problem by using numbers, objects, signs and symbols, or simple diagrams (see also standard NA 4.18).

Mohamed had 3 pots with 5 pencils in each pot. How many pencils did he have altogether? Which one of these would you use to work out the answer to the question?

A. 5 + 5 B. 3 + 3 C. 5 × 3 D. 3 + 5

1.2 Explain orally in own words or by using numbers, objects, signs and symbols or simple diagrams the method used to solve a simple problem.

A jug has 500 millilitres of water in it.

150 millilitres of water are poured out of the jug. How much water is left in the jug? Explain how you worked out your answer.

1.3 Explain a simple line of reasoning.

Look at these digits.

Use all the digits to make the number nearest to 600.

Explain how you arrived at your answer.

Grade 2

Key standards


Cross-references




Grade 2 students are not necessarily expected to read the examples. It is assumed that the teacher asks many of these and similar questions orally.


Number and algebra

By the end of Grade 2, students understand and use place value in numbers up to 1000. They know by heart addition and subtraction facts to 20. They use these facts and their knowledge of the inverse relationship between addition and subtraction to calculate with larger numbers and to check results. They choose, use and explain mental methods to add and subtract multiples of 1, 10 or 100 to two- and three-digit numbers and written column methods to add and subtract numbers with two digits. They understand the operations of multiplication and division. They know by heart multiplication tables for 2, 5 and 10, and other facts to 5 × 5, and use them to multiply and divide, recording results in a number sentence. They use the four operations to solve routine and non-routine one-step problems. They find the unknown number in problems such as + 6 = 11 and 15 – = 8 and solve real-life word problems involving money or a standard unit of measurement.

Students should:

2 Understand and use place value in whole numbers to 1000

2.1 Read and write whole numbers up to 1000 in numerals and words.

2.2 Represent the place value of three-digit numbers (hundreds, tens, ones) using models and expanded notation, e.g. 574 = 500 + 70 + 4.

What number is the arrow pointing to?

2.3 Compare and order a set of three-digit numbers and position them on a number line.

Write these numbers in order of size.

3 Recognise properties of numbers and number sequences

3.1 Count on or back from a given three-digit number in 1s, 10s, 100s.

3.2 Count in steps 2, 3, 4, 5 and 10 from zero to the 10th multiple.

Fill in the two missing numbers in this sequence.

28 24 20 … 12 8 … 0

4 Add, subtract, multiply and divide whole numbers and apply these skills to solving routine and non-routine problems


4.1 Know by heart addition and subtraction facts to 20.

Place value

Stress the use of 0 as a place holder.

Represent numbers using models such as a number line or abacus, or using place value cards.

Counting on

Include finding missing terms in number sequences.

Include counting backwards.


4.2 Use the commutative and associative laws of addition to simplify mental calculations and check results.

Example of commutative law 5 + 18 = 18 + 5

Examples of associative law 36 + 9 = 36 + (4 + 5) = (36 + 4) + 5 = 40 + 5 = 45

4.3 Add three one-digit numbers mentally.

Write a number in the box to make this correct. 3 + + 9 = 17

4.4 Use and explain mental methods, supported at first by a model such as a number line or 100-square, to add and subtract multiples of 1, 10 or 100, including crossing the tens boundary:

• a two-digit number and ones e.g. 46 + 9, 82 – 7

• a two-digit number and tens e.g. 73 + 50, 94 – 60

• a three-digit number and ones e.g. 457 + 6, 312 – 7

• a three-digit number and tens e.g. 253 + 60, 242 – 70

• a three-digit number and hundreds e.g. 546 + 300, 695 – 400

4.5 Use and explain written column methods to add and subtract whole numbers with two digits.

Add: 63 + 48 24 + 37 + 46 Subtract: 86 – 57

4.6 Know and use the inverse relationship between addition and subtraction to calculate and to check results.

Given that 79 + 63 = 142, what is 142 – 63?

Multiplication and division

4.7 Relate multiplication to repeated addition or counting in multiples and division to repeated subtraction, sharing or forming equal groups.

One box holds 3 cakes. How many cakes are there in 5 boxes?

Laws of arithmetic

Exclude names of the laws.

Mental methods

Include revision of the mental methods from Grade 1.


Include crossing the tens boundary.

Inverse relationship between + and –

For example, since 8 + 6 = 14, deduce that 14 – 6 = 8.


Model with number line, a 100-square and rectangular grids.

Exclude remainders.


Divide 18 into 3 equal groups. How many are there in each group?

Divide 18 into groups of 3. How many groups are there?

4.8 Understand the meaning of the symbols × and ÷ and use them to record multiplication and division calculations in number sentences.

4.9 Know that the commutative law applies to multiplication but not division (e.g. 4 × 2 = 2 × 4 but 8 ÷ 4 ≠ 4 ÷ 8).

4.10 Understand the inverse relationship between multiplication and division (e.g. since 3 × 10 = 30, 30 ÷ 10 = 3).

4.11 Multiply and divide within the multiplication tables for 2, 5 and 10, and other facts to 5 × 5.

4.12 Know by heart multiplication tables for 2, 5 and 10.

4.13 Know by heart multiplication tables for 3 and 4.

4.14 Know by heart doubles of whole numbers to 15 and corresponding halves.

4.15 Recognise half, quarter and three quarters of simple shapes and small whole numbers.

Problem solving with whole numbers

4.16 Given a number sentence involving a single addition, subtraction, multiplication or division, create a problem ‘story’ that might lead to it.

4.17 Solve missing-number problems using inverse relationships.

Write the missing number in the box. 5 × 4 = 10 ×

Wadha thinks of a number. She says: ‘If I subtract 14 from it, I get 29.’ What is Wadha’s number?

4.18 Given a word problem involving one-step addition, subtraction, multiplication or division, identify the relevant operation, write a related number sentence and do the required calculation (see also standard RP 1.1).

There are 58 oranges in one basket and 82 oranges in another basket. What is the total number of oranges in the baskets?

One box will hold 5 apples. How many boxes are needed to hold 35 apples?

Commutative law

Exclude the name of the law.


Limit to numbers in the 2, 5 and 10 times tables.

Word problems

Include money and measurement problems expressed in the same unit.

Limit multiplication and division to numbers in the 2, 5 and 10 times tables.


4.19 Solve non-routine problems involving small whole numbers.

Put all the numbers 20, 30, 50 and 60 in the circles. The numbers along each side must add up to 90.

5 Solve problems involving money

5.1 Show how to pay an exact number of riyals using smaller bank-notes.

5.2 Find the total value of a mixed set of half and quarter riyal coins, up to QR 2.

5.3 Find a total or amount of change up to QR 500.

A tennis racket costs QR 325. A box of tennis balls costs QR 50. What is the total cost of the tennis racket and the tennis balls? How much change would you get from QR 500?

5.4 Find the cost of several items, given the cost of one item and the number of items.

A ticket for the zoo costs QR 5. What is the cost of 8 tickets for the zoo?


By the end of Grade 2, students identify common 2-D and 3-D shapes, straight and curved lines and flat and curved surfaces. They measure and compare length, weight and capacity using standard metric units, reading scales to the nearest division. They use a ruler to measure and draw lines to the nearest centimetre. They tell the time to the nearest 5 minutes and calculate a simple time difference.

Students should:

6 Name common 2-D and 3-D shapes and describe their properties using everyday language

6.1 Identify straight and curved lines and flat and curved surfaces.

Two of these letters from the English alphabet are made from straight lines only.

E P A C S Draw a circle around each of them.

6.2 Use a ruler to measure and draw a straight line of a given length to the nearest centimetre.

Draw a new line 4 cm longer than this one. Use a ruler.

Finding totals

Add up the value of mixed coins mentally.

Limit totals up to QR 500 to whole numbers of riyals.

Costs of items

Limit to numbers in the 2, 5 and 10 times tables.

Straight and curved lines

Include forming figures with straight lines and curves.


6.3 Identify and name the cube, cuboid, sphere, cylinder, cone and pyramid, and the pentagon, hexagon and octagon; describe simple properties of these shapes using everyday language.

Write the missing numbers in the two empty boxes.

Use the dots to draw a different pentagon. Use a ruler.

6.4 Complete geometric patterns made from solid or flat shapes, according to shape, size, colour or orientation, or two of these attributes.

Use a structured set of shapes (e.g. four shapes, in three colours, two sizes and two thicknesses). Create a pattern in which there are exactly two differences between each shape and the next one. Describe the differences as each shape is placed in the pattern.

7 Use standard metric units to measure and compare length, weight, capacity and time

7.1 Use a single standard unit (metre, centimetre; kilogram, gram; litre, millilitre) to estimate, measure and compare length, mass/weight or capacity, recording readings using a single unit.

Use a ruler. Measure the longest side of this shape to the nearest centimetre.

7.2 Choose and use appropriate measuring equipment, reading scales to the nearest division.

How heavy is Aziz? How much water is in the jug?

Shapes

Use a square-based pyramid.

Include the term polygon.

Include properties such as cubes pack, or cylinders roll in a straight line when placed on a curved face but are stable when placed on an end face.

Standard units

Include abbreviations (m, cm; kg, g; l, ml)

Weight


Measuring equipment

Use everyday equipment such as rulers, metre sticks and measuring tapes; bathroom scales, kitchen scales and spring balances; measuring jugs, cups and cylinders.

Measurements should involve a single unit only.


7.3 Read the time on a clock with hands to the nearest 5 minutes; understand and use the notation 6:35, reading this as six thirty-five.

The bowling alley closes at 7:45. Draw the hands on the clock face to show this time.

7.4 Calculate a time interval of less than 1 hour (a multiple of 5 minutes), or of a whole number of hours.

The time is 6:30 in the morning. What time was it 20 minutes ago?

A family left at 9 o’clock to go to the Al Bida Park. They arrived 45 minutes later. Draw a ring around the time they got to the Al Bida Park.

9:15 11:15 9:45 10:45 10:15

The time is 5:15. What time will it be 4 hours later? What time was it 3 hours earlier?

Data handling

By the end of Grade 2, students make and interpret pictograms where the symbol represents a group of 2, 5 or 10 units. They answer questions by using data from simple graphs and tables.

Students should:

8 Make and interpret pictograms of simple sets of data using a scale representation

8.1 Represent a given set of data in a pictogram using a symbol representing 2, 5 or 10 units.

Look at this pictogram.

12 children like dates best. Show this on the pictogram.

Time

Include the phrases quarter to and quarter past the hour.

Time intervals

Exclude time intervals of less than 1 hour that cross the hour.

Pictograms

Exclude graphs where an incomplete symbol represents part of a unit, unless the part is specified in the key to the graph.



8.2 Read and interpret pictograms where the symbol represents 2, 5 or 10 units.

Look at the pictogram. How many more books were borrowed on Wednesday than on Monday?

8.3 Solve problems by using data from simple graphs and tables.

Roza asked some children which fruit they like best.

The fruit we like best

fruit number of children

apples grapes bananas pears

7 4 6 3

How many children did Roza ask altogether?





Students model or represent mathematical problems using calculations, mathematical symbols, diagrams, graphs, charts and tables. They explain orally in their own words, or in writing or by using diagrams, the method used to solve a problem, or why an answer is correct. They justify their reasoning in simple cases.

Number and algebra

Students use their knowledge of place value in numbers up to 10 000 to multiply and divide whole numbers by 10 or 100. They compare numbers using the symbols <, = and >. They identify odd and even numbers. They know by heart multiplication facts to 10 × 10 and use the inverse relationship between multiplication and division to derive corresponding division facts. They choose, use and explain mental methods to add three one-digit numbers, to add and subtract numbers with two digits and to multiply and divide a two-digit number by a one-digit number in simple cases. They use written column methods to add and subtract numbers with three or four digits. They solve two-step problems involving addition and subtraction and one-step problems involving multiplication or division of whole numbers, including simple division problems that give rise to remainders. Their problem solving extends to both non-routine and routine problems, including real-life problems involving money or measures. They recognise and compare simple fractions represented by diagrams and find simple unit fractions of small whole numbers.


Students understand angle as a measure of turn and recognise right angles. They identify equilateral, isosceles and right-angled triangles and regular and irregular polygons, and complete simple symmetrical patterns. They use and know the relationships between standard units of length (kilometre, metre, centimetre), weight (kilogram, gram) and capacity (litre, millilitre), and standard units of time (hour, minute, second). They record estimates and measurements of length, weight and capacity, including readings from scales, in mixed units. They read the time to the nearest minute, use the 12-hour clock and the notation 6:45, and use simple timetables. They find the perimeters of simple shapes, compare areas by counting squares and calculate the areas of squares and rectangles.

Data handling

Students collect data systematically. They record and read data in simple tally charts. They represent and interpret data in bar charts, reading simple scales on the axes. They solve problems by asking and answering questions related to the data.

Grade 3






Number and algebra


Data handling

60% 30% 10%






By the end of Grade 3, students model or represent mathematical problems using calculations, mathematical symbols, diagrams, graphs, charts and tables. They explain orally in their own words, or in writing or by using diagrams, the method used to solve a problem, or why an answer is correct. They justify their reasoning in simple cases.

Students should:


1.1 Model or represent a problem using calculations, mathematical symbols, diagrams, graphs, charts and tables.

Sharifa has read the first 78 pages of a book that is 130 pages long. Which of these number sentences could Sharifa use to find the number of pages she must read to finish the book?

A. 130 + 78 = B. – 78 = 130 C. 130 ÷ 78 = D. 130 – 78 =

TIMSS Grades 3 and 4

1.2 Explain orally in own words, or in writing or by using diagrams, the method used to solve a problem, or why an answer is correct.

Use a number line to show why 382 – 179 = 203.

1.3 Justify reasoning in simple cases.

The outside of this circle has 11 squares fitted around part of it.

About how many squares could be fitted around the whole circle? Tick ( ) the answer.

80 40 25 100 65

Explain how you worked out your answer.

When you subtract one of the numbers below from 900, the answer is greater than 300. Which number is it?

A. 823 B. 712 C. 667 D. 579

Explain how you know.


Grade 3

Key standards


Cross-references





Number and algebra

By the end of Grade 3, students use their knowledge of place value in numbers up to 10 000 to multiply and divide whole numbers by 10 or 100. They compare numbers using the symbols <, = and >. They identify odd and even numbers. They know by heart multiplication facts to 10 × 10 and use the inverse relationship between multiplication and division to derive corresponding division facts. They choose, use and explain mental methods to add three one-digit numbers, to add and subtract numbers with two digits and to multiply and divide a two-digit number by a one-digit number in simple cases. They use written column methods to add and subtract numbers with three or four digits. They solve two-step problems involving addition and subtraction and one-step problems involving multiplication or division of whole numbers, including simple division problems that give rise to remainders. Their problem solving extends to both non-routine and routine problems, including real-life problems involving money or measures. They recognise and compare simple fractions represented by diagrams and find simple unit fractions of small whole numbers.

Students should:

2 Understand and use place value in whole numbers to 10 000

2.1 Read and write whole numbers up to 10 000 in numerals and words.

Use each of the digits 7, 9, 0 and 5 exactly once. What is the smallest number you can make? Write your number in words.

2.2 Represent the place value for whole numbers with up to four digits (thousands, hundreds, tens, ones) using models and expanded notation.

What number goes where the box is to make this number sentence true?

2000 + + 30 + 8 = 2538

Draw an arrow (↑) to show the position for 7500 on this number line.

2.3 Compare and order numbers using the symbols <, = and >.

Use these signs: = < >. Write the correct sign in each box.

8 × 7 9 × 6 5 × 7 5 × 5 10 × 6 6 × 10

2.4 Round two-digit whole numbers to the nearest 10.

3 Recognise properties of numbers and number sequences

3.1 Count on or back from a given four-digit number in 1s, 10s, 100s, 1000s.

3.2 Count on from and back to zero in steps of 6, 7, 8, 9.

3.3 Identify odd and even numbers.

Choose three of these number cards.

Make an even number that is greater than 400.


Use models such as a number line, an abacus, or place value cards.

Ordering

Include ordering on the number line.

Counting on

Include completing number sequences.


4 Add, subtract, multiply and divide whole numbers and apply these skills to solving routine and non-routine problems


4.1 Add three one-digit numbers mentally.

4.2 Use and explain mental methods to add and subtract any pair of two-digit whole numbers.

In a group of sixty-four children, twenty-nine wear glasses. How many do not wear glasses?

Join two more pairs of numbers that total 71.

50 18

19 43

28 5321

4.3 Use and explain written column methods to add and subtract whole numbers with up to four digits.


4.4 Derive and learn by heart multiplication facts to 10 × 10.

4.5 Understand the special properties of 0 and 1 in multiplication and division.

4.6 Use the commutative, associative and distributive laws of multiplication, and the distributive law of division, to multiply and divide.

Example of the distributive law of multiplication over addition (4 + 3) × 5 = (4 × 5) + (3 × 5) = 20 + 15 = 35

Example of the distributive law of division over addition (20 + 8) ÷ 4 = (20 ÷ 4) + (8 ÷ 4) = 5 + 2 = 7

4.7 Use and explain mental methods to multiply and divide a two-digit number by a one-digit number in simple cases, supported where necessary by models (no remainder for division), e.g. 24 × 3, 52 ÷ 4.

4.8 Find remainders after division.

4.9 Round the quotient up or down after division, depending on the context.

One box holds 8 cakes. There are 62 cakes. How many boxes can be filled with cakes?

70 boys go camping. They sleep in tents. Each tent takes up to 6 boys. What is the least number of tents the boys will need?

4.10 Use and explain written methods to multiply and divide two-digit numbers by a one-digit number (including a remainder for division).

Multiply: 43 × 6

Divide: 84 ÷ 7

Write in the missing digit: 2 × 8 = 184

4.11 Multiply or divide whole numbers by 10 or 100.

4.12 Multiply or divide whole numbers by 1000.

4.13 Use the inverse relationship between multiplication and division (e.g. since 8 × 9 = 72, 72 ÷ 9 = 8) to calculate and check results.


Include the use of the terms sum and difference.

Multiplication/division

Include the terms product, quotient, remainder.

Laws of arithmetic

Exclude names of the laws.

Mental methods

Include doubling two-digit numbers.

Use informal recording to support or explain mental calculations, e.g.

Exclude crossing 100.

Division

Limit division by 10, 100 or 1000 to whole-number answers.



4.14 Solve word problems involving whole numbers, measurements or money using addition and subtraction (up to two steps) or multiplication and division (one step).

Noura is 109 cm tall. Muna is 137 cm tall. Huda is 53 cm taller than Noura. How much taller is Huda than Muna? Apples are sold in packs of 5. How many apples are there in 72 packs? There are 440 drinking straws in a box. The straws are red, yellow, blue and green. There is the same number of each colour.

How many red straws are in the box?

4.15 Solve problems in which a symbol represents an unknown number, including problems involving inverse operations.

Write the missing numbers in the boxes.

+ 28 = 113 × 3 = 45 65 + = 402 30 × = 90

– 42 = 650 ÷ 2 = 18 91 – = 37 48 ÷ = 3

4.16 Model a problem ‘story’ or ‘situation’ by writing an equation.

4.17 Solve non-routine problems involving whole numbers less than 100.

Here is a row of numbers.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Find three numbers next to each other that add up to 39. Draw a ring round them.

5 Solve problems involving money

5.1 Find a total (up to QR 10 000) and calculate change.

It costs QR 3540 for an adult to fly from Doha to New York. It costs QR 1855 for a child under 12 to fly from Doha to New York. What does it cost for two adults and one child to fly from Doha to New York?

5.2 Solve money problems involving simple proportional reasoning:

• determine the total cost of a number of items, given the unit cost (up to QR 100) and number of units (up to 9 units);

• determine the unit cost when given the total cost (up to QR 100) and number of units (up to 9).

A box of six tennis balls costs QR 84. What does one tennis ball cost?

6 Recognise simple fractions of shapes and numbers

6.1 Use diagrams, including number lines, to recognise and name:

• unit fractions to 112 ;

• simple fractions that are several parts of a whole, e.g. 35 .

Word problems

Include money and units of measurement.

Measurements in a problem should be expressed in the same unit.

Finding totals and change

Include two-step problems.

Finding the unit cost

Limit to whole numbers, riyals only.

Fractions

Include the terms numerator, denominator.


Shade three quarters of the rectangle. What fraction of the circle is shaded?

Two squares on the snake are filled in. Fill in more squares to cover half of the snake.

6.2 Use fraction notation.

6.3 Know that when all fractional parts are included, the result equals one whole, so that 5

5 1= .

6.4 Use diagrams, including number lines, to:

• identify the equivalence of simple fractions, e.g. show that half a circle is equivalent to two quarters of a circle of the same size;

• compare two unit fractions, or two simple fractions with the same denominator, e.g. show that 7 5

8 8> .

Tick ( ) the shape that is more than half shaded.

6.5 Find a unit fraction of a whole number up to 30.

Hana ate 14 of these sweets.

How many sweets did she eat?

One third of a number is 4. What is the number?

What fraction of these squares is circled?

6.6 Solve simple word problems involving fractions.



By the end of Grade 3, students understand angle as a measure of turn and recognise right angles. They identify equilateral, isosceles and right-angled triangles and regular and irregular polygons, and complete simple symmetrical patterns. They use and know the relationships between standard units of length (kilometre, metre, centimetre), weight (kilogram, gram) and capacity (litre, millilitre), and standard units of time (hour, minute, second). They record estimates and measurements of length, weight and capacity, including readings from scales, in mixed units. They read the time to the nearest minute, use the 12-hour clock and the notation 6:45, and use simple timetables. They find the perimeters of simple shapes, compare areas by counting squares and calculate the areas of squares and rectangles.

Students should:

7 Identify simple properties of shapes, understand angle as a measure of turn and recognise right angles

7.1 Know that angle measures the amount of turn and recognise whole, half and quarter turns; know that an angle that measures a quarter turn is called a right angle.

Draw the arrow after a quarter turn clockwise.

7.2 Identify right angles in the environment and contained in 2-D shapes.

Two of the shapes are hexagons and have two right angles. Put a tick ( ) on each of the two shapes.

7.3 Use the eight points of the compass.

7.4 Describe and visualise 2-D and 3-D shapes.

7.5 Identify simple properties of triangles, such as two equal sides for an isosceles triangle, three equal sides for an equilateral triangle, a right angle for a right-angled triangle.

Use a ruler to draw two more lines to make an isosceles triangle.

7.6 Identify regular and irregular polygons.

Angles

Include clockwise and anticlockwise turns.

Exclude use of the terms acute, obtuse, reflex.

Include the shapes identified in earlier grades and the tetrahedron and heptagon.


7.7 Draw a line of symmetry and complete simple symmetrical patterns.

Complete this pattern so that it is symmetrical about the mirror line.

8 Choose and use appropriate units and measurement tools to estimate and measure length, weight, capacity, time

8.1 Estimate measurements using a single unit.

Estimate the capacity of a kitchen bucket, an egg cup, … the weight of a shoe, of a pencil, … the height of a classroom, the length of a pencil, …

8.2 Know the relationships between:

• kilometres, metres and centimetres;

• kilograms and grams;

• litres and millilitres.

8.3 Choose suitable measuring equipment to measure the length, weight and capacity of given objects.

8.4 Record estimates and measurement, including readings from scales, using mixed units where appropriate.

What is the total weight of these apples? (1 kg 500 g)

8.5 Convert metres to centimetres and centimetres to metres, expressing the result in mixed units where appropriate.

8.6 Measure minutes or seconds using a stopwatch or clock.

8.7 Read the time to the minute from digital and analogue clocks and watches; use the 12-hour clock notation 6:45, specifying morning or afternoon.

One of these watches is 3 minutes fast. The other watch is 4 minutes slow. What is the correct time?

Weight


Units of measurement

Include abbreviations (km, m, cm; kg, g; l, ml).

Exclude conversions between units, with the exception of the units in standard GM 8.5.

Time

Include abbreviations (h, min, s).

Include the use of past and to, as in 23 minutes past 6, 14 minutes to 9.


8.8 Read and make simple calculations from timetables.

These are the opening times at a museum.

Opening times

morning afternoon

Sunday 8:00 to 6:00

Monday 10:30 to 5:30

Tuesday 10:30 to 8:30

Wednesday 10:30 to 9:00

Thursday Museum closed

Friday Museum closed

Saturday 11:00 to 4:00

How many hours is the museum open on a Monday? Which day has the latest closing time?

Abdulla arrives at the museum at 3:15 on Saturday afternoon. How many minutes is this before the museum closes?

9 Find the perimeter of a regular polygon and the perimeter and area of a square and rectangle

9.1 Know that perimeter is the distance measured around the boundary of a figure.

9.2 Measure and calculate the perimeter of squares, rectangles and regular polygons with whole-number sides.

Here is a centimetre square grid. On the grid, draw a rectangle with a perimeter of 10 cm.

Each side of a pentagon is 12 cm. What is the perimeter of the pentagon?

9.3 Know that area is the size of a surface, or of the space enclosed by the boundary of a plane figure; find and compare the areas of irregular plane shapes by estimating/counting the number of unit squares that cover them.

Match each shape on the left to one with equal area on the right. One has been done for you.

Timetables

Limit calculations of time intervals from timetables to intervals of:

• less than 1 hour (multiples of 5 minutes, including crossing the hour);

• more than 1 hour (whole numbers of hours).

Perimeters

Include estimating and measuring perimeters.


Newsha has a square piece of card.

She cuts along the dashed line to make two pieces of card. Do the two pieces of card have the same area? Explain your answer.

9.4 Derive and use the formula area = length × width for calculating the area of a square or rectangle.

Here is a centimetre square grid. On the grid, draw two different rectangles, each with an area of 12 cm2.

Data handling

By the end of Grade 3, students collect data systematically. They record and read data in simple tally charts. They represent and interpret data in bar charts, reading simple scales on the axes. They solve problems by asking and answering questions related to the data.

Students should:

10 Make and interpret tables and bar charts of sets of data and solve problems using the information

10.1 Collect and record data systematically, keeping track of what has been counted (e.g. in a tally chart).

This table shows how many people went into shops in one hour.

How many people went into the fish shop in the hour? How many more people went into the post office than the book shop?

10.2 Record and interpret information in simple Carroll diagrams.

Area

Include square centimetres (cm2) and square metres (m2).


10.3 Represent and interpret data in a bar chart with a scale numbered in intervals of 2, 4, 5 or 10; label the axes and give the graph a title.

Here is a bar chart of the information shown in the tally chart in standard 10.1. Draw in the missing bar.

10.4 Ask and answer questions related to data in tables and bar charts.

Some girls made a bar chart showing how far they could swim.

How far can Haya swim? Sara can swim further than Alia. How much further?

What other questions could you ask about the data?

The table shows the times when some boys start their swimming lessons.

Name Time

Fahad Tamin Jassim Saad Suood Mishal

9:15 9:45 8:45 7:15 8:15 7:45

Which two boys have a swimming lesson after Saad and before Jassim?

Bar charts






Students represent and interpret routine and non-routine mathematical problems using calculations, mathematical symbols, diagrams, graphs, charts and tables. They explain orally in their own words, or in writing or by using diagrams, the method used to solve a problem, or why an answer is correct. They check that results are appropriate in the context of the problem. They justify their reasoning in simple cases.

Number and algebra

Students represent whole numbers and decimals to two places in expanded form and use their understanding of place value to order numbers and to multiply and divide by multiples of 10 and 100. They round whole numbers to the nearest 10 or 100, and decimals to the nearest whole number, and estimate answers to calculations. They identify multiples of one-digit numbers and extend and find missing numbers in a simple linear sequence. They know multiplication and division facts to 10 × 10 and use factors to simplify mental multiplication and division calculations. They choose, use and explain written column methods to multiply and divide three-digit by one-digit whole numbers, multiply three-digit by two-digit whole numbers, add and subtract decimals to two places, and multiply a decimal with up to two places by a one-digit whole number. They add and subtract two simple fractions with the same denominator or where one denominator is a multiple of the other, expressing the answer as a mixed number. They solve both routine and non-routine problems (up to two steps with whole numbers or one step with decimals), including real-life problems related to money or measures.


Students identify parallel and perpendicular lines, recognise lines of symmetry and complete symmetrical figures. They identify angles as greater than or less than a right angle and put a set of acute and obtuse angles in order of size. They identify simple properties of squares, rectangles and parallelograms. They construct squares and rectangles on grids and by using a set square and ruler, drawing lines to the nearest millimetre. They solve simple problems involving scale. They find the perimeters of irregular polygons and perimeters and areas of shapes that can be split into squares and rectangles. They choose and use suitable units to estimate and measure and read scales with increasing accuracy. They convert centimetres to metres or millimetres, using decimal notation. They calculate a time interval of up to 1 hour in minutes, and larger time intervals that are multiples of 15 minutes.

Data handling

Students complete, extract and interpret information presented in lists, two-way tables and simple Carroll diagrams. They solve problems using data presented in bar charts and tables.

Grade 4






Number and algebra

Geometry and measures Data

handling

60% 30% 10%






By the end of Grade 4, students represent and interpret routine and non-routine mathematical problems using calculations, mathematical symbols, diagrams, graphs, charts and tables. They explain orally in their own words, or in writing or by using diagrams, the method used to solve a problem, or why an answer is correct. They check that results are appropriate in the context of the problem. They justify their reasoning in simple cases.

Students should:



Class survey of favourite fruit drinks

flavour number of children

pineapple 2

orange 10

blackcurrant 8

grapefruit 6

apple 9

Complete the pictogram for the class.

How many children altogether chose the three most popular flavours? Write another question you can ask someone about the results of the survey.

The table shows the number of shirts in a shop.

How many shirts are white? How many shirts are there altogether?

Grade 4

Key standards


Cross-references





Write the correct number shown by the arrow.

This shape is made from four cubes stuck together. Two circles are drawn on the shape.

Draw the circles on the shape in its new position.

1.2 Explain orally in own words, or in writing or by using diagrams, the method used to solve a problem, or why an answer is correct.

Yassim wanted to add 1463 and 319. She added 1263 and 319 by mistake. What could she do to correct her answer?

A. Add 200. B. Add 2. C. Subtract 2. D. Subtract 200.


1.3 Justify reasoning in simple cases.

Sara is older than Aida. Aida is older than Huda. Which statement must be true?

A. Sara is older than Huda. B. Sara is younger than Huda. C. Sara is the same age as Huda. D. We cannot tell who is older from this information.


1.4 Check that results are appropriate in the context of the problem.

Some chocolates are put into boxes. One box holds 7 chocolates. How many boxes are needed to hold 50 chocolates? Explain your reasoning.


Number and algebra

By the end of Grade 4, students represent whole numbers and decimals to two places in expanded form and use their understanding of place value to order numbers and to multiply and divide by multiples of 10 and 100. They round whole numbers to the nearest 10 or 100, and decimals to the nearest whole number, and estimate answers to calculations. They identify multiples of one-digit numbers and extend and find missing numbers in a simple linear sequence. They know multiplication and division facts to 10 × 10 and use factors to simplify mental multiplication and division calculations. They choose, use and explain written column methods to multiply and divide three-digit by one-digit whole numbers, multiply three-digit by two-digit whole numbers, add and subtract decimals to two places, and multiply a decimal with up to two places by a one-digit whole number. They add and subtract two simple fractions with the same denominator or where one denominator is a multiple of the other, expressing the answer as a mixed number. They solve both routine and non-routine problems (up to two steps with whole numbers or one step with decimals), including real-life problems related to money or measures.

Students should:

2 Understand place value in and order whole numbers

2.1 Read and write whole numbers in numerals and words; identify the place value for each digit in whole numbers.

Write in figures the number two and a half million.

2.2 Round three- or four-digit whole numbers to the nearest 10 or 100.

Circle the number that is nearest in value to 750. 570 699 810 852 1050

3 Identify factors and multiples

3.1 Know all facts to 10 × 10 for multiplication and division.

3.2 Recognise multiples up to 100 of one-digit numbers; find all the factors of whole numbers to 100, recognising that:

• many whole numbers can be factorised in different ways; e.g. 12 = 4 × 3 = 2 × 6 = 2 × 2 × 3

• prime numbers (2, 3, 5, 7, 11, 13, …) have exactly two factors, the number itself and 1.

The two-digit number 98 has 2 and 7 among its factors. Write another two-digit number that has 2 and 7 among its factors.

Use the four digits 7, 5, 2 and 1. Choose two digits each time to make the following two-digit numbers.

a multiple of 9

a prime number

a factor of 96

Whole numbers

Include large numbers involving millions.

Facts to 10 × 10

Learn division facts by heart.

Factors

Include finding a common factor of two numbers.

Exclude finding the highest common factor (HCF).

Prime numbers

Stress that the first prime number is 2.


3.3 Find missing terms in a number pattern, and describe the relationship between one term and the next, e.g. ‘add 4’.

The rule for this number sequence is ‘double and subtract 1’. Write in the missing number.

2 → 3 → 5 → 9 →

Here is part of another sequence with the same rule. Write in the missing number.

→ 13 → 25 → 49

3.4 Generate pairs of numbers that follow a given rule, such as ‘multiply the first number by 3 and add 2 to get the second number’; given some pairs of numbers that satisfy a relationship, identify a rule for the relationship.

This number machine multiplies all numbers by 2, and then adds 1.

IN OUT 5 11

13

Write the missing numbers in the table.

117

4 Calculate with whole numbers and apply these skills to solve routine and non-routine problems

Addition, subtraction, multiplication and division

4.1 Add and subtract two or more whole numbers, choosing and using mental or written methods as appropriate.

Draw a line to join two other numbers that have a total of 700.

4.2 Use and explain mental methods to multiply or divide a two-digit number by a one-digit number, supported as necessary by informal jottings.

4.3 Use factors to multiply and divide whole numbers by multiples of 10 or 100.

e.g. 45 × 60 = (45 × 6) × 10 = 270 × 10 = 2700 45 × 600 = (45 × 6) × 100 = 270 × 100 = 27 000

720 ÷ 80 = (720 ÷ 10) ÷ 8 = 72 ÷ 8 = 9 7200 ÷ 800 = (7200 ÷ 100) ÷ 8 = 72 ÷ 8 = 9

4.4 Use and explain written column methods to:

• multiply or divide numbers with two or three digits by a one-digit number; e.g. 53 × 7, 428 × 6, 865 ÷ 5

• multiply numbers with up to three digits by a two-digit number.

e.g. 437 × 28

Dividing by multiples of 10 or 100

Limit division to whole-number answers.


4.5 Estimate answers to whole-number calculations by using approximations.

Ahmed worked 57 hours in March, 62 hours in April, and 59 hours in May. Which is the BEST estimate of the total number of hours he worked for the 3 months?

A. 50 + 50 + 50 B. 55 + 55 + 55 C. 60 + 60 + 60 D. 65 + 65 + 65

TIMSS Grade 4


4.6 Model a problem involving an unknown number by writing an equation.

4.7 Solve problems with up to two steps using addition, subtraction, multiplication or division of whole numbers, measurements or money.

Some fruit is put in baskets. There are 3 oranges and 2 apples in every basket. There are 45 pieces of fruit altogether. How many apples are there?

Najla use 5 tomatoes to make half a kilogram of tomato sauce. How much sauce can she make from 15 tomatoes?

A. One and a half kilograms B. Two kilograms C. Two and a half kilograms D. Three kilograms

TIMSS Grade 4

4.8 Solve missing-number problems involving inverse operations.

What number goes in the box to make this equation true?

87 + = 129

Badriya thought of a number. She multiplied it by 2 and added 4. The answer was 16. What was the number Badriya first thought of?

Adapted from TIMSS Grade 4

4.9 Solve non-routine problems, including those involving more than one step.

A soft drink and a cake together cost QR 9. Two soft drinks and a cake together cost QR 14. What does a cake cost? Explain how you got your answer.

Khalid makes a sequence of five numbers. The first number is 2. The last number is 18. His rule is to add the same amount each time. Write in the missing numbers.

2 18

Write in the missing digits to make this calculation correct. 4 × 6 = 2052

5 Understand and use simple fractions

5.1 Recognise that a proportion of a number of objects can be described by a fraction.

5 out of the 12 children in a group wear glasses. What fraction of the children wear glasses?

Estimating answers

Include checking the reasonableness of an answer.

Problems

Include:

• problems involving proportional reasoning;

• money and measures.

Measurements in a problem should be expressed in the same unit (no conversions).


5.2 Identify simple equivalent fractions.

Tick ( ) two cards that have the same value.

5.3 Add and subtract two proper fractions:

• with the same denominator;

• where one denominator is a multiple of the other.

Tick ( ) two cards that give a total of seven twelfths.

5.4 Express an improper fraction as a mixed number and vice versa.

5.5 Calculate the product of a proper fraction and a whole number.

5.6 Solve simple word problems involving fractions. 38 of the books on a shelf are story books.

What fraction of the books on the shelf are not story books?

6 Understand the place value of decimals to two places and how decimals relate to simple fractions

6.1 Read and write decimals with one or two places; represent place value in decimals with one or two places in words, models or expanded form.

e.g. Know that 5 45410 100 1000.45 0.4 0.05= + = + =

Express 4.62 as: 4 0.6 0.02+ +

or as: 6 210 1004 + +

6.2 Understand that fractions and decimals are two different representations of the same concept; recognise the fraction and decimal equivalents for one half, one quarter, three quarters, tenths and hundredths.

6.3 Order a set of decimals with one or two places and position them on a number line.

2.1 is marked on the number line.

Mark 0.65 on the number line.

6.4 Round decimals to the nearest whole number or one decimal place.

On the number line, which of these numbers would be closest to 1?

0.1 0.9 1.2 1.9

TIMSS Grade 4

Write each of these to the nearest whole number.

13.7 is nearest to ............... 388 is nearest to ...............

3.38 is nearest to ...............


Include the terms numerator, denominator.

Mixed numbers

Include expressing the improper fraction or mixed number in its simplest form.

Decimal place value

Include identifying values of digits as fractions or decimals.

Fraction and decimal equivalents

When writing hundredths as fractions, exclude reducing the fraction to its lowest terms, other than for one half, one quarter, three quarters and tenths.

Rounding decimals

Include units of measure and money.


7 Extend understanding of calculations with whole numbers to decimals and apply these skills to solve problems

Addition, subtraction, multiplication and division

7.1 Multiply or divide decimals with up to two places by 10 and 100 and recognise the effect.

Jamal is thinking of a number. He multiplies it by 10, then adds 25. The answer is 100. What is Jamal’s number?

7.2 Read and write money in decimal form; convert riyals in decimal form to dirhams, and dirhams to riyals in decimal form.

Mental and written calculations

7.3 Use and explain mental methods to:

• add and subtract two-digit decimals;

e.g. 3.5 + 0.6 4.8 – 1.9

• multiply or divide a simple decimal by a one-digit number.

e.g. 0.7 × 8 0.03 × 6 5.4 ÷ 9 0.45 ÷ 5

7.4 Use and explain written column methods to:

• add and subtract decimals with up to two places;

e.g. 36.4 + 5.62 4.78 − 3.9

• multiply decimals with up to two places by a one-digit whole number.

e.g. 1.47 × 9

7.5 Estimate answers to calculations involving decimals by using approximations.

Circle the number that is about the same as the correct answer to 4.9 + 4.8.

1 5 4 10 7 20

Solving problems

7.6 Solve one-step problems involving decimals, including rounding answers to a specified degree of accuracy.

Circle two numbers with a total of 0.12.

0.1 0.5 0.05 0.7 0.07 0.2

Bader is 1.35 metres tall. Salman is 1.4 metres tall. How much taller than Bader is Salman?

Some boys go camping for 6 nights. It costs QR 5.50 for each boy to camp each night. How much does it cost for each boy to camp for the 6 nights?

A box of 7 pineapples costs QR 31.50. How much does each pineapple cost?

Division by 10, 100

Limit decimal quotients to no more than two places.

Include units of measure and money.


Keep within multiplication table facts.

Written methods

Include adding and subtracting numbers with different numbers of decimal places.

Include money and decimal measurements, e.g. 4.6 m.

Solving problems

Include money and decimal measurements.

Include checking reasonableness of answer.



By the end of Grade 4, students identify parallel and perpendicular lines, recognise lines of symmetry and complete symmetrical figures. They identify angles as greater than or less than a right angle and put a set of acute and obtuse angles in order of size. They identify simple properties of squares, rectangles and parallelograms. They construct squares and rectangles on grids and by using a set square and ruler, drawing lines to the nearest millimetre. They solve simple problems involving scale. They find the perimeters of irregular polygons and perimeters and areas of shapes that can be split into squares and rectangles. They choose and use suitable units to estimate and measure and read scales with increasing accuracy. They convert centimetres to metres or millimetres, using decimal notation. They calculate a time interval of up to 1 hour in minutes, and larger time intervals that are multiples of 15 minutes.

Students should:

8 Identify and use simple properties of shapes

8.1 Identify parallel and perpendicular lines; draw parallel and perpendicular lines using a ruler and set square.

ABCD is a rectangle.

Which of these statements is true?

• Line CD is parallel to line BC.

• Line AB is perpendicular to line AD.

8.2 Classify angles as greater than, equal to, or less than a right angle; compare given angles and put them in order of size.

8.3 Know simple side and angle properties of:

• squares: four equal sides, four right angles, opposite sides parallel;

• rectangles: opposite sides equal, four right angles, opposite sides parallel;

• parallelograms: opposite sides equal, opposite angles equal, opposite sides parallel.

Here are five shapes on a square grid.

Which two shapes fit together to make a square? Explain why.

Parallel and perpendicular lines

Include use of the terms horizontal, vertical.

Angles

Include use of the terms acute and obtuse.

Exclude reflex angles.


Explain which of the following statements are true and why.

• All squares are rectangles.

• All rectangles are squares.

• All rectangles are parallelograms.

• Some parallelograms are squares.

• All parallelograms are rectangles.

8.4 Use knowledge of properties of squares and rectangles to:

• construct squares and rectangles of given dimensions using a ruler and set square;

• construct squares and rectangles on grids.

The line on the grid is one side of a square. On the grid, draw the other three sides of the square. Use a ruler.

Draw two more straight lines to make a rectangle. Use a ruler.

8.5 Identify lines of symmetry in 2-D shapes and complete a 2-D shape to make it symmetrical about a given line of symmetry.

Complete this shape so that it is symmetrical about the mirror line. Use a ruler.

8.6 Solve simple problems involving properties of lines, squares and rectangles.

9 Choose appropriate units and measurement tools to estimate and measure

9.1 Choose and use suitable units to estimate and measure.

9.2 Read measurements from scales with increasing accuracy, e.g. read a measure on a scale marked in intervals of 100 g to the nearest 0.1 kg.

9.3 Convert metres to centimetres, centimetres to millimetres, millimetres to centimetres, and centimetres to metres, using decimal notation.

ICT opportunity

Use Logo to construct squares and rectangles.

Symmetry

Include identifying and visualising line symmetry in the environment.

Include finding more than one line of symmetry in a given shape.

Exclude rotation symmetry.

Measuring scales Include linear scales and circular dials.


What is 76 millimetres to the nearest centimetre?

Children get points for how far they jump. Standing long jump

over 80 cm 1 point

over 100 cm 2 points





Hassan jumped 1.25 metres. How many points does he get?

9.4 Know the fraction and decimal equivalents for one half, one quarter, three quarters, one tenth and one hundredth of 1 km, 1 m, 1 kg and 1 l, e.g. know that 0.75 kg, or three quarters of a kilogram, is 750 grams.

9.5 Measure and draw lines to the nearest millimetre.

Join the dots which are 40 millimetres apart. Use a ruler.

9.6 Solve simple problems involving:

• scale;

• proportional reasoning.

A map has a scale of 6 cm to 1 km. A road on the map is 15 cm long. What is the length of the road in kilometres?

Ana placed 12 identical blocks on the scales as shown in the figure.

How many blocks should be removed to reduce the weight to 1000 g?

TIMSS Grade 4

9.7 Convert weeks to days and vice versa; know the number of days in each month; use a Western calendar to find a time interval in days and weeks; be aware of the Hijri calendar and lunar months.

9.8 Calculate a time interval of up to 1 hour in minutes, and larger time intervals that are multiples of 15 minutes.

10 Understand perimeter and area

10.1 Solve simple problems involving the area and/or perimeter of squares and rectangles.

A thin wire 20 centimetres long is formed into a rectangle. The width of the rectangle is 4 centimetres. What is its length?

A. 5 centimetres B. 6 centimetres C. 12 centimetres D. 16 centimetres

TIMSS Grade 4

Area problems

Include simple problems requiring application of the formula for the area of a rectangle or square.


Rafeya has a 6 cm by 6 cm square mat.

How many 2 cm by 2 cm square tiles will cover the mat?

10.2 Recognise that shapes with the same area can have different perimeters and that shapes with the same perimeter can have different areas.

10.3 Find the perimeter of a regular or irregular polygon with whole-number sides.

This ring is made of regular pentagons, with sides of 5 centimetres.

What is the length of the outer edge of the ring?

Data handling

By the end of Grade 4, students complete, extract and interpret information presented in lists, two-way tables and simple Carroll diagrams. They solve problems using data presented in bar charts and tables.

Students should:

11 Solve problems by collecting, organising, representing and interpreting data and drawing conclusions

11.1 Complete a table from given information.

11.2 Answer questions by collecting data systematically and:

• recording and interpreting information in lists, two-way tables and Carroll diagrams;

• representing and interpreting information in one- or two-variable bar charts with a scale numbered in intervals of 2, 4, 5, 10, 20 or 100.

Here is a Carroll diagram for sorting numbers. Write one number in each box of the diagram.

less than 100 100 or more

multiples of 20

not multiples of 20

Perimeters

Include estimating and measuring perimeters.

ICT opportunity

Use data from the Internet.

ICT opportunity

Use a spreadsheet with graphs and charts to represent data in tables and bar charts.


Here is a table of the numbers of stamps used each day in school. Value of stamp Sunday Monday Tuesday Wednesday Thursday

QR 1 22 11 14 32 13 QR 2 8 17 4 6 19 QR 3 6 0 2 8 4 QR 4 6 0 6 1 0 QR 5 6 0 2 12 3

How many QR 2 stamps did the school use on Wednesday?

This is a graph of one kind of stamp that the school used.

Which stamp is it?

This chart shows the results of a survey. All of the students in Class 4 said which ice cream they like best. Each of them chose one kind of ice cream.

How many students are in Class 4? What fraction of Class 4 likes chocolate ice cream best?






Students represent and interpret routine and non-routine mathematical problems using calculations, mathematical symbols, diagrams, graphs, charts and tables. They describe and explain their problem solving strategies and reasoning, orally and in writing. They find examples that match a general statement and present their results and conclusions systematically. They use ICT to support their work, including deciding when it is appropriate to use a calculator to solve a number problem and how to interpret the display in the context of the problem.

Number and algebra

Students recognise properties of numbers up to 100 such as factor, multiple or prime. They add mentally several one-digit numbers and use factors to multiply and divide two-digit numbers mentally. They use their understanding of place value to multiply and divide whole numbers and decimals with up to three places by 10, 100 or 1000, and to order decimals to three places. They divide whole numbers with up to three digits by a two-digit number, giving the answer as a fraction or a decimal. They solve routine and non-routine problems, with and without a calculator, using addition, subtraction, multiplication or division of whole numbers or decimals, including inverse operations. In solving problems, students check the reasonableness of results by estimating answers and by referring to the context. They round answers to a suitable degree of accuracy. They order related and unrelated fractions with denominators up to 12, identify simple equivalent fractions and compare simple fraction, decimal and percentage equivalents. They recognise proportions of a whole and use simple fractions and percentages to describe these and to solve problems. They begin to use simple formulae expressed in words. They find pairs of numbers related by a given rule, extend and find missing terms in number sequences, and describe in words the relationship between one term and the next.


Students recognise a quarter turn as 90° or a right angle, and estimate, measure and draw acute angles in degrees. They identify equilateral, isosceles and right-angled triangles. They know the sum of angles at a point, on a straight line and in a triangle, and use these properties to find unknown angles. They identify the nets of a cube and cuboid. They use coordinates in the first quadrant. They convert one metric unit of measurement to another using decimal notation and interpret with appropriate accuracy readings on a range of measuring instruments. They use the 24-hour clock. They solve simple problems involving finding the areas and perimeters of shapes related to rectangles and squares, and the volumes of cuboids.

Data handling

Students collect discrete data, grouping them where appropriate, and represent and interpret data in frequency diagrams. They construct and

Grade 5


interpret simple line graphs and Venn diagrams. They solve problems by asking and answering their own questions related to data, and drawing and analysing graphs, charts and tables, including those generated by ICT.





Number and algebra


Data handling

60% 30% 10%






By the end of Grade 5, students represent and interpret routine and non-routine mathematical problems using calculations, mathematical symbols, diagrams, graphs, charts and tables. They describe and explain their problem solving strategies and reasoning, orally and in writing. They find examples that match a general statement and present their results and conclusions systematically. They use ICT to support their work, including deciding when it is appropriate to use a calculator to solve a number problem and how to interpret the display in the context of the problem.

Students should:



This four-digit number is a square number. Write in the missing digits.

9 9

What size is the angle between the hands of a clock at 5 o’clock?

1.2 Describe and explain their problem solving strategies and reasoning, orally and in writing.

Here is a number sequence. Write the missing number.

1 3 6 10

Explain how you worked it out.

I have a square made out of paper. The square measures 20 cm by 20 cm. I keep folding it in half until I have a rectangle that is 5 cm by 10 cm.

How many times did I fold it?

Sabbah needs half a litre of lemon juice to make a fruit drink. She squeezes 14 lemons. Each lemon gives her 35 ml of juice. Does Sabbah have enough lemon juice for her recipe?

Grade 5

Key standards


Cross-references

Standards are referred to using the notation RP for reasoning and problem solving, NA for number and algebra, GM for geometry and measures, and DH for data handling, e.g. standard NA 2.3.




1.3 Find examples that match a general statement.

Circle the numbers in this list that are 1 more than a multiple of 6.

13 16 23 31 46 55 63

Fill in three missing numbers.

Which of these triangles has an obtuse angle and is not isosceles?

1.4 Present results and conclusions in an organised way.

Adel, Essa, Haya and Mai played some games at home. They played two different games. Each child played each of the others in each game.

Mai recorded how many Adel recorded who won games each person won. each game.

Adel / / / Game 1 Game 2

Mai / / / Adel Mai

Haya / / / / Haya Essa

Essa / / Essa

Adel Mai

Adel Haya

Haya Mai

Adel forgot to put one of the names on his table. Use Mai’s table to work out what the missing name is.

Give one reason why Mai’s table is a good way of recording the results. Give one reason why Adel’s table is a good way of recording the results.

1.5 Decide when it is appropriate to use a basic calculator to solve a numerical problem; know how to interpret the display in the context of the problem.

There is 60 g of rice in one portion. How many portions are there in a 3 kg bag of rice?

1.6 Use ICT to support their mathematics.


Number and algebra

By the end of Grade 5, students recognise properties of numbers up to 100 such as factor, multiple or prime. They add mentally several one-digit numbers and use factors to multiply and divide two-digit numbers mentally. They use their understanding of place value to multiply and divide whole numbers and decimals with up to three places by 10, 100 or 1000, and to order decimals to three places. They divide whole numbers with up to three digits by a two-digit number, giving the answer as a fraction or a decimal. They solve routine and non-routine problems, with and without a calculator, using addition, subtraction, multiplication or division of whole numbers or decimals, including inverse operations. In solving problems, students check the reasonableness of results by estimating answers and by referring to the context. They round answers to a suitable degree of accuracy. They order related and unrelated fractions with denominators up to 12, identify simple equivalent fractions and compare simple fraction, decimal and percentage equivalents. They recognise proportions of a whole and use simple fractions and percentages to describe these and to solve problems. They begin to use simple formulae expressed in words. They find pairs of numbers related by a given rule, extend and find missing terms in number sequences, and describe in words the relationship between one term and the next.

Students should:

2 Use their knowledge of place value in the decimal system to round numbers and estimate answers to calculations

2.1 Read and write decimals with up to three places; represent place value in decimals with up to three places (tenths, hundredths, thousandths) in words, models or expanded form.

e.g. 34.627 30 4 0.6 0.02 0.007= + + + +

6 7210 100 100034.627 30 4= + + + +

2.2 Compare and order decimals with one, two or three places.

Write these numbers in order of size, starting with the smallest.

2.3 Multiply or divide whole numbers and decimals by 10, 100 or 1000.

2.4 Round:

• whole numbers to the nearest 10, 100 or 1000;

• decimals to the nearest whole number, or to one or two decimal places.

2.5 Use approximations to estimate answers to addition, subtraction, multiplication and division calculations involving whole numbers or decimals, with and without a calculator.

2.6 Round answers to calculations, including those done on a calculator, to a given degree of accuracy.

Decimal place value

Include identifying values of digits as fractions or decimals.

Dividing by 10, 100, 1000

Limit decimal quotients to up to three places.

Rounding decimals

Include measures and money.

Estimating and rounding

Include checking whether answers are reasonable.


3 Calculate with whole numbers and apply these skills to solving routine and non-routine problems

3.1 Add mentally several one-digit numbers.

3.2 Know the squares of whole numbers 1 to 12 and the corresponding square roots; use the notation 52 for 5 × 5.

3.3 Use factors to multiply and divide two-digit numbers mentally.

e.g. 12 × 15 = (12 × 5) × 3 = 180

72 ÷ 18 = (72 ÷ 2) ÷ 9 = 36 ÷ 9 = 4

3.4 Identify properties of numbers less than 100 such as multiple, factor or prime; use factors to multiply and divide whole numbers by multiples of 10, 100 or 1000.

e.g. 45 × 6000 = (45 × 6) × 1000 = 270 × 1000 = 270 000

720 ÷ 800 = (720 ÷ 8) ÷ 100 = 90 ÷ 100 = 0.9

3.5 Express the result of division by a whole number as a fraction or decimal.

e.g. 3548 5 9 or 9.6÷ =

3.6 Use and explain written column methods to multiply and divide three-digit numbers by a two-digit number.

e.g. 683 × 37, 437 ÷ 28


3.7 With and without a calculator, solve word problems with up to three steps using addition, subtraction, multiplication or division of whole numbers, including real-life problems involving money or measures.

Work out the number halfway between 86 and 145. Write it in the box.

12 adults and 20 children visited a zoo. It cost QR 8.50 for an adult and QR 4.50 for a child. How much did they pay altogether?

3.8 With and without a calculator, solve missing-number problems involving inverse operations.

Write in the missing digit.

92 ÷ 14 = 28

3.9 Model with an equation a problem situation involving an unknown number.

Mohammed thought of a number. He doubled it, then added 4. The answer was 88. Write an equation to show this.

3.10 With and without a calculator, solve non-routine problems involving large or small numbers, or decimals.

Use two + signs and two – signs. Put them in the boxes to make this equation correct.

5 4 3 2 1 = 5

A large cake costs QR 9 and a small cake costs QR 7.50. Noura bought some cakes. Altogether, she spent QR 42. How many cakes did Noura buy?

Square roots

Exclude the √ symbol.

Dividing by multiple of 10, 100, 1000

Limit decimal quotients to decimals with up to three places.

Word problems

Include related but different units in problems involving money, cm and mm, cm and m, days and weeks.

Include problems involving proportional reasoning.


4 Understand the equivalence of fractions and use fractions to solve problems

4.1 Compare and order related and unrelated fractions with denominators up to 12 by using diagrams.

Here are some number cards.

Use two of the cards to make a fraction which is less than 12 .

4.2 Add and subtract two proper fractions where one denominator is a multiple of the other.

4.3 Relate fractions to division (see standard NA 3.5); find fractions of quantities.

4.4 Know that the value of a fraction does not change if the numerator and denominator are multiplied or divided by the same number. Use this to identify equivalent fractions by:

• reducing a fraction to its simplest form by cancelling a common factor;

• listing the first eight equivalent fractions, given a unit fraction with a denominator not greater than 10.

4.5 Solve up to two-step problems involving fractions, including finding the whole given a fractional part.

Three fifths of a number is 12. What is the number?

5 Calculate with and solve problems involving decimals

5.1 Know equivalent fractions and decimals in simple cases, such as 0.25 = 1⁄4, 0.75 = 3⁄4, 0.125 = 1⁄8, 0.333 ≈ 1⁄3, 0.667 ≈ 2⁄3.

5.2 Convert fractions expressed as tenths, hundredths or thousandths to decimals, and vice versa.

5.3 Use written column methods to:

• add and subtract decimals with up to three places; e.g. 6.475 + 5.625, 4.375 – 3.125, 4.06 – 1.275

• multiply and divide decimals with up to two places by a one-digit whole number. e.g. 5.83 × 9, 3.47 ÷ 4

5.4 With and without a calculator, solve problems with up to two steps using addition, subtraction, multiplication or division of decimals, including rounding the answer to a specified degree of accuracy (see standard NA 2.6).

5.5 Use a calculator to solve problems involving inverse operations.

What is the missing number?

6669 ÷ = 38

Ordering fractions

The number of fractions to order should not be greater than three.

Include both increasing and decreasing order.

Use diagrams to support ordering of fractions.

Converting decimals

Include reducing the fraction to its simplest form.

Decimals

Include adding and subtracting decimals with different numbers of places.

Solving problems

Include measurements and money.

Include checking whether answers are reasonable.


6 Use a basic calculator effectively

6.1 Use a calculator for calculations involving several digits, including decimals, interpreting the display. Know how to:

• clear the display before starting a calculation;

• use the [+], [–], [×] and [÷] keys, the [=] key and decimal point;

• change an accidental wrong entry by using the [clear entry] key;

• key in and interpret calculations involving money or measures, e.g. – interpret 0.75 as 75 dirhams in the context of money; – key in 4.65 m + 3.85 m as 4.65 [+] 3.85 [=], and interpret the

outcome of 8.5 as 8.5 m, or 8 metres 50 centimetres; • interpret a rounding error such as 6.9999999 as 7.

7 Understand the meaning of percentage and the relationship between fractions, decimals and percentages

7.1 Understand that percentage means the number of parts per 100 and that it is used for comparisons.

7.2 Find and compare fraction, decimal and percentage equivalents for halves, quarters, tenths and hundredths.

Tick each card that shows more than a half.

7.3 Calculate a given percentage of a whole, e.g. 25% of 30 kg.

Shade 10% of this grid.

7.4 Without a calculator, solve up to two-step word problems involving percentages.

Ten students chose which subject they liked best.

Number of students

Arabic 4

Mathematics 2

Science 3

English 0

Geography 1

Total 10

Which subject did 20% of the students choose? What percentage of the students chose mathematics or science?

8 Write and evaluate simple formulae in words

8.1 Express simple unit conversions in words and numbers.

e.g. number of days = number of weeks × 7 number of weeks = number of days ÷ 7

Using a calculator

Include:

• estimating answers to check reasonableness;

• giving a result to a given degree of accuracy, e.g. the nearest metre;

• discussing when and when not to use a calculator.

Percentages

Include the use of the percentage symbol (%).

Solving problems

Exclude percentage increase/decrease and percentage profit and loss.

Conversions

Relate to standards GM 11.1 and GM 11.2.


8.2 Evaluate simple formulae by substituting numbers for words.

The total cost in riyals of a number of pens is given by the formula: cost of pens = 5 × number of pens What is the cost of 11 pens?

9 Use rules to generate ordered pairs and simple sequences

9.1 Find pairs of numbers related by a given rule.

9.2 Given a rule, generate a simple sequence.

The rule for a number sequence is: ‘The next number is the sum of the two previous numbers.’ Use the rule to write in the three missing numbers.

, , , 5, 8, 13, 21, …

9.3 Describe in words the relationship between one term of a sequence and the next.


By the end of Grade 5, students recognise a quarter turn as 90° or a right angle, and estimate, measure and draw acute angles in degrees. They identify equilateral, isosceles and right-angled triangles. They know the sum of angles at a point, on a straight line and in a triangle, and use these properties to find unknown angles. They identify the nets of a cube and cuboid. They use coordinates in the first quadrant. They convert one metric unit of measurement to another using decimal notation and interpret with appropriate accuracy readings on a range of measuring instruments. They use the 24-hour clock. They solve simple problems involving finding the areas and perimeters of shapes related to rectangles and squares, and the volumes of cuboids.

Students should:

10 Identify properties of and relationships in geometric shapes

Angles and geometrical reasoning

10.1 Associate 360° with one whole turn, 270° with a three quarters turn, 180° with a half turn or a straight line, and 90° with a quarter turn or right angle.

10.2 Identify rotation symmetry in 2-D shapes.

Shade in one more square so that this design has rotation symmetry of order 4.

Rules

Include showing why a number pair satisfies a given rule.

Use coordinates in the first quadrant to plot pairs that satisfy a given rule (see also standard GM 10.11).

ICT opportunity

Use a spreadsheet to generate simple sequences.

Rotation symmetry

Include the term order of rotation symmetry.

Use ICT to show examples of rotation symmetry.


10.3 Estimate and compare the size of acute angles.

Which of these angles is closest to 45°?

10.4 Use the labelling conventions for angles, e.g. ∠ABC.

10.5 Find unknown angles in a straight line or around a point.

Look at this diagram. Calculate the size of angle y.

10.6 Recognise and use the property that the angle sum of a triangle is 180°.

10.7 Know the angle and side properties of isosceles, equilateral and right-angled triangles; classify triangles according to these properties.

10.8 Find unknown angles involving the angle properties of an isosceles, equilateral or right-angled triangle, and rectangles and squares.

Here is an equilateral triangle inside a rectangle. Calculate the value of angle x.

10.9 Identify nets of open and closed boxes in the shape of a cube or cuboid.

A cube has shaded triangles on three of its faces.

Here is the net of the cube. Draw in the two missing shaded triangles.

Constructions

10.10 Use a protractor to measure acute angles in degrees and to draw a given acute angle.

Measure accurately the smallest angle in this shape. Use a protractor.

Angles

Include the terms acute and obtuse and the notation ∠ABC or ∠x to name angles.

Labelling angles

Include identifying given angles on a labelled diagram.


10.11 Use coordinates in the first quadrant to represent points; understand that the length of a horizontal line segment equals the difference of the x-coordinates and the length of a vertical line segment equals the difference of the y-coordinates.

Here is a graph.

Six points on the line are marked with dots (•). They are equally spaced. What are the coordinates of the point A?

Haya says: ‘The point B has coordinates (11, 5).’ Use the graph to explain why she cannot be correct.

10.12 Construct 2-D shapes on grids.

Use the dots to draw a quadrilateral with no right angles. Use a ruler.

11 Measure and compare length, mass, capacity and time

11.1 Convert between km, m, cm and mm, kg and g, and l and ml, using decimal notation.

This table shows the mass of some fruits and vegetables. Complete the table.

grams kilograms

potatoes 3500 3.5

apples 1.2

grapes 250

ginger 0.03

All the water in the first two containers is poured into the third empty container. Draw where the water level is in the third container.

11.2 Convert standard units of time, including years to months or weeks, weeks to days, days to hours, hours to minutes, minutes to seconds, and vice versa (see also standard NA 8.1).

Coordinates and ordered pairs

Link to work on NA 9.1, 9.2.

Constructing shapes on grids

Include triangles (isosceles, equilateral, right-angled, scalene), squares, rectangles and other quadrilaterals.

Mass/weight

In earlier grades, weight and mass are treated as the same. From Grade 5 mass (measured in kg) is used to distinguish from weight, which is a force.

Units of measurement

Include abbreviations (km, m, cm, mm, kg, g, l, ml).

Units of time

Include the abbreviations h, min, s.


11.3 Use 24-hour clock times; use timetables and calculate a time interval in hours and minutes.

A flight from London arrived in Doha at 20:47. It departed again for London at 23:25. For how many hours and minutes was the aeroplane on the ground at Doha?

11.4 Use rulers, measuring cylinders, weighing scales and stopwatches to make measurements; read measurements from scales with appropriate accuracy.

What is the total mass of this fruit? (1.7 kg)

12 Understand and calculate perimeter, area and volume

12.1 Know the formula for the area of a rectangle and use it to find:

• the area of squares and rectangles;

• the side of a square, given its area or perimeter; The area of a square is 64 cm2. What is the length of its perimeter?

• the side of a rectangle, given its length or width and its area or perimeter.

12.2 Find the perimeter or area of shapes formed from two or more squares or rectangles.

Two rectangular tiles are used to make an L-shape. What is the perimeter of the L-shape?

What is the area of this shape?

12.3 Build solids with unit cubes and compare their volumes by counting cubes.

12.4 Derive and use the formula for calculating the volume of a cuboid.

This cuboid is made from centimetre cubes. What is its volume?

Another cuboid is made from centimetre cubes. It has a volume of 30 cubic centimetres. What could its length, height and width be?

24-hour clock

Include conversions from the 12-hour clock.

Reading measurements from scales

Include circular dials.

Areas of squares

Exclude use of √ sign.

Compound areas

Include finding an area by subtraction of areas.

Volume

Include abbreviations (m3, cm3).

Exclude conversions between m3 and cm3.


12.5 Solve simple problems involving the area and perimeter of squares and rectangles, or the volume of cuboids.

Data handling

By the end of Grade 5, students collect discrete data, grouping them where appropriate, and represent and interpret data in frequency diagrams. They construct and interpret simple line graphs and Venn diagrams. They solve problems by asking and answering their own questions related to data, and drawing and analysing graphs, charts and tables, including those generated by ICT.

Students should:

13 Solve problems by collecting, organising, representing and interpreting data and drawing conclusions

13.1 Pose questions and answer them by collecting and analysing data.

13.2 Represent and interpret discrete data in a frequency diagram.

This graph shows the number of letters in each word in 50 words from the Gulf Times.

What fraction of the 50 words have more than 6 letters?

13.3 Represent a given set of data, or data from an experiment, in a line graph; interpret line graphs.

Khalid was ill in March. This is his temperature chart.

For how many days was his temperature marked as more than 37°C? Which date showed the largest change in temperature from the day before?

Estimate Khalid’s highest temperature shown on the graph. Give your answer to 1 decimal place.

ICT opportunities

Include collecting data from the Internet.

Frequency diagrams

Include the use of ICT. For example, use a spreadsheet with graphs and charts to produce frequency diagrams/bar charts.

Line graphs

Exclude distance–time graphs.

Stress when it is appropriate and when it is not appropriate to join plotted points with a straight line; e.g. rise and fall in temperature from one reading to another is not necessarily represented by a straight line joining the two points.

Include the use of ICT. For example, use graph plotting software or a spreadsheet to generate line graphs.


This graph shows the cost of phone calls in the daytime and in the evening.

Estimate how much it costs to make a 9-minute call in the evening.

How much more does it cost to make a 6-minute call in the daytime than in the evening?

13.4 Use ICT to generate graphs, charts and tables, including frequency diagrams and line graphs.

13.5 Represent and interpret data in Venn diagrams.

Write these numbers in the correct places on the Venn diagram. Some numbers are already placed.

99 170 221





Students represent and interpret routine and non-routine mathematical problems in a range of contexts. They change from one representation to another when it would help to solve a problem. They explain their methods and reasoning, orally and in writing. They check the reasonableness of calculated results, for example, by using estimation. They search systematically for all possibilities in a given situation. They identify patterns in data or results and generalise where appropriate. They use ICT to support their mathematical work.

Number and algebra

Students calculate efficiently with whole numbers and decimals, without and with a calculator, and explain and justify their chosen methods. They use a range of strategies to calculate mentally. They use written column methods to add and subtract decimals and to multiply and divide them by a one- or two-digit whole number. They estimate answers to calculations and check results using inverse operations. They find and use factors and multiples of numbers, and identify prime numbers. They cancel common factors to reduce a fraction to its simplest form, and add and subtract proper fractions with different denominators. They change fractions to decimals or percentages, and vice versa, and calculate fractional or percentage parts of quantities and measurements, using a calculator where appropriate. They use simple ratios and apply proportional reasoning. Students simplify and evaluate simple algebraic expressions, and construct and solve simple linear equations. Given a rule, they generate the terms of a sequence. They use ordered coordinate pairs to plot points that follow a simple rule. They apply their skills to solving a range of routine and non-routine problems, including mathematical and real-life problems.


Students reflect a 2-D shape in a given mirror line and rotate a 2-D shape about one of its vertices. They recognise vertically opposite angles, angles on a straight line and around a point. They know angle sum of a triangle and use this property to find the angle sum of a quadrilateral. They use these and other properties to identify equal sides or to find the values of angles in geometric figures. They use a ruler and protractor to construct triangles, given two sides and the included angle, or two angles and the included side. They solve problems involving the area of rectangles, triangles and parallelograms, and the volume and surface area of cuboids. They find the volume of liquid in cuboid containers.

Data handling

Students answer questions by collecting data and representing them in tables and bar charts. They interpret and draw conclusions from bar charts and pie charts (circle graphs), including those generated by ICT. They understand and use the mean and median of discrete data, and describe characteristics of a data set using the range and the mode.

Grade 6






Number and algebra


Data handling

60% 30% 10%






By the end of Grade 6, students represent and interpret routine and non-routine mathematical problems in a range of contexts. They change from one representation to another when it would help to solve a problem. They explain their methods and reasoning, orally and in writing. They check the reasonableness of calculated results, for example, by using estimation. They search systematically for all possibilities in a given situation. They identify patterns in data or results and generalise where appropriate. They use ICT to support their mathematical work.

Students should:

1 Use mathematical reasoning to solve problems

1.1 Model or represent problems from a range of contexts; change from one representation to another.

Five children sorted themselves into these sets.

Complete this diagram to show the same information.

1.2 Explain their methods and reasoning, orally and in writing.

This graph shows how far a truck had travelled through the desert at different times.

Adel says: ‘The truck travelled further in the first hour than in the second hour’. Explain how the graph shows this.

Grade 6

Key standards


Cross-references

Standards are referred to using the notation RP for reasoning and problem solving, NA for number and algebra, GM for geometry and measures, and DH for data handling, e.g. standard NA 2.3.




Imagine a square of paper. Imagine folding it in half, then in half again, then cutting out one small square.

Then unfold your paper. Circle the diagram below that shows what your paper looks like now.

Explain how you know.

Ghada thinks of a number. She says: ‘Halve my number and then add 17. The answer is 83.’ What is Ghada’s number? Explain how you worked out your answer.

Leila knows that 65 × 3 = 195. Explain how she can use this information to find the answer to 165 × 3.

1.3 Check the reasonableness of calculated results, for example, by estimating.

Circle the best estimate of the answer to 32.7 × 0.48.

1.2 1.6 12 16 120 160

1.4 Search systematically for all possibilities in a given situation.

Use the digits 5, 0, 4, 7. How many different three-digit numbers can you make? In each number, each digit must be different.

An isosceles triangle has a perimeter of 12 cm. One of its sides is 5 cm. What could the length of each of the other two sides be? Two different answers are possible. Give both answers.

1.5 Identify patterns in data or results, generalising where appropriate.

The rule for this sequence of numbers is ‘add 3 each time’.

1 4 7 10 13 16 …

The sequence continues in the same way. Mary says: ‘There will never be a multiple of 3 in the sequence.’ Is she correct? Circle YES or NO. Explain how you know.

Hissa is working with whole numbers. She says: ‘If you add a two-digit number to a two-digit number you cannot get a four-digit number.’ Is she correct? Circle YES or NO. Explain why.

1.6 Use ICT to support mathematical work.


Number and algebra

By the end of Grade 6, students calculate efficiently with whole numbers and decimals, without and with a calculator, and explain and justify their chosen methods. They use a range of strategies to calculate mentally. They use written column methods to add and subtract decimals and to multiply and divide them by a one- or two-digit whole number. They estimate answers to calculations and check results using inverse operations. They find and use factors and multiples of numbers, and identify prime numbers. They cancel common factors to reduce a fraction to its simplest form, and add and subtract proper fractions with different denominators. They change fractions to decimals or percentages, and vice versa, and calculate fractional or percentage parts of quantities and measurements, using a calculator where appropriate. They use simple ratios and apply proportional reasoning. Students simplify and evaluate simple algebraic expressions, and construct and solve simple linear equations. Given a rule, they generate the terms of a sequence. They use ordered coordinate pairs to plot points that follow a simple rule. They apply their skills to solving a range of routine and non-routine problems, including mathematical and real-life problems.

Students should:

2 Calculate efficiently with whole numbers and decimals and apply these skills to solve problems

2.1 Find and use factors and multiples of numbers, and identify prime numbers; find the prime factorisation of a number, and the highest common factor (HCF) and lowest common multiple (LCM) of two numbers.

Write in the boxes the three prime numbers which multiply to make 231.

× × = 231

2.2 Know and use tests of divisibility for 2, 3, 4, 5, 6 and 10.

2.3 Use and explain mental methods to multiply and divide whole numbers and decimals in special cases:

• multiply by 25 by multiplying by 100 and dividing by 4; e.g. 48 × 25 = 48 × 100 ÷ 4 = 4800 ÷ 4 = 1200

• divide by a number with one-digit factors; e.g. 864 ÷ 24 = (864 ÷ 4) ÷ 6 = 216 ÷ 6 = 36

• multiply by a one-digit decimal; e.g. 23 × 0.7 = 23 × 7 ÷ 10 = 161 ÷ 10 = 16.1

• divide by a one-digit decimal. e.g. 48 ÷ 0.3 = 48 × 10 ÷ 3 = (48 ÷ 3) × 10 = 160

2.4 Use and explain written column methods to add and subtract decimals.

2.5 Use and explain written column methods to multiply and divide:

• whole numbers with up to four digits by a two-digit number; e.g. 2386 × 35, 1596 ÷ 38

• decimals with up to two places by a two-digit whole number. e.g. 4.17 × 24, 2.52 ÷ 42

Factors and multiples

Include common factors and common multiples.

Mental methods

Exclude decimals with more than one decimal place.


2.6 Know the order of operations for carrying out calculations with more than one step.

2.7 Use calculator methods to multiply and divide by decimals (other than the special cases in standard NA 2.3 above), to convert fractions to decimals, and to calculate with numbers involving several digits. Know how to:

• use the [clear] and [clear entry] keys, all operation keys, the [=] key and decimal point, and the [√] key, to calculate with realistic data;

• enter a negative number;

• key in and interpret the outcome of calculations involving money;

• key in fractions, recognise the equivalent decimal form, and use this to compare and order fractions;

• read the display of, say, 0.3333333 as point three recurring, know that it represents one third, and that 0.6666666 represents two thirds;

• select the correct key sequence to carry out routine calculations involving more than one step: for example, 8 × (37 + 58);

• use the memory.

Put a tick ( ) in the correct box for each calculation. Use a calculator.

2.8 Check answers for accuracy by using inverse operations.

2.9 Check answers for reasonableness by using an estimate based on approximations and by considering the context of the problem.

2.10 Round answers to calculations to a given degree of accuracy.

Problem solving with whole numbers and decimals

2.11 Solve problems with up to three steps using addition, subtraction, multiplication or division of whole numbers or decimals, with and without a calculator.

Aisha buys a pack of 24 cans of lemon drink for QR 60. She sells five of the cans for QR 4 each. How much profit did she make on the five cans?

2.12 Solve missing-number problems involving inverse operations, with and without a calculator.

Use a calculator to find the missing number.

568.1 ÷ = 24.7

2.13 Solve routine and non-routine problems involving whole numbers or decimals, including word problems based on real-life contexts, and problems in mathematical contexts, with and without a calculator.

Order of operations

Exclude brackets.

Using a calculator

Include:

• checking the reasonableness of an answer by estimating;

• giving a result to a specified degree of accuracy, e.g. the nearest metre;

• discussing when and when not to use a calculator.

Exclude the use of the % key (see NA 4.5).

Word problems

Include money and units of measurement.


3 Work with fractions and use them to solve problems

3.1 Recall that the value of a fraction does not change if the numerator and denominator are multiplied or divided by the same number and use this method to find equivalent fractions:

• reduce a fraction to its simplest form by cancelling a common factor;

• given a proper fraction, and either the numerator or denominator of an equivalent fraction, write the equivalent fraction.

3.2 Express a decimal with up to three places as a fraction, reducing it to its simplest form; know the fraction and decimal equivalents for one thousandth, one eighth, three eighths, five eighths and seven eighths.

3.3 Convert a proper fraction or mixed number to a decimal using division:

• with a calculator;

• without a calculator.

3.4 Recognise and use the notation for recurring decimals, e.g. 4.3 , for 4.33333… or 1

34 , and 0.18 , for 0.181818… or 211 .

3.5 Compare and order unrelated fractions by converting them to decimals (if necessary, with a calculator) and positioning them on a number line.

Which one of these fractions is closest in value to 13 ?

10 20 30 40 5031 61 91 121 151

3.6 Add and subtract proper fractions with different denominators by writing them with a common denominator.

This square is divided into three parts.

Part A is 13 of the area of the square. Part B is 2

5 of the area of the square. What fraction of the area of the square is part C?

3.7 Multiply a proper fraction by a proper fraction.

3.8 Know that, since 1 ÷ a = 1a = 1 × 1

a , division is equivalent to multiplication by the reciprocal, and use this principle to divide a proper fraction by a whole number.

3.9 Solve up to two-step word problems involving fractions.

There are 24 coloured cubes in a box. Three quarters of the cubes are red, four of the cubes are blue and the rest are green. How many green cubes are in the box?

One more blue cube is put into the box. What fraction of the cubes in the box are blue now?

4 Calculate and use percentages

4.1 Change fractions and decimals to percentages, and vice versa.

4.2 Calculate a percentage of a whole.

Find 45% of QR 400.

Converting fractions

Exclude denominators greater than 10, other than 100 and 1000.

Ordering fractions

Include both ascending and descending order.

Percentages

Exclude percentage increase/decrease and percentage profit and loss.


4.3 Express one quantity as a percentage of another.

In a survey, 34 out of 40 people liked swimming. What percentage of people liked swimming?

4.4 Find the whole, given a part and the percentage, in simple cases.

10% of a number is 21.5. What is the number?

4.5 With and without a calculator, solve word problems involving percentages.

Jamal had 75 riyals. He gave 60 riyals to his sister. What percentage of his riyals did Jamal give to his sister?

5 Use simple ratios and direct proportion

5.1 Use simple ratios to show the relative sizes of two quantities.

The ratio of the distance from A to B to the distance from B to C is 3 : 1. The distance from A to C is 60 centimetres. Calculate the distance from A to B.

5.2 Recognise equivalent ratios and reduce a ratio to its lowest terms.

5.3 Recognise simple cases when two quantities are in direct proportion.

A recipe for a fruit drink for 6 people is:

chopped oranges 300 g lemonade 1500 ml orange juice 750 ml

Noor wants to make enough fruit drink for 10 people. How many millilitres of lemonade will she need?

5.4 Without a calculator, solve simple problems involving ratios or direct proportion.

Students voted whether they preferred the Lunar Park or the Zoo. The result was a ratio of 10 : 3 in favour of the Lunar Park. 40 students voted in favour of the Lunar Park. How many students voted for the Zoo?

Samira paid 45 dollars for a present in the USA. 1 dollar equals QR 3.2. What would the present cost in QR?

6 Use and interpret letters to write, simplify and evaluate simple expressions and solve simple equations

6.1 Use a letter to represent an unknown number or variable and write a simple algebraic expression in one variable to model a given situation.

6.2 Simplify algebraic expressions with a single variable by collecting terms.

Simplify: a. 7 + 2t + 3t b. b + 7 + 2b + 10

6.3 Evaluate simple algebraic expressions and formulae by substituting numbers for words or letters.

The formula c = 5n represents the cost in riyals of n pens at QR 5 each. What is the cost of 12 pens?

Finding the whole

Limit to parts of 25%, 50%, 75%, and multiples of 10%.

Calculators

Exclude the use of the % key so that students understand how the calculation is carried out.

Ratio

Include the notation a : b.

Include simple scale drawings such as maps or plans.

Direct proportion

Include converting riyals to US dollars ($) or British pounds (£), and vice versa.

ICT opportunity

Use a spreadsheet to generate conversion tables.

Simplification

Exclude brackets.


6.4 Solve simple problems by constructing and solving simple linear equations with integer coefficients (unknown on one side only), using inverse operations; verify the solution by substituting in the original equation.

The length of a photograph frame is twice its height. It is made with 126 cm of wood. What is the length of the frame?

7 Generate sequences and plot graphs of functions

7.1 Given a rule, generate a simple sequence:

• find a term from the previous term, given a rule such as ‘add 6’;

• find a term given its position in the sequence.

This sequence follows the rule: ‘to get the next number, add the two previous numbers’. Write in the next two numbers in the sequence.

2.1 2.2 4.3 6.5 … …

Each number in this sequence is double the previous number. Write in the missing numbers.

… 3 6 12 24 48 …

7.2 Use ordered coordinate pairs to plot points that follow a simple rule.

Write the coordinates of the next triangle in the sequence.

7.3 Plot the graphs of simple linear functions (y given explicitly in terms of x); use the graph to find the value of y, given the value of x, and vice versa.

The diagram shows the graph of y = 7 – x.

Write the coordinates of one point on the line between A and B.

Equations

Include the forms: ax = b ax + b = c a – bx = c

Exclude brackets.

Sequences

Include common integer sequences, such as odd and even numbers, square numbers.

Include extending sequences to negative numbers.

ICT opportunity

Use a spreadsheet to generate sequences.

ICT opportunity

Use graph plotting software.



By the end of Grade 6, students reflect a 2-D shape in a given mirror line and rotate a 2-D shape about one of its vertices. They recognise vertically opposite angles, angles on a straight line and around a point. They know angle sum of a triangle and use this property to find the angle sum of a quadrilateral. They use these and other properties to identify equal sides or to find the values of angles in geometric figures. They use a ruler and protractor to construct triangles, given two sides and the included angle, or two angles and the included side. They solve problems involving the area of rectangles, triangles and parallelograms, and the volume and surface area of cuboids. They find the volume of liquid in cuboid containers.

Students should:


Transformations

8.1 Draw the reflection of a 2-D shape in a given mirror line.

The shaded triangle is a reflection of the white triangle in the mirror line.

Write the coordinates of point A and point B.

8.2 Draw the rotation of a 2-D shape about one of its vertices.

Here is a shaded shape on a grid. The shape is rotated 90° clockwise about point A. Draw the shape in its new position on the grid.

Angles, shapes and geometrical reasoning

8.3 Use the labelling conventions for angles, lines and geometric figures.

8.4 Identify angles in a straight line, at a point and vertically opposite angles.

8.5 Know that the angle sum of a triangle is 180°; derive and use the property that the angle sum of a quadrilateral is 360°.

Two of the angles in a quadrilateral add up to 280°. The other two angles are equal. What is the size of one of these other two angles?

Transformations

Keep to the first quadrant.

ICT opportunity

Use a dynamic geometry system (DGS) to explore reflections and rotations.

The programming language Logo can also be used to explore rotations.

Vertically opposite angles

Include a proof that vertically opposite angles are equal.

Angle sum of a triangle

Exclude a formal proof. Include a demonstration, e.g. by tearing off the corners of a paper triangle.


8.6 Identify equal lengths or find unknown angles in geometric figures, involving:

• angles in a straight line, around a point or vertically opposite angles;

• the angle sum of a triangle;

• side or angle properties of an isosceles, equilateral, right-angled and scalene triangle;

• side or angle properties of a square, rectangle or parallelogram.

Triangle ABD is the reflection of triangle ABC in the line AB.

Explain why ACD is an equilateral triangle. What is the size of angle x?

Constructions

8.7 Use ruler and protractor to construct a triangle:

• given two angles and the included side;

• given two sides and the included angle.

Here is a sketch of a triangle. It is not drawn to scale. Construct the triangle using a ruler and protractor.

8.8 Use ruler, protractor and set square to measure and draw angles, perpendicular and parallel lines, rectangles and squares.

8.9 Use ICT to draw shapes such as rectangles and squares.

9 Solve problems involving perimeter, area and volume

9.1 Derive and use formulae for the area of a triangle and the area of a parallelogram.

On the grid draw a triangle with the same area as the shaded rectangle. Use a ruler.

This is a centimetre grid. Draw three more lines to make a parallelogram with an area of 10 cm2. Use a ruler.

Unknown angles

Exclude:

• properties related to diagonals of 2-D shapes;

• identifying congruent triangles;

• the need for extra construction lines.

ICT opportunity

Use a dynamic geometry system to explore constructions.

Angles

Include reflex angles.

Area of triangle and parallelogram

Include identifying the base and the corresponding height.

Exclude finding the base or height given the area.


9.2 Find:

• the surface area of a cube, given the length of an edge;

• the edge of a cube, given its volume or surface area;

• one dimension of a cuboid, given its volume and the other two dimensions.

9.3 Recognise the equivalence of 1 litre or 1000 ml and 1000 cm3; know and use the formula for the volume of a cuboid to find the volume of liquid in a cuboid container.

9.4 Solve problems involving:

• the area of rectangles, triangles and parallelograms;

• the volume of cubes and cuboids and of liquids in cuboid containers.

Boxes measure 2.5 cm wide by 4.5 cm long by 6.2 cm deep. A shopkeeper puts them in a tray.

Work out the largest number of boxes that can lie flat in the tray.

Data handling

By the end of Grade 6, students answer questions by collecting data and representing them in tables and bar charts. They interpret and draw conclusions from bar charts and pie charts (circle graphs), including those generated by ICT. They understand and use the mean and median of discrete data, and describe characteristics of a data set using the range and the mode.

Students should:

10 Collect, process, represent and interpret data and draw conclusions

10.1 Collect data to answer a question, including from the Internet.

10.2 Represent data in tables and bar charts.

Here is a chart of the maximum temperature each day for a week in Cairo in January. The temperature on Saturday was 14°C. Draw in the bar for Saturday.

Make a table showing the same information as on the bar chart.

Volumes of cubes

Limit to cubes and cuboids with integer dimensions.

Exclude use of 3√ sign.

Volume of liquid

Include conversions between l, ml and cm3.

Include problems about the height of water in cuboid containers.

Link to work in science but exclude finding volume of a solid by displacing liquid.

Data handling and ICT

Data handling provides many opportunities to use spreadsheets and graph drawing packages to present tables, charts and graphs. The Internet is an excellent source of real data of interest to students.


10.3 Read and interpret graphs and charts, including bar charts and pie charts.

This pie chart shows the different ways that wood is used in the world.

Use the pie chart to estimate the percentage of wood that is used for paper.

54% of the wood is used for fuel. Calculate the angle for the fuel sector on the pie chart.

10.4 Use ICT to generate graphs, charts and tables, including bar charts and pie charts.

Pie charts

Exclude construction of pie charts, other than those generated using ICT.

ICT opportunity

Use a spreadsheet and graph drawing package.


10.5 Calculate a mean and find the median of a set of data; draw conclusions about a set of data based on the shape of the graph and the mode or range.

An article from the Gulf Times has 50 words in it. Here is a bar chart of the number of letters in each word.

What is the range of the number of letters in the words used? What is the most common number of letters used in a word?

Bashir runs 100 metres ten times. These are his times in seconds.

13.413.5

13.014.0

13.914.4

13.713.8

13.314.0

What is his mean time?

Alia counts the matches in nine matchboxes. Here are her results for the nine boxes.

Number of matches in a box

48 49 50 51 52 53 54

What is the median number of matches in a box? What is the mode for the number of matches in a box? What is the range of the number of matches in a box?

Alia counts the matches in one more box. She works out that the mean number of matches in all ten boxes is 51. Calculate how many matches are in the tenth box.





Students represent and interpret routine and non-routine mathematical problems in a range of contexts, changing from one representation to another as appropriate. They choose and use appropriate mathematical techniques and tools to solve problems, including ICT. They present and explain their solutions and conclusions in the context of the original problem, orally and in writing. They reason logically to establish the truth of a statement. They make general statements using words and symbols.

Number and algebra

Students calculate accurately with whole numbers, decimals, fractions and percentages using mental, written and calculator methods. They solve a range of routine and non-routine problems. They estimate and round answers, checking results to see if they are reasonable. They know the order of operations and use brackets appropriately. They find positive and negative square roots and order, add, subtract, multiply and divide positive and negative numbers, using the symbols <, ≤, >, ≥, =, ≠ correctly. They calculate with fractions using the four operations, including combined operations for addition and subtraction. They know which number to consider as 100%, or a whole, in problems involving comparisons, and use this to evaluate one number as a fraction or percentage of another. They understand and use the equivalences between fractions, decimals and percentages, calculate using ratios, and use the unitary method to solve problems involving direct proportion. They describe mathematical situations by using words, symbols and diagrams. They express in symbolic form and use formulae involving up to two operations, simplify and evaluate linear expressions, and construct and solve linear equations with integer coefficients. They find the rule for the next term or the nth term in a linear sequence that relates to a spatial pattern. They use coordinates in all four quadrants to plot the graphs of simple linear functions.


Students identify alternate, supplementary and corresponding angles and know angle properties related to diagonals of squares, rectangles, parallelograms and rhombuses. They use these and other properties to find the values of unknown angles in geometric figures. They use a ruler and compasses to construct angle bisectors and perpendicular bisectors and, together with a protractor, to construct simple geometric figures from given data. They identify and describe properties of solid shapes, and recognise their nets. They use a range of measurements, including rate and speed, to solve problems. They choose suitable units to make estimates of measurements. They find the area of trapeziums and other rectilinear figures. They know common estimates for π and apply formulae to estimate the circumference and area of a circle.

Grade 7


Data handling

Students answer questions by constructing, analysing and drawing conclusions from tables, pictograms, bar charts and line graphs drawn on paper or generated using ICT. They interpret pie charts. They compare different representations of the same set of data. They understand and use the probability scale from 0 to 1, and find probabilities of single events in simple contexts. They know that the total probability of all mutually exclusive outcomes is 1 and use this to solve problems.




For Grades 7 to 9, the proportion of algebra in the number and algebra strand increases as the proportion of number decreases, and so is shown separately in the table below. The weightings of the content strands relative to each other are as follows:

Number Algebra Geometry and

measures* Data

handling

Grade 7 30% 25% 27.5% 17.5%

Grade 8 25% 30% 27.5% 17.5%

Grade 9 15% 40% 27.5% 17.5% * including trigonometry in Grade 9






By the end of Grade 7, students represent and interpret routine and non-routine mathematical problems in a range of contexts, changing from one representation to another as appropriate. They choose and use appropriate mathematical techniques and tools to solve problems, including ICT. They present and explain their solutions and conclusions in the context of the original problem, orally and in writing. They reason logically to establish the truth of a statement. They make general statements using words and symbols.

Students should:


1.1 Model or represent mathematical problems from a range of contexts, changing from one representation to another as appropriate.

What number goes in the box to make this equation true?

32.45 × = 253.11

The number 6 is halfway between 4.5 and 7.5.

Fill in the missing numbers below. The number 6 is halfway between 2.8 and ............ The number 6 is halfway between –12 and ............

1.2 Choose and use appropriate mathematical techniques and tools to solve a problem, including ICT.

Use your calculator to explore these calculations: 7 × 9, 7 × 99, 7 × 999, … Predict the answer to 7 × 9 999 999.

Explore 9 × 9, 9 × 99, 9 × 999, … Find a general rule.

In Class 7, 80% of the students like dates. 75% of the students who like dates also like chocolate. In Class 7, what percentage of the students like both dates and chocolate?

Triangle ABC is isosceles and has a perimeter of 20 centimetres. Sides AB and AC are each twice as long as BC. Calculate the length of the side BC.

Grade 7

Key standards


Cross-references





1.3 Present and explain solutions and conclusions in the context of the original problem, orally and in writing.

Which is larger, 13 or 2

5 ? Explain how you know.

A teacher needs 220 booklets. The booklets are in packs of 16. How many packs must the teacher order?

A newspaper prints a graph predicting what the ages of teachers will be in 6 years time. It says that the total number of male teachers will be about the same as the total number of female teachers.

Asif says: ‘Generally, male teachers will tend to be younger than female teachers.’ Is he correct? Circle YES or NO. Explain how you know.

1.4 Use logical reasoning to establish the truth of a statement.

Huda has these eight rods.

She can use five of her rods to make a rectangle.

Huda can make a square with all eight of her rods. Show how she can do this.

1.5 Make general statements using words and symbols.

P stands for a multiple of 3. Q stands for a different multiple of 3. Tick ( ) each statement to show whether it is always true, sometimes true or never true.

always true

sometimes true

never true

The sum of P and Q is a multiple of 6.

The difference between P and Q is a multiple of 3.

The product of P and Q is a multiple of 9.


Number and algebra

By the end of Grade 7, students calculate accurately with whole numbers, decimals, fractions and percentages using mental, written and calculator methods. They solve a range of routine and non-routine problems. They estimate and round answers, checking results to see if they are reasonable. They know the order of operations and use brackets appropriately. They find positive and negative square roots and order, add, subtract, multiply and divide positive and negative numbers, using the symbols <, ≤, >, ≥, =, ≠ correctly. They calculate with fractions using the four operations, including combined operations for addition and subtraction. They know which number to consider as 100%, or a whole, in problems involving comparisons, and use this to evaluate one number as a fraction or percentage of another. They understand and use the equivalences between fractions, decimals and percentages, calculate using ratios, and use the unitary method to solve problems involving direct proportion. They describe mathematical situations by using words, symbols and diagrams. They express in symbolic form and use formulae involving up to two operations, simplify and evaluate linear expressions, and construct and solve linear equations with integer coefficients. They find the rule for the next term or the nth term in a linear sequence that relates to a spatial pattern. They use coordinates in all four quadrants to plot the graphs of simple linear functions.

Students should:

2 Calculate efficiently with whole numbers and decimals and apply these skills to solve problems

2.1 Use the symbols <, ≤, >, ≥, =, ≠ correctly to compare numbers or expressions.

Explain why 8 < √ 70 < 9.

2.2 Round whole numbers to any given power of 10, and decimals, including measures, to the nearest whole number or given decimal place; use the approximation sign ≈.

Circle the number below that is closest in value to 0.1. 0.01 0.05 0.11 0.2 0.9

A family uses about 6000 litres of water per week. Approximately how many litres of water do the family use each year? A. 30 000 B. 240 000 C. 300 000 D. 2 400 000 E. 3 000 000

2.3 Multiply and divide integers and decimals by 0.1, 0.01, 0.001.

2.4 Use and explain mental methods to add, subtract, multiply and divide whole numbers and decimals in simple cases.

Circle the two numbers that add up to 1. 0.1 0.65 0.99 0.45 0.35

Calculate 15.05 – 14.84.

Khalid knows that 137 × 28 = 3836. Explain how he can use this information to work out 138 × 28.

Use of ICT

Function graph plotters, graphics calculators and spreadsheets help to explore ideas in number and algebra.

Rounding

Include rounding answers to calculations.


2.5 Use and explain written column methods (algorithms) to:

• multiply by a decimal with up to two decimal places, understanding where to position the decimal point by considering equivalent fractions; e.g. 0.13 × 0.7 = 0.091, since 13 7 91

100 10 1000× =

• divide by a decimal with up to two decimal places by transforming the calculation to one with an integer divisor. e.g. 2.6 ÷ 0.14 = 260 ÷ 14

2.6 Consolidate adding and subtracting decimals with up to three places. e.g. calculate 2.201 − 0.75.

2.7 Know the order of operations and work out the value of expressions containing more than two terms.

2.8 Use brackets to show which operation to perform first when writing expressions containing more than two terms.

2.9 Use a scientific calculator, including the memory, for calculations with whole numbers and decimals, including combined operations, working efficiently, interpreting the display and rounding answers in the context of the problem. (See also NA 4.7.)

Use a calculator to work out 49.3 × (2.06 + 8.5).

What number goes in the box? 404.09 ÷ = 8.5

A shop sells sheets of sticky labels. On each sheet there are 36 rows and 18 columns of labels. How many labels are there altogether on 45 sheets?



In a country there are sixty-six point eight million people. 51.5% of them are female. How many females are there to the nearest million?

2.12 Round answers to calculations to a given degree of accuracy, or to a degree of accuracy appropriate to the context of the problem.

Every day a machine makes 100 000 paper clips which go into boxes. A full box contains 120 paper clips. How many boxes can be filled using 100 000 paper clips?

2.13 With and without a calculator, solve routine and non-routine problems involving whole numbers, decimals, money or measures.

Write the missing digits. 323 × 7 = 15 18

In a discus throwing competition, the winning throw was 61.60 m long. The second-place throw was 59.72 m long. How much longer was the winning throw than the second-place throw?

A rubber ball rebounds to half the height it drops. The ball is dropped from a rooftop 18 m above the ground. What is the total distance travelled by the time the ball hits the ground the third time?

A. 31.5 m B. 40.5 m C. 45 m D. 63 m

TIMSS Grade 8

Written column methods

The word algorithm is derived from the name of the mathematician Abu Ja’far Muhammad ibn Musa al-Khwarizmi, c. 780–850.

Calculations

Exclude tedious calculations when use of a calculator is not allowed.

Problems

Include combined operations.


3 Calculate with positive and negative numbers and evaluate powers and roots

3.1 Represent, compare and order positive and negative numbers using words and models, including a number line.

3.2 Add, subtract, multiply and divide positive and negative numbers and solve problems involving them.

Sharifa makes a sequence of numbers starting with 100. She subtracts 45 each time. Write the next two numbers in the sequence. 100 55 10

Using negative numbers only, fill in the missing numbers in the boxes. – = 5 – = –5

3.3 Evaluate mentally positive integer powers of whole numbers, and positive and negative square roots of perfect squares to 144; use the square root sign √; establish upper and lower bounds for square roots of numbers to 100.

Look at the numbers below. Which is the largest? Which is equal to 92?

16 25 34 43 52 61

If √81 < n < √144, then n could be which of the following numbers?

A. 9 B. 11 C. 12 D. 13

TIMSS Grade 8

3.4 Use the x2, √x and xy keys of a scientific calculator.

4 Calculate with fractions and use them to solve problems

4.1 Convert terminating decimals to fractions, expressing them in their simplest form.

Write 0.28 as a fraction reduced to its lowest terms.

4.2 Convert fractions to decimals, using division or a scientific calculator, and represent them on a number line.

4.3 Order fractions by creating a common denominator or by converting them to decimals.

4.4 Relate operations with fractions to situations and models (e.g. relate the multiplication of one fraction by another to rectangular areas).

Add 610 and 6

5 . Use an arrow (↓) to show the result on the number line.

4.5 Use mental methods to find a fraction of a number or quantity, and to compare, add, subtract, multiply and divide proper fractions in simple cases.

4.6 Use written methods to add, subtract, multiply and divide mixed numbers, including combined operations for addition and subtraction.

Calculate 8 435 15÷ .

4.7 Use the fraction key on a scientific calculator to calculate with fractions.

4.8 Solve problems involving the use of fractions in a range of contexts.

Positive and negative numbers

Include mental calculations.

Powers and roots

Exclude laws of indices.

Converting decimals to fractions

Exclude recurring decimals.

Scientific calculator

See also NA 2.9.


5 Calculate percentages and use them to solve problems

5.1 Find fraction, decimal and percentage equivalents.

5.2 Estimate and calculate a given percentage of a quantity.

5.3 Express one quantity as a percentage of another.

5.4 Find the whole, given a percentage part.

40% of a class of students own a bicycle. If 16 students own a bicycle, how many students are there in the class?

5.5 Calculate the outcome of a percentage increase or decrease, and find a percentage profit or loss, given the cost price and selling price.

A small bottle of fruit juice holds 450 ml. A large bottle of fruit juice holds 35% more than the small bottle. How much fruit juice will the large bottle hold?

5.6 Use mental methods to work out calculations of percentages in simple cases.

5.7 Solve routine and non-routine problems involving percentages, including with a calculator.

6 Use ratios and proportions to solve problems

6.1 Relate ratios to fractions; given a ratio:

• express one value as a fraction of another;

• state how many times larger one value is than another.

What is the ratio of the side of a square to its perimeter? A. 1

1 B. 12 C. 1

3 D. 14

6.2 Identify and find equivalent ratios and reduce a ratio to its lowest terms.

6.3 Divide a quantity in a given ratio to show the relative sizes of two or more quantities.

6.4 Find the ratio of two or more quantities.

2½ cartons of apple juice and 1½ cartons of orange juice are poured into a big jug. What is the ratio of apple juice to orange juice in the jug?

6.5 Compare two ratios.

Fruit drink A is made from 1 part orange juice and 9 parts pineapple juice. Fruit drink B is made from 1 part orange juice and 4 parts pineapple juice. Which has more orange juice, 1 litre of fruit drink A or 1 litre of fruit drink B?

6.6 Solve problems involving ratios.

Class 7 has 12- and 13-year-old students. There are 28 students in the class. The ratio of 13-year-old students to 12-year-old students is 4 : 3. How many 13-year-old students are in the class?

Purple paint is made by mixing 3 parts of blue paint and 2 parts of red paint. For 25 litres of purple paint, how many litres of blue paint do you need?

6.7 Use the unitary method to solve problems involving direct proportion.

Assume that one British pound (£) is worth QR 4.50. In London, a magazine costs £3.50. In Doha, the same magazine costs QR 15.50. In which city is the magazine cheaper, London or Doha?

Problems

Include sales discount, sales tax, service charge. Discuss Islamic alms.

Ratios and fractions

Include writing x : y = a : b

as =x ay b .

Direct proportion

Include currency conversions, and scales.


Here is a recipe for strawberry ice cream for 8 people.

0.5 litre cream 1 kg strawberries 250 g sugar

Faraj makes enough strawberry ice cream for 12 people. How much cream does he use?

Najib makes strawberry ice cream using 2.5 kg strawberries. How much sugar does he use?

7 Write, simplify and evaluate linear expressions and solve linear equations

7.1 Write simple linear expressions and formulae to model a situation.

Write an expression for each missing length in this rectangle. Write each expression as simply as possible.

7.2 Simplify algebraic expressions with one or two variables by collecting like terms and multiplying a single term over a bracket.

m represents a positive number. Which of these is equivalent to m + m + m + m? A. m + 4 B. 4m C. m4 D. 4(m + 1)

Write each expression in its simplest form.

7 + 2t + 3t

b + 7 + 2b + 10

(3d + 5) + 4(d – 2)

3m – (–m)

7.3 Evaluate formulae and linear expressions by substituting integers for letters and using the correct order of operations.

n stands for a number. Complete this table of values.

n 5n – 2

20

38

p = qr. If p = 12, and q = 3, then r is equal to: A. 3

4 B. 3 C. 4 D. 12 E. 36

7.4 Solve linear equations or inequalities with integer coefficients (unknown on one or both sides, including brackets), and verify the solution.

Find y if 2(y − 3) = 16.

Find x if 10x − 15 = 5x − 10.

TIMSS Grade 8

7.5 Solve problems by writing and solving linear equations.

The perimeter of a rectangular swimming pool is 116 metres. The length of the swimming pool is 22 metres longer than the width. What is the width of the swimming pool?

Ali thinks of a number. He says: ‘Multiplying my number by 2 and then adding 5 gives the same answer as subtracting my number from 23.’ What is Ali’s number?

Algebra

The book Hisab al-jabr w'al-muqabala is the most famous of al-Khwarizmi's works. The word algebra is derived from the title. The book is about ‘what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects are concerned’. The section on algebra was the first of its kind, although the algebra was done entirely in words with no symbols used.

Linear equations

Include the forms ax = b + cx + d a(x + b) = c ax + b = cx + d

Include solving a linear equation by inverse operations, and by transforming both sides in the same way.



8.1 Extend and find missing terms in numeric or geometric patterns or sequences using words, diagrams or symbols (term-to-term or position-to-term rules).

Some number chains start like this: 1 → 5 → Show three different ways to continue this number chain. For each of your chains, write down the next three numbers. Then write down the rule you are using.

8.2 Use the [ANS] and [ENTER] keys on a graphics calculator to generate term-to-term and position-to-term sequences.

8.3 Generalise the relationship between one term of a sequence and the next, or between the number of the term and the term, using words or symbols.

Here is a pattern of L-shapes made with sticks.

The rule is: ‘Find the number of sticks for a shape by first multiplying the shape number by 4, then adding 3.’ Work out the number of sticks for the shape that has shape number 10.

Aisha uses 59 sticks to make an L-shape in this pattern. What is its shape number?

Write a formula to work out the number of sticks for any L-shape. Use S for the number of sticks and N for the shape number.

Explain why the number of sticks for a shape increases by 4 from one shape to the next.

8.4 Use coordinates in all four quadrants to plot graphs of y = mx + c, on paper and using ICT.

Suggest possible equations for these straight line graphs.

8.5 Use a straight line graph to estimate the value of y given the value of x, and vice versa; identify intercepts on axes.

Does the point (25, 75) lie on the straight line y = 3x? Explain how you know.

The straight line y = 4x + 2 crosses the y-axis at point A and the x-axis at point B. Write the coordinates of each of the points A and B.

Sequences

Include the sequence of square numbers, and powers of 2 and 10.

ICT opportunity

Investigate patterns using a spreadsheet or a graphics calculator.

ICT opportunity

Use graph plotting software or a graphics calculator.



By the end of Grade 7, students identify alternate, supplementary and corresponding angles and know angle properties related to diagonals of squares, rectangles, parallelograms and rhombuses. They use these and other properties to find the values of unknown angles in geometric figures. They use a ruler and compasses to construct angle bisectors and perpendicular bisectors and, together with a protractor, to construct simple geometric figures from given data. They identify and describe properties of solid shapes, and recognise their nets. They use a range of measurements, including rate and speed, to solve problems. They choose suitable units to make estimates of measurements. They find the area of trapeziums and other rectilinear figures. They know common estimates for π and apply formulae to estimate the circumference and area of a circle.

Students should:


Angles, shapes and geometric reasoning

9.1 Identify, sketch, label and describe angle, side, diagonal and symmetry properties of plane shapes:

• triangles (isosceles, equilateral, right-angled, acute- and obtuse-angled scalene triangle);

• quadrilaterals (square, rectangle, parallelogram, rhombus, trapezium, kite);

• polygons (pentagon, hexagon, octagon, decagon).

What are the coordinates of vertex P?

A quadrilateral must be a parallelogram if it has: A. one pair of adjacent sides equal B. one pair of parallel sides C. a diagonal as an axis of symmetry D. two adjacent angles equal E. two pairs of parallel sides

TIMMS Grade 8

9.2 Calculate unknown angles in geometric figures, involving:

• angles in a straight line, around a point or vertically opposite angles;

• corresponding, alternate and supplementary angles;

• side or angle properties of isosceles, equilateral, right-angled and scalene triangles, including the angle sum and exterior angle properties;

• angle properties of squares, rectangles, parallelograms and rhombuses, including angle properties related to their diagonals;

• angle bisectors and perpendicular bisectors.

Use of ICT

Geometry is enhanced with use of a dynamic geometry system, or DGS, which provides an interactive means for investigating results that can then be shown to be true.

Plane shapes

Include these terms: plane; acute, obtuse, reflex, base angles; interior, exterior angles; vertex/vertices, diagonal, perpendicular, parallel, hypotenuse, bisector; regular/irregular polygon.

Unknown angles

Exclude the need for extra construction lines.


Triangle ABC is equilateral. Calculate the size of angle x.

The shape ABCD is a rectangle. BD is parallel to EF. Calculate the sizes of the angles x and y.

9.3 Identify solid shapes and describe their properties, including planes of symmetry; identify and sketch the nets of cubes, cuboids, prisms, pyramids and cones.

I cut a sector from a paper circle.

I join the two radii to make a 3-D shape. Which of the 3-D shapes below do I make?

Constructions

9.4 Use a ruler, set square, protractor and compasses to:

• measure and draw line segments and angles;

• draw parallel and perpendicular lines;

• construct rectangles of given dimensions;

• draw circles and arcs, and use circles and arcs to construct Islamic patterns;

• construct angle bisectors and perpendicular bisectors;

• construct simple geometric figures from given data.

Here is a rough sketch of a sector of a circle.

Make an accurate full-size drawing of this sector.

9.5 Use ICT to generate and explore constructions.

3-D shapes

Include cubes, cuboids, prisms, cylinders, pyramids, cones and spheres.

Construction of figures

Include constructing triangles given three sides, or given a right angle, hypotenuse and side.

Using ICT

Use a dynamic geometry system (DGS).


10 Use a range of measures, including rate and speed, to solve problems

10.1 Calculate the area of triangles, rectangles, parallelograms, trapeziums and related shapes.

A rectangular picture is pasted on a sheet of white paper as shown.

What is the area of the white paper not covered by the picture?

This trapezium has an area of 20 cm2. Give three possible sets of values of a and b.

10.2 Name the parts of a circle; know common estimates for π, and formulae for the circumference and area of a circle, and use these to estimate the circumference and area of circles; use a calculator and the π key to work out decimal equivalents to appropriate degrees of accuracy.

This shape is made from wire. Each semicircle has a diameter of 15 cm.

Calculate the total length of the wire.

The diagram shows a right-angled triangle inside a circle of radius 5 centimetres.

Calculate the area of the shaded part of the diagram.

10.3 Estimate length, circumference/perimeter, area, volume, capacity, mass, time and angle in problem situations, choosing suitable units.

Which of these is closest to the mass of a hen’s egg?

A. 5g B. 50 g C. 250 g D. 500 g

10.4 Understand and use measures of rate; solve problems involving calculating an average rate.

A person’s heart beats 72 times per minute. At this rate, about how many times does it beat in an hour?

A. 420 000 B. 42 000 C. 4200 D. 420

TIMSS Grade 8

Circles

Include the terms centre, circumference, radius, diameter, arc, sector, segment, chord, semicircle.

Include giving the area or circumference in terms of π, e.g. area = 49π cm2, and:

• the use of approximations such as π ≈ 3, or π ≈ 22⁄7 to give estimates, e.g. area ≈ 150 cm2;

• the use of the π key on a scientific calculator to work out decimal equivalents.

Rate and speed

Link to work in science.


10.5 Know that average speed = distance/time; solve problems involving calculating average speed, distance or time.

Majed drove 55 km from Doha to Al Khor. The journey took 75 minutes. What was his average speed?

A goods train travels 300 km at an average speed of 40 km/h. For how many hours does the train travel?

A truck travels for 21/2 hours at an average speed of 36 km/h. How far does the truck travel?

Data handling

By the end of Grade 7, students answer questions by constructing, analysing and drawing conclusions from tables, pictograms, bar charts and line graphs drawn on paper or generated using ICT. They interpret pie charts. They compare different representations of the same set of data. They understand and use the probability scale from 0 to 1, and find probabilities of single events in simple contexts. They know that the total probability of all mutually exclusive outcomes is 1 and use this to solve problems.

Students should:


Statistics

11.1 Answer questions by collecting and classifying data, and constructing and interpreting:

• tables,

• bar charts and pictograms,

• line graphs,

• pie charts,

on paper and using ICT.

This graph shows how high two rockets go during their flight.

Estimate: a. how much higher rocket A reaches than rocket B; b. the time after the start when the two rockets are at the same height; c. how long rocket A was more than 200 m above the ground.

Average speed

Include distance–time graphs.


Data handling provides many opportunities to use spreadsheets and graph drawing packages to present tables, charts and graphs. The Internet is an excellent source of real data of interest to students.

Graphs and charts

Exclude the construction of pie charts requiring calculation of angles.

Include both horizontal and vertical formats for pictograms and bar charts.


80 people were asked which sport they liked best. 30 liked football. 25 liked swimming. 10 liked horse riding. 15 liked tennis. Complete the pie chart to show this information.

11.2 Compare different representations of the same set of data and determine which are the most useful for a given purpose.

11.3 Calculate the mean of a small set of data; find the mode, median and range; distinguish between the purposes for which these are used.

Sara has three number cards. The mode of the three numbers is 5. The mean of the three numbers is 8. What are the three numbers?

Ali played three games in a competition. His mean score was 3 points. His range was 4 points. What points might Ali have scored in his three games?

Which two numbers have a mean of 10 and a range of 8?

There are four people in Sharifa’s family. Their shoe sizes are 4, 5, 7 and 10. What is the median shoe size in Sharifa’s family?

The median of these five numbers is 12.

5 5 12 13 20

Write a set of four numbers that has a median of 12.

Here is a bar chart showing rainfall in New York.

There is a dotted line on the bar chart. Explain why it cannot show the mean rainfall for the four months.

ICT opportunity

Compare representations using a spreadsheet and graph drawing package.


Probability

11.4 Judge the likelihood of an event as certain, more likely, equally likely, less likely, or impossible.

Here are two spinners, P and Q. Each one is a regular hexagon.

For each statement, put a tick ( ) if it is true or a cross ( ) if it is not true.

a. Scoring 1 is more likely on P than on Q. b. Scoring 2 is more likely on P than on Q. c. Scoring 3 is as equally likely on P as on Q.

11.5 Use the probability scale, and represent probability as a fraction, proportion, decimal or percentage.

Here is a spinner.

Estimate the probability that the arrow stops in sector E. Show this probability by putting a cross ( ) on the probability scale below.

11.6 Know that the total probability of all mutually exclusive outcomes is 1, and use this to solve problems.

In a box of potato chips there are cheese, chicken and plain packets. The probability of picking a chicken packet is 7 out of 10. The probability of picking a cheese packet is 1 out of 5. What is the probability of picking a plain packet?

11.7 Solve simple problems based on equally likely outcomes for a single event.

Here are two spinners. Each outcome on each spinner is equally likely.

What is the probability of spinning a 4 on spinner A? Write your answer as a fraction.

On which spinner is it more likely to get a 1? Give a reason for your answer.





Students solve routine and non-routine mathematical problems in a range of contexts. They represent and interpret problems and solutions in numeric, algebraic, geometric or graphical form, using correct terms and notation. They choose and use appropriate mathematical techniques and tools to solve a problem, including ICT. They use diagrams and explanatory text to explain the solution to a problem and support it with evidence. They present a concise, reasoned argument orally and by using symbols. They use step-by-step reasoning to deduce properties or relationships in a given geometrical figure. They find a counter-example to show that a conjecture is false and begin to consider special cases. They find alternative solutions to problems.

Number and algebra

Students solve routine and non-routine problems by calculating accurately with positive and negative whole numbers, decimals and fractions, and with percentages, ratios and proportions. They select mental, written or calculator methods as appropriate, applying the commutative, associative or distributive laws. They estimate and calculate positive integer powers of numbers, and square and cube roots, using the power and root keys of a scientific calculator where appropriate. They simplify and evaluate algebraic expressions and formulae, and find the sum or difference of simple algebraic fractions with integer denominators. They formulate and use linear expressions to model situations. They construct and solve linear equations, including those with simple fraction coefficients, and determine whether given values satisfy an equation. They extend and find missing terms in numeric, geometric or algebraic sequences, and generalise the relationship between one term and the next, or describe the nth term, using symbols. They interpret and sketch graphs of proportional or linear functions representing practical situations, including distance–time and conversion graphs. Given the graph of a function, they identify intercepts on axes and intervals where the function increases, decreases or is constant.


Students identify all the symmetries of 2-D shapes. They calculate interior and exterior angles of polygons. They solve problems using angle and symmetry properties of polygons and angle properties of parallel and intersecting lines. They identify the reflection, rotation or translation of a 2-D shape, and draw simple transformations, including a combination of two transformations. They recognise similar shapes and enlarge shapes by a positive integer scale factor. They construct 2-D shapes from given information, including scale drawings. They visualise and describe 3-D shapes in different orientations. They convert measurements within systems of units. They solve problems involving speed or density, or the volume and surface area of cubes, cuboids, prisms and cylinders, using a calculator where appropriate. They recognise that measurements are not precise.

Grade 8


Data handling

Students solve problems by selecting and using an appropriate method of data collection, including from secondary sources. They collect and record continuous data using equal class intervals. They recognise that inappropriate grouping of data can be misleading. They construct bar charts, frequency diagrams and pie charts. They compare two data sets, using the range, median or mean, and the shape of the corresponding frequency distributions. They interpret data sets by drawing conclusions, making predictions, and estimating values between and beyond given points. They use data from experiments to estimate probability for favourable outcomes and understand that different outcomes may result from repeating an experiment. They use problem contexts to calculate theoretical probabilities for possible outcomes.






measures* Data

handling

Grade 7 30% 25% 27.5% 17.5%

Grade 8 25% 30% 27.5% 17.5%







By the end of Grade 8, students solve routine and non-routine mathematical problems in a range of contexts. They represent and interpret problems and solutions in numeric, algebraic, geometric or graphical form, using correct terms and notation. They choose and use appropriate mathematical techniques and tools to solve a problem, including ICT. They use diagrams and explanatory text to explain the solution to a problem and support it with evidence. They present a concise, reasoned argument orally and by using symbols. They use step-by-step reasoning to deduce properties or relationships in a given geometrical figure. They find a counter-example to show that a conjecture is false and begin to consider special cases. They find alternative solutions to problems.

Students should:


1.1 Represent and interpret problems and solutions in numeric, algebraic, geometric or graphical form, using correct terms and notation.

Put brackets into this expression to make it correct.

102 ÷ 10 ÷ 10 ÷ 10 ÷ 10 = 100 This is a centimetre grid. On the grid draw a triangle that has an obtuse angle and an area of 7.5 cm2. Use a ruler.

A child is having a bath. The simplified graph shows the depth of the water in the bath.

From A to B both taps are turned full on. What might be happening at point B? Which part of the graph shows the child getting into the bath?

Grade 8

Key standards


Cross-references





1.2 Choose and use appropriate mathematical techniques and tools to solve a problem, including ICT.

Use a graphics calculator to draw some straight line graphs passing through (4, –6).

1.3 Use diagrams and explanatory text to explain the solution to a problem and support it with evidence.

All the sections on this rectangular spinner have equal areas. Maryam says: ‘All the numbers on the spinner have the same probability of coming up.’ Explain why Maryam is not correct.

12 rectangles, all the same size, are arranged to make a square, as shown in the diagram. Calculate the area of one of the rectangles.

1.4 Present a concise, reasoned argument orally, in writing and by using symbols.

Explain why the sum of two consecutive odd numbers is even.

1.5 Use step-by-step reasoning to deduce properties or relationships in a given geometrical figure.

The diagram shows a rectangle that just touches an equilateral triangle.

Find the size of the angle marked x. Explain how you worked out your answer.

1.6 Find a counter-example to show that a conjecture is false and begin to consider special cases.

‘The sum of four even numbers is divisible by 4.’ Is this statement always true, sometimes true or never true? Justify your answer.

1.7 Find and explain alternative solutions to problems.

Two numbers are in the ratio 3 : 2. One of the numbers is 0.6. There are two possible answers for the other number. What are the two possible answers?


Number and algebra

By the end of Grade 8, students solve routine and non-routine problems by calculating accurately with positive and negative whole numbers, decimals and fractions, and with percentages, ratios and proportions. They select mental, written or calculator methods as appropriate, applying the commutative, associative or distributive laws. They estimate and calculate positive integer powers of numbers, and square and cube roots, using the power and root keys of a scientific calculator where appropriate. They simplify and evaluate algebraic expressions and formulae, and find the sum or difference of simple algebraic fractions with integer denominators. They formulate and use linear expressions to model situations. They construct and solve linear equations, including those with simple fraction coefficients, and determine whether given values satisfy an equation. They extend and find missing terms in numeric, geometric or algebraic sequences, and generalise the relationship between one term and the next, or describe the nth term, using symbols. They interpret and sketch graphs of proportional or linear functions representing practical situations, including distance–time and conversion graphs. Given the graph of a function, they identify intercepts on axes and intervals where the function increases, decreases or is constant.

Students should:

2 Solve numerical problems, including problems involving measures

2.1 Estimate and calculate positive integer powers of whole numbers and decimals; know cube roots of perfect cubes to ±216; use the cube root sign 3√; find approximate values of square roots of whole numbers to 100.

√12 = 3.46 to two decimal places

2.2 Use the x2, √x and xy keys of a scientific calculator.

Jamal thinks of a number. He uses his calculator to multiply the number by itself and then adds 10. His answer is 34.01. What is Jamal’s number?

2.3 Know the general principles of the commutative, associative and distributive laws and how they underpin mental and written calculations.

Which of the following is FALSE when a, b and c are different numbers?

A. (a + b) + c = a + (b + c) B. ab = ba C. a + b = b + a D. (ab)c = a(bc) E. a − b = b − a

TIMSS Grade 8

Use of ICT


Powers and roots

Include positive and negative square roots and cube roots.

Exclude laws of indices.

Use a calculator or spreadsheet to find approximate values of square roots.


2.4 Use mental, written and calculator methods for the four operations to solve problems involving whole numbers, decimals, money or measures, interpreting the calculator display and rounding answers appropriately.

A stack of 200 identical sheets of paper is 2.5 cm thick. What is the thickness of one sheet of paper?

A. 0.008 cm B. 0.0125 cm C. 0.05 cm D. 0.08 cm

Which of these is 89.0638 rounded to the nearest hundredth?

A. 100 B. 90 C. 89.1 D. 89.06 E. 89.064

TIMSS Grade 8



3 Calculate with fractions, percentages, ratios and proportions, and use them to solve problems

3.1 In appropriate cases, use mental methods to:

• compare, add, subtract, multiply and divide proper fractions;

• calculate using percentages;

• divide a quantity in a given ratio, or find the ratio of two or more quantities.

Write a proper fraction that is larger than 2324 .

The price of a sack of rice is raised from QR 60 to QR 75. What is the percentage increase in the price?

A. 15% B. 20% C. 25% D. 30%

Majed combines 5 litres of red paint, 2 litres of blue paint and 2 litres of yellow paint. What is the ratio of red paint to the total amount of paint?

A. 52 B. 9

4 C. 54 D. 5

9


3.2 Use written methods for combined operations with mixed numbers, including brackets.

Calculate:

1. 3 324 3 82 1+ −

2. ( )3 2 14 3 4+ ×

TIMSS Grade 8

3.3 Use a scientific calculator to calculate with fractions, percentages, ratios or proportions.

3.4 Solve routine and non-routine problems involving fractions, percentages, ratios or proportions.

Two groups of tourists each have 60 people. 34 of the first group and 2

3 of the second group travel by air. How many more people in the first group than the second group travel by air?

Calculations


Ratios and fractions

Include writing x : y = a : b

as =x ay b .

Problems

Include problems involving scale drawings.


Last year there were 1175 students at a school. This year there are 16% more students than last year. Approximately how many students are at the school this year?

A. 1700 B. 1600 C. 1500 D. 1400 E. 1200

The table shows the values of x and y, where x is proportional to y. x 3 6 P

y 7 Q 35

What are the values of P and Q?

A. P = 14 and Q = 31 B. P = 10 and Q = 14 C. P = 10 and Q = 31 D. P = 14 and Q = 15 E. P = 15 and Q = 14

TIMSS Grade 8

4 Write, simplify and evaluate linear expressions and formulae, and solve linear equations

4.1 Add, subtract, multiply and divide simple expressions containing variables; add and subtract simple algebraic fractions with integer denominators.

Simplify:

1. 3a × 5ab

2. 3m3n ÷ 9m

3. 43 2x x −+

4. 3( 5)23 2

xx −−

4.2 Simplify and compare algebraic expressions to determine their equivalence.

4.3 Evaluate linear expressions or formulae by substituting given integer values of the variables.

4.4 Formulate linear expressions or equations to model a situation.

Write an equation to represent the following sentence. ‘When half of x is added to 10, the result is 24.’

Ibrahim wanted to find three consecutive whole numbers that add up to 81. He wrote the equation (n – 1) + n + (n + 1) = 81. What does the n stand for?

A. The least of the three whole numbers B. The middle whole number C. The greatest of the three whole numbers D. The difference between the least and the greatest of the three whole numbers

TIMSS Grade 8

4.5 Factorise simple algebraic expressions by removing common factors.

Factorise: 1. ax + ay 2. 2m2 + 6mn

4.6 Determine whether given integer values satisfy a given linear equation.

Algebraic expressions

Include use of the terms coefficient, variable, constant, linear, expression.


4.7 Solve linear equations or inequalities with fractional coefficients (whole-number denominators), and simple cases of decimal coefficients; verify the solution.

72x < is equivalent to:

A. x < 72 B. x < 5 C. x < 14 D. x > 5 E. x > 14

TIMSS Grade 8 Solve:

1. 1 12 35x x+ = −

2. 2 + 0.6x = 2x

3. 2 33 4x x −+ =

4.8 Solve problems by writing and solving linear equations or inequalities.

There are 54 kilograms of apples in two boxes. The second box of apples weighs 12 kilograms more than the first. How many kilograms of apples are in each box?

TIMSS Grade 8


5.1 Use a graphics calculator to generate sequences and plot graphs.

5.2 Extend and find missing terms in numeric, algebraic or geometric patterns or sequences (term-to-term or position-to-term rules).

The numbers in the sequence 2, 7, 12, 17, 22, … increase by fives. The numbers in the sequence 3, 10, 17, 24, 31, … increase by sevens. The number 17 occurs in both sequences. If the two sequences are continued, what is the next number that will be seen in both sequences?

TIMSS Grade 8

5.3 Generalise the relationship between one term of a sequence and the next, or describe the nth term, using symbols.

Enter the terms and sum for row 7 in the table below. Row Terms Sum

Row 1 1 1 Row 2 1 + 3 4 Row 3 1 + 3 + 5 9 Row 4 1 + 3 + 5 + 7 16

: Row 7

:

Without writing out all the terms, what is the sum for row 20? What is the value of the sum for row n?

TIMMS Grade 8

5.4 Know that a function is a relation between two sets called the domain and the range in which each member of the domain is related to precisely one member of the range, called its image.

5.5 Recognise that functions can be represented as ordered pairs, tables, graphs, words or equations; given a function in one representation, generate an equivalent representation.

Linear equations

Include the forms

x/a = b x/a = b/c x/a + x/b = c x/a + b = x/c + d

where a, b, c and d are integers.

Sequences

Include the sequence of triangular numbers, and cube numbers.

ICT opportunity

Include the extension of number sequences using a spreadsheet or graphics calculator.


5.6 Interpret and sketch graphs of proportional and linear functions using data from practical situations, including distance–time and conversion graphs.

I went for a walk. The distance–time graph shows information about my walk.

Which of these statements describes my walk? A. I was walking faster and faster. B. I was walking slower and slower. C. I was walking north-east. D. I was walking at a steady speed. E. I was walking uphill.

5.7 Given the graph of a function, identify attributes such as points that lie on the graph, intercepts on axes and intervals where the function increases, decreases or is constant.

A straight line graph passes through the points (3, 2) and (4, 4). Which of these points also lies on the line?

A. (1, 1) B. (2, 4) C. (5, 6) D. (6, 3) E (6, 5)

TIMSS Grade 8


By the end of Grade 8, students identify all the symmetries of 2-D shapes. They calculate interior and exterior angles of polygons. They solve problems using angle and symmetry properties of polygons and angle properties of parallel and intersecting lines. They identify the reflection, rotation or translation of a 2-D shape, and draw simple transformations, including a combination of two transformations. They recognise similar shapes and enlarge shapes by a positive integer scale factor. They construct 2-D shapes from given information, including scale drawings. They visualise and describe 3-D shapes in different orientations. They convert measurements within systems of units. They solve problems involving speed or density, or the volume and surface area of cubes, cuboids, prisms and cylinders, using a calculator where appropriate. They recognise that measurements are not precise.

Students should:



6.1 Calculate interior and exterior angles of polygons.

Prove that the interior angle of a regular hexagon is 120°.

In a quadrilateral, two of the angles are 110°, and a third angle is 90°. What size is the remaining angle?

A. 50° B. 90° C. 130° D. 140° E. None of these

TIMSS Grade 8

Use of ICT

Geometry is enhanced with use of a dynamic geometry system, or DGS, which provides an interactive means for investigating and hypothesising results that can then be proved.


6.2 Identify reflection and rotation symmetry properties of 2-D shapes, including triangles, quadrilaterals and regular polygons, and draw 2-D symmetrical figures.

Which shows all the lines of symmetry for a rectangle?

TIMSS Grade 8

6.3 Use knowledge of angle properties of intersecting and parallel lines, and of the angle, side and symmetry properties of triangles, quadrilaterals and polygons, to conjecture or deduce properties in a given figure.

In the figure below, F is the centre of a regular pentagon. Work out the value of angle x. Explain how you worked out your answer.

In the figure below, ∠AOB is 70°, ∠COD is 60° and ∠AOD is 100°. What is the size of ∠COB?


6.4 Identify similar shapes; know that corresponding sides of similar shapes are in the same ratio.

Transformations

6.5 Draw transformations of a simple 2-D shape, including:

• reflection in lines parallel to or at 45° to the axes;

• rotation about the origin, or a vertex of the shape, or a mid-point of a side, through multiples of 90°;

• translation parallel to one of the axes;

• enlargement by a whole-number scale factor using a given centre of enlargement;

• the combination of two of the above transformations.

The diagram shows triangle ABC and the line y = x. Draw triangle PQR, which is the reflection of triangle ABC in the line y = x.

Symmetry

Include the terms line of symmetry, centre and order of rotation symmetry.

Similarity

Link to scale drawings and enlargements (GM 6.9 and 6.5).

Transformations

Include the terms line of reflection (mirror line), centre and angle of rotation, centre of enlargement, scale factor.

Include defining the scale factor as the ratio of two corresponding line segments.

Include the use of Cartesian coordinates.

ICT opportunity

Transformations are best developed through the use of a dynamic geometry system (DGS).


Here is a shape on a square grid. The shape is rotated 90° clockwise about point B and enlarged by a scale factor of 2. Use a ruler to draw the enlarged shape in its new position.

B

6.6 Identify the reflection, rotation, translation or enlargement of a 2-D shape.

A half turn about point T is applied to the shaded figure.

T

Which of these shows the result of the half turn?

TIMSS Grade 8

6.7 Use ICT to explore transformations and to explain or establish geometrical properties.

Constructions

6.8 Use ruler, set square, protractor and compasses to construct geometrical figures from given data, on paper and using ICT.

6.9 Construct and interpret scale drawings.

1 centimetre on the map represents 8 kilometres on the land.

About how far apart are Dukhan and Al Jumailiyah on the land?

A. 4 km B. 16 km C. 35 km D. 50 km


Transformations

Exclude finding a line of reflection, centre or angle of rotation, scale factor or centre of enlargement.

ICT opportunity

Use a dynamic geometry system (DGS).

Constructions

Include drawing parallel lines, given the distance between them.

ICT opportunity

Use a dynamic geometry system (DGS) to explore constructions and scale drawings.


Here is a drawing of a model car. 2 centimetres on the model represents 1 metre on the real car.

What is the length of the model? Give your answer in centimetres, correct to one decimal place.

The height of the model is 2.8 centimetres. What is the height in metres of the real car?

6.10 Visualise, describe and draw 3-D shapes in different orientations.

This solid will be turned to a different position.

Which of these could be the solid after it is turned?

TIMSS Grade 8

7 Use a range of measures, including compound measures, to solve problems

7.1 Use relationships among units for conversions within systems of units.

Which of these is the longest time? A. 15 000 seconds B. 1500 minutes C. 10 hours D. 1 day

A paper clip is made from 9.2 centimetres of wire. What is the greatest number of paper clips that can be made from 10 metres of wire?

A bag of rice contains 1.5 kilograms. Sara uses 60 grams of rice each day. How many days does the bag of rice last?

The number of 750 ml bottles that can be filled from 600 l of water is: A. 8 B. 80 C. 800 D. 8000


7.2 Solve problems involving average speed, distance or time, using a calculator as appropriate.

Noura went on a 40-kilometre cycle ride. This is a graph of how far she had travelled at different times.

How many minutes did Noura take to travel the last 10 km of the ride? What was her average speed in km per hour for the last 10 km of the ride?

Use the graph to estimate the distance travelled in the first 20 minutes of the ride. What was her average speed in km per hour for the first 20 minutes of the ride?

ICT opportunity

Use a graphics calculator to generate conversion tables and graphs.


7.3 Use a calculator to:

• enter and interpret numbers in time calculations in which parts of an hour need to be entered as decimals or fractions;

• convert one unit of speed to another (e.g. convert km/h to m/s).

7.4 Know that density = mass/volume; solve problems involving calculating density.

The table shows some data (July 2003) about five countries. Country Land area in km2 Estimated population

Bahrain 620 645 000 Iraq 437 072 23 331 000 Qatar 11 437 817 000 Saudi Arabia 1 960 582 22 757 000 Jordan 92 300 5 154 000

Which country has the most people for each km2? Which country has the fewest people for each km2?

Look at the information for Qatar. Imagine that the area of land was shared out equally among all the people. Calculate how much land, in m2, each person would get. 1 km2 = 1 000 000 m2.

7.5 Find the volumes and surface areas of cubes and cuboids and related solids.

The edge of a small cube is 1.5 centimetres. A larger cube is made out of small cubes. The volume of the larger cube is 216 cm3. How many small cubes are used?

A cuboid has a square base. It is twice as tall as it is wide. Its volume is 250 cubic centimetres. Calculate the width of the cuboid.

7.6 Recall the equivalence of 1 litre and 1000 cm3 (a cubic decimetre).

A rectangular tank is 1.22 m wide by 1.85 m long. Find the increase in the depth of oil when 1750 litres of oil are added.

7.7 Recognise that measurements are not precise (e.g. give the upper and lower bounds of a measurement recorded as 15 cm to the nearest centimetre).

15 pupils measured two angles. Here are their results.

Use the results to decide what each angle is most likely to measure. How did you decide?

Anglemeasured as

Anglemeasured as

Number ofpupils

Number ofpupils

36°

37°

38°

39°

45°

134°

135°

136°

1

2

10

2

5

3

4

3

Angle A Angle B

The mass of a dolphin was reported as 170 kg, to the nearest 10 kg. Write down a mass that might have been the actual mass of the dolphin.

TIMSS Grade 8

Density

Include problems involving population density.

Link to work on density in science.


Data handling

By the end of Grade 8, students solve problems by selecting and using an appropriate method of data collection, including from secondary sources. They collect and record continuous data using equal class intervals. They recognise that inappropriate grouping of data can be misleading. They construct bar charts, frequency diagrams and pie charts. They compare two data sets, using the range, median or mean, and the shape of the corresponding frequency distributions. They interpret data sets by drawing conclusions, making predictions, and estimating values between and beyond given points. They use data from experiments to estimate probability for favourable outcomes and understand that different outcomes may result from repeating an experiment. They use problem contexts to calculate theoretical probabilities for possible outcomes.

Students should:


Statistics

8.1 Recognise possible sources of error in collecting and organising data and plan how to minimise the effect (e.g. bias, inappropriate grouping).

Some students wanted to find out if people liked a new biscuit. They decided to do a survey and wrote a questionnaire. One question was: ‘How old are you (in years)?’

19 or younger 20 to 30 30 to 40 40 to 50 50 or older

Amna said: ‘The labels for the middle three boxes need changing.’ Explain why Amna was right.

A different question was: ‘How much do you usually spend on biscuits each week?’

a lot a little nothing don’t know

Amna said: ‘Some of these labels need changing too.’ Write new labels for any boxes that need changing.

.................. .................. .................. ..................

The students decide to give their questionnaire to 50 people. Amna said: ‘Let’s ask 50 students in our school.’ Give one disadvantage of Amna’s suggestion. Give one advantage.

8.2 Choose and use an appropriate method of data collection to answer a given question (e.g. survey, experiment, data logging); design a suitable questionnaire or data collection sheet.

8.3 Collect data from secondary sources, including tables and lists from ICT-based sources such as CD-ROMs and the Internet.


Data handling provides many opportunities to use ICT applications to present statistical tables and graphs. The Internet is an excellent source of real data of interest to students.

ICT opportunity

Link to data logging in science.


8.4 Represent data in charts and graphs, on paper and using ICT, including:

• frequency tables;

• bar charts and frequency diagrams for discrete and continuous data;

• pie charts.

Use the data in the table below to construct a bar graph. Indicate the scale for the percentage increase and label each bar.

Sales Increases at Super Value Stores 2003 to 2004

Type of item Percentage increase

CDs Food Toys TVs

Clothes

80% 15% 25% 40%

120%

TIMSS Grade 8

8.5 Compare in general terms two data sets using, for example, the mean, median or range, or the shape of distribution.

These graphs show the range in the midday temperature in Miami and Orlando each month; e.g. in January the midday temperature in Miami ranges from 17°C to 24°C.

In which city is there the greatest variation each month in the midday temperature? In which three months is the maximum temperature in Miami greater than the maximum temperature in Orlando? What is the range in the minimum monthly temperature in Miami? In Orlando?

8.6 Answer questions by interpreting data sets (e.g. draw conclusions, make predictions, and estimate values between and beyond given points).

80 students from a school took part in a sponsored swim. The numbers of lengths swum are shown on this graph.

Did Grade 10 have fewer pupils taking part in the swim than Grade 7? A. Yes B. No C. Cannot tell Explain your answer.

What is the mean number of lengths swum by each of the 80 students?

Frequency tables and diagrams

Include using equal class intervals.

Pie charts

Include constructing pie charts by calculating angles.

ICT opportunity

Include the use of a spreadsheet with graphs and charts.

Comparing data sets

Use examples from science.


Five songs are recorded on tape. The playing time of each song is shown in the table. Song Amount of time

1 2 minutes 41 seconds



4 3 minutes


Estimate to the nearest minute the total time taken for all five songs to play. Explain how this estimate was made.

TIMSS Grade 8

Probability

8.7 Use data from experiments to estimate probability for favourable outcomes.

8.8 Understand that different outcomes may result from repeating an experiment.

8.9 Use problem conditions to calculate theoretical probabilities for possible outcomes.

In a bag of cards, 16 are green, 1

12 are yellow, 12 are white and 1

4 are blue. Someone takes a card from the bag without looking. Which colour is it most likely to be?

A drawer contains 28 pens, some white, some blue, some red and some yellow. The probability of selecting a blue pen is 2

7 . How many blue pens are in the drawer?

TIMSS Grade 8

8.10 List systematically all the possible outcomes of an experiment.

Nabil and Soud each have three cards, numbered 2, 3 and 4. They each choose one of their own cards. They then add together the numbers on the two chosen cards. The table shows all the possible answers.

What is the probability that their answer is an even number? What is the probability that their answer is a number greater than 6?

Another time Nabil and Soud multiply together the numbers on the two chosen cards. Draw a table to show all possible answers.

8.11 Compare experimental and theoretical probability in simple cases.





Students solve routine and non-routine mathematical problems in a range of contexts. They represent, interpret, analyse and synthesise information presented in numeric, algebraic, geometric or graphical form. They solve more complex problems by breaking them down into smaller tasks. They choose and use appropriate mathematical techniques and tools to solve each part of a problem, including ICT. They explain and justify the steps taken to solve a problem or arrive at a conclusion, orally and in writing. They recognise when an exact solution to a problem is required and when an approximate solution is sufficient, and give answers to a specified degree of accuracy. They develop simple proofs. They generalise where possible and identify exceptional cases. They seek alternative solutions to problems.

Number and algebra

Students choose and use appropriate strategies to solve a range of routine and non-routine problems, using a calculator efficiently and appropriately. They use the four operations and numbers of any size, estimating by rounding to one significant figure and calculating mentally. They use the laws of indices and write numbers in the standard form A × 10n, where n is an integer and 1 ≤ A < 10. They calculate simple interest. They add, subtract, multiply and divide simple algebraic fractions and evaluate formulae and expressions, including quadratic expressions, by substituting integers. They change the subject of a simple formula. They multiply expressions of the form (x ± a)(x ± b). They factorise linear expressions by removing common factors and recognise the factors of expressions of the form a2x2 – b2y2 and x2 ± 2ax + a2. They write and solve simple quadratic equations, and simultaneous linear equations with two unknowns, including finding approximate solutions by graphical methods. They use trial and improvement methods to solve equations such as x3 + x = 20. They find the gradient of the lines y = mx + c, and of lines parallel and perpendicular to y = mx + c. They interpret and sketch graphs of functions representing practical situations.


Students solve problems by identifying congruent or similar triangles and their corresponding angles or sides. They use their knowledge of angles and properties of 2-D shapes to deduce properties in a given plane figure. They use Cartesian coordinates to find the mid-point and length of a line segment, and the point that divides a line segment in a given ratio. They identify a single transformation mapping a shape onto its image, and enlarge shapes by a fractional scale factor, recognising the similarity of the resulting shape. They draw and use plans and elevations of 3-D objects. They solve simple problems in two dimensions by applying Pythagoras’ theorem and finding the side or angle of a right-angled triangle using trigonometric ratios. They calculate areas of 2-D shapes related to circles and volumes and surface areas of right prisms and cylinders.

Grade 9


Data handling

Students solve problems by selecting, using and evaluating methods of collecting, organising, representing, analysing and interpreting data. They represent continuous data in frequency diagrams, choosing appropriate class intervals, on paper and using ICT. They calculate the mean, range and median of small sets of continuous data; they identify the modal class and estimate the mean, median and range for sets of grouped data, choosing the statistic that is most appropriate to their enquiry. They draw conclusions from scatter diagrams, have a basic understanding of correlation, and draw a line of best fit on a scatter diagram, by inspection. They know that if A and B are mutually exclusive, the probability of A or B is the sum of the probabilities of A and of B. They understand relative frequency as an estimate of probability and use this to compare outcomes of experiments. They compare experimental and theoretical probability in different contexts.



The reasoning and problem solving strand cuts across the other three strands and should be integrated with them in teaching and assessments. For Grade 9, at least 70% of the teaching and assessment of each of the other three strands should be devoted to reasoning and problem solving.



measures* Data

handling

Grade 7 30% 25% 27.5% 17.5%

Grade 8 25% 30% 27.5% 17.5%







By the end of Grade 9, students solve routine and non-routine mathematical problems in a range of contexts. They represent, interpret, analyse and synthesise information presented in numeric, algebraic, geometric or graphical form. They solve more complex problems by breaking them down into smaller tasks. They choose and use appropriate mathematical techniques and tools to solve each part of a problem, including ICT. They explain and justify the steps taken to solve a problem or arrive at a conclusion, orally and in writing. They recognise when an exact solution to a problem is required and when an approximate solution is sufficient, and give answers to a specified degree of accuracy. They develop simple proofs. They generalise where possible and identify exceptional cases. They seek alternative solutions to problems.

Students should:


1.1 Represent, interpret, analyse and synthesise information presented in numeric, algebraic, geometric or graphical form.

The shaded shape ABCD is a square. What are the coordinates of D?

1.2 Solve more complex problems by breaking them down into smaller tasks.

1.3 Choose and use appropriate mathematical techniques and tools to solve each part of a problem, including ICT.

Find an approximate solution to x3 + x = 20.

Grade 9

Key standards


Cross-references





1.4 Explain and justify the steps taken to solve a problem or arrive at a conclusion, orally and in writing.

Here is a spinner with five equal sections.

Rasha and Sahar play a game. They spin the pointer many times. If it stops on an odd number, Rasha gets 2 points. If it stops on an even number, Sahar gets 3 points. Is this a fair game? Circle YES or NO. Explain your answer.

1.5 Recognise when an exact solution to a problem is required and when an approximate solution is sufficient, and give answers to a specified degree of accuracy.

A bus company has 62 minibuses. On average, each minibus travels 19 km on 5 litres of fuel and goes 284 km each day. The company says it needs about 5000 litres of fuel every day. Is what the company says about right? Explain how you got your answer.

Circle the best estimate of the answer to 32.7 × 0.48.

1.2 1.6 12 16 120 160

1.6 Develop a simple proof.

Prove that any two-digit whole number is divisible by 9 if the sum of its digits is divisible by 9.

In the diagram, ABDF is a rectangle and triangle BCE is equilateral. ABC and DEC are straight lines.

Prove that triangle BED is isosceles.

1.7 Generalise where possible and identify exceptional cases.

Equations may have different numbers of solutions. Tick ( ) the correct box for each algebraic statement below.

Correct for no values of x

Correct for one value of x

Correct for all values of x

3x + 7 = 8

3(x + 1) = 3x + 3

x + 3 = x – 3

5 + x = 5 – x

1.8 Seek alternative solutions to problems.


Number and algebra

By the end of Grade 9, students choose and use appropriate strategies to solve a range of routine and non-routine problems, using a calculator efficiently and appropriately. They use the four operations and numbers of any size, estimating by rounding to one significant figure and calculating mentally. They use the laws of indices and write numbers in the standard form A × 10n, where n is an integer and 1 ≤ A < 10. They calculate simple interest. They add, subtract, multiply and divide simple algebraic fractions and evaluate formulae and expressions, including quadratic expressions, by substituting integers. They change the subject of a simple formula. They multiply expressions of the form (x ± a)(x ± b). They factorise linear expressions by removing common factors and recognise the factors of expressions of the form a2x2 – b2y2 and x2 ± 2ax + a2. They write and solve simple quadratic equations, and simultaneous linear equations with two unknowns, including finding approximate solutions by graphical methods. They use trial and improvement methods to solve equations such as x3 + x = 20. They find the gradient of the lines y = mx + c, and of lines parallel and perpendicular to y = mx + c. They interpret and sketch graphs of functions representing practical situations.

Students should:

2 Solve numerical problems

2.1 Round whole numbers and decimals, including measures, to a given number of significant figures; use rounding to make mental estimations of calculations.

Estimate the answer to 8.62 22.15.23

+ .

Give your answer to one significant figure.

2.2 Use index notation and the laws of indices to evaluate expressions with integral powers, including positive and negative powers of 10.

Given that 64 = 82 = 4k = 2m, what are the values of k and m?

Given that 215 = 32 768, what is 214?

2.3 Read and write numbers in the standard form A × 10n, where n is a positive or negative integer and 1 ≤ A < 10; interpret numbers in standard form on a calculator display; use standard form in calculations and to estimate.

2.4 Use the x2, √x, xy and x1/y keys of a scientific calculator, distinguishing between the root and the decimal approximation.

The diagram shows a circle with radius 12 mm and a square.

The ratio of the area of the circle to the area of the square is approximately 2 : 1. What is the area of the square to the nearest square millimetre? What is the side length of the square?

Use of ICT


Indices

Include the terms power (index or exponent) and base. Include:

am × an = am + n am ÷ an = am – n (am)n = amn (ab)m = ambm a0 = 1 (a ≠ 0) a1 = a


2.5 Find the value of a square root or cube root to a given degree of accuracy using a calculator or spreadsheet. (See also standard NA 3.9.)

2.6 Use the four operations to solve problems involving whole numbers, decimals, money or measures.

The diameter of a red blood cell is 0.000 714 cm and the diameter of a white cell is 0.001 243 cm. Use a calculator to work out the difference between the diameter of a red blood cell and the diameter of a white cell. Give your answer in millimetres.

2.7 Solve problems involving fractions, percentages, ratios and proportions.

What is 70 increased by 9%? A. 70 × 0.9 B. 70 × 1.9 C. 70 × 0.09 D. 70 × 1.09

The table shows the land area of each of the world’s continents.

Continent Land area (in 1000 km2)

Africa 30 264 Antarctica 13 209

Asia 44 250

Europe 9 907

North America 24 398

Oceania 8 534

South America 17 793

World 148 355

Which continent is approximately 12% of the world’s land area? What percentage of the world’s land area is Antarctica?

About 30% of the world’s area is land. The rest is water. The amount of land in the world is about 150 million km2. What is the approximate total area (land and water) of the world?

Here are the labels from two pots of yoghurt.

A boy eats the same amount of yoghurt A and yoghurt B. Which yoghurt provides him with more carbohydrate?

2.8 Calculate simple interest.

2.9 Check answers for accuracy and reasonableness, and round answers appropriately.

3 Write, simplify and evaluate algebraic expressions and formulae and solve equations

3.1 Expand expressions of the form a(x ± b), (x + a)2, (x – a)2, (x + a)(x – a), (x ± a)(x ± b), where a and b are positive or negative integers.

Multiply out and simplify these expressions. 3(x – 2) – 2(4 – 3x) (x + 2)(x +3) (x + 4)(x – 1) (x – 2)2

Calculations


Algebraic expressions

Include use of the terms coefficient, variable, constant, linear, quadratic.


3.2 Evaluate algebraic expressions and formulae for given integer values of the variables.

Find the values of a and b when p = 10. 33

2pa =

22 ( 3)7

p pb p−=

3.3 Factorise algebraic expressions:

• by removing common factors from expressions such as: ax ± ay ax + bx + ay + by

• by recognising the factors of expressions such as: a2x2 – b2y2 x2 ± 2ax + a2

where a and b are positive or negative integers.

3.4 Add, subtract, multiply and divide algebraic fractions.

Simplify:

1. 3 54 3a ab× 2. 3 9

4 10a a÷ 3. 1 2

2 3x x+− −

3.5 Change the subject of a simple formula.

Rearrange this equation to make e the subject. p = 2(e + f )

Rearrange this equation to make R the subject. V = IR

3.6 Write and solve linear equations, including simple cases of fractional linear equations, and apply these skills to solving problems; verify the solution.

Solve:

1. 2 33 4x x −+ = 2. 5 10x = 3. 3 62x =−

3.7 Write and solve simultaneous linear equations with two unknowns by elimination and by substitution, and apply these skills to solving problems; verify the solution. (See also standard NA 4.3.)

Solve these simultaneous equations to find the values of x and y.

x + 8y = 48 4x + 4y = 52

3.8 Solve quadratic equations of the form a2x2 – b2 = 0 or x2 ± 2ax + a2 = 0 by factorisation; verify the solutions by substituting in the original equation.

3.9 Find the approximate solutions of equations such as x3 + x = 20 using ICT and trial and improvement methods. (See also standard NA 2.5.)

A rectangle has an area of 34 cm². Its sides are x cm and (12 – x) cm, so that x(12 – x) = 34. Between which numbers (to one decimal place) does x lie? Use the table.

x 12 – x Area 3 9 27

Evaluating expressions

Include evaluating quadratic and cubic expressions.

Algebraic fractions

Include simple cases of linear algebraic denominators.

Changing the subject of a formula

Link to work on Ohm’s law V = IR in science.

Linear equations

Include the forms

a(bx + c) + d(ex + f) = gx x/a + x/b = c (x + a)/b +(x + c)/d = e a/x = b a/(x + b) = c

where a, b, c, d, e, f, g are integers.

ICT opportunity

Use a spreadsheet or graphics calculator.


4 Plot and interpret graphs of functions

4.1 Use a graphics calculator and a function graph plotter to plot graphs.

4.2 Find the gradients of lines given by y = mx + c; understand the idea of slope; find the gradients of lines parallel and perpendicular to y = mx + c.

Write the equation of the straight line which goes through the point (0, –1) and is parallel to the straight line y = 3x.

Use a graphics calculator to draw these quadrilaterals.

4.3 Use graphical methods to find the approximate solution of a pair of simultaneous linear equations with two unknowns, on paper and using ICT. (See also standard NA 3.7.)

Look at this graph.

Show that the equation of line A is y = 8 – 2x. Write the equation of line B. On the graph, draw the line whose equation is y = 2x + 1. Label this line C.

Use the graph to find the approximate solution of these simultaneous equations.

y = 2x + 1 y = 8 – 2x

4.4 Generate points and plot graphs of simple quadratic and cubic functions, e.g. y = 3x2 + 4, y = x2 –2x + 1; use graphical methods to find approximate solutions of quadratic equations, on paper and using ICT.

Create a display like this using a function graph plotter.

Find a possible equation for this curve.

Gradient and slope

Link to work on ratio.

Use a graphics calculator or graph plotting software for plotting graphs of functions.

ICT opportunity


4.5 Sketch and interpret graphs of functions based on practical situations.

Water is poured at a constant rate into three of these five containers.

The graphs show the depth of water as the containers fill up.

Fill in the gaps below to show which container matches each graph. Graph 1 matches container ............... Graph 2 matches container ............... Graph 3 matches container ...............


By the end of Grade 9, students solve problems by identifying congruent or similar triangles and their corresponding angles or sides. They use their knowledge of angles and properties of 2-D shapes to deduce properties in a given plane figure. They use Cartesian coordinates to find the mid-point and length of a line segment, and the point that divides a line segment in a given ratio. They identify a single transformation mapping a shape onto its image, and enlarge shapes by a fractional scale factor, recognising the similarity of the resulting shape. They draw and use plans and elevations of 3-D objects. They solve simple problems in two dimensions by applying Pythagoras’ theorem and finding the side or angle of a right-angled triangle using trigonometric ratios. They calculate areas of 2-D shapes related to circles and volumes and surface areas of right prisms and cylinders.

Students should:



5.1 Use knowledge of angles and properties of 2-D shapes to conjecture or deduce properties in a given plane figure.

The diagram shows two overlapping squares and a straight line.

Calculate the value of angle x and the value of angle y.

5.2 Find coordinates of points determined by geometric information; given the coordinates of A and B, find:

• the mid-point of line segment AB;

Use of ICT

Geometry is enhanced with use of a dynamic geometry system, or DGS, which provides an interactive means for investigating and hypothesising results that can then be proved.


• the length of line segment AB;

• the point that divides line segment AB in a given ratio.

5.3 Identify similar triangles and their corresponding angles and sides.

In the diagram, PQ is parallel to BC.

Calculate length PQ.

5.4 Identify congruent triangles and their corresponding angles and sides; know the conditions of congruence and determine whether two triangles are congruent.

The diagram shows five triangles. All lengths are in centimetres.

Write the letters of two triangles that are congruent to each other. Explain how you know they are congruent.

Write the letters of two triangles that are mathematically similar to each other but not congruent. Explain how you know they are mathematically similar.

5.5 Use the properties of congruence or similarity of triangles to solve problems, e.g. find unknown sides or angles of similar or congruent triangles.

Transformations

5.6 Identify a single transformation mapping a 2-D shape onto its image: reflection, rotation, translation or enlargement by a positive integer scale factor; find a line of reflection, centre or angle of rotation, scale factor or centre of enlargement in simple cases.

5.7 Identify and draw, on paper and using ICT, the enlargement of a simple plane figure by a positive fractional scale factor; identify the scale factor as the ratio of two corresponding line segments.

Reema has a photograph that measures 21.25 cm by 13.75 cm. She wants a smaller copy that will fit exactly in a 8.5 cm by 5.5 cm photograph frame. What scale factor should she use to make the copy?

5.8 Use ICT to explore transformations.

Constructions

5.9 Recognise 3-D objects from 2-D representations; draw the plan and elevation of a 3-D object from sketches and models; sketch or build a 3-D object given its plan and elevation.

Congruence

Stress that the conditions of congruence are:

SSS three corresponding sides

SAS two sides and the included angle

ASA two angles and the included side

RHS right angle, hypotenuse, side

Transformations

Include the terms line of reflection (mirror line), angle and centre of rotation, centre of enlargement, scale factor.

ICT opportunity

Transformations are best developed through the use of a dynamic geometry system (DGS).


Here is a model made from 10 cubes. From direction B, it looks like this.

On the grid, draw how the model looks from direction A.

6 Solve problems involving area and volume

6.1 Find the area of plane shapes related to circles.

A very large round table has a radius of 2.75 metres. Assume that to sit at the table one person needs 45 cm around the circumference. Is it possible for 50 people to sit around the table?

Assume that people sitting around the table can reach 1.5 m.

Calculate the area of the table that can be reached.

The diagram shows a square and a circle. The circle touches the edges of the square.

What percentage of the diagram is shaded?

6.2 Find the volume and surface area of right prisms and cylinders and related solids.

This door wedge is the shape of a prism.

The shaded face of the door wedge is a trapezium. Calculate the volume of the door wedge.

A cylinder has a radius of 2.5 cm. The volume of the cylinder is 4.5 cm3.

What is the height of the cylinder?

Volume and surface area of prisms and cylinders

Use the formula that the volume is:

area of base × height


7 Solve problems involving right-angled triangles

7.1 State and apply Pythagoras’ theorem (not proof).

ABC and ACD are both right-angled triangles.

Explain why the length of AC is 10 cm. Calculate the length of AD.

Look at this triangle.

Explain why angle x must be a right angle.

7.2 Solve problems involving finding a side of a right-angled triangle.

A boat sails from the harbour to the buoy. The buoy is 6 km to the east and 4 km to the north of the harbour.

Calculate the shortest distance between the buoy and the harbour. Give your answer to one decimal place.

7.3 Know the sine, cosine and tangent ratios for a right-angled triangle.

7.4 Use a scientific calculator to:

• find the values of trigonometric ratios;

• find an angle using the inverse trigonometric function keys.

Calculate the value of y. Calculate the value of angle m.

ABC is an isosceles triangle drawn on a square grid, with AC = AB.

Find the size of angle CAB. Calculate angle ABC.

Pythagoras’ theorem

Include finding a side of a right-angled triangle, and using the relationship between the three sides of a triangle to show than an angle is a right angle.

Right-angled triangle problems

Exclude angles expressed in degrees and minutes.

Exclude problems involving bearings or angles of elevation and depression.

Trigonometric ratios

Exclude angles measured in radians.


Data handling

By the end of Grade 9, students solve problems by selecting, using and evaluating methods of collecting, organising, representing, analysing and interpreting data. They represent continuous data in frequency diagrams, choosing appropriate class intervals, on paper and using ICT. They calculate the mean, range and median of small sets of continuous data; they identify the modal class and estimate the mean, median and range for sets of grouped data, choosing the statistic that is most appropriate to their enquiry. They draw conclusions from scatter diagrams, have a basic understanding of correlation, and draw a line of best fit on a scatter diagram, by inspection. They know that if A and B are mutually exclusive, the probability of A or B is the sum of the probabilities of A and of B. They understand relative frequency as an estimate of probability and use this to compare outcomes of experiments. They compare experimental and theoretical probability in different contexts.

Students should:


Statistics

8.1 Identify questions or problems that can be answered or solved by collecting, organising, representing, analysing and interpreting data.

8.2 Construct and interpret scatter diagrams, and lines of best fit by eye, understanding what these represent.

The scatter diagram shows the heights and masses of some horses. It also shows a line of best fit.

What does the scatter diagram show about the relationship between the height and mass of horses?

A horse has a mass of 625 kg. Use the line of best fit to estimate the height of the horse.

A teacher asks his class to investigate this statement: ‘The length of the back leg of a horse is always less than the length of the front leg of a horse.’ What might a scatter graph look like if the statement is correct? Use the axes below to show your answer.


Data handling provides many opportunities to use ICT applications to present statistical tables and graphs. The Internet is an excellent source of real data of interest to students.

Scatter diagrams

Include the use of ICT, e.g. using a spreadsheet and graph drawing package.


8.3 Calculate the mean, range and median of small sets of discrete or continuous data; identify the modal class and estimate the mean, median and range for sets of grouped data.

The table shows the numbers of peas in a sample of 50 pea pods.

Number of peas in a pod

Number of pods

3 4 5 6 7 8

2 7 14 12 10 5

What is the mode for the number of peas in a pod in the sample? What is the median number of peas in a pod in the sample?

Work out the mean number of peas in a pod in the sample. Estimate the number of peas in 200 pods. Explain how you made your estimate.

About how many pods out of 200 would you expect to have 3 or 4 peas?

8.4 Construct and interpret frequency diagrams, choosing appropriate equal class intervals.

Here are some long jump competition results, measured to the nearest centimetre.

Ahmed jumped 315 cm. He says: ‘Only 2 people jumped further than me.’ Could Ahmed be correct? Circle YES or NO. Explain your answer.

Tarik says: ‘The median jump was 275 cm.’ Explain how the graph shows that Tarik is not correct.


Hussain collects information about how long the phone calls are in his house. He makes a frequency table using class intervals of 30 seconds. Here is part of the table.

Length of call in seconds 0–29 30–59 60–89 90–119

Number of calls 3 25 35 19

The longest call was 175 seconds. Which class interval does this fit into?

Altogether Hussain recorded 91 calls. Hussain makes a rough estimate that half the calls lasted less than 75 seconds. Explain how he could make this estimate.

Probability

8.5 Use relative frequency as an estimate of probability and use this to compare outcomes of experiments.

The manager records the waiting times of 100 customers at a supermarket checkout.

Use the graph to estimate the probability that a customer chosen at random will:

• wait for 2 minutes or longer; • wait for 2.5 minutes or longer.

The manager wants to improve the survey. She records the waiting times of more customers. Give a different way the manager could improve the survey.

8.6 Know that if A and B are mutually exclusive, the probability of A or B is the sum of the probabilities of A and of B.

A special dice has the numbers 1 to 6 on it. It is biased so that a 6 or a 1 is less likely to come up than a 2, 3, 4 or 5. The probability of rolling a 6 is 0.1. The probability of rolling a 1 is 0.1. The numbers 2, 3, 4 or 5 each have an equal probability of coming up. Calculate the probability of rolling a 5 with this dice.

8.7 Compare experimental and theoretical probability in different contexts.

The manufacturers of a computer game claim that the probability of winning each game is 0.65. Sara plays this game 200 times and wins 124 times. She says: ‘The manufacturers must be wrong.’ Do you agree with her? Circle YES or NO. Explain your reasons.

Relative frequency

Include grouped data.


Some students threw three fair dice, each numbered from 1 to 6. They recorded how many times the numbers on the dice were the same.

ResultsName

Numberof

throws all different 2 the same all the same

Masood

Saleh

Walid

Ali

40

140

20

100

26

81

10

54

12

56

10

42

2

3

0

4

Which student’s data are most likely to give the best estimate of the probability of getting each result? Explain your answer.

The next table shows all the students’ results collected together. Use these data to estimate the probability of throwing three different numbers.

Numberof

throws

300 171 120 9

2 the same all the sameall different

Results

The theoretical probability of each result is shown below:

Probability


59

512

136

Use the theoretical probabilities to calculate, for 300 throws, how many times you would expect to get each result.

Numberof

throws

300 ............... ............... ...............


Theoretical results

Explain why the students’ experimental results are not the same as the theoretical results.

181 | Qatar mathematics standards | Grade 10 foundation © Supreme Education Council 2004




Students solve routine and non-routine problems in a range of mathematical and other contexts, and use mathematics to model and predict the outcomes of real-world applications. They identify and use connections between mathematical topics. They break down complex problems into smaller tasks, and set up and perform appropriate manipulations and calculations. They develop and explain short chains of logical reasoning, using correct mathematical notation and terms. They generate simple mathematical proofs and identify exceptional cases. They aim to generalise. They approach problems systematically, knowing when it is important to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They recognise when to use ICT and do so efficiently.

Number and algebra

Students identify and use number sets and set notation. They calculate with any real numbers, including powers, roots, and numbers expressed in standard form. They use proportional reasoning to solve a range of problems involving scale, ratios and percentages. They are aware of the role of symbols in algebra. They generate and manipulate algebraic expressions, including algebraic fractions, equations and formulae. They sum arithmetic sequences and investigate the growth of simple patterns, generalising relationships to model the behaviour of the patterns. They find the solution of any linear equation, and a pair of simultaneous linear equations, and plot straight line and simple quadratic graphs. They use function notation. Through their study of linear and simple quadratic functions and their graphs, and the solution of the related equations, students begin to appreciate numerical and algebraic applications in the real world. They use realistic data and ICT to analyse problems.


Students use their knowledge of geometry, Pythagoras’ theorem and the trigonometry of right-angled triangles to solve practical and theoretical problems relating to shape and space. They understand congruence and similarity. They carry out straight edge and compass constructions and determine the locus of an object moving according to a rule. They use a range of SI units and measures, including bearings and compound measures. They use formulae to calculate: the circumference and area of a circle; the perimeter and area of any triangle, or trapezium, parallelogram or quadrilateral with perpendicular diagonals; the surface area and volume of a right prism, cylinder, square-based pyramid and cone; and the volume of a sphere. They use ICT to explore pattern, similarity, congruence and constructions.

Grade 10 Foundation



Students know that statistical data are collected from observation or measurement on samples taken from a larger population, and that, by analysing the sample data, inferences can be made about the population as a whole. They distinguish between qualitative (or categorical) data and quantitative data, and between discrete and continuous data. They plan simple surveys, design simple questionnaires and plot histograms in which the height of each bar is proportional to the frequency of that class. They calculate means and medians, and understand mode and modal class. They plot and interpret simple scatter diagrams between two random variables, and draw a line of best fit where there appears to be correlation. They use relevant statistical functions on a calculator and ICT applications to present statistical tables and graphs.


The foundation mathematics standards are grouped into four strands: reasoning and problem solving; number and algebra; geometry and measures; and probability and statistics.

The reasoning and problem solving strand cuts across the other three strands. Reasoning, generalisation and problem solving should be an integral part of the teaching and learning of mathematics in all lessons.

The weightings of the content strands relative to each other are as follows:

Foundation Number and

algebra Geometry and

measures* Probability

and statistics

Grade 10 55% 30% 15%

Grade 11 55% 30% 15%

Grade 12 50% 25% 25%

* including trigonometry

The standards are numbered for easy reference. Those in shaded rectangles, e.g. 1.2, are the performance standards for all foundation students. The national tests for foundation mathematics will be based on these standards.

Many of the Grade 10 foundation standards have been introduced in earlier grades. Teachers should review and consolidate these standards, moving through them as quickly or as slowly as befits the students.




By the end of Grade 10, students solve routine and non-routine problems in a range of mathematical and other contexts, and use mathematics to model and predict the outcomes of real-world applications. They identify and use connections between mathematical topics. They break down complex problems into smaller tasks, and set up and perform appropriate manipulations and calculations. They develop and explain short chains of logical reasoning, using correct mathematical notation and terms. They generate simple mathematical proofs and identify exceptional cases. They aim to generalise. They approach problems systematically, knowing when it is important to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They recognise when to use ICT and do so efficiently.

Students should:


1.1 Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.

1.2 Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation.

1.3 Identify and use interconnections between mathematical topics.

1.4 Break down complex problems into smaller tasks.

1.5 Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

1.6 Develop short chains of logical reasoning, using correct mathematical notation and terms.

1.7 Explain their reasoning, both orally and in writing.

1.8 Generate simple mathematical proofs, and identify exceptional cases.

1.9 Learn to generalise and begin to understand the importance of generalisation in mathematics.

1.10 Approach a problem systematically, recognising when it is important to enumerate all outcomes.

1.11 Conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions.

Grade 10 Foundation

Key standards


Cross-references

Standards are referred to using the notation RP for reasoning and problem solving, NA for number and algebra, GM for geometry and measures, and PS for probability and statistics, e.g. standard NA 2.3.




Reasoning, generalisation and problem solving should be an integral part of the teaching and learning of mathematics in all lessons.

Proofs

Relate to the mathematics in the other strands.


1.12 Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms.

1.13 Work to expected degrees of accuracy, and know when an exact solution is appropriate.

1.14 Recognise when to use ICT and when not to, and use it efficiently.

Number and algebra

By the end of Grade 10, students identify and use number sets and set notation. They calculate with any real numbers, including powers, roots, and numbers expressed in standard form. They use proportional reasoning to solve a range of problems involving scale, ratios and percentages. They are aware of the role of symbols in algebra. They generate and manipulate algebraic expressions, including algebraic fractions, equations and formulae. They sum arithmetic sequences and investigate the growth of simple patterns, generalising relationships to model the behaviour of the patterns. They find the solution of any linear equation, and a pair of simultaneous linear equations, and plot straight line and simple quadratic graphs. They use function notation. Through their study of linear and simple quadratic functions and their graphs, and the solution of the related equations, students begin to appreciate numerical and algebraic applications in the real world. They use realistic data and ICT to analyse problems.

Students should:

2 Identify and use number sets

2.1 Identify the number sets:

the set of all real numbers;

the set of all integers; + the set of all positive integers {1, 2, 3, 4, …}; – the set of all negative integers;

the set of all rational numbers, i.e. all the different numbers that can be expressed in the form a/b, where a and b are integers with b ≠ 0;

the set of all non-negative integers, called the set of natural numbers {0, 1, 2, 3, 4, …}.

Is a subset of ?

To what set does √2 belong? How do you know?

2.2 Know when a real number is irrational, i.e. when it is not a member of .

2.3 Use and understand the following symbols associated with set theory: E for ‘the universal set’; ∅ for ‘the null set’; ∈ for ‘is a member of’; ∉ for ‘is not a member of’; ∀ for ‘for all’; use brace notation to denote a set.

A = {x: x ∈ and 1 ≤ x < 10} denotes ‘the set A, whose members are all real numbers

greater than or equal to 1 and less than 10’.

Algebra

Students should learn that algebra enables generalisation and the establishment of relationships between quantities and/or concepts. They should understand the nature and place of algebraic reasoning and proof, and link it to geometric concepts wherever possible.

Natural numbers

In some texts, is taken to be the same as +.


List the elements of each of the following sets: A = {x: x is a colour of the Qatar flag}; B = {x: x is a state in the GCC}; C ={x: x is a member of the Arab League}.

Is the statement that √2/3 ∈ true or false?

What is the solution set of the equation x(x + 3) = x(x – 3) + 6x + 1? Explain your answer.

2.4 Understand the meaning of the union of two sets A and B and that this is denoted by A ∪ B, and the meaning of the intersection of two sets A and B, denoted by A ∩ B, and represent these sets in a Venn diagram; represent the complement of set A as A′ and know that A ∪ A′ = E.

Use a Venn diagram to decide whether A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C.

Describe the set ∪ ′.

In a school of 650 students, everyone studies Arabic, English, mathematics and science. They all have to choose to study at least one of art, French, or history. 195 students choose only French. Three times as many students study French and history as study all three subjects, and five times as many study French and art as study all three subjects. 30 students study French and history, 65 do art and history, and 200 do art but not French or history. How many students study history but not art or French?

2.5 Know from definitions that every even number can be written in the form 2m, where m is an integer, and that every odd number can be written in the form 2n + 1, where n is an integer; understand and use the words factor, multiple, divisor, prime number, prime factor, prime factor decomposition, least common multiple, highest common factor and lowest common denominator.

Prove that the product of two odd numbers is an odd number.

What is the largest prime number you can think of? How do you know it is prime?

Is there a largest prime number? Justify your answer.

What is the highest common factor of a3b2c and c3b2a?

3 Use index notation and solve numerical problems

3.1 Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents; use the xy key on a calculator.

Without using a calculator, evaluate (53)4 ÷ 510.

Use a calculator to evaluate 79.

Simplify 81/3 × 2–1.

3.2 Know that a root that is irrational is an example of a surd, as are expressions containing the addition or subtraction of an irrational root; perform exact calculations with surds.

Calculate (√2 – 1)(√3 – √2).

3.3 Use standard form in appropriate situations: for exact calculations, to estimate results of calculations and to make comparisons.

To four significant figures, the speed of light is 299 800 000 metres per second. Write this in standard form.

Laws of exponents

For a > 0: ax × ay = ax+y ax ÷ ay = ax–y (ax)y = axy (a1/n)n = a a0 = 1

Standard form

Use examples drawn from science, geography or other real world applications.


A car has a mass of 1200 kilograms and a length of 4.5 metres. The Earth has a mass of 5.98 × 1024 kg and a radius of approximately 6400 kilometres. Estimate the ratio of the mass of the Earth to the mass of the car and the ratio of the radius of the Earth to the length of the car.

Sir Isaac Newton (1642–1727) was a mathematician, physicist and astronomer.

a. In his work on the gravitational force between two bodies, Newton found that he needed to multiply their masses together.

Work out the value of the mass of the Earth multiplied by the mass of the Moon. Give your answer in standard form.

Mass of Earth = 5.98 × 1024 kg Mass of Moon = 7.35 × 1022 kg

b. Newton also found that he needed to work out the square of the distance between the two bodies.

Work out the square of the distance between the Earth and the Moon. Give your answer in standard form.

Distance between Earth and Moon = 3.89 × 105 km

c. Newton’s formula to calculate the gravitational force (F) between two bodies is 1 22

Gm mFR

=

where G is the gravitational constant, m1 and m2 are the masses of the two bodies, and R is the distance between them.

Work out the gravitational force (F) between the Sun and the Earth using this formula with the information given below. Give your answer in standard form.

m1m2 = 1.19 × 1055 kg2 R2 = 2.25 × 1016 km2 G = 6.67 × 10–20

3.4 Calculate with any real numbers, including mental calculations in appropriate cases.

Calculate mentally the value of 999 × 33.

3.5 Add, subtract, multiply and divide any two fractions and understand how to use a unit fraction as a multiplicative inverse.

3.6 Understand the multiplicative nature of proportional reasoning; form, simplify and compare ratios, and apply these in a range of problems, including mixtures, map scales and enlargements in one, two or three dimensions.

A recipe for six people includes a quarter of a kilogram of figs. How many kilograms of figs would be needed if the recipe were made in the same proportion for eight people?

A map is drawn to scale 1 : 190 000. Two places A and B are 3 cm apart on the map. How far apart are A and B?

3.7 Perform percentage calculations, including finding a percentage of a percentage and inverse percentages.

After Haya’s salary is increased by 15% and Abdullah’s salary is decreased by 27%, Haya and Abdullah both end up with a salary of QR 36 000. What were their original salaries? What percentage of Abdullah’s original salary was Haya’s original salary?


The diagram shows water flowing through some pipes. The water starts at A. At each junction the percentage of the inflowing water flowing out through the pipes is indicated.

What percentage of the original water flows out at B?

What percentage flows out at C?

Due to inflation, the price of a television in a store is increased by 15%. In the sales at the end of the year, the price is then reduced by 15%. Does the television revert to its original pre-inflation price? Or is it more, or less? Explain your reasoning.

4 Generate and manipulate algebraic expressions and formulae, and solve algebraic equations

4.1 Solve any linear equation with one unknown.

4.2 Generate sequences from term-to-term and position-to-term definitions; investigate the growth of simple patterns, generalising algebraic relationships to model the behaviour of the patterns.

Each term of a sequence is 3 times the preceding term. The first term is 5. Set up a term-to-term definition for this sequence. Give an expression for the nth term in terms of n. Write down, but do not simplify, the 50th term.

The table shows the first six triangular numbers.

Position 1 2 3 4 5 6

Term 1 3 6 10 15 21

Investigate diagrammatic ways of representing triangular numbers.

The diagram shows some ways in which this might be done.

Set up a relationship to describe the nth term in terms of its position value n. What is the 100th triangular number? What is the 1000th triangular number?

4.3 Sum arithmetic sequences, including the first n consecutive integers, and give a ‘geometric proof’ for the formulae for these sums.

The diagram is a useful representation of an arithmetic series.

How could you use this diagram to find the sum of the arithmetic series?

Find the sum of the first n consecutive positive integers, and hence the sum of any set of n consecutive positive integers.

Find the sum of all numbers between 1 and 100 that are exactly divisible by 3.

Sequences

Link to geometric concepts where possible.

Include quadratic sequences and second order differences.

ICT opportunity

Include the use of spreadsheets or graphics calculators to explore arithmetic sequences.


4.4 Identify number patterns contained in Pascal’s triangle.

Describe carefully in words how any entry in Pascal’s triangle is related to entries in the row above. Set up an algebraic relationship to describe this. Where are the triangular numbers located in Pascal’s triangle? What other patterns can you spot?

Look at the numbers in an early row of Pascal’s triangle. Sum the squares of these numbers. In what row is the answer located? Identify where to find the sum of the squares of the numbers in any row of Pascal’s triangle. Explain your reasoning.

4.5 Distinguish the different roles played by letter symbols in algebra, and understand that the transformation of algebraic objects generalises the well-defined rules of arithmetic. Recognise that letters are used to represent:

• the solution set of initially unknown numbers in equations;

• defined variables in formulae;

• generalised independent numbers in identities;

• new equations, expressions or functions in terms of known, or given, expressions or functions.

Is (x + 4)2 = x(x + 12) – 4(x – 4) an equation or an identity? Explain your reasoning.

4.6 Use brackets and correct order of precedence of operations when performing numerical or algebraic calculations.

4.7 Multiply any combinations of monomial and binomial expressions, collecting and simplifying similar terms.

4.8 Simplify and combine numeric or algebraic fractions, including by cancelling common factors; rationalise a denominator of a fraction when the denominator contains simple combinations of surds.

Rationalise the expression 21 3+

.

Simplify the expression 3 2 2 3

2 2a b a b

a b− .

4.9 Generate formulae from a physical context; rearrange formulae connecting two or more variables.

The three different edges of a solid cuboid have lengths x, 2x and y, as shown. All the lengths are measured in centimetres.

The total surface area of the cuboid is 800 cm2. Find a formula for y in terms of x.

What is the total length of all the edges of the cuboid? Give the answer in terms of x.

Make b the subject of the formula 2a bA += .

Make l the subject of the formula 2 lT gπ= .

Make x the subject of the formula xw z v= − + .

Algebraic symbols

Include references and Internet research on the contributions to algebra of Arab scholars such as Al-Khwarizmi.

Formulae

Include examples drawn from science.


5 Generate and solve problems with functions and graphs

5.1 Use function notation; investigate a range of mathematical and physical situations to develop the concepts of function, domain and range, recognising one-to-one and many-to-one mappings as functions and a one-to-many mapping as not a function.

If p is a person, state with reasons whether each of the following maps are functions: a. p maps to the place of birth of p; b. p maps to brother of p; c. p maps to nationality of p; d. p maps to teacher of p; e. p maps to mother of p.

A firm rents out cars by the day or by the week. The daily charge rate is QR 170 with 150 km free and then QR 2 for every additional kilometre. The weekly charge is QR 1400 with no additional charges. A man needs to hire a car for five days. How many kilometres will he have to drive to make it worthwhile to hire the car for a week?

Look up any country in an atlas and pick six towns from it. Which of these maps represents a function and which does not: towns → country; country → towns? Justify your answer. What are the domain and range for the mapping that represents a function?

5.2 Understand and use the concept of related variables and, in special cases, set up appropriate functional relationships between them.

In an electric circuit, V = IR, where V is the voltage in volts, I is the current in amps and R is the resistance in ohms. The electrical power in watts is P = VI. Find a formula connecting the variables P, V and R.

5.3 Plot a graph to show the relationship between two variables given quantitative information between the variables in tabular or algebraic form.

5.4 Use a graphics calculator to plot a range of simple functional relationships, some continuous and others discontinuous, arising in familiar contexts.

Draw a graph showing the functional relationship between postage rate in Qatar and the weight of package to be posted.

5.5 Recognise when a graph represents a functional relationship between two variables and when it does not.

Direct proportion

5.6 Translate the statement ‘y is proportional to x’ into the symbolism y ∝ x and into the equation y = kx, and know that the graph of this equation is a straight line through the origin and that the constant of proportionality, k, is the gradient of this line.

Functions

The notation y = f(x) denotes that y is a function of x.

Functional relationships


ICT opportunity

Graphics calculators or graph plotters can be used to explore a range of functional relationships.


5.7 Know that if two coordinate variables are connected by a straight line graph that passes through the origin of coordinates, then each coordinate variable is proportional to the other; use relevant information to find k.

5.8 Identify common examples of two linear quantities varying in direct proportion to each other.

Straight lines and linear functions

5.9 Know that a straight line in the explicit form y = mx + c represents a function; plot the graphs of such functions, relating the gradient of the line and intercept on the x- or y-axis to the coefficients m and c.

Are there straight lines that do not represent functions? Justify your answer.

5.10 Construct the Cartesian equation of a straight line from its graph alone, or from the knowledge of the coordinates of two points on the line, or from the coordinates of one point on the line and the gradient of the line.

What is the equation of the straight line through the points (5, –2) and (–4, 3)? What is the gradient of this line? Where does it cross the y-axis? Where does it cross the x-axis?

A triangle has vertices at the points (1, 1), (5, –4) and (–3, 2). Find the equation of each of its sides.

5.11 Know the condition for two straight lines to be parallel or perpendicular, including the special cases of one of the lines being parallel to either axis.

Give equations of lines parallel and perpendicular to the line y = 5x – 3.

5.12 Read off the coordinates of the point of intersection, given the graphs of two intersecting straight lines; use algebraic means to find exactly the coordinates of the point of intersection of two straight lines, given their equations.

Find the intersection point of the line y = 4x + 2 with the line y = 9 – 3x.

Discuss whether two lines have no intersection, a unique intersection point, or infinitely many intersection points.

Discuss whether non-parallel lines in two dimensions must intersect. What happens in three dimensions?

5.13 Interpret the solution set of the simultaneous equations E1 and E2, where E1 and E2 are the equations of two straight lines.

Here are the equations of some straight lines: y = 2x – 7; y = 7 – 2x; y = 2x + 9; y = 14 – 4x; y= –10; y = –10 + 2x; x = 1; y = –0.5x + 8.

List all the pairs of lines that are: a. parallel to each other; b. perpendicular to each other; c. different representations of the same line. From the list, find pairs of lines that intersect in a unique point and find the intersection point in each case.

Quadratic functions

5.14 Recognise a simple second-order polynomial in one variable, y = ax2 + c, as a quadratic function; plot graphs of such functions (recognising that these are all parabolas), and pick out the intercepts with the coordinate axes, the axis of symmetry and the coordinates of the maximum or minimum point.

5.15 Model a range of situations with quadratic functions of the form f(x) = ax2 + c.

Direct proportion

Use real examples, e.g.

• currency conversions;

• for a given resistance, the voltage in an electric circuit is directly proportional to the current;

• the pressure of gas in a constant volume is directly proportional to its temperature.

Gradients

Include the terms slope and rate of change.

ICT opportunity

Include the use of graphics calculators or graph plotters.



By the end of Grade 10, students use their knowledge of geometry, Pythagoras’ theorem and the trigonometry of right-angled triangles to solve practical and theoretical problems relating to shape and space. They understand congruence and similarity. They carry out straight edge and compass constructions and determine the locus of an object moving according to a rule. They use a range of SI units and measures, including bearings and compound measures. They use formulae to calculate: the circumference and area of a circle; the perimeter and area of any triangle, or trapezium, parallelogram or quadrilateral with perpendicular diagonals: the surface area and volume of a right prism, cylinder, square-based pyramid and cone: and the volume of a sphere. They use ICT to explore pattern, similarity, congruence and constructions.

Students should:

6 Develop geometrical reasoning and proof, and solve geometric problems

Congruence and similarity: properties of angles, straight lines and triangles

6.1 Use knowledge of angles at a point, angles on a straight line, and alternate and corresponding angles between parallel lines and a transversal line to present formal arguments to establish the congruency of two triangles.

Prove that each of the angles in an equilateral triangle is 60°.

6.2 Establish the congruency of two triangles to generate further knowledge and theorems about triangles, including proving that the base angles of an isosceles triangle are equal and that the line joining the mid-points of two sides of a triangle is parallel to the remaining side.

6.3 Understand similarity of two triangles and other rectilinear shapes, knowing that similarity preserves shape and angles, but not size; make inferences about the lengths of sides and about the areas of similar figures; prove that if two triangles are similar, then the ratio of the areas of the two triangles is the square of the ratio of any pair of corresponding sides of the two triangles chosen in the same order; in three dimensions, calculate the ratio of the volume of a scale model to the volume of the actual object.

A goldsmith has a block of gold in the shape of a cube. He wants to make another gold cube that has exactly twice the volume of the first cube. What scale factor must he use?

The diagram shows two triangles ADE and BCE. Side AD is parallel to side BC. Explain why the two triangles are similar to each other. Calculate the missing lengths for triangle BCE.


Students should develop an appreciation of the importance and range of geometrical applications in the real world, and the aesthetic qualities of geometric models. They should understand the nature and place of geometric reasoning and proof, and how geometry may be related to algebraic concepts, and vice versa.

Use of ICT

Geometry is enhanced with use of a dynamic geometry system, or DGS, which provides an interactive focus to investigate and conjecture results which could then be proved as theorems.


Calculate the length of CD in the diagram.

Two similar shaped gas-filled balloons are made of a special material. The area of material used in one balloon is 100 cm2. The material for the other balloon has an area of 225 cm2. Calculate the ratio of the volume of the larger balloon to the volume of the smaller balloon. Give this ratio in its simplest form.

A scale model of a dhow has a volume of 300 cm3. The length of the actual dhow is 100 times longer than the length of the model. What is the volume of the dhow? Give the answer in appropriate units.

6.4 Calculate the interior and exterior angles of regular polygons; name polygons with up to ten sides.

Trigonometry, Pythagoras’ theorem and the solution of triangles

6.5 Know the standard trigonometric ratios, and their standard abbreviations, for sine of θ, cosine of θ and tangent of θ, given an angle θ in a right-angled triangle; use these ratios to find the angles of a right-angled triangle given two sides, or to find the remaining sides given one side and one angle.

Show that sintan cosθθ θ= .

6.6 Know at least two different proofs of Pythagoras’ theorem.

Explain how you could use the diagram to prove Pythagoras’ theorem.

6.7 Use Pythagoras’ theorem to find the distance between two points and to solve right-angled triangles; set up the Cartesian equation of a circle of radius r, centred at the origin of an xy-coordinate system.

Show that a triangle with sides of length m2 – n2, 2mn and m2 + n2 respectively is always right-angled. Find some right-angled triangles using this result.

Solve the triangles shown. Give all the angles and all the sides.

Each side of a cube is 5 cm. Calculate the length of a diagonal of the cube from one vertex on the ‘base’ to the opposite vertex on the ‘top face’. What is the angle between this diagonal and the base?


There are many websites devoted to proofs of this fundamental theorem in geometry.


Constructions

6.8 Perform and justify straight edge and compass constructions, including those to bisect a line, to construct an equilateral triangle with a side of given length, to drop a perpendicular from a point to a line, and to bisect an angle.

Construct a square. You may only use a straight edge, a pencil and a pair of compasses.

Use the construction to bisect an angle several times over to construct an angle of 22.5°.

Explain why the construction to bisect an angle works.

Loci

6.9 Determine the locus of an object moving according to a rule, including those arising in simple physical situations.

A goat is on a rope attached at one corner of a rectangular enclosure. The enclosure measures 10 m by 4 m. The rope is 6 m long. Draw a scale drawing of the enclosure and shade in the locus in which the goat can move.

Find the locus of all points 3 cm from a circle of radius 5 cm. Discuss how the locus is changed if three dimensions are allowed.

Transformations

6.10 Investigate Islamic patterns and describe their features.

The picture shows a pattern from a mosque in Isfahan, in Iran.

Use this and other Islamic patterns to discuss key features of the pattern (its construction, reflection symmetries, translations, and so on).

Use of ICT

6.11 Use ICT to explore geometrical relationships.

7 Use a range of measures and compound measures to solve problems

7.1 Use formulae to calculate: the circumference and area of a circle; the perimeter and area of any triangle, or trapezium, parallelogram or quadrilateral with perpendicular diagonals; the surface area and volume of a right prism, cylinder, square-based pyramid and cone; and the volume of a sphere.

The diagram shows a circle drawn inside a square. The radius of the circle is 5 cm.

Find the area of the square.

Calculate the exact ratio of the area of the square to the area of the circular region.

Transformations

Transformations are best developed through use of DGS.


The solid is a prism, with dimensions as shown. The cross-section is shaded.

Calculate the volume of this prism.

The diagram shows the frame of a kite. The frame is made of sticks and forms a right prism with octagonal ends.

The lengths of some sticks are marked on the diagram.

What is the total length of all the sticks?

The volume of a pyramid is 13 (base area × perpendicular height).

Calculate the volume of a pyramid with a square base of side 5 cm and perpendicular height 6 cm.

Another pyramid has a square base with side 4 cm and a volume of 48 cm3. What is its perpendicular height?

The diagram shows a pyramid with a triangular base that is an isosceles right-angled triangle.

Write down a formula for the volume of this pyramid.

7.2 Use bearings.

An oil tanker sails 350 km from Doha towards Dubai on a bearing of 090° and then from Dubai towards Al Kuwayt on a bearing 310°. Al Kuwayt is about 600 km from Doha. Approximately, how far is it from Dubai to Al Kuwayt?

7.3 Work with SI units and compound measures: rates such as cost per litre, kilometres per litre, litres per kilometre; and average speed and density, including population density (number of people per unit area).

A satellite passes over both the north and south poles, and it travels 800 km above the surface of the Earth. The satellite takes 100 minutes to complete one orbit.

Assume the Earth is a sphere and that the diameter of the Earth is 12 800 km.

Calculate the speed of the satellite, in kilometres per hour.

Compound measures

Use appropriate SI units and dimensions. Stress how units are calculated in compound measures.



By the end of Grade 10, students know that statistical data are collected from observation or measurement on samples taken from a larger population, and that, by analysing the sample data, inferences can be made about the population as a whole. They distinguish between qualitative (or categorical) data and quantitative data, and between discrete and continuous data. They plan simple surveys, design simple questionnaires and plot histograms in which the height of each bar is proportional to the frequency of that class. They calculate means and medians, and understand mode and modal class. They plot and interpret simple scatter diagrams between two random variables, and draw a line of best fit where there appears to be correlation. They use relevant statistical functions on a calculator and ICT applications to present statistical tables and graphs.

Students should:

8 Collect, process, represent, analyse and interpret data and reach conclusions

Introductory statistical techniques

8.1 Know that different types of data can be collected from samples – qualitative or categorical data (e.g. eye colour, male, female) and quantitative data (e.g. age, height, lifespan, mortality rates) – and that quantitative data may be discrete (e.g. number of defective items in a production process) or continuous (e.g. weight); understand the concept of a random variable.

8.2 Plan simple surveys and design questionnaires to collect meaningful primary data from samples in order to test simple hypotheses about, or estimate, characteristics of the population as a whole; formulate problems using secondary data from published sources, including the Internet.

8.3 Plot simple histograms in which the height of the bars is proportional to the frequency of each class interval and use related vocabulary, including frequency, range and mode, modal class and modal frequency.

8.4 Calculate measures of central tendency such as the arithmetic mean and the median.

A company makes breakfast cereal containing nuts and raisins. They counted the number of nuts and raisins in 100 small packets.


Students should know that statistics is the branch of mathematics used to predict the outcomes of large numbers of events when these outcomes are uncertain, and that probability lies at the heart of statistics. They should be aware of the uses of statistics in society and recognise when statistics are used sensibly and when they are misused or likely to be misunderstood.

Statistical techniques

Include primary data collected in other subjects, such as science and the social sciences.

Include secondary data downloaded from the Internet.


a. Calculate an estimate of the mean number of nuts in a packet. You may complete the table below to help you with the calculation.

Number of nuts

Mid-point of bar (x)

Number of packets (f)

f x

4–6 5 26 130

7–9 8 33

10–12 11 20

13–15 14 15

16–18 17 6

100

b. Calculate an estimate of the number of packets that contain 24 or more raisins.

8.5 Make simple inferences and draw conclusions from the formulation of a problem to the analysis of data in a range of simple situations.

Compare the television viewing habits of students in different grades at school.

9 Simple correlation

9.1 Draw scatter diagrams between two random variables associated with some common context; identify through elementary qualitative discussion positive and negative correlation; where there appears to be correlation, draw a line of best fit, judging by eye the line about which the data points are most evenly distributed.

Compare the examination marks for all students in a class for: a. mathematics and science; b. Arabic and English; c. mathematics and art. Discuss whether there appears to be correlation or not.

The scatter diagram shows the total amounts of sunshine and rainfall for 12 seaside towns in the UK during one summer. Each town has been given a letter. The dashed lines go though the mean amounts of sunshine and rainfall.

Which town’s rainfall was closest to the mean?

Draw a line of best fit on the scatter diagram. Use your line to find an estimate of the hours of sunshine for a seaside town that had 30 cm of rain.

10 Use of ICT

10.1 Use a calculator with statistical functions to aid the analysis of large data sets, and ICT applications to present statistical tables and graphs.

ICT opportunity

Use a graphics calculator to draw a scatter plot, or a spreadsheet application with graphs.





Students solve routine and non-routine problems in a range of mathematical and other contexts, and use mathematics to model and predict the outcomes of real-world applications. They identify and use connections between mathematical topics. They break down complex problems into smaller tasks, and set up and perform appropriate manipulations and calculations. They develop and explain chains of logical reasoning, using correct mathematical notation and terms. They generate mathematical proofs and identify exceptional cases. They aim to generalise. They approach problems systematically, knowing when it is important to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They recognise when to use ICT and do so efficiently.

Number and algebra

Students use the laws of exponents, proportional reasoning and harder percentage calculations to solve problems, including compound interest problems. They find the sums of geometric sequences and convert any recurring decimal to a fraction. They simplify and combine numeric and algebraic fractions and multiply any two monomial, binomial or trinomial expressions, collecting and simplifying similar terms. They factorise quadratic expressions, relating the factorisations to geometric representations. They generate formulae from physical contexts and rearrange formulae connecting two or more variables. Through their continued study of linear, quadratic, reciprocal and other functions and their graphs, and the solution of associated equations, students appreciate a range of numerical and algebraic applications in the real world. They solve simple problems represented by regions of linear inequality. They recognise when quadratic functions are increasing, decreasing or stationary. They model situations with quadratic functions and find exact and approximate solutions of quadratic equations, and a pair of simultaneous equations, one linear and one quadratic. They solve problems involving inverse proportion. They find the tangent at a point on the graph of a function. They continue to use realistic data and ICT to analyse problems.


Students continue to use their knowledge of geometry and trigonometry to solve practical and theoretical problems relating to shape and space. They solve right-angled triangles in two and three dimensions using the standard trigonometric ratios. They know and use the sine rule and cosine rule, and calculate the area of a triangle using ½ ab sin C. They use Pythagoras’ theorem to show that sin2 θ + cos2 θ = 1 for any angle θ, to find the distance between two points and to set up the Cartesian equation of a circle. They find the points of intersection of a straight line with a circle. They plot the graphs of circular functions and solve simple problems modelled by these functions. They prove standard circle theorems. They continue to use SI units and a range of measures to solve problems, including radian measure

Grade 11 Foundation


to calculate sector areas and arc lengths and compound measures. They use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth’s surface. They use ICT to explore geometry.


Students plan questionnaires and surveys to collect meaningful primary data from samples. They know the importance of representative samples, and can locate sources of bias. They collect data from secondary sources, including the Internet, and ask and answer questions related to the data. They group data and plot histograms and other frequency and relative frequency distributions. They draw stem-and-leaf diagrams and box-and-whisker plots. They continue to calculate and use measures of central tendency. They analyse results to draw conclusions and use a range of graphs, charts and tables to present their findings.








and statistics

Grade 10 55% 30% 15%

Grade 11 55% 30% 15%

Grade 12 50% 25% 25%







By the end of Grade 11, students solve routine and non-routine problems in a range of mathematical and other contexts, and use mathematics to model and predict the outcomes of real-world applications. They identify and use connections between mathematical topics. They break down complex problems into smaller tasks, and set up and perform appropriate manipulations and calculations. They develop and explain chains of logical reasoning, using correct mathematical notation and terms. They generate mathematical proofs and identify exceptional cases. They aim to generalise. They approach problems systematically, knowing when it is important to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They recognise when to use ICT and do so efficiently.

Students should:







1.6 Develop chains of logical reasoning, using correct mathematical notation and terms.


1.8 Generate mathematical proofs, and identify exceptional case.

1.9 Aim to generalise.




Grade 11 Foundation

Key standards


Cross-references






Proofs





Number and algebra

By the end of Grade 11, students use the laws of exponents, proportional reasoning and harder percentage calculations to solve problems, including compound interest problems. They find the sums of geometric sequences and convert any recurring decimal to a fraction. They simplify and combine numeric and algebraic fractions and multiply any two monomial, binomial or trinomial expressions, collecting and simplifying similar terms. They factorise quadratic expressions, relating the factorisations to geometric representations. They generate formulae from physical contexts and rearrange formulae connecting two or more variables. Through their continued study of linear, quadratic, reciprocal and other functions and their graphs, and the solution of associated equations, students appreciate a range of numerical and algebraic applications in the real world. They solve simple problems represented by regions of linear inequality. They recognise when quadratic functions are increasing, decreasing or stationary. They model situations with quadratic functions and find exact and approximate solutions of quadratic equations, and a pair of simultaneous equations, one linear and one quadratic. They solve problems involving inverse proportion. They find the tangent at a point on the graph of a function. They continue to use realistic data and ICT to analyse problems.

Students should:


2.1 Use as appropriate the language of number sets from Grade 10.

What is the solution set in of the quadratic equation 4x2 + 3x – 1 = 0?

What is the solution set in of the quadratic equation (x – 2)(3x + 1) = 0?

What is the complement set of the set A ∪ (B ∪ C)? Show this on a Venn diagram.


3.1 Understand and use the laws of exponents to calculate and simplify problems, including mental calculations in appropriate cases. Calculate mentally the value of 9999 × 0.033.

In this question, you should not use a calculator. An elastic band is fixed on four pins on a pinboard, as shown in the diagram below. Show that the total length of the band in this position is 14√ 2 units.

Algebra

Students should learn that algebra enables generalisation and the establishment of relationships between quantities and/or concepts. They should understand the nature and place of algebraic reasoning and proof, and how algebra may be related to geometric concepts, and vice versa.

Laws of exponents



Use standard form to estimate the value of 4350 × 237.8 × π2.

The Earth is approximately a sphere of radius 6378 kilometres. Without using a calculator estimate the circumference at the equator.

The mass of the Earth is 5.98 × 1024 kg. A typical man has a mass of about 70 kg. Approximately how many men would have a total mass equal to that of the Earth?

Light travels at about 300 000 kilometres per second. Use standard form to find the distance away from the Earth of a light-emitting body whose light signal is received at Earth one year after it is emitted.

The Earth completes its orbit around the Sun in 365 days. The Earth is 148.8 million kilometres from the Sun. Assume that the Earth’s orbit is circular and that it travels around the Sun with constant speed. Calculate the Earth’s speed in kilometres per hour.

3.2 Solve a range of problems using the multiplicative nature of proportional reasoning.

3.3 Perform harder percentage calculations, including taking a percentage of a percentage, inverse percentage and compound interest problems.

QR 10 000 has to be invested in deposit accounts. There is a choice of two accounts. One account pays an annual interest of 4.6%. The other account pays interest of 1.5% three times a year. What is the AER of the second account? Which is the better account to invest in and how much more interest will there be after one year in this account than in the other account?

3.4 Investigate the problem of compounding interest more and more frequently and note that this tends to a limiting value; use this context to learn about the number e.


4.1 Know the properties of geometric sequences and the conditions under which an infinite geometric series can be summed.

Grains of rice are placed on each square of a chessboard. The board has 64 squares. One grain is placed on the first square, two on the second, four on the third, eight on the fourth, and so on. Calculate the total number of grains of rice on the chessboard.

1 kilogram of rice contains approximately 16 000 grains of rice. Estimate the weight of all the rice on the chess board.

The sum of the infinite geometric series 1 – 1/2 + 1/4 – 1/8 + … is A. 5/8 B. 2/3 C. 3/5 D. 3/2

Investigate compound interest problems as examples of geometric series.

4.2 Convert any recurring decimal to an exact fraction.

Explain why 4330.12 = .

4.3 Develop further a sense of working with symbols, understanding that the transformation of all such algebraic objects generalises the well-defined rules of arithmetic, and knowing that letters are used to represent:




• new equations, expressions or functions defined in terms of known, or given, expressions or functions.

Percentages

Include problems involving the annual equivalent rate (AER).


Is (x – a)(x2 + ax + a2) = x3 – a3 an equation or an identity? Discuss what happens to this mathematical statement when a is replaced throughout by –a.

Give examples of what is meant by an associative law and a distributive law.

Calculate (√5 + √3)(√5 – √3).

4.4 Combine numeric or algebraic fractions, and multiply combinations of monomial, binomial and trinomial expressions, collecting and simplifying similar terms.

Use Pascal’s triangle to read off the coefficients of the powers of x in the expansion of (1 + x)n for different values of the positive integer n. Check the results for n = 3 by expanding (1 + x)3.

Simplify (2x – 3)(x2 + x – 10).

4.5 Factorise expressions of the form a2x2 – b2y2, and quadratic expressions; conceptualise geometric representations for these factorisations and other similar quadratic expressions. Without using a calculator, find the exact value of 7.922 – 2.082.

Explain why (a + b)2 ≠ a2 + b2 .

Draw a diagram to represent the identity (a + b)2 = a2 + 2ab + b2.

Draw a diagram to represent the identity (a – b)2 = a2 – 2ab + b2.

Construct some quadratic expressions from two linear factors in a and b and draw geometric representations for them.

4.6 Simplify numeric and algebraic fraction expressions by cancelling common factors; rationalise a denominator of a fraction when the denominator contains simple combinations of surds.

Rationalise the expression 1/(√2 + √3).

4.7 Generate further formulae from a physical context, and rearrange formulae connecting two or more variables; substitute an expression for a given variable into a different formula containing this variable.

Melons cost QR 1.5 each and apples cost QR 3.75 per kilogram. A man buys apples and melons at the supermarket. Write a formula to describe the total cost of his purchase. Investigate how many melons and how many kilograms of apples he could buy for QR 30.

Find R in terms of R1 and R2 when 1/R = 1/R1 + 1/R2 .

The volume of a solid cylinder of length h and radius r is V. Find a formula for the curved surface area, A, of the cylinder in terms of r and h. Use this formula to find a formula expressing V in terms of A and r.


5.1 Use a graphics calculator to plot and interpret a range of functional relationships, some continuous and others discontinuous, arising in familiar contexts.

Plot the graph of y = 1/x2 for the domain set {x: x ∈ and 1 ≤ x ≤ 4}. Discuss whether the domain could be extended.

Plot the curve y = √ x on a suitably defined domain. Discuss why the domain cannot be the set . Compare this curve with the curve of y = x2, drawn on the same axes.

Quadratic expressions

Include the forms:

x2 + (α + β)x + αβ

x2 − (α + β)x + αβ

x2 ± (α − β)x − αβ

ICT opportunity



Ahmed does a parachute jump. He jumps out of the plane and falls faster towards the ground. After a few seconds his parachute opens. He slows down and then falls to the ground at a steady speed. Which of these graphs shows Ahmed’s parachute jump? Explain why each of the other graphs is wrong.

Invent examples of functions with different definitions on different subdomains, for example, electricity charges as a function of the number of units of electricity used.

The Int x function, written as [x], maps x to the greatest integer less than or equal to x. Find [5.9], [6] and [–4.7].

Plot on the same axes the curves y = 2x and y = 2–x for –3 ≤ x ≤ 3. Describe the features of the two curves. Discuss situations that could be modelled by these equations.

A rectangular enclosure has a wall on one side and the other three sides are made of metal fencing. The side parallel to the wall has length d, measured in metres. The enclosure has an area of 600 m2. Show that the total length, L metres, of fencing is given by L = d + 1200/d. Plot this function using a graphics calculator. Find from the graph the value of d that makes L as small as possible.


Discuss whether or not the graphs of a circle and a semicircle represent functions. Identify any special cases.

5.3 Draw the tangent line at a point on the graph of a function, calculate the slope of this line and interpret the behaviour of the function at that point, knowing whether the function is increasing or decreasing at the point, or stationary.

Direct proportion

5.4 Translate the statement y is proportional to x2 into the symbolism y ∝ x2 and into the equation y = kx2; know that the graph of this equation is a parabola through the origin.

A body falling from rest under the force of gravity falls a distance s metres in time t seconds where s = 4.9t2. Find the distance fallen after 5 seconds. How long does it take the body to fall 30 metres?

Discuss how to plot a linear graph s = 4.9z, by defining the variable z = t2.

5.5 Recognise some other common examples of proportional variation.



5.6 Know that a straight line in the explicit form y = mx + c represents a function, but that a straight line in the implicit form ax + by + d = 0 may, or may not, be a function; know that any straight line in the xy-plane can be represented in this implicit form, but that only certain lines in the plane can be represented by the explicit form; work with both of these forms.

Look at this octagon. The line through D and B has the equation 3y = x + 25. The line through G and H has the equation x = y + 15.

Solve the simultaneous equations 3y = x + 25 x = y + 15 to find the point of intersection of these two lines.

5.7 Plot the graphs of equations in 5.6 above; know the meanings of gradient of the line (and be familiar with alternative wordings such as slope or rate of change of y with respect to x) and intercept on the x- or y-axis, and relate these to the coefficients a, b and d, or to the coefficients m and c.

What is the gradient of the line 3x + 2y – 5 = 0? Find an equation of a line that is perpendicular to this line. Draw the two lines on a graph.

A triangle has its vertices at the points (1, 3), (2, 5) and (3, 4.5). Find the equations of the lines containing each side. Is the triangle a right-angled triangle? Explain how you know.

What angle does the line y = √ 3x + 1 make with the positive x-axis?

5.8 Graph regions of linear inequality and solve simple problems (e.g. elementary linear programming) represented by such regions; understand simple quadratic inequalities.

The shaded region is bounded by the curve y = x2 and the line y = 2. What two inequalities together fully describe the shaded region?

A company delivers new cars to Doha. It has a contract to deliver at least 65 cars each day. The company owns 7 carriers that can each carry 8 cars and 5 carriers that can each carry 10 cars. The company employs 8 drivers. Each carrier can make only one journey with a full load each day. What is the maximum numbers of cars that can be delivered each day? What is the minimum number of drivers needed to fulfil the contract?


Quadratic functions

5.9 Recognise a second-order polynomial in one variable, y = ax2 + bx + c, as a quadratic function; plot graphs of such functions (recognising that these are all parabolas), and pick out the intercepts with the coordinate axes, the axis of symmetry and the coordinates of the maximum or minimum point; understand when such functions are increasing, when they are decreasing and when they are stationary.

5.10 Model a range of situations with appropriate quadratic functions.

A fountain at ground level sprays out jets of water. Each jet is a parabola. The jet that sprays the farthest has equation y = –x2 + 8x – 15. Factorise this expression. Hence find: a. where the fountain jet is positioned in this xy-coordinate system; and b. how far from the fountain jet the water hits the ground. Calculate the greatest height that the water reaches.

Huda throws a ball to Mariam who is standing 20 m away. The ball is thrown and caught at a height of 2.0 m above the ground.

The ball follows the curve with equation y = 6 + c(10 – x)2, where c is a constant. Calculate the value of c by substituting x = 0, y = 2 into the equation.

Curve A is the reflection in the x-axis of y = x2. What is the equation of curve A?

An n-sided polygon has 12 ( 3)n n − diagonals. How many diagonals has an octagon?

A polygon has 104 diagonals. How many sides does it have?

5.11 Solve quadratic equations exactly, by factorisation, by completing the square and by using the quadratic formula.

5.12 Find the axis of symmetry of the graph of a quadratic function and the coordinates of its turning point by algebraic manipulation; understand the effect of varying the coefficients a, b and c in the expression ax2 + bx + c.

y = (x – 3)2 + 5 is a quadratic function of x. What is the minimum value of this function and for what value of x does it occur? What is the maximum range of the function? Give the equation of the axis of symmetry of the function. Write an alternative form for the equation defining the function. Sketch the graph of this function.

5.13 Find approximate solutions of the quadratic equation ax2 + bx + c = 0 by reading from the graph of y = ax2 + bx + c the x-coordinate(s) of the intersection point(s) of the graph of this function and the x-axis.

ICT opportunity

Include the use of a graphics calculator.

ICT opportunity



The graph below shows the curve y = x2 + 4x.

a. Solve the equation x2 + 4x – 2 = 0 using the graph.

Give your answers to two decimal places.

b. The equation x2 + 4x + 5 = 0 cannot be solved using the graph. Why not?

5.14 Find exactly by algebraic means, and approximately from the points of intersection of a straight line with the graph of a quadratic function, the solution set of two simultaneous equations L1 and Q1, where L1 represents a linear relation for y in terms of x, and Q1 a quadratic function of y in terms of x.

5.15 Solve physical problems modelled simultaneously by two such functions.

Inverse proportion and the reciprocal function

5.16 Understand the statement y is inversely proportional to x and set up the corresponding equation y = k/x; know some characteristics, including that x ≠ 0 and that x = 0 is an asymptote to the curve, as is y = 0; study examples of inverse proportionality.

Three people working flat out complete a job in sixteen hours. How many hours would it take eight people to do the same job? Explain any assumptions you have made.

Look at the graphs below.

a. One of the graphs shows the equation y = kx – x2 (k is a constant). Which graph is it?

b. One of the graphs shows the equation y = k/x, where k is a positive constant. Which graph is it?

ICT opportunity


ICT opportunity



The average speed for a fixed distance journey is inversely proportional to the time taken to complete the journey. A family travels in Europe by car. They travel exactly half their journey in 2 hours, then stop for lunch for 1 hour, and then take 3 hours over the second half of the journey. How were the average speeds related on each part of the journey? If the average speed for the first half of the journey was 72 kilometres per hour what was the average speed for the whole journey?

Explain why the function y = k/x cannot be defined on the domain set . What is the largest domain the function can be defined on? Sketch the graph of the function for this domain. Does the function have a greatest or least value? Is there anywhere where the function increases?


By the end of Grade 11, students continue to use their knowledge of geometry and trigonometry to solve practical and theoretical problems relating to shape and space. They solve right-angled triangles in two and three dimensions using the standard trigonometric ratios. They know and use the sine rule and cosine rule, and calculate the area of a triangle using ½ ab sin C. They use Pythagoras’ theorem to show that sin2 θ + cos2 θ = 1 for any angle θ, to find the distance between two points and to set up the Cartesian equation of a circle. They find the points of intersection of a straight line with a circle. They plot the graphs of circular functions and solve simple problems modelled by these functions. They prove standard circle theorems. They continue to use SI units and a range of measures to solve problems, including radian measure to calculate sector areas and arc lengths and compound measures. They use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth’s surface. They use ICT to explore geometry.

Students should:

6 Develop geometrical reasoning and proof and solve geometric problems


6.1 Use dynamic geometry systems to conjecture results and to explore geometric proof.

In the diagram below, the altitudes BN and CM of the triangle ABC intersect at S. ∠ MSB is 40° and ∠ SBC is 20°. Prove that triangle ABC is an isosceles triangle.

TIMSS Grade 12



Use of ICT

Geometry is enhanced with the use of dynamic geometry systems, or DGS, which provide an interactive means for investigating and hypothesising results which can then be proved as theorems.



6.2 Solve right-angled triangles using the standard trigonometric ratios, including tan θ = sin θ /cos θ, and/or Pythagoras’ theorem.

6.3 Know and use the sine rule and the cosine rule to solve triangles.

Show that Pythagoras’ theorem is a special case of the cosine rule.

A ship sails 50 kilometres in a direction 032° and then 29 kilometres in a direction 315°. How far is the ship from its starting point? What is its bearing from its starting point?

A helicopter at airfield A received a distress call from a boat. The position of the boat, B, was given as 147 km from the airfield, on a bearing of 072°. A man on the boat is flown to hospital. Calculate the distance the helicopter travelled from the boat to the hospital at H.

On another occasion the helicopter travelled from the airfield on a bearing of 218° to fly a pregnant woman at W to the hospital. The helicopter then flew on a bearing of 081° to the hospital, H. Calculate the distance the helicopter travelled from where it picked up the woman to the hospital.

A triangle has its three angles in the ratio 2 : 3 : 4. Find to two significant figures the ratio of the lengths of its sides.

6.4.1 6.4 Solve triangle problems in two and three dimensions.

The two sides of a canal are straight, parallel and the same height above the water level. Jana and Shrifa want to find the width of the canal. They measure 100 m on the canal bank and stand facing each other at the points J and S. Jana measures the angle she turns through to look at the post, P, as 25°. Shrifa measures the angle she turns through to look at the post as 15°. Calculate the width of the canal.

Triangle problems

Include the terminology angle of inclination and angle of declination.


The Great Pyramid of Cheops in Egypt is built on a square base with side 230 metres. Each face of the pyramid is at 52° to the horizontal. Calculate the height of the pyramid. Calculate the inclination of an edge of the pyramid to the horizontal.

The Great Pyramid of Cheops at Giza. Source: www:kingtutshop.com

6.5 Calculate the area of a triangle using 12 sinab C .

6.6 Use Pythagoras’ theorem to find the distance between two points in the Cartesian plane; set up the Cartesian equation of a circle of radius r, centred at the point (α, β) .

Find the equation of a circle of radius 5 units, centred at the point (5, –3).

Find the exact distance between the point (1, 4) and the point (–2, 5).

Two sides of a right-angled triangle are of length 21 cm and 29 cm. What are the possible lengths of the remaining side?

6.7 Find the points of intersection of a straight line with a circle by using algebraic substitution from the equation of the straight line into the equation of the circle. Find the points where the line 4x – 3y = 0 cuts the circle x2 + y2 = 100.

Circular functions

6.8 Use the unit circle x2 + y2 = 1 to plot graphs of the circular functions sin θ° and cos θ° for any angle θ° , where 0° ≤ θ° ≤ 360°; know that any point on this circle has coordinates (cos θ°, sin θ°), where θ° is the angle the radius to the point makes with the positive x-axis.

Explain why sin (180° – θ°) = sin θ° .

Give the exact value of cos 225°. What other angle has the same cosine value?

6.9 Derive and recall the exact values for the sine, cosine and tangents of 0°, 30°, 45°, 60°, 90° and use these in relevant calculations.

Calculate the exact area of an equilateral triangle with sides of length 6 cm.

6.10 Use a calculator to find sine and cosine values of a given angle and to find the angle corresponding to a given value of the sine or cosine of that angle, and know that these are inverse functions defined on a restricted domain.

Find the angle whose sine is 0.9063. What is the cosine of this angle? For this angle, verify the result in GM 6.11.

6.11 Use Pythagoras’ theorem to show that sin2 θ° + cos2 θ° = 1 for any angle θ° .

Verify this result for the angles 30°, 45° and 60°. What happens when θ° = 90°?

6.12 Solve simple problems modelled by circular functions.


Many interesting websites are devoted to proofs and applications of this key theorem in geometry.

Powers of (co)sines

Note that (cos θ ° )2 is written as cos2 θ ° and that (sin θ ° )2 is written as sin2 θ ° .


Radian measure and circle geometry

6.13 Use radian measure to calculate sector areas and arc lengths.

A satellite is 1500 km above the Earth. It has a camera with a 50° angle of view with which it surveys the Earth below. Draw a diagram to represent the satellite and its camera in relation to the Earth. Calculate how far apart the two furthest points on the Earth are that can be photographed by the satellite at any one time. Take the Earth to be a sphere of radius 6378 kilometres.

A manufacturer makes party hats shaped like hollow cones. To make the hats she cuts pieces of card that are sectors of a circle, radius 24 cm. The angle of the sector is 135°.

a. Show that the arc length of the sector is 18π cm.

b. The sector is joined edge to edge to make a cone. The edges of the sector meet exactly with no overlap. Calculate the vertical height of the completed hat.

6.14 Prove and use the theorems:

• The perpendicular from the centre of a circle to a chord bisects the chord.

• The two tangents from an external point to a circle are of equal length.

• The angle subtended by an arc at the centre of the circle is twice the angle subtended by the arc at a point on the circle, including, as a special case, the angle in semicircle is a right angle.

• Angles in the same segment subtended by a chord are equal.

• The angle subtended by a chord at the centre of a circle is twice the angle between the chord and the tangent to the circle at an end point of the chord.

• When two chords BC and DE in a circle intersect at A then AB × AC = AD × DE.

• Opposite angles of a cyclic quadrilateral are supplementary.

Two circles with centres at A and B have radii of 7 cm and 10 cm as shown in the diagram. The length of the common chord PQ is 8 cm. Calculate the length of AB.

TIMSS Grade 12

Sectors and arcs

Include terms associated with a circle: centre, radius, diameter, circumference, arc length, sector, segment, chord.

Circles

Include terms associated with a circle: centre, radius, diameter, circumference, arc length, sector, segment, chord, tangent, inscribed circle, circumcircle, cyclic quadrilateral.

Circle theorems

Include the use of dynamic geometry systems (DGS).


In the diagram below, AD is a tangent to the circle with centre O. ∠ ABC is 63° and AC is a chord of the circle. AB is 18 cm and BC is 3 cm.

Calculate the values of ∠ AOC, ∠ OCA and ∠ CAD. Calculate the area of triangle ABC. Calculate the length of AC.


7.1 Calculate lengths, areas and volumes of geometrical shapes.

The diagram models a rectangular rear windscreen of a car. The windscreen wiper can rotate through 160°. Calculate the percentage of the rear window that is cleaned by the wiper.

On the pinboard, draw a trapezium that has a perimeter of 6 + 4√ 2.

This shape is designed using three semicircles.

The radii of the semicircles are 3a, 2a and a.

a. Find the area of each semicircle, in terms of a and π, and show that the total area of

the shape is 6πa2.

b. Find a when the area is 12 cm2, leaving your answer in terms of π.

7.2 Use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth’s surface.

How would you find the shortest distance between Doha and London?


A plane flies from Doha to Karachi almost along the line of latitude 25 degrees north. Doha is at longitude 51 degrees east approximately and Karachi is at longitude 67 degrees east approximately. How far is it from Doha to Karachi along this route?

What is a great circle of the Earth?


7.3 Work with SI units and compound measures including density, average speed and acceleration, measures of rate, and population density (number of people per unit area), using appropriate units and dimensions.

Wafa recorded the speed of a car every 10 seconds throughout a short journey from her home to school. She used the data to draw a speed–time graph.

a. Find a point during the journey when the car’s speed was increasing most quickly.

Mark this point as P. By drawing appropriate lines on the graph, find the acceleration of the car at point P.

b. Find a point during the journey when the car’s speed was decreasing most quickly. Mark this point as Q. By drawing appropriate lines on the graph, find the acceleration of the car at point Q.

c. The car uses least fuel when it travels at speeds between 20 m/s and 25 m/s. Find an approximate value for the area under the graph for the period when the car was travelling at between 20 m/s and 25 m/s. What does this area represent? Give the correct units.

A cable car takes passengers to the top of a volcano. It starts from station A and takes 16 minutes to reach station B at the top of the volcano. The average speed of the cable car is 2 metres per second. The cable car is at an angle 25° to the horizontal. Find, to the nearest metre, the height of the volcano as measured from A.

TIMSS Grade 12

Compound measures

Use appropriate SI units and dimensions. Stress how units are calculated in compound measures.

Link where appropriate to work in science and technology, using compound measures such as rate of doing work or momentum.



By the end of Grade 11, students plan questionnaires and surveys to collect meaningful primary data from samples. They know the importance of representative samples, and can locate sources of bias. They collect data from secondary sources, including the Internet, and ask and answer questions related to the data. They group data and plot histograms and other frequency and relative frequency distributions. They draw stem-and-leaf diagrams and box-and-whisker plots, and continue to calculate and use measures of central tendency. They analyse results to draw conclusions and use a range of graphs, charts and tables to present their findings.

Students should:


Sampling

8.1 Know that:

• it is important to choose representative samples;

• in a random sample there are chance variations;

• in a biased sample there are systematic differences between the sample and the population from which it is drawn;

and locate obvious sources of bias within a sample.

An article in a newspaper claimed that 93% of us drop litter every day. It does not say how the journalist knows that 93% of people drop litter every day. Some students think the percentage of people who drop litter every day is much lower than 93%. They decide to do a survey.

a. Jabor plans to ask 10 people if they drop litter every day. Give two different reasons why Jabor’s method might not give very good data.

b. Layla plans to go into Doha on Sunday morning. She will stand outside a shop and record how many people walk past and how many of those drop litter. Give two different reasons why Layla’s method might not give very good data.

Mosa wants to investigate whether more people are born in the winter than in the summer. He plans to ask 30 students in his grade whether they were born in the winter or the summer. Discuss ways in which Mosa could improve his survey.


8.2 Plan surveys and design questionnaires to collect meaningful primary data from samples in order to test hypotheses about or estimate characteristics of the population as a whole; formulate problems using secondary data from published sources, including the Internet.

8.3 Calculate and use measures of central tendency such as the arithmetic mean and the median.

Investigate life expectancy in a range of countries including Qatar, Iran, Turkey, India, Brazil, China, Russia, Italy, the United Kingdom and the United States of America.




These lay the foundations for later development in the advanced mathematics course on quantitative methods.


8.4 Construct (relative frequency) histograms and plot cumulative frequency distributions, grouping continuous data when necessary.

A scientist wanted to investigate the lengths of an egg from a particular breed of hen. Taking a sample of 80 eggs, she measured the length of each one and grouped the data as follows:

Length (l) in cm

4.4 ≤ l < 5.0 5.0 ≤ l < 5.4 5.4 ≤ l < 5.8 5.8 ≤ l < 6.3 6.3 ≤ l < 6.5

Frequency 4 20 36 16 4

Complete the histogram to show this information. Write the frequency density on each part of the histogram.

Calculate the mean length of the eggs in her sample. Discuss how to calculate best estimates for the modal value and the median value of the lengths of the eggs in the sample.

304 people took part in a swimming contest. They swam 1.5 km. The histogram shows the distribution of their times for the event.

a. The histogram is constructed using frequency densities.

The table shows the frequency densities. Complete the table to show the frequencies.

Time t (minutes)

Frequency density

Frequency

17 ≤ t < 22 16.0 80

22 ≤ t < 27 28.0

27 ≤ t < 32 12.4

32 ≤ t < 52 1.1

b. 304 people took part. Calculate an estimate of the mean time for swimming.

c. Explain why the median time for swimming must be between 22 and 27 minutes.

d. Calculate an estimate of the median time for swimming.

Histograms and cumulative frequency distributions

Include the terms frequency, frequency distribution, frequency density, relative frequency and relative frequency distribution, and range, percentile, interquartile range, semi-interquartile range, and mode, modal class, modal frequency.


Sulaiman did a survey of the age distribution of 160 people at a theme park.

The cumulative frequency graph shows his results.

a. Use the graph to estimate the median age of the 160 people at the theme park.

b. Use the graph to estimate the interquartile range of the age of the 160 people at the theme park.

8.5 Draw stem-and-leaf diagrams and box-and-whisker plots and use them in presenting conclusions.

A hospital clinic records the number of patients seen each day in a stem-and-leaf diagram.

12 5 6 7 8 9

13 0 1 2 2 4 7 8 9

14 0 1 2 3 5 9

Key: 12 | 5 means 125 patients.

Find the range, the median and the mode.

The diagram shows a box-and-whisker plot of examination marks for a class of students.

L represents the lowest mark scored and H is the highest mark scored. LH then represents the range of marks. Q1 is the first quartile mark, Q3 is the third quartile mark and Q1Q3 is the interquartile range. M is the median mark.

A school for boys and a school for girls each enters students for the same mathematics examination. The girls’ marks were:

97 98 57 45 63 75 87 34 56 28 67 89 45 61 53 49 81 32 23 45 47 72 34 54 23 100 76 47.

The boys’ marks were:

67 87 83 92 34 31 23 25 29 39 89 91 54 47 41 50 77 18 89 10 26 62 39 14 90.

Draw back-to-back stem-and-leaf diagrams to represent these scores. Compare the performance of the girls and the boys, explaining your methodology and findings.

Using the above data, plot a cumulative frequency graph for the marks of the girls. What was the median score? What was the interquartile range of the distribution of marks? Draw a box-and-whisker plot to represent the girls’ marks.

Draw a relative frequency histogram for these data, explaining how the data were grouped and the meaning of each bar of the histogram.

8.6 Make inferences and draw conclusions from the formulation of a problem to the collection and analysis of data in a range of situations; use a range of statistics and graphs, charts and tables to present and justify findings.

Stem plots and box plots

Stem-and-leaf diagrams and box-and-whisker plots are also known as stem plots and box plots.


9 Use of ICT

9.1 Use a calculator with statistical functions to aid the analysis of large data sets, and ICT applications to present statistical tables and graphs.

ICT opportunity

A range of ICT applications can support data handling.





Students solve routine and non-routine problems in a range of mathematical and other contexts and use mathematics to model and predict the outcomes of substantial real-world applications. They identify and use connections between mathematical topics. They break down complex problems into smaller tasks, and set up and perform appropriate manipulations and calculations. They develop and explain longer chains of logical reasoning, using correct mathematical notation and terms, and generate mathematical proofs. They aim to generalise. They approach problems systematically, knowing when it is important to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They recognise when to use ICT and do so efficiently.

Number and algebra

Students appreciate a wide range of numerical and algebraic applications in the real world. They rearrange harder formulae connecting two or more variables and generate further formulae from physical contexts. They generate recursive sequences to model the behaviour of real-world situations. They use physical contexts to plot and interpret the graphs of linear, quadratic, cubic, reciprocal, exponential and logarithm functions, the sine and cosine functions, the modulus function and other simple non-standard functions. They solve a range of problems using inverse and composite functions. They apply combinations of transformations to the graph of the function y = f(x). They use realistic data and ICT to analyse problems.


Students use approximation methods to calculate the area of an irregular two-dimensional flat surface and the volume of a prism with a constant, but irregular-shaped, cross-section. They draw and use plans and elevations, and interpret maps and scale drawings. They translate, reflect, rotate and enlarge two-dimensional geometric objects. They begin to use vectors to solve physical problems. They solve a range of problems involving compound measures, using appropriate units and dimensions. They explore aspects of geometry using ICT.


Students arrive at conclusions from the formulation of a problem to the collection and analysis of data in a range of situations. They use secondary data from published sources, including the Internet. They use ICT to calculate statistical quantities and to produce a range of graphs, charts and tables to present and justify their findings. They calculate measures of spread, including the variance and standard deviation. They construct histograms and plot cumulative frequency distributions, using grouped continuous data if necessary. They understand that a random variable has a

Grade 12 Foundation


range of values that cannot be predicted with certainty and investigate common examples. They measure the empirical probability (relative frequency) of obtaining a particular value of a random variable. They use a simple mathematical model to calculate the theoretical probability of obtaining a particular outcome for a random variable associated with a set of events. They calculate probabilities of single and combined events, and understand risk as the probability of the occurrence of an adverse event. They use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another. They use simple simulations and consider trends over time using a moving average.








and statistics

Grade 10 55% 30% 15%

Grade 11 55% 30% 15%

Grade 12 50% 25% 25%



Grade 12 teachers should review and consolidate Grades 10 and 11 standards where necessary.




By the end of Grade 12, students solve routine and non-routine problems in a range of mathematical and other contexts, and use mathematics to model and predict the outcomes of substantial real-world applications. They identify and use connections between mathematical topics. They break down complex problems into smaller tasks, and set up and perform appropriate manipulations and calculations. They develop and explain longer chains of logical reasoning, using correct mathematical notation and terms, and generate mathematical proofs. They aim to generalise. They approach problems systematically, knowing when it is important to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They recognise when to use ICT and do so efficiently.

Students should:



1.2 Use mathematics to model and predict the outcomes of substantial real-world applications; compare and contrast two or more given models of a particular situation.




1.6 Develop longer chains of logical reasoning, using correct mathematical notation and terms.



1.9 Generalise when appropriate.



Grade 12 Foundation

Key standards


Cross-references






Proofs






Number and algebra

By the end of Grade 12, students appreciate a wide range of numerical and algebraic applications in the real world. They rearrange harder formulae connecting two or more variables and generate further formulae from physical contexts. They generate recursive sequences to model the behaviour of real-world situations. They use physical contexts to plot and interpret the graphs of linear, quadratic, cubic, reciprocal, exponential and logarithm functions, the sine and cosine functions, the modulus function and other simple non-standard functions. They solve a range of problems using inverse and composite functions. They apply combinations of transformations to the graph of the function y = f(x). They use realistic data and ICT to analyse problems.

Students should:


2.1 Make appropriate use of their knowledge of number sets from Grades 10 and 11.


3.1 Develop further confidence in all the calculation skills established in Grades 10 and 11.

Farida is making a scale model of the Earth and the Moon for a museum. She has found out the diameters of the Earth and the Moon, and the distance between them in metres.

Diameter of the Earth 1.28 × 107 m Diameter of the Moon 3.48 × 106 m Distance between Earth and Moon 3.89 × 108 m

a. How many times bigger is the diameter of the Earth than the diameter of the Moon?

b. In Farida’s scale model the diameter of the Earth is 50 cm. What should be the distance between the Earth and the Moon in Farida’s model?

Look at the table.

Earth Mercury

Mass (kg) 5.98 × 1024 3.59 × 1023

Atmospheric pressure (N/m2) 2 × 10–8

The atmospheric pressure on Earth is 5.05 × 1012 times as great as the atmospheric pressure on Mercury. Calculate the atmospheric pressure on Earth.

Algebra




4.1 Rearrange harder formulae connecting two or more variables and generate further formulae from physical contexts.

The mathematician Johannes Kepler set out three laws of planetary motion in his famous book ‘The Harmony of the World’, published in 1619. Kepler’s third law of planetary motion states that the square of the period of revolution of a planet about the Sun is proportional to the cube of the mean distance of the planet from the Sun. Write this statement as a mathematical equation.

The value of a new car depreciates by 20 per cent at the end of the first year and then loses value at the rate of 10 per cent for every subsequent year. Set up a formula to describe the value V of the car t years after purchase. After how many years will the car be worth one quarter of its purchase price?

4.2 Generate recursive sequences from term-to-term and position-to-term definitions to model the behaviour of real-world situations, for example population growth.

In a certain country, there is a net increase in population from one year to the next of 5 per cent. Set up a recurrence relation to describe the population in year n + 1 in terms of the population in year n. Find the population in year n + 4 compared to the population in year n. Use your formula to find the number of years it takes to double the population from year n.

A woman buys a car and pays in monthly instalments. The car costs QR 60 500 and interest is charged on any outstanding debt at a monthly rate of r%. The woman pays back a fixed amount each month of QR M. Set up a recurrence relation connecting the amount owed, An+1, after n + 1 months in terms of the amount owed, An, at the end of the nth month. How many months will it take to repay the debt if M = QR 1200 and r = 1.2%? How much will the woman have then paid for the car? Investigate repayments for different values of M and r.


Functions and their inverses

5.1 Use a graphics calculator, including the trace function, to show approximate solutions to physical problems requiring the location and physical interpretation of the intersection points of two or more graphs.

5.2 Use physical contexts to plot and interpret:

• graphs of linear, quadratic and cubic functions;

• graphs of the reciprocal function y = k/x (x ≠ 0);

• graphs of the sine and cosine functions;

• graphs of the modulus function and a range of simple non-standard functions.

Which grows faster for x ≥ 0: the power function y = x3 or the exponential function y = ex? Justify your answer. (See also NA 5.8.)

Investigate physical examples of inverse square laws.

Find physical examples that are modelled by circular functions.

A big wheel makes one complete revolution every 90 seconds. The wheel has a diameter of 20 metres. The bottom of the wheel is 2 metres above the ground. Two people get on the wheel and sit in a seat, and then the wheel starts to rotate. T seconds later their height above the ground is given by h = 2 +8 sin 4T°. Explain why this is an appropriate formula to use. At what two consecutive times are they 12 m above the ground?

ICT opportunity

Use spreadsheets in examples like these.

Modelling with functions and their inverses

This aspect of mathematics adds realism, shows the importance of the subject through application and motivates students. Where possible, the use of real data and its analysis through ICT should be encouraged.

Modelling with circular functions

Examples could include oscillations on a spring, bungee jumping, pulse rate, blood pressure, alternating currents, daylight hours.


A ship can only enter a harbour when the tide is in; it must have a minimum depth of water of 8 metres. The tide follows a daily sinusoidal variation given by the formula d = 5 sin 15t° + 8, where t is the time in hours from midnight onwards, measured on the 24-hour clock. At how many times in a day will the depth of water in the harbour be exactly 8 m? For how many hours a day can the ship enter the harbour? Sketch how the level of the tide varies with the time of the day.

5.3 Find, graph and use the inverse function of those functions in NA 5.2 given by a one-to-one mapping or restricted to such mappings; know that the graph of the inverse function may be found by reflecting the graph of the function in the line y = x; solve a range of problems using inverse functions.

The cost of production of q silver bracelets is C = 200 + 15q. Find the inverse function and interpret its meaning.

5.4 Add, subtract and multiply two functions given in the form y1 = f1(x) and y2 = f2(x); write down, without simplification, the mathematical form for one function divided by another.

5.5 Understand the concept of a composite function and use the notation y = f(g(x)).

5.6 Deconstruct a composite function into its constituent functions, using inverse functions.

Calculate the inverse function of f(x) = 5x – 8.

Starting from the function y = x, describe how the function y = (5x – 3)2 is constructed. Show how to deconstruct this function back to the original function.

Transformation of functions

5.7 Understand the transformations of the function y = f(x) given by:

• y = f(x) + a, representing a translation by a in the positive y-direction;

• y = f(x – a), representing a translation by a in the positive x-direction;

• y = af(x), representing a stretch with scale factor a parallel to the y-axis;

• y = f(ax), representing a stretch with scale factor 1/a parallel to the x-axis;

use these and combinations of these transformations to sketch, stage by stage, the transformation of the graph of y = f(x) into the graph of the transformed function.

The straight line y = mx + c is the straight line y = mx translated parallel to itself a distance c in the y-direction. When y = mx, the variable y is directly proportional to the variable x. By redefining the origin to the point (0, c) the straight line y = mx + c implies that the variable (y – c) is directly proportional to the variable x, since y – c = mx and this passes through the point (0, c).

The diagram shows the graph with equation y = x2. On the same axes, sketch the graph with equation y = 2x2.

Curve B is the translation, one unit up the y-axis, of y = x2. What is the equation of curve B? Translate curve B two units to the left. What is the equation of this new curve?

Composite functions

Use a ‘function machine’ to introduce the idea of a composite function and its inverse.


A function is defined by f(x) = x2. Describe the functions a. f(x – 2) and b. f(x + 1), stating how the graphs of each function relate to the graph of y = f(x), and give the defining equation for each function.

Transform the curve y = x3 into the curve y = 5x3. Describe the effect of the transformation. The curve is then translated one unit in the positive x-direction. What is the equation of this new curve?

Describe in words how the graph of y = 1/x is transformed into the graph y = 4 + 5/x. Sketch each graph on the same set of axes.

Explain the difference between a. the functions y = cos x° and y = cos (x + 45)° and b. the functions y = cos x° and y = 2 cos x°.

Modelling with exponential functions

5.8 Understand the ideas of exponential growth and decay and the forms of the associated graphs y = ax, where a > 0; use a graphics calculator to plot the graphs of the exponential function, ex, and the natural logarithm function, ln x; know that one is the inverse function of the other and use this to find solutions to physical problems; solve for x the equation y = ax and use this in problems; use the log function (logarithm in base 10) on a calculator.

The growth of the Internet since 1990 has been modelled by the function N = 0.2(1.8)t, where N is the number of users, counted in millions, t years from 1990. Plot the graph of this function. How many Internet users does the model predict for the year 2006?

When living organisms die the amount of carbon-14 present in the dead matter decays exponentially according to the formula N = N0e–0.000121t, where N0 is the initial quantity and t is the time in years. A bone uncovered at an archaeological site has 35% of its original carbon-14. Estimate the age of the bone. After how many more years will the bone have only 25% of its carbon-14?

The number of bacteria in a colony of bacteria grows exponentially. At 1300 hours yesterday the number of bacteria was 1000 and at 1500 hours it was 4000. How many bacteria were there at 1800 hours yesterday? How many bacteria will there be at 1000 hours today?

The Global Report estimated the population of the world in 1975 as 4.1 billion people and that it was growing at the rate of 2% per year. Set up an equation to predict the world population t years from 1975. Use this model to predict the world’s population in 2020. Discuss any assumptions you have made.

Earthquakes produce oscillations in the ground. The strength, S, of the quake is measured on the Richter scale and is given by S = log A, where A is the measured amplitude of the oscillation as measured in millimetres on a calibrated seismograph. What amplitude of oscillation corresponds to a major earthquake with a Richter scale value of 7.8? What is the Richter scale value of an earthquake with an oscillation that has an amplitude of 2000 mm?


Include further examples to illustrate population growth and decay, radioactivity, cooling, drug absorption, spread of an epidemic, and compound interest.



By the end of Grade 12, students use approximation methods to calculate the area of an irregular two-dimensional flat surface and the volume of a prism with a constant, but irregular-shaped, cross-section. They draw and use plans and elevations, and interpret maps and scale drawings. They translate, reflect, rotate and enlarge two-dimensional geometric objects. They begin to use vectors to solve physical problems. They solve a range of problems involving compound measures, using appropriate units and dimensions. They explore aspects of geometry using ICT.

Students should:


Congruence and similarity: properties of angles, straight lines, triangles and circles

6.1 Use ICT to investigate a range of geometrical situations, including:

• the generation of geometric patterns, including Islamic patterns;

• similarity and congruence;

• constructions;

• plans and elevations.

Each side of the regular hexagon ABCDEF is 10 cm long. Find the length of the diagonal AC.

TIMSS Grade 12

7 Work with transformations and projections

7.1 Transform rectilinear figures using combinations of translations, rotations about a centre of rotation, enlargements about a centre of enlargement, and reflections about a line; understand the meanings of positive, negative and fractional scale factors in enlargements.

An equilateral triangle ABC has side length 10 cm. It rotates around the inside of a square of side length 20 cm.

a. Triangle ABC rotates about C to the position shown as CA1B1. What is the angle of

rotation?

b. Calculate the distance along the path travelled by point A in turning from A to A1.

c. Calculate the distance along the path travelled by point A in turning from A1 to A2.



Use of ICT

Geometry is enhanced with the use of a dynamic geometry system, or DGS, which provides an interactive means for investigating and hypothesising results in geometrical situations.

Transformations



d. The triangle continues rotating around the inside of the square in the same way until it is back at the original position. Which of the original points A, B or C will point A land on when it has completed its rotations around the inside of the square?

A triangle has vertices at the points (4, 5), (6, 1) and (8, 11). The triangle is enlarged by a factor of 2 about a centre of enlargement at the point (3, –3). Draw the enlarged triangle in its correct position on a coordinate grid.

The line segment OA is 3.0 cm long. The line segment OB is √ 7 cm long. OB can rotate in a horizontal plane about the point O.

a. Find the maximum possible distance B can be from A. Explain whether your answer

is a rational number or an irrational number.

b. Find the minimum possible distance B can be from A. Explain whether your answer is a rational number or an irrational number.

c. Sketch a different position for the line segment OB so that the distance from A to B, AB, is a rational number. Confirm by calculation that your answer is a rational number.

d. OB is reduced in length to 2.6 cm. OA is still 3.0 cm long. Calculate the distance AB when angle AOB is 120°.

e. The lengths of 2.6 cm and 3.0 cm are accurate to one decimal place. The 120° angle is accurate to the nearest degree. Calculate the greatest and least possible values of AB.

The diagram shows two rectangles, P and Q.

The rectangle Q in the diagram CANNOT be obtained from the rectangle P by means of:

A a reflection about an axis in the plane of the page

B a rotation in the plane of the page

C a translation

D a translation followed by a reflection

Circle the correct answer.

TIMSS Grade 12

7.2 Interpret maps and scale drawings.

7.3 Visualise the effect of transformations on a plane figure; know that the image of a planar figure under rotation or reflection is congruent to the original figure before rotation or reflection and that every circle is similar to any other circle. (See also GM 7.1.)


The diagram shows parts of two circles, sector A and sector B.

a. Which sector has the bigger area?

b. The perimeter of a sector is made from two straight lines and an arc. Which sector has the bigger perimeter?

A semicircle, of radius 4 cm, has the same area as a complete circle of radius r cm. What is the radius of the complete circle?

7.4 Draw the plan and elevation of two-dimensional projections of three-dimensional rectilinear objects. (See also GM 6.1.)

8 Use vectors

8.1 Consider coordinate systems as grids for moving around space in two or three dimensions; understand position vector, unit vector and components of a vector.

A particle is at the point (6, 2). What is its position vector in terms of the unit vectors i and j in the x- and y-directions respectively? Calculate the length (magnitude) of this vector.

8.2 Interpret a translation as a vector displacement; know that a vector displacement from A to B depends only on the starting point A and the finish point B and not on intermediate steps from A to C to D to … to B, and that the vector sum of all these separate displacements from A to B is equivalent to the resultant displacement from A to B directly.

8.3 Add and subtract two vectors in up to three dimensions and draw corresponding vector diagrams.

Four vectors a, b, c and d are given by a = 2i – 3j, b = 5j + k, c = 4i – 7k and d = 3i + j. Find a + b, b – c, a – b – c. Draw vector diagrams to represent a + d and a – d. What are the components of these two vectors in the i and j directions?

8.4 Multiply a vector by a scalar and know that this amounts to stretching the vector; calculate the magnitude and direction of a vector; use vectors to calculate displacement and velocity in a range of contexts.

A particle moves with constant velocity from A to B. Its position vector at A is a = i + j and its position vector at B is b = 5i – 7j. Calculate the vector displacement from A to B. If distance is measured in metres, show that the distance from A to B is 4√5 metres. The particle takes 2 seconds to move from A to B. What is its velocity?

Vectors

In three dimensions, vectors provide the natural language to place and displace figures in space. They also link with Cartesian coordinate systems.

Unit vectors

Unit vectors in three mutually perpendicular directions are usually written as i, j and k.


8.5 Use the scalar product of two vectors to calculate the angle between the vectors and the scalar product of a vector with itself to find the magnitude of the vector.

Find the magnitude of each of the vectors a and d in the example in GM 8.3 above. Calculate the angles between these vectors.

8.6 Solve physical problems using vectors.

i and j are unit vectors in the east and north direction respectively. A ship has position r = 3i + 4j at 1200 hours. It then moves with constant velocity v = 4i – 5j. The velocity is measured in kilometres per hour. What is the speed of the ship (the magnitude of its velocity)? What is the position of the ship at 1500 hours?

A particle of mass m kilograms is moving with constant acceleration a, measured in metres per second per second. The total external force F acting on the particle is measured in newtons, and is the vector sum of the individual forces acting on the particle. The relationship between F and a is given by Newton’s second law of motion and is F = ma.

A particle of mass 2 kg is acted upon by two forces F1 = i – j and F2 = 3j. Find the acceleration of the particle and give its magnitude.



A light shade is in the form of a frustum of a right cone. The radius at the top of the shade is 10 cm and the radius at the bottom is 25 cm. Find the surface area of the material used for the light shade.

9.2 Use approximation methods to calculate the area of an irregular two-dimensional flat surface and the volume of a prism with a constant, but irregular-shaped, cross-section.

9.3 Solve problems involving compound measures, using appropriate SI units and dimensions.

In one and a half hours a car uses 8 litres of petrol when travelling at a speed of 70 kilometres per hour. What is the petrol consumption in litres per kilometre?

Compound measures

Include measures of power, average speed and acceleration, measures of rate (such as rate of growth of income), and population density. Link where relevant to work in science, technology and the social sciences.



By the end of Grade 12, students arrive at conclusions from the formulation of a problem to the collection and analysis of data in a range of situations. They use secondary data from published sources, including the Internet. They use ICT to calculate statistical quantities and to produce a range of graphs, charts and tables to present and justify their findings. They calculate measures of spread, including the variance and standard deviation. They construct histograms and plot cumulative frequency distributions, using grouped continuous data if necessary. They understand that a random variable has a range of values that cannot be predicted with certainty and investigate common examples. They measure the empirical probability (relative frequency) of obtaining a particular value of a random variable. They use a simple mathematical model to calculate the theoretical probability of obtaining a particular outcome for a random variable associated with a set of events. They calculate probabilities of single and combined events, and understand risk as the probability of the occurrence of an adverse event. They use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another. They use simple simulations and consider trends over time using a moving average.

Students should:


Sampling

10.1 Know that:



• in a biased sample there are systematic differences between the sample and the population from which it is drawn;

and locate obvious sources of bias within a sample.


10.2 Plan surveys and design questionnaires to collect meaningful primary data from samples (including data collected in other subjects, such as science, geography or history) in order to make estimates of, or test hypotheses about, quantities or attributes characteristic of the population as a whole.

10.3 Formulate problems using secondary data from published sources, including the Internet.

10.4 Calculate measures of central tendency such as the arithmetic mean and the median.

10.5 Calculate measures of spread, including the variance and standard deviation.




Find the mean and median salaries of the group of workers in Qatar whose weekly salaries in riyals are given in the table below.

Salary (QR) 250 300 350 400 450 500 550 600

Frequency 5 11 20 31 18 12 7 3

Which average is the most representative for these workers? Justify your answer. Use statistical functions on a calculator to calculate the standard deviation for the salaries in this group. What information does this convey?

10.6 Construct (relative frequency) histograms and know that the area of each block of the histogram represents the frequency of occurrence of the respective class interval associated with the block; plot cumulative frequency distributions, using grouped continuous data if necessary.

The table below shows the number of cars leaving a car park during the periods given.

Number of minutes after 1700 h

0 ≤ n < 5 5 ≤ n < 10 10 ≤ n < 20 20 ≤ n < 50 50 ≤ n < 60

Number of cars leaving

74 115 248 1174 189

Complete the histogram to show the information in the table. Write the frequency density above each rectangle of the histogram.

The value 14.8 on the histogram is the frequency density for the period 0 ≤ n < 5 minutes. Explain what is meant by frequency density with regard to cars leaving the car park.

10.7 Make inferences and arrive at conclusions from the formulation of a problem to the collection and analysis of data in a range of situations; use a range of statistics and graphs, charts and tables to present and justify findings.

11 Understand random variables and calculate probability

11.1 Know that all probability values lie between 0 and 1, and that the extreme values correspond respectively to impossibility and certainty of occurrence.

11.2 Understand that a random variable has a range of values that cannot be predicted with certainty, and investigate common examples of random variables; measure the empirical probability (relative frequency) of obtaining a particular value of a random variable.

11.3 Use a simple mathematical model to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events.

Two six-sided dice, each numbered from 1 to 6, are thrown and the total score on the two dice is found. Assuming that either dice is equally likely to show any of its six faces, what is the probability that the total score is greater than 4 and less than 10?

Histograms and cumulative frequency distributions

Include the terms frequency, frequency distribution, frequency density, relative frequency and relative frequency distribution.

Include also the terms range, percentile, interquartile range, semi-interquartile range, and mode, modal class, modal frequency.

Distributions

Include as models the rectangular distribution and the binomial distribution, and simple applications of these.


Assume that it is equally likely for a woman to give birth to a girl as it is to give birth to a boy. What is the probability that a woman with six children has four girls and two boys? What is the probability that if another woman has four children they are all boys? A woman with three daughters is going to have a fourth child. What is the probability that the fourth child will be boy?

A company makes computer disks. It tested a random sample of disks from a large batch. The company calculated the probability of any disk being defective as 0.025. Naima buys two disks.

a. Calculate the probability that both disks are defective.

b. Calculate the probability that only one of the disks is defective.

c. The company found three defective disks in the sample they tested. How many disks were likely to have been tested?

100 students were asked whether they studied French or German. 27 students studied both French and German.

a. What is the probability that a student chosen at random will study only one of the

languages?

b. What is the probability that a student who is studying German is also studying French?

A piece of string is 12 centimetres long. It is randomly cut into two pieces. What is the probability that one piece has length greater than 9 cm?

11.4 Understand risk as the probability of occurrence of an adverse event; investigate some instances of risk in everyday situations, including in insurance and in medical and genetic matters.

The probability of dying of cancer is 1/3. What is the probability that if three people are chosen at random two of them will die of cancer? What is the probability that none of them will die of cancer?

Probability of combined events

11.5 Understand when two events are mutually exclusive, and when a set of events is exhaustive; know that the sum of probabilities for all outcomes of a set of mutually exclusive and exhaustive events is 1, and use this in probability calculations.

11.6 Know that when two events A and B are mutually exclusive the probability of A or B, denoted by P(A ∪ B), is P(A) + P(B), where P(A) is the probability of event A alone and P(B) is the probability of event B alone.

Mona has a chance of 1 in 4 of passing on a particular genetic condition to one of her children. Mona has three children. Calculate the probability that two of the children will inherit the condition. What is the probability that none of her children will inherit the condition? What is the probability that at least one her children will inherit the condition?


A computer game has nine circles arranged in a square. The computer chooses circles at random and shades them black.

a. At the start of the game, two circles are to be shaded black. Show that the probability that both circles J and K will be shaded black is 1/36.

b. Halfway through the game, three circles are to be shaded black. Here is one

example of the three circles shaded black in a straight line. Show that the probability that the three circles shaded black will be in a straight line is 8/84.

c. At the end of the game, four circles are to be shaded black. Here is one example of

the four circles shaded black forming a square. What is the probability that the four circles shaded black form a square?

11.7 Know that two events A and B are independent if the probability of A and B occurring together, denoted by P(A ∩ B), is the product P(A) × P(B).

In a class of 35 students, the probability that a student picked at random is taller than 1.8 metres is 0.2 and the probability that the student wears spectacles is 0.3. What is the probability when three students are chosen at random that two are over 1.8 metres in height and that one of them wears spectacles?

Show that two events A and B are mutually exclusive when P(A ∩ B) = 0.

11.8 Use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another.

On a tropical island the probability of it raining on the first day of the rainy season is 2/3. If it does not rain on the first day, the probability of it raining on the second day is 7/10. If it rains on the first day, the probability of it raining more than 10 mm on the first day is 1/5. If it rains on the second day but not on the first day, the probability of it raining more than 10 mm is 1/4.

You may find it helpful to fill in this tree diagram before answering the questions below.


a. What is the probability that it rains more than 10 mm on the second day, and does not rain on the first?

b. What is the probability that it has rained by the end of the second day of the rainy season?

c. Why is it not possible to work out the probability of rain on both days from the information given?

20 per cent of the population of a country has a particular disease. A test can be given to members of the population to help determine whether or not they have the disease. The probability that the test gives a positive identification for those that have the disease is 0.7. But there is a 0.1 chance that a patient who does not have the disease still registers positive on the test. Find the probability that an individual selected at random tests positive, but does not have the disease.

Another person is chosen at random. Calculate the probability that the test result for this person is positive.

12 Calculate moving averages

12.1 Consider trends over time and calculate a moving average.

Find out about the cost, in Qatar, of a barrel of crude oil over the period January 2000 to March 2004. Analyse the data over periods of three months and compare the moving average price per barrel. Discuss your findings.

13 Simulation

13.1 Use coins, dice or random numbers to generate models of random data.

Do an investigation using random numbers to simulate waiting times at a doctor’s surgery.

14 Use of ICT

14.1 Use a calculator with statistical functions to aid the analysis of large data sets, and ICT packages to present statistical tables and graphs.

Random numbers

These can be generated on scientific or a graphics calculator using the RND and RAN function keys.

ICT opportunity

A range of ICT applications can support data handling.

233 | Qatar mathematics standards | Grade 10 advanced © Supreme Education Council 2004




Students solve routine and non-routine problems in a range of mathematical and other contexts. They use mathematics to model and predict outcomes of real-world applications. They break down complex problems into smaller tasks, and set up and perform appropriate calculations and manipulations. They identify and use connections between mathematical topics. They develop and explain short chains of logical reasoning, using correct mathematical notation and terms. They generate mathematical proofs and generalise when possible. They approach problems systematically, knowing when to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They recognise when to use ICT and do so efficiently.

Number and algebra

Students identify and use number sets and set notation. They calculate with any real numbers, including powers, roots, and numbers expressed in standard form. They use proportional reasoning to solve problems involving scale, ratios and percentages, including compound interest. They are aware of the role of symbols in algebra. They generate and manipulate algebraic expressions, including algebraic fractions, equations and formulae. They multiply any two polynomial expressions and factorise quadratic expressions. They sum arithmetic and geometric sequences and investigate the growth of patterns, generalising relationships to model the behaviour of the pattern. They convert recurring decimals to exact fractions. They use algebraic methods to solve linear and quadratic equations, and a pair of simultaneous linear equations. They plot and interpret straight line and quadratic graphs, and graph regions of linear inequality. They use function notation and find the tangent at a point on the graph of a function. Through their study of linear and quadratic functions and their graphs, and the solution of the related equations, students begin to appreciate numerical and algebraic applications in the real world. They use realistic data and ICT to analyse problems.


Students use their knowledge of geometry, Pythagoras’ theorem and the trigonometry of right-angled triangles to solve practical and theoretical problems relating to shape and space. They understand congruence and similarity. They prove that the perpendicular from the centre of a circle to a chord bisects the chord and that the two tangents from an external point to a circle are of equal length. They carry out straight edge and compass constructions and determine the locus of an object moving according to a rule. They use radians as a measure of angle, and dimensionally correct units for length, area and volume. They solve problems involving rates and compound measures. They use formulae to calculate the length of an arc and the area of a sector of a circle, the area of any triangle, trapezium, parallelogram or quadrilateral with perpendicular diagonals, and the surface area and volume of a right prism, cylinder, cone, sphere and pyramid. They

Grade 10 Advanced


use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth’s surface. They use ICT to explore geometrical relationships.


Students distinguish between qualitative or categorical data and quantitative data, and between discrete and continuous data. They understand the concept of a random variable. They can locate sources of bias. They plan questionnaires and surveys to collect meaningful primary data from representative samples. They collect data from secondary sources, including the Internet, and formulate and solve problems related to the data. They group data and plot histograms and other frequency and relative frequency distributions. They calculate measures of central tendency and measures of spread, including variance and standard deviation. They draw stem-and-leaf diagrams and box-and-whisker plots. They plot and interpret simple scatter diagrams between two random variables, and draw a line of best fit where there appears to be correlation. They use relevant statistical functions on a calculator and ICT applications to present statistical tables and graphs.


The advanced mathematics standards have four strands: reasoning and problem solving; number, algebra and calculus; geometry and measures; and probability and statistics. Calculus is introduced in Grade 12.



Advanced Number, algebra

and calculus Geometry, measures

and trigonometry Probability and

statistics

Grade 10 55% 30% 15%

Grade 11 55% 30% 15%


40% – 60%


75% 25% –

The standards are numbered for easy reference. Those in shaded rectangles, e.g. 1.2, are the performance standards for all advanced students. The national tests for advanced mathematics will be based on these standards.

Many of the Grade 10 advanced standards have been introduced in earlier grades. Teachers should review these standards briefly and devote a greater proportion of time to the work that is new to students.




By the end of Grade 10, students solve routine and non-routine problems in a range of mathematical and other contexts. They use mathematics to model and predict outcomes of real-world applications. They break down complex problems into smaller tasks, and set up and perform appropriate calculations and manipulations. They identify and use connections between mathematical topics. They develop and explain short chains of logical reasoning, using correct mathematical notation and terms. They generate mathematical proofs and generalise when possible. They approach problems systematically, knowing when to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They recognise when to use ICT and do so efficiently.

Students should:



1.2 Use mathematics to model and predict the outcomes of real-world applications.




1.6 Develop short chains of logical reasoning, using correct mathematical notation and terms.



1.9 Generalise whenever possible.



1.12 Synthesise, present, interpret and criticise mathematical information presented in various mathematical forms.

Grade 10 Advanced

Key standards


Cross-references






Proofs


Generalisation

Students should appreciate that generalisation is important in mathematics.




Number and algebra

By the end of Grade 10, students identify and use number sets and set notation. They calculate with any real numbers, including powers, roots, and numbers expressed in standard form. They use proportional reasoning to solve problems involving scale, ratios and percentages, including compound interest. They are aware of the role of symbols in algebra. They generate and manipulate algebraic expressions, including algebraic fractions, equations and formulae. They multiply any two polynomial expressions and factorise quadratic expressions. They sum arithmetic and geometric sequences and investigate the growth of patterns, generalising relationships to model the behaviour of the pattern. They convert recurring decimals to exact fractions. They use algebraic methods to solve linear and quadratic equations, and a pair of simultaneous linear equations. They plot and interpret straight line and quadratic graphs, and graph regions of linear inequality. They use function notation and find the tangent at a point on the graph of a function. Through their study of linear and quadratic functions and their graphs, and the solution of the related equations, students begin to appreciate numerical and algebraic applications in the real world. They use realistic data and ICT to analyse problems.

Students should:


2.1 Identify the number sets:

the set of all real numbers;

the set of all integers; + the set of all positive integers {1, 2, 3, 4, …}; – the set of all negative integers;

the set of all rational numbers, i.e. all the different numbers that can be expressed in the form a/b, where a and b are integers with b ≠ 0;

the set of all non-negative integers, called the set of natural numbers {0, 1, 2, 3, 4, …}.

Is a subset of ?

To what set does √2 belong? How do you know?

2.2 Know when a real number is irrational, i.e. when it is not a member of .

2.3 Use and understand the following symbols associated with set theory: E for ‘the universal set’; ∅ for ‘the null set’; ∈ for ‘is a member of’; ∉ for ‘is not a member of’; ∀ for ‘for all’; use brace notation to denote a set.

A = {x: x ∈ and 1 ≤ x < 10} denotes ‘the set A, whose members are all real numbers greater than or equal to 1 and less than 10’.

Algebra


Natural numbers

In some texts, is taken to be the same as +.


List the elements of each of the following sets: A = {x: x is a colour of the Qatar flag}; B = {x: x is a state in the GCC}; C = {x: x is a member of the Arab League}.

Is the statement that √2/3 ∈ true or false?

What is the solution set of the equation x(x + 3) = x(x – 3) + 6x + 1? Explain your answer.

2.4 Understand the meaning of the union of two sets A and B and that this is denoted by A ∪ B, and the meaning of the intersection of two sets A and B, denoted by A ∩ B, and represent these sets in a Venn diagram; represent the complement of set A as A′ and know that A ∪ A′ = E.

Use a Venn diagram to decide whether A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C.

Describe the set ∪ ′.

In a school of 650 students, everyone studies Arabic, English, mathematics and science. They all have to choose to study at least one of art, French, or history. 195 students choose only French. Three times as many students study French and history as study all three subjects, and five times as many study French and art as study all three subjects. 30 students study French and history, 65 do art and history, and 200 do art but not French or history. How many students study history but not art or French?

2.5 Know from definitions that every even number can be written in the form 2m, where m is an integer, and that every odd number can be written in the form 2n + 1, where n is an integer; understand and use the words factor, multiple, divisor, prime number, prime factor, prime factor decomposition, least common multiple, highest common factor and lowest common denominator.

Prove that the product of two odd numbers is an odd number.

What is the largest prime number you can think of? How do you know it is prime?

Is there a largest prime number? Justify your answer.

What is the highest common factor of a3b2c and c3b2a?


3.1 Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents; use the xy key on a calculator.

Without using a calculator, evaluate (53)4 ÷ 510.

Use a calculator to evaluate 79.

Simplify 81/3 × 2–1.

3.2 Know that a root that is irrational is an example of a surd, as are expressions containing the addition or subtraction of an irrational root; perform exact calculations with surds.

Calculate (√2 – 1)(√3 – √2).

3.3 Use standard form in appropriate situations: for exact calculations, to estimate results of calculations and to make comparisons.

To four significant figures, the speed of light is 299 800 000 metres per second. Write this in standard form.

Laws of exponents


Standard form

Use examples from science or geography lessons or real-world applications.


The Earth is approximately a sphere of radius 6378 kilometres. Without using a calculator estimate the circumference at the equator.

The mass of the Earth is 5.98 × 1024 kg. A typical man has a mass of about 70 kg . Approximately how many men would have a total mass equal to that of the Earth?

Light travels at about 300 000 kilometres per second. Use standard form to find the distance away from the Earth of a light-emitting body whose light signal is received at Earth one year after it is emitted.

The Earth completes its orbit around the Sun in 365 days. The Earth is 148.8 million kilometres from the Sun. Assume that the Earth’s orbit is circular and that it travels around the Sun with constant speed. Calculate the Earth’s speed in kilometres per hour.

Sir Isaac Newton (1642–1727) was a mathematician, physicist and astronomer.

a. In his work on the gravitational force between two bodies, Newton found that he needed to multiply their masses together.

Work out the value of the mass of the Earth multiplied by the mass of the Moon. Give your answer in standard form.

Mass of Earth = 5.98 × 1024 kg Mass of Moon = 7.35 × 1022 kg

b. Newton also found that he needed to work out the square of the distance between the two bodies.

Work out the square of the distance between the Earth and the Moon. Give your answer in standard form.

Distance between Earth and Moon = 3.89 × 105 km

c. Newton’s formula to calculate the gravitational force (F) between two bodies is 1 22

Gm mF R=

where G is the gravitational constant, m1 and m2 are the masses of the two bodies, and R is the distance between them.

Work out the gravitational force (F) between the Sun and the Earth using this formula with the information given below. Give your answer in standard form.

m1m2 = 1.19 × 1055 kg2 R2 = 2.25 × 1016 km2 G = 6.67 × 10–20

3.4 Perform calculations with any real numbers, including mental calculations in appropriate cases. Calculate mentally the value of 9999 × 0.033.

Use standard form to estimate the value of 4350 × 237.8 × π2.

3.5 Add, subtract, multiply and divide any two fractions and understand how to use a unit fraction as a multiplicative inverse.

3.6 Understand the multiplicative nature of proportional reasoning; form, simplify and compare ratios, and apply these in a range of problems, including mixtures, map scales and enlargements in one, two or three dimensions.

A recipe for six people includes a quarter of a kilogram of figs. How many kilograms of figs would be needed if the recipe were made in the same proportion for eight people?

A map is drawn to scale 1 : 190 000. Two places A and B are 3 cm apart on the map. How far apart are A and B?


3.7 Perform percentage calculations involving taking a percentage of a percentage, inverse percentage, and compound interest.

The diagram shows water flowing through some pipes. The water starts at A. At each junction the percentage of the inflowing water flowing out through the pipes is indicated. What percentage of the original water flows out at B? What percentage flows out at C?

After Haya’s salary is increased by 15% and Abdullah’s salary is decreased by 27%, Haya and Abdullah both end up with an annual salary of QR 72 000. What were their original salaries? What percentage of Abdullah’s original salary was Haya’s original salary?

Due to inflation, the price of a television in a store is increased by 15%. In the sales at the end of the year, the price is then reduced by 15%. Does the television revert to its original pre-inflation price? Or is it more, or less? Explain your reasoning.

QR 100 00 has to be invested in deposit accounts. There is a choice of two accounts. One account pays an annual interest of 4.6%. The other account pays interest of 1.5% three times per year. What is the AER of the second account? Which is the better account to invest in and how much more interest will there be after one year in this account than in the other account?


4.1 Solve any linear equation with one unknown.

4.2 Generate sequences from term-to-term and position-to-term definitions; investigate the growth of simple patterns, generalising algebraic relationships to model the behaviour of the patterns.

Each term of a sequence is 3 times the preceding term. The first term is 5. Set up a term- to-term definition for this sequence. Give an expression for the nth term in terms of n. Write down, but do not simplify, the 50th term.

The table below shows the first six triangular numbers.

Position 1 2 3 4 5 6

Term 1 3 6 10 15 21

Investigate diagrammatic ways of representing these numbers.

Set up a relationship to describe the nth term in terms of its position value n. What is the 100th triangular number? What is the 1000th triangular number?

4.3 Identify and sum arithmetic sequences, including the first n consecutive positive integers, and give a ‘geometric proof’ for the formulae for these sums.

Percentages

Include real-world financial examples such as taxes, mortgage rates, interest rates, including the annual equivalent rate (AER).

Generalising

Link algebraic reasoning to geometric concepts where possible.

ICT opportunity

Include the use of spreadsheets or a graphics calculator to explore arithmetic sequences.


The diagram is a useful representation of an arithmetic series.

How could you use this diagram to find the sum of the arithmetic series?

Find the sum of the first n consecutive positive integers, and hence the sum of any set of n consecutive positive integers.

Find the sum of all numbers between 1 and 100 that are exactly divisible by 3.

4.4 Identify and sum geometric sequences and know the conditions under which an infinite geometric series can be summed.

Grains of rice are placed on each square of a chess board. The board has 64 squares. One grain is placed on the first square, two on the second, four on the third, eight on the fourth, and so on. Calculate the total number of grains of rice on the chessboard. Given that 1 kilogram of rice contains approximately 16 000 grains of rice, estimate the weight of all the rice on the chess board.

The sum of the infinite geometric series 1 – 1/2 + 1/4 – 1/8 + … is A. 5/8 B. 2/3 C. 3/5 D. 3/2

Investigate compound interest problems as examples of geometric series.

4.5 Convert any recurring decimal to an exact fraction.

Explain why 4330.12 = .

4.6 Identify number patterns contained in Pascal’s triangle.

Describe carefully in words how any entry in Pascal’s triangle is related to entries in the row above. Set up an algebraic relationship to describe this. Where are the triangular numbers located in Pascal’s triangle? What other patterns can you spot?

Look at the numbers in an early row of Pascal’s triangle. Sum the squares of these numbers. In what row is the answer located? Identify where to find the sum of the squares of the numbers in any row of Pascal’s triangle. Explain your reasoning.

4.7 Develop confidence and accuracy in working with symbols, understanding that the transformation of all such algebraic objects generalises the well-defined rules of arithmetic. Recognise that letters are used to represent:




• new equations, expressions or functions in terms of known, or given, expressions or functions.

Is (x + 4)2 = x(x + 12) – 4(x – 4) an equation or an identity? Explain your reasoning.

Is (x – a)(x2 + ax + a2) = x3 – a3 an equation or an identity? Discuss what happens to this mathematical statement when a is replaced throughout by –a.

Give examples of what is meant by an associative law and a distributive law.

Calculate (√5 + √3)(√5 – √3).

Geometric sequences

Include consideration of compounding interest more and more frequently.


4.8 Use brackets and correct order of precedence of operations when performing numerical or algebraic calculations.

4.9 Combine numeric or algebraic fractions, and multiply combinations of monomial, binomial and trinomial expressions; multiply any two polynomials, collecting up and simplifying similar terms.

Use Pascal’s triangle to identify the coefficients of the powers of x in the expansion of (1 + x)n for different values of the positive integer n. Check the results for n = 3 by expanding out (1 + x)3.

Simplify (2x – 3)(x2 + x – 10).

4.10 Factorise quadratic expressions; conceptualise geometric representations for these factorisations and other similar quadratic expressions. Without using a calculator, find the exact value of 7.922 – 2.082.

Explain why (a + b)2 ≠ a2 + b2.

Draw a diagram to represent the identity (a + b)2 = a2 + 2ab + b2.

Draw a diagram to represent the identity (a − b)2 = a2 − 2ab + b2.

Construct some quadratic expressions from two linear factors in a and b and draw geometric representations for them.

4.11 Solve quadratic equations exactly, by factorisation, by completing the square and by using the quadratic formula.

4.12 Simplify numeric and algebraic fractions by cancelling common factors; rationalise a denominator of a fraction when the denominator contains simple combinations of surds.

Rationalise the expression 21 3+

.

Simplify the expression 3 2 2 3

2 2a b a b

a b− .

4.13 Generate formulae from a physical context, and rearrange formulae connecting two or more variables.

Make b the subject of the formula 2a bA += .

Make l the subject of the formula 2 lT gπ= .

Make x the subject of the formula xw z v= − + .

Find R in terms of R1 and R2 when 1 2

1 1 1R R R= + .

The three different edges of a solid cuboid have lengths x, 2x and y, as shown. All the lengths are measured in centimetres. The total surface area of the cuboid is 800 cm2. Find a formula for y in terms of x. What is the total length of all the edges of the cuboid? Give the answer in terms of x.

Quadratic expressions

Include the forms:

x2 + (α + β)x + αβ

x2 − (α + β)x + αβ

x2 ± (α − β)x − αβ

a2x2 – b2y2


Melons cost QR 1.5 each and apples cost QR 3.75 per kilogram. A woman buys apples and melons at the supermarket. Set up a formula to describe the total cost of her purchase. Investigate how many melons and how many kilograms of apples she could buy for QR 30.

The mathematician Johannes Kepler set out three laws of planetary motion in his famous book ‘The Harmony of the World’, published in 1619. Kepler’s third law of planetary motion states that the square of the period of revolution of a planet about the Sun is proportional to the cube of the mean distance of the planet from the Sun. Write this statement as a mathematical equation.

The value of a new car depreciates by 20 per cent at the end of the first year and then loses value at the rate of 10 per cent for every subsequent year. Set up a formula to describe the value V of the car t years after purchase. After how many years will the car be worth one quarter of its purchase price?

4.14 Substitute an expression for a given variable into a different formula containing this variable.

The volume of a solid cylinder of length h and radius r is V. Find a formula for the curved surface area, A, of the cylinder in terms of r and h. Use this formula to find a formula expressing V in terms of A and r.


5.1 Investigate a range of mathematical and physical situations to develop the concepts of function, domain and range, recognising one-to-one and many-to-one mappings as functions and a one-to-many mapping as a relation but not a function.

If p is a person, state with reasons whether each of the following maps are functions: a. p maps to the place of birth of p; b. p maps to brother of p; c. p maps to nationality of p; d. p maps to teacher of p; e. p maps to mother of p.

A firm rents out cars by the day or by the week. The daily charge rate is QR 170 with 150 km free and then QR 2 for every additional kilometre. The weekly charge is QR 1400 with no additional charges. A man needs to hire a car for five days. How many kilometres will he have to drive to make it worthwhile to hire the car for a week?

Look up any country in an atlas and pick six towns from this country. Which of these maps represents a function and which does not: towns → country; country → towns? Justify your answer. What are the domain and range for the mapping that represents a function?

5.2 Understand and use the concept of related variables and, in special cases, set up appropriate functional relationships between them.

In an electric circuit, V = IR, where V is the voltage in volts, I is the current in amps and R is the resistance in ohms. The electrical power in watts is P = VI. Find a formula connecting the variables P, V and R.

5.3 Plot a graph to show the relationship between two variables given quantitative information between the variables in tabular or algebraic form.

5.4 Use a graphics calculator or graph plotter and pencil and paper methods to plot and interpret a range of functional relationships, some continuous and others discontinuous, arising in familiar contexts.

Functions

Include use of the notation y = f(x) to denote that y is a function of x.

Functional relationships


ICT opportunity

Graphics calculators can be used to explore a range of functional relationships.


Draw a graph showing the functional relationship between postage rate in Qatar and the weight of package to be posted.

Plot the graph of y = 1/x2 for the domain set {x: x ∈ and 1 ≤ x ≤ 4}. Discuss whether the domain could be extended.

Plot the curve y = √ x on a suitably defined domain. Discuss why the domain cannot be the set . Compare this curve with the curve of y = x2, drawn on the same axes.

Ahmed does a parachute jump. He jumps out of the plane and falls faster and faster towards the ground. After a few seconds his parachute opens. He slows down and then falls to the ground at a steady speed.

Which of these graphs shows Ahmed’s parachute jump? Explain why each of the other graphs is wrong.

Invent examples of functions with different definitions on different subdomains, for example, electricity charges as a function of the number of units of electricity used.

The Int x function, written as [x], maps x to the greatest integer less than or equal to x. Find [5.9], [6] and [–4.7].

Plot on the same axes the curves y = 2x and y = 2–x for –3 ≤ x ≤ 3. Describe the features of the two curves. Discuss situations that could be modelled by these equations.

A rectangular enclosure has a wall on one side and the other three sides are made of metal fencing. The side parallel to the wall has length d, measured in metres. The enclosure has an area of 600 m2. Show that the total length, L metres, of fencing is given by L = d + 1200/d. Plot this function using a graphics calculator. Find from the graph the value of d that makes L as small as possible.


Discuss whether or not the graph of i. a circle and ii. a semicircle represents a function. Look at any special cases that may arise.

Direct proportion

5.6 Translate the statement y is proportional to x into the symbolism y ∝ x and into the equation y = kx, and know that the graph of this equation is a straight line through the origin and that the constant of proportionality, k, is the gradient of this line.

5.7 Know that if two coordinate variables are connected by a straight line graph that passes through the origin of coordinates, then each coordinate variable is proportional to the other; use relevant information to find k.

5.8 Identify common examples of two linear quantities varying in direct proportion to each other.

ICT opportunity


Direct proportion

Use examples from science and the real world, e.g.

• conversion rates;

• Ohm’s law, or V = IR;

• the pressure of gas in a constant volume is directly proportional to its temperature.



5.9 Know that a straight line in the explicit form y = mx + c represents a function, but that a straight line in the implicit form ax + by + d = 0 may, or may not, be a function; know that any straight line in the xy-plane can be represented in this implicit form, but that only certain lines in the plane can be represented by the explicit form; work with both of these forms.

5.10 Plot the graphs of the equations in 5.9; know the meanings of gradient of the line and intercept on the x- or y-axis and relate these to the coefficients a, b and d, or to the coefficients m and c.

What is the gradient of the line 3x + 2y – 5 = 0? Find an equation of a line that is perpendicular to this line. Draw the two lines on a graph.

A triangle has its vertices at the points (1, 3), (2, 5) and (3, 4.5). Find the equations of the lines containing each side. Is the triangle a right-angled triangle? Explain how you know.

What angle does the line y = √ 3x +1 make with the positive x-axis?

5.11 Construct the Cartesian equation of a straight line from its graph alone, or from the knowledge of the coordinates of two points on the line, or from the coordinates of one point on the line and the gradient of the line.

What is the equation of the straight line through the points (5, –2) and (–4, 3)? What is the gradient of this line? Where does it cross the y-axis? Where does it cross the x-axis?

A triangle has vertices at the points (1, 1), (5, –4) and (–3, 2). Find the equation of each of its sides.

5.12 Know the condition for two straight lines to be parallel or perpendicular, including the special cases of one of the lines being parallel to either axis.

Give equations of lines parallel and perpendicular to the line y = 5x – 3.

5.13 Read off the coordinates of the point of intersection, given the graphs of two intersecting straight lines; find exactly, by algebraic means, the coordinates of the point of intersection of two lines, given their equations.

Find the intersection point of the line y = 4x + 2 with the line y = 9 – 3x.

Discuss whether two lines have no intersection, a unique intersection point, or infinitely many intersection points.

In two dimensions discuss whether non-parallel lines must intersect. What happens in three dimensions?

5.14 Interpret the solution set of the simultaneous equations E1 and E2, where E1 and E2 are the equations of two straight lines.

Look at this octagon. The line through D and B has the equation 3y = x + 25. The line through G and H has the equation x = y + 15.

Gradients

Include the terms slope and rate of change.


Solve the simultaneous equations y = 2x – 6 y + 3x = 5 to find the point of intersection of these two lines.

Here are the equations of some straight lines: y = 2x – 7; y = 7 – 2x; y = 2x + 9; y = 14 – 4x; y = –10; y = –10 + 2x; x = 1; y = –0.5x + 8.

List all the pairs of lines that are: a. parallel to each other; b. perpendicular to each other; c. different representations of the same line. From the lists, find pairs of lines that intersect in a unique point and find the intersection point in each case.

5.15 Draw the tangent line at a point on the graph of a function, calculate the slope of this line and interpret the behaviour of the function at that point, knowing whether the function is increasing or decreasing at the point, or stationary.

Direct proportion (continued)

5.16 Translate the statement y is proportional to x2 into the symbolism y ∝ x2 and into the equation y = kx2 and know that the graph of this equation is a parabola through the origin.

A body falling from rest under the force of gravity falls a distance s metres in time t seconds where s = 4.9t2. Find the distance fallen after 5 seconds. How long does it take the body to fall 30 metres?

Discuss how to plot a linear graph s = 4.9z, by defining the variable z = t2.

5.17 Identify some other common examples of proportional variation.

Linear and quadratic inequalities

5.18 Graph regions of linear inequality and solve simple problems (e.g. elementary linear programming) represented by such regions; understand simple quadratic inequalities.

A firm delivers new cars to Doha. It has a contract to deliver at least 65 cars each day. The firm owns 7 carriers that can each carry 8 cars and 5 carriers that can each carry 10 cars. The firm employs 8 drivers and each carrier can only make one journey with a full load each day. What is the maximum numbers of cars that can be delivered each day? What is the minimum number of drivers needed to fulfil the contract?

The shaded region is bounded by the curve y = x2 and the line y = 2. What two inequalities together fully describe the shaded region?


Quadratic functions

5.19 Recognise a second-order polynomial in one variable, y = ax2 + bx + c, as a quadratic function; plot graphs of such functions (recognising that these are all parabolas) and identify the intercepts with the coordinate axes, the axis of symmetry and the coordinates of the maximum or minimum point; understand when quadratic functions are increasing, when they are decreasing and when they are stationary.

Create a display like this with your graphics calculator.

5.20 Model a range of situations with quadratic functions of the form y = ax2 + c.


By the end of Grade 10, students use their knowledge of geometry, Pythagoras’ theorem and the trigonometry of right-angled triangles to solve practical and theoretical problems relating to shape and space. They understand congruence and similarity. They prove that the perpendicular from the centre of a circle to a chord bisects the chord and that the two tangents from an external point to a circle are of equal length. They carry out straight edge and compass constructions and determine the locus of an object moving according to a rule. They use radians as a measure of angle, and dimensionally correct units for length, area and volume. They solve problems involving rates and compound measures. They use formulae to calculate the length of an arc and the area of a sector of a circle, the area of any triangle, trapezium, parallelogram or quadrilateral with perpendicular diagonals, and the surface area and volume of a right prism, cylinder, cone, sphere and pyramid. They use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth’s surface. They use ICT to explore geometrical relationships.

Students should:



6.1 Use knowledge of angles at a point, angles on a straight line, and alternate and corresponding angles between parallel lines and a transversal line to present formal arguments to establish the congruency of two triangles.

Prove that each of the angles in an equilateral triangle is 60°.

ICT opportunity

Include the use of a graphics calculator or graph plotter.



Use of ICT

Geometry is enhanced with use of a dynamic geometry system, or DGS, which provides an interactive focus to investigate and conjecture results which could then be proved as theorems.


6.2 Establish the congruency of two triangles to generate further knowledge and theorems about triangles, including proving that the base angles of an isosceles triangle are equal and that the line joining the mid-points of two sides of a triangle is parallel to the remaining side.

6.3 Understand similarity of two triangles and other rectilinear shapes, knowing that similarity preserves shape and angles, but not size; make inferences about the lengths of sides and about the areas of similar figures; prove that if two triangles are similar, then the ratio of the areas of the two triangles is the square of the ratio of any pair of corresponding sides of the two triangles chosen in the same order; in three dimensions, calculate the ratio of the volume of a scale model to the volume of the actual object.

A goldsmith has a block of gold in the shape of a cube. He wants to make another gold cube that has exactly twice the volume of the first cube. What scale factor must he use?

Two similar shaped gas-filled balloons are made of a special material. The area of material used in one balloon is 100 cm2. The material for the other balloon has an area of 225 cm2. Calculate the ratio of the volume of the larger balloon to the volume of the smaller balloon. Give this ratio in its simplest form.

The diagram shows two triangles ADE and BCE. Side AD is parallel to side BC. Explain why the two triangles are similar to each other.

Calculate the missing lengths for triangle BCE.

Calculate the length of CD in the diagram on the right.

A scale model of a dhow has a volume of 300 cm3. The length of the actual dhow is 100 times longer than the length of the model. What is the volume of the dhow? Give the answer in appropriate units.

6.4 Calculate the interior and exterior angles of regular polygons; name polygons with up to ten sides.


6.5 Know the standard trigonometric ratios, and their standard abbreviations, for sine of θ, cosine of θ and tangent of θ, given an angle θ in a right-angled triangle, and use these ratios to find the remaining sides of a right-angled triangle given one side and one angle or to find the angles given two sides.

Show that sintan cosθθ θ= .

6.6 Derive and recall the exact values for the sine, cosine and tangents of 0°, 30°, 45°, 60°, 90° and use these in relevant calculations.

Calculate the exact area of an equilateral triangle with sides of length 6 cm.

Trigonometric ratios

Use a calculator to find sine and cosine values of a given angle and to find the angle corresponding to a given value of the sine or cosine of that angle.

Students should know that these are inverse functions defined on a restricted domain.


Show that cos2 60° + sin2 60° = 1. Show that a similar result is true when the angle is replaced by any of 0°, 30°, 45°, 90°.

On a particular day, the depth of the water, d metres, in a harbour is given by:

d = 7 + 2 sin (30t)°

where t is the time, in hours, since midnight. What is the maximum depth of water? At what time is the depth of the water at its maximum?

6.7 Discuss at least two proofs of Pythagoras’ theorem.

Explain how you could use the diagram below to prove Pythagoras’ theorem.

6.8 Use Pythagoras’ theorem to find the distance between two points, to solve triangles, to find Pythagorean triples, and to set up the Cartesian equation of a circle of radius r, centred at the point (α, β).

Find the equation of a circle of radius 5 units, centred at the point (5, –3).

Find the exact distance between the point (1, 4) and the point (–2, 5).

Two sides of a right-angled triangle are of length 21 cm and 29 cm. What are the possible lengths of the remaining side?

In this question, you should not use a calculator.

An elastic band is fixed on four pins on a pinboard, as shown in the diagram.

Show that the total length of the band in this position is 14√ 2 units.

Show that a triangle with sides of length m2 – n2, 2mn and m2 + n2 respectively is always right-angled. Find some right-angled triangles using this result.

Solve the triangles shown, giving all the angles and all the sides.

Each side of a cube is 5 cm. Calculate the length of a diagonal of the cube from one vertex on the ‘base’ to the opposite vertex on the ‘top face’. What is the angle between this diagonal and the base?

Powers of (co)sines

Note that (cos θ ° )2 is written as cos2 θ ° and that (sin θ ° )2 is written as sin2 θ ° .


There are many interesting websites devoted to proofs and applications of this fundamental theorem in geometry.


Suggest possible equations for these straight lines. Find the shortest distance between them.

Circle theorems

6.9 Prove the circle theorems:

• The perpendicular from the centre of a circle to a chord bisects the chord.

• The two tangents from an external point to a circle are of equal length.

Constructions

6.10 Perform and justify straight edge and compass constructions, including those to bisect a line, to construct an equilateral triangle with a side of given length, to drop a perpendicular from a point to a line, and to bisect an angle.

Construct a square. You may only use a straight edge, a pencil and a pair of compasses.

Use the construction to bisect an angle several times over to construct an angle of 22.5°.

Explain why the construction to bisect an angle works.

Loci

6.11 Determine the locus of an object moving according to a rule, including those arising in simple physical situations.

A goat is on a rope attached at one corner of a rectangular enclosure. The enclosure measures 10 m by 4 m. The rope is 6 m long. Draw a scale drawing of the enclosure and shade in the locus in which the goat can move.

Find the locus of all points 3 cm from a circle of radius 5 cm. Discuss how the locus is changed if three dimensions are allowed.

Transformations

6.12 Investigate Islamic patterns and describe their features.

This pattern is from a mosque in Isfahan, in Iran. Use this and other Islamic patterns to discuss key features of the pattern (its construction, reflection symmetries, translations, and so on).

Use of ICT

6.13 Use ICT to explore geometrical relationships.

Circle theorems

Include terms associated with a circle: centre, radius, diameter, circumference, arc length, sector, segment, chord, tangent.

Include the use of a dynamic geometry system (DGS).

Transformations




7.1 Find perimeters and areas of rectilinear shapes and volumes of rectilinear solids; find the circumference and area of a circular region, and the surface area and volume of a right prism, cylinder, cone and pyramid, and a sphere, using dimensionally correct units.

The volume of a pyramid is (base area × perpendicular height). Calculate the volume of a pyramid with a square base with side 4 cm and a volume of 48 cm3. What is its perpendicular height?

This prism was made from three cuboids.

Show that the area of the cross-section of the prism is 24x2 + 3xy. The volume of the prism is 3x2(8x + y). What is the depth of the prism?

A round peg, of radius r, just fits into an equilateral triangular hole.

What proportion of the hole is filled by the peg?

7.2 Use radians to calculate sector areas and arc lengths.

A manufacturer makes party hats shaped like hollow cones. To make the hats she cuts pieces of card which are sectors of a circle, radius 24 cm. The angle of the sector is 135°.

a. Show that the arc length of the sector is 18π cm.

b. The sector is joined edge to edge to make a cone. The edges of the sector meet exactly with no overlap. Calculate the vertical height of the completed hat.

Sectors and arcs

Include terms associated with a circle: centre, radius, diameter, circumference, arc length, sector, segment, chord.

Define 1 radian as the angle that an arc of length 1 unit subtends at the centre of a circle of radius 1 unit.


A satellite is 1500 km above the Earth. It has a camera with a 50° angle of view with which it surveys the Earth below. Draw a diagram to represent the satellite and its camera in relation to the Earth. Calculate how far apart the two furthest points on the Earth are that can be photographed by the satellite at any one time. Take the Earth to be a sphere of radius 6378 kilometres.

7.3 Use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth’s surface.

How would you find the shortest distance between Doha and London?

A plane flies from Doha to Karachi almost along the line of latitude 25 degrees north. Doha is at longitude 51 degrees east approximately and Karachi is at longitude 67 degrees east approximately. How far is it from Doha to Karachi along this route?

What is a great circle of the Earth?


7.4 Solve problems involving compound measures, including average speed, such as cost per litre, kilometres per litre, litres per kilometre, population density (number of people per unit area), density (mass per unit volume), pressure (force per unit area) and power (energy per unit time).

A satellite passes over both the north and south poles, and it travels 800 km above the surface of the Earth. The satellite takes 100 minutes to complete one orbit.

Assume the Earth is a sphere and that the diameter of the Earth is 12 800 km. Calculate the speed of the satellite, in kilometres per hour.

A water tank is filled through a hosepipe connected to a tap. The rate of flow through the hosepipe can be varied. A tank of capacity 4000 litres fills at a rate of 12.5 litres per minute. How long in hours and minutes does it take to fill the tank?

Another tank takes 5 hours to fill at a different rate of flow. How long would it take to fill this tank if this rate of flow is increased by 100%? How long would it have taken to fill this tank if the rate of low had been increased by only 50%?

This tank, measuring a by b by c, takes 1 hour 15 minutes to fill.

How long does it take to fill 2a by 2b by 2c, at the same rate of flow?

Compound measures

Use appropriate units and dimensions. Stress how units are calculated in compound measures.

Draw on examples from science.



By the end of Grade 10, students distinguish between qualitative or categorical data and quantitative data, and between discrete and continuous data. They understand the concept of a random variable. They can locate sources of bias. They plan questionnaires and surveys to collect meaningful primary data from representative samples. They collect data from secondary sources, including the Internet, and formulate and solve problems related to the data. They group data and plot histograms and other frequency and relative frequency distributions. They calculate measures of central tendency and measures of spread, including variance and standard deviation. They draw stem-and-leaf diagrams and box-and-whisker plots. They plot and interpret simple scatter diagrams between two random variables, and draw a line of best fit where there appears to be correlation. They use relevant statistical functions on a calculator and ICT applications to present statistical tables and graphs.

Students should:


Sampling

8.1 Know that:



• in a biased sample there are systematic differences between the sample and the population from which it is drawn.

8.2 Locate obvious sources of bias within a sample.

8.3 Know that different types of data can be collected from samples – qualitative/categorical data (e.g. eye colour, male, female) and quantitative data (e.g. age, height, lifespan, mortality rates) – and that quantitative data may be discrete (e.g. number of defective items in a production process) or continuous (e.g. weight); understand the concept of a random variable.


8.4 Plan surveys and design questionnaires to collect meaningful primary data from representative samples in order to test hypotheses about, or estimate, characteristics of the population as a whole.

8.5 Formulate and solve problems using secondary data from published sources, including the Internet.

8.6 In analysing data, calculate and use measures of central tendency such as the arithmetic mean and the median.

Investigate life expectancy in a range of countries, including Qatar, Iran, Turkey, India, Brazil, China, Russia, Italy, the United Kingdom and the United States of America.



Data collection

Include data collected in other subjects, such as science or social science.


A company makes breakfast cereal containing nuts and raisins. They counted the number of nuts and raisins in 100 small packets.

a. Calculate an estimate of the mean number of nuts in a packet. b. Calculate an estimate of the number of packets that contain 24 or more raisins.

8.7 Calculate measures of spread, including the variance and standard deviation.

A fisherman kept a record of the mass, M, of each of the fish he caught in one season.

Mass (kg) Frequency

0.5 < M ≤ 1 14

1 < M ≤ 1.5 29

1.5 < M ≤ 3 45

3 < M ≤ 4 16

4 < M ≤ 6 10

Total 114

The fisherman exaggerated the mass of fish caught. He added 0.5 kg to the mass of each fish before he recorded it. State what effect this would have on the estimate of the mean. State what effect this would have on the estimate of the standard deviation.

Another fisherman doubled the mass of each fish before he recorded it. Comment on the effect this would have on the mean and standard deviation of the mass of fish caught.

Find the mean and median salaries of the group of workers in Qatar whose weekly salaries in riyals are given in the table below.

Salary (QR) 250 300 350 400 450 500 550 600

Frequency 5 11 20 31 18 12 7 3

Which average is the most representative for these workers? Justify your answer.

Use statistical functions on a calculator to calculate the standard deviation for the salaries in this group. What information does this convey?

8.8 Construct histograms, grouping continuous data when necessary.

A scientist wanted to investigate the lengths of eggs from a particular breed of hen. Taking a sample of 80 eggs, she measured the length of each one and grouped the data as follows:

Length (l) in cm 4.4 ≤ l < 5.0 5.0 ≤ l < 5.4 5.4 ≤ l < 5.8 5.8 ≤ l < 6.3 6.3 ≤ l < 6.5


Complete a histogram to show this information. Write the frequency density on each part of the histogram.

Calculate the mean length of the eggs in her sample. Discuss how to calculate best estimates for the modal and median values of the lengths of the eggs in the sample.

Histograms

Include the terms frequency, frequency distribution and frequency density, relative frequency and relative frequency distribution.


304 people took part in a swimming contest. They swam 1.5 km. The histogram shows the distribution of their times for the event.

a. The histogram is constructed using frequency densities. The table shows the

frequency densities. Complete the table to show the frequencies.

Time t (minutes)

Frequency density

Frequency

17 ≤ t < 22 16.0 80

22 ≤ t < 27 28.0

27 ≤ t < 32 12.4

32 ≤ t < 52 1.1 b. 304 people took part. Calculate an estimate of the mean time for this event.

c. Explain why the median time for the event must be between 22 and 27 minutes.

d. Calculate an estimate of the median time for this event.

The table below shows the number of cars leaving a car park during the periods given.

Number of minutes after 1700 h

0 ≤ n < 5 5 ≤ n < 10 10 ≤ n < 20 20 ≤ n < 50 50 ≤ n < 60

Number of cars leaving

74 115 248 1174 189

Complete the histogram to show the information in the table. Write the frequency density above each rectangle of the histogram.

The value 14.8 on the histogram is the frequency density for the period 0 ≤ n < 5 minutes. Explain what is meant by frequency density with regard to cars leaving the car park.


8.9 Plot cumulative frequency distributions, grouping continuous data when necessary.

Sulaiman did a survey of the age distribution of 160 people at a theme park. The cumulative frequency graph below shows his results.

a. Use the graph to estimate the median age of the 160 people at the theme park.

b. Use the graph to estimate the interquartile range of the age of the 160 people at the theme park.

8.10 Draw stem-and-leaf diagrams and box-and-whisker plots and use them in presentations of findings.

The diagram shows a box-and-whisker plot of examination marks for a class of students.

L represents the lowest mark scored and H is the highest mark scored. LH then represents the range of marks. Q1 is the first quartile mark, Q3 is the third quartile mark and Q1Q3 is the interquartile range. M is the median mark.

A school for boys and a school for girls each enter students for the same mathematics examination. The girls’ marks were: 97 98 57 45 63 75 87 34 56 28 67 89 45 61 53 49 81 32 23 45 47 72 34 54 23 100 76 47. The boys’ marks were: 67 87 83 92 34 31 23 25 29 39 89 91 54 47 41 50 77 18 89 10 26 62 39 14 90.

Draw back-to-back stem-and-leaf diagrams to represent these scores. Compare the performances of the girls and the boys, explaining your methodology and findings.

Using the above data, plot a cumulative frequency graph for the marks of the girls. What was the median score? What was the interquartile range of the distribution of marks? Draw a box-and-whisker plot to represent the girls’ marks.

Draw a relative frequency histogram for these data, explaining how the data were grouped and the meaning of each bar of the histogram.

8.11 Make inferences and draw conclusions from the formulation of a problem to the collection and analysis of data in a range of situations; select statistics and a range of charts, graphs and tables to present findings.

Compare the television viewing habits of students in different grades at school.

Cumulative frequency

Include the terms range, percentile, interquartile range, semi-interquartile range, and mode, modal class, modal frequency.


9 Simple correlation

9.1 Draw scatter diagrams between two random variables associated with some common context; identify through elementary qualitative discussion positive and negative correlation; where there appears to be correlation, draw a line of best fit, constructing it to pass through the point representing the arithmetic means of the two variables in the chosen samples, judging by eye the line about which the data points are most evenly distributed.

Compare the examination marks for all students in a class for: a. mathematics and science; b. Arabic and English; c. mathematics and art. Discuss whether there appears to be correlation or not.

Scientists have observed that insects called crickets move their wings faster in warm temperatures than in cold temperatures. Below is a graph showing 13 observations of cricket ‘chirps’ per second and the associated air temperatures.

a. On the graph, draw the estimated line of best fit for this data.

b. Using your line, estimate the air temperature when cricket chirps of 22 per second are heard.

TIMSS Grade 12

10 Use of ICT


ICT opportunity

A range of applications can be used to support data handling.





Students solve routine and non-routine problems in a range of mathematical and other contexts. They use mathematics to model and predict outcomes of real-world applications, and compare and contrast two or more given models of a particular situation. They break down complex problems into smaller tasks, and set up and perform appropriate calculations and manipulations. They identify and use connections between mathematical topics. They develop and explain chains of logical reasoning, using correct mathematical notation and terms. They generate mathematical proofs and generalise when possible. They approach problems systematically, knowing when to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They recognise when to use ICT and do so efficiently.

Number and algebra

Students use proportional reasoning and harder percentage calculations to solve problems, including inverse proportion problems. They use sigma notation and generate and sum simple recursive sequences, including arithmetic and geometric series. They find combinations and permutations, and use the binomial theorem expansion of (1 + x)n, where n is a positive integer. They form and manipulate algebraic expressions and formulae, simplify and combine algebraic fractions, rearrange harder formulae and generate further formulae from physical contexts. They find approximate solutions of quadratic equations, and a pair of simultaneous equations, one linear and one quadratic. They use physical contexts to plot and interpret the graphs of linear, quadratic, cubic, reciprocal, exponential and logarithm functions, the sine and cosine functions, and the modulus function. They recognise symmetry properties of functions, when they are even or odd, and when they are increasing, decreasing or stationary. They solve problems using inverse and composite functions. They apply combinations of transformations to the graph of the function y = f(x). They understand the concept of a limit and find derivatives of functions. Through their study of functions and their graphs, and the solution of associated equations, students appreciate a range of numerical and algebraic applications in the real world. They continue to use realistic data and ICT to analyse problems.


Students use geometry and trigonometry to solve practical and theoretical problems. They know and use the sine and cosine rules, and calculate the area of a triangle using ½ ab sin C. They use Pythagoras’ theorem to show that sin2 θ + cos2 θ = 1 for any angle θ. They find the points of intersection of a straight line with a circle. They plot the graphs of circular functions and solve simple problems modelled by these functions. They prove standard circle theorems. They draw and use plans and elevations. They translate, reflect, rotate and enlarge two-dimensional geometric objects. They begin to use vectors to solve physical problems. They solve a range of problems involving compound measures, using appropriate units and dimensions. They explore

Grade 11 Advanced


geometry using ICT.


Students use a simple mathematical model to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events. They calculate probabilities of single and combined events, and understand risk as the probability of the occurrence of an adverse event. They use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another. They use simple simulations and consider trends over time using a moving average. They analyse results to draw conclusions and use a range of graphs, charts and tables to present their findings. They use relevant statistical functions on a calculator and ICT applications to present statistical tables and graphs.


The advanced mathematics standards have four strands: reasoning and problem solving; number, algebra and calculus; geometry and measures; and probability and statistics. Calculus is introduced in Grade 12 in the advanced standards.






statistics

Grade 10 55% 30% 15%

Grade 11 55% 30% 15%


40% – 60%


75% 25% –

The standards are numbered for easy reference. Those in shaded rectangles, e.g. 1.2, are the performance standards for all advanced students. The national tests for advanced mathematics will be based on these standards. Grade 11 teachers should review and consolidate Grade 10 standards where necessary.





By the end of Grade 11, students solve routine and non-routine problems in a range of mathematical and other contexts. They use mathematics to model and predict outcomes of real-world applications, and compare and contrast two or more given models of a particular situation. They break down complex problems into smaller tasks, and set up and perform appropriate calculations and manipulations. They identify and use connections between mathematical topics. They develop and explain chains of logical reasoning, using correct mathematical notation and terms. They generate mathematical proofs and generalise when possible. They approach problems systematically, knowing when to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They recognise when to use ICT and do so efficiently.

Students should:







1.6 Develop chains of logical reasoning, using correct mathematical notation and terms.


1.8 Generate mathematical proofs, and identify exceptional cases.




Grade 11 Advanced

Key standards


Cross-references

Standards are referred to using the notation RP for reasoning and problem solving, NA for number and algebra, GM for geometry and measures and PS for probability and statistics. e.g. standard NA 2.3


Examples of problems in italics are intended to clarify the standards, not to represent the full range of possible problems.



Proofs


Generalisation

Students should appreciate that generalisation is important in mathematics.





Number and algebra

By the end of Grade 11, students use proportional reasoning and harder percentage calculations to solve problems, including inverse proportion problems. They use sigma notation and generate and sum simple recursive sequences, including arithmetic and geometric series. They find combinations and permutations, and use the binomial theorem expansion of (1 + x)n, where n is a positive integer. They form and manipulate algebraic expressions and formulae, simplify and combine algebraic fractions, rearrange harder formulae and generate further formulae from physical contexts. They find approximate solutions of quadratic equations, and a pair of simultaneous equations, one linear and one quadratic. They use physical contexts to plot and interpret the graphs of linear, quadratic, cubic, reciprocal, exponential and logarithm functions, the sine and cosine functions, and the modulus function. They recognise symmetry properties of functions, when they are even or odd, and when they are increasing, decreasing or stationary. They solve problems using inverse and composite functions. They apply combinations of transformations to the graph of the function y = f(x). They understand the concept of a limit and find derivatives of functions. Through their study of functions and their graphs, and the solution of associated equations, students appreciate a range of numerical and algebraic applications in the real world. They continue to use realistic data and ICT to analyse problems.

Students should:


2.1 Make appropriate use of knowledge of number sets.


3.1 Develop further confidence in using the rules for indices.

The atmospheric pressure on Mercury is 2 × 10–8 N/m2. The atmospheric pressure on Earth is 5.05 × 1012 times as great as the atmospheric pressure on Mercury. Calculate the atmospheric pressure on Earth.

Farida is making a scale model of the Earth and the Moon for a museum. She has found out the diameters of the Earth and the Moon, and the distance between them in metres.

Diameter of the Earth 1.28 × 107 m Diameter of the Moon 3.48 × 106 m Distance between Earth and Moon 3.89 × 108 m

a. How many times bigger is the diameter of the Earth than the diameter of the Moon?

b. In Farida’s scale model the diameter of the Earth is 50 cm. What should be the distance between the Earth and the Moon in Farida’s model?

Algebra


Laws of exponents



4 Work with sequences, series, recurrence relations and arrangements

4.1 Generate sequences from term-to-term definitions and from position-to-term definitions, including recursive sequences, to model the behaviour of the real-world situations, for example population growth.

In a TV quiz show a contestant can triple her winnings if she survives from one round to the next. Write down a formula for her prize Pn+1 in the (n + 1)th round in terms of her prize Pn in the nth round. Write an alternative formula for Pn+1 in terms of n. The prize for winning in the first round is QR 1000. What is the minimum number of rounds that will have to be contested to win at least QR 700 000?

A sequence is defined by un+2 = un+1 – un with u1 = 5 and u2 = –4. Write down the first eight terms of the sequence.

A sequence is defined by un = n(n – 1) + 41. Write down the first twelve terms of this sequence. What do you notice about these terms? Form a conjecture about this sequence and carry out further tests to see if your conjecture is correct.

In a certain country, there is a net increase in population from one year to the next of 5 per cent. Set up a recurrence relation to describe the population in year n + 1 in terms of the population in year n. Find the population in year n + 4 compared to the population in year n. Use your formula to find the number of years it takes to double the population from year n.

A woman buys a car and pays in monthly instalments. The car costs QR 60 500 and interest is charged on any outstanding debt at a monthly rate of r%. The woman pays back a fixed amount each month of QR M. Set up a recurrence relation connecting the amount owed, An+1, after n + 1 months in terms of the amount owed, An, at the end of the nth month. How many months will it take to repay the debt if M = QR 1200 and r = 1.2%? How much will the woman have then paid for the car? Investigate repayments for different values of M and r.

4.2 Understand and use sigma notation for summing the terms of a sequence.

Find 10

1( 1)r +∑ .

Find 2r∑ for integer values of r from 1 to 10.

Write out in full 4

1

( 1)( 1)

r

r r−

+∑ and use partial fractions to evaluate this sum.

Rewrite these sums using sigma notation. 2 2 2 2

1 1 1 1 1121 144 169 196 225

7 8 9 10+ + +

− + − +

4.3 Recognise an arithmetic progression (AP); sum an arithmetic series and know the formula for the rth term of the series in terms of the first term and the common difference between terms.

The fourth term of an AP is 25. The sum of the first six terms of the AP is 120. Find the eleventh term.

4.4 Recognise a geometric progression (GP); generate term-to-term and position-to term definitions for the terms of a GP in terms of the common ratio between terms; sum a finite geometric series.

At the end of every year a car loses 30 per cent of its value at the start of the year. Construct a formula, in terms of the original purchase price, to give the value of the car n years after purchase. After how many years will the car first be worth less than 90 per cent of its original value?

Recurrence relations

Term-to-term definitions of sequences are also known as recurrence relations.

ICT opportunity

Use spreadsheets in examples like these.

Sigma notation

The start and end values of the positions of the terms to be summed are commonly indicated below and above the sigma sign respectively.


4.5 Sum to infinity a convergent geometric series and know the condition on the common ratio for an infinite geometric series to be convergent.

The first term of an infinite geometric series is 1 and the common ratio is ½. Write down the first six terms and find the sum of the series.

In another infinite geometric series, the first term is 1 and the common ratio is –½. Write down the first six terms and find the sum of the series.

4.6 Understand and use formulae for the sum of: the squares of the first n positive integers; the cubes of the first n positive integers.

4.7 Understand and use factorial notation and know that 0! = 1; know the binomial theorem expansion of (1 + x)n for positive integer n and that the

term in xr has coefficient nCr, where !C( )! !

nr

nn r r

=−

; know how to use

Pascal’s triangle to find nCr; find permutations and combinations.

Show that ( 1)( 2)...( 1)C !n

rn n n n r

r− − − += .

Explain why 0

( ) Cn

n n n r rra b a b−+ =∑ for any positive integer n.

Use the binomial theorem to expand (x – 2y)4.


5.1 Understand the symmetry properties of functions, and when a function is even or odd; sketch and describe the features of polynomial functions up to order 3.

Use symmetry properties to help sketch graphs of polynomial functions up to order 3. What other considerations are helpful in sketching curves?

State whether the following functions are even, or odd, or neither even nor odd. 3 2

3

2

f ( ) 2 5

g( )1

x x xxx

x

= + −

=−

Quadratic functions

5.2 Model a range of situations with appropriate quadratic functions.

A fountain at ground level sprays out jets of water. Each jet is a parabola. The jet that sprays the farthest has equation y = –x2 + 8x – 15. Factorise this expression. Hence find a. where the fountain jet is positioned in this xy-coordinate system and b. how far from the fountain jet the water hits the ground. Calculate the greatest height that the water reaches.

Curve A is the reflection in the x-axis of y = x2.

What is the equation of curve A?

Binomial theorem

The theorem has important applications in probability theory.

An alternative notation

The notation ⎛ ⎞⎜ ⎟⎝ ⎠nr is often

used for the expression − − − +n n n n r

r( 1)( 2)...( 1)

!

when n is not a positive integer.

Functions

Include use of the notation x xf : f( ) , as well as

y = f(x) or f(x) = …

Modelling with functions and their inverses

This aspect of mathematics adds realism, and shows the importance of the subject through application.

ICT opportunity

Where possible, use real data and analyse the data using ICT.


Huda throws a ball to Mariam who is standing 20 m away. The ball is thrown and caught at a height of 2.0 m above the ground.

The ball follows the curve with equation y = 6 + c(10 – x)2, where c is a constant. Calculate the value of c by substituting x = 0, y = 2 into the equation.

An n-sided polygon has 12 ( 3)n n − diagonals. How many diagonals has an octagon?

A polygon has 104 diagonals. How many sides does it have?

5.3 Find the axis of symmetry of the graph of a quadratic function, and the coordinates of its turning point by algebraic manipulation; understand the effect of varying the coefficients a, b and c in the expression ax2 + bx + c.

y = (x – 3)2 + 5 is a quadratic function of x. What is the minimum value of this function and for what value of x does it occur? What is the maximum range of the function? Give the equation of the axis of symmetry of the function. Write an alternative form for the equation defining the function. Sketch the graph of this function.

5.4 Given a quadratic equation of the form ax2 + bx + c = 0, know that:

• the discriminant ∆ = b2 − 4ac must be non-negative for the exact solution set in to exist;

• there are two distinct roots if ∆ is positive and one repeated root if ∆ is zero.

5.5 Find approximate solutions of the quadratic equation ax2 + bx + c = 0 by reading from the graph of y = ax2 + bx + c the x-coordinate(s) of the intersection point(s) of the graph of this function and the x-axis.

The graph shows the curve y = x2 + 4x.

a. Solve the equation x2 + 4x – 2 = 0 using the graph. Give your answers to two decimal places.

b. The equation x2 + 4x + 5 = 0 cannot be solved using the graph. Why not?

5.6 Solve equations and inequalities using algebra or a combination of algebra and graphical representation.

Find the solution set of (x – 1)2 ≥ 9.

Solve the inequality | 2x – 3 | ≤ x + 3.

5.7 Use the graph of the function f(x) = ax2 + bx + c to determine regions where ax2 + bx + c is greater than or less than zero.

Solve the inequality x2 + x – 6 > 0.

ICT opportunity



5.8 Find exactly by analytical methods and approximately by graphical methods, the solution set of two simultaneous equations L1 and Q1, where L1 represents a linear relation for y in terms of x, and Q1 a quadratic function of y in terms of x.

How can you determine how many points of intersection a given straight line y = mx + c has with a given quadratic curve y = ax2 + bx + c?

Find the intersection points of the straight line 4y = 3x and the circle (x – 4)2 + y2 = 9.

5.9 Solve physical problems modelled simultaneously by two such functions.

Inverse proportion and the reciprocal function

5.10 Understand the statement y is inversely proportional to x and set up the corresponding equation y = k/x; know some characteristics, including that x ≠ 0 and that x = 0 is an asymptote to the curve, as is y = 0; study examples of inverse proportionality. Look at these graphs.

a. One of the graphs shows the equation y = kx – x2 (k is a constant). Which graph is it?

b. One of the graphs shows the equation y= k/x, where k is a positive constant. Which graph is it?

Explain why the function y = k/x cannot be defined on the domain set . What is the largest domain the function can be defined on? Sketch the graph of the function for this domain. Does the function have a greatest or least value? Is there anywhere where the function increases?

Three people working flat out complete a job in sixteen hours. How many hours would it take eight people to do the same job? Explain any assumptions you have made.

The average speed for a fixed distance journey is inversely proportional to the time taken to complete the journey. A family travels in Europe by car. They travel exactly half their journey in 2 hours, then stop for lunch for 1 hour, and then take 3 hours over the second half of the journey. How were the average speeds related on each part of the journey? If the average speed for the first half of the journey was 72 kilometres per hour, what was the average speed for the whole journey?

5.11 Use a graphics calculator, including use of the trace function, to show approximate solutions to physical problems requiring the location and physical interpretation of the intersection points of two or more graphs.

5.12 Use physical contexts to plot and interpret graphs of functions, recognising when the functions are increasing, decreasing or stationary, including:

• linear, quadratic and cubic functions;

• the reciprocal function y = k/x (x ≠ 0);

• the sine and cosine functions;

• the modulus function and a range of simple non-standard functions.

Which grows faster for x ≥ 0: the power function y = x3 or the exponential function y = ex? Justify your answer.

ICT opportunity


ICT opportunity



A big wheel makes one complete revolution every 90 seconds. The wheel has a diameter of 20 metres. The bottom of the wheel is 2 metres above the ground. Two people get on the wheel and sit in a seat, and then the wheel starts to rotate. T seconds later their height above the ground is given by h = 2 + 8 sin 4T° . Explain why this is an appropriate formula to use. At what two consecutive times are they 12 m above the ground?

A ship can only enter a harbour when the tide is in; it must have a minimum depth of water of 8 metres. The tide follows a daily sinusoidal variation given by the formula d = 5 sin 15t° + 8, where t is the time in hours from midnight onwards, measured on the 24-hour clock. At how many times in a day will the depth of water in the harbour be exactly 8 m? For how many hours a day can the ship enter the harbour? Sketch how the level of the tide varies with the time of the day.

Find physical examples that are modelled by circular functions.

5.13 Find, graph and use the inverse function of those functions given by a one-to-one mapping or restricted to such mappings; know that the graph of the inverse function may be found by reflecting the graph of the function in the line y = x; solve a range of problems using inverse functions.

The cost of production of q silver bracelets is C = 200 + 15q. Find the inverse function and interpret its meaning.

5.14 Add, subtract and multiply two functions given in the form y1 = f1(x) and y2 = f2(x); write down, without simplification, the mathematical form for one function divided by another.

5.15 Understand the concept of a composite function and use the notation y = f(g(x)).

5.16 Deconstruct a composite function into its constituent functions, using inverse functions.

Find the inverse function of f(x) = 5x – 8.

Starting from the function y = x, describe how the function y = (5x – 3)2 is constructed. Show how to deconstruct this function back to the original function.

Transformation of functions

5.17 Understand the transformations of the function y = f(x) given by:

• y = f(x) + a, representing a translation by a in the positive y-direction;

• y = f(x + a), representing a translation by –a in the positive x-direction;

• y = af(x), representing a stretch with scale factor a parallel to the y-axis;

• y = f(ax), representing a stretch with scale factor 1/a parallel to the x-axis.

Use these and combinations of these transformations to sketch, stage by stage, the transformation of the graph of y = f(x) into the graph of the transformed function.

The straight line y = mx + c is the straight line y = mx translated parallel to itself a distance c in the y-direction. When y = mx, the variable y is directly proportional to the variable x. By redefining the origin to the point (0, c) the straight line y = mx + c implies that the variable (y – c) is directly proportional to the variable x, since y – c = mx and this passes through the point (0, c).

A function is defined by f(x) = x2. Describe the functions a. f(x – 2) and b. f(x + 1), stating how the graphs of each function relate to the graph of y = f(x), and give the defining equation for each function.

Modelling with circular functions

Examples could include oscillations on a spring, bungee jumping, pulse rate, blood pressure, alternating currents, daylight hours.

Composite functions

Use a ‘function machine’ to introduce a composite function and its inverse.


The diagram shows the graph with equation y = x2. On the same axes, sketch the graph with equation y = 2x2.

Curve B is the translation, one unit up the y-axis, of y = x2. What is the equation of curve B?

Translate curve B two units to the left. What is the equation of this new curve?

Transform the curve y = x3 into the curve y = 5x3. Describe the effect of the transformation. The curve is then translated one unit in the positive x-direction. What is the equation of this new curve?

Describe in words how the graph of y = 1/x is transformed into the graph y = 4 + 5/x. Sketch each graph on the same set of axes.

Explain the difference between a. the functions y = cos x° and y = cos (x + 45)° and b. the functions y = cos x° and y = 2 cos x°.


5.18 Understand the ideas of exponential growth and decay and the forms of the associated graphs y = ax, where a > 0; use a graphics calculator to plot the graphs of the exponential function, ex, and the natural logarithm function, ln x; know that one is the inverse function of the other and use this to find solutions to physical problems; solve for x the equation y = ax and use this in problems; use the log function (logarithm in base 10) on a calculator.

The growth of the Internet since 1990 has been modelled by the function N = 0.2(1.8)t, where N is the number of users, counted in millions, t years from 1990. Plot the graph of this function. How many Internet users does the model predict for the year 2006?

When living organisms die the amount of carbon-14 present in the dead matter decays exponentially according to the formula N = N0e–0.000121t, where N0 is the initial quantity and t is the time in years. A bone uncovered at an archaeological site has 35% of its original carbon-14. Estimate the age of the bone. After how many more years will the bone have only 25% of its carbon-14?

The number of bacteria in a colony grows exponentially. At 1300 hours today the number of bacteria was 1000 and at 1500 hours it was 4000. How many bacteria were there at 1800 hours today? How many bacteria will there be at 1000 hours tomorrow?

The Global Report estimated the population of the world in 1975 as 4.1 billion people and that it was growing at the rate of 2% per year. Set up an equation to predict the world population t years from 1975. Use this model to predict the world’s population in 2020. Discuss any assumptions you have made.

Earthquakes produce oscillations in the ground. The strength, S, of the quake is measured on the Richter scale and is given by S = log A, where A is the measured amplitude of the oscillation as measured in millimetres on a calibrated seismograph. What amplitude of oscillation corresponds to a major earthquake with a Richter scale value of 7.8? What is the Richter scale value of an earthquake with an oscillation that has an amplitude of 2000 mm?


Use examples of population growth and decay, cooling, radioactivity, spread of an epidemic, drug absorption, and compound interest.

Use a graphics calculator to support modelling.


6 Understand the concept of a limit

6.1 Understand the concept of a limiting value; for example, explain that:

• the sequence of terms 2, 4, 9, 25, …, n2, … gets larger and larger as n gets larger and larger and that this sequence diverges as n tends to infinity;

• the sequence of terms 1 2 3 4, , , , ..., , ...2 3 4 5 1

nn +

gets closer and closer to 1

as n gets larger and larger, and that this sequence converges to 1, the limit of the sequence, as n tends to infinity;

• the function 1f ( )xx

= tends to minus infinity as x tends to zero from the

negative side, but tends to positive infinity as x tends to zero from the positive side. At x = 0 the function is not defined. When x tends to infinity the function tends to zero from the positive side. When x tends to minus infinity the function tends to zero from the negative side. The lines x = 0 and y = 0 are asymptotes of the function;

• the function f(x) = x tends to 2 when x tends to 2 from just below 2 and also tends to 2 when x tends to 2 from just above 2. For this function we can write

2limf ( ) 2x

x→

= .

The function 2 1f ( ) 1

xx x−= + is defined for all x except x = –1, and for these values is the

same as the function g(x) = x – 1. Explain why this is so. For each function find the limit as x tends to –1. What, if any, is the distinction between the functions f and g?

The value of 0

2 2limh

hh→

+ − is

A. 0 B. 12 2

C. 12 D. 1

2 E. ∞

TIMSS Grade 12

6.2 Consider a chord across the graph of the function y = f(x) between the points with coordinates (x, f(x)) and (x + h, f(x + h)), and show that if θ is the angle this chord makes with the positive x-axis then

f ( ) f ( ) f ( ) f ( )tan( )x h x x h xx h h h

θ + − + −= =+ −

.

6.3 Evaluate 0

f ( ) f ( )limh

x h xh→

+ − for the following functions:

f(x) = x2 f(x) = x3

f(x) = x–1.

6.4 Understand that the limit 0

f ( ) f ( )limh

x h xh→

+ − for the function y = f(x) is

represented geometrically as the slope of the tangent to the curve for this function at the point (x, f(x)).

Discuss how this limiting process corresponds to finding the slopes of chords drawn across a smaller and smaller interval of the curve from this point, and that in this process the chord itself tends to the tangent line at the point.

Limits

Include discussion of

→

⎛ ⎞+⎜ ⎟⎝ ⎠

n

n n0

1lim 1 . It could be

argued that this is equal to

1, since n1 tends to zero as

n gets larger and larger. On the other hand, it could be

argued that + n11 is always

greater than 1 and so, taken to power n, gets larger and larger as n increases. Neither argument is correct. In fact, the expression tends to the irrational value e = 2.718 281 8…, the basis for the exponential function and the number base for natural logarithms.


7 Calculate the derivative of a function

7.1 Identify the tangent at the point (x, f(x)) on the function y = f(x) as the derivative of the function at this point, and denote the derivative by either

of the two common notations ddyx and f ′(x); interpret the derivative as a rate

of change.

7.2 Calculate derivatives of simple powers of x from first principles, using the definition in NA 6.4 above.

Find f ′(x) for f(x) = x2. How does the mathematics change if f(x) = 3x2?

Find f ′(x) for f(x) = xn. Use the definition of the derivative applied to this function and the binomial expansion for (x + h)n, remembering that, since h is itself very small, h2 and higher powers of h are negligibly small.

7.3 Know the general result that if f(x) = axn, where a is constant, then f ′(x) = anxn–1 for all real values of n.

The force of gravitational attraction between the Earth of mass M and a satellite of mass m is given by Newton’s law of gravitation as F = GMm / r2, where r is the distance between the centre of the Earth and the satellite and G is the universal constant of gravitation. Find F′(r) and interpret its meaning.

7.4 Know that if f(x) = axn ± bxm, where a and b are constants, then f ′(x) = anxn–1 ± bmxm–1.

Reading derivatives

Read the derivative dy / dx as ‘dee-y by dee-x’ or as ‘the derivative of y with respect to x’ and the form f′(x) as ‘f-dash of x’.

Differentiation

The process of forming the derivative is called differentiation, so named because the method involves setting up a difference equation.



By the end of Grade 11, students use geometry and trigonometry to solve practical and theoretical problems. They know and use the sine and cosine rules, and calculate the area of a triangle using ½ ab sin C. They use Pythagoras’ theorem to show that sin2 θ + cos2 θ = 1 for any angle θ. They find the points of intersection of a straight line with a circle. They plot the graphs of circular functions and solve simple problems modelled by these functions. They prove standard circle theorems. They draw and use plans and elevations. They translate, reflect, rotate and enlarge two-dimensional geometric objects. They begin to use vectors to solve physical problems. They solve a range of problems involving compound measures, using appropriate units and dimensions. They explore geometry using ICT.

Students should:


8.1 Use a dynamic geometry system to conjecture results and to explore geometric proof.

In triangle ABC, the altitudes BN and CM of the triangle ABC intersect at S. ∠ MSB is 40° and ∠ SBC is 20°. Prove that triangle ABC is an isosceles triangle.

TIMSS Grade 12

Each side of the regular hexagon ABCDEF is 10 cm long. Find the length of the diagonal AC.

TIMSS Grade 12


8.2 Know and use the sine rule and the cosine rule to solve triangles.

Show that Pythagoras’ theorem is a special case of the cosine rule.

A triangle has its three angles in the ratio 2 : 3 : 4. Find to two significant figures the ratio of the lengths of its sides.

A ship sails 50 kilometres in a direction 032° and then 29 kilometres in a direction 315°. How far is the ship from its starting point? What is its bearing from its starting point?



Use of ICT

Geometry is enhanced with use of a dynamic geometry system, or DGS, which provides an interactive means for investigating and hypothesising results that can then be proved as theorems.


A helicopter at airfield A received a distress call from a boat. The position of the boat, B, was given as 147 km from the airfield, on a bearing of 072°. A man on the boat is flown to hospital. Calculate the distance the helicopter travelled from the boat to the hospital at H.

On another occasion the helicopter travelled from the airfield on a bearing of 218° to fly a pregnant woman at W to the hospital. The helicopter then flew on a bearing of 081° to the hospital, H. Calculate the distance the helicopter travelled from where it picked up the woman to the hospital.

6.4.1 8.3 Solve triangle problems in two and three dimensions.

The two sides of a canal are straight, parallel and the same height above the water level. Jana and Sharifa want to find the width of the canal. They measure 100 m on the canal bank and stand facing each other at the points J and S. Jana measures the angle she turns through to look at the post, P, as 25°. Sharifa measures the angle she turns through to look at the post as 15°. Calculate the width of the canal.

The Great Pyramid of Cheops in Egypt is built on a square base with side 230 metres. Each face of the pyramid is at 52° to the horizontal. Calculate the height of the pyramid. Calculate the inclination of an edge of the pyramid to the horizontal.

The Great Pyramid of Cheops at Giza. Source: www:kingtutshop.com

Triangle problems

Include the terminology angle of inclination and angle of declination.


8.4 Calculate the area of a triangle using ½ ab sin C.

8.5 Find the points of intersection of a straight line with a circle by using algebraic substitution from the equation of the straight line into the equation of the circle. Find the points where the line 4x – 3y = 0 cuts the circle x2 + y2 = 100.

Circular functions

8.6 Use the unit circle x2 + y2 = 1to plot and describe the features of the graphs of the circular functions f(θ) = sin θ f(θ) = cos θ f(θ) = tan θ where θ is measured in radians and 0 ≤ θ ≤ 2π; know the symmetries and periodicities of these functions; know that any point on the unit circle has coordinates (cos θ, sin θ), where θ is the angle the radius to the point makes with the positive x-axis.

Explain why sin (2π – θ) = sin θ.

Give the exact value of cos 5π⁄4. What other angle has the same cosine value?

Draw the tangent line at the origin on the curve y = sin x. Use this to estimate the value of sin x for small values of x. Explain why it is important to use radian measure in this context.

8.7 Use Pythagoras’ theorem to show that sin2 θ° + cos2 θ° = 1 for any angle θ° .

Verify this result for the angles 30°, 45° and 60°. What happens when θ° = 90°?

8.8 Solve simple problems modelled by circular functions.

Circle geometry

8.9 Prove the circle theorems:

• The angle subtended by an arc at the centre of the circle is twice the angle subtended by the arc at a point on the circle, including, as a special case, the angle in semicircle is a right angle.

• Angles in the same segment subtended by a chord are equal.

• The angle subtended by a chord at the centre of a circle is twice the angle between the chord and the tangent to the circle at an end point of the chord.

• When two chords BC and DE in a circle intersect at A then AB × AC = AD × DE.

• Opposite angles of a cyclic quadrilateral are supplementary.

Two circles with centres at A and B have radii of 7 cm and 10 cm as shown in the diagram. The length of the common chord PQ is 8 cm.

Calculate the length of AB.

TIMSS Grade 12

Trigonometric functions

The full names of these functions are sine, cosine and tangent. For the calculus of these functions it will be necessary to work with the reciprocal functions (sec), cosecant (cosec) and cotangent (cot).

Include use of the term amplitude.

Trigonometric identities

Include deriving simple related identities.

Circle theorems

Include terms associated with a circle: inscribed circle, circumcircle, cyclic quadrilateral.

Include the use of a dynamic geometry system (DGS).


In the diagram on the right, AD is a tangent to the circle with centre O. ∠ ABC is 63° and AC is a chord of the circle. AB is 18 cm and BC is 3 cm.

Calculate the values of ∠ AOC, ∠ OCA and ∠ CAD.Calculate the area of triangle ABC. Calculate the length of AC.

9 Work with transformations and projections

9.1 Transform rectilinear figures using combinations of translations, rotations about a centre of rotation, enlargements about a centre of enlargement, and reflections about a line; understand the meanings of positive, negative and fractional scale factors in enlargements.

An equilateral triangle ABC has side length 10 cm. It rotates around the inside of a square of side length 20 cm.

a. Triangle ABC rotates about C to the position shown as CA1B1. What is the angle of

rotation?

b. Calculate the distance along the path travelled by point A in turning from A to A1.

c. Calculate the distance along the path travelled by point A in turning from A1 to A2.

d. The triangle continues rotating around the inside of the square in the same way until it is back at the original position. Which of the original points A, B or C will point A land on when it has completed its rotations around the inside of the square?

The rectangle Q in the diagram below CANNOT be obtained from the rectangle P by means of a

A. reflection about an axis in the plane of the page B. rotation in the plane of the page C. translation D. translation followed by a reflection

Circle the correct answer.

TIMSS Grade 12

A triangle has vertices at the points (4, 5), (6, 1) and (8, 11). The triangle is enlarged by a factor of 2 about a centre of enlargement at the point (3, –3). Draw the enlarged triangle in its correct position on a coordinate grid.

Transformations

Transformations are best developed through use of a dynamic geometry system (DGS).


The line segment OA is 3.0 cm long. The line segment OB is √ 7 cm long. OB can rotate in a horizontal plane about the point O.

a. Find the maximum possible distance B can be from A. Explain whether your answer

is a rational number or an irrational number.

b. Find the minimum possible distance B can be from A. Explain whether your answer is a rational number or an irrational number.

c. Sketch a different position for the line segment OB so that the distance from A to B, AB, is a rational number. Confirm by calculation that your answer is a rational number.

d. OB is reduced in length to 2.6 cm. OA is still 3.0 cm long. Calculate the distance AB when angle AOB is 120°.

e. The lengths of 2.6 cm and 3.0 cm are accurate to one decimal place. The 120° angle is accurate to the nearest degree. Calculate the greatest and least possible values of AB.

9.2 Visualise the effect of transformations on a plane figure; know that the image of a planar figure under rotation or reflection is congruent to the original figure before rotation or reflection and that every circle is similar to any other circle.

The diagram shows parts of two circles, sector A and sector B.

a. Which sector has the bigger area?

b. The perimeter of a sector is made from two straight lines and an arc. Which sector has the bigger perimeter?

A semicircle, of radius 4 cm, has the same area as a complete circle of radius r cm. What is the radius of the complete circle?

9.3 Draw the plan and elevation of three-dimensional rectilinear objects.


10 Use vectors

10.1 Consider coordinate systems as grids for moving around space in two or three dimensions; understand position vector, unit vector and components of a vector.

A particle is at the point (6, 2). What is its position vector in terms of the unit vectors i and j in the x- and y-directions respectively? Calculate the length (magnitude) of this vector.

10.2 Interpret a translation as a vector displacement; know that a vector displacement from A to B depends only on the starting point A and the finish point B and not on intermediate steps from A to C to D to … to B, and that the vector sum of all these separate displacements from A to B is equivalent to the resultant displacement from A to B directly.

10.3 Add and subtract two vectors in up to three dimensions and draw corresponding vector diagrams.

Four vectors a, b, c and d are given by a = 2i – 3j, b = 5j + k, c = 4i – 7k and d = 3i + j. Find a + b, b – c, a – b – c. Draw vector diagrams to represent a + d and a – d. What are the components of these two vectors in the i and j directions?

10.4 Multiply a vector by a scalar and know that this amounts to stretching the vector; calculate the magnitude and direction of a vector; use vectors to calculate displacement and velocity in a range of contexts.

A particle moves with constant velocity from A to B. Its position vector at A is a = i + j and its position vector at B is b = 5i – 7j. Calculate the vector displacement from A to B. If distance is measured in metres, show that the distance from A to B is 4√5 metres. The particle takes 2 seconds to move from A to B. What is its velocity?

10.5 Use the scalar product of two vectors to calculate the angle between the vectors and the scalar product of a vector with itself to find the magnitude of the vector.

Find the magnitude of each of the vectors a and d in the example in GM 10.3 above. Calculate the angles between these vectors.

10.6 Solve physical problems using vectors.

i and j are unit vectors in the east and north direction respectively. A ship has position r = 3i + 4j at 1200 hours. It then moves with constant velocity v = 4i – 5j. The velocity is measured in kilometres per hour. What is the speed of the ship (the magnitude of its velocity)? What is the position of the ship at 1500 hours?

A particle of mass m kilograms is moving with constant acceleration a, measured in metres per second per second. The total external force F acting on the particle is measured in newtons, and is the vector sum of the individual forces acting on the particle. The relationship between F and a is given by Newton’s second law of motion and is F = ma.

A particle of mass 2 kg is acted upon by two forces F1 = i – j and F2 = 3j. Find the acceleration of the particle and give its magnitude.



Vectors

In three dimensions, vectors provide a natural language to place and displace figures in space. They also link with Cartesian coordinate systems.

Unit vectors

Unit vectors in three mutually perpendicular directions are usually written as i, j and k.


On the pinboard, draw a trapezium that has a perimeter of 6 + 4√ 2.

This shape is designed using three semicircles.

The radii of the semicircles are 3a, 2a and a.

a. Find the area of each semicircle, in terms of a and π, and show that the total area of

the shape is 6πa2.

b. Find a when the area is 12 cm2, leaving your answer in terms of π.

A light shade is in the form of a frustum of a right cone. The radius at the top of the shade is 10 cm and the radius at the bottom is 25 cm. Find the surface area of the material used for the light shade.

11.2 Work with compound measures including density, average speed and acceleration, measures of rate (such as rate of growth of income), and population density (number of people per unit area), using appropriate units and dimensions.

Jabor sees a flash of lightning. 25 seconds later he hears the sound of thunder. Speed of light is about 1.1 × 109 km per hour. Speed of sound is about 1.2 × 103 km per hour. Calculate how far away Jabor is from the lightning.

A cable car takes passengers to the top of a volcano. It starts from station A and takes 16 minutes to reach station B at the top of the volcano. The average speed of the cable car is 2 metres per second. The cable car is at an angle 25° to the horizontal. Find, to the nearest metre, the height of the volcano as measured from A.

TIMSS Grade 12

Compound measures

Reinforce links with science and technology, using power or rate of doing work, momentum, average speed and acceleration.

Rates

Students who will study quantitative methods in Grade 12 should work with a range of rates to support later work in probability and statistics.


Wafa recorded the speed of a car every 10 seconds throughout a short journey from her home to school. She used the data to draw a speed–time graph.

a. Find a point during the journey when the car’s speed was increasing most quickly.

Mark this point as P. By drawing appropriate lines on the graph, find the acceleration of the car at point P.

b. Find a point during the journey when the car’s speed was decreasing most quickly. Mark this point as Q. By drawing appropriate lines on the graph, find the acceleration of the car at point Q.

c. The car uses least fuel when it travels at speeds between 20 m/s and 25 m/s. Find an approximate value for the area under the graph for the period when the car was travelling at between 20 m/s and 25 m/s. What does this area represent? Give the correct units.


By the end of Grade 11, students use a simple mathematical model to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events. They calculate probabilities of single and combined events, and understand risk as the probability of the occurrence of an adverse event. They use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another. They use simple simulations and consider trends over time using a moving average. They analyse results to draw conclusions and use a range of graphs, charts and tables to present their findings. They use relevant statistical functions on a calculator and ICT applications to present statistical tables and graphs.

Students should:


12.1 Know that all probability values lie between 0 and 1, and that the extreme values correspond respectively to impossibility and certainty of occurrence.

12.2 Understand that a random variable has a range of values that cannot be predicted with certainty, and investigate common examples of random variables; measure the empirical probability (relative frequency) of obtaining a particular value of a random variable.




12.3 Use a simple mathematical model to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events.

A company makes computer disks. It tested a random sample of disks from a large batch. The company calculated the probability of any disk being defective as 0.025. Naima buys two disks.

a. Calculate the probability that both disks are defective.

b. Calculate the probability that only one of the disks is defective.

c. The company found three defective disks in the sample they tested. How many disks were likely to have been tested?

100 students were asked whether they studied French or German. 27 students studied both French and German.

a. What is the probability that a student chosen at random will study only one of the

languages?

b. What is the probability that a student who is studying German is also studying French?


A piece of string is 12 centimetres long. It is randomly cut into two pieces. What is the probability that one piece has length greater than 9 cm?


The probability of dying of cancer is 1⁄3. What is the probability that if three people are chosen at random two of them will die of cancer? What is the probability that none of them will die of cancer?


12.5 Understand when two events are mutually exclusive, and when a set of events is exhaustive; know that the sum of probabilities for all outcomes of a set of mutually exclusive and exhaustive events is 1, and use this is in probability calculations.

12.6 Know that when two events A and B are mutually exclusive the probability of A or B, denoted by P(A ∪ B), is P(A) + P(B), where P(A) is the probability of event A alone and P(B) is the probability of event B alone.

Mona has a chance of 1 in 4 of passing on a particular genetic condition to one of her children. Mona has three children. Calculate the probability that two of the children will inherit the condition. What is the probability that none of her children will inherit the condition? What is the probability that at least one of her children will inherit the condition?

Distributions

Include as mathematical models the rectangular distribution and the binomial distribution, and simple applications of these.


A computer game has nine circles arranged in a square. The computer chooses circles at random and shades them black.

a. At the start of the game, two circles are to be shaded black. Show that the probability that both circles J and K will be shaded black is 1/36.

b. Halfway through the game, three circles are to be shaded black. Here is one

example of the three circles shaded black in a straight line. Show that the probability that the three circles shaded black will be in a straight line is 8/84.

c. At the end of the game, four circles are to be shaded black. Here is one example of

the four circles shaded black forming a square. What is the probability that the four circles shaded black form a square?

12.7 Know that two events A and B are independent if the probability of A and B occurring together, denoted by P(A ∩ B), is the product P(A) × P(B).

In a class of 35 students, the probability that a student picked at random is taller than 1.8 metres is 0.2 and the probability that the student wears spectacles is 0.3. What is the probability when three students are chosen at random that two are over 1.8 metres in height and that one of them wears spectacles?

Show that two events A and B are mutually exclusive when P(A ∩ B) = 0.

On a road there are two sets of traffic lights. The traffic lights work independently. For each set of traffic lights, the probability that a driver will have to stop is 0.7.

a. A woman is going to drive along the road. What is the probability that she will have to stop at both sets of traffic lights? What is the probability that she will have to stop at only one of the two sets of traffic lights?

b. In one year, a man drives 200 times along the road. Calculate an estimate of the number of times he drives through both sets of traffic lights without stopping.

Adel makes two clay pots. Each pot is fired independently. The probability that a pot cracks while being fired is 0.03.

a. Calculate the probability that both of Adel’s pots crack while being fired.

b. Calculate the probability that only one of Adel’s pots cracks while being fired.

c. Adel has enough clay for 80 pots. He receives an order for 75 pots. Does he have enough clay to make 75 pots without cracks? Explain your answer.

A warning system installation consists of two independent alarms having probabilities of operating in an emergency of 0.95 and 0.90 respectively. Find the probability that at least one alarm operates in an emergency.

A. 0.995 B. 0.975 C. 0.95 D. 0.90 E. 0.885

TIMSS Grade 12



On a tropical island the probability of it raining on the first day of the rainy season is 2⁄3. If it does not rain on the first day, the probability of it raining on the second day is 7⁄10. If it rains on the first day, the probability of it raining more than 10 mm on the first day is 1⁄5. If it rains on the second day but not on the first day, the probability of it raining more than 10 mm is 1⁄4.

You may find it helpful to fill in this tree diagram before answering the questions below.

a. What is the probability that it rains more than 10 mm on the second day, and does

not rain on the first?

b. What is the probability that it has rained by the end of the second day of the rainy season?

c. Why is it not possible to work out the probability of rain on both days from the information given?

20 per cent of the population of a country has a particular disease. A test can be given to help determine whether or not people have the disease. The probability that the test is positive for those that have the disease is 0.7. But there is a 0.1 chance that a patient who does not have the disease registers positive on the test.

a. Find the probability that an individual selected at random tests positive, but does not have the disease.

b. Another person is chosen at random. Calculate the probability that the test result for this person is positive.

12.9 Know that in general if event B is dependent on event A, then the probability of A and B both occurring is P(A ∩ B) = P(A) × P(B|Α), where P(B|Α) is the conditional probability of B given that A has occurred.

13 Calculate moving averages

13.1 Consider trends over time and calculate a moving average.

Find out about the cost, in Qatar, of a barrel of crude oil over the period January 2000 to March 2004. Analyse the data over periods of three months and compare the moving average price per barrel. Discuss your findings.

14 Simulation

14.1 Use coins, dice or random numbers to generate models of random data.

Reema passes through two sets of traffic lights each morning on her way to work. She has timed both of them.

a. Reema has found that the first set shows green for 30 out of 50 seconds. What is the probability that Reema does not stop at the first set of traffic lights? Describe a method for simulating this probability using random numbers.

Random numbers

These can be generated on a scientific or graphics calculator using the RND and RAN function keys.


b. The second set of traffic lights shows green for 30 out of 90 seconds. What is the probability that Reema does not stop at the second set of traffic lights. Describe a method for simulating this probability using random numbers.

c. To simulate Reema’s journey through both sets of traffic lights you must choose two random numbers. Repeat this simulation 20 times. Use the simulation to estimate the probability that Reema will have to stop:

– at both sets of traffic lights;

– at only one set of traffic lights;

– at neither set of traffic lights.

Do an investigation using random numbers to simulate waiting times in a doctor’s surgery.

15 Use of ICT


ICT opportunity

A range of applications can support data handling.

281 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods © Supreme Education Council 2004




Students analyse problems in a range of mathematical and statistical contexts. They break problems into smaller tasks, and set up and perform relevant manipulations, calculations and tests. They identify and use connections between mathematical topics and appropriate statistical techniques. They develop and explain chains of logical reasoning, using correct mathematical notation and terms, and generalise when possible. They approach problems systematically, knowing when and how to enumerate all outcomes. They identify exceptional cases and statistical outliers, and conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They collect, organise, analyse and interpret relevant and realistic data, using statistical functions on a calculator and ICT. They work to expected degrees of accuracy.

Number, algebra and calculus

Students use number, algebra and calculus to further their understanding of statistics. They use the series expansion of ex and the rules of logarithms. They use the remainder theorem and the factor theorem and find permutations and combinations. They sketch and interpret the graphs of linear, quadratic, cubic, reciprocal, exponential and logarithm functions, the sine and cosine and tangent functions (using radian measure for angles), and the modulus function. They solve related equations (except cubic equations) on specified domains. They recognise when functions are increasing, decreasing or stationary. They calculate and interpret the derivative of powers of x, of polynomial functions, and of the exponential function, including second and higher order derivatives. They calculate the derivative of the sum, difference and product of any two of these functions. They know that integration is the inverse of differentiation. They use integration to calculate areas under curves.


Students apply and use the work on geometry and measures learned in earlier grades to solve problems.


Students collect data, organise data and make inferences from data. They plan questionnaires and surveys and design experiments to collect primary data from samples, distinguishing a sample from its parent population. They know the significance of a simple random sample and the effect of bias in a sample. They understand the importance of a random variable. They distinguish a parameter for a population from a statistic for a sample. They formulate problems based on primary data, or on secondary data from published sources, including government statistics and the Internet. They calculate measures of central tendency and of spread. They construct histograms and frequency distributions, using box-and-whisker plots and associated vocabulary in presenting their findings and conclusions. They distinguish between nominal, ordinal and interval or ratio scales. They look for correlation between two random variables and calculate the rank order

Grade 12 Advanced (quantitative

methods)


correlation coefficient and the product moment coefficient of correlation and interpret their meaning. They draw lines of best fit where linear correlation is exhibited. They calculate probabilities of single and combined events, and use and understand vocabulary associated with the probabilities of occurrence of different events. They use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another. They compare probabilities derived from sampling with theoretical models of probability, including both discrete and continuous models. They use the chi-squared test to compare observation with theoretical expectation. They use significance levels to accept or reject a null hypothesis and to make inferences from data. They use random numbers to choose random samples, to assign individual units to samples, and to simulate a range of statistical situations. They present findings and conclusions using a range of graphs, charts and tables. They use relevant statistical functions on a calculator and ICT.


The advanced mathematics standards for Grade 12 have two pathways: mathematics for science and quantitative mathematics, to support the social sciences and economics. Each pathway includes reasoning and problem solving, and number, algebra and calculus. The mathematics for science standards include substantial work on calculus but no new work on probability and statistics, whereas the quantitative methods standards include substantial work on probability and statistics, less calculus, and no new work on geometry and measures.

The reasoning and problem solving strand cuts across the other strands. Reasoning, generalisation and problem solving should be an integral part of the teaching and learning of mathematics in all lessons.





statistics

Grade 10 55% 30% 15%

Grade 11 55% 30% 15%


40% – 60%


75% 25% –

The standards are numbered for easy reference. Those in shaded rectangles, e.g. 1.2, are the performance standards for all advanced quantitative mathematics students. The national tests for advanced quantitative mathematics will be based on these standards.

Grade 12 teachers should consolidate earlier standards as necessary.




By the end of Grade 12, students analyse problems in a range of mathematical and statistical contexts. They break problems into smaller tasks, and set up and perform relevant manipulations, calculations and tests. They identify and use connections between mathematical topics and appropriate statistical techniques. They develop and explain chains of logical reasoning, using correct mathematical notation and terms, and generalise when possible. They approach problems systematically, knowing when and how to enumerate all outcomes. They identify exceptional cases and statistical outliers, and conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They collect, organise, analyse and interpret relevant and realistic data, using statistical functions on a calculator and ICT. They work to expected degrees of accuracy.

Students should:


1.1 Solve routine and non-routine problems in a range of mathematical and statistical contexts, including open-ended and closed problems.

1.2 Use statistical techniques to model and predict the outcomes of statistical situations, including real-world applications; work to definitions and perform appropriate tests.



1.5 Use a range of strategies to solve problems, including working the problem backwards and redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation; and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

1.6 Develop chains of logical reasoning, using correct terminology and mathematical notation.


1.8 Identify exceptional cases and statistical outliers.


1.10 Approach complex problems systematically, recognising when and how it is important to enumerate all outcomes.



Grade 12 Advanced (quantitative

methods)

Key standards


Cross-references

Standards are referred to using the notation RP for reasoning and problem solving, NAC for number, algebra and calculus, GM for geometry and measures and PS for probability and statistics, e.g. standard NAC 2.3.







1.14 Check statistical data for reliability, internal consistency, arithmetic errors and plausibility.

There is an error in the following table, which appeared in a report called ‘Smoking and Health Now’ published by the Royal College of Physicians in the United Kingdom. The table shows the number of men aged 35 or more who died due to smoking related diseases. Locate the error and correct it.

Lung cancer Chronic

bronchitis Coronary heart

disease All causes

Number of deaths 26 973 24 976 85 892 312 537

Percentage 8.6% 8.0% 2.75% 100%

Find an article of a statistical nature in a newspaper. Check the article for lack of consistency, faulty arithmetic, implausible numbers and omission of relevant information.

1.15 Use statistical functions on a calculator to analyse real data sets.

1.16 Recognise when to use ICT and when not to, and use it efficiently; use ICT to present findings and conclusions.


By the end of Grade 12, students use number, algebra and calculus to further their understanding of statistics. They use the series expansion of ex and the rules of logarithms. They use the remainder theorem and the factor theorem and find permutations and combinations. They sketch and interpret the graphs of linear, quadratic, cubic, reciprocal, exponential and logarithm functions, the sine and cosine and tangent functions (using radian measure for angles), and the modulus function. They solve related equations (except cubic equations) on specified domains. They recognise when functions are increasing, decreasing or stationary. They calculate and interpret the derivative of powers of x, of polynomial functions, and of the exponential function, including second and higher order derivatives. They calculate the derivative of the sum, difference and product of any two of these functions. They know that integration is the inverse of differentiation. They use integration to calculate areas under curves.

Students should:

2 Manipulate algebraic expressions

2.1 Multiply, factorise and simplify expressions and divide a polynomial by a linear or quadratic expression.

Write down in ascending powers of x the expansion of (2 – 3x)3.

Factorise 27x3 + 8y3. Use your factorisation to write an identity for 27x3 – 8y3.

Simplify 3

211

xx

+−

.

Solve the equation (x + 3)(3x –1) – (x + 3)(5x + 2) = 0.


The standards for number, algebra and calculus are designed, in the main, to support the statistical techniques developed in the probability and statistics strand.

Factorisation

Include the sum and difference of two cubes.


2.2 Combine and simplify rational algebraic fractions; decompose a rational algebraic fraction into partial fractions (with denominators not more complicated than repeated linear terms).

Simplify 25 3 2

1 1x

x x−−+ −

.

Show that 1 1 1( 1) 1r r r r≡ −+ + .

Find the values of A, B and C in the identity 2 21

3 1 1(3 1)( 1) ( 1)≡ + ++ ++ + +

A B Cr rr r r

.

2.3 Understand and use the remainder theorem.

3x3 – 2x2 – ax – 28 has a remainder of 5 when divided by x – 3. What is the value of a?

2.4 Understand and use the factor theorem.

Show that 2x3 + x2 – 13x + 6 is divisible by x – 2 and find the other factors. Hence find the solution set in of the equation 2x3 + x2 – 13x + 6 = 0. What is the solution set in ?

3 Use index notation and logarithms to solve numerical problems

3.1 Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents; use the xy key and its inverse on a calculator.

Without using a calculator, simplify 5

28 2

4n n

n× .

Plot a graph of y = 1000 × 2–t for values of t from 0 to 1, increasing in steps of 0.2.

3.2 Know the definition of a logarithm in number base a (a > 0), and the rules of combination of logarithms, including change of base.

Give the value of log10 1000.

Evaluate log2 64.

Express log 27 – 2 log 3 as a single logarithm.

Prove that loglog logc

ac

bb a= and use this result to show that 1log logab

b a= .

Given logb 2 = 1/3, logb

32 is equal to

A. 2 B. 5 C. –3/5 D. 5/3 E. 2

3log 32

TIMSS Grade 12

Explain why the number base of a logarithm must be positive, but why the logarithm itself may take any value.

3.3 Use the ln and log keys on a calculator and the corresponding inverse function keys.

Use a calculator to evaluate log5 4 correct to three decimal places.

In 1916, two scientists each named du Bois derived a formula to estimate the surface area S m2 of human beings in terms of their mass M kg and their height H cm. The formula is S = 0.000 718 4 × M0.425 × H0.725. A boy has a mass of 60 kg and a height of 160 cm. Calculate the boy’s surface area to three significant figures.

The intensity of sound, N, is measured in decibels (dB) and is defined by the formula N = 10 log (I / 10–16), where I is the power of sound measured in watts. Find N for normal speech with a power of 10–10 watts. Find the power of a jet aircraft with a sound intensity of 150 dB.

Logarithms

In addition to formulae for the sum and difference of two logarithms, include:

=

=== >

=

a

xa

x

a

an

a a

a x

a xa

ax n x

log

log

log 1log 1 0 for any 0

log log


In chemistry, the pH value of a solution is defined as pH = – log [H], where [H] is the concentration of hydrogen ions in the solution. Solutions with pH < 7 are acidic and solutions with pH > 7 are alkaline. Solutions with pH = 7 are neutral. A solution has [H] = 5 × 10–7. Find its pH value and state whether it is acid, neutral or alkaline.


4.1 Understand that nCr is the number of combinations of r different objects from n different objects and that the number of permutations of r different objects from n different objects is r! nCr, which is denoted by nPr.

How many different committees of four people can be chosen from a group of nine people?

A committee is to be chosen from five men and three women. The committee will have two men and two women. In how many ways may the committee be selected?

An examination consists of 13 questions. A student must answer only one of the first two questions and only nine of the remaining ones. How many choices of question does the student have? A. 13C10 = 286 B. 11C8 = 165 C. 2 × 11C9 = 110 D. 2 × 11P2 = 220 E. some other number TIMSS Grade 12

Evaluate 9C6 and 10P5.

Prove that nCr = nCn–r and that nCr + nCr+1 = n+1Cr+1.

In how many ways can 5 thick books, 4 medium sized books and 3 thin books be arranged on a bookshelf so that the books of the same size remain together?

A. 5! 4! 3! 3! = 103 680 B. 5! 4! 3! = 17 280 C. (5! 4! 3!) × 3 = 51 840 D. 5 × 4 × 3 × 3 = 180 E. 212 × 3 = 12 288 TIMSS Grade 12

4.2 Investigate properties of nCr; expand the binomial series (1 + x)n for any rational value of n.

Show that nC0 + nC1 + nC2 + …+ nCn = 2n.

5 Work with functions and their graphs

5.1 Use a graphics calculator to plot exponential functions of the form y = ekx; describe these functions, distinguishing between cases when k is positive or negative, and the special case when k is zero.

5.2 Plot and describe the features of the natural logarithm function y = ln x; understand that the natural logarithm function is inverse to the exponential function.

A radioactive element decomposes according to the formula y = y0e–kt, where y is the mass of the element remaining after t days and y0 is the value of y for t = 0. Find the value of the constant k for an element whose half-life (i.e. the time taken to decompose half of the material) is 4 days. A. 1

e4 log 2 B. 1e 2log C. 2log e D. 1

4e(log 2) E. 42e

TIMSS Grade 12

Permutations

The order of selection distinguishes a permutation from a combination.

Binomial coefficients

The properties of these coefficients can form interesting extension work.

Functions

Include use of the notation x xf : f( ) , as well as

y = f(x) or f(x) = …


5.3 Understand the modulus function y = | x | and sketch its graph.

Sketch the graph of the function f(x) = | x – 2 | + | x | for –3 ≤ x ≤ 3.

5.4 Form composite functions and use the notation y = g(f(x)).

The function f(x) = 4x – 7 is defined on . A second function g(x) = (x + 1)2 is also defined on . Find f(g(x)) and g(f(x)). Comment on why these functions are not the same.

5.5 Form inverse functions (on a restricted domain, if necessary) and use the notation 1f ( )y x−= .

Show that the function f ( ) 1xx x= − , where x ≠ 1, is its own inverse.

Find the inverse of the function f : 2t t + , defined for t ≥ 0.

5.6 Know that 1f (f ( ))x x− = and that the graph of 1f ( )y x−= is the reflection of the graph of f ( )y x= about the straight line y = x.

Show that lne x x= .

Sketch the graph of y x= for non-negative values of x.

6 Solve equations associated with functions

6.1 Given a quadratic equation of the form ax2 + bx + c = 0, know that if the discriminant ∆ = b2 − 4ac is negative, there are two complex roots, which are conjugate to each other.

6.2 Solve exponential and logarithmic equations of the form ekx = A, where A is a positive constant, and ln kx = B, where B is constant.

6.3 Solve trigonometric equations of the form: sin (ax + b) = A, where –1 ≤ A ≤ 1; cos (ax + b) = A, where –1 ≤ A ≤ 1; tan (ax + b) = A, where A is constant; and find solutions in the interval 0 ≤ x ≤ 2π.

6.4 Find approximate solutions for the intersection of any two functions from the intersection points of their graphs, and interpret this as the solution set of pairs of simultaneous equations.

An approximate solution of the equation x = ex can be found by plotting on the same axes the curves y = x and y = ex and finding the x-coordinate of their point of intersection.


7.1 Know that the derivative of f ′(x) is called the second derivative of the

function y = f(x) and that this can also be written in the forms f′′(x) or 2

2

dd

yx

;

know that higher derivatives may be taken in the same way.

For the function f(x) = x3 – 6x, show that f′′(x) = 6x. What is the value of f′′(x) at the points where f′(x) is zero?

7.2 Interpret the numerical value of the derivative at a point on the curve of the function; know that: when the derivative is positive the function is increasing at the point; when the derivative is negative the function is decreasing at the point; when the derivative is zero the function is stationary at the point.

Composite functions

Study of the functions in standards NAC 5.4−5.6 can form extension work for the most able students.

Complex numbers

Complex numbers could form extension work for the most able students.

Reading the second derivative

f′′(x) is read as ‘f-double-dash of x’ and d2y/dx2 is read as ‘dee-two-y by dee-x-squared’.

Higher derivatives

In general an nth order derivative is denoted by

n

ny

xdd

or f(n)(x).


Discuss how knowledge of when a function increases, when it decreases and when it is stationary gives important information about the function as a whole and helps to analyse what it looks like.

Discuss the derivative function associated with the function f : | |x x .

Is there anywhere on this function where the derivative does not exist? Justify your answer.

7.3 Understand that stationary points of any function may correspond to a local maximum or minimum of the function, or may be a point of inflexion; understand how the derivative changes as the point at which the derivative is calculated moves through the local maximum or minimum, or through an inflexion; understand that not all points of inflexion are stationary points.

7.4 Understand and use the second derivative to test whether a stationary point is a local maximum, or a local minimum, or a point of inflexion.

Which of the following graphs has these features: f′(0) > 0, f′(1) < 0, and f′′(x) is always negative?

TIMSS Grade 12

7.5 Know that of all exponential functions, the exponential function y = ex is defined as the one in which the slope at the y-intercept point (0, 1) has the value 1.

7.6 Know that the function f(x) = ex is the only non-zero function in mathematics for which the derivative f′(x) = ex gives back the original

function, and that 2 3

e 1 ... ...2! 3! !

= + + + + + +n

x x x xxn

(ad infinitum).

It can be shown that the infinite series 2 3 4

1 ...2! 3! 4!x x xx+ + + + + represents the number ex.

Show that if you differentiate the series term by term and add all these terms together, you get back to what you started with.

7.7 Know that the derivative of the natural logarithm function ln x is 1/x.

Derivative of combinations of functions

7.8 Understand that given any function f(x) = f1(x) + f2(x) then the derivative of this sum of two functions is f′(x) = f1′(x) + f2′(x).

Find the derivative of the function given by y = 5x + ex.

7.9 Understand that given a function f(x) = u(x) v(x) then the derivative of this product of two functions is given by f′ = uv′ + vu′; use this result in calculating the derivative of the product of two functions; know the special case of this result that if y = a f(x), where a is constant, then y′ = a f′.

Find ddyx when y = 10x ex.

7.10 Understand that given a function f(x) = u(x) / v(x) then the derivative of this quotient of two functions is given by f′ = (uv′ – vu′) / v2; use this result in calculating the derivative of the quotient of two functions.

Maxima and minima

These are sometimes referred to as turning points.

The function ex This expansion of ex is important for the Poisson distribution in PS 12.2.


7.11 Understand that given a composite function h(x) = g(f(x)) then the derivative of this composite function is given by h′(x) = g′(f(x))f′(x); use this result in calculating the derivative of the composite of two functions.

Use the chain rule to show that if y = 10 e–2t then 2d 20edty

x−= − .

Find the derivative with respect to x of y = (3x + 1)2 (2 – x).

Prove that d(ln ) 1d

xx x= .

Show that d 1d d

d

xy y

x

= .

7.12 Recognise that the derivative of f(x) = A ekx, where A and k are constants, is f′(x) = kA ekx.

7.13 Find the derivative of a function defined implicitly.

Find ddyx for the implicit function x2 + y2 = 25 for y ≥ 0.

Applications using derivatives

7.14 Use the first and second derivatives of functions to analyse the behaviour of functions and to sketch curves.

Sketch the curve 1( 2)y x x= − , showing clearly its turning points and asymptotes.

Sketch the curve y = xe–x for x ≥ 0, showing clearly its turning points and any points of inflexion.

7.15 Use the derivative to explore a range of problems in which a function is maximised or minimised.

8 Reconstruct a function from its derivative

The indefinite integral

8.1 Understand integration as the inverse process to differentiation.

8.2 Understand and use the notation for indefinite integrals, knowing that f ( )d f ( )x x x c′ = +∫ , where c is any constant, and that there is a whole family

of curves y = f(x) + c, each member of which has derivative function f′(x).

Discuss how the individual members of the family of curves represented by y = f(x) + c are related to each other.

The graph of the function g passes through the point (1, 2). The slope of the tangent to the graph at any point (x, y) is given by g′(x) = 6x – 12. What is g(x)? Show all your work.

TIMSS Grade 12

8.3 Know the integrals of the functions: xn, where n ≠ –1 1/x, with x ≠ 0 ekx; write the integrals of multiples of these functions and of linear combinations of these functions.

Composite functions

An alternative and equally acceptable way of writing

this is = ×y y zx z x

d d dd d d ,

where the composite function is formed by first mapping x to z and then mapping z to y. This rule is often called the chain rule because it extends for composite functions formed in more than two stages.

Integration

The word integration comes from integrating, i.e. adding, the contributions of many small parts. The symbol

∫ x...d denotes summation

with respect to x. In the

definite integral ∫b

a, a and b

are called the limits of the integral, or the limits of integration.


The definite integral

8.4 Use the definition of the definite integral: f ( )d f ( ) f ( )

b

ax x b a′ = −∫ , where f(x) is a function of x and a ≤ x ≤ b;

interpret this as ‘the integral of a rate of change of a function is the total change of that function’; understand the effect of interchanging the limits of integration; know that .

b c b

a a c= +∫ ∫ ∫

8.5 Use summation of areas of rectangles to calculate lower and upper bounds for the area between the x-axis and a curve y = f(x) with y > 0, bounded on either side by lines x = constant; understand that as the width δx of each of the rectangles tends to zero the sums Σ f(x) δx for the lower and upper bounds on the area under the curve tend to the same value, and that this value is called the area under the curve.

8.6 Understand that the area bounded by a positive function y = f(x), the x-axis and the lines x = a and x = b, with a ≤ x ≤ b, is the definite integral

f ( )db

ax x∫ .

The line l in the figure is the graph of y = f(x). 3

2f ( )dx x

−∫ is equal to

A. 3 B. 4 C. 4.5 D. 5 E. 5.5

TIMSS Grade 12

8.7 Use the trapezium rule to find an approximation to the area represented by

the definite integral of a particular function when it is not easy or possible to integrate the function.

8.8 Understand that if a curve y = f(x) lies entirely below the x-axis, so that its

y-value is always negative, then the definite integral f ( )db

ax x∫ over the

interval a ≤ x ≤ b has a negative value.

Calculate the area between the curve y = x2 + 5, the x-axis and the lines x = –2 and x = 3.

Find the area between the curves y = x2 – 4 and y = 4 – x2.

Find the area between the curves y = x3 and y = x.

This figure shows the graph of y = f(x). S1 is the area enclosed by the x-axis, x = a and y = f(x); S2 is the area enclosed by the x-axis, x = b and y = f(x); where a < b and 0 < S2 < S1.

The value of f ( )db

ax x∫ is

A. S1 + S2

B. S1 – S2 C. S2 – S1 D. | S1 – S2

| E. ( )1

1 22 S S+

TIMSS Grade 12

Areas under curves

Students should work with ‘area-so-far’ for the area under a curve y = f(x), using definite integrals as in NAC 8.6, or the trapezium rule as in NAC 8.7, to reinforce work in the probability and statistics strand.


8.9 Understand that the integration by parts formula uv d uv vu dx x′ ′= −∫ ∫

reverses the derivative of the product of two functions.

Find e dxx x∫ .

8.10 Understand that g (f ( )) f ( )d g(f ( ))x x x x c′ ′ = +∫ reverses the derivative of a

composite function; recognise ‘simple’ functions for which this formula can be instantly applied.

Explain why 222 d ln ( 1)

1x x x c

x= + +

+∫ .

8.11 Use partial fractions to integrate.

Find 1 d( 1)( 2) xx x+ +∫ .

Integration by parts

Extension work on probability distributions could make use of standards 8.9, 8.10 and 8.11.



By the end of Grade 12, students collect data, organise data and make inferences from data. They plan questionnaires and surveys and design experiments to collect primary data from samples, distinguishing a sample from its parent population. They know the significance of a simple random sample and the effect of bias in a sample. They understand the importance of a random variable. They distinguish a parameter for a population from a statistic for a sample. They formulate problems based on primary data, or on secondary data from published sources, including government statistics and the Internet. They calculate measures of central tendency and of spread. They construct histograms and frequency distributions, using box-and-whisker plots and associated vocabulary in presenting their findings and conclusions. They distinguish between nominal, ordinal and interval or ratio scales. They look for correlation between two random variables and calculate the rank order correlation coefficient and the product moment coefficient of correlation and interpret their meaning. They draw lines of best fit where linear correlation is exhibited. They calculate probabilities of single and combined events, and use and understand vocabulary associated with the probabilities of occurrence of different events. They use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another. They compare probabilities derived from sampling with theoretical models of probability, including both discrete and continuous models. They use the chi-squared test to compare observation with theoretical expectation. They use significance levels to accept or reject a null hypothesis and to make inferences from data. They use random numbers to choose random samples, to assign individual units to samples, and to simulate a range of statistical situations. They present findings and conclusions using a range of graphs, charts and tables. They use relevant statistical functions on a calculator and ICT.

Students should:

9 Collect, organise and analyse data, and make inferences from data

Measuring and sampling

9.1 Know the difference between categorical data, discrete data and continuous data.

9.2 Understand what it means to measure a property of a person or thing in a statistical sense.

Explain the difference between a scientific measurement (e.g. the measurement of the amount of heat generated in a chemical reaction) and a statistical measurement (e.g. the number of homeless people in a big city like London).

Why is measuring ‘authoritarian personality’ different from measuring height?

9.3 Distinguish between nominal, ordinal, interval and ratio scales.

Explain the difference between nominal, ordinal, interval and ratio scales, giving an example of each type.


Students should know that probability lies at the heart of statistics. They should be aware of the uses of statistics in society and recognise when statistics are used sensibly and when they are misused or likely to be misunderstood.


What sort of measurement scale is used for categorical data? Give an example of such a scale.

A badly designed opinion poll poses the proposition: ‘The price of oil should be linked to the political situation in the Middle East.’ Subjects are asked whether they • strongly agree • agree • are undecided • disagree • strongly disagree with the proposition. Identify the type of scale used for this variable response.

An oil company keeps records on its employees. The database includes the age of the employees, the date on which they were hired, their gender, their grouping by country of origin (Qatari, Palestinian, Syrian, Indian, etc.), their type of job, their salary. Name the scale on which each of these variables is measured

9.4 Distinguish between population, sample and census; know the importance of choosing a representative sample; locate obvious sources of bias within a sample.

Explain, with examples, the important differences between a sample from a population and a census of the population.

Describe the effect of bias in a sample.

Explain why it is often in the interest of politicians, those that espouse good causes, telephone interviewers, advertisers and others to use questionable statistics and biased sampling methods.

Two common types of sampling are convenience sampling (when the easiest option is chosen to select units for sampling, e.g. stopping any shopper in a Suq) and voluntary response samples (where people choose themselves, e.g. to respond to a television poll). Explain why both convenience sampling and voluntary response sampling are likely to be biased.

9.5 Know the distinction between bias and precision.

Give examples of samplings with: high bias and high precision; low bias and low precision; high bias and low precision; low bias and high precision.

9.6 Understand and use the concept of a random variable; understand the meaning and properties of random numbers; know how to generate random numbers using the random number function(s) on a calculator; know how to assign random numbers in a variety of situations; use tables of random numbers.

Explain to someone who knows no statistics what a random variable is.

Collection and organisation of data

9.7 Understand how to collect a simple random sample using random numbers.

A class wishes to make a complaint to the school principal. The girls decide to select a simple random sample of five girls from the class to go to the principal with the complaint. Show how to use a random number table to select the five girls from the class of the class of 28 whose names are given below: Aida Nada Naima Haifa Maram Farha Zahra Muna Inas Roza Safa Majda Sharifa Farida Badriya Halima Hayat Deena Huwaida Haya Mariam Zalikha Amina Anissa Noor Fathiya Habiba Salma

Vocabulary of sampling

Know the meanings of vocabulary associated with sampling, including population, census, sample, unit, sampling frame, variable, parameter and statistic.

Random numbers

Random numbers can be used in a variety of ways, including choosing a simple random sample, assigning subjects to different treatments in a randomised trial and in carrying out simulations.


9.8 Distinguish different types of sampling: simple random sampling, stratified sampling and cluster sampling.

Write a short essay comparing different types of sampling techniques.

9.9 Plan surveys and design questionnaires to collect meaningful primary data from samples in order to test hypotheses about, or estimate, characteristics of the population as a whole.

9.10 Formulate problems based on secondary data from published sources, including the Internet.

9.11 Display categorical data with a pie chart or bar graph.

What is the essential difference between a pie chart and a bar graph?

The table below, from the 1995 Statistical Abstract of the United States, shows the percentage of females who were awarded doctorates in a number of subjects

Subject Percentage of doctorates obtained by females

Education 59.5 Psychology 59.7 Life sciences 38.3 Computer science 13.3 Engineering 9.6

Explain why a pie chart cannot be used to represent this information. Display the information in a bar chart.

9.12 Construct and interpret frequency tables and histograms for continuous grouped data, using equal and unequal class intervals; know that the frequency of occurrence in each class interval of a histogram is represented by the area of the rectangle constructed on that class interval; display discrete data in a vertical line chart; comment on how outliers might affect these distributions.

9.13 Calculate measures of central tendency: the arithmetic mean, the median and the mode; distinguish between these measures.

Why is the median often a more useful statistic than the mean?

A frequency diagram for a set of data is shown in the diagram below. No scale is given on the frequency axis, but summary statistics are given for the distribution:

Σ f = 50, Σ fx = 100, Σ fx2 = 344

a. State the mode and the mid-range value of the data.

b. Identify two features of the distribution.

c. Calculate the mean and standard deviation of the data and explain why the value 8, which occurs just once, may be regarded as an outlier.

d. Explain how you would regard the outlier if the diagram represents:

i. the difference of the scores obtained when throwing a pair of ordinary dice,

ii. the number of children per household in a neighbourhood survey.

e. Find new values for the mean and standard deviation if the single outlier is removed.

MEI

Data collection and analysis

Use primary data collected in other subjects, such as science or social science, and also secondary sources of data, e.g. from government statistics and from the Internet.

Relative frequency

Include the terms frequency and frequency distribution, relative frequency and relative frequency distribution.

Mean

Include the terms mode, modal class, modal frequency.

The sample mean x is defined as

= =∑

∑∑

n

i in

i n

i

f xx xn

f

1

1

1

1

The population mean is usually denoted by µ.


A group of seven professional tennis players compare their financial winnings in one year. The amounts they win in US dollars, to the nearest thousand dollars, are 138 000 2 597 000 155 000 146 000 369 000 199 000 283 00 Find the mean and median earnings of these players. Explain which is the better statistic to use as an average measure of their winnings? What would be the new mean and median if an eighth player, with winnings of US$ 538 000, joins the group?

A scientist took a sample of 80 eggs. She measured the length of each one and grouped the data as follows and then plotted frequency density against length:

Length (l) in cm 4.4 ≤ l < 5.0 5.0 ≤ l < 5.4 5.4 ≤ l < 5.8 5.8 ≤ l < 6.3 6.3 ≤ l < 6.5


In grouped data, one way to estimate the mode is to use similar triangles.

a. Explain why AB = 40 and DE = 58. Use similar triangles to calculate an estimate of the mode.

b. The best estimate of the median is 5.6 (to one decimal place). The estimate is calculated like this:

16.55.4 (0.4)35+

5.4 is the initial value of the class containing the median. Explain what the other three numbers in the formula stand for.

QCA, modified

9.14 Calculate measures of spread, including the variance and standard deviation; know the distinction between population and sample variance, and the corresponding standard deviations.

9.15 Use calculator function keys for mean, standard deviation and variance.

9.16 Understand how mean, variance and standard deviation are affected by the linear coding i iy a bx= + .

Find the mean and standard deviation of the set of numbers 5, 7, 4, 3, 8, 7, 8, 6. Add 3 to each number. Repeat the calculation of the mean and standard deviation. Explain how these values relate to the original values for the mean and standard deviation?

A golf tournament is taking place. For each round, the players’ scores are recorded relative to a fixed score of 72. [For example, a true score of 69 would be recorded as –3.] The recorded scores, x, for the ten players to complete the first round were: –4 –3 –7 6 2 0 0 3 5 7

a. Calculate the mean and standard deviation of the values of x.

b. Deduce the mean and standard deviation of the true scores.

In the second round of the tournament, the recorded scores, x, for the same ten golfers produced a mean of –0.3 and standard deviation 2.9.

c. Comment on how the performance of the golfers has changed from the first to the second round.

d. Calculate the mean and the standard deviation of the twenty true scores for the two rounds.

MEI

Sample variance

This is denoted by s2. The definition of s2 is

= −− ∑

n

is x xn2 2

1

1 ( )1

The sample variance is s, the square root of s2.

The population variance is denoted by σ 2 , and the standard deviation is given by its square root, σ.


9.17 Plot and interpret cumulative frequency distributions and box-and-whisker plots, using grouped continuous data as necessary; use the vocabulary range, percentile, interquartile range, semi-interquartile range.

Explain why making a comparison of the performance ratings of two different types of motor vehicle is easily made using box-and-whisker plots.

9.18 Draw stem-and-leaf diagrams.

Compare the advantages and disadvantages of histograms and stem-and-leaf diagrams.

9.19 Interpret, in qualitative terms, the skewness of a frequency distribution and understand the importance of a symmetric distribution.


10.1 Know that all probability values lie between 0 and 1, and that the extreme values correspond respectively to impossibility and certainty of occurrence; calculate probabilities.

The eighteenth-century French scientist Count Buffon did many experiments on probability. One of these was to estimate the value of π by dropping a needle between two parallel lines drawn exactly one needle length apart. Buffon showed that the probability that the needle lands on or across one of the parallel lines is 2/π. Estimate the probability of needles falling on a line by throwing similar length needles in the air and recording how they land. Use this estimate to calculate π. Can you prove Buffon’s result?

In a group of 36 blood donors, 16 are male and 20 are female. Four of these people are chosen for an interview.

a. In how many ways can they be chosen? b. Find the probability that they are all of the same sex.

MEI

10.2 Know that all possible outcomes for an experiment form the sample space for that experiment; use the sample space to calculate probabilities for each outcome.

Two fair six-sided dice are thrown and their total is recorded. Give a diagrammatic representation of the sample space. Calculate the probability that when the dice are thrown the total is at least 8.

10.3 Understand that a random variable has a range of values that cannot be predicted with certainty; investigate common examples of random variables; measure the empirical probability (relative frequency) of obtaining a particular value of a random variable.

Comment on this statement: ‘The behaviour of a random variable is not randomly chaotic, but represents a kind of order that emerges only in the long run’.

Repeatedly throw an ordinary six-sided dice. Investigate the number of throws ‘on average’ to first record a five on the uppermost face.

Here are two statements: An unbiased coin is tossed many times; the probability of heads will be close to one half. An unbiased coin is tossed very many times; the number of heads will be close to half the number of tosses. Are both statements true? Is only one of the statements true? Are both statements false? Explain the reasoning behind your answer.

Use computer generation of random numbers to simulate the tossing of a coin. What is the long-term probability of tossing heads?

What does ‘the law of averages’ really mean?

Discrete random variables

The probability that a discrete random variable X has the value x is denoted by P(X = x).


10.4 Know that a probability distribution for a random variable assigns the probabilities of all the possible values of the variable and that these values total to 1; use a simple mathematical probability distribution to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events.

In a ‘heads or tails’ experiment a coin that is believed to be unbiased is repeatedly tossed in the air and the uppermost face on landing (H or T) is recorded. The first six throws of the experiment give TTTTTT. Is this evidence that the coin is biased? If not, explain how this can happen. Is the next throw likely to be a head?

Use a random number table to simulate the tossing of a fair coin. Estimate the probability of getting a run of four heads or four tails.


A couple wants to have children. They would like to stop having children once they have a girl, but they do not want to have more than four children. Use a random number table to simulate the birth of children using the couple’s strategy. Assume that boys and girls are equally likely to be born. Estimate the probability that the couple will have a girl. Carry out your own simulation for this model. Use the data collected to estimate the mean number of children for families using this model of child bearing.

A quiz consists of six multiple choice questions, each with four possible answers. The questions are very unusual and no one is expected to know any of the answers. So everyone has to guess the answers. What is the probability that someone guesses four correctly? What is the probability that someone guesses at least two correctly?


Huntingdon’s disease is a serious condition that may be passed on by women to their children. Waafaa has a probability of 0.5 of passing on the disease to her daughter Moza, but her daughter has been killed in a car crash and it is not known whether or not she was a carrier for the disease. What is the probability that Moza’s daughter will inherit the disease?

A person who is a carrier for cystic fibrosis has a 0.5 probability of passing on the gene to his or her child. Since cystic fibrosis is a recessive disorder, any children that inherit the disease must inherit the gene for cystic fibrosis from both their parents. If both parents are carriers, what is the probability that their child will have cystic fibrosis?

Further properties of discrete random variables

10.6 Use and understand expected value, or expectation, of a quantified random variable as the sum of the products of each possible value and the probability of obtaining that value, and that this is the mean value.

A state lottery offers the following 100 prizes for every 100 000 tickets sold: 1 prize of US $5000, 9 prizes of US $500, and 90 prizes of US $50. A man buys one ticket for US $1.What is his probability of winning nothing? What is the expectation for his winnings? Is it worth the man’s trouble?

The number, X, of occupants of cars coming into a city centre is modelled by the

probability distribution P( ) kX r r= = for r = 1, 2, 3, 4.

a. Tabulate the probability distribution and determine the value of k.

b. Calculate E(X) and Var(X).

MEI

Probability models

Include as mathematical models for discrete random variables the discrete uniform distribution and the binomial distribution, and simple applications of these.

Expectation and variance

= ∑n

i iX x x1

E( ) P( )

µ= −X X 2 2Var( ) E( )

or

µ= −X X 2Var( ) E[( ) ]


10.7 Use alternative formulae for the variance of a discrete random variable.

Prove that 2 2Var( ) E( )X X µ= − is equivalent to 2Var( ) E[( ) ]X X µ= − .


10.8 Understand when two events are mutually exclusive, and when a set of events is exhaustive; know that the sum of probabilities for all outcomes of a set of mutually exclusive and exhaustive events is 1, and use this in probability calculations.

10.9 Know that:

• when two events A and B are mutually exclusive the probability of A or B, denoted by P(A ∪ B), is P(A) + P(B), where P(A) is the probability of event A alone and P(B) is the probability of event B alone;

• two events A and B are independent if the probability of A and B occurring together, denoted by P(A ∩ B), is the product P(A) × P(B);

• when two events A and B are not mutually exclusive the probability of A or B, denoted by P(A ∪ B), is P(A ∪ B) = P(A) + P(B) – P(A ∩ B), where P(A) is the probability of event A alone, P(B) is the probability of event B alone and P(A ∩ B) is the probability of both A and B occurring together.

The probability that A occurs is 0.5. The probability that B occurs is 0.35. The probability that neither A nor B occurs is 0.3. Find the probability that both A and B occur.

MEI, modified


An unbiased dice is thrown until a five is recorded. Calculate the probability of winning after one throw, after two throws, after three throws, and so on. Imagine this process keeps on building up for ever-increasing numbers of throws to first record a five. Now calculate the expectation of the number of throws needed to throw a five. If you have set this up correctly the probability distribution that you obtain is called a geometric distribution. Can you see why?

45 per cent of the population of a country has a particular disease. A screening test can be given to help determine whether or not people have the disease. The probability that the test is positive for those that have the disease is 0.7. But there is a 0.1 chance that a patient who does not have the disease registers positive on the test. Find the probability that an individual selected at random tests positive, but does not have the disease. Another person is chosen at random. Calculate the probability that the test result for this person is positive.

10.11 Know that in general if event B is dependent on event A, then the probability of A and B both occurring is P(A ∩ B) = P(A) × P(B|A), where P(B|A) is the conditional probability of B given that A has occurred.

In a set of 28 dominoes each domino has from 0 to 6 spots at each end. Each domino is different from every other and the ends are indistinguishable so that, for example, the two diagrams below represent the same domino.


A domino which has the same number of spots at each end, or no spots at all, is called a ‘double’. A domino is drawn at random from the set. The sample space diagram on the right represents the complete set of outcomes, each of which is equally likely.

6

5

4

3

2

1

00 1 2 3 4 5 6

Let the event A be ‘the domino is a double’, event B be ‘the total number of spots on the domino is 6’ and event C be ‘at least one end of the domino has 5 spots’.

The diagram on the right shows the sample space with event A marked.

a. Write down the probability that event A occurs.

b. Find the probability that either B or C or both occur.

c. Determine whether or not events A and B are independent.

d. Find the conditional probability P(A|C). Explain why events A and C are not independent.

e. After the first domino has been drawn, a second domino is chosen at random from the remainder. Find the probability that at least one end of the first domino has the same number of spots as at least one end of the second domino. [Hint: Consider separately the cases where the first domino is a double and where it is not.]

MEI

In the United Kingdom, pregnant women are screened to see if there is a high risk that their baby has Down syndrome. The screening test indicates if the risk is high enough to warrant the woman having further investigations. As with all screening tests, some women with Down syndrome pregnancies will fail to be detected as being in the high risk group while a number of normal pregnancies will be identified as high risk. The result is false positive if the baby tests positive but does not have the syndrome, and a false negative if the baby has the syndrome but the test result is negative. The true incidence of the syndrome can be found from other tests and after the babies are born. The table below gives the result of the screening test on 1400 babies.

Number of babies

Number of positive results

Number of negative results

Down syndrome 20 14

Not Down syndrome 1380 1310

Complete the table. What percentage of babies had false positive results? What is the probability that a baby selected at random will have Down syndrome and give a positive test result?

In the fictitious country of Virtualia there are three prisoners who cannot communicate with each other. They have been told that next day two of them will be executed, but that the choice made by random selection will be revealed next morning. Each prisoner calculates his probability of being executed. One of the prisoners begs the jailor to reveal the name of one of the other prisoners that will be executed and the jailor finally agrees. The jailer thinks he has given nothing away. The prisoner thinks his chances of survival have increased from 1/3 to 1/2. Which of them is correct?


11 Understand and use the binomial distribution to make inferences from data

11.1 Recognise when to use the binomial distribution and know how to identify the probability of success, p, and the probability of failure, (1 – p); know the notation X ~ B(n, p) for a random variable X modelled by the binomial distribution.

State the conditions to assume that a random variable has a binomial distribution.

A binomial distribution has n = 5 and p = 0.1. Plot a vertical line graph of the probability of success against the number of successes. Comment on the skewness of this distribution.

11.2 Know that the sum of all probabilities in a binomial distribution totals to 1. Calculate binomial probabilities and expected frequencies for different numbers of successes.

1 in 20 people are believed to be left-handed. What size sample is needed so that the expected number of left-handers in the sample is 3?

Over the years two football teams play each other six times. Calculate the probability that one team wins the toss 4 times. Calculate the probability that one team wins the toss at least 3 times.

11.3 Calculate the mean and variance of the binomial distribution as µ = np and σ2 = np(1 – p); use the mean and variance to model sample data expected to have a binomial distribution.

A random variable is X ~ B(n, p). Find the expected value of X and its variance. Calculate P(µ – σ < X < µ + σ).

[Extension example] Consider the random variable X ~ B(n, p). Let the random variable Yi (i = 1, 2, …, n) represent the number of successes on the ith trial. Find E(Yi) and Var(Yi). Use the fact that

1E( ) E( )=∑n

iX Y and that 1

Var( ) Var( )niX Y=∑ to show

that µ = np and σ2 = np(1 – p).

11.4 Understand the principle of a hypothesis test involving a null hypothesis or alternative hypothesis, and use the related vocabulary of significance level, one-tail or two-tail test, critical value, critical region, acceptance region.

Explain the meaning of a significance level to someone who knows no statistics.

11.5 Set up and perform a hypothesis test on a binomial probability distribution model, identifying the null hypothesis and the alternative hypothesis, and make correct inferences from the test.

A road safety team examines the tyres of a large number of commercial vehicles. They find that 20% of vans have defective tyres. Following a campaign to reduce the proportion of vehicles with defective tyres, 18 vans are stopped at random and their tyres are inspected. Just one of the vans has defective tyres. Carry out a suitable hypothesis test to examine whether the campaign appears to have been successful.

a. State your hypotheses clearly, justifying the form of the alternative hypothesis.

b. Carry out the test at the 5% significance level, stating your conclusions clearly.

c. State, with reason, the critical value for the test.

d. Give a level of significance such that you would come to the opposite conclusion for your test. Explain your reasoning.

MEI

The binomial distribution

The binomial distribution is the distribution in which the probability of r successes in n trials is −−n r n r

r p pC (1 )

for r = 0, 1, 2, …, n.

Mean and variance

No proofs of these results will be required.

Hypothesis testing

Use the notation H0 for the null hypothesis and H1 for the alternative hypothesis.


12 Understand and use the Poisson distribution to make inferences from data

12.1 Recognise that the Poisson distribution is used when single events in space or time occur independently of each other at a constant rate.

12.2 Use the probability distribution

eP( )!

r

X rr

λλ−

= =

to calculate values of P(X) when r = 0, 1, 2, 3, … (ad infinitum), and know that the sum to infinity of all these probabilities is 1; know that λ is the parameter of the distribution.

12.3 Know the notation X ~ P(λ) describes the Poisson distribution for a discrete random variable representing the number of events that occur at random in a certain interval of space or time, where λ is the mean number of events that occur in the interval.

If X ~ P(4), calculate P(X = 0), P(X = 1) and P(X ≤ 2).

The number of goals per game scored by football teams playing at home or away in a football competition are modelled by independent Poisson distributions with means 1.63 and 1.17 respectively.

a. Find the probability that in a game chosen at random: i. the home team scores at least 2 goals; ii. the result is a 1–1 draw; iii. the teams score 5 goals between them.

b. Give two reasons why the proposed model might not be suitable.

c. The number of goals scored per game by the Alpha team is modelled by the Poisson distribution with mean 1.63. In a season they play 19 home games. Use a suitable approximating distribution to find the probability that Alpha will score more than 35 goals in their home games.

MEI

12.4 Know that both the mean and variance of X ~ P(λ) are equal to λ.

12.5 Know without proof that the Poisson distribution X ~ P(λ) can be used as an approximation to the binomial distribution X ~ B(n, p) when n tends to infinity and the mean, np, is kept constant.

13 Understand and use the normal distribution to make inferences from data

13.1 Understand and describe the main features of a normal distribution for a continuous random variable.

Explain the main features of the normal distribution to someone who knows no statistics.

13.2 Use the notation X ~ N(µ, σ2) for a continuous random variable modelled by a normal distribution with mean µ and variance σ2.

13.3 Standardise a normally distributed continuous random variable.

13.4 Use statistical tables to read off probabilities for a standardised normal distribution; know that the total area under the standardised normal distribution curve is 1; know probabilities for obtaining a result 1, 2 or 3 standard deviation units either side of the mean.

Explain what you think the ‘68–95–99.7’ rule means in relation to the standard normal distribution.

The Poisson distribution

This is named after the French mathematician Simeon Denis Poisson (1781–1840).

Normal distribution

The normal distribution is the distribution of many naturally occurring variables, such as the heights of adult men in a city, the masses of carrots in a field, and so on.


13.5 Use the normal distribution as an approximation for the binomial distribution X ~ N(np, npq) ≈ X ~ B(n, p) when n is large and where q = 1 – p.

13.6 Use the normal distribution as an approximation for the Poisson distribution X ~ N(λ, λ) ≈ X ~ P(λ).

13.7 Know that if a population distribution is normal the sampling distribution of the mean is also normal; know that if the population distribution is not normal the sampling distribution of the mean is approximately normal for large samples; know the mean and variance of the sampling distribution of means in terms of the mean and variance (or estimated variance) of the population distribution.

13.8 Perform a hypothesis test on a population mean using the standardised normal distribution in situations where the population variance is known or where the population variance is unknown but the sample size is large.

Extralite are testing a new long-life light bulb. The lifetimes, in hours, are assumed to be normally distributed with mean µ and standard deviation σ. After extensive tests, they find that 19% of bulbs have a lifetime exceeding 5000 hours, while 5% have a lifetime under 4000 hours.

a. Illustrate this information on a sketch.

b. Show that σ =396 and find the value of µ.

In the remainder of this question take µ to be 4650 and σ to be 400.

c. Find the probability that a bulb chosen at random has a lifetime between 4250 and 4750 hours.

d. Find the probability that a bulb has a lifetime of over 4500 hours.

e. Extralite wish to quote a lifetime which will be exceeded by 99% of bulbs. What time, correct to the nearest 100 hours, should they quote?

f. A new school classroom has 6 light fittings, each fitted with an Extralite long-life bulb. Find the probability that no more than one bulb needs to be replaced within the first 4250 hours of use.

MEI

The lengths of metal rods used in an engineering structure is specified as being 40 cm. It does not matter if they are slightly longer, but they should not be any shorter. These rods are made by a machine in such a way that their lengths are normally distributed with standard deviation 0.2 cm. The mean, µ cm, of the lengths is set to a value slightly above 40 cm to give a margin for error.

To examine whether the specification is being met, a random sample of 12 rods is taken. Their lengths, in cm, are found to be 40.43 40.49 40.19 40.36 40.81 40.47 40.46 40.63 40.41 40.27 40.34 40.54

It is desired to test whether µ = 40.5.

a. State a suitable alternative hypothesis for the test.

b. Carry out the test at the 5% level of significance, stating your conclusions carefully.

MEI

13.9 Calculate the standard error for a population mean and give a confidence interval for the mean after applying the confidence test described in PS 13.8 above.

Standard error

If samples of size n are taken from a population with distribution N(µ, σ 2 ) then the distribution of the

sample mean is N(µ, σn

2

).

σn

is often called the

standard error of the mean.


14 Test for association in bivariate data

14.1 Understand the distinction between an independent variable and a dependent variable.

14.2 Draw a scatter diagram to suggest strength of relationship between two variables with interval or ratio scales of measurement and with each measured on the same subject; know that two variables have a positive association if larger values of one seem to link to larger values of the other, and a negative association if larger values of one seem to link with smaller values of the other.

14.3 Know and use the term linear correlation to indicate if scatter points on a scatter diagram are clustered around a straight line.

14.4 Calculate the product moment correlation coefficient from bivariate data, and know that the value lies between –1 and 1; understand the distinction between positive and negative correlation, and the special cases when r = –1, 0 or 1.

14.5 Understand that the value of r may be severely affected by outliers.

14.6 Test for evidence for a null hypothesis of no correlation using the calculated value of r from data and from tables of critical values.

A medical statistician wishes to carry out a hypothesis test to see if there is any correlation between the head circumference and body length of newly born babies.

a. State appropriate null and alternative hypotheses for the test.

A random sample of 20 newly born babies have had their head circumference, x cm, and body length, y cm, measured. This bivariate sample is illustrated below.

Summary statistics for this data set are as follows.

n = 20 Σ x = 691 Σ y = 1018 Σ x2 = 23 917 Σ y2 = 51 904 Σ xy = 35 212.5

b. Calculate the product-moment correlation coefficient for the data. Carry out the hypothesis test at the 1% significance level, stating the conclusion carefully. What assumption is necessary for the test to be valid?

Originally, the point x = 34, y = 51 had been recorded incorrectly as x = 51, y = 34.

c. Calculate the values of the summary statistics if this error had gone undetected. Use the uncorrected summary statistics to show that the value of the product-moment correlation coefficient would be negative.

d. How is it that this one error produces such a large change in the value of the correlation coefficient and also changes its sign?

MEI

14.7 Calculate the line of best fit for linear correlation using least squares regression of dependent variable y on independent variable x.

14.8 Calculate and use Spearman’s coefficient of rank correlation.

Variables

In some statistical texts an independent variable is called an explanatory variable and a dependent variable is called a response variable.

Product moment correlation coefficient

This is usually denoted by r.


14.9 Test for evidence of the null hypothesis no association using the rank correlation coefficient and from tables of critical values.

14.10 Appreciate the difference between association and causation.

15 Understand and apply the chi-squared test

15.1 Know that the chi-squared (χ2) test is used to test whether two or more population proportions are independent of each other and that this is done using observed frequencies from a sample and expected frequencies from a probability model to calculate the value of χ2 statistic.

15.2 Interpret the result of a χ2 test applied to a contingency table in which one basis for classification is across the columns and the other basis for classification is along the rows.

Data are extracted from medical records of a random sample of patients of a large clinic, showing for part of a particular year the frequencies of contracting or not contracting influenza for patients who had not had influenza inoculations.

Influenza

Yes No

Yes 8 18 Inoculated

No 35 17

State the null hypothesis for a suitable test of independence of inoculation and occurrences of influenza. Carry out the test at the 5% level of significance.

MEI

16 Simulation

16.1 Use coins, dice or random numbers to generate models of events described by random variables and to calculate probabilities and frequencies.

Do an investigation using random numbers to investigate the building up of a queue of vehicles at a set of traffic lights.

Scientists have invented a fictitious beetle, called the stochastic beetle, that reproduces in the following manner: • Different females reproduce independently. • 50% of the females have two offspring. • 30% of the females have one offspring. • The remaining females die out.

Use random numbers to simulate the growth of the population of stochastic beetles, stating any assumptions that are made in carrying out the simulation and stating clearly how the random numbers are used in the simulation. What conclusions can be made about the population of stochastic beetles? How do these conclusions change if you vary the percentages of the female beetles in the above categories?

You wish to find the least number of people in a gathering so that the probability that two of them have the same birthday (date of month only, not year of birth) exceeds 0.5. Plan and carry out a simulation to do this.

17 Use of ICT


Chi-squared test

This is denoted as the χ2 test.

Random numbers

These can be generated on scientific or graphics calculators using the RND and RAN function keys. Large-scale simulations are best done with computer software. Random number tables are also very useful.

ICT opportunity

A range of ICT applications can support data handling. Random numbers can be rapidly generated on a computer and programs developed to simulate particular situations. Secondary data sets are readily available on the Internet. Statistical calculations are rapidly carried out using statistical software packages or statistical functions on a calculator. Statistical charts and graphs can be drawn using appropriate software and graphic calculators.

305 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science © Supreme Education Council 2004




Students solve a wide range of problems in mathematical and other contexts. They use mathematics to model and predict outcomes of substantial real-world applications. They break problems into smaller tasks, and set up and perform relevant manipulations and calculations. They identify and use interconnections between mathematical topics. They develop and explain chains of logical reasoning, using correct mathematical notation and terms, including logic symbols. They generate mathematical proofs. They generalise when possible and remark on special cases. They approach problems systematically, knowing when and how to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information. They work to expected degrees of accuracy, and understand error bounds. They recognise when to use ICT efficiently and use it efficiently.


Students continue to develop skills of algebraic manipulation through further work on factorisation, exponents and logarithms, partial fractions, summation of series and combinatorics. They understand and use the remainder theorem and the factor theorem. They expand and use the binomial series (1 + x)n for any rational value of n. They work with a range of functions and their inverses, including polynomial functions up to order four; the reciprocal, exponential and logarithmic functions, and the modulus function. They plot and describe the features of the circular functions. They understand the more detailed behaviour of these functions through their awareness of the associated differential and integral calculus. They find higher order derivatives of functions, and work out approximations. They find indefinite and definite integrals and solve simple differential equations. They use these functions and the calculus to model a range of substantial real-world scenarios. They use realistic data and ICT to analyse problems.


Students are aware of links between geometry and algebra, which deepens their understanding of space and movement. They understand the roles that trigonometry and circular functions play in modelling and in mathematical transformation. They use trigonometric identities to solve trigonometric equations. They use vectors to extend the study of space and motion into three dimensions, and they are familiar with curves represented by parametric equations. They use dimensionally correct units for length, area and volume and for a range of measures, including velocity, acceleration and other compound measures. They find areas and volumes by integration and volumes of revolution. They use ICT to explore geometrical relationships.

Grade 12 Advanced

(mathematics for science)



Students apply and use the work on probability and statistics learned in earlier grades to solve problems.


The advanced mathematics standards for Grade 12 have two pathways: mathematics for science and quantitative mathematics, to support the social sciences and economics. Each pathway includes reasoning and problem solving, and number, algebra and calculus. The mathematics for science standards include substantial work on calculus but no new work on probability and statistics, whereas the quantitative methods standards include substantial work on probability and statistics, less calculus, and no new work on geometry and measures.

The reasoning and problem solving strand cuts across the other strands. Reasoning, generalisation and problem solving should be an integral part of the teaching and learning of mathematics in all lessons.

The weightings of the strands relative to each other are as follows:




statistics

Grade 10 55% 30% 15%

Grade 11 55% 30% 15%


40% – 60%


75% 25% –

The standards are numbered for easy reference. Those in shaded rectangles, e.g. 1.2, are the performance standards for all mathematics for science students. The national tests for advanced mathematics for science will be based on these standards. Grade 12 teachers should consolidate earlier standards as necessary.




By the end of Grade 12, students solve a wide range of problems in mathematical and other contexts. They use mathematics to model and predict outcomes of substantial real-world applications. They break problems into smaller tasks, and set up and perform relevant manipulations and calculations. They identify and use interconnections between mathematical topics. They develop and explain chains of logical reasoning, using correct mathematical notation and terms, including logic symbols. They generate mathematical proofs. They generalise when possible and remark on special cases. They approach problems systematically, knowing when and how to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information. They work to expected degrees of accuracy, and understand error bounds. They recognise when to use ICT efficiently and use it efficiently.

Students should:



1.2 Use mathematics to model and predict the outcomes of substantial real-world applications, and to compare and contrast two or more given models of a particular situation.



1.5 Use a range of strategies to solve problems, including working the problem backwards and redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation; and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

1.6 Develop chains of logical reasoning, using correct terminology and mathematical notation, including symbols for logical implication.

State whether the following statements are true or false.

x2 = 16 ⇒ x = 4

x2 ≤ 16 ⇒ –4 ≤ x ≤ 4

43

4 3≥ ⇒ ≤xx

Grade 12 Advanced

(mathematics for science)

Key standards


Cross-references

Standards are referred to using the notation RP for reasoning and problem solving, NAC for number, algebra and calculus, GM for geometry and measures, e.g. standard GM 2.3






The Al Huda sisters made these statements.

Inas: ‘If the rug is in the car, then it is not in the garage.’ Safa: ‘If the rug is not in the car, then it is in the garage.’ Roza: ‘If the rug is in the garage, it is in the car.’ Farida: ‘If the rug is not in the car, then it is not in the garage.’

If Roza told the truth, who else must have told the truth?

A. Inas B. Safa C. Farida D. None need have told the truth.

TIMSS Grade 12


1.8 Understand and generate mathematical proofs, and discuss exceptional cases, knowing the importance of a counter-example.


1.10 Approach complex problems systematically, recognising when it is important to enumerate all outcomes.




1.14 Identify error bounds on measurements.



By the end of Grade 12, students continue to develop skills of algebraic manipulation through further work on factorisation, exponents and logarithms, partial fractions, summation of series and combinatorics. They understand and use the remainder theorem and the factor theorem. They expand and use the binomial series (1 + x)n for any rational value of n. They work with a range of functions and their inverses, including polynomial functions up to order four; the reciprocal, exponential and logarithmic functions, and the modulus function. They plot and describe the features of the circular functions. They understand the more detailed behaviour of these functions through their awareness of the associated differential and integral calculus. They find higher order derivatives of functions, and work out approximations. They find indefinite and definite integrals and solve simple differential equations. They use these functions and the calculus to model a range of substantial real-world scenarios. They use realistic data and ICT to analyse problems.

Algebra and calculus

Students should know that algebra enables generalisation and the establishment of relationships between quantities and/or concepts. They should understand the nature and place of algebraic reasoning and proof, and how algebra may be related to geometric concepts, and vice versa. They should appreciate how calculus furthers the study of functions and of mathematical applications.


Students should:

2 Manipulate algebraic expressions

2.1 Multiply, factorise and simplify expressions and divide a polynomial by a linear or quadratic expression.

Write down in ascending powers of x the expansion of (2 – 3x)3.

Factorise 27x3 + 8y3. Use your factorisation to write an identity for 27x3 – 8y3.

Simplify 3

211

xx

+−

, given that x ≠ ±1.

Solve the equation (x + 3)(3x –1) – (x + 3)(5x + 2) = 0.

2.2 Combine and simplify rational algebraic fractions.

Simplify 25 3 2

1 1x

x x−−+ −

.

2.3 Decompose a rational algebraic fraction into partial fractions (with denominators not more complicated than repeated linear terms).

Show that 1 1 1( 1) 1r r r r≡ −+ + .

Find the values of A, B and C in the identity 2 21

3 1 1(3 1)( 1) ( 1)A B C

r rr r r≡ + ++ ++ + +

.

2.4 Understand and use the remainder theorem. 3x3 – 2x2 – ax – 28 has a remainder of 5 when divided by x – 3. What is the value of a?

2.5 Understand and use the factor theorem.

Show that 2x3 + x2 – 13x + 6 is divisible by x – 2 and find the other factors. Hence find the solution set in of the equation 2x3 + x2 – 13x + 6 = 0. What is the solution set in ?

3 Use index notation and logarithms to solve numerical problems

3.1 Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents; use the xy key and its inverse on a calculator.

Without using a calculator, simplify 5

28 2

4n n

n× .

Plot a graph of y = 1000 × 2–t for values of t from 0 to 1, increasing in steps of 0.2.

3.2 Know the definition of a logarithm in number base a (a > 0), and the rules of combination of logarithms, including change of base.

Give the value of log10 1000.

Evaluate log2 64.

Express log 27 – 2 log 3 as a single logarithm.

Prove that loglog logc

ac

bb a= and use this result to show that 1log logab

b a= .

Given logb 2 = 1/3, logb

32 is equal to

A. 2 B. 5 C. –3/5 D. 5/3 E. 2

3log 32

TIMSS Grade 12

Factorisation

Include the sum and difference of two cubes.

Logarithms

In addition to formulae for the sum and difference of two logarithms, include:

==== >

=

a

xa

x

a

an

a a

a xa x

aa

x n x

log

log

log 1log 1 0 for any 0log log


Explain why the number base of a logarithm must be positive, but why the logarithm itself may take any value.

3.3 Use the ln and log keys on a calculator and the corresponding inverse function keys.

Use a calculator to evaluate log5 4 correct to three decimal places.

In 1916, two scientists each named du Bois derived a formula to estimate the surface area S m2 of human beings in terms of their mass M kg and their height H cm. The formula is S = 0.000 718 4 × M0.425 × H0.725. A boy has a mass of 60 kg and a height of 160 cm. Calculate the boy’s surface area to three significant figures.

The intensity of sound, N, is measured in decibels (dB) and is defined by the formula N = 10 log (I / 10–16), where I is the power of sound measured in watts. Find N for normal speech with a power of 10–10 watts. Find the power of a jet aircraft with a sound intensity of 150 dB.

In chemistry, the pH value of a solution is defined as pH = – log [H], where [H] is the concentration of hydrogen ions in the solution. Solutions with pH < 7 are acidic and solutions with pH > 7 are alkaline. Solutions with pH = 7 are neutral. A solution has [H] = 5 × 10–7. Find its pH value and state whether it is acid, neutral or alkaline.


4.1 Find permutations and combinations.

An examination consists of 13 questions. A student must answer only one of the first two questions and only nine of the remaining ones. How many choices of question does the student have? A. 13C10 = 286 B. 11C8 = 165 C. 2 × 11C9 = 110 D. 2 × 11P2 = 220 E. some other number

TIMSS Grade 12

Evaluate 9C6 and 10P5.

Prove that nCr = nCn–r and that nCr + nCr+1 = n+1Cr+1.

A committee of 6 people is to be chosen from 6 men and 4 women. In how many ways can the committee be chosen to include 3 men and 3 women?

Show that nC0 + nC1 + nC2 + …+ nCn = 2n.

In how many ways can 5 thick books, 4 medium sized books and 3 thin books be arranged on a bookshelf so that the books of the same size remain together? A. 5! 4! 3! 3! = 103 680 B. 5! 4! 3! = 17 280 C. (5! 4! 3!) × 3 = 51 840 D. 5 × 4 × 3 × 3 = 180 E. 212 × 3 = 12 288

TIMSS Grade 12

Using exponentials

Further examples of modelling with exponential functions are given in the margin note at NAC 8.18.

Permutations

The order of selection distinguishes a permutation from a combination.

Students should know that nCr is the number of combinations of r different objects from n different objects and that the number of permutations of r different objects from n different objects is r! nCr, which is denoted by nPr.


4.2 Expand the binomial series (1 + x)n for any rational value of n.

Without using a calculator, find the value of (1.01)6 to four decimal places.

By writing 3.75 as 3(1 + 0.25), apply the binomial series to evaluate √3.75 correct to three decimal places.

Show that, for small values of x, 22

1 1 2 3(1 )

x xx

≈ − ++

.

5 Work with functions and their graphs

5.1 Use a graphics calculator to plot exponential functions of the form y = ekx; describe these functions, distinguishing between cases when k is positive or negative, and the special case when k is zero.

Investigate the behaviour of the tangent lines to these curves and observe that these follow a similar pattern of increase or decrease as the function itself.

5.2 Plot and describe the features of the natural logarithm function y = ln x; understand that the natural logarithm function is inverse to the exponential function (see NAC 5.5 below).

A radioactive element decomposes according to the formula y = y0e–kt, where y is the mass of the element remaining after t days and y0 is the value of y for t = 0. Find the value of the constant k for an element whose half-life (i.e. the time taken to decompose half of the material) is 4 days. A. 1

e4 log 2 B. 1e 2log C. 2log e D. 1

4e(log 2) E. 42e

TIMSS Grade 12

5.3 Understand the modulus function y = | x | and sketch its graph; sketch the modulus of the functions in NAC 5.2–5.5.

Sketch the graph of the function f(x) = | x – 2 | + | x | for –3 ≤ x ≤ 3.

Sketch the curve with equation y = | sin x |.

5.4 Form composite functions and use the notation y = g(f(x)).

The function f(x) = 4x – 7 is defined on . A second function g(x) = (x + 1)2 is also

defined on . Find f(g(x)) and g(f(x)). Comment on why these functions are not the

same.

5.5 Form inverse functions (on a restricted domain, if necessary) and use the notation 1f ( )y x−= .

Show that the function f ( ) 1xx x= − , where x ≠ 1, is its own inverse.

Find the inverse of the function f : 2t t + , defined for t ≥ 0.

5.6 Know that 1f (f ( ))x x− = and that the graph of 1f ( )y x−= is the reflection of the graph of f ( )y x= about the straight line y = x.

Show that lne x x= .

Sketch the graph of y x= for non-negative values of x.

5.7 Understand that some functions are continuous everywhere and that some are piecewise continuous.

Sketch the graph of y = 3 + (x + 2)–1. Show clearly the asymptotes of the graph and the intercepts with each axis.

The function f(n) = n is defined on +. Sketch the graph of this function.

Binomial theorem

The theorem is used to expand positive integer powers of binomial terms. It has applications in probability. The binomial series is commonly used to generate approximations.

Functions

Include use of the notation Use the notation

x xf : f( ) , as well as

y = f(x) or f(x) = …

Discontinuous functions

Some exceptional functions are discontinuous everywhere.


Comment on the difference between the possible domains of the functions defined implicitly by the equations y – y1 = m(x – x1) and m = (y – y1) / (x – x1). How does this difference affect the graphs of the two functions?

Rewrite the function f(x) = (x – 1) / (x + 1) in the form A + B / (x + 1) and find the values of A and B. Hence sketch the curve y = f(x). Show clearly the values of the intercepts on each axis and give the equations of each of its asymptotes.

The function f is defined on . Sketch the function f(x) = [ x ], where [ x ] means the

greatest integer less than or equal to x.

5.8 Understand that functions which repeat at regular intervals are called periodic functions, and that the smallest of these intervals is the period of the function.

A function on is defined by f(x) = 3x for 0 ≤ x < 3 with f(x) = f(x + 3). Sketch the graph

of this function and state its period.

A circular function is given as 135sin (2 )y θ π= + . State the amplitude of this function

and its periodicity. Sketch the graph of this function.

5.9 Understand that relations (one-to-many mappings) that represent looped curves are often described in terms of a parameter; consider simple examples of this type.

A curve is described by the parametric equations x = t2 and y = 2t. By eliminating t between these equations find the Cartesian equation of the curve. Sketch the curve.

Find ddyt and d

dxt . Verify that d d d

d d dy y xx t t= ÷ .

6 Solve equations associated with functions

6.1 Given a quadratic equation of the form ax2 + bx + c = 0, know that if the discriminant ∆ = b2 − 4ac is negative, there are two complex roots, which are conjugate to each other.

6.2 Solve exponential and logarithmic equations of the form ekx = A, where A is a positive constant, and ln kx = B, where B is constant.

6.3 Solve trigonometric equations of the form: sin (ax + b) = A, where –1 ≤ A ≤ 1; cos (ax + b) = A, where –1 ≤ A ≤ 1; tan (ax + b) = A, where A is constant; and find all the solutions in a stated interval.

6.4 Find approximate solutions for the intersection of any two functions from the intersection points of their graphs, and interpret this as the solution set of pairs of simultaneous equations.

An approximate solution of the equation x = ex can be found by plotting on the same axes the curves y = x and y = ex and finding the x-coordinate of their point of intersection.

7 Understand and use complex numbers

7.1 Understand that a complex number z = x + iy, where i2 = –1, consists of a real part x and an imaginary part y.

7.2 Know the rules for the addition, subtraction and multiplication of two complex numbers z1= x1 + iy1 and z2 = x2 + iy2; know that the complex conjugate of z is z* = x – iy and that zz* = x2 + y2; use this to divide one complex number by another.

Complex numbers

Complex numbers could be an extension topic for the most able students.


7.3 Represent a complex number as a point in the complex plane or as a vector in this plane; understand and use Argand diagrams to add or subtract two complex numbers; understand that addition or subtraction on an Argand diagram is analogous to the addition or subtraction of two component vectors.

7.4 Know the polar form of a complex number, using the modulus r and the argument θ, and the results that x = r cos θ and y = r sin θ and r2 = x2 + y2 = zz*.

7.5 Understand and use De Moivre’s theorem that z = r eiθ and zn = reinθ.

7.6 Perform multiplication and division of complex numbers using polar form, and apply this to representing multiplication and division of complex numbers on an Argand diagram.

7.7 Use de Moivre’s theorem to calculate roots of complex numbers.

7.8 Know that every polynomial of order n with real coefficients can be factorised into n linear factors in which complex number factors always occur in pairs that are complex conjugate to each other.

7.9 Use complex numbers to generate trigonometric identities.

7.10 Use complex numbers to investigate functions of a complex variable.


8.1 Know that the derivative of f ′(x) is called the second derivative of the

function y = f(x) and that this can also be written in the forms f′′(x) or 2

2

dd

yx

;

know that higher derivatives may be taken in the same way.

For the function f(x) = x3 – 6x, show that f′′(x) = 6x. What is the value of f′′(x) at the points where f′(x) is zero?

8.2 Interpret the numerical value of the derivative at a point on the curve of the function; know that:

• when the derivative is positive the function is increasing at the point;

• when the derivative is negative the function is decreasing at the point;

• when the derivative is zero the function is stationary at the point.

Discuss how knowledge of when a function increases, when it decreases and when it is stationary gives important information about the function as a whole and helps to analyse what it looks like.

Discuss the derivative function associated with the function f : | |x x .

Is there anywhere on this function where the derivative does not exist? Justify your answer.

8.3 Understand that stationary points of any function may correspond to a local maximum or minimum of the function, or may be a point of inflexion; understand how the derivative changes as the point at which the derivative is calculated moves through the local maximum or minimum, or through an inflexion; understand that not all points of inflexion are stationary points.

8.4 Understand and use the second derivative to test whether a stationary point is a local maximum, or a local minimum, or a point of inflexion.

Reading the second derivative

f′′(x) is read as ‘f-double-dash of x’ and d2y/dx2 is read as ‘dee-two-y by dee-x-squared’.

Higher derivatives

In general an nth order derivative is denoted by

n

ny

xdd

or f(n)(x).

Cusps

More able students might discuss examples of curves with cusp points, and show why a derivative cannot exist at the cusp point.

Maxima and minima

These are sometimes referred to as turning points.


Which of the following graphs has these features: f′(0) > 0, f′(1) < 0, and f′′(x) is always negative?

TIMSS Grade 12

8.5 Know that of all exponential functions, the exponential function y = ex is defined as the one in which the slope at the y-intercept point (0, 1) has the value 1.

8.6 Use the definition in NAC 8.5 and properties of exponents to find the derivative of y = ex from first principles.

8.7 Know that the function f(x) = ex is the only non-zero function in mathematics for which the derivative f′(x) = ex gives back the original function.

It can be shown that the infinite series 2 3 4

1 ...2! 3! 4!x x xx+ + + + + represents the number ex.

Show that if you differentiate the series term by term and add all these terms together, you get back to what you started with.

8.8 Know that the derivative of the natural logarithm function ln x is 1/x.

It can be shown that, for small values of z, 2 3 4

ln (1 ) ...2 3 4z z zz z+ = − + − + (where

−1 < z ≤ 1). Use this expansion and the properties of logarithms to calculate the derivative of f(x) = ln x from first principles.

Discuss, with examples, the process of logarithmic differentiation.

8.9 Know, without proof, the derivatives of the circular functions sin θ, cos θ and tan θ, and that the domain set for these functions must be in radians.

Derivative of combinations of functions

8.10 Understand that given any function f(x) = f1(x) + f2(x) then the derivative of this sum of two functions is f′(x) = f1′(x) + f2′(x).

Find the derivative of the function given by y = 5x + ex.

8.11 Understand that given a function f(x) = u(x) v(x) then the derivative of this product of two functions is given by f′ = uv′ + vu′; use this result in calculating the derivative of the product of two functions; know the special case of this result that if y = a f(x), where a is constant, then y′ = a f′.

Find ddyx when y = 10x ex.

8.12 Understand that given a function f(x) = u(x) / v(x) then the derivative of this quotient of two functions is given by f′ = (uv′ – vu′) / v2; use this result in calculating the derivative of the quotient of two functions.


8.13 Understand that given a composite function h(x) = g(f(x)) then the derivative of this composite function is given by h′(x) = g′(f(x))f′(x); use this result in calculating the derivative of the composite of two functions.

Use the chain rule to show that if y = 10 e–2t then 2d 20edty

x−= − .

Find the derivative with respect to x of y = (3x + 1)2 (2 – x).

Prove that d(ln ) 1d

xx x= .

Show that d 1d d

d

xy y

x

= .

Differentiate ln (3x2 + 1).

Take natural logarithms to find the derivative of y = ax.

Given f : sin5θ θ find df( )d

θθ when 6

πθ = .

Differentiate 2cos 2 2πθ +

with respect to θ.

8.14 Recognise that the derivative of f(x) = A ekx, where A and k are constants, is f′(x) = kA ekx.

8.15 Find the derivative of a function defined implicitly.

Find ddyx for the implicit function x2 + y2 = 25 for y ≥ 0.

Applications using derivatives

8.16 Use the first and second derivatives of functions to analyse the behaviour of functions and to sketch curves.

Sketch the curve 1( 2)y x x= − , showing clearly its turning points and asymptotes.

Sketch the curve y = xe–x for x ≥ 0, showing clearly its turning points and any points of inflexion.

8.17 Use the derivative to explore a range of optimisation problems in which a function is maximised or minimised.

8.18 Analyse a range of problems using exponential functions.

8.19 Analyse a range of problems involving periodicity or oscillation using circular functions.

8.20 Use polynomial and other functions to model a range of phenomena, including some relating to mechanics and motion, knowing that the derivative of distance with respect to time is a speed (or velocity) and that the derivative of speed (or velocity) with respect to time is acceleration.

9 Perform numerical approximation

9.1 Understand the error bounds on measurements recorded to a given number of significant figures.

9.2 Understand and use the tangent line approximation of f(x) near x = a in the form f(x) ≈ f(a) + f′(a) (x – a) and in the special case near the origin when a = 0.

Composite functions

An alternative and equally acceptable way of writing

this is = ×y y zx z x

d d dd d d ,

where the composite function is formed by first mapping x to z and then mapping z to y. This rule is often called the chain rule because it extends for composite functions formed in more than two stages.

Exponential functions

Students should explore exponential growth or decay, through a range of problems such as population growth, interest on loans, radio-carbon dating, cooling, half-life of radioactive elements, and the absorption of a medical drug into the body. See also NAC 11.2.


9.3 Know the approximations sin θ ≈ θ and cos θ ≈ 2121 θ− for small values of

θ in radians.

Find 0

sinlimθ

θθ→

.

9.4 Understand and use the Taylor series expansion 2 3 ( )f (0) f (0) f (0)f ( ) f (0) f (0) ... ...

2! 3! !

n nx x xx xn

′′ ′′′′≈ + + + + + +

to approximate functions and numerical values.

9.5 Perform simple iterations to find roots of equations, including xn+1 = f(xn)

and the Newton–Raphson iteration 1f ( )f ( )

nn n

n

xx x

x+ = − ′ , where f′(xn) ≠ 0.

10 Reconstruct a function from its derivative

The indefinite integral

10.1 Understand integration as the inverse process to differentiation; use the notation for indefinite integrals, knowing that f ( )d f ( )x x x c′ = +∫ , where c

is any constant, and that there is a whole family of curves y = f(x) + c, each member of which has derivative function f′(x).

Discuss how the individual members of the family of curves represented by y = f(x) + c are related to each other.

The graph of the function g passes through the point (1, 2). The slope of the tangent to the graph at any point (x, y) is given by g′(x) = 6x – 12. What is g(x)? Show all your work.

TIMSS Grade 12

10.2 Know the integrals of the functions: xn, where n ≠ –1 1/x, with x ≠ 0 ekx sin kx , cos kx and sec2 kx, where k is constant; write the integrals of multiples of these functions and of linear combinations of these functions.

The definite integral

10.3 Use the definition of the definite integral: f ( )d f ( ) f ( )

b

ax x b a′ = −∫ , where f(x) is a function of x and a ≤ x ≤ b;

interpret this as ‘the integral of a rate of change of a function is the total change of that function’; understand the effect of interchanging the limits of integration; know that .

b c b

a a c= +∫ ∫ ∫

Evaluate 3

6

cos dx xπ

π∫ .

10.4 Use summation of areas of rectangles to calculate lower and upper bounds for the area between the x-axis and a curve y = f(x) with y > 0, bounded on either side by lines x = constant; understand that as the width δx of each of the rectangles tends to zero the sums Σ f(x) δx for the lower and upper bounds of the area under the curve tend to the same value, and that this value is called the area under the curve.

Integration

The word integration comes from integrating, i.e. adding, the contributions of many small parts. The symbol

∫ x...d denotes summation

with respect to x. In the

definite integral ∫b

a, a and b

are called the limits of the integral, or the limits of integration.


10.5 Understand that the area bounded by a positive function y = f(x), the x-axis and the lines x = a and x = b, with a ≤ x ≤ b, is the definite integral

f ( )db

ax x∫ .

The line l in the figure is the graph of y = f(x). 3

2f ( )dx x

−∫ is equal to

A. 3 B. 4 C. 4.5 D. 5 E. 5.5

TIMSS Grade 12

10.6 Use the trapezium rule to find an approximation to the area represented by

the definite integral of a particular function when it is not easy or possible to integrate the function.

10.7 Understand that if a curve y = f(x) lies entirely below the x-axis, so that its

y-value is always negative, then the definite integral f ( )db

ax x∫ over the

interval a ≤ x ≤ b has a negative value.

Calculate the area between the curve y = x2 + 5, the x-axis and the lines x = –2 and x = 3.

Find the area between the curves y = x2 – 4 and y = 4 – x2.

Find the area between the curves y = x3 and y = x.

This figure shows the graph of y = f(x). S1 is the area enclosed by the x-axis, x = a and y = f(x); S2 is the area enclosed by the x-axis, x = b and y = f(x); where a < b and 0 < S2 < S1.

The value of f ( )db

ax x∫ is

A. S1 + S2

B. S1 – S2 C. S2 – S1 D. | S1 – S2

| E. ( )1

1 22 S S+

TIMSS Grade 12

10.8 Interpret and use an integral of velocity with respect to time as distance travelled, and an integral of acceleration with respect to time as velocity.

10.9 Solve other physical problems in which the integral of the rate of change of a physical quantity has to be interpreted as a total change in that quantity.

10.10 Use the integration by parts formula uv d uv vu d′ ′= −∫ ∫x x

and understand that it reverses the derivative of the product of two functions.

Find e dxx x∫ .

Find cos dx x x∫ .

Areas under curves

Students should work with ‘area-so-far’ for the area under a curve y = f(x), using definite integrals as in NAC 10.5, or the trapezium rule as in NAC 10.6.

Physical integrals

Include integrating force with respect to distance and integrating momentum with respect to velocity.


10.11 Understand that g (f ( )) f ( )d g(f ( ))x x x x c′ ′ = +∫ reverses the derivative of a

composite function; recognise ‘simple’ functions for which this formula can be instantly applied.

Explain why 222 d ln ( 1)

1x x x c

x= + +

+∫ .

10.12 Use the terminology that if z is a function of x then the derivative of z with

respect to x is dd

zx

and the differential of z is the symbolic expression

dd dd

zz xx

= ; understand that z and its differential can be used to replace the

variable of integration in an integral.

10.13 Perform simple cases of integration by substitution to undo the ‘chain rule’; perform integration with a given substitution.

Evaluate 21

0e dxx x∫ using the substitution w = x2.

Use the substitution z = 2 – 3x to evaluate ( 4) 2 3 dx x x+ −∫ .

10.14 Use partial fractions to integrate.

Find 1 d( 1)( 2) xx x+ +∫ .

10.15 Analyse simple instances of convergent definite integrals in which the upper limit tends to infinity.

Find 30

e dx x∞

−∫ . [Hint: replace the upper limit by b and let b tend to infinity.]

11 Solve simple differential equations

11.1 Recognise when an equation is a differential equation, and how such an equation can be formed; solve a differential equation that can be solved by separation of variables.

Find the general solution of the equation ddy kyx = , where k is a constant.

Show that the solution of the differential equation ddy xx y= − is the family of circles

x2 + y2 = c, where c is a positive constant.

11.2 Solve a range of physical problems involving simple differential equations for exponential growth and decay.

11.3 Know that the differential equation 2

2

dd

y kyx

= − , where k > 0, represents

simple harmonic motion (SHM) and that the solution of this equation has the form y = A sin x + B cos x; investigate some common cases of SHM.

Show that the function y = A sin x + B cos x satisfies the differential equation 2

2dd

y kyx

= − .

Exponential models

See the note at NAC 8.18.



By the end of Grade 12, students are aware of links between geometry and algebra, which deepens their understanding of space and movement. They understand the roles that trigonometry and circular functions play in modelling and in mathematical transformation. They use trigonometric identities to solve trigonometric equations. They use vectors to extend the study of space and motion into three dimensions, and they are familiar with curves represented by parametric equations. They use dimensionally correct units for length, area and volume and for a range of measures, including velocity, acceleration and other compound measures. They find areas and volumes by integration and volumes of revolution. They use ICT to explore geometrical relationships.

Students should:

12 Extend their understanding of circular functions

Sum or difference of two angles

12.1 Know, but not prove, identities for: sin (A + B); sin (A – B); cos (A + B); cos (A – B); tan (A + B); tan (A – B).

Show that sin 2A = 2 sin A cos A.

Find an exact expression for 512sin π .

By writing 7 sin θ + 5 cos θ in the form R sin (θ + α) find R and α, and hence the greatest value of the expression.

Show that cos 3A = 4 cos3 A – 3 cos A.

12.2 Know corresponding identities for double or half angles.

Sum or difference of two sines or cosines

12.3 Use the relevant identities from GM 12.1 to find the ‘sum–product’ identity

sin sin 2sin cos2 2

X Y X YX Y + −+ ≡ ; and corresponding identities for

sin X – sin Y; cos X + cos Y; cos X – cos Y.

Solution of trigonometric equations

12.4 Use trigonometric identities to solve trigonometric equations over specified angle domains.

Solve the equation cos 2θ + 3 sin θ = 2 for 0 ≤ θ ≤ 2π.

Solve the equation sin 2x = cos x for 0 ≤ x ≤ 2π.


Students should appreciate the importance and range of geometrical applications in the real world. They should understand the nature and place of geometric reasoning and proof, and how geometry may be related to algebraic concepts, and vice versa. They should know how a dynamic geometry system, or DGS, can be used to investigate results.


13 Use vectors to study position, displacement and motion

13.1 Use vectors in up to three dimensions; identify the components of the vector in relation to three orthogonal directions; use unit vectors i, j and k in these directions; use column matrix form for vectors, including unit vectors; use the notation AB to denote the vector from point A to point B; use and understand the terms position vector and displacement vector.

13.2 Know the rules for the addition and subtraction of two vectors; represent addition and subtraction of two vectors diagrammatically; know that there exists a null vector 0 such that a – a = 0 for any vector; know that vector addition is commutative and associative.

13.3 Find the magnitude | a | of any vector a and the direction of a in relation to specified axes.

13.4 Know the distinction between a vector and a scalar; know that any vector can be multiplied by a positive scalar to rescale it, or by a negative scalar to rescale it and reverse its direction.

13.5 Know the notation a.b for the scalar product of two vectors a and b; form and calculate the scalar product, and interpret the scalar product in terms of the magnitudes of the two vectors and the angle between them; know that a.a is the square of the magnitude of a.

Find the angle between the two vectors a = 3i + j – 2k and b = 2i – 5j – k.

Show that | a + b |2 = a2 + b2 + 2a.b and use this result to prove the cosine rule.

13.6 Know that if a and b are two non-zero vectors and a.b = 0 then a and b are perpendicular to each other.

Prove that the diagonals of any rhombus are perpendicular to each other.

13.7 Find the mid-point of a line segment AB given the position vectors of A and B.

13.8 Find the vector equation of a straight line in the form r = a + λb, where r is the position vector of any point on the line, a is the position vector of a given point on the line, b is a vector in the direction of the line and λ is a variable scalar.

13.9 Use vectors to represent velocity and know that speed is the magnitude of velocity; use vectors to represent acceleration, force and momentum.

13.10 Solve dynamical problems by differentiating or integrating vectors that are functions of position, or time, or velocity.


14.1 Find areas and volumes by integration; find volumes of revolution.

14.2 Solve problems using a range of compound measures using appropriate units and dimensions: for example, density (mass per unit volume), pressure (force per unit area) and power (energy per unit time).

15 Use ICT to explore geometric relationships

15.1 Use ICT to explore geometric relationships.

Variable scalars

These are called parameters.

Compound measures

Reinforce links with physics, using compound measures such as pressure, power, velocity and acceleration.

4 Appendix

323 | Qatar mathematics standards | Appendix © Supreme Education Council 2004

Sources used for international comparisons for mathematics 1 The PISA 2003 assessment framework – mathematics, reading, science and

problem solving knowledge and skills, Programme for International Student Assessment (PISA)

2 Mathematics assessment framework, TIMSS 2003, IEA 3 TIMSS-R mathematics items: released set for population 2 (seventh and

eighth grades), IEA 4 TIMSS mathematics items: released set for population 2 (third and fourth

grades), IEA 5 TIMSS example mathematics items 2003, Grade 4 and Grade 8, IEA 6 National contexts for mathematics and science education, Robitaille, 1997,

British Columbia 7 International review of curriculum and assessment frameworks: comparative

tables and factual summaries, Metais and Tabberer, NFER, 1997 8 Mathematics in the school curriculum: an international perspective,

Ruddock, NFER, 1998 9 Programmes and tests for mathematics, International Baccalaureate

Organisation (IBO) 10 A review of New Zealand’s school curriculum, Donnelly, 2002 11 Standards for mathematics, British Columbia Department of Education 12 Mathematics framework for Californian public schools, California

Department of Education, 2000 13 Mathematics curriculum guide P1 to P6, 2000, Secondary mathematics

syllabus S1 to S5, 2001, Hong Kong Ministry of Education 14 Mathematics program in Japan, Japan Society of Mathematical Education,

2000 15 Primary and secondary mathematics syllabus, 2000, Singapore Ministry of

Education 16 Western Australia outcomes and standards framework for mathematics,

2000, Department of Education and Training, Government of Western Australia

17 Curriculum and standards framework for mathematics, Victorian Curriculum and Assessment Authority, 2002

18 National curriculum for mathematics for maintained schools in England, 2000, Department for Education and Skills, UK

19 Framework for teaching mathematics from Reception to Year 6, 1999, and Framework for teaching mathematics: Years 7, 8 and 9, 2000, Department for Education and Skills, London, England

20 A comparative study of algebra curricula, Sutherland, Qualifications and Curriculum Authority, London, 2002

21 A comparative study of geometry curricula, Hoyles et al., Qualifications and Curriculum Authority, London, 2002

22 Matrix of range and sequence of the contents of mathematics curricula in the State of Qatar, Ministry of Education, Qatar

curriculum standards for the state of qatar - international · pdf file ·...

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