mathematics syllabus shs 1-3

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R E P U B L I C O F G H A N A

MI N I S T R Y O F E D U C A T I O N , S C I E N C E A N D S P O R T S

TEACHING SYLLABUS FOR MATHEMATICS

( SENIOR HIGH SCHOOL 1 – 3 )

Enquiries and comments on this syllabus should be addressed to:

The DirectorCurriculum Research and Development Division (CRDD)

P. O. Box GP 2739,Accra.Ghana.

September, 2007

Republic of Ghana



ii

TEACHING SYLLABUS FOR MATHEMATICS(SENIOR HIGH SCHOOL)

RATIONALE FOR TEACHING MATHEMATICS

Development in almost all areas of life is based on effective knowledge of science and mathematics. There simply cannot be any meaningful

development in virtually any area of life without knowledge of science and mathematics. It is for this reason that the education systems of countriesthat are concerned about their development put great deal of emphases on the study of mathematics. The main rationale for the mathematics syllabus

is focused on attaining one crucial goal: to enable all Ghanaian young persons to acquire the mathematical skills, insights, attitudes and values that

they will need to be successful in their chosen careers and daily lives. The new syllabus is based on the premises that all students can learn

mathematics and that all need to learn mathematics. The syllabus is therefore, designed to meet expected standards of mathematics in many parts of the world. Mathematics at the Senior High school (SHS) in Ghana builds on the knowledge and competencies developed at the Junior High School

level. The student is expected at the SHS level to develop the required mathematical competence to be able to use his/her knowledge in solving real

life problems and secondly, be well equipped to enter into further study and associated vocations in mathematics, science, commerce, industry and a

variety of other professions.

GENERAL AIMS

To meet the demands expressed in the rationale, the SHS Core Mathematics syllabus is designed to help the student to:

1. Develop the skills of selecting and applying criteria for classification and generalization.

2. Communicate effectively using mathematical terms, symbols and explanations through logical reasoning.

3. Use mathematics in daily life by recognizing and applying appropriate mathematical problem-solving strategies.

4. Understand the process of measurement and use appropriate measuring instruments.

5. Develop the ability and willingness to perform investigations using various mathematical ideas and operations.

6. Work co-operatively with other students in carrying out activities and projects in mathematics.

7. Develop the values and personal qualities of diligence, perseverance, confidence, patriotism and tolerance through the study of

mathematics8. Use the calculator and the computer for problem solving and investigations of real life situations

9. Develop interest in studying mathematics to a higher level in preparation for professions and careers in science, technology,

commerce, industry and a variety of work areas.10. Appreciate the connection among ideas within the subject itself and in other disciplines, especially Science, Technology, Economics

and Commerce



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GENERAL OBJECTIVES

By the end of the instructional period students will be able to:

1. Develop computational skills by using suitable methods to perform calculations;

2. Recall, apply and interpret mathematical knowledge in the context of everyday situations;3. Develop the ability to translate word problems (story problems) into mathematical language and solve them with related mathematical

knowledge;4. Organize, interpret and present information accurately in written, graphical and diagrammatic forms;

5. Use mathematical and other instruments to measure and construct figures to an acceptable degree of accuracy;6. Develop precise, logical and abstract thinking;

7. Analyze a problem, select a suitable strategy and apply an appropriate technique to obtain it’s solution;

8. Estimate, approximate and work to degrees of accuracy appropriate to the context;

9. Organize and use spatial relationships in two or three dimensions, particularly in solving problems;

10. Respond orally to questions about mathematics, discuss mathematics ideas and carry out mental computations;

11. Carry out practical and investigational works and undertake extended pieces of work;

12. Use the calculator to enhance understanding of numerical computation and solve real life problems

GENERAL OBJECTIVES

The student will:

13. Work co-operatively with other students and develop interest in mathematics.

14. Read and write numbers.

15. Use appropriate strategies to perform number operations.16. Recognize and use patterns, relationships and sequences and make generalizations.

17. Recognize and use functions, formula, equations and inequalities.18. Identify and use arbitrary and standard units of measure.

19. Make and use graphical representations of equations and inequalities.20. Use the appropriate unit to estimate and measure various quantities.

21. Relate solids and plane shapes and appreciate them in the environment.22. Collect, analyze and interpret data and find probability of events.

23. Use the calculator to enhance understanding of numerical computation and solve real life problems.24. Manipulate learning material to enhance understanding of concepts and skills.



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SCOPE OF SYLLABUS

This syllabus is based on the notion that an appropriate mathematics curriculum results from a series of critical decisions about three

inseparable linked components: content, instruction and assessment. Consequently, the syllabus is designed to put great deal of emphases onthe development and use of basic mathematical knowledge and skills. The major areas of content covered in all the Senior High School classes

are as follows:

1. Numbers and Numeration.2. Plane Geometry

3. Mensuration4. Algebra

5. Statistics and Probability

6. Trigonometry

7. Vectors and Transformation in a Plane

* Problem solving and application (mathematical processes).

“Numbers and Numeration” covers reading and writing numerals in base two through twelve and the four basic operations on them as well as

ratio, proportion, and parentages. Fractions, integers and rational and irrational numbers and four operations on them are treated extensively.

Plane geometry covers angles of a polygon, Pythagoras’ theorem and its application and circle theorem including tangents. Mensuration covers perimeters and areas of plane shapes, surface areas and volumes of solid shapes. In addition, the earth as a sphere is also treated under

mensuration. “Algebra” – Algebra is a symbolic language used to express mathematical relationships. Students need to understand how

quantities are related to one another, and how algebra can be used to concisely express and analyze those relationships. “Statistics and

Probability” – are important interrelated areas of mathematics. Each provides students with powerful mathematical perspectives on everyday

phenomena and with important examples of how mathematics is used in the modern world. Statistics and probability should involve students in

collecting, organizing, representing and interpreting data gathered from various sources, as well as understanding the fundamental concepts of

probability so that they can apply them in everyday life. Trigonometry covers the trigonometry ratios and their applications to angles of elevation and depression. Drawing and interpretation of graphs of trigonometric functions is also covered under trigonometry.

Topics treated under vectors include, representation, operations on vectors, equal and parallel vectors as well as magnitude of vectors and bearing. Transformation deals with rigid motion and enlargement including scale drawing and its application.

“Problem solving and application” has not been made a topic by itself in the syllabus since nearly all topics include solving word problems as

activities. It is hoped that teachers and textbook developers will incorporate appropriate problems that will require mathematical thinking rather than mere recall and use of standard algorithms. Other aspects of the syllabus should provide opportunity for the students to work co-

operatively in small groups to carry out activities and projects which may require out-of-school time. The level of difficulty of the content of the syllabus has been designed to be within the knowledge and ability range of Senior High School students.



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STRUCTURE AND ORGANIZATION OF THE SYLLABUS

The syllabus is structured to cover the four years of Senior High School. Each year’s work has been divided into units. SHS 1 has 13 units; S HS 2has 10 units, SHS 3 has 6 units while SHS 4 has 3 units of work. The unit topics for each year have been arranged in a suggested teaching sequence.

No attempt has been made to break the year’s work into terms. This is deliberate because it is difficult to predict, with any degree of certainty, therate of progress of students in each year. Moreover, the syllabus developers wish to discourage teachers from forcing the instructional pace but

would rather advise teachers to ensure that students progressively acquire a good understanding and application of the material specified for eachyear’s class work. It is hoped that no topics will be glossed over for lack of time because it is not desirable to create gaps in students’ knowledge.

The unit topics for the four years' course are indicated on the table below.

SHS 1 2 3 4

UNITS1

2

3

4

5

6

7

8

9

10

11

12

13

Sets and Operations on set

Real number system

Algebraic expressions

Number Bases

Plane Geometry

Linear equations and inequalities

Relations and Functions

Vectors in a plane

Simultaneous linear equation

Rigid motion I

Statistics I

Ratio and Rates

Percentages I

Modular arithmetic

Indices and logarithms

Surds

Percentages II

Variation

Quadratic functions and equations

Plane geometry II (Circles)

Mensuration I

Trigonometry I

Probability

-

-

-

Sequences and Series

Bearings

Constructions

Statistics II

Interpretation of linear and quadratic

graphs

Mensuration II

-

-

-

-

-

-

-

Logical reasoning

Percentage III

Rigid motion IIand Enlargement

Trigonometry II

-

-

-

-

-

-

-

-



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TIME ALLOCATION

Mathematics is allocated eight periods a week, each period consisting of thirty (40) minutes. The eight periods should be divided into four double

periods, each of one-hour twenty minutes duration for each day of the week. Suggested time allocation for Core Mathematics and other subjects/items for the four years are as follows:

Year 1 Year 2 Year 3 Year 4

Ø Core Mathematics 8 6 4 3

Ø Physical Education 3 2 2 2

Ø Library Work (Reading and Research) 3 2 2 2

Ø SBA Project 3 2 2 2

Ø Worship 2 2 2 2

Ø Free Period 1 1 1 2

SUGGESTIONS FOR TEACHING THE SYLLABUS

General Objectives

General Objectives for this syllabus have been listed on page iii of the syllabus. The general objectives flow from the general aims of mathematics

teaching listed on the first page of this syllabus. The general objectives form the basis for the selection and organization of the units and their topics.

Read the general objectives very carefully before you start teaching. After teaching all the units for the year, go back and read the general aims and

general objectives again to be sure you have covered both of them adequately in the course of your teaching.

Years and Units

The syllabus has been planned on the basis of Years and Units. Each year's work is covered in a number of units that have been sequentially

arranged to meet the teaching and learning needs of teachers and students.

Syllabus Structure

The syllabus is structured in five columns: Units, Specific Objectives, Content, Teaching and Learning Activities and Evaluation. A description of the contents of each column is as follows:



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Column 1 - Units: The units in Column 1 are the major topics for the year. The numbering of the units is different in mathematics from thenumbering adopted in other syllabuses. The unit numbers consist of two digits. The first digit shows the year or class while the second digit shows

the sequential number of the unit. A unit number like 1.2 is interpreted as unit 2 of SHS1. Similarly, a unit number like 3.5 means unit 5 of SHS3.The order in which the units are arranged is to guide you plan your work. However, if you find at some point that teaching and learning in your class

will be more effective if you branched to another unit before coming back to the unit in the sequence, you are encouraged to do so.

Column 2 - Specific Objectives: Column 2 shows the Specific Objectives for each unit. The specific objectives begin with numbers such as1.2.5 or 3.4.1. These numbers are referred to as "Syllabus Reference Numbers". The first digit in the syllabus reference number refers to the

year/class; the second digit refers to the unit, while the third refer to the rank order of the specific objective. For instance 1.2.5 means Year 1 or SHS1, Unit 2 (of SHS1) and Specific Objective 5. In other words 1.2.5 refers to Specific Objective 5 of Unit 2 of SHS1. Using syllabus reference

numbers provides an easy way for communication among teachers and other educators. It further provides an easy way for selecting objectives for test construction. For instance, Unit 4 of SHS3 may have three specific objectives 3.4.1 - 3.4.3. A teacher may want to base his/her test

items/questions on objectives 3.4.2 and 3.4.3 and not use the other first objective. The teacher would hence be able to use the syllabus reference

numbers to sample objectives within units and within the year to be able to develop a test that accurately reflects the importance of the various skills

taught in class.

You will note also that specific objectives have been stated in terms of the students i.e. what the students will be able to do during and after

instruction and learning in the unit. Each specific objective hence starts with the following "The student will be able to", this in effect, means that

you have to address the learning problems of each individual student. It means individualizing your instruction as much as possible such that the

majority of students will be able to master the objectives of each unit of the syllabus.

Column 3 - Content: The "content" in the third column of the syllabus shows the mathematical concepts, and operations required in the teaching

of the specific objectives. In some cases, the content presented is quite exhaustive. In some other cases, you could add some more information

based upon your own training and based also on current knowledge and information.

Column 4 - Teaching/Learning Activities (T/LA): T/LA activities that will ensure maximum student participation in the lessons are presented inColumn 4. The General Aims of the subject can only be most effectively achieved when teachers create learning situations and provide guidedopportunities for students to acquire as much knowledge and understanding of mathematics as possible through their own activities. Students’

questions are as important as teacher's questions. There are times when the teacher must show, demonstrate, and explain. But the major part of astudents’ learning experience should consist of opportunities to explore various mathematical situations in their environment to enable them make

their own observations and discoveries and record them. Avoid rote learning and drill-oriented methods and rather emphasize participatory teachingand learning in your lessons. You are encouraged to re-order the suggested teaching/learning activities and also add to them where necessary in order

to achieve optimum students learning. Emphasize the cognitive, affective and psychomotor domains of knowledge in your instructional systemwherever appropriate.

A suggestion that will help your students acquire the capacity for analytical thinking and the capacity for applying their knowledge to problems and

issues is to begin each lesson with a practical and interesting problem. Select a practical mathematical problem for each lesson. The selection must be made such that students can use knowledge gained in the previous lesson and other types of information not specifically taught in class.



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Column 5 - Evaluation: Suggestions and exercises for evaluating the lessons of each unit are indicated in Column 5. Evaluation exercises can be

in the form of oral questions, quizzes, class assignments, essays, project work, etc. Try to ask questions and set tasks and assignments, etc. that willchallenge students to apply their knowledge to issues and problems as we have already said above, and that will engage them in developing solutions,

and in developing observational and investigative skills as a result of having undergone instruction in this subject. The suggested evaluation tasks arenot exhaustive. You are encouraged to develop other creative evaluation tasks to ensure that students have mastered the instruction and behaviours

implied in the specific objectives of each unit.

Lastly, bear in mind that the syllabus cannot be taken as a substitute for lesson plans. It is necessary that you develop a scheme of work and lesson plans for teaching the units of this syllabus.

DEFINITION OF PROFILE DIMENSIONS

The concept of profile dimensions was made central to the syllabuses developed from 1998 onwards. A 'dimension' is a psychological unit for

describing a particular learning behaviour. More than one dimension constitutes a profile of dimensions. A specific objective may be stated with an

action verb as follows: The student will be able to describe….. etc. Being able to "describe" something after the instruction has been completed

means that the student has acquired "knowledge". Being able to explain, summarize, give examples, etc. means that the students has understood the

lesson taught.

Similarly, being able to develop, plan, solve problems, construct, etc. means that the students can "apply" the knowledge acquired in some new

context. Each of the specific objectives in this syllabus contains an "action verb" that describes the behaviour the students will be able to demonstrate

after the instruction. "Knowledge", "Application", etc. are dimensions that should be the prime focus of teaching and learning in schools. It has been

realized unfortunately that schools still teach the low ability thinking skills of knowledge and understanding and ignore the higher ability thinking

skills. Instruction in most cases has tended to stress knowledge acquisition to the detriment of the higher ability behaviours such as application,

analysis, etc. The persistence of this situation in the school system means that students will only do well on recall items and questions and perform

poorly on questions that require higher ability thinking skills such as application of mathematical principles and problem solving. For there to be anychange in the quality of people who go through the school system, students should be encouraged to apply their knowledge, develop analytical

thinking skills, develop plans, generate new and creative ideas and solutions, and use their knowledge in a variety of ways to solve mathematical problems while stil l in school. Each action verb indicates the underlying profile dimension of each particular specific objective. Read each objective

carefully to know the profile dimension toward which you have to teach.

In Mathematics, the two profile dimensions that have been specified for teaching, learning and testing at the SHS level are:

Knowledge and Understanding 30% Application of knowledge 70%



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Each of the dimensions has been given a percentage weight that should be reflected in teaching, learning and testing. The weights indicated on theright of the dimensions, show the relative emphasis that the teacher should give in the teaching, learning and testing processes at Senior High School.

Explanation and key words involved in each of the profile dimensions are as follows:

Knowledge and Understanding (KU)

Knowledge the ability to, read, count, identify, define, describe, list, name, locate, match, state principles, facts and concepts. Knowledge

is simply the ability to remember or recall material already learned and constitutes the lowest level of learning.

Understanding the ability to explain, distinguish, factorize, calculate, expand, measure, predict, give examples, generalize, estimate or predictconsequences based upon a trend. Understanding is generally the ability to grasp the meaning of some material that may be

verbal, pictorial, or symbolic.

Application of Knowledge(AK)

The ability to use knowledge or apply knowledge, as implied in this syllabus, has a number of learning/behaviour levels. These levels

include application, analysis, synthesis, and evaluation. These may be considered and taught separately paying attention to reflect each of them

equally in your teaching. The dimension "Application of Knowledge" is a summary dimension for all four learning levels.

Details of each of the four sub-levels are as follows:

Application the ability to apply rules, methods, principles, theories, etc. to concrete situations that are new and unfamiliar. It also involves

the ability to produce, solve, plan, demonstrate, discover, etc.

Analysis the ability to break down material into its component parts; to differentiate, compare, distinguish, outline, separate, identify

significant points, etc.; ability to recognize unstated assumptions and logical fallacies; ability to recognize inferences fromfacts, etc.

Synthesis the ability to put parts together to form a new whole. It involves the ability to combine, compile, compose, devise, plan,

revise, design, organize, create, generate new ideas and solutions, etc.

Evaluation the ability to appraise, compare features of different things and make comments or judgments, compare, contrast criticize, justify, support, discuss, conclude, make recommendations, etc. Evaluation refers to the ability to judge the worth or value of

some material based on some criteria.



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FORM OF ASSESSMENT

It is important that both instruction and assessment be based on the specified profile dimensions. In developing assessment procedures, first selectspecific objectives in such a way that you will be able to assess a representative sample of the syllabus objectives. Each specific objective in the

syllabus is considered a criterion to be mastered by the students. When you develop a test that consists of items and questions that are based on arepresentative sample of the specific objectives taught, the test is referred to as a “Criterion-Referenced Test”. It is not possible to test all specific

objectives taught in the term or in the year. The assessment procedure you use i.e. class test, homework, projects etc. must be developed in such away that it will consist of a sample of the important objectives taught over the specified period.

End-of-Term Examination

The end-of-term examination is a summative assessment system and should consist of a sample of the knowledge and skills students have acquired inthe term. The end-of-term test for Term 3 should be composed of items/questions based on the specific objectives studied over the three terms, using

a different weighting system such as to reflect the importance of the work done in each term in appropriate proportions. For example, a teacher may

build an end of Term 3 test in such a way that it would consist of the 20% of the objectives studied in Term 1, 20% of the objectives studied in Term

2, and 60% of the objectives studied in Term 3.

The diagram below shows a recommended examination structure for end of term examination in Senior High School following the structure of

WAEC examination papers. The structure consists of two examination papers. Paper 1 is the objective test paper essentially testing knowledge and

understanding. The paper may also contain some items that require application of knowledge. Paper 2 will consist of questions that essentially test

“application of knowledge”. The application dimension should be tested using questions that call for reasoning. Paper 2 could also contain somequestions that require understanding of mathematical principles etc. The SBA should be based on both dimensions. The distribution of marks for

Paper 1, Paper 2 and the SBA should be in line with the weights of the profile dimensions as shown in the last column of the table below.



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Distribution of Examination Paper Weights and Marks

Dimensions Paper 1 Paper 2 SBA Total Marks Total Marks

scaled to 100

Knowledge and

Understanding

30 20 10 60 30

Application of knowledge

10 80 50 140 70

Total Marks 40 100 60 200

% Contribution of

Examination Papers

20 50 30 100

Paper 1 or Section A, will be marked out of 40, while Paper 2, the more intellectually demanding paper, will be marked out of 100. The mark

distribution for Paper 2 or Section B, will be 20 marks for “knowledge and understanding” and 80 marks for “application of knowledge”. SBA will

be marked out of 60. The last row shows the percentage contribution of the marks from Paper 1/Section A, Paper 2/Section B, and the School BasedAssessment on total performance in the subject tested.

For testing in schools, the two examination sections could be separate where possible. Where this is not possible, the items/questions for Papers 1 and

2 could be in the same examination paper as two sections; Sections A and B as shown in the example above. Paper 1 or Section A will be an

objective-type paper/section testing knowledge and understanding, while Paper 2 or Section B will consist of application questions with a few

questions on knowledge and understanding.

Combining SBA marks and End-of-Term Examination Marks

The new SBA system is important for raising students’ school performance. For this reason, the 60 marks for the SBA will be scaled to 50. The total

marks for the end of term test will also be scaled to 50 before adding the SBA marks and end-of-term examination marks to determine students’ endof term results. The SBA and the end-of-term test marks will hence be combined in equal proportions of 50:50. The equal proportions will affect onlyassessment in the school system. It will not affect the SBA mark proportion of 30% used by WAEC for determining examination results at the BECE.



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GUIDELINES FOR SCHOOL BASED ASSESSMENT

A new School Based Assessment system (SBA), formally referred to as Continuous Assessment, will be introduced into the school system from

September 2008. SBA is a very effective system for teaching and learning if carried out properly. The new SBA system is designed to provideschools with an internal assessment system that will help schools to achieve the following purposes:

o Standardize the practice of internal school-based assessment in all schools in the country

o Provide reduced assessment tasks for each of the primary school subjectso Provide teachers with guidelines for constructing assessment items/questions and other assessment tasks

o Introduce standards of achievement in each subject and in each class of the school system

o Provide guidance in marking and grading of test items/questions and other assessment tasks

o Introduce a system of moderation that will ensure accuracy and reliabili ty of teachers’ marks

o Provide teachers with advice on how to conduct remedial instruction on difficult areas of the syllabus to improve students performance

The new SBA system will consist of 12 assessments a year instead of the 33 assessments in the previous continuous assessment system. This will

mean a reduction by 64% of the work load compared to the previous continuous assessment system. The 12 assessments are labeled as Task 1, Task

2, Task 3 and Task 4. Task 1-4 will be administered in Term 1; Tasks 5-8 will be administered in Term 2, and Tasks 9-12 administered in Term 3.

Task 1 will be administered as an individual test coming at the end of the first month of the term. The equivalent of Task 1 will be Task 5 and Task 9

to the administered in Term 2 and Term 3 respectively. Task 2 will be administered as a Group Exercise and will consist of two or three instructionalobjectives that the teacher considers difficult to teach and learn. The selected objectives could also be those objectives considered very important and

which therefore need students to put in more practice. Task 2 will be administered at the end of the second month in the term. Task 3 will also be

administered as individual test under the supervision of the class teacher at the end of the 11 th or 12 week of the term.

Task 4 (and also Task 8 and Task 12) will be a project to be undertaken throughout the term and submitted at the end of the term. Schools will be

supplied with 9 project topics divided into three topics for each term. A student is expected to select one project topic for each term. Projects for the

second term will be undertaken by teams of students as Group Projects. Projects are intended to encourage students to apply knowledge and skillsacquired in the term to write an analytic or investigative paper, write a poem 9 (as may be required in English and Ghanaian Languages), use science

and mathematics to solve a problem or produce a physical three-dimensional product as may be required in Creative Arts and in Natural Science.

Apart from the SBA, teachers are expected to use class exercises and home work as processes for continually evaluating students’ class performance,and as a means for encouraging improvements in learning performance.

Marking SBA Tasks

At the SHS level, students will be expected to carry out investigations involving use of mathematics as part of their home work assignments and as

part of the SBA. The suggested guideline for marking such assignments and projects is as follows:



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1. Introduction 20%2. Main text – descriptions, analysis, charts etc. 40%

3. Conclusion and evaluation of results/issues 20%4. Acknowledgement and other references 20%

In writing a report on an experiment or any form of investigation, the students has to introduce the main issue in the investigation, project or report. Theintroduction carries a weight of 20%. The actual work, involving description of procedures and processes, use of charts and other forms of diagrammed, andthe analysis of data is given a weight of 40%. Conclusions and generalizations from the investigation, project etc. is weighted 20%. The fourth item, that is,

acknowledgement and references is intended to help teach young people the importance of acknowledging ones source of information and data. The students

should provide a list of at least three sources of references for major work such as the project. The references could be books, magazines, the internet orpersonal communication from teacher or from friends. This component is given a weight of 20%.

The marks derived from projects, the end of month SBA tests and home work specifically designed for the SBA should together constitute the School BasedAssessment component and weighted 60 per cent. The emphasis is to improve students learning by encouraging them to produce essays, poems, and artisticwork and other items of learning using appropriate process skills, analyzing information and other forms of data accurately and make generalizations andconclusions. The SBA will hence consist of:

Ø End-of-month testsØ Home work assignments (specially designed for SBA)Ø

Project

Other regulations for the conduct of SBA will reach schools from GES.

GRADING PROCEDURE

To improve assessment and grading and also introduce uniformity in schools, it is recommended that schools adopt the following grade

boundaries for assigning grades.

Grade A: 80 - 100% - Excellent

Grade B: 70 - 79% - Very GoodGrade C: 60 - 69% - GoodGrade D: 45 - 59% - Credit (Satisfactory)

Grade E: 35 - 44% - PassGrade F: 34% - Fail

In assigning grades to students’ test results, you may apply the above grade boundaries and the descriptors which indicate the meaning of each grade.

For instance, a score of 75% and above is considered "Excellent"; a score of 66% is within the grade boundary of 65-74% and is considered



xiv

"Very Good". Writing 60% for instance, without writing the meaning of the grade does not provide the students with enough information to evaluatehis/her performance on the assessment. It is therefore important to write the meaning of the grade alongside the score you write. The grade

descriptors, Excellent, Very Good etc do not provide enough feedback to students. You should therefore provide short diagnostic information alongside the grade descriptor or write other comments such as:

o Good work, keep it up

o Could do better

o Hard working students

o Not serious in class; more room for improvement etc.

The grade boundaries are also referred to as grade cut-off scores. When you adopt a fixed cut-off score grade system as in this example, you areusing the criterion-referenced grading system. By this system a students must make a specified score to earn the appropriate grade. This system of

grading challenges students to study harder to earn better grades. It is hence very useful for achievement testing and grading.



1

SENIOR HIGH SCHOOL 1

UNIT SPECIFICOBJECTIVES

CONTENT TEACHING AND LEARNINGACTIVITIES

EVALUATION

UNIT 1.1SETS ANDOPERATIONS ON SETS

The student will be able to:

1.1.1 determine and writethe number of subsetsin a set

1.1.2 identify the propertiesof operations on sets.

Finding the number of subsetsin a set with n elements

Properties of Set Operations-Commutativity

Associativity

Distributivity

Review with students description of sets; - words/set builder notation - listing. - Venn diagrams.

Guide students to deduce the number of subsetsin a set with elements. i.e. the number of subsets = 2n

Guide students to determine the commutativeproperty of sets involving given sets A and B

i.e. A B = B A ; A U B = B U A

Guide students to determine the associativeproperty of sets involving three given sets A, Band C,i.e.(A U B) U C = A U (B U C) and (A B) C = A (B C)

Guide students to determine the distributiveproperty of sets involving three given sets A, Band C,i.e. A U (B C) = (A U B) (A U C) A (B U C) = (A B) U (A C)Illustrate the properties with Venn diagrams.

Let students

find the number of subsets in a given set.

verify commutative,associative anddistributive properties of operations on sets

Illustrate properties of set operations usingVenn diagrams



2



EVALUATION

1.1 (CONTD)

SETS ANDOPERATIONS ONSETS


1.1.3 describe the regionsof a Venn diagram interms of the setoperations- union,complement

1.1.4 find solution topractical problemsinvolvingclassifications usingVenn diagrams

Description and identification of the regions of Venn diagramsusing set operations

Three-set problems using Venndiagrams

Guide students to develop the concept of thecomplement of a set in a Venn diagram,E.g.

Guide students to(I) describe regions of up to three (3) intersecting sets(ii) shade regions corresponding to given descriptions

Assist students to verify (using Venn diagrams)that given any two intersectingsets, A and B,(A U B)/ = A/

B/

(A B)/ = A/ U B/

Review two-set problems.

Guide students to solve problems involving threesets

Let students

draw 2 intersecting setsand shade givenregions

describe shadedregions of 3intersecting sets

solve 3 set problemsinvolving real lifesituations

A1

A



3


CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION

UNIT 1.2REALNUMBERSYSTEM

Students will be able to:

1.2.1 distinguishbetween rational andirrational numbers.

1.2.2 represent realnumbers on thenumber.

Rational andirrational numbers

Real Numbers onthe number line

Guide students to revise natural numbers, whole numbers and integers.

Guide students distinguish between rational and irrational numbers i.e.

rational numbers can be expressed as ba , where a and b are real numbers

and b ≠ 0

Guide students to draw Venn diagrams to illustrate the relationship betweenthe members of the real number system.

i.e.

Assist students to locate points for real numbers onthe number line.E.g. =1, 0, 1, 1, 1.5, , 3.5 2 2Guide students to(i) graph given sets of real numbers on the number line;

(ii) find the range of values for a given graph. E.g. -1≤ x ≤ 2

Let students:

Identify regions representinggiven types of real numbersfrom Venn diagrams.

illustrate a given range of numbers on the number line

R

NZ

Q I



4

UNIT SPECIFIC OBJECTIVES CONTENT TEACHING AND LEARNINGACTIVITIES

EVALUATION

UNIT 1.2 (CONTD)

REAL NUMBERSYSTEM


1.2.1 compare and order

rational numbers

1.2.2 approximate byrounding off decimalnumbers to a givennumber of placevalues

1.2.3 approximate adecimal number to agiven number of significant figures

1.2.4 express recurringdecimals as commonfractions.

Comparing and ordering

rational numbers

Approximating and roundingoff numbers

Significant figures

Recurring decimals

Assist students to compare and order different

types of rational numbers E.g. common fractions,whole numbers, percentages, decimal fractionsand integers using <,> and the number line.

E.g. arrange the fractions; 0.3, 41 , 45% and 28 in

ascending order.

Guide students to approximate decimal numbersto given place valuesE.g. 587.3563 to 2 decimal places (nearesthundredth) is 587.36 and 5873456 rounded to thenearest thousand is 5873000.

Guide students to approximate given decimalnumbers to given number of significant figuresE.g. 46.23067 approximated to 5 significantfigures is 46.231.

Review changing common fractions to decimalsand vice versa.

Guide students to realize that a recurring decimalhas a digit or a block of digits which keeprepeating

E.g. 1.666 or 1..

6

0.727272 or 0...27

Guide students to express recurring decimals as

fractions of the formba where b 0

E.g. 0..

7 =97 and 0.

..

81 =112

9918 =

verify results by using the calculator.

Let students:

arrange sets of rational

numbers in order of magnitude.

round off numbers togiven number of placevalues.

approximate numbers togiven number of significant figures.

express given recurringdecimal fractions ascommon fractions.



5


EVALUATION

UNIT 1.2(CONTD)

REAL NUMBERSYSTEM


1.2.5 express very large or

very small numbers instandard form.

1.2.6 state and use propertiesof operations on realnumbers.

1.2.7 interpret given binaryoperations and apply

them to real numbers

Standard form

Properties of operations

Commutative property

Associative Property

Distributive Property

Binary operations

Guide students to express a very large number in

the form; k x 10n

, where 1 ≤ k <10 and n is aninteger.E.g. 14835 = 1.4835 x 104

0.0034678 = 3.4678 x 10-3

Guide students to investigate the commutativeproperties of addition and multiplication.i.e. a + b = b + a and ab = ba

Guide students to investigate the associativeproperty of addition and multiplication.

i.e. (a + b) + c = a + (b + c), and (ab)c = a(bc)

Guide students to investigate the distributiveproperty of multiplication over addition andsubtraction.i.e. a(b + c) = ab + ac, and

a(b c) = ab ac.

Guide students to interpret and carry out binaryoperations on real numbers such as a * b = 2a + b ab

p * q = p + q 2pq.Use calculator to compute/perform binaryoperations

Let students:

express given numbers

in standard forms andvice versa

State the properties of operations applied ingiven mathematicalsentences.

apply the appropriateproperties to evaluateexpressionsE.g. 18 x 22 = 18 (20+2) 90 x 95 = 90 (100 5).

carry out defined binaryoperations over real

numbersE.g. if m * n = m + n +5,

find 8 * 4



6



EVALUATION

UNIT 1.3ALGEBRAICEXPRESSIONS

The students will be able to:

1.3.1 express statements inmathematical symbols

1.3.2 add and subtractalgebraic

expressions

1.3.3 multiply two binomial Expressions

1.3.4 factorize algebraic expressions

1.3.5 apply difference of twosquares to solveproblems

Algebraic expressions

Operations on algebraicexpressions

Binomial expressions

Factorization

Difference of two squares

Assist students to express simple statementsinvolving algebraic expressions in mathematicalsymbols.

Guide students to add, subtract and simplifyalgebraic expressions involving the four basicoperations.

Assist students to multiply two binomialexpressions and simplifyE.g. (a + b) (c + d) = c (a + b) + d (a + b) = ac + bc + ad + bd

Guide students to identify common factors inalgebraic expressions and factorize (index of thevariable not exceeding 2).

Assist students to develop the rule of differenceof two squares i.e. a2

b2 = (a + b) (a b)

Guide students to apply the idea of difference of two squares to evaluate algebraic expressionsE.g. x² y² = (x + y) (x y) i.e. 52

42

= (5 + 4)(5 4) = 9 x 1 = 9.

Let students

translate statementsinvolving algebraicexpressions inmathematical symbols.

add and subtractalgebraic expressions..

expand and simplifyproduct of two binomialexpressions.

factorize given algebraicexpressions withvariable index notexceeding 2.

apply difference of twosquares to simplifyalgebraic expressions



7


EVALUATION

UNIT 1.3 (CONTD)

ALGEBRAICEXPRESSIONS


1.3.6 perform operations onsimple algebraicfractions.

1.3.7 determine the conditions

under which algebraicfraction is undefined or zero..

Operations on algebraicfractions with monomialdenominators

Operation on algebraicfractions with binomialdenominators

Undefined or Zero algebraic

fractions

Guide students to add and subtract algebraicfractions with monomial denominators.

E.g. axa x

xa+=+ 212

Assist students to add and subtract algebraicfractions with binomial denominators.E.g.

+− b x

1

a x −1

=))((

2

a xb x

ba x

−−−−

,

where x ≠ a, x ≠ b

Discuss with students the condition under which

an algebraic expression is zero.E.g. in 3a 5y , is zero. When 3a = 0 i.e. a = o

Assist students to determine the condition under which an algebraic expression is undefinedE.g. 1 is undefined when 2 2x2 2x = 0. or when x=1

Let students

solve problemsinvolving addition andsubtraction of algebraicfractions with monomialand binomialdenominators.

solve for the value of variables in algebraic

fractions for which thefractions is zero.

find the value of avariable for which analgebraic fraction isundefined.



8



EVALUATION

UNIT 1.4

NUMBER BASES


1.4.1 convert base tennumerals to other bases and vice-versa

1.4.2 solve simple equationsinvolving number bases

1.4.3 perform operations onnumber bases other than base ten

Converting base ten numeralsto numerals in other base andvice versa.

Operations on numbersinvolving number bases other than base ten.

• Addition and subtraction

• Multiplication

Guide students to revise number bases byconverting base ten numerals to bases two andfive and vice-versa.

Guide students to convert given numerals frombase ten to numerals in other bases up to basetwelve.

Guide students to solve equations involvingnumber basesE.g. 132x = 42ten

Guide students to add and subtract numbers inbases other than base ten.

Guide students to construct addition tables for bases other than base ten.

Guide students to find the product of twonumbers and construct multiplication table in agiven base other than base ten.

Let students:

convert numerals inbase ten to numerals inother bases and vice-versa.

solve for the base of anumber in equationsinvolving number bases.

construct addition tablein given base other thanbase ten and use it tosolve problems

solve problems onnumber bases (other than base ten) involvingaddition and subtraction



9

∠ ABC + ∠BAC =∠ ACD



EVALUATION

UNIT 1.5PLANE GEOMETRY I


1.5.1 calculate the angles ata point

1.5.2 state and use theproperties of parallellines

1.5.3 state and use theexterior angle theoremof a triangles

Angles at a point

Parallel linesRelationships betweencorresponding angles, verticallyopposite angles, alternateangles and adjacent angles,supplementary angles

Exterior angle theorem

Revise with students the sum of angles on astraight line by measuring using a protractor.

Assist students to use protractors to measureangles at a point to verify that they add up to3600.

In groups let students draw parallel lines and atransversal and, measure all the angles todiscover the relationships between;corresponding angles, vertically opposite angles,alternate angles, adjacent angles, andsupplementary angles.

Perform practical activities with students to revisethe sum of the interior and the sum of the exterior angles of a triangle.Guide students to apply this knowledge to find thevalue of missing angles in a triangle

Guide students to perform practical activities toverify the exterior angle theorem of a triangle,students to measure the interior and the exterior angles of a triangle.

E.g.

Let students:

find missing angles ingiven diagrams

find missing anglesbetween parallel linesand a transversal

find missing angles of triangles from givendiagrams

D

A

B C



10



EVALUATION

UNIT 1.5 (CONTD)

PLANE GEOMETRY I


1.5.4 identify variousproperties of specialtriangles

Special trianglesIsosceles and equilateraltriangles

Right angled triangle

Revise different types of triangles includingscalene, isosceles equilateral and right-angledtriangles.

Guide students to establish the properties of isosceles and equilateral triangles.

E.g. (i) the line of symmetry of an isoscelestriangle bisects the base and the angle oppositeit, and is perpendicular to the base, (ii) equilateral triangles have three lines of symmetry in the plane and the lines arecongruent

Guide students to use practical activities,including the use of the geoboard to identify the

right-angled triangle and discover the relationshipbetween the hypotenuse and the other two sides.

Let students:

Use properties of special triangles to findmissing angles intriangles

Solve problemsinvolving the application

of the Pythagorastheorem.

use Pythagorastheorem to find amissing side of a right angled triangle whentwo sides are given

c

b

ac

2 = a

2 + b

2



11



EVALUATION

UNIT 1.5 (CONTD)

PLANE GEOMETRY I


1.5.5 state the properties of quadrilaterals

1.5.6 calculate the sums of interior angles andexterior angles of apolygon

Quadrilaterals

Polygons

Guide students to use cut-out shapes and fold toestablish congruent sides, congruent angles and,lines of symmetry of quadrilaterals such asparallelograms, kites, rectangles, etc.

Guide students to establish the relation betweenthe number of sides and the number of trianglesin any polygon with sides

. Assist students to complete the table below.

Polygon No. of sides

No. of angles

Sum of angles

Triangle

Quadrilateral

Pentagon

Hexagon...n

3

4

1

2

180º

360º

Guide them to use their results to find the formulafor finding the sum of the interior angles of apolygon.i.e. sum of angles = (n 2)180º

Guide students to perform activities to find thesum of the exterior angles of a polygon.

Let students:

find the number of linesof symmetry of givenquadrilaterals.E.g. Rhombus Parallelogram, etc.

calculate the sum of interior angles of givenpolygon

find an interior or exterior angle of apolygon using the ideasof the sum of interior angles and exterior angles of a polygon.



12


EVALUATION

UNIT 1.6LINEAR EQUATIONSAND INEQUALITIES


1.6.1 find solution sets for linear equations in one variable.

1.6.2 solve word problemsinvolving linear equationsin one variable

1.6.3 solve linear inequalities inone variable and i llustratethe result on the number

line

1.6.4 solve word problemsinvolving linear inequalities in one

variable

Solution sets of linear equations inone variable

Word problems involving linear equations in one variable

Linear inequalities in one variable

Word problems involving linear inequalities in one variable

Guide students to find solution sets of givenlinear equations in one variable

E.g. 5x 2 = 3x + 2 4 2

T = {x : x = - 6}

Guide students to solve word problemsinvolving linear equations in one variable.

Find and illustrate truth sets of linear inequalities in one variable on the number line.E.g. 0 3x 1 2

131 ≤≤ x

Guide students to solve word problemsinvolving linear inequalities in one variable

Let students

find the truth sets of linear equations in one variable.

solve word problemsinvolving linear equationsin one variable.

solve problems involvinglinear inequalities in onevariable and show thesolution on the number

line.

solve real life problemsinvolving linear inequalities

3

1− 03

1

3

2 13

4



13


EVALUATION

UNIT 1.7RELATIONS ANDFUNCTIONS


1.7.1 distinguish between thevarious types of relations.

1.7.2 Identify functions from

other relations.

1.7.3 determine the rule for agiven mapping

Types of relations

Functions

Mapping

Review with students, relations between two sets;arrow diagrams; ordered pairs; domain; co-domain;range.

Guide students to use arrow diagrams to illustratetypes of relations including one-to-one, one-to-many, many- to-one and many-to-manyrelations.

Use expressions of real life relations such as is thefather of , is the wife of to describe relations.

(Encourage students to develop the sense of belongingness)

Use arrow diagrams to guide students to identify the

relations one-to-one and many-to-one asfunctions.

Assist students to determine the rule for a givenmapping on the set of real numbers

The rule is: f(x) = 2x 3.

Let students

determine the type of relation described in agiven arrow diagram.

give reasons why a givenrelation is or is not a

function

determine the rule for agiven function.

-2-10123

-7-5-3-113

X f(x)



14


EVALUATION

1.7 (CONTD)

RELATIONS ANDFUNCTIONS


1.7.4 draw linear and quadratic

graphs for given functions

1.7.5 find the gradient of astraight line, given the co-ordinates of two points onthe line

1.7.6 find the equation of astraight line

.

1.7.7 find the distance betweentwo points

1.7.8 change the subject of arelation

Graphs of Functions

- linear functions

- quadratic functions

Gradient of a straight line

Equation of a straight line

Magnitude of a linesegment

Change of subject of arelation

Guide students to form table of values for a given

linear function defined on the set of real numbers for agiven domain.

Guide students to use completed tables to plot points,draw graphs and read values from the graphs.

Guide students to draw table of values for quadraticfunctions defined on the set of real numbers for agiven domain.

Assist students to use the table of values to drawquadratic graphs and also to read values from thegraphs.

Assist students to develop the ratio12

12 x x y y −− as the

gradient of a straight line joining the points (x1, y1) and(x2, y2)

Guide students to derive the equation of a line from

12

12

1

1

x x

y y

x x

y y

−−

−− =

where (x, y) is an arbitrary point on the line

Discuss with the students different forms of equationof a straight line i.e. (i) y = mx + c

(ii) ax + by + c = 0Guide students to find the distance between twopoints with given coordinates (x1, y1) and (x2, y2) as

d = 212

212 )()( y y x x −+−

Guide students to find one variable in terms of theothers in a relation.

Let students

draw graphs of given

linear functions and readvalues of the function for agiven pre-image and viceversa

draw the graphs of givenquadratic functions

read the value of thefunction for a given pre-image and vice versa

find the gradient of a linefrom given coordinates of points on the line

find the equation of a linefrom given points on theline

find the length of a line joining two given points

change the subject of arelation.



15


EVALUATION

UNIT 1.8VECTORS IN A PLANE


1.8.1 distinguish between scalar and vector quantities

1.8.2 represent vectors invarious forms

1.8.3 add and subtract vectors,

1.8.4 multiply a vector by ascalar

Scalar and vector quantities

Vector notation andrepresentation

Addition and subtraction of vectors

Multiplying a vector by a scalar

Guide students to use diagrams to illustratethe idea of scalar and vector quantities.

Guide students to identify the following:

(a) free vector notation; a, u, etc.

(b) position vector notation; OP , OB ,etc.

(c) representation of vectors in

component form

y

x

(d) in bearing-magnitude form (r, θ0)

Guide students to add and subtract vectors.Eg.

(a)

+

2

2

1

1

y

x

y

x

++

=21

21

y y

x x

(b)

−

2

2

1

1

y

x

y

x =

−−

21

21

y y

x x

Guide students to multiply a vector by a scalar k.

E.g.

y

xk =

ky

kx

Let students

distinguish between scalar and vector quantities

express given vectors withthe appropriate notations

find the sum and differenceof given vectors

multiply given vectors by

given scalars



16


EVALUATION

1.8 (CONTD)

VECTORS IN A PLANE


1.8.5 express the componentsof a vector in column form

1.8.6 add two vectors using thetriangle law of vector addition.

1.8.7 state the conditions for two vectors to be equal or parallel

Column vectors

Triangle law of vectors

Equal and Parallel vectors

Guide students to use graph to determinecomponents of vectors in column form for given coordinates

E.g. A (x1, y1) and B (x2 , y2) in the Oxy plane;

(a)

=

=

2

2

1

1,

y

xOB

y

xOA

(b)

−−

=12

12

y y

x x AB

Using graphs guide students to deduce thetriangle law of vectors addition.

i.e. AB + BC =AC where A, B and C are pointsin the Oxy plane.

Assist students to establish conditions for vectors to be equal or parallel :

i.e. If.

=

2

2

1

1

y

x

y

x

, then x 1 =x 2 and y 1 = y 2 .

If

1

1

y

x is parallel to

2

2

y

x then

=

2

2

1

1

y

xk

y

x, where k is a scalar.

Let students

find the vector AB givencoordinates of A and B

find the coordinates of B, given the vector AB andthe coordinates of A.

find the diagonals of aquadrilateral ABCD invector component form,given the coordinates of the vertices

add two given vectors

use the idea of equal andparallel vectors to solverelated problems



17


EVALUATION

1.9 (CONTD)VECTORS IN A PLANE

UNIT 1.9

SIMULTANEOUSLINEAR EQUATION


1.8.8 find the negative vector of a given vector

.

1.8.9 find the magnitude anddirection of a vector.

1.9.1 find solution sets of twogiven linear equations in

two variables

1.9.2 solve word problemsinvolving simultaneous

linear equations

Negative vectors

Magnitude and direction of avector

Solving Simultaneous linear equations in two variables

Elimination and substitutionmethod

Graphical method

Solving word problems involvingsimultaneous linear equations in

two variables

Assist students to find the negative vector of agiven vector. E.g. the negative vector of

→

B = −2

3

is =− B

→

BA = − 2

3

Guide students to find the magnitude anddirection of a given vector.

i.e. if

=

→

y

x AB , then

→|| AB = ²² y x +

and direction is given by θ = tan-1 )( x

y

Guide students to find the solution sets of twolinear equations in two variables

simultaneously using(i) the elimination method and(ii) the substitution method.

E.g. 2x + 5y = 10 x - 2y = 4

Guide students to use graphical approach tofind the solution sets of two linear equations intwo variables.

Pose word problems involving simultaneouslinear equations in two variables for students

to solve.E.g. A family of three adults and two childrenpaid GH¢8.00 for a journey. Another family of four adults and three children paid GH¢11.00as the fare for the same journey. Calculate thefare for i. an adult ii. a child iii. a family of four adults and five children

Let students

find negative vectors of given vectors incomponent form

find the magnitude anddirection of given vectors.

find the solution set of two

linear equations in twovariables

use graphical method tofind the solution set of twolinear equations in twovariables

solve word problemsinvolving simultaneous

linear equations



18


EVALUATION

UNIT 1.10

RIGID MOTION I


1.10.1 identify and translate an

object or point by atranslating vector anddescribe the image

1.10.2 identify and explain thereflection of an object in amirror line

1.10.3 describe the image pointsof shapes in a reflection

Translation by a vector.

Reflection in a line.

Characteristics of reflection.

Revise congruency through the use of lines of

symmetries of plane figures.Guide students to identify some Ghanaiansymbols that are symmetricalE.g. Gye Nyame symbol.

Guide students to slide shapes todemonstrate translation as movement in astraight line.

Ask students to give real life examples of translation.

Assist students to translate points and planefigures by given vectors on graph sheets and

describe the images.

E.g. Translate P(3, 4) by vector

=

→

1

2 PQ ,

Guide students to carry out different activitieswith concrete objects to identify the images of an object in a given mirror.

Assist students to draw shapes on graphsheets and find their images under reflectionsin given mirror lines.

i.e. x = k y = k y = kx, where k is an integer.

Let students

translate plane figures andpoints by given vectors andstate the image points

draw images of geometricshapes under reflection in

given mirror lines and statethe points



19


EVALUATION

UNIT 1.11

STATISTICS I


1.11.1 draw frequencydistribution tables for

ungrouped and groupeddata

1.11.2 represent data on asuitable graph

1.11.3 calculate the mean usingappropriate formula

1.11.4 estimate the mode from ahistogram

Frequency distribution tables.

Graphical representation of data.

Mean of a distribution.

Estimating the mode from ahistogram.

Guide students to prepare frequency tablesfor grouped and ungrouped distributions.

(exclude unequal class intervals)

Guide students to use appropriate graph torepresent data from real life situations like testscores, rainfall, health records, imports,exports etc.

(Use pie charts and bar charts for ungroupeddata and histograms for grouped data).

Assist students to calculate the mean usingthe formula:

(i)n

x x

∑= _

for ungrouped data

(ii)∑∑=

f

fx x _

for grouped data.

Guide students to use a histogram to estimatethe mode.

Let students:

construct frequency tablesfor grouped and ungrouped

data and represent themon suitable diagrams.

calculate the mean of agiven data using the

appropriate formula

draw histogram for a givendata and use it to estimatethe mode



20


EVALUATION

UNIT 1.12

RATIO AND RATES


1.12.1 divide a quantity in a

given ratio.

1.12.2 interpret scales usedin drawing plans andmaps and use themto calculate distancesbetween two points

1.12.3 convert foreigncurrencies into Ghanacedis and vice versa

Ratio

Scales

Foreign exchange

Guide students to revise the idea of ratio byfinding how many times one quantity is of the

other E.g. 7 and 21 are in the ratio 1 : 3

Assist students to share given quantities ingiven ratios.E.g. Share GH ¢2.5m in the ratio 3 : 2.

Guide students to examine maps, planstoposheets and identify the scales used.E.g. a scale of 1 : 125000 means 1cm on themap represents 125000 cm on the ground.

Guide students to use given scales to drawplans of given areas in the locality or in theschool and let them draw geometric shapesusing scales.

Guide students to use rates obtained fromForex Bureau or banks to convert givenamounts in foreign currencies to Ghana cedisand vice versa.[the use of calculator and spread sheet(Computer) is recommended]E.g. If $1 = GH ¢0.96find in cedis$25.60

Assist students with samples of water bills,electricity bills, telephone bills and populationcharts to explain the idea of rates.

Let students

solve word problemsinvolving division of

quantities in given ratios

draw plans of given placesand shapes

find actual distances

between two points on amap for a given scale

convert given amounts of foreign currencies intoGhana cedis and viceversa



21


EVALUATION

1.12 (CONTD)

RATIO AND RATES


1.12.4 explain and usecommon rates suchas kmh 1, ms 1

1.12.5 draw travel graphsand interpret them..

1.12.6 calculate andcompare populationdensities

Rates.

Travel Graphs.

Population Density.

Guide students to solve problems involving rates, E.g. speed of a vehicles, wages and salaries

Guide students to draw a distance time graph from agiven data and use it to calculate average speed, totaldistance traveled, total time taken, etc.

Assist students to calculate population density aspopulation per square kilometer (collect data fromstatistical services department or the internet).

Students:

solve practical problemsinvolving rates

draw distance time graphfor a given data andinterpret.

calculate and comparepopulation densities incities urban and rural areas



22


EVALUATION

UNIT 1.13

PERCENTAGES I


1.13.1 compare two amounts or

quantities by expressingone as a percentage of the other.

1.13.2 do money makingcalculations that applypercentages.

1.13.3 Do money spendingcalculations that apply hirepurchase.

Comparison by percentages.

Discount, Commission, SimpleInterest.

Hire Purchase.

Assist students to express one quantity as

percentage of another.

Guide students to calculate percentageincrease or decrease on prices of goods andservices.

Guide students to calculate discount asmoney saved on goods bought andcommission as money earned in atransaction.

Assist students calculate the price of goodswhen discount and commission are given.

Guide students to use current bank rates tocalculate interests on savings and loans

Guide students to explain and performcalculations involving Hire Purchase.

NB: the use of calculator to checkcomputation should be encouraged

Let students:

calculate the value of one

quantity as a percentage of another.

solve problems involvingdiscount and commission.

work out hire purchasepayments over givenperiods



23



EVALUATION

UNIT 2.1

MODULARARITHMETIC


2.1.1 calculate the value of numbers for a givenmodulo.

2.1.2 add and multiply numbers ina given modulo.

Calculation of a number for a given modulo.

Addition (⊕) andmultiplication(⊗) tables ingiven modulo.

Guide students to use the clock face todetermine the modulus of a number.

Use the idea of remainders to assiststudents to determine the modulo of anumber.E.g. 27 = 2 mod 5 8 = 2 mod 6

Guide students to construct addition ⊕ andmultiplication (⊗) tables in a given modulo.

Let students:

calculate the value of numbersfor a given modulo.

find the sum and product of anytwo given numbers in givenmodulo

construct addition (⊕) andmultiplication (⊗) tables in givenmodulo; use the tablesconstructed to find the truth setsof statements



24


EVALUATION

UNIT 2.2

INDICES ANDLOGARITHMS


2.2.1 write in exponent form therepeated factors of a number Laws of indices. Revise with students the first two laws of indices

i.e. ax x ay = a(x + y)

a x ÷ ay = a(x-y)

Guide students to discover further rulesgoverning indices

i.e. (am) n = amn

a -m = 1 am

n maa n

m

=

Guide students to deduce the value for anon-zero number with zero exponent

i.e. ao = 1

Let students:

solve problems involvingrepeated factors



25


EVALUATION

2.2 (CONTD)INDICES ANDLOGARITHMS


2.2.2 solve equations involvingindices.

2.2.3 relate indices to logarithms.

.

Solving equationsinvolving indices.

Relating indices tologarithms in base ten

Assist students to solve simple equationsinvolving indices.

E.g. 8x = 32 23x 25

3x = 5

3

5= x

Guide students to express given numbers aspowers of 10.

E.g. 100 = 102

10,000 = 104

Guide students to identify the relationbetween indices and logarithms.

i.e. x = 10n ⇔ log10 x = n

Let students:

solve equations involving indices.

write an expression in indicesusing logarithm

E.g. x = 10y ⇒ log10 x = y



26


EVALUATION

2.3 (CONTD)

INDICES ANDLOGARITHMS


2.2.4 deduce the rules of logarithmsand apply them.

2.2.5 find the anti-logarithm of a givennumber.

Rules of logarithmsand their application.

Find the anti-logarithms of givennumbers.

Guide students to discover the rules of logarithms.

E.g. log10 (A x B) = log10 A + logn B

log10 B A = log10 A - log10 B

log10 Ax = x log10 A

Assist students to use the rules to simplifylogarithmic expressions and solve problems.

Assist students to read the anti-logarithm of given numbers using (i) tables.

(ii) calculators.

Let students:

use the rules to solve logarithmicproblems.

find the anti-logarithms of givennumbers using tables andcalculators.



27


EVALUATION

UNIT 2.3SURDS.


2.3.1 simplify surds

2.3.2 carry out operations involvingsurds

2.3.3 rationalize a surd with monomialdenominator

.Simplifying surds.

Addition, subtractionand multiplication of surds.

Rationalization of surds with monomialdenominators.

Guide students to simplify surds of the form

a

E.g. 8 2 2=

27 3 3=

=72 26

Assist students simplify the product of surds.i.e

a x b = ab

( a )2 = a

b

a =

b

a

Guide students to find sums, differences andproducts of surds.

Guide students to rationalize surds withmonomials denominators.

E.g.3

32

3

3

3

2

3

2=×=

Let students:

simplify surds of the form aand products of surds.

use the relations in surds to solveproblems

solve problems involvingaddition, subtraction andmultiplication of surds

simplify and rationalize thedenominators of surds



28


EVALUATION

UNIT 2.4PERCENTAGES II


2.4.1 solve real life problems involving compound interest

2.4.2 determine the depreciation of an item over a specified period

Compound interestfor a given period.

(up to 4 years)

Depreciation.

Guide the students to revise simple interestand other applications of percentages withstudents

Guide students to calculate compoundinterest of any given amount.(formula is accepted but not required)

Discuss with students examples of thingsthat lose their values with passage of time.E.g. cars, fridges, etc.

calculate the depreciation of an item for agiven period.(Formula is accepted but not required)

Note use of calculator to check computationshould be encouraged

Let students:

calculate the compound intereston a given amount for a givennumber of years

solve practical problems:E.g.

A car bought for GH¢5000.00depreciates at 10% per annum.Calculate the value after 5 years.



30


EVALUATION

2.5 (CONTD)VARIATION

UNIT 2.6QUADRATIC

FUNCTIONS ANDEQUATIONS.


2.5.5 solve problems involving partial variation.

2.6.1 identify and solve quadratic equations by factorization.

Partial variation.

Solving quadraticequations byfactorization.

Discuss with students what partial variationis.

Guide students to write an equationinvolving partial variation.

Guide students to solve problems involvingpartial variations.

Guide the students to identify a quadraticequation as an equation of the formax2 + bx + c = 0 where a, b and c areconstants.

E.g. 2x2 + x - 3 = 0 3x2 - 27 = 0

Guide students to solve quadratic equationsby factorization.

E.g. find the truth set of

2x2 + 5x = 12 = 0

(x + 4) (2x 3) = 0

T = {x : x = -4, 23 }

Let students:

solve problems involving partialvariations.

identify and write quadraticequations

solve given quadratic equationsby factorization.



31


EVALUATION

2.6 (CONTD)QUADRATICFUNCTIONS ANDEQUATIONS


2.6.2 solve quadratic equations by graphical method.

2.6.3 find the minimum and maximum values and points from graphs.

2.6.4 identify the line of symmetry and

write its equation.

Graphical solution of quadratic equations.

Minimum andmaximum values andpoints of quadraticgraphs.

Line of symmetry.

Guide students to complete tables of valuesfor given quadratic functions and draw

graphs of the functions on graph sheets.

Assist students to find the truth sets of quadratic equations from graphs.

Guide students to find the maximum andminimum values from graphs and state thecoordinates of the points where these occurs

Assist students to establish that the

quadratic graph is symmetrical about avertical line and write its equation as x = k,where k is a real number.

Let students:

solve given quadratic equations

graphically.

find and state themaximum/minimum points andvalues of graphs they draw.

find the line of symmetry from a

quadratic graph and write itsequation



32


EVALUATION

UNIT 2.7

PLANE GEOMETRY II

(CIRCLES)


2.7.1 draw circles for given radii.

2.7.2 state and use the circle theorems

2.7.3 identify the tangent as perpendicular to the radius at the point of contact.

2.7.4 verify that the angle between the tangent and the chord at the point of contact is equal to the angle in the alternate segment.

The Circle as a Locus.

Circle Theorems.

Perpendicularity of Tangent and Radius of aCirlce

Angle between Tangentand a Chord.

Guide students to find all points which are a

given distance from a fixed point.E.g. fix a point O and find all points whichare 5cm from O.

Assist students to find the relationshipbetween the angle subtended at the centreand that at the circumference by an arc.

Guide students to find the value of the anglesubtended by a diameter at thecircumference.

Guide students to find the relationshipbetween opposite angles of a cyclicquadrilateral.

Guide students to verify that the tangent isperpendicular to the radius at the point of contact.

Assist students to verify the alternate angletheorem by drawing.

Let students:

draw circles of varying radii.

find missing angles using circletheorems.

construct a tangent to a circleusing the property of the tangentand radius.

find missing angles using thealternate angle theorem



33


EVALUATION

UNIT 2.7 (CONTD)PLANE GEOMETRY II(CIRCLES)


2.7.5 verify that tangents drawn from an external point to

the same circle are equal when measured from their points of contact

Tangents from anExternal Point.

Guide students to verify that two tangentsdrawn from an external point, T, to a circle at

points A and B are equal in length

i.e. |AT| = |BT|

A

B T

Let students:

solve for missing angles in agiven diagram.



34


EVALUATION

UNIT 2.8

MENSURATION I


2.8.1 find the length of an arc of a

circle

2.8.2 calculate the perimeter of planefigures.

Length of an arc.

Perimeter of planefigures.

Revise parts of the circle with students.

Guide students to deduce the formula for thelength of an arc of a circle.

i.e. 2360

×θ

π r

θ = the angle subtended at the centre of thecircle by the arc.r = radius of the circle.

Revise the perimeters of rectangles andsquares with students

Guide students to deduce the formula for the perimeter of sectors

i.e.360

θ× 2πr + 2r

Guide students to find the perimeter of other plane figures with various sides.

Let students:

calculate the lengths of arc of given circles

find the perimeter of given planefigures



35


EVALUATION

2.8 (CONTD)MENSURATION I


2.8.3 calculate the areas of sectorsand segments

2.8.4 find the areas of quadrilaterals.

.

Areas of sectors andsegments.

Areas of quadrilaterals.

Guide students to revise the area of a circleand triangle with students.

Guides students to establish the formulae for the areas of a sector and a segment.

i.e. Area of sector = ×360

θ πr 2

Area of segment = (area of sector area of triangle).

Guide students to find the areas of givenquadrilaterals. E.g. trapezium, rhombus, etc

Let students:

calculate the areas of sectorsand segment of given circles

find the areas of givenquadrilaterals.



36


EVALUATION

UNIT 2.9

TRIGONOMETRY I


2.9.1 define and compute the sine,

cosine and tangent of an acute angle in degrees.

.

Sine, cosine and

tangent of acuteangles.

Guide students to use appropriate diagrams to

define trigonometric ratios.

E.g. B3

B2

B1

B

O A A 1 A2 A3

Sin θ =3

33

2

22

1

11

OB

B A

OB

B A

OB

B A

OB

AB===

Cos θ =3

3

2

2

1

1

OB

OA

OB

OA

OB

OA

OB

OA===

Tan θ =3

33

2

22

1

11

OA

B A

OA

B A

OA

B A

OA

AB===

Guide students to read the values of giventrigonometric ratios of acute angles from tablesand calculators.

Let students:

express the sine, cosine and

tangent in relation to the sides of a given acute angle in a right-angled triangle

read values of giventrigonometric ratios of acuteangles from tables andcalculators

θ



37


EVALUATION

2.9 (CONTD)TRIGONOMETRY I


2.9.2 calculate the values of trigonometric ratios of 30º

45ºand 60º

The trigonometricratios of 30º, 45ºand

60º.

Guide the students to draw an equilateraltriangle of dimensions 2-units and use it to

derive the ratios for 30º and 60º.

E.g.

300 300

2 2

3

600 600

1 1

sin 30º = 2

1

cos 30º= 2

3

sin 60º =2

3 cos 60º =

2

1

Assist students to draw a square of side oneunit, draw one of the diagonals and use thediagonal and two sides to derive the value of trigonometric ratios of 450

450 sin 450 = 1 = √2

√ 2 √2 2 1 Cos 450 = 1 = √2

√2 2 450

1 tan 450 = 1

Let students:

find the trigonometric ratios of theangles 30º ,60ºand 45º



38


EVALUATION

2.9 (CONTD)TRIGONOMETRY


2.9.3 use the calculator to read thevalues of sine, cosine and

tangent of angles up to 360º

2.9.4 find the inverse of trigonometricratios

2.9.5 calculate angles of elevationand angles of depression

.

The use of calculatorsto read sine, cosine

and tangent of anglesbetween 0º and 360º.

Inverse of trigonometric ratios.

Angles of elevationand depression.

Guide the students to use their calculators tofind given angles from 00 and 3600.

Assist students to find the inverse of giventrigonometric ratios using tables or calculators.

Discuss with students what angles of elevation and angles of depression areusing diagrams.E.g.

θ d

α

a a

s α

b

θ is the angle of elevation =

−

b

d 1tan

α is the angle of depression =

−

b

a1tan

Let students:

find the values of sine, cosineand tangent of given angles

using calculators

find the inverse of given angles

explain what angles of elevationand depression are

solve problems involving anglesof elevation and angles of depression.



39


EVALUATION

2.9 (CONTD)TRIGONOMETRY

UNIT 2.10

PROBABILITY


2.9.6 apply the use of trigonometric ratios to calculate distances and heights

2.10.1 determine the sample space of a simple experiment.

2.10.2 determine the sample space of a compound experiment.

2.10.3 calculate the probability of an event

2.10.4 interpret OR in probability as addition.

2.10.5 interpret AND in probability as multiplication.

Application of trigonometric ratios.

Sample Space of simple experiments

Sample Space of compoundexperiment.

Probability of anevent

Addition law for mutually exclusive

events.

Multiplication law for independent events.

Pose problems of real life situationsinvolving trigonometric ratios for students tosolve.

Guide students to perform simpleexperiments such as tossing a coin once,throwing a die once, etc. and list the samplespaces as the set of all possible outcomes,E.g. sample space for throwing a die once isS = {1, 2, 3, 4, 5, 6}

Guide students to perform compoundexperiments such as tossing two coins,tossing a coin and throwing a die, etc. andlist the sample spaces.

Assist students to calculate the probability of an event

i.e. P(E) =)(

)(

S n

E n

Guide students to establish the followingfacts;P(S) = 1P(φ) = 00 P(A) 1P(Á/ ) = 1 P(A)

Guide students to realize that mutuallyexclusive events do not have anything in

common.i.e. P(A or B) = P(A) + P(B)

Guide students to realize that for twoindependent events, the probability of event

A and event B happening together isP(A and B) = P(A) x P(B)

Let students:

apply trigonometric ratios tosolve problems on real lifesituations

make a list of all possibleoutcomes of an experiment

calculate the probability of givenevents

apply the addition law tocalculate probabilities of naturally

exclusive events

apply the multiplication law tocalculate the probability of independent events



40


UNIT SPECIFIC OBJECTIVES CONTENT TEACHING AND LEARNING

ACTIVITIES

EVALUATION

UNIT 3.1

SEQUENCES ANDSERIES


3.1.1 continue a sequence with more terms.

3.1.2 recognize an arithmetic progression (AP) and find the nth term or general term

3.1.3 find the sum of n terms of an AP

3.1.4 recognise a geometric

progression(GP)

3.1.5 find an expression for thegeneral term of a GP

Patterns of sequence

Arithmetic Progression

Sum of n terms of an AP.

Geometric Progression

General term of a GP

Guide students to examine and continue asequence of numbers.

Guide students to identify commondifference (or constant) and find the nth termof an A.P.

i.e. Un = a + (n - 1)d

Assist students to deduce and use the rulefor finding the sum (Sn ) of an AP.

i.e. Sn =2n {a + Un }

=2n {2a + (n 1)d}

Guide students to use practical situations toillustrate a GP. E.g. Depreciation

Guide students to deduce the general term

of a GP as Un = ar n-1 ; where a is the first

term and r , the common ratio

Let students:

write the next two or more termsof a given sequence.

write the nth term of givenarithmetic progressions for givenvalues of n.

find the sum of n terms of an AP.

solve everyday problems usingthe concept of GP



41


EVALUATION

UNIT 3.2

BEARINGS

UNIT 3.3CONSTRUCTION


3.2.1 interpret bearing asdirection of one point from

another.

3.2.2 write bearing of one point from another as (r, θ).

3.2.3 find the bearing of a point A from another point B, given the bearing of B from A.

3.3.1 construct 75o 105o 1350and 150o

3.3.2 construct a triangle or quadrilateral under given conditions.

3.3.3 construct a particular loci

Bearing of a point fromanother.

Distance-bearing form

Reverse bearing

Construction of 75o 105o

135o and 150o.

Construction of Trianglesand Quadrilaterals.

Constructing loci

Guide students to measure angles betweenpoints in clockwise direction taking North (N)

as direction of reference.

Guide students to deduce and write thebearing of one point from another in thedistance-bearing form (r, θ).

Assist students to deduce reverse bearingsi.e. if B is θº from A, then A is(i) (180 + θ )0 from B for 00 < θ < 1800

(ii) (θ - 180)0 from B for 1800 < θ < 3600

Review the construction of 30o , 45o , 60o

and 90º with the students.

Guide students to construct angles 75 o ,105 o , and 135 o .

Assist students to use a pair of compassesand ruler only to construct;

1. a triangle, given two sides and anincluded angle;

2. a triangle, given two angles and aside.

3. a quadrilateral under givenconditions.

Guide students to construct the locus of points equidistant from two or more fixedpoints and two or more intersecting straightlines

Let students:

record angle measure in 3-digits.

find the bearing of a point C from A, given the bearing of B from Aand the bearing of C from B.

find the bearing of one point fromanother, given the reversebearing.

construct some given angles.

construct triangles andquadrilaterals under givenconditions

solve loci related problemsthrough construction



42


EVALUATION

UNIT 3.4STATISTICS II


3.4.1 draw a histogram for given

data

3.4.2 use appropriate formulacalculate the mean

3.4.3 draw cumulative frequency curves (ogive) and interpret them.

Histogram.

Mean.

Cumulative FrequencyCurves (ogive).

Guide students to revise the drawing of

frequency table for ungrouped and groupeddata; use it to draw histogram (Restrict togroups of equal intervals) and estimate themode from the histogram.

Guide students to find the mean of ungrouped and grouped data using;

(i) x =n

x∑

(ii) x = ∑∑ f

fx where x is the class

mid-point (in case of grouped data)

(accept assumed mean method but notrequired)

Let students:

represent a given data on a

histogram

estimate the mode from ahistogram

calculate mean of given data

draw a cumulative frequencycurve and use it to estimate;

(i) lower and upper quartiles;

(ii) median;(iii) given deciles and

percentiles;



43


EVALUATION

3.4 (CONTD)STATISTICS II


3.4.4 calculate and interpretstandard deviation andvariance of ungrouped

data.

.

Standard deviation andVariance

Guide students to draw cumulativefrequency curves using data and use thecurves to estimate:

(i) lower and upper quartiles;(ii) median;(iii) deciles and percentiles, etc

Guide students to calculate and interpretstandard deviation and variance of anungrouped data.

Eg method of ungrouped data

Sd = ( )n

x x∑ −2

Variance = Sd2

Eg The scores from an English test are 30,50, 70, 76, 26, 60, 42, 38, 92, and 49. x&& = 53.3: Sd = 21.1

Mathematics scores: 80, 48, 60, 40, 32, 54,62, 31, 86 and 55. x = 53.0 and Sd = 15.7

Therefore, English scores are more spread

around the mean; this implies thatperformance in mathematics is better thanthat of English.

NB: Students may use spreadsheet to drawgraphs and calculate mean, mode, medianand standard deviation

Let student:

Calculate and interpret standarddeviation and variance.



44


EVALUATION

UNIT 3.5

INTERPRETATION OFLINEAR AND

QUADRATIC GRAPHS.


3.5.1 solve simultaneous

equations one linear and one quadratic using graphs

3.5.2 use quadratic graph to solve related equations

3.5.3 find the range of values of x for which y is increasing or decreasing.

3.5.4 find the range of values of x for which y is positive or negative

Solving linear and

quadratic equations usinggraphs.

Solving related quadraticequations.

Increasing/Decreasingvalues of quadraticgraphs.

Positive/Negative valuesof quadratic graph.

Guide students to solve simultaneous

equations, one linear, one quadratic bydrawing the two graphs on the same axes.

Assist students to solve related equationsusing the quadratic graph;

i.e. use the graph of y = ax2 + bx + c to solve ax2 + dx + k = 0 where a, b, c, d and k are constants.

Assist students to determine the range of values of x for which the graph is increasingor decreasing.

Guide students to determine the range of values of x for which a quadratic graph ispositive or negative. (i.e. above or below thex-axis)

Let students:

draw a quadratic and linear

graphs on the same axes andfind the values of x which satisfythe two equations simultaneouslyand find the correspondingvalues of y

use graph of y = ax2 + bx + c tosolve ax2 + dx + k = 0 for various values of k

find the range of values of x for which a given graph is increasingor decreasing.

find the range of values of x for which y is positive or negative.



45


EVALUATION

UNIT 3.6

MENSURATION II

(SURFACE AREA,VOLUME OF SOLIDSAND THE EARTH AS AEARTH)


3.6.1 draw nets of prisms

3.6.2 calculate the surfacearea of prisms

3.6.3 calculate the volume of prisms

3.6.4 calculate the totalsurface area of cone.

Nets of prisms.

Surface Area of Prisms.

Volume of prisms

Surface Area of Cone

Guide students to identify solids with uniform

cross-section as prisms E.g. triangular prisms, rectangular prism, square prism, etc.

Guide students to use cut-out shapes toform the nets of open/close prisms andidentify the faces.

Guide students to discover that the totalsurface area is the sum of the areas of allthe faces.E.g. Cuboid - Area = 2bl + 2bh + 2lh Closed cylinder - A = 2πr (r + h)

Assist students to calculate volume of prisms by multiplying the area of uniformcross-section by the height or length.

Let students open a cone and examine thenet.

Guide students to draw the net and measurethe angle of the sector.

Guide students to deduce the formula for finding the surface area of a cone as

A = Curved Surface + Base Area

×360θ πl

2 +πl

2

= π

+× 22

360r l

θ

Let students:

draw nets of given prisms

calculate the total surface Area of prisms of given dimensions.

calculate the volume of prisms of given dimensions.

use the formula to calculate thetotal surface area of given coneof given dimensions.



46


EVALUATION

3.5 (CONTD)

MENSURATION II


3.6.5 calculate the volume of a cone

3.6.6 calculate the totalsurface area of apyramid

3.6.7 calculate the volumesof Pyramids

3.6.8 calculate surface area of a sphere

3.6.9 calculate the volume of a Sphere

3.6.10 calculate distance along a given latitude and longitude

Volume of a Cone.

Surface Area of aPyramid.

Volume of a pyramid.

Surface area of a sphere

Volume of a sphere

Distances of arcs of spheres

Guide students to:

Assist student to establish the formula for finding the volume of a cone. i.e. V =

31 π r 2 h

Guide students to calculate the total surface Area of a pyramid as the sum of thetriangular faces and the base.

Guide students to deduce the formula for the volume of a pyramid

i.e. Volume =31 x base Area x h

Guide students to find the surface areas of spheres of given radii using the formula

A = 4πr²

Guide students to establish the formula for finding the volume of a sphere. i.e.

V =34 πr 3

Guide students to draw a sphere andindicate two points on the same latitude or the same longitudes (great circles).

Guide students to draw a sphere andillustrate two points on the same latitude butdifferent longitudes

Guide students to calculate distancesbetween two towns on the earth surface.

Let students:

use the formula to find volume of a given cone.

calculate the total surface areasof pyramids of given dimensions

calculate the volumes of givenpyramids

calculate surface area of givenradii

calculate the volume of spheresof given radii

solve real life applicationproblems

E. g .time taken for aeroplanes tofly between two towns etc



47



EVALUATION

UNIT 4.1

LOGICAL REASONING


4.1.1 identify and form true or false statements.

4.1.2 form the negation of simple statements.

4.1.3 draw conclusions using the implication sign statements made.

Statements.

.

Negation of statement.

Implication signs⇒ , ⇔

Guide students to identify true or falsestatements.

Guide students to write statements innegation form.E.g. Kofi is not a lazy boy is the negation of Kofi is a lazy boy.

Assist students to use the implication sign todraw conclusions from statements made.

E.g. 41023 =⇒=− x xDiscuss the use of the symbol, ⇔ with

studentsE.g. 41023 =⇒=− x x

and if 10234 =−⇒= x x

so, 41023 =⇔=− x x

Let students:

Identify true or false statements.

negate given statements

draw conclusion from statementsmade using the implication sign

Use the symbol ⇔, iff (if and only

if) to draw conclusions from givenstatements



48


EVALUATION

4.1 (CONTD)LOGICAL REASONING


4.1.4 use Venn diagrams to

determine the validity or otherwise of implications or conclusions.

Use of Venn Diagrams

:

Guide students to draw Venn diagrams to

illustrate given statements E.g. consider thestatement: P : All students are hardworking

S = {students}

H = {hardworking people}

U= {People}

S

Assist students to determine whether givenconclusions are valid or not

Let students:

use Venn diagrams to determinethe validity or otherwise of givenstatements

H

S

U



50


EVALUATION

4.2 (CONTD)

PERCENTAGES III


4.2.4 calculate taxes paid on

goods and services.

4.2.5 calculate and explain thevalue added tax. (VAT)

4.2.6 calculate electricity, water and telephone bills.

Income Tax.

Value Added Tax (VAT).

Household bills.

Guide students to identify the government

agencies responsible for collecting incometaxes.

NB.: Encourage students to appreciate theneed for people to pay taxes.

Assist students to calculate income taxusing a given tax schedule.

Assist students to identify some goods andservices that attract VAT and calculate theVAT on them.

Guide students to identify the varioushousehold bills such as electricity bills, water bills and telephone bills.

Guide students to identify how these billsare calculated.

Note: Emphasize the need for students tobe prudent in the use of these utilities.

Let students:

calculate the income tax for a

given income.

find the VAT on a bill for servicesor sales

calculate the total bill paid by ahousehold at the end of themonth at a given rate



51


EVALUATION

UNIT 4.3

RIGID MOTION II ANDENLARGEMENT


4.3.1 find the image of an object under rotation

4.3.2 carry out an enlargement of a plane shape given a scale factor

4.3.3 identify a scale drawing asan enlargement/reduction

of a plane figure (shape).

4.3.3 establish the relationship between the areas and volumes of plane figures and solids and their images

Rotation

Enlargement

Scale drawing

Areas and Volumes of similar figures.

:

Revise examples of turning in everyday lifesituation to explain rotation

Guide students through construction to findthe images of plane figures under rotation.

Guide students to derive the rules for rotation using graphical method E.g. Anticlockwise about the origin (0,0)through 90o ; (x, y) (-y, x) 180 o ; (x, y) (-x, -y), etc.

Include clockwise rotation about the originand rotation about a point other than theorigin

Guide students to find images of planefigures under enlargement from the origin for given scale factors.

Guide students to use scale drawing toenlarge or reduce plane figures.

Assist students to discover the relationshipbetween the areas and volumes of similar figures and solids.

i.e. Area of image : Area of object = K2

: 1 and Volume of Image : Vol. of solid = K3 : 1 where k is the scale factor

Let students:

identify a rotation among a set of movements

draw a given plane figure on agraph paper and rotate itthrough given angles about theorigin and about a given point

draw the images of plane figuresunder enlargement from theorigin for given scale factors.

use scale drawing to enlarge or reduce plane figures given thescale

calculate areas and volumes of similar plane figures and solidsgiven corresponding dimensions




EVALUATION

UNIT 4.4

TRIGONOMETRY II


4.4.1 draw the graphs of simple trigonometric functions and identify maximum and minimum values

4.4.2 draw the graphs of trigonometric functions and interpret

.

Graphs of trigonometric functions

Trigonometric equations

:

Guide students to prepare tables for giventrigonometric functions for :

xa y sin= and xb y cos= in the range00 3600 ≤≤ x

Guide students to use their tables to drawthe graphs of the functions and find themaximum and minimum values.

Guide students to draw simple graphs of trigonometric functions of the form :

xb xa x f cossin)( += in the range

0° ≤ x ≤ 360°

Guide students to use their graphs to solveequations such as : ,0cossin =+ xb xa

k xb xa =+ cossin , etc.

Let students:

draw graphs of giventrigonometric functions and usethem to solve related problems

mathematics syllabus shs 1-3

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