atlantic canada mathematics curriculum guide: grades...

Atlantic Canada Mathematics

Curriculum Guide:

Grades 4–6

Atlantic Canada Mathematics Curriculm: Grades 4-6

© Crown Copyright, Province of Nova Scotia 1999Prepared by the Department of Education and Culture

Contents of this publication may be reproduced in whole or in part provided the intended use isfor non-commercial purposes and full acknowledgement is given to the

Nova Scotia Department of Education and Culture.

Cataloguing-in-Publication Data

Main entry under title.

Atlantic Canada mathematics curriculm: grades 4-6 / Nova Scotia.Department of Education and Culture. English Program Services.-

ISBN: 0-88871-553-6

1. Curriculum planning -- Atlantic Provinces. 2. Mathematics --Study and teaching -- Atlantic Provinces. 3. Education, Elementary --Atlantic Provinces. I. Nova Scotia. Department of Education andCulture. II. Atlantic Provinces Education Foundation.

372.7043--dc21 1999

Acknowledgements

The departments of education of New Brunswick, Newfoundland and Labrador, Nova Scotia, and PrinceEdward Island gratefully acknowledge the contribution of the regional mathematics curriculum committee tothe development of the grade primary/kindergarten–6 mathematics curriculum guides. Current and pastmembers of the grade primary/kindergarten–6 mathematics curriculum committee include the following:

New Brunswick Nova Scotia

John Hildebrand, Mathematics Consultant Ken MacInnis, Elementary TeacherDepartment of Education Sir Charles Tupper Elementary School

Joan Manuel, Mathematics/Science Richard MacKinnon, Mathematics ConsultantSupervisor, School District 10 Department of Education and Culture

David McKillop, Curriculum ConsultantDepartment of Education and Culture

Newfoundland and Labrador Prince Edward Island

Patricia Maxwell, Mathematics Consultant Clayton Coe, Mathematics/Science ConsultantDepartment of Education Department of Education

Sadie May, Mathematics Teacher Bill MacIntyre, Elementary Mathematics/Deer Lake-St. Barbe South Integrated Science ConsultantSchool Board Department of Education

Donald Squibb, Mathematics TeacherSt. James Regional High School

The regional mathematics curriculum committee gratefully acknowledges the input, feedback, suggestions,and other contributions to the curriculum guide of many educators from across the Atlantic region.

ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4–6 iii

ACKNOWLEDGEMENTS

Contents

Background and

Rationale

Program Design and

Components

Background ......................................................................... 1Rationale ............................................................................. 2

Program Organization ......................................................... 3Unifying Ideas ..................................................................... 4Learning and Teaching Mathematics .................................... 5Adapting to the Needs of All Learners .................................. 6Support Resources ............................................................... 6Role of Parents ..................................................................... 7

Assessing Student Learning .................................................. 9Program Assessment ............................................................. 9

Curriculum Outcomes ....................................................... 11Specific Curriculum Outcomes: Grade 4

Number Concepts/Number and Relationship Operations:Number Sense and Number Concepts ..........................4-1

Number Concepts/Number and Relationship Operations:Operation Sense and Number Operations .................. 4-17

Patterns and Relations .....................................................4-45Shape and Space: Measurement ......................................4-51Shape and Space: Geometry ............................................4-67Data Management and Probability:

Data Management ...................................................... 4-83Data Management and Probability: Probability ............4-101

Specific Curriculum Outcomes: Grade 5Number Concepts/Number and Relationship Operations:

Number Sense and Number Concepts ..........................5-1Number Concepts/Number and Relationship Operations:

Operation Sense and Number Operations .................. 5-25Patterns and Relations ....................................................5-49Shape and Space: Measurement ......................................5-59Shape and Space: Geometry ...........................................5-75Data Management and Probability:

Data Management ...................................................... 5-93Data Management and Probability: Probability............5-105

Assessment and

Evaluation

Curriculum

Outcomes

ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4–6 v

CONTENTS

Bibliography

CONTENTS

ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4–6vi

Specific Curriculum Outcomes: Grade 6Number Concepts/Number and Relationship Operations:

Number Sense and Number Concepts ..........................6-1Number Concepts/Number and Relationship Operations:

Operation Sense and Number Operations ..................6-23Patterns and Relations ....................................................6-49Shape and Space: Measurement ......................................6-63Shape and Space: Geometry ...........................................6-77Data Management and Probability:

Data Management ......................................................6-95Data Management and Probability: Probability............ 6-113

Bibliography .................................................................... 7-1

1ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4–6

Mathematics curriculum reform in Atlantic Canada is shaped by avision that fosters the development of mathematically literate studentswho can extend and apply their learning, and who are effectiveparticipants in an increasingly technological society. Curriculumreform has been motivated by a desire to ensure that students inAtlantic Canada benefit from world-class curriculum and instructionin mathematics as a significant part of their total school-learningexperience.

The Foundation for the Atlantic Canada Mathematics Curriculum usedthe Curriculum and Evaluation Standards for School Mathematics(National Council of Teachers of Mathematics, 1989) as a guidingbeacon for pursuing this vision. These publications endorse theprinciples of students learning to value mathematics and of theirbeing active “doers”; they advocate a meaningful curriculum focussingon four unifying ideas—mathematical problem solving,communication, reasoning and connections. The foundationdocument subsequently establishes a framework for the developmentof detailed grade-level guides describing mathematics curriculum,assessment, and instructional techniques.

Mathematics curriculum development has taken place under theauspices of the Atlantic Provinces Education Foundation (APEF), anorganization sponsored and managed by the governments of the fourAtlantic provinces. APEF brought together teachers and provincialdepartment of education officials to plan and develop co-operativelythe curricula in mathematics, science, and language arts in bothofficial languages.

Each of these subject initiatives has produced a curriculum using alearning-outcomes framework, outlined in Figure 1, that supports the

Background and Rationale

BACKGROUND AND RATIONALE

Background

2 ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4–6

regionally developed essential graduation learnings (EGLs) inaesthetic expression, citizenship, communication, personaldevelopment, problem solving, and technological competence. (Seethe Foundation for the Atlantic Canada Mathematics Curriculum for adetailed presentation of these EGLs and the contribution of themathematics curriculum to their achievement.)

The Foundation for the Atlantic Canada Mathematics Curriculumprovides an overview of the philosophy and goals of the mathematicscurriculum, presenting broad curriculum outcomes and addressing avariety of issues with respect to the learning and teaching ofmathematics. The curriculum is based upon several beliefs aboutmathematics learning that have grown out of research and practice.These beliefs include: mathematics learning is an active andconstructive process; learners are individuals who bring a wide rangeof prior knowledge and experiences, and who learn via various stylesand at different rates; learning is most likely to occur when placed inmeaningful contexts and in an environment that supportsexploration, risk-taking, and critical thinking and that nurturespositive attitudes and sustained effort; and learning is most effectivewhen standards of expectation are made clear with ongoingassessment and feedback.

Rationale

BACKGROUND AND RATIONALE


Program Design and Components

PROGRAM DESIGN AND COMPONENTS

While the mathematics curriculum does contribute to students’achievement of each EGL, the communication and problem solvingEGLs are particularly well addressed by the curriculum’s unifyingideas—mathematical problem solving, reasoning, communication,and connections. The Foundation for the Atlantic Canada MathematicsCurriculum describes the mathematics curriculum in terms of a seriesof outcomes—general curriculum outcomes (GCOs) that relate tosubject strands and key-stage curriculum outcomes (KSCOs) thatfurther articulate the GCOs for the end of grades 3, 6, 9, and 12.

This curriculum guide and those for other grade levels provide greaterspecificity and clarity for the classroom teacher by relating grade-levelspecific curriculum outcomes (SCOs) to each KSCO. As illustrated inFigure 2, these outcomes represent the means by which students worktoward accomplishing the KSCOs, the GCOs and, ultimately, theEGLs. It is important to emphasize, however, that the presentation ofthe SCOs follows the outcomes structure established in theFoundation for the Atlantic Canada Mathematics Curriculum and doesnot represent a suggested teaching sequence. While some outcomeswill need to be addressed before others, a great deal of flexibility existsas to the structuring of the program. It is expected that teachers willmake decisions regarding the sequencing of outcomes. Manyoutcomes across a number of strands may very well be addressed inone lesson or series of lessons; some outcomes such as those inpatterns and data management may be addressed on an ongoing basisthroughout a number of strands.

Program

Organization


Decisions on sequencing of outcomes will depend on a number offactors, including the nature and interests of the students themselves.For instance, what might serve well as a “kickoff ” strand for onegroup of students might be less effective in that role with a secondgroup. Another consideration will be coordinating the mathematicsprogram with other aspects of the students’ school experience. Forexample, they could study facets of measurement in connection withtopics in science, data management with social studies issues, andaspects of geometry with physical education. Sequencing could bealso influenced by other factors such as major events in thecommunity or province like elections, exhibitions, or fairs.

While the GCOs are organized around content strands, everyopportunity has been taken to infuse the KSCOs with one or more ofthe unifying ideas—mathematical problem solving, communication,reasoning, and connections. (See Figure 3.)

These unifying ideas serve to link the content to methodology. Theymake it clear that mathematics is to be taught in a problem-solvingmanner and that classroom activities and student assignments must bestructured to provide opportunities for students to communicatemathematically. Furthermore, through teacher encouragement andquestioning, students must explain and clarify their mathematicalreasoning and the mathematics with which students are involved onany given day must be connected to other mathematics, otherdisciplines, and/or the world around them.

Unifying Ideas



Students will be expected to solve routine and/or non-routinemathematical problems on a daily basis. Over time, numerousproblem-solving strategies should be modelled and discussed.

Learning and

Teaching

Mathematics


Students should be encouraged to employ various strategies in manydifferent problem-solving situations. While choices with respect tothe timing of the introduction of any given strategy will vary,strategies such as try-and-adjust, look for a pattern, draw a picture,act it out, use models, make a table or chart, and make an organizedlist should all become familiar to students during their early years ofschooling. Other strategies such as working backward, logicalreasoning, trying a simpler problem, changing point of view, andwriting an open sentence or equation would be part of a student'srepertoire upon leaving elementary school.

The unifying ideas of the mathematics curriculum suggest quiteclearly that the mathematics classroom needs to be one in whichstudents are actively engaged each day in the “doing of mathematics.”Often students have come to view mathematics as a set of conceptsand algorithms that the teacher transmits for them to practise.Instead, students must come to see mathematics as a vibrant anduseful tool for helping them understand the world, and as a disciplinethat lends itself to multiple strategies, student innovation, and, quiteoften, multiple solutions. (See Contexts for Learning and TeachingMathematics in the Foundation for the Atlantic Canada MathematicsCurriculum.)

The learning environment will be one in which students and teachersmake regular use of manipulative materials and technology, activelyparticipate in discourse, conjecture, verify reasoning, and sharesolutions. This environment will be one in which respect is given to


all ideas and in which reasoning and sense-making are valued abovegetting the right answer. Students will have access to a variety oflearning resources, will balance the acquisition of procedural skillsand the attainment of conceptual understanding, will estimateroutinely to verify the reasonableness of their work, will compute in avariety of ways while continuing to place emphasis on basic mentalcomputation skills, and will engage in homework as a useful extensionof their classroom experience.

The Foundation for the Atlantic Canada Mathematics Curriculumstresses the need to deal successfully with a wide variety of equity anddiversity issues. Not only must teachers adapt instruction toaccommodate differences in student readiness, but also they mustavoid gender and cultural biases. Ideally, every student should findhis/her learning opportunities maximized in the mathematicsclassroom.

The reality of individual student differences must not be ignoredwhen making instructional decisions. While this curriculum guidepresents grade-level SCOs, it must be acknowledged that all studentswill neither progress at the same pace nor will attain a given outcomeat a given time. The SCOs represent, at best, a reasonable frameworkfor assisting students to ultimately achieve the KSCOs and GCOs.

As well, teachers should design instruction to accommodatedifferences in student learning styles. Different instructional modesare clearly appropriate, for example, for those students who areprimarily visual learners versus those learners who learn best by doing.Designing classroom activities to support a variety of learning stylesmust also be reflected in assessment strategies.

This curriculum guide for grades 4 to 6 represents the principalreference for teachers of mathematics. This guide is central to daily,unit, and yearly planning, as well as a reference point to determine theextent to which the outcomes have been met.

Texts and other resources will have significant roles in themathematics classroom in as much as they support the SCOs. Manymanipulative materials need to be readily at hand, and technologicalresources (e.g., software and videos) should be available. Calculatorswill be an integral part of many learning activities. Also, professionalresources will need to be available to teachers as they seek to broadentheir instructional and mathematical understandings. Key amongthese are the Curriculum and Evaluation Standards for School

Mathematics (NCTM), Addenda Series (NCTM), various Yearbooks(NCTM), Elementary School Mathematics: Teaching Developmentally(Van de Walle 1994), Elementary and Middle School Mathematics:Teaching Developmentally (Van de Walle 1998), Developing Number

Adapting to the

Needs of All

Learners

Support

Resources



Concepts Using Unifix Cubes (Richardson), and About TeachingMathematics: A K-8 Resource (Burns 1992).

Societal change dictates that students’ mathematical needs today arein many ways different from those of their parents. These differencesare both in respect to mathematical content and in instructionalapproach. As a consequence, it is important that educators take everyopportunity to discuss with parents changes in mathematicalpedagogy and why these changes are significant. Parents whounderstand the reasons for changes in instruction and assessment willbe better able to support their children in mathematical endeavours.They can foster positive attitudes towards mathematics, stress theimportance of mathematics in their children’s lives, assist theirchildren with mathematical activities at home, and, ultimately, helptheir children to become confident, independent learners ofmathematics.

Role of Parents



Assessment and evaluation are integral to learning and teaching.Ongoing assessment and evaluation not only are critical for clarifyingstudent achievement and thereby motivating student performance,but also for providing a basis upon which teachers can makemeaningful instructional decisions. (See the Assessment andEvaluating Student Learning section in the Foundation for the AtlanticCanada Mathematics Curriculum.)

Characteristics of good student assessment include the use of a widevariety of assessment strategies and tools, the alignment of assessmentstrategies and tools with the curriculum and instructional techniques,and the assurance of fairness both in application and scoring. ThePrinciples for Fair Student Assessment Practices for Education in Canada(University of Alberta 1993) which elaborates good assessmentpractices served as a guide for student assessment in the Foundationfor the Atlantic Canada Mathematics Curriculum.

Program assessment will serve to provide information to educators onthe relative success of the mathematics curriculum and itsimplementation. It will address whether or not students are meetingthe curriculum outcomes, whether or not the curriculum is beingequitably applied across the region, whether or not the curriculumreflects a proper balance between procedural knowledge andconceptual understanding, and whether or not technology is fulfillingits intended role.

Program

Assessment

Assessment and Evaluation

ASSESSMENT AND EVALUATION

Assessing

Student

Learning


Curriculum Outcomes

This guide provides details regarding SCOs for each grade. Asindicated earlier, the order of presentation neither prescribes apreferred order of presentation nor suggests an isolated treatment ofeach outcome; rather, it organizes the SCOs in terms of the broadframework of GCOs and KSCOs.

The SCOs are presented on two-page spreads (see Figure 4). At thetop of each page the overarching GCO is presented, while theappropriate KSCO and SCO(s) are displayed in the left-handcolumn. As well, the bottom of many left-hand columns contains arelevant quotation. The second column of the layout, entitledElaboration–Instructional Strategies/Suggestions, provides aclarification of the SCO(s) with suggestions for possible teachingstrategies and activities that could be used to help students achieve theoutcome(s). While these strategies and activities are only suggestionsand are not intended to be rigidly applied, they should help toestablish the nature and extent of the development of the SCO(s).Background theory based on research findings is also often includedin this column.

The third column of the two-page spread, entitled Worthwhile Tasksfor Instruction and/or Assessment, serves several purposes. While thesample tasks presented may be used for assessment, they will alsofurther clarify the SCOs and will often represent useful instructionalactivities. As well, they regularly incorporate one or more of the fourunifying ideas of the curriculum. While these tasks have assessmentheadings—performance, paper and pencil, interview, observation,presentation, and portfolio—, teachers should treat these headings

CURRICULUM OUTCOMES


CURRICULUM OUTCOMES

only as suggestions. These sample tasks are intended as examples only:teachers will want to tailor them to meet the needs and interests oftheir students.

The fourth column of each display, entitled Suggested Resources, isprovided for teachers to use to collect useful references to resourcesthat are particularly valuable in achieving the outcome(s).

Grade 4

Number Concepts/

Number and Relationship

Operations:Number Sense and

Number Concepts

ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4–64-2

GCO A: Students will demonstrate number sense and apply number-theory concepts.

SPECIFIC CURRICULUM OUTCOMES: GRADE 4

Elaboration—Instructional Strategies/Suggestions

KSCO

By the end of grade 6, studentswill have achieved the outcomesfor entry–grade 3 and will also beexpected to

i) demonstrate an understand-ing of number meaningswith respect to whole num-bers, fractions, and decimals

SCO

By the end of grade 4, studentswill be expected to

A1 identify and model frac-tions and mixed numbers

A1 It is important that students develop visual images for fractionsand be able to tell about how much of a whole a particular fractionrepresents. Students should model fractions and mixed numbersusing a variety of materials such as - fraction circles - fraction squares and/or rectangles

21

2

green red

- pattern blocks - geoboards and grid paper

21

2

Students should understand that fractions can describe

(a) part of a whole 1

3(b) part of a group

2

3

• Invite students to determine what fraction of the letters in theirnames are vowels, e.g., TARA EDAM has

1

2 vowels.

To strengthen the fraction concept, change the size of the wholeregularly so students have to adjust their thinking.

• Show students the yellow pattern-block and say, "If this representsone, what does this blue block represent?" Continue by showingthe red block and asking, "If this is one, show me a block torepresent

1

3. If this is one, what does the hexagon represent?"

• Break egg cartons into sections (1 through 11), and use completecartons as well. Distribute at least one of the sections to eachstudent and ask, “If this (whole carton) is one, what is

1

2? If this 9-

section piece is one whole, show me one third. If this 2-sectionpiece is one, show me 2

1

2.” (Note: Students should realize that

any one section can have many different names depending on thesize of the whole. It would also be beneficial for students to framesimilar questions for their classmates.)

In their work with fractions, students should recognize that somemodels will show fractional parts that make a whole; others will showfractional parts that make more than one whole, e.g., 3 thirds

3

3

FHG

IKJ

makes a whole and 6 thirds 6

3

FHG

IKJ makes 2 wholes. It is quite appropri-

ate to use these fractions. This should be extended to mixed numbers,e.g.,

7

3 = 2

1

3 that would be modelled as

Many pairs of fractions can be comparedwithout using a formal algorithm, such asfinding a common denominator orchanging each fraction to a decimal.Children need informal ordering schemesto estimate fractions quickly or to judgethe reasonableness of answers. They canbe led to discover these relationships ifthey have had experiences in construct-ing mental images of fractions. (NCTM1989a, 162)

2

31

3

1

4

ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4–6 4-3

Worthwhile Tasks for Instruction and/or Assessment Resources/Notes



PerformanceA1.1 Ask students to flip coins 10 times and name the fraction thatdescribes the frequency of “tails.”

A1.2 Have students “shake and spill” a number of two-colouredcounters and ask them to name the fraction that represents the redcounters.

A1.3 Provide pattern blocks for the students. Ask them to make and

describe a design that shows 42

3 . Have them model

8

3 and provide

another name to describe it.

Paper and PencilA1.4 Ask students to divide cakes such as the one below into thirdsin two different ways:

A1.5 Provide students with shapes. Ask, If your shape represents awhole, draw a shape that would show 2

1

2.

InterviewA1.6 Ask students to explain how both diagrams below show twothirds.

A1.7 Tell students that Martin said that the green pattern-blockrepresented

1

3. Stephen said that Martin must be wrong because he

knew the blue block represented 1

3. Ask students to provide explana-

tions for this difference of opinion.

PortfolioA1.8 Ask students to design flags that can be described using thirds.

A1.9 Ask students to decide whether they would rather have, 4

5 of a

pizza or 4

3 of a pizza. Have them explain their choice in writing.





KSCO

By the end of grade 6, studentswill have achieved the outcomesfor entry–grade 3 and will alsobe expected to

i) demonstrate an under-standing of numbermeanings with respect towhole numbers, fractionsand decimals

SCO


A2 interpret and modeldecimal tenths andhundredths

A2 Because decimals are fractional parts, it is essential that therelationship between decimals and fractions be regularly addressed.Students should use a variety of materials to model and interpretdecimal tenths and hundredths. Models could include:

- base-10 blocks

- grids or decimal squares - metre sticks

- 10 x 10 geoboards

- hundredths circles or disks

- money = 0.23

Money is perhaps the least effective for conceptual developmentbecause most students do not think of cents as hundredths of adollar, i.e., They are not thinking of the decimals as fractions.

Most students may see the relationship between 0.01, 0.1 and 1.0better if analogies are made to real-life objects that are sized propor-tionally.

Models must be used at all grade levelsto develop fraction concepts adequately.Further, ... children should haveexperiences with a wide assortment ofmodels. (Van de Walle 1994, 222-23)





PerformanceA2.1 Ask students to show 0.2 if represents one whole;

if represents one whole; if represents one whole.

A2.2 Ask students, About where would you place 1.76 on thisnumber line? Have their explain their choices.

Paper and PencilA2.3 Have students use hundred grids to show a capital “T” thattakes up more than 0.2 of the grid and one that takes less than 0.20of the grid. Ask them what decimal part they could be given thatwould make the task very difficult.

InterviewA2.4 Ask students to select two different models with which to show0.38.

A2.5 Ask students to use models of their choice to explain why 0.40is the same as 0.4.

A2.6 Have students shade in their estimates of 0.36 of a circle.Provide an acetate overlay of a hundredths circle/disk so that theirestimates can be checked.

A2.7 Ask students, How long is 0.25 metres? How do you know?

A2.8 Ask students if 0.53 would be read as “53 hundredths” or as“5 tenths and 3 hundredths.” Also ask them to describe situations whenthis number might be used.

A2.9 Ask students to give the numbers that are 0.01 more than, andless than, three and twenty-four hundredths.

A2.10 Ask students to identify situations in which 0.36 might referto a large amount.

PortfolioA2.11 Ask students to draw pictures and write everything they knowabout 0.45 including examples of situations where it could be used.





KSCO


iii) read and write wholenumbers and decimals anddemonstrate an understand-ing of place value (tomillions and to thou-sandths)

SCO


A3 model and record numbersto 99 999

A3 Students should recognize the value represented by each digit ina number, as well as what the number means as a whole. Includesituations in which students use - base-10 blocks, e.g., To model 10 000 have the class make a long rod with 10 big cubes. It will be a 10 thousand rod. - money, e.g., How many $100 bills are there in $12 347? - place-value charts Thousands

Students should have opportunities to(a) model numbers including thosecontaining zeros

e.g., 1003 means 1 thousand 3 ones

(b) read numbers several ways, e.g., 12 347 is read 12 thousand,three hundred forty-seven but might also be expressed as 12 thou-sands, 34 tens, 7 ones; 12 thousands, 33 tens, 17 ones; 123 hun-dreds, 47 ones.(c) record numbers, e.g., Ask students to write twenty-eight thou-sand sixty; a number which is eighty less than ninety thousand.

• Invite students to investigate the length of a line comprising10 000 pennies.Encourage students to share the various strategies they used toinvestigate this problem. It is also important to have them sharestrategies that they considered, but rejected.

• Provide 10-sided dice, prepared cards marked 0–9, or playingcards ace (1) to 9 (with the joker as the zero). Shuffle the cards.Have students select 5 cards each (or toss a die 5 times) and makethe greatest (least) possible number. Have students lay out thecards (leaving a space after the thousands) and read their numbersto their groups. Have some students write their numbers on theboard and read them. Ask, Who thinks they might have thegreatest number? How far from the greatest possible number isyours? Would it be possible for someone to have the greatest andthe least possible numbers when the cards are rearranged? Whatdigits would you want in order to have the greatest differencebetween your greatest and your least numbers?





PerformanceA3.1 Provide decks of four sets of cards numbered 0–9. Ask studentsto shuffle the deck, select five cards, and arrange them to make thelargest possible number. Ask them to record and read the numberand to rearrange the cards to make the smallest possible number.Have them record this number under their larger number. Ask themto estimate the differences between their two numbers. (Perhaps theycould practise the front-end subtraction strategy.)

A3.2 Ask students to use base-10 blocks to model 2046 in threedifferent ways. Have them explain the models.

Paper and PencilA3.3 Tell students that a number is represented by using 10 largebase-10 cubes and some flats. Ask, What might this number be?

A3.4 Tell students that a number has five digits. The digit in the tenthousands place is greater than the digit in the tens place. Ask, Whatis the greatest number this could be? the least?

A3.5 Ask students to record a series of numbers that have been readto them, such as forty-six thousand eighty-two, ninety thousand five,and twelve thousand one. (Include numbers such as the greatest 5-digit number or a number one hundred less than the greatest 5-digitnumber.)

InterviewA3.6 Ask students to imagine flats placed on top of each other toform a tower. Ask, How many hundred flats would be required toconstruct a tower representing 75 000? How high would this be?

A3.7 Ask students, For how many $100 bills could $15 000 befairly exchanged?

A3.8 Ask students to describe the base-10 blocks that they wouldneed to show 75 089.

A3.9 Tell student that the number 13 420 has 134 hundreds and 2tens. Ask them to describe three other ways to express this number.

A3.10 Tell students that a car costs $16 135. Ask, If you were to payfor it in $10 bills, how many would you need?

A3.11 Put 1003 and 10 003 on the board. Ask, How are these twonumbers different? similar?





KSCO


iv) order whole numbers,fractions and decimals andrepresent them in multipleways

SCO


A4 compare and order wholenumbers

A4 Students should investigate meaningful contexts to compare andorder two or more numbers both with and without models. It isexpected that they be able to explain why one number is greater orless than another, e.g., 2542 < 3653 because 2542 is less than 3thousand while 3653 is more than 3 thousand.

• Provide students with opportunities to practise comparingnumbers such as 32 998 and 33 010 asking them to explaintheir reasoning.

• Prepare cards with numbers for students to order from least togreatest, e.g.,

• Assign pairs of students the task of making challenging numbercards for their classmates to order.

• Ask students to name numbers which are greater than or less thana number you give them. In some cases the amount greater or lesscould be specified, such as 29 more or 3000 less. Ask them toname numbers which are between two numbers you give them.

• Invite students to decide which of the following is worth more:11 356 quarters, 27 462 dimes or 99 999 pennies.Have students predict first, then use calculators to help solvethe problem.

• Display a 5-digit number on an overhead calculator (or on a cardor on the board). Have students enter on their calculators anumber which differs by 1 digit. Have them read their numbersand ask others to determine if their numbers are greater than orless than the number on the overhead. Collect five of their num-bers and ask the students to order them. Ask for an explanation ofthe order.

Number sense is the ability to under-stand and use numbers and operationson numbers in computation, measure-ment, and estimation situations. Thisability takes many years to develop and iswell worth the investment; it is valuableto both children and adults as theyencounter mathematical situations inand out of school. When children haveexperiences that encourage them tomodel and describe numbers in a varietyof settings, they will readily learn toapply mathematical understandings inappropriate and efficient ways. (NCTM1992a, 9)





PerformanceA4.1 Have students use reference books to find the populations oftwo communities. Then ask them to find another population that isgreater than that of one of the communities, but less than the other.

A4.2 Give students some number cards and ask them to order thenumbers from largest to smallest.

Paper and PencilA4.3 Ask students to find three ways to fill in the boxes to make thestatement true: 2 245 > 15 8 4

A4.4 Ask student to record two numbers: the first has 3 in thethousands place, but is less than the second that has 3 in the hun-dreds place.

A4.5 Ask students to each write a number that has 5210 tens.

A4.6 Ask students to each write a number that would fall about halfway between 95 987 and 100 000.

A4.7 Tell students that you are thinking of a 5-digit number thathas 4 thousands, a greater number of tens, and an even greaternumber of ones. Ask them to give three possibilities.

A4.8 Have students create numbers that are greater then 98 950using the same digits. Ask, How many of these numbers are there?

InterviewA4.9 Tell students that Jodi’s number had 9 hundreds and Fran’shad only 6 hundreds but Fran’s number was greater. Ask, How wasthis possible?

A4.10 Ask students, Which number below must be greater? Explainwhy. 4 2 9 3

A4.11 Ask students how many whole numbers are greater than 8000but less than 8750.

A4.12 Ask students how they might explain to younger students theway to determine which of two numbers is greater.

PortfolioA4.13 Have students find large numbers from newspapers andmagazines. Ask them to create a collage that would illustrate theorder of the numbers from least to greatest.





KSCO



SCO


A5 compare and orderfractions

A5 Students should recognize that there are various ways to comparefractions. It is important that these early experiences be embedded ininvestigations with various models. It is through these investigationsthat students will develop a visual image of fractions which isessential for fraction number sense.

Provide situations in which students will explore and comparefractions using(a) area models (part of a whole area)

1

3>

1

4

(b) length models (part of a length measurement)

(c) set models (part of a set of like objects)

Students should be able to compare fractions - having the same denominator, e.g.,

2

6 <

5

6 because if an item is cut

into 6 equal pieces, 2 of those pieces are less than 5 of them.

- having the same numerator, e.g., 1

3 >

1

4 because if 3 people share

1 item, they will each get more than if 4 people share this item.

- using reference points such as 1

2 or 1, e.g.,

2

5 <

7

8 because

2

5 is less than

1

2, while

7

8 is more than

1

2.

Encourage students to use various comparison methods, dependingon the situation.

• Invite students to create problems for others to solve, e.g., Ted ate2

5 of a cheese pizza, and Lee ate

2

3 of a mushroom pizza. Each ate

two pieces. Who ate more? (Note: It is important that theyrecognize that this can be solved only if they know the sizes of thepizzas.)

Many pairs of fractions can be comparedwithout using a formal algorithm, such asfinding a common denominator orchanging each fraction to a decimal.Children need informal ordering schemesto estimate fractions quickly or to judgethe reasonableness of answers. They canbe led to discover these relationships ifthey have had experiences in construct-ing mental images of fractions. (NCTM1989a, 162)





PerformanceA5.1 Ask students to estimate where the following fractions wouldbe on a number line marked only with a 0 and a 1:

2

3,

1

5,

7

8, and

1

3

A5.2 Give pairs of students cards with the following fractions onthem:

3

6

2

3

1

4

7

8

5

12

5

6

Ask them to arrange the cards in order from smallest to largest and togive reasons for their arrangements.

Paper and PencilA5.3 Ask students to list three fractions between

1

2 and 1.

A5.4 Ask students, What possible numerators could be used in thestatement below if both fractions are less than 1?

InterviewA5.5 Ask students to explain why

2

3 is greater than

2

5.

A5.6 Ask students, Why is it easy to compare 2

5 and

2

12?

PresentationA5.7 Ask pairs of students to work together to provide the class withexplanations for how they know that 7

8 is closer to 1 than

5

6.

PortfolioA5.8 Have students explain in writing how it is possible for

2

3 of one

pizza to be less than 1

2 of another pizza.

A5.9 Ask students to write letters to a younger child explaining whyhaving

2

3 of a number of dimes is better than having

3

6 of the same

number of dimes.





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A6 rename fractions with andwithout the use of models

The use of manipulatives is crucial indeveloping students' understanding offraction ideas. Manipulatives helpstudents construct mental referents thatenable them to perform fraction tasksmeaningfully. (NCTM 1989a, 158)

A6 Students should be able to find equivalent fractions,

e.g., 1

2 =

2

4 =

3

6.

To develop conceptual understanding of equivalency, it is importantthat models be used to generate the different representations of afraction. The “rule” for finding equivalent fractions should not betaught or used until the students understand the reason for applyingthis rule. They must understand why a fraction can have anothername, e.g.,

2

3 =

4

6 and yet have the same value. Students should be

encouraged to look for patterns in equivalency. It is recommendedthat the term “reduce” be avoided since it suggests to the studentsthat the size actually changes.

Students must develop number sense for fractions. The use of visualsis a key component in this development.

In this example, if each third is cut in half, there will then be twice asmany total pieces and twice as many shaded pieces than before.

Pattern blocks work well to show equivalence, e.g., if the hexagonis one whole unit, then

• Invite students to make designs with pattern blocks and givevalues to their design based on one of the blocks being 1.If = 1, then = 3

2

3.

Students may present their designs for others to rename by

covering them with other combinations of pieces, e.g.,11

3 = 3

2

3

Paper folding is also an effective way of showing equivalency.

• Ask students to fold pieces of paper in half, open them and shadein one part. Have them refold the paper and once again fold inhalf. When opened, the folds now provide the coloured visual for1

2 =

2

4. Continue to fold to show

1

2 =

2

4 =

4

8 =

8

16.





PerformanceA6.1 Ask students to use rectangles on geoboards to determine othernames for

1

4.

A6.2 Ask student to use models of their choice to show that 3

4 =

6

8.

A6.3 Ask students to fold pieces of paper to show that 1

3 =

2

6.

Paper and PencilA6.4 Have students use diagrams to show that

2

5 =

4

10.

A6.5 Ask students, What equivalent fractions does this diagramshow? Have them list all the pairs they can.

InterviewA6.6 Tell students that Sally ate

2

3 of a large pizza and that Billy ate

4

6 of a medium-sized pizza. Sally said that they ate the same amount

because she had learned that 2

3 =

4

6. Ask students to respond to

Sally’s comment.

A6.7 Ask students, Why is there always another name for a fraction?

PresentationA6.8 Ask students to work in pairs to provide explanations for theirclassmates as to why

G

10 cannot be another name for

3

4.

PortfolioA6.9 Ask students to write what they know about equivalent frac-tions. Advise them that they could use pictures to help.

A6.10 Have students draw visual representations and explain in

writing how they know that 32

3 =

11

3.





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A7 compare and order deci-mals with and withoutmodels

A7 Students should compare and order decimals with and withoutusing models while investigating relevant situations, e.g., sports eventswith measurement of times, distances, and scores and capacities ofvarious containers such as 25 mL, 0.5 L, 500 mL.

• Invite students to make tablesof distances (expressed inmetres) that represents howfar each person can kick atissue, flip a coin, and othersuch tasks. They might thenlist each set of distances fromleast to greatest.

The number line and metre stick both provide good models fordecimals.

• Have students work together to locate 0.5 m and 0.6 m on a metrestick or measuring tape and to discover how they would name thepoints between the two.

It is important that work with decimals not be distinct from work withfractions. Decimals are fractional parts, and the connection between thetwo should be a major part of exploration.

Hundredths disks and base-10 blocks are helpful to make this connec-tion.





PerformanceA7.1 Ask students to use base-10 blocks to explain why 1.02 < 1.2.

Paper and PencilA7.2 Have students record three decimal numbers between 0.4 and0.5.

A7.3 Have students arrange the digits 4, 2, 9, and 0 in the boxesbelow to make the greatest and least amounts.

.

A7.4 Ask students to find at least three ways to make the followingstatement true: 1.3 < 1. 2

Ask, Can the statement be true if the first box contains a 2? If thesecond contains a 3? Explain.

InterviewA7.5 Provide the times for four sprinters in a 100-metre race. Askstudents to determine the first-, second-, and third-place winners.

A7.6 Ask students to order the following measurements from least togreatest and to provide an explanation for the order.

1.24 m, 2.01 m, 0.97 m, 2.20 m, 3 m, and 108 m

A7.7 Ask students to name situations in which a contest result of0.23 might actually beat 0.26.

PortfolioA7.8 Have students use each of the digits 0–9 once to fill in the tenboxes to make the statements true.

Ask students, How many different ways can be found?

A7.9 Ask students to write about situations in which they wouldprefer to use fractions and situations where decimals would be theirchoice.

Number Concepts/


Operations:Operation Sense and

Number Operations

4-18 ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4–6


GCO B: Students will demonstrate operation sense and apply operation principles andprocedures in both numeric and algebraic situations.


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i) model problem situationsinvolving whole numbersand decimals by selectingappropriate operations andprocedures

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By the end of grade 4, students willbe expected to

B1 add and subtract decimalsinvolving tenths and hun-dredths, and whole numbersto five digits

B1 Students should recognize that adding or subtracting tenths,e.g., 3 tenths and 4 tenths are 7 tenths, is analogous to adding or sub-tracting quantities of other items, e.g., 3 apples and 4 apples are7 apples. The same is true with hundredths. Rather than simply tellingstudents to line up decimals vertically, or suggesting that they “addzeros,” you should suggest that they visualize what each digitrepresents in base-10 blocks and what blocks would be combined.

To determine solutions to questions such as 1.62 + 0.3, students mightthink, I have 1 whole, 9 (6 + 3) tenths and 2 hundredths, or 1.92. Tosubtract 1.4 from 3, they might think, 3 - 1 or 2 wholes, subtracting 4tenths from 2 is 1 and 6 tenths, or 1.6. Alternatively, they might think30 tenths - 14 tenths = 16 tenths, or 1.6.

Base-10 blocks and hundredths grids continue to be useful modelsfor students to visualize the addition or subtraction of decimals. If a flatrepresents one whole unit, then 3.2 + 1.54 would be modelled as

Students need to recognize that all the properties and techniques estab-lished for the addition and subtraction of whole numbers also apply todecimals.

Provide students with story problems that require the addition and/or subtraction of whole numbers and decimals. Particularly appropriatecontexts are money and measurement problems, e.g., (a) If an insecttravels 3.45m while a car travels 721.6m, how much farther has the cartravelled? (b) The chef purchased 12.4kg steak to be ready for the lunchhour at the diner. If he already had 7.25kg steak in the deep freeze, howmuch steak does he have altogether? (c) If the length of a file card is20.3cm and its width is 12.6cm, what is its perimeter?





PerformanceB1.1 Ask students to use calculators, the digits 7, 5, 1, and 2, and thesymbols +, =, and . (decimal point) to produce 7.8 on the displays.

B1.2 Tell students that Alan’s solution to an addition question is shownbelow. Ask them to write to Alan explaining what he did wrong and touse pictures of base-10 blocks to show how to solve the problem.

5.2 + 3.4 5 3.9 7

B1.3 Ask students to use hundredths grids or base-10 blocks to model2.3 - 1.8.

Paper and PencilB1.4 Ask students to estimate and then calculate the following:

4268.73 6.473 + 12.89 + 4.06 = - 79.45 52790.3 - 801.43 =

InterviewB1.5 Ask students to select from the following questions the one theythink is the easiest to solve and to explain why:

6 - 0.531.43 - 0.872.64 - 0.99

B1.6 Ask students to explain how solving 0.4 + 0.5 is like solving 4 + 5.

B1.7 Ask students how they might solve 4.97 and 6.99 mentally.





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B2 demonstrate an understand-ing of multiplicationmeanings and applications

B2 Students should continue to recognize the array, set, andrepeated addition meanings for multiplication. They should alsoexplore three other meanings for multiplication.(a) Multiplication is used to find combinations where eachelement in one group is matched with each element inanother group, e.g., Toni has 3 shirts and 4 pairs of pants.How many different outfits are there? 3 x 4

(b) Multiplication is often used in rate problems, e.g., Jane biked 6 km/h. At this rate, how far did she bike in four hours?

(c) Multiplication can be used in making comparisons, e.g., Moira hassaved $8. Morgan has saved 6 times as much. How much has Morgansaved?

Students should know and understand some fundamental principlesabout mulitplication; namely, the order (commutative) property, thedistributive property, the role of zero, and the role of one.

The order (commutative) property involves recognizing that multiplica-tions such as 3 x 4 and 4 x 3 have the same product but differentmeanings. 3 x 4 is read as 3 sets of 4 while 4 x 3 is 4 sets of 3; however,both products are 12. Pictorially, this array shows 3rows of 4. If it is turned a quarter turn, 4 rows of 3would be seen.

The distributive property is illustrated byx x x x x x x xx x x x x x x x 4 x 8 = (4 x 5) + (4 x 3)x x x x x x x x = 20 + 12x x x x x x x x = 32

The roles of zero and one in multiplication need special attention—theyare often sources of error for students. To help them develop goodunderstandings use contexts and pictures, e.g., On a number line theycan see that 3 hops of 0 or 0 hops of 3 leaves them still on 0 or that 1hop of 3 moves them to 3, the same as 3 hops of 1.

Provide opportunities for students not only to solve multiplicationproblems, but also to create their own problems. Encourage them to usea variety of different meanings.





PerformanceB2.1 Ask students to use counters to show 5 x 8.

B2.2 Have students use products and models of their choice to explainthe order property, e.g., 4 x 9 = 9 x 4.

Paper and PencilB2.3 Ask students to illustrate two different ways to think about6 x 7.

B2.4 Tell students that an ice-cream store sells waffle cones and regularcones and that it has only four flavours of ice cream. Ask them toillustrate how to determine the number of different choices they have atthis store.

InterviewB2.5 Ask students, How does an array model show repeated addition?

B2.6 Ask students, What multiplication number sentence does thispicture show?

25 cm 25 cm 25 cm

B2.7 Ask students to explain how knowing 6 x 5 might help them tofind 12 x 5.

PortfolioB2.8 Ask students to explain how multiplying could be involved infinding the perimeters of squares.

B2.9 Have students pose and answer three problems that could besolved by multiplication. Ask them to solve these problems in differentways, one of which does not involve multiplying.

B2.10 Tell students that you have eight boxes, each of which holds sixchalkboard erasers, and one other box that has only five erasers in it.Ask students to describe at least two ways one could find the totalnumber of erasers. Ask them to explain which way they would preferand why.





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B3 demonstrate an understand-ing of the various meaningsof division

B3 Students have had some experiences with division, particularly insituations involving sharing fairly and finding how many groups.

(a) sharing, e.g., Bill has 30 candies. He wants to share themequally among his 5 friends. How many candies should eachfriend receive? (Note: A special case of sharing involves finding means(averages), in which a total is theoretically shared fairly among a givennumber of individuals. This specific type of sharing is likely to be newfor your students.)

• Ask students to model the following situation with counters:Jeremy has 8 marbles, Maggie has 10 marbles, Joshua has 13 marbles,and Simon has 9 marbles. If they put the marbles together and redis-tribute them evenly, how many will they each get?Provide many similar opportunities for students to find means (aver-ages) using concrete materials.

(b) finding how many groups, e.g., Bill had 30 candies. He wanted to put6 of them into each bag. How may bags did Bill need?

Other meanings of multiplication—such as rate, comparison, andcombinations—have corresponding division problems.

(c) rate, e.g., Jane left the tap dripping. She wasted 120 mL ofwater in 30 minutes. How much water was wasted in 1 minute?15 minutes? in 10 minutes?

(d) comparison, e.g., Erica earned four times as much as herbrother raking leaves. She earned $24. How much did herbrother earn?

(e) combinations, e.g., Kevin says he has 8 outfits made up ofpants and shirts. He only has 2 pairs of pants. How many shirtsmust he have?

... There are two different concepts ofdivision, depending on which factor isunknown. ... If a quantity is to beseparated evenly into a given number ofsubsets [i.e., fair sharing], then divisionexpresses the number in each subset. ... Ifa quantity is to be measured out into setsof a specified size, then the divisionexpresses the number of such sets thatcan be made [i.e., how many groups].(Van de Walle 1994, 124-25)





PerformanceB3.1 Give a number of students different lengths of paperclip chains.Have them work together to find ways to give everyone chains of equallength.

B3.2 Ask four students to find their heights and report on theiraverage (mean) height.

B3.3 Ask students to use counters to model two meanings for 36 4÷ .Ask them to use stories about 36 candies to show these two meanings.

Paper and PencilB3.4 Tell students that an amusement park has rides that are priced asfollows:

$1 for the Ferris wheel,$2 for the Bullet, and$3 for the Twister.

Ask, How many rides, and of which kind, can you have for $13? Arethere other answers?

B3.5 Tell students that the corner store offers twelve different kinds ofsandwiches and that the customer has a choice of white or whole-wheatbread. Ask, How many different kinds of sandwich fillings must thecorner store have?

InterviewB3.6 Ask students to describe two different ways to think about

42 7÷ .

B3.7 Show students this diagram and ask them to create a story prob-lem that would be modelled inthis way.

B3.8 Ask students, Why can’tyou have a remainder of 4 whendividing by 3?

B3.9 Tell students that you have divided 49 by a number and there wasa remainder of 1. Ask, What might the number have been?





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B4 multiply 2- or 3-digitnumbers by single-digitnumbers concretely, pictori-ally, and symbolically

B4 It is important that students understand multiplication and notsimply follow a procedure to obtain a product. Encourage students toexplore their own methods–ways that make sense to them–for findingproducts. These “invented algorithms” are often the procedure of choiceeven when a more traditional algorithm has been presented.

Students should recognize the correspondence of the symbols with aphysical multiplication model, e.g., For the problem: Three cases of 34bottles are delivered. How many bottles are there altogether?, studentalgorithms could include the following:

This algorithm leads to the front-end mentalalgorithm.

This method is a precursor for the more tradi-tional algorithm.

The tradional algorithm: Here, 10 onesmake 1 ten and 2 ones areleft. 9 tens and 1 more from theregrouping make 10 tens.

The base-10 blocks serve as a tool for understanding themultiplication operation. It is important that students usebase-ten language as they manipulate the materials and record thecorresponding symbols for the products, e.g., To describemultiplying 34 by 3 a student might say, “3 groups of 4 ones is 12 ones.I’ll regroup 10 ones for 1 ten. That leaves 2 ones to place in the onesplace. 3 groups of 3 tens is 9 tens and I’ll add the regrouped ten to make10 tens to place in the tens place. The answer is 102.”

When students are comfortable with the process, they should beencouraged to use the front-end mental multiplication strategy forquestions such as 3 x 125 = 375 (3 x 100 + 3 x 25).





PerformanceB4.1 Ask students to use base-10 blocks to determine how far a bluewhale can swim in 7 hours if it swims 37 km/hour.

B4.2 Have students construct a 2-digit number, using six base-10blocks of any size. Ask, If your number were to be repeated four moretimes, what would the total be? Have them record the multiplicationsentences that would describe their situations.

B4.3 Have students use grid papers to draw array pictures for 6 x 17.Ask them to partition their arrays to show the two steps in the multipli-cation procedure to calculate 6 x 17 and to explain these steps.

Paper and PencilB4.4 Ask students to fill in the boxes with 3, 4, and 5 in three differentways and to calculate the different answers.

x

B4.5 Ask students, Can the units digit in the answer of the multiplica-tion sentence below be 5? Explain.

4 x

B4.6 Ask students to determine the missing digits in multiplicationssuch as the following:

5 23 2 x 8 x 5 x 4 52 26 3300

InterviewB4.7 Ask students to explain how they know that 2 x 152 is greaterthan 300.

B4.8 Tell students that to compute 5 x 63, Jason first said, “5 x 60 is300.” Ask, What would he say next?

PresentationB4.9 Have students prepare reports on the speeds of various animals,calculating how long it would take one animal to travel the same dis-tance as another.





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By the end of grade 6, studentswill have achieved the otucomesfor entry–grade 3 and will also beexpected to


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B5 divide 2- and 3-digit wholenumbers by a single-digitdivisor

B5 Students may use more than one algorithm to divide by 1-digitnumbers, e.g., They may perform 63 ÷ 3 by

(a) sharing

“If I share 6 rodsamong 3, each gets 2.”

“I shared 6 rods altogether, or 60.I have 3 units left to share.”

(b) identifying how many groups

“10 groups of 3 uses up 30.”

“I have enough for 10 more groups of 3.”

“I can make 1 more group of 3.”

Students should be encouraged to record theirprocedures, using notation that makes sense tothem, e.g., When dividing 269 by 4, students mayuse their knowledge that four groups of 50 is 200,leaving 69 to share. Four additional groups of 10 is40, with 29 left to share among the four, which is 7more for each, or a total of 4 groups of 67, andthere will be 1 left over.

Present division questions in context to identify either the sharing (howmany in each group) or the measurement (how many groups) meaning.An example of a sharing problem would be:

Brent no longer wants his collection of 556 pogs. He wishes togive them to his 3 brothers and 1 cousin. How can he dividethem fairly?

A student may write 100 + 25 + 10 + 4 = 139 and say, “I can give eachof them 100 pogs, using 400 of them, and then 25 more will use upanother 100. That will leave 56. I’ll give each of them another 10 andhave 16 left. That’s easy—each will get 4 more—for a total of 139.”

... There are two different concepts ofdivision, depending on which factor isunknown ... If a quantity is to beseparated evenly into a given number ofsubsets [i.e., fair sharing], then divisionexpresses the number in each subset ... Ifa quantity is to be measured out into setsof a specified size, then the divisionexpresses the number of such sets thatcan be made [i.e., how many groups].(Van de Walle 1994, 124-25)





PerformanceB5.1 Have students draw models to show 343 ÷ 3.

B5.2 Provide sets of base-10 blocks. Ask students to model threedifferent division questions of their choice and to write the divisionsentence for each.

B5.3 Ask students to determine how to divide $7.47 equally among 3people. Ask them to share different methods they have used.

Paper and PencilB5.4 Ask students to fill in the boxes to make this calculation true.

B5.5 Tell students that Chris wishes to make the frame for a sandboxfrom a plank 455 cm in length. If the plank is cut into four pieces, whatis the largest square sandbox that Chris can make?

B5.6 Have students calculate 48 ÷ 2, 48 ÷ 4, and 48 ÷ 8. Ask, Whatpattern do you observe? Could you write another division sentencewithout having to do the division process?

B5.7 Have students explain in writing why the answer to 69 ÷ 3 has tobe 10 more than the answer to 39 ÷ 3.

Interview

B5.8 Ask students, Are there any hundreds in the quotient of 4 3 5 7 ?

Explain how you know.

PortfolioB5.9 Invite students to describe division questions that they find easyand those that they find more difficult. Have them provide explanationsfor their choices.





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ii) model problem situationsinvolving the addition andsubtraction of simplefractions

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B6 use models informally toadd simple fractions withcommon denominators

Have students add two fractions using afraction model. The results should comecompletely from the use of the modeleven if some of the students have beenexposed to symbolic rules and the idea ofa common denominator. (Van de Walle1994, 244)

B6 Students should not work in any formal way with adding or sub-tracting fractions. It makes sense, however, to capitalize on students’intuitive knowledge that

- a half and a half are a whole- a fourth and a fourth are a half- an eighth and an eighth are two-eighths- a tenth and another two-tenths are three-tenths

When working with tenths, a connection to decimal computationsshould be made.

Manipulatives such as pattern blocks, square tiles, fraction pieces, andgeoboards can all be used to model addition and subtraction of frac-tions, e.g.,

Opportunities should be provided for students to use models such aspattern blocks to investigate other fractional combinations, e.g.,

if = 1, = 1

2 +

1

6 +

1

6 +

1

6 = 1 .

Note: Paper and pencil fraction computations are not necessary at thispoint.





PerformanceB6.1 Have students use base-10 blocks to show why 3 tenths and 2tenths equal 5 tenths.

B6.2 Ask students to use fraction pieces to show that three fourths isone half more than one fourth.

B6.3 Instruct students to cover hexagonal pattern blocks with red andgreen blocks, and ask them to describe the addition that is shown.

B6.4 Tell students that Bill and Ann each used a model to add 3

5 and

4

5. Bill said that the answer was

7

5, while Ann found the result to be

12

5. Ask students to use models to show that both answers are correct.

InterviewB6.5 Ask students how they know that three fifths and two fifths equala whole.

B6.6 Ask students how many and what arrangement of square tiles theymight use in a display that would show that

1

3 +

1

3 =

2

3.

B6.7 Show students the following rectangle that is made up of 4 greentiles, 2 blue tiles, and 2 yellow tiles. Ask them to describe the addition offractions that is modelled.

Portfolio

B6.8 Ask students to design rectangular mats that are 1

4 blue,

1

4red,

1

8yellow, and the rest white.





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iii) explore algebraic situationsinformally

iv) apply computational factsand procedures (algorithms)in a wide variety of problemsituations involving wholenumber and decimals

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B7 demonstrate an understand-ing of the use of the openframe as a place holder for adigit on some occasions andfor a number on otheroccasions

B8 relate multiplication anddivision facts, using princi-ples of these operations

B7 Students have been using the open frame ( ) for several years as aplaceholder. It is worthwhile to have students explicitly discuss what theframe means. Sometimes the open frame will represent a number, as in x 4 = 24, and sometimes it will represent a single digit, as in5 x 3 = 168.

B8 No formal use of variables or algebraic procedures is expected;however, it is not too early to point out to students that some of therelationships they use in relating multiplication and division facts can bedescribed in terms of general “rules”, e.g., Students might realize thatwhen multiplying two numbers, one can be halved and the other dou-bled without changing the product such as in 3 x 16 being the same as 6x 8 since twice as many groups will have half as many in each group.

Similarly, students might realize that they can double both numbers in adivision calculation without changing the result, e.g., Consider that iftwice as many people share twice as much material, everyone will stillhave the same size share that they would have had initially such as 12shared among 4 is the same as 24 is shared among 8 with each getting 3.

Person Number12 3 4 5 6 7 8

x x x x x x x xx x x x x x x xx x x x x x x x

You should ensure that students recognize that multiplication anddivision are two ways of looking at the same situation—this is very clearwhen they examine models or pictures. Most students think, “What do Imultiply 3 by to get 18?” when asked to find 18 ÷ 3. Students who havelearned their multiplication facts have automatically learned theirdivision facts.





PerformanceB8.1 Provide students with counters. Ask them to rearrange models for12 sets of 5 (12 x 5) and to show why it has to have the same answer as4 x 15.

InterviewB8.2 Ask students to explain why doubling both numbers in a divisionquestion leaves the answer unchanged, while doubling both numbers ina multiplication question produces a different product.

B8.3 Tell students that to calculate 5 x 14 Leigh said, "That’s10 x 7,” while Connie said, “That’s 50 + 20.” Ask students to explainthese strategies.

B7.1 Have students discuss the difference in their interpretations ofthese two open sentences:

+ = 12 and + = 8

B7.2 Ask students to find a value to put in the boxes so that thissentence is true: 3 ÷ = . Ask them why there is only one possibleanswer.

B8.4 Tell students that Melissa had trouble remembering multiplicationfacts involving 8. Have them offer suggestions on how Melissa mightsolve 8 x 15 without having to know the 8 times table.

PortfolioB8.5 Ask students to create sets of “rules” that describe some of therelationships between groups of multiplication facts.

B8.6 Tell students that Jasmine said that she was supposed to divide130 by 5, but found it easier to divide 260 by 10. Ask them to explainJasmine’s method. Have students give examples for which it is mucheasier to double both numbers before dividing.

B8.7 Tell students that the “6” button on the calculator is not working.Have them suggest ways to use the calculator to solve “6 x 64” withoutusing this button.





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iv) apply computational factsand procedures (algorithms)in a wide variety of problemsituations involving wholenumbers and decimals

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B9 demonstrate a knowledge ofmultiplication facts to 9 x 9

B9 Students should have developed an understanding of the concept ofmultiplication through story problems, manipulatives, and pictorialrepresentations as well as symbolic representation. Some students willhave developed efficient strategies for remembering some facts. Demon-strating knowledge of a multiplication fact, however, means giving aquick response—less than three seconds for most students— withoutresorting to non-efficient techiques such as counting. This should not beconstrued, however, as an endorsement of timed pencil-paper fact tests.Most rehearsal sessions for facts should be oral in nature with as muchemphasis on discussion of strategies used as on answers.

Allow time for both strategy development and the practice of thesestrategies to ensure that students can demonstrate their knowledge offacts. Often fact strategies rely on previously-developed strategies orconcepts, e.g., Students might need to be reminded of the order (com-mutative) property, e.g., 3 x 9 = 9 x 3, stressing that while 3 x 9 refers to3 nines and 9 x 3 refers to 9 threes, their products are the same; so, ifthey want 9 x 3, it may be mentally easier to find 3 x 9.

The recommended approach for strategy practice and retrieval is tointroduce the strategy with the use of materials to develop understand-ing of why the strategy works, to practise the strategy with the appropri-ate facts, and then combine it with other strategies that have beenlearned and practised. When students have two or more strategies, it isimportant to focus on strategy selection. Strategy selection involveschoosing the strategy that will be most useful to determine a particularfact or, if a fact has more than one possible strategy, to choose the onethat each student prefers. This fact learning should take place 3–5minutes a day over several months.

Some strategies for multiplication facts include(a) doubles—2 times a number is another form of known additionfacts, e.g., 2 x 7 = 7 + 7 = 7 x 2.(b) fives—It is useful to make the connection between the 5 times tableand the number pointed to by the minute hand of a clock whick indi-cates the number of minutes after the hour, e.g., When the minute handis on the nine, think 45.(c) nifty nines—Students enjoy the patterns found in the nine times table(see question B9.2) or multiplying by 10 and subtracting one set, e.g.,For 9 x 4, think 10 x 4, or 40, subtract 4 is 36. (continued next page)





PerformanceB9.1 Randomly point to a number on a clock. Ask students to respondwith the number of minutes after the hour which corresponds to it, e.g.,35 for 7, 20 for 4. Pairs of students can practise together.

B9.2 Have students examine the “9” facts. Ask, Whatdo you notice about the first digit of the products, e.g.,in 9 x 7 = 63? What do you notice about the sum of thedigits in the products? Use flash cards such as the oneshown to help students practise these strategies.

B9.3 Place students in pairs to practise the “double and double again”strategy for the “4” facts such as 4 x 7, e.g., 4 x 7 is double 2 x 7 which is7 + 7. Since two sevens is 14, 4 x 7 is 28. Students should take turnsasking facts and providing answers by repeated doubling.

B9.4 Have students play the “Target Game.”

3 multiplied by what number is closest to the target number withoutgoing over?

Paper and PencilB9.5 Give students worksheets of facts. Ask them to circle those factsthat they could readily determine using one strategy and underline thosefor which they would use a second strategy. Have students then answeronly these circled and underlined items, e.g., In a list such as this onethey might circle “double and double again” facts and underline “ninetimes” facts and provide answers for these.

____________________________________________________

Strategies (continued from previous page):(d) decomposition—breaking a fact up into parts, calculating the parts,and recombining, e.g., 6 x 8 = 5 x 8 + 1 x 8 = 40 + 8 = 48

It is recommended that you see Van de Walle 1994 or 1997 which has achapter devoted to helping students master their basic facts. This chapterdiscusses these strategies and others more fully. Fact learning is theimportant foundation for estimation and other mental strategies.





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B10 demonstrate an understand-ing of various treatments ofremainders in divisionsituations

B11 solve and create wordproblems involving wholenumber computations’

B12 solve and create wordproblems involving addingand subtracting decimals (tohundredths)

B10 Students should understand that when solving divisionproblems, remainders are handled differently, depending on the context.They should recognize when a remainder

(a) is best described as a fraction, e.g., When 3 children share7 licorice pieces, each gets 2

1

3 licorice pieces.

(b) needs to be ignored, e.g., When you want to know how many 75¢notebooks can be bought with $3.25, the answer is four since there isnot enough money for 5.

(c) needs to be rounded up, e.g., When you want to know how many 4-passenger cars are needed to transport 17 children, the answer is fivesince you cannot leave anyone behind.

(d) must be addressed specifically, e.g., When 91 students are to betransported in 3 buses, there may be 30 on 2 buses, 31 on the other.

B11 Students should have many opportunities to solve and create wordproblems for the purpose of answering real-life questions, preferablychoosing topics of interest to them. These opportunities providestudents with a chance to practise their computational skills and clarifytheir mathematical thinking.

• Ask students to solve this problem: A rumour is started. On Mondayit is told to 2 people. On Tuesday each person who knows tells 3people. On Wednesday each person who knows tells 4 people, and soit continues. How many people know by Sunday?

• Ask students to make up division problems about situations in theclassroom and write the division sentences for them. Have them posttheir division sentences and invite others to try to guess what thedivision situations might be.

B12 Students should have ample opportunity to solve and createdecimal problems within relevant contexts. Contexts which lend them-selves well to decimals include money; measurement, e.g., time, length,capacity, and mass; and fractional parts, e.g., about 0.6 of the studentsin our class are girls.

• Ask students to create perimeter problems about shapes that they havedesigned. Have them exchange problems for solution.





PerformanceB11/12.1 Provide several food packages on which the mass of each ismarked. Ask student to select two packages with a total mass lessthan a particular amount.

Paper and PencilB11/12.2 Ask students to create word problems about a sum of moneyreceived as a birthday present.

B10.1 Ask students, How many grocery bags are needed to carry 34loaves of bread if each bag can hold a maximum of 5 loaves?

B12.1 Ask students to find the perimeter of the figure below:

B11.1 Tell students that there are 228 children in Sharon’s school.Sharon was in charge of buying juice for the school for the field day.She said that each student would need at least two, and maybe three,drinks during the day. Sharon ordered 500 containers of juice. Ask, Willshe have enough? (Encourage an estimate first.)

InterviewB11.2 Ask students to create word problems which 65 + 12 =could represent.

B10.2 Ask students to describe situations where 9, rather than 9R2,might be an appropriate answer to 38 ÷ 4.

B11.3 Ask students to describe multiplication situations that wouldinvolve 144 and 7.

PortfolioB11/12.3 As students create various word problems, have them placethem in a working portfolio. After a group of problems has been created,have students select their five best and provide criteria for their choices.Have them solve one another’s problems.





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v) apply estimation techniquesto predict, and justify thereasonableness of, results inrelevant problem situationsinvolving whole numbersand decimals

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B13 estimate sums and differ-ences of whole numbersand decimals

One of the best ways of working onestimation skills seems to be to integratethem with other areas of the mathemat-ics curriculum. (Van de Walle 1994, 203)

B13 Students need to recognize that estimation is a useful skill in theirlives. To be efficient when estimating sums and differences mentally,students must be able to access a strategy quickly and they need a varietyfrom which to choose. Strategies include among others

(a) using referents, e.g., 27 + 128 might be described as being more than150 (25 + 125), but less than 160 (30 + 130); subtracting 126 from 207would give an answer between 75 (200 - 125), and 85 (210 - 125).

(b) rounding, e.g., 439 + 52 is approximately 440 + 50; 35 + 57 mightbe estimated as 30 + 60 rather than 40 + 60. Often it is just as easy tofind the actual answer using a mental strategy, e.g., To find 439 + 52,adding 1 to 439 and subtracting 1 from 52 might be the strategy ofchoice to get 440 + 51.

(c) front-end addition, e.g., For the addition below, a student might32.2 think, “30 and 30 (20 + 10) is 60 and the ones24.5 and tenths clustered together make about another

14.1 10 for a total of 70.” (Note that this estimate iscloser to the actual answer than if one were toround and add.)

(d) front-end subtraction, e.g.,247 Students may subtract the hundreds first,

- 129 followed by the tens, and then choose to ignorethe difference in the ones place, for a good estimateof 120.

(e) clustering of near compatibles, e.g.,

The 46 and 55 together make about 100; the 134and 68 make about another 200 for a total of 300. +The tenths and hundredths would be ignored.





Paper and PencilB13.1 Ask students to find two numbers with a difference of about 150and a sum of about 500; to find two numbers with a difference of about80 and a sum of about 200.

InterviewB13.2 Provide a list of chores and the time it takes to complete each.Ask students which combination of chores could be completed in about30 minutes.

B13.3 Ask students to explain how they would estimate the differencebetween the costs of two items, e.g., a $599 item and a $378 item.

B13.4 Ask students to estimate what they might subtract in each casebelow so that the answer is close to, but not exactly, 50:

384 - 219 - 68 -

B13.5 Ask students to describe how they would estimate the sum of4.97 + 6.99 + 3.

B13.6 Tell students that to estimate 583 - 165, Jeff said, “575 subtract175.” Ask them if the estimate will be high or low, and to explain whyJeff might have chosen to estimate in this way.

B13.7 Ask students, How can you know that 465 + 593 > 1000 with-out actually completing the addition?

B13.8 Tell students that Jason saved the following amounts over a 4-month period:

June $12.45July $36.57August $54.15September $19.05

Ask them to give an estimate of the total amount saved. Have themexplain how their estimates were determined.





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B14 estimate the product orquotient of 2- or 3-digitnumbers and single-digitnumbers

B14 Students should be expected to estimate products of 2- or3-digit numbers and single-digit numbers before calculating the exactanswers. They might approximate the greater factor with a near multipleof 10 or 100 and either use the single-digit number itself or an approxi-mation of it, such as 5 or 10, e.g., Students might decide that 42 x 8 isabout 40 x 8= 320 or about 40 x 10 = 400 or about 42 x 10 = 420.

Students might use a similar approach when estimating quotients, e.g.,491 ÷ 8 is about 500 ÷ 10 = 50 or about 480 ÷ 8 = 60. Encouragestudents to discuss which of the estimates is closer and why. Numbersthat are easy to work with in estimation situations are sometimes re-ferred to as “nice” or “friendly” numbers. They are often multiples of 5,10 or 100, but may also be chosen for other reasons, e.g., In the lastexample, 480 is chosen because it is easily divided mentally by 8.

When it is easy to decide, students should recognize that their estimatesare likely to be high or low, e.g., Estimating 37 x 6 as 40 x 10 is clearlyhigh, as is estimating 57 ÷ 3 as 60 ÷ 3. On the other hand, estimating420 ÷ 8 as 400 ÷ 8 is clearly low. It is difficult to be sure, however,whether an estimate of 400 ÷ 5 for 432 ÷ 6 is too high or too low.

It is important to remember that computational estimation should be amental activity. Regular oral practice, accompanied by the sharing ofstrategies, must be provided. When assessing the development of theskill, the amount of time provided must be controlled. An alternative toa timed assessment would be an individual interview. The goal is forstudents to become so competent with the skill that they automaticallyestimate in any problem situation, not just when asked to do so by thetext or the teacher.

One of the best ways of working onestimation skills seems to be to integratethem with other areas of the mathemat-ics curriculum. (Van de Walle 1994, 203)





Paper and PencilB14.1 Have students estimate 425 ÷ 6 and indicate whether theirestimates are too high or too low and why they know. Ask them tosuggest other division questions for which the same estimate would beappropriate.

InterviewB14.2 Tell students that Mark estimated 47 x 7 as 500. Ask them whatstrategy they think Mark was using and if they would estimate it differ-ently.

B14.3 Ask students to give examples of pairs of factors for which itwould be advisable to round one up and the other down in order toestimate the product.

B14.4 Ask students to give estimates for each of the following:79 x 6374 ÷ 9215 x 7997 ÷ 5

B14.5 Tell students that Jocelyn said that it was just as easy for her tomentally solve 4 x 525 as to give an estimate. Ask, How might she havefound the answer mentally?

PortfolioB13/14.1 Have students choose one of 100, 200, 300, ... 900. Askthem to list at least five each of addition, subtraction, multiplication,and division questions that would have estimated answers equal to theirchosen number.

B14.6 Ask students to explain in writing the estimation strategies givenfor each question below and to determine which estimation is closer tothe actual product.

79 x 9 as 80 x 10 or 80 x 9

17 x 15 as 8 x 30 or 20 x 10

B14.7 Have students write reports discussing situations where estimat-ing products is worthwhile.





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vi) select and use appropriatecomputational techniques(including mental, paper-and-pencil, and technologi-cal) in given situations

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B15 mentally solve appropriateaddition and subtractioncomputations

B15 When a problem requires an exact answer, students should firstdetermine if they are able to calculate it mentally—this should becomean automatic response.

Provide opportunities to introduce and practise a variety of mental mathstrategies. It is suggested that you introduce a strategy by giving studentsa question for which that strategy would be efficient to use, ask yourstudents to calculate it mentally, and have them share the strategies theyused. Very often the strategy you would like to highlight is already beingused by some student(s) and you can have them explain it. Then have allstudents practise this strategy on a number of relevant questions.

Strategies for mentally adding and subtracting include among others

(a) front-end strategy , e.g., To find 523 + 245, students begin at the leftand think, “7 hundred (500 + 200), sixty (40 + 20), eight (5 + 3).” Thisstrategy is almost as easy even if there is some regrouping involved, e.g.,When adding 1636 and 247, students think “1thousand, 8 hundred,seventy, no, that’s eighty-three.” Most students will find it easier to dowhen the numbers are written vertically, e.g.,

2263 Beginning at the left, a student might think,3145 “5 thousand (2000 + 3000), 3 hundred

(200 + 100), no, that’s 400 (60 + 40) 8. Theanswer is 5408.”

(b) compensation, e.g., For 347 + 18, 347 + 20 is 367, subtract 2 is 365.To find 238 + 297, think of 238 + 300 with 3 taken from the total. For722 - 197, the student might think, “722 - 200 is 522 and I’ll add 3, foran answer of 525.” This could also be solved by the constant differencestrategy—adding the same amount to each number and, therefore,preserving the difference—by adding 3 to both to get 725 - 200.

(c) counting on/counting back, e.g., For 243 - 197, they would say, “3more makes 200 and 43 added to that gives an answer of 46.” Somemight prefer to count back and say, “43 gets me back to 200, and I’lladd 3 more for a difference of 46.”

(d) compatible numbers, e.g., Recognize numbers that go together tomake multiples of ten and hundred such as 40 and 60, 300 and 700, 25and 75, and 750 and 250 as well as less obvious number pairs such as 47and 53, 28 and 72, and 16 and 84. Ultimately, for a calculation such as225 + 68 + 75, a student should recognize the “25, 75” pair and think,“225 and 75 makes 300 and 68 more is 368.”





Paper and PencilB15.1 From a group of computation questions ask students to circle theones that they could solve mentally.

B15.2 Ask students to write only the answers to computations that youshow on an overhead for a brief period of time—5 to 10 seconds perquestion depending upon the level of difficulty.

InterviewB15.3 Show students $44.98 + $3.98 + $10.99. Ask them to calculatethe answer mentally.

B15.4 Tell students that Rebecca said that for 217 - 180, she wouldrather add than subtract to determine the difference. Ask, What didRebecca mean?

B15.5 Ask students to explain what “nice numbers” are and to givesome examples.

B15.6 Ask students to determine how best to calculate each of thefollowing without a calculator. If they decide to use mentalstrategies, have them compute and share their strategies.

34 + 256 + 92 + 386 532 1775$4.99 + 1.98 + 0.99 124 368874 + 968 + 1245 33 977129 - 90500 - 295436 + 258732 - 89

PortfolioB15.7 Ask students to make up five questions for which thecalculations can be done mentally. Ask them to include questions thatwould use different strategies. Have them pair up and exchange theirquestions with their partners. Ask them to take turns solving each other’squestions and explaining their strategies.





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By the end of grade 6, studentswill have achived the outcomesfor entry–grade 3 and will also beexpected to


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B16 mentally multiply 2-digitnumbers by 10 or 100

B17 use technology for compu-tations involving manydecimal places or largewhole numbers

B16 Students should be aware that if a whole number is multiplied by10 or 100, it could be represented by that number of rods or flats,respectively, e.g., 43 x 10 means 43 rods or 43 tens with no units andsimilarly 43 x 100 means 43 hundreds with no tens or units. Studentsshould understand this rather than simply memorizing a rule about“adding zeros.” In multiplication questions involving 100, such as43 x 100, the answers are most often read using the word “hundred” asin 43 hundred, rather than 4 thousand, 3 hundred.

Practical contexts for multiplying by multiples of 10 are situations inwhich students change SI units, e.g., 16m = 16 x 100cm.

B17 Investigating real data may necessitate that students performcalculations involving very large whole numbers or numbers with manydecimal places. In fact, students should not be limited to problems thatinvolve numbers which they can compute mentally or manually. The useof calculators should be encouraged in these situations.

It is important that students be taught not only “how” to use a calcula-tor, but “when” it is appropriate to use one. Provide examples of a varietyof computations, some complex ones for which a calculator would berecommended, some lending themselves to a mental calculation, andsome relatively easy ones that could be solved with a paper and pencilalgorithm. A set of examples might be

785 x 28.3 56 x 4 80 x 90 532 ÷ 4

35 + 25 + 25 + 25 257 + 739 + 89 + 457

768.1 - 86.4 1000 - 395 3748.5 ÷ 147 699 ÷ 3

Students must continue to practise computational estimation, as thisskill is expected whether using a mental strategy, a pencil-and-paperalgorithm, or a calculator. You could reinforce this by always modellingestimation yourself whenever you use a calculator or demonstrate analgorithm and by always expecting this from your students. Alertness tothe reasonableness of answers no matter the mode of computation is amajor goal for all students.





PerformanceB16/17.1 Provide questions that involve the multiplication of 2-digitnumbers by 10 or 100. Ask students to work in pairs with one studentdoing the calculations mentally and the other using a calculator. Askthem to keep track of who was able to the calculations more quickly.

B17.1 Ask students to use the digits 3, 4, 5, and 6 to make the largestpossible product for x . (Use calculators.)

Paper and PencilB16.1 Give a number of questions orally, all involving the multiplica-tion of 2-digit numbers by 10 or 100. Have students record theiranswers to these questions. Include examples of measurement questions,such as (a) 25 metres is how many centimetres? (b) 16 decimetres is howmany centimetres? (c) How many millimetres is 53 cm?

InterviewB16.2 Ask students to complete the following in more than one way:

______ x _____ = 400.

B16.3 Have students explain using base-10 blocks how their knowingthe first multiplication helps them to complete the others.

6 60 600 x 7 x 7 x 7 42

B16.4 Tell students that Christine multiplied 58 by 100 and got ananswer of 58 hundred and that Marty’s answer to the same question was5 thousand 8 hundred. Ask, Can you explain this?

PortfolioB17.2 Ask students to use data from the newspaper to make up prob-lems requiring calculations for their solution. Ask them to be sure toinclude some multi-step problems.

Patterns and

Relations

GCO C: Students will explore, recognize, represent, and apply patterns and relation-ships, both informally and formally.




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By the end of grade 6, studentswill have achieved the outcomesfor entry-grade 3 and will also beexpected to

i) describe, extend, and createa wide variety of patternsand relationships to modeland solve problems involv-ing real-world situationsand mathematical concepts

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C1 demonstrate an understand-ing of the relationshipbetween adding decimalsand adding whole numbers

C2 apply the pattern identifiedwhen multiplying byincreasing powers of ten

C3 use patterns to solve com-putation problems

The study of patterns affords students anopportunity to make conjectures aboutrelationships ... By continuing to providea broad variety of opportunities toexplore and use patterns, we helpstudents move from a basic recognitionof patterns to a more sophisticated use ofpatterns as a problem-solving strategy.(NCTM 1992a, 1)

C1 Students should recognize that adding and subtracting tenths orhundredths of parts of items, e.g., 3 tenths + 4 tenths or 3 hun-dredths + 4 hundredths, involves the same process as adding andsubtracting whole number quantities of other items, e.g., 3 apples +4 apples, such as in 3.5 + 1.8 which can be thought of as 35 tenthsplus 18 tenths (35 + 20 - 2 = 53 tenths) or 5.3.

C2 Students should identify and apply patterns produced whenmultiplying by 10, 100, or 1000, e.g.,

4 x 10 = 40 4.5 x 10 = 454 x 100 = 400 4.5 x 100 = 4504 x 1000 = 4000 4.5 x 1000 = 4500

C3 Many complex computational problems can be solved usingpatterns based on simple numbers.

• Work with students to generate the first entries in this list:37 x 3 = 11137 x 6 = 222

Ask students to predict 37 x 9 and then check. Ask them to predict37 x 18 and to identify the tenth entry in the pattern.

• In order to predict the answer to 999 999 999 + 999 999 999have students observe these simpler additions.

9 99 999 + 9 + 99 +999

18 198 1998

• Students could explore “the trick” for multiplying 2-digit numbersby 11. Have them notice

42 53 62 x 11 x 11 x 11

462 583 682

Calculators would facilitate identifying patterns for larger 2-digitnumbers, e.g., 78 x 11, 3-digit numbers, e.g., 243 x 11, andmultiples of 11, e.g., 49 x 22.



GCO C: Students will explore, recognize, represent, and apply patterns and relationships,both informally and formally.


Paper and PencilC2.1 Ask students to record the product for each of thefollowing without using a pencil-and-paper algorithm or a calculator:

37 x 104.2 x 1001532 x 1052.6 x 1000

C3.1 Have pairs of students use a calculator to complete the calcula-tions and identify the pattern that emerges:

5 x 5 = 2515 x 15 =25 x 25 =35 x 35 =

Ask students to predict 75 x 75 and then check with a calculator.

InterviewC1.1 Ask students to explain how they know that 33 hundredthssubtract 22 hundredths has to be 11 hundredths.

C3.2 Ask students to describe computational situations in whichpatterns might be used to help solve them.

PortfolioC3.3 Ask students to make collections of computationalproblems they have solved by using patterns. Have them each choosea favourite and explain why it was selected.

GCO C: Students will explore, recognize, represent, and apply patterns and relation-ships, both informally and formally.




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ii) explore how a change in onequantity in a relationshipaffects another and

iii) represent mathematicalpatterns and relationships ina variety of ways (includingrules, tables, and one- andtwo-dimensional graphs)

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C4 understand how a change ineither a or b in a + b, a - b, ax b, or a ÷ b will affect theresult of the computation

C5 represent multiplicationfacts either in a table orgraphically

C6 complete open sentences ofthe forms a x b = ,

a x = c, a ÷ b = anda ÷ = c

Many fourth graders will be able togeneralize and verbalize number patternsas part of small-group explorations andclass discussions. (NCTM 1992a, 1)

C4 Students should begin to explicitly articulate some of the “rules”describing how a change in one variable in a computation affects theresults, e.g.,

in +10, as increases by 1, so does the sumin 10 - , as increases by 1, the difference decreases by 1in 10 x , as increases by 1, the product increases by 10in ÷10, as increases by 10, the quotient increases by 1.

C5 Students should be familiar with tables that list either all themultiplication facts or some portion of them, e.g., The 3-times tablemight be shown as

Students should use a bar graph or coordinate graph to show the 3-times table.

C6 Students should be familiar with the use of open sentences inaddition and subtraction situations and should be ready to extendtheir use to multiplication and division, e.g.,

10 x 2 = 10 sets of 2 are how many?10 x = 20 10 sets of how many are 20?10 ÷ 2 = 10 divided into groups of 2 are

how many groups?10 ÷ = 5 10 shared by how many are 5 each?

0 3 6 9 12 15 18 21 24 27

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9



GCO C: Students will explore, recognize, represent, and apply patterns and relationships,both informally and formally.


PerformanceC5.1 Ask students to draw graphs that show the 4-facts.

Paper and PencilC4.1 Ask students, In each of the following, does the answer increaseor decrease? by how much?

50 - , as increases by 120 + , as increases by 1 2 x , as increases by 1

InterviewC4.2 Ask students why the expression 10 x + 1 increases by 20as increases by 2.

C6.1 Ask students to explain how to find the missing number in4 x = 100.

C5.2 Provide the graph below. Ask students, What multiplicationfacts does the graph show?

Second Factor

Prod

uct

Shape and Space:Measurement




GCO D: Students will demonstrate an understanding of and apply concepts and skillsassociated with measurement.

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i) extend understanding ofmeasurement concepts andattributes to include vol-ume, temperature, perim-eter and angle

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D1 recognize and demonstratethat objects of variousshapes can have the samearea

D2 recognize and demonstratethat objects of the same areacan have different perim-eters

D1 It is important for students to explore not only the areas ofrectangles, but also areas of other shapes. Through these investiga-tions students should recognize that objects of different shapes canhave the same area.

• Show students the geoboard shape (A) and ask them to determineits area. Have them construct other shapes with the same area ontheir geoboards and record their shapes on geopaper. Encouragestudents to find shapes using partial squares such as shown in B.

A. B.

5 units 5 units• Invite the students to select 16 pieces of the same pattern block,

e.g., 16 green triangles. Have them make different shapes, all ofwhich have an area of these 16 units. Ask them to find, amongothers, the most compact shape and the longest shape, using therule that at least one side of a block must abut a congruent side ofanother block.

D2 Students should recognize that area and

perimeter are independent of one another,e.g., These two figures have the same area eventhough the perimeters are different.

Pentominoes may also be used to illustrate this concept. Pentominoesare shapes each made up of fivesquares, all of which must have at leastone side abutting a side of another,e.g.,

• Provide pairs of students with 10 colour tiles. Have them investi-gate to determine how many different perimeters they can pro-duce by making shapes with areas of 10 square units. At least oneside of each tile must abut the side of another tile. Ask that theiranswers be coloured in on, or cut out of, square grid paper.




GCO D: Students will demonstrate an understanding of and apply concepts and skills associated with measurement.

PerformanceD1.1 Show students pictures of four or five shapes with differentareas. Ask them to order them from least area to greatest area by sightand to find a way to check their answers.

D2.1 Ask students to use geoboards to create rectangles with an areaof 9 square units and a perimeter of 12 units. (Note: Some studentsthink that a diagonal segment between two pins on a geoboard has alength measurement of one unit. They may need an investigation toshow that this diagonal is greater than one unit.)

Paper and PencilD2.2 Tell students that a certain rectangle on a geoboard has an area of12 square units. Ask, What is its perimeter? Could there be anotherrectangle with the same area but with a different perimeter?

D1.2 Ask students to circle the letters of the shapes that have thesame area as the one on the left.

D2.3 Ask students to use centimetre grid paper to create shapes withan area the same as that of a 4cm x 4cm square, but with a perimetergreater than this square. Ask them to state the perimeters of theirshapes.

D2.4 Ask students to use centimetre grid paper to construct fourshapes, each with an area of 20cm2, but with varying perimeters.

InterviewD1/2.1 Ask students to explain how a long, skinny shape and asquare shape could have the same area. Then ask, Which has thegreater perimeter?

PortfolioD2.5 Ask the students to explore the concept of figures having thesame perimeters but different areas. Ask them to write about theirfindings.





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D3 measure volume, usingnon-standard units

D4 estimate and determine thevolume of rectangularprisms, using centimetrecubes

"Volume" and "capacity" are both termsfor measures of the "size" of three-dimensional regions. Standard units ofvolume are expressed in terms of lengthunits, such as ... cubic centimetres.Capacity units are generally applied toliquids or the containers that holdliquids. (Van de Walle 1994, 302)

D3 Students should recognize volume as the amount of space takenup by an object, or the amount of material required to build it(assuming that it is not hollow).

It is important for students to have numerous opportunities to buildsolids with many types of materials, e.g., interlocking cubes, base-10blocks, blocks, and Lego. Students should realize that volume can bedescribed using a wide variety of units, e.g., A box might have avolume of 20 juice boxes or 75 pattern block triangles or 25 pieces ofLego.

• Have students investigate their home environment to find thevolume of various items, e.g., a loaf of bread (as the number ofslices), a package of spaghetti, and a box of crackers.

Be sure always to ask students to give estimates before doing anymeasurement.

D4 Students should estimate the volumes of rectangular prisms,then verify by building them with centimetre cubes or base-10blocks. Provide opportunities for students to (a) determine thevolumes of rectangular prisms and (b) build prisms with specifiedvolumes.

Sometimes it is useful to provide only a small number of cubes tohelp students develop estimation strategies. Also, students shouldexperience situations which help them see that prisms of differentlinear dimensions can have the same volume, e.g.,

• Have students work in pairs to build prisms which each measure16cm3, then compare the structures for similarities and differ-ences.

Students must have personal referents when dealing with any meas-urement. The base-10 blocks are useful referents for volume becausethe small cube is 1cm3; the large cube has a volume of 1000cm3. Therelationship of volume to capacity is also easily made since the centi-metre cube has a potential capacity of 1mL, and the 1000cm3 block acapacity of 1000mL, or 1L.





PerformanceD3.1 Ask students to find the volume of packages of pencils or boxesof chalk.

D4.1 Have students use centimetre cubes to build various “animals”and to compare their volumes.

Paper and PencilD3.2 Tell students that a farmer stores his baled hay in the barn instacks. Each stack is 5 bales across, 3 bales high, and 4 bales deep.

Ask, What is the volume (in bales) of each stack?

D3.3 Point out to students that the volume of the structure shownbelow is 44 blocks. Ask, If another layer is added, what is the newvolume?

InterviewD3/4.1 Ask students, What might be a good unit to use whenestimating the volume of a shoe box? What might the volume be ifyou were to use this unit? How did you determine your answer?What would you expect the volume to be if the unit were half thevolume? twice the volume?

D3.4 Tell students that the volume of one box is 8 cubic units. Thevolume of another is 4 of the same unit. Ask, In what way might thesecond box be bigger than the first?

PortfolioD3/4.2 Have students describe in writing several ways they mightdetermine the volume of an empty box.





A unit for measuring an angle must bean angle. Nothing else has the sameattribute of "spread" that we want tomeasure. (Contrary to popular opinion,it is not necessary to use degrees tomeasure angles.)(Van de Walle 1994, 305)

D5 You should avoid the frequently-given definition of angles as “themeeting of two rays at a common vertex” in favour of having studentsconceptualize an angle as a turn and the measure of the angle as theamount of this turn.

It is important for students to understand that a larger angle corre-sponds to a greater turn from the starting position (as shown on theleft below) and that the length of the arms of the angle does notaffect the turn amount and, therefore, does not affect angle size (seediagram below right)

• Provide students with both short and long straws; have themsuper-impose the straws and turn them simultaneously. Thestudents can then compare the amounts of turn, noticing that thelengths of the arms differ, but not the amount of turn.

Students should recognize that the angle associated with a quarterturn is called a right angle, and differentiate among angles that areright angles and those that are acute or obtuse.

Pipe cleaners and geo-strips are good materials for exploring themeasure of angles.

• Have students bend pipe cleaners to make right angles. Ask themif it matters where they bend them.

• Have students make increasingly smaller angles, until the twoarms of their pipe cleaners are coincident, i.e., are on top of eachother. Ask them to describe what is happening.

• Ask half the class to make and show an acute angle that is almostright. Ask the other half to make an obtuse angle that is almostright. Ask students to physically compare their angles to describethe comparisons.

• Ask students to start with a right angle and make increasinglylarger angles until the pipe cleaners are straight.

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i) extend understanding ofmeasurement concepts andattributes to includevolume, temperature,perimeter and angle

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D5 recognize that the measureof an angle indicates anamount of turn





PerformanceD5.1 Ask students to find ways to order the three angles below fromgreatest to least. (Ensure that the arms are different lengths.)

D5.2 Ask students, How many right angles are there in a full turn?

PresentationD5.3 Ask students to convince their partners that the length of thearms of an angle does not affect its measure.

PortfolioD5.4 Give students tracing paper and sheets containing a number ofangles of various sizes and arm lengths and in different positions,such as

Have them use the tracing paper to compare angle sizes. Ask them towrite a report on their findings.





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D6 estimate and measureangles, using non-standardunits

D7 use a thermometer to readtemperatures

D6 Provide students with many opportunities to estimate anglesizes and to use various non-standard means of measurement in realsituations. Although some students may be curious about protractors,at this point the term “degree” holds very little meaning withoutnon-standard measuring experiences. Later, the degree can be concep-tualized as a very thin wedge. Students need to understand that thesmaller the wedge size, the greater the number of units required.

Students can create paper and/orcardboard “wedges” with whichto measure angles, e.g.,

Students should measure thesame angle with wedges ofvarious sizes to notice theeffect of the wedge size.

D7 Students have probably had experiences at home with variouskinds of thermometers. Ensure that they know how to read thethermometer. Also, ensure that they associate some Celsius tempera-tures with familiar situations, e.g.,

0o C is the temperature at which water freezes10o C is jacket weather20o C is a comfortable room temperature30o C is a hot summer day.

As with all measurement activities, students should estimate beforereading the thermometers.





PerformanceD6.1 Tell students that you measured the angle below with a wedgeand the measure was 4 units. Ask them to create the wedge you used.

D6.2 Provide students with non-standard angle units such as wedgescut from cardboard. Ask them to use these units to create angles witha measure of 2.5 units.

D7.1 Ask students to read thermometers to tell the temperatures ofa variety of liquids in containers.

InterviewD6.3 Tell students that you have an angle unit and that there are 5of these angle units in a right angle. Ask students to describe howthey could make this angle unit.

D7.2 Ask students to describe how cold or warm a day is if thetemperature is 15o C.

PortfolioD7.3 Have students select ten times during the week to record thetemperature. Ask them to choose times at which they think thetemperatures will vary the most. Have them present their findings insuch a way that there is an explanation for the variance.





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ii) communicate using stand-ard units, demonstrate anunderstanding of the rela-tionship among commonlyused SI units (e.g., mm, cm,m, km) and select appropri-ate units in given situationsand

iii) estimate and apply measure-ment concepts and skills inrelevant problem situa-tions and select and useappropriate tools and units

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D8 estimate and measure inmillimetres, centimetres,decimetres, metres andkilometres

D8 Students should estimate and measure in millimetres, centime-tres, decimetres, metres, and kilometres for the purpose of answeringquestions of interest to them.

Through estimating, and then verifying by measuring, studentsdevelop a better sense of the sizes of units. It is important thatstudents have a personal referent for each of these units of length.The centimetre referent might be the width of their little fingers and/or the side length of a base-10 small cube. If they make an “O” withtheir thumbs and index fingers, a millimetre is about the distancejust before they touch. The decimetre referent could be the length ofa base-10 rod. The metre referent could be the height of the door-knob or the length of the metre stick. Students should find a personalreferent for kilometre, e.g., the distance from the school to a certainlandmark. As well, it is important for students to have a sense of alonger distance, such as 100km from their home town to a nearbycity.

• Ask students to show with fingers/hands an estimate for each ofthe following measurements: 8 millimetres, 7 decimetres, 40centimetres, 800 millimetres, and 0.6 metre.

• Cut from fluorescent paper a 10cm x 2cm rectangle. This is theclass decimetre. Put it in a different place each day and ask thestudents to make a “decimetre search” each morning when theyenter the classroom. This helps establish a referent in the students’minds for the length of a decimetre.

Students should realize that a millime-tre is 0.1 of a centimetre or 0.01of adecimetre, and that it takes 1000mmto make a metre.

• Invite students to investigate the possible side lengths of polygonsmade with a 40cm loop of string.

• Ask students, How many students would be needed to run a relayrace covering 5km? 10km? Ask, What factors must be considered?How might the answer differ for older students and for those whohave trained to run long distances?

• Invite students to investigate distances between towns and cities inNova Scotia, using string and the scale on a map.





PerformanceD8.1 Ask students to find objects with one of the dimensionsmeasuring approximately 4mm.

D8.2 Ask student to estimate, then measure accurately, the length ofborder needed to frame the bulletin board.

Paper and PencilD8.3 Ask students to complete the table below and explain anypatterns they notice.

InterviewD8.4 Ask students to describe how long 0.4 metre is.

D8.5 Ask students, About how long is the average couch?

D8.6 Ask students, About how long would it take to walk 1 km?

D8.7 Ask students to share their personal referents for centimetres,decimetres, and metres.

PortfolioD8.8 Ask students, About how many cars, bumper to bumper,would there be in a line of cars one kilometre long? Have themexplain how they determined their solutions.

D8.9 Ask students to estimate, then check, how many 50m rolls offencing would be needed to surround the playground. Have themwrite a report on the assignment for their portfolios.

mm cm dm

400 40 4

4

4





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iii) estimate and apply measure-ment concepts and skills inrelevant problem situationsand select and use appropri-ate tools and units

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D9 estimate and measure areain square centimetres

D10 solve relevant problemsinvolving millilitres andlitres, grams and kilograms

D9 Students should understand that the area of a shape is expressedas the number of units required to cover a certain surface. Provideopportunities for students to estimate and calculate the area ofvarious surfaces. A possible referent for square centimetres is one ofthe faces of the small base-10 cube; the long surface of a rod is apossible referent for 10cm2; a standard sheet of paper is about600cm2. Laying an acetate centimetre grid over objects is helpfulwhen determining surface area in square centimetres.

Students might investigate the area of shapesdrawn on centimetre dot paper. Strategies for doingthis include adding squares and half squares withinthe figure and placing a rectangle around theshape, determining its area, and subtracting thearea of the "extra" pieces.

• Invite students to use centimetre dot paper to create coloureddesigns comprising at least five different quadrilat-erals joined together as in the diagram to theright. Have the students estimate and calculate thearea of each quadrilateral and make up a questionabout their design for another student to solve.

D10 Students should continue to solve problems involving the useof millilitres/litres and grams/kilograms, and verify their solutions.These problems should be of interest to the students and generateuseful information.

• Ask students to bring in juice containers from home. After theyestimate and find the capacities of the containers, ask them todetermine the most common container size.

• Ask students to determine the number of grams of various foods itwould take to ingest 100 calories.

Be sure to continue to emphasize the importance of having a referentfor these measurements—base-10 blocks are helpful for this.

1000 cm3 1 cm3

1 L 1 mL





PerformanceD9.1 Provide centimetre grid paper and ask students to estimate andthen measure the area of the covers of their mathematics texts.

D10.1 Have a group of students find the masses of their lunches inorder to provide an estimate for the average mass of the students’lunches.

D10.2 Ask students to match pictures of objects, such as atelephone, a book, a pencil, and a child, to the list of masses below:

120 kg, 25 kg, 2.5 g, 550 g, 80 kg, 5 kg

Paper and PencilD10.3 Ask students, How large a pitcher is needed to make orangejuice if there are 350mL of concentrate and it takes 3 cans of waterfor each can of concentrate?

InterviewD10.4 Have students describe, or select from a collection of, con-tainers that would hold about

3 L 0.5 L 5 L 2500 ml 10 L

D10.5 Have students determine the heaviest mass they can comfort-ably lift.

PresentationD9.2 Make the design below on an overhead geoboard and askstudents to explain how to find its area.

Have students alter the shape on their own geoboards to increase thearea by 1 square unit.

PortfolioD10.6 Ask students to compare the dimensions of 1-L and 2-L milkcartons and to write about their observations.





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iv) develop and apply rules andprocedures for determiningmeasures (using concreteand graphing models)

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D11 relate dimensions and areasof rectangles to factors andproducts

D11 Students should relate the area of a rectangle to the product ofthe numbers describing its length and width. Conversely, any factorof the number representing the area of a rectangle can be one dimen-sion of a rectangle with that area, e.g., Consider rectangles with anarea of 8 square units.

2 x 4 and 1 x 8 are the dimensions of rectangles with an area(product) of 8 square units. Since there are no other ways tomake a rectangle with 8 squares, there are no other factors of 8.

• Have students use colour tiles or grid paper to investigate thenumbers from 1 to 30 to see how many different rectangles can bemade for each. Students should record their results in a table andlook for patterns, e.g., Number Rectangles

Note: While students have had considerable experience working withrectangles, and have associated their dimensions and areas withmultiplication, it is not an outcome at grade four to have themmemorize the area formula for rectangles and use it to calculate theirareas.





PerformanceD11.1 Ask students to use colour tiles or grid paper to explain why7 cannot be a factor of 39.

Paper and PencilD11.2 Have students draw rectangles to show that 9 is a factorof 63.

InterviewD11.3 Tell students that a rectangle made up of completecentimetre squares has a width of 8cm. Ask, What might its area be?

D11.4 Ask students, How does this diagram show that 2 is a factorof 10?

PortfolioD11.5 Ask students to determine and record on geopaper all possi-ble rectangles that can be made with an area of 36 square units.

D11.6 Have students colour a number of different-sized squaresusing centimetre grid paper. Ask, Can the area ever be 50cm2? Whyor why not? What are some possibilities for the dimensions of arectangle with an area of 50cm2?

Shape and Space:

Geometry



GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties,and relationships.


E1 Students’ previous experiences have been with cutting out andassembling prepared nets. It is now expected that students drawtheir own nets for rectangular prisms, square prisms, and cubes.They should also consider the various possibilities for these nets.

• Have students trace on paper the various faces of cubes orrectangular prisms to make their nets. Have them cut out thenets and fold them around the shapes to see if they work. Askthem to record these nets on grid paper. Have them cut one ofthe faces off and investigate the possible places it could bereattached to make a new net. Have them record each new neton grid paper.

E2 Have students cut out prepared nets for cylinders and cones.Have them make skeletal models for prisms and pyramids usingrolled newspapers and tape, straws and string, or toothpicks andminiature marshmallows. There are also commercial materials thatallow students to snap faces together to build 3-D shapes.

E3 In addition to having students build from drawings of common3-D shapes, you could also have them build from drawings ofstructures in which the representations allow for “hidden” cubes,e.g.,

E4 Design explorations in which students will discuss suchmatters as - shapes that have the same number of faces, edges, or vertices - how a cone is like/different from any pyramid - how a cylinder is like/different from any prism - how sizes of two rectangular prisms or two square prisms compare (informally)

Students should be able to discuss the similarities and differencesbetween the prisms and pyramids that share the same name, e.g.,square prism and square pyramid or hexagonal prism and hexagonalpyramid.

The study of geometry helps studentsrepresent and make sense of the world.Geometric models provide a perspectivefrom which students can analyse and solveproblems, and geometric interpretations canhelp make an abstract (symbolic) representa-tion more easily understood. (NCTM1989b, 112)

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i) identify, draw, and buildphysical models of geometricfigures and

iv) solve problems using geomet-ric relationships and spatialreasoning

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By the end of grade 4, students will beexpected to

E1 draw various nets for rectangu-lar prisms and cubes

E2 construct models for variouscylinders, cones, prisms, andpyramids

E3 construct shapes givenisometric drawings

E4 explore relationships among3-D shapes





PerformanceE1.1 Tell students that this diagram is part of a netfor a square prism. Ask them to complete the net bydrawing the additional faces that would be needed.

E1.2 Provide students with square or rectangular prisms and 11-pin x 11-pin geoboards. Ask them to use elastics to construct a netfor the prism. Ask them to discuss how they might move one ofthe faces to make a new net for the same prism. Have them checkby recording the new nets on square dot paper and cutting themout.

E1.3 Provide students with a pentomino puzzle piece—a 2-Dshape made by joining 5 squares along full sides—that would foldto make a box with no top. Ask them to trace this piece and thenadd a square for the top of the box. Ask, In how many places canthis square be added? One such pentomino piece:

(Note: These pieces could be cut from grid paper.)

E2.1 Ask students to build a skeletal model of the shape that hasthese faces:

E2.2 Ask students to build prisms that will use nine toothpicksfor edges.

E2/4.1 Ask students to build skeletal models of two differenttriangular pyramids. Ask them to explain how they are the sameand how they are different.

E3.1 Ask students to use cubes to build two possible structuresthat would have this drawing:

InterviewE4.1 Ask students, What shapes are we? We both have six verti-ces. We both have some triangular faces.





E5 Students need many hands-on experiences putting shapes togetherto make new shapes before they are able to visualize these results.They should be encouraged to predict the results before they doany combining and to follow these activities with a pulling apartand visually recombining exercise.

• Show students two isosceles right triangles, e.g., two smalltriangles in a tangram set, on an overhead. Have them predictwhat polygons could be made by combining them with equalsides abutted. Check their predictions and/or explore otherpossibilities.

• Have students investigate all the distinct shapes that can bemade from four congruent squares with sides abutted. Manystudents might recognize these as the shapes in Tetris, a compu-ter game.

• Have groups of students investigate all the distinct shapes thatcan be made from three pattern blocks with equal sides abutted.These could be put on display or traced on paper to record.

• Have students find the midpoints of three sides of a square.Have them join consecutive midpoints so that the square isdivided into two right triangles and a pentagon. Have them cutthe square into these three pieces and investigate all the shapesthat can be made from them.

• Have students investigate the distinct shapes that can be madefrom four cubes with at least one face of each cube abuttinganother. Put these on display. Compare to the four-squareactivity above.

Note: These activities involve many aspects of spatial sense, par-ticularly discrimination, position-in-space, perception of spatialrelations, and perceptual constancy. Experiences such as these areuseful for students to develop visualization ability.

A good way to explore shapes ... is to usesmaller shapes or tiles to create largershapes. Different criteria or directions canprovide the intended focus to the activity.Among the best materials for this purposeare pattern blocks, but many teacher-madematerials can be used. (Van de Walle 1994,331)

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i) identify, draw, and buildphysical models of geometricfigures

iv) solve problems using geomet-ric relationships and spatialreasoning

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E5 find all possible compositefigures that can be made froma given set of figures





PerformanceE5.1 Give students pictures of two congruent isosceles (not right-angled) triangles side by side. Ask them to predict the shapes thatcould be made by combining them with equal sides fully joined. Askthem to cut out the triangles, check their predictions, and, ifnecessary, investigate other possibilities. Compare these shapeswith those made by two congruent isosceles right triangles (seeElaboration section).

E5.2 Provide students with four squares—two each of twocolours—and one square four times larger than these four. Havethem cut each square along one diagonal. Ask them to arrange theeight triangles inside the large square to make a quilt block design.

E5.3 Ask students to investigate all the distinct shapes that can bemade by combining the square and two small triangles from atangram puzzle (with equal sides fully joined). Have them recordall the shapes they find by tracing them on paper.

E5.4 Ask students to investigate the different 3-D shapes that canbe formed by combining two common 3-D shapes (with congru-ent faces fully joined), e.g., If they used two congruent rectangularprisms, they could make three different new rectangular prisms bymatching different faces.

Paper and PencilE5.5 Provide students with square dot paper and ask them to drawall the shapes that can be made from

Interview

E5.6 Provide students with a variety of polygons made by outlin-ing three combined pattern blocks. Ask them to predict whichthree blocks are in each new shape and check their predictions byusing the blocks.





E6 Students have had experiences recognizing and naming rightangles as square corners in shapes and recognizing angles greater orless than right angles.

• Have students investigate angles in various shapes, using thecorner of a piece of paper as a reference for right angle. (Does itfit the angle of the shape or is the angle greater/less than thecorner of the paper?)

Students should have experiences that introduce the names acuteangle—one less than a right angle—and obtuse angle—one greaterthan a right angle but less than a straight line—by examiningangles in shapes and angles as separate entities. They should learnto recognize these angles by their overall appearance, not by theirmeasurements.

Students often focus on the length of arms of angles rather than thespread of the arms; therefore, activities should include angles withshort, long, and a combination of short and long arms. They willneed to understand that the arms of the angle could be continuedto any length without changing its size. These activities should beconnected to measuring activities with angles—see SCOs, D5 andD6.

Angles should be presented in a variety of contexts, e.g., anglesformed by the two hands of a clock, by the intersection of tworoads, and by the blades of scissors or hedge clippers.

• Have students make various acute angles with pipe cleaners, e.g.,almost a right angle, about half a right angle, almost no angle.

• Show students acute angles, with arms of varying lengths, invarious positions and of different sizes. Ask them to estimateeach as “almost right”, “almost no angle” or “about half a rightangle.”

• Have students find acute angles in various 2-D polygons and onfaces of 3-D shapes.

Similar activities should be carried out with obtuse angles. Theyshould check their estimates. Folding a corner of a piece of paper inhalf could help visualize half a right angle.• Have students stand with their arms out straight and then make

types of angles according to your instructions.

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ii) describe, model, and compare2- and 3-D figures and shapes,explore their properties andclassify them in alternativeways

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E6 recognize, name, describe,and construct acute andobtuse angles





PerformanceE6.1 Tell students: Jeri went on a trip with her parents. To amuseherself, she sketched the ways roads met at interesections. Thefollowing are some of her drawings. Ask, How many of the anglesformed by the roads were right angles? acute? obtuse?

E6.2 Ask students to classify the angles found on each of the facesof a 3-D shape, e.g., a hexagonal pyramid.

E6.3 Have students investigate and name the angles formed whenthey print various capital letters, e.g., A, E, L, M, W.

E6.4 Have students explore the angles in the six different patternblocks. Ask, Which blocks have only acute angles? only obtuseangles? both acute and obtuse angles? only right angles?

E6.5 On overhead clocks, prepare 6–8 different times. Displayingthem one at a time on an overhead, ask students to name anddescribe the angle made by the hands of each clock.

Paper and PencilE6.6 Provide students with toothpicks of two sizes. Ask them tomake a display of three angles—one that uses two short tooth-picks, one that uses two long toothpicks, and one that uses a shortand a long toothpick—for each of the following: (a) an acuteangle that is almost right, (b) an obtuse angle that is almost astraight line, (c) an acute angle that is almost half a right angleand (d) an obtuse angle that is almost a right angle.

E6.7 Ask students to combine two or more pattern blocks tomake examples of acute, right, and obtuse angles. Have themrecord the angles by tracing them on paper.





E7 Students should have guided explorations to discover patternsfor different types of triangles—these patterns are their properties.

• Prepare cutouts of several examples of these three kinds oftriangles. Ask students to sort them into three groups. Ask themto explain their sorting rule. Often, they will sort them by howtheir sides look, without knowing the actual names. If so, thiswill lead to a focus on measuring and comparing the sides, andnoting common properties to which the names equilateral, isosce-les, and scalene can be attached. (If not, you could sort themand ask students to determine your sorting rule and to do otherexplorations.)

• Mix up a set of triangles and have students sort them by num-bers of lines of reflective symmetry. Folding or using mirasshould be part of this exploration. Ask students what theynotice about the resulting sets of triangles.

Sorting and exploring for lines of symmetry should lead to theseproperties: (a) equilateral triangles have three lines of symmetry, (b)isosceles triangles have one line of symmetry, and (c) scalene tri-angles have no symmetry.

Exploring triangles by folding to compare the angles should lead todiscovering these properties: (a) all angles in equilateral triangles areequal, (b) two angles in isosceles triangles are equal and (c) allangles in scalene triangles are different.

• Provide various lengths of strips of paper or geo-strips for stu-dents to make triangles that can be classified. Have studentsdiscuss their properties.

• Provide students with straws, scissors, and string to threadthrough straws. Ask them to assemble examples of each kind oftriangle.

Note: A property of a set of shapes is a characteristic that all membersof the set have in common, e.g., If a shape is classified as an isosce-les triangle, it can only be assumed that it has one line of reflectivesymmetry and two equal sides. While an equilateral triangle couldbe placed under the isosceles heading because it has these twoproperties, it would be better classified as equilateral where moreproperties can be assumed. It is recommended that students seethese as two distinct shapes at this point in their development.

A teacher’s questioning techniques andlanguage in directing students’ thinking arecritical to the students’ development of anunderstanding of geometric relationships.Students should be challenged to analyzetheir thought processes and explanations.(NCTM 1989b, 113)

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E7 recognize, name, describe,and construct equilateral,isosceles, and scalene triangles





PerformanceE7.1 Ask students to draw, on squared grid paper, an example ofeach member of the quadrilateral family: square, rectangle, rhom-bus, parallelogram, trapezoid, and kite. Have them make a copy ofeach of these figures. Ask them to draw in one diagonal in eachfigure in one set of drawings. Have them describe and name thetriangles formed in each figure and ask, Are they congruent? Didyou get the same results as your neighbour? Ask them to draw inthe other diagonal of each figure in the other set of drawings. Havethem describe and name the triangles formed in each figure andask, Are the triangles the same ones you got for the first diagonaldrawn?

E7.2 Provide students with two congruent equilateral triangles,two congruent isosceles triangles, and two congruent scalenetriangles. Ask them to predict what polygons could be made bycombining two congruent equilateral triangles. Have them checktheir prediction. Repeat with the other pairs. Ask, Which pair ofcongruent triangles produced the greatest variety of polygons?

E7.3 Ask students to explore a variety of 3-D shapes to findtriangular faces. Have them trace around them on paper and classifythe triangles drawn.

E7.4 Ask students to draw right triangles. Ask them to draw theimages of the right triangles using a mira on one of the sides form-ing the right angle. Ask, What type of triangle is formed by the twotriangles?

PresentationE7.5 Provide groups of three students with 2m of yarn. Ask themto stand in formation to make each kind of triangle. Ask them toexplain how they are sure that each one they are showing is correct.

PortfolioE7.6 Provide students with ten toothpicks, all the same length.Ask them to investigate how many different triangles can bemade using from as few as three up to as many as ten tooth-picks for each triangle. Toothpicks may only meet at theirendpoints. Have students draw to record and put thename of each type of triangle under its drawing, e.g.,Isosceles





E8/E9 Initially, students identify shapes by their overall appear-ance. While many of their properties have been implied, it is nowthe goal to make the properties of some shapes explicit. Squares,rectangles, parallelograms, rhombi, trapezoids, and kites shouldeach be analysed separately so that students, through hands-oninvestigations, can see and describe the patterns that are the prop-erties of these quadrilaterals. You should prepare a series of ques-tions to guide the students’ investigations.

• Provide students with pictures or cutouts of a variety of rectan-gles. Ask them to compare opposite sides by sight and byfolding over to compare directly. Ask, What do you notice? Havethem describe the four angles of their rectangles. Ask, Are any ofthe sides parallel in your rectangle? Which ones? Is this the samefor other students’ rectangles? Show students a cutout of a largerectangle and ask them what they could say about the sides andangles. Have them write a summary, including some examples,of what they know about rectangles. Students should then begiven application problems that require the use of these proper-ties. (See the note in E6 for a discussion of “property.”) Thereflective symmetry property of rectangles should be also ex-plored at this same time—see SCO, E12.

Some Properties of the Quadrilateral Family

At this level students begin to appreciatethat the reason a collection of shapes goestogether has something to do with proper-ties. It makes sense to define and sort shapesby these properties rather than by appear-ances. (Van de Walle 1994, 325)

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E8 make generalizations about theangle, side length, and parallelside properties of the variousquadrilaterals

E9 sort quadrilaterals underproperty headings





PerformanceE8.1 Provide students with four toothpicks or straws, all the samelength. Ask, What different quadrilaterals can be made using allfour? Using your knowledge of properties, what can you say aboutthe sides and angles of each of these shapes?

E8.2 Provide students with four toothpicks or straws, two each oftwo different lengths. Ask, What different quadrilaterals can bemade using all four? What can you say about the sides and angles ofeach of these shapes?

E9.1 Provide students with cutouts of several quadrilaterals. Askthem to select a property card (use the property headings on thechart in the Elaboration section) and place any/all of their cutoutsthat have this property with the card. Select another propertycard. Ask, Do any of them also have this property? Explain.

E8.3 Have a variety of lengths of geostrips available. Assignstudents a particular quadrilateral to make. Observe how theyselect materials and build it. Ask them to describe the propertiesof the shape.

Paper and PencilE8.4 Tell students that a farmer wants to know the distancearound his rectangular field. Ask them to explain the fewestnumber of measurements he would need to make. Ask, If the fieldis square, then how many measurements would he need to make?

InterviewE8.5 Ask students, What is the same and what is different aboutthe sides of a kite and of a parallelogram?

E9.2 Ask students, Who am I? I am a quadrilateral. All of mysides are equal. None of my angles is a right angle.

PresentationE8.6 Have lengths of rope or string available, four in one length,two each in two other lengths. Have groups of four studentschoose a quadrilateral to make, have them select the ropes theywill need, and have them stand, holding these ropes, to make thedesired quadrilateral. Ask them to describe the properties of theshape.





E10 Through guided explorations, students should begin toidentify some properties of 3-D shapes. Prior experiences “stacking”2-D shapes to make prisms should have established the uniformnature of these shapes. In the same way, “stacking” circles wouldresult in a cylinder which has this same uniformity.

• Have students make, using toothpicks and marshmallows,skeletal models of prisms—triangular, rectangular, pentagonal,hexagonal—and record their findings in table form.

By looking for patterns in this table and thinking about how theskeletal models were made, students should find these patterns forprisms: (a) The number of vertices for any prism is two times thenumber associated with its name, e.g., For an octagonal prism, 2x8or 16 is its number of vertices because the vertices come from thetwo bases. (b) The number of edges is three times the numberassociated with its name because the edges come from the two basesplus the edges that join the two bases. (c) The number of faces istwo more than the number associated with its name because of itstwo bases plus one each for the faces that join corresponding sidesof the bases.

Similar explorations with pyramids and their skeletons shouldresult in finding these patterns: (a) The number of faces is equal tothe number of vertices—both are one more than the numberassociated with its name because a triangular face starts from eachside of the base and the base itself is a face; the vertices are thevertices of the base plus the single vertex , e.g., A pentagonalpyramid has 6(5+1) vertices and faces. (b) The number of edges istwo times the number associated with its name because each side ofthe base is an edge and there is an edge from each vertex of the baseto the single vertex.

Cylinders have two faces, one surface, no edge, and no vertex; coneshave one face, one surface, no edge, and one vertex.

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E10 make generalizations about thenumbers of vertices, edges, and facesof various prisms, pyramids, cones,and cylinders





PerformanceE10.1 Ask students to build skeletal models of 3-D shapes thatsatisfy each of these conditions: (a) a prism with 12 edges, (b) apyramid with seven faces, (c) a shape with eight faces and eightvertices, and (d) a shape with eight faces and 16 vertices.

Paper and PencilE10.2 Place an octagonal prism and a cylinder side by side. Askstudents to write a comparison of them mentioning things that arethe same and that are different.

E10.3 Ask students to explain why the number of faces andvertices of hexagonal pyramids is one more than the numberassociated with its name.

InterviewE10.4 Ask students to use the terms vertices, edges, and faces todescribe an octagonal prism.

E10.5 Ask students, What shape am I? I have five faces and fivevertices. One of my faces is different from the other four.

E10.6 Ask students, (a) What prism am I? I have 12 vertices. (b)What prism am I? I have 15 edges. (c) What prism am I? I have 12faces. (d) What pyramid am I? I have 11 faces. (e) What pyramidam I? I have 12 edges.





E11 Encouraging students to think in terms of sliding, flipping, andturning shapes is a very useful strategy to help with their visualizationin geometry.

• Provide students with right triangles and havethem try to visualize the images of these trianglesreflected in each of their three sides. Ask, Whatshape do this triangle and its image make to-gether in each case? (Answers: Two different isosceles trianglesand a kite).

• Have students, working in pairs, place two geoboards side by side.Have them make a triangle on the left-hand geoboard. Explain thatthis geoboard will be rotated a quarter turn clockwise. Ask them totry to make a triangle on the right-hand geoboard that will looklike the other triangle after it is rotated. Have them check theirresults by doing the rotation. Repeat this activity several times,using other shapes. Include half turns for some of the rotations.

• Have students draw squares on squared dot paper. Ask, Whatshape would result from combining this square and its imageafter a slide completely along any of its sides?

• Have students use shapes from pattern blocks, tangrams, logicblocks, and other sources to predict and confirm the results ofperforming various transformations.

• Give the students pictures of shapes and their images under varioustransformations. Have them predict what the transformationalrelationships are and then confirm by using tracing paper.

E12

• Have students draw on squared dot paper examples of thedifferent quadrilaterals—square, rectangle, rhombus, parallelo-gram, trapezoid, and kite—and cut them out. Have them foldthe shapes to find lines of symmetry. Ideally, there would be anumber of different examples of each shape in the class. Ask,Do all quadrilaterals of the same type appear to have the samenumber of lines of reflective symmetry?

You could have students undertake another investigation usingmiras and pictures of the different quadrilaterals to locate and drawin lines of reflective symmetry.

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By the end of grade 6, students willhave achieved the outcomes forentry–grade 3 and will also beexpected to

iii) investigate and predict theresults of transformations andbegin to use them to compareshapes and explain geometricconcepts (e.g., symmetry andsimilarity)

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E11 predict and confirm the resultsof various 2-D figures underslides, reflections, and quarter/half turns

E12 make generalizations about thereflective symmetry propertyof the various quadrilaterals





PerformanceE11.1 Have students explore the different polygons that can bemade with each of the six different pattern blocks under reflections intheir sides and under half-turns about the midpoint of each side.Have them predict by visualizing; then, using two of each block, dothe transformations, and trace around each result. Ask, Whichblocks produced only one shape under all these transformations?Which block produced the most shapes? Have them examine allthe shapes made under these transformations and ask, Could anyof them also be described as a pattern block under a slide?

E11.2 Have students work in pairs, with two geoboards betweenthem. Ask one student to make a shape on one geoboard and askthe other student to visualize and make its slide, reflected, orturned image on the second geoboard. Have them check by usinga mira (for reflection) or turning the original (for a turn) after thevisualized image has been placed on the geoboard. Have the pairsof students change roles.

E11.3 Place the large triangle from the tangram set on theoverhead. Ask students to visualize and sketch each shape formedby reflecting this triangle in each of its sides. How many differentshapes were made?

InterviewE12.1 Tell students that someone says she has a quadrilateralwith three lines of symmetry. Ask students to explain how thiscould happen.

E12.2 Ask students to name each shape below, to state howmany lines of symmetry each has, and to show where they are.

PortfolioE12.3 Ask students to fold pieces of paper in half and in halfagain, and to cut out from the corner of the double fold a polygonthat will unfold to make a rhombus. Have them repeat to make arectangle. Ask them to explain how they did these cuts and whythey work.

Data Management

and Probability:Data Management




GCO F: Students will solve problems involving the collection, display, and analysis ofdata.

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i) collect, organize and de-scribe relevant data inmultiple ways

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F1 recognize and use a varietyof methods for the collec-tion and organization ofdata

The collection of the data requiresdecisions to be made about what isneeded and how to get it. Before anydata are collected, data analysis beginswith decisions about what constraintsand freedoms will guide the collectionprocess. These decisions profoundlyaffect the later stage of interpretation.(NCTM 1989a, 135)

F1 Students have had many experiences collecting data in their earlyyears in school, e.g., Most will have found the favourite foods of otherstudents, lengths of names, or what pets their classmates have. Thistype of data collection should continue.

Students should be aware that there are many ways to collect dataand that these various methods may provide slightly different results,e.g., Students could consider the difference in data collected aboutfavourite foods if they simply ask each classmate to list his/her favour-ite, as opposed to offering a choice of three foods and asking studentswhich of the three they prefer; they might also consider the differencein results concerning favourite foods if they collect data right beforelunch rather than at another time of the day.

Many of the data collection experiences in the earlier grades includedsurveys of entire populations. Now students should recognize thatsampling procedures might be necessary in collecting data. Discus-sion should address introductory aspects of the concept of bias insampling, the idea of a sample being representative of the entirepopulation, and the ways sample size might affect the data.

• Ask students, Would data collected from surveying the basketballteam about their favourite sport probably look the same as datacollected from hockey players? How might a sample be selected sothat results are not biased? How many people should the sampleinclude? Ask students to give other examples of situations forwhich there could be bias in the sample used.

When provided with a set of data, students should take time toconsider the best way to organize it, e.g., If the information collectedis about pets, they might have to consider whether to have a categoryfor each different exotic pet, or one category called “other”; theymight have to decide whether to list the number of different petowners as opposed to the number of different pets, depending on theproposed use of the data. Choices must also be made about theformat of presentation, e.g., tables, graphs or descriptive displays,which might also influence the way students decide to organize thedata in the first place.





InterviewF1.1 Ask students to consider what questions they might ask inorder to determine what their parents’ attitudes toward school werewhen they were attending.

F1.2 Ask students to describe situations in which they might expectto get very different responses depending upon the people ques-tioned, e.g., The favourite hobbies of a group of 10-year-olds andthose of a group of senior citizens are likely to be different.

F1.3 Ask students, How reliable would data on average height ofgrade 4 students be if five students were measured?

PresentationF1.4 Ask groups of students to choose some facet of students’ hob-bies in which they are interested. Then ask them to determine waysto collect the information on the hobbies, to organize it, and todescribe it. Finally, have them present the information to theirclassmates.





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i) collect, organize and de-scribe relevant data inmultiple ways

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F2 describe data maxima,minima, range and fre-quency

F2 Often, with numerical data, it is possible for students to describemaxima, minima, range, and frequency, e.g., If each student in theclass goes home and counts the number of pennies he/she has savedand reports that information, it would be possible to describe suchvalues as

- the greatest number of pennies saved- the least number of pennies saved- the difference between the greatest and least- the most frequently cited number of pennies saved

Students could explore situations in which it would be useful toknow the greatest amount in a set of data called the maximum, theleast amount in a set of data called the minimum, and the differencebetween the least and greatest amounts called the range, e.g., Theycould consider situations when it is important to know the highestprices of items, the lowest prices of items, or the range in the prices.

Most students will relate frequency to the heights of bar graphs thatrepresent the number of items in different categories. They shouldexamine these graphs and describe which data values have particularlyhigh or low frequencies. Students could also think about topics to begraphed, and the categories to be used, for which the frequenciesmight be fairly equal, as well as ones for which the frequencies mightbe quite different.





Paper and PencilF2.1 Provide class lists for your students. Ask them to examine thelast names and find a way to describe the lengths of the last names ina concise format without listing each value.

F2.2 Tell students that a survey of numbers of pets in a class had arange of 6. Ask them to state what the data might have been.

InterviewF2.3 Ask students to suggest data for which

- the maximum might be 100- the minimum might be 100- the range might be 100- one of the frequencies might be 100.

F2.4 Ask students to tell why the frequency of “yes” and “no”responses to “Do you like mosquitoes?” might not be equal. Then askthem to think of other questions for which variable frequencies ofresponses might be expected.

PresentationF2.5 Have groups of students collect, organize, and describeinformation about the books they have read over the past six months.





F3 Students should continue to be given opportunities to read andinterpret bar graphs for which one bar unit or one symbol representsa quantity greater than one. They should also be asked to read andinterpret line graphs and stem-and-leaf plots, e.g., This line graphshowing the change in high temperatures over a period of twoweeks could be thesource of manyquestions thatwould requirestudents to read andinterpret the datapresented.

This will probably be students’ first experience with a stem-and-leafplot. This is a good way to present numerical data so that the data iswell organized for us to see trends and patterns yet all the data canstill be seen and worked with. When collected data is organized ingroups, e.g., 10-19, 20-29, 30-39, and 40-49, the stems would bethe tens digits and the leaves would be the ones digits of all the datain each group. For the data collected on a paper clip toss (measure-ments in centimetres): 12, 12, 13, 19, 20, 20, 25, 36, 36, 36, 40,42, 47, 47, 48, and 49, the stem-and-leaf plot would be constructedas 1 2 2 3 9

2 0 0 5 Distance Covered by Paper Clip

3 6 6 6 (in centimetres)

4 0 2 7 7 8 9

Similarly, data on student heights (in cm) couldbe shown on a stem-and-leaf plot (thestem is the first two digits of theheight) or on a connected bar graphwhere only the frequency of data ineach interval is shown. Because all thespecific data is still seen, the stem-and-leaf allows more specific questions to be answered or interpretationsto be made. Student Heights 13 8 9

14 1 2 6 6 9 9 15 2 3 8 9

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iii) read, interpret, and makeand modify predictionsfrom displays of relevantdata

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F3 read and interpret bargraphs, line graphs, picto-graphs and stem-and-leafplots





InterviewF3.1 Ask students, What questions might be answered by interpret-ing the graph below?

Favourite Types of Television Programs

F3.2 Ask students to interpret the graph below which shows dataabout district track meet participants and their homeroom classes.

Birchmount 4A 4B 5A 5A 5A 5B

Magnetic Hill 4A 4A 5A 5A 5B 5B

F3.3 Have students describe something this graph might be aboutand label the bars accordingly.

F3.4 Provide students with pictographs and bar graphs of the samedata. Ask them which display they prefer and why.

F3.5 Ask students to think of their favourite 2-digit number andwrite it down. Collect these numbers and prepare a stem-and-leafplot to display for students. Ask, In what interval do you find themost favourite numbers? What is the largest favourite number? thesmallest? What is the range of favourite numbers? Was there anumber picked by students more frequently than other numbers?





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ii) construct a variety of datadisplays (including tables,charts and graphs) andconsider their relativeappropriateness

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F4 display position, usingordered pairs on a grid

F4 Students should be introduced to the first quadrant of thecoordinate grid. They should be aware of the method for namingpoints and know why the order is significant, e.g.,(4,5) is not the same point as (5,4). It is appropri-ate to introduce, but not emphasize, terminologysuch as “axes,” “coordinates,” “plot,” and “origin.”

A grid of intersecting city streets isa useful context, e.g., John lives on3rd Avenue between 4th Street and5th Street. Place his house on thegrid. If the house were named bygiving its coordinates, what wouldthey be?

Geoboards can be used to introduce coordinate pairs, with pegsrepresenting the coordinates. On some commercial geoboards, thevarious vertical and horizontal positions are numbered. If they arenot, teachers can place number stickers on the bottom and left sidesof the boards.

The emphasis or goal of ourinstruction should be to help childrensee how graphs and charts tell aboutinformation. (Van de Walle 1994, 392)





PerformanceF4.1 Ask students to plot the points (2,3) and (6,9) on a coordinategrid. Then ask them to plot and name a point that they think mightbe considered to be between the other two.

F4.2 Ask students, What do you notice when you plot (1,2), (2,4),(3,6), and (4,8)? Ask students to name other points which go withthese.

Paper and PencilF4.3 Ask students to describe in writing the characteristics of thecoordinates of points that are (a) close to the origin, (b) well to theright of the origin, and (c) relatively high up above the origin.

F4.4 Ask students, If two vertices of a parallelogram are at (3, 5) and(4, 8), what are the possible coordinates of the other vertices?

InterviewF4.5 Ask students, Why do you need two numbers to locate a pointon a grid?

F4.6 Ask students, What do you know about the points on the gridfor which the first coordinate is equal to the second coordinate?

PortfolioF4.7 Have students plot the points (1,8), (2,7), (3,6), (4,5), (5,4),(6,3), (7,2) and (8,1). Ask, What is true about all these coordinates?What shape do you see after plotting them? Then ask students todesign a similar shape using different coordinates.





F5 Previously students had experiences interpreting graphs in whichpictures are used as symbols to represent a group of objects, e.g., acircle might represent a group of four people. Students should beginto make decisions about what symbol to use and what that symbolshould represent. Their decision will be based on the data, e.g.,Suppose student pet ownership totals 30 dogs, 15 cats, and 5 birds.Students might choose that a symbol for the pet graph represent 5animals.

Pet GraphDogs XXXXXXCats XXXBirds X

X represents 5 animals

Sometimes the numbers do not lend themselves to a simple choice,e.g., If the data were 23 dogs, 8 cats, and 2 birds, students mightdecide to let a circle represent 2 animals and use a half circle ( )where required. Students should make similar decisions regardinghow much each square in a bar graph is to represent.

Students will construct stem-and-leaf plots for the first time. Stem-and-leaf plots are considered valuable because they preserve the actualpieces of numeric data, e.g., If high temperatures (°C) over twelvedays were 26, 24, 19, 28, 24, 27, 30, 32, 29, 24, 25 and 28, astem-and-leaf plot would show the tens digit as the stem and onesdigit as each leaf as shown below.

High Temperatures

1 92 6, 4, 8, 4, 7, 9, 4, 5, 83 0, 2

• Tell students that you rolled a red die to get a tens digit and awhite die to get a ones digit and recorded the 2-digit numbersyou got as follows: 43, 12, 62, 25, 54, 15, 42, 36, 52, 21, 37,54, 61, 33, 13, 46, 25, 52, 66, 43, 36, 12, 65, 32, 44. Askthem to organize and present this data as a stem-and-leaf plot.

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ii) construct a variety of datadisplays (including tables,charts and graphs) andconsider their relativeappropriateness

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F5 construct bar graphs,pictographs and stem-and-leaf plots

Fourth grade is a wonderful time forchildren to see the connections amongtheir school subjects. There is no betterway to do this than by making sense ofall the information around them. Theycan use the techniques of collecting,analysing, and presenting data in manysituations that arise from various subjectareas. (NCTM 1989a, 17)





PerformanceF5.1 Ask students to collect information about the birthdays ofstudents in the class and display the data in a stem-and-leaf plot.Then ask them to generate two questions which could be answeredusing the graph.

F5.2 Ask students to draw a bar graph to display the number ofstudents travelling on each different school bus in the morning. Theyshould have each step along a bar represent more than one student.

F5.3 Present students with the following data:Favourite Subjects

Subject No. StudentsMath 100Language Arts 60Physical Education 80Music 75

Ask students to construct both a bar graph and a pictograph to showthe information. They should have each unit or symbol representingmore than one student.

F5.4 Present the stem-and-leaf plot below showing the amount oftelevision time for different students.

Minutes of Television Watching Time 1 5

2

3 0 0 0

4

5

6 0 0 0 0 0

7 5

8

9 0 0

Ask students to describe this graph. Ask, What does the shape of thisgraph tell you about the amount of time most students watch televi-sion? How could you use multiplication to help determine the meantelevision watching time?





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F6 interpolate data from adisplay

F6 Students looking at a graph or table can sometimes “interpolate”information—find information about a situation which was notspecifically provided in the data or on the graph, e.g., The graphbelow shows the multiples of 4. Use of the graph makes it clear that4 groups of 2

1

2 must be 10, even though the student has not studied

multiplication involving fractions and it was not a point used toconstruct the line.

Determining the amount of liquid in graduated measuring cupsprovides a real-life context that is related to students’ interpolation ofinformation from bar graphs, e.g., A student reading the containerbelow would have to infer that there must be about 175mL of liquiddisplayed.





Paper and PencilF6.1 Ask students to explain how the graph below shows that thedistance they would drive in 2.5 hours is 200 kilometres. Ask themto provide other hour-kilometre relationships that could be takenfrom this display.

F 6.2 Show students this graph of distances that a marble rolled off a30cm long ramp from different heights. Ask them to tell five thingsabout it.

(NCTM Addenda Series, Grade 4)





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iv) develop and apply measuresof central tendency (mean,median and mode)

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F7 describe data, using themean

For smaller sets of data ... the mean isperhaps a more meaningful statistic [thanthe median]. [Also] ... the mean is usedin the computation of other statisticssuch as the standard deviation. There-fore, it remains important that studentshave a good concept of what the meantells them about a set of numbers. (Vande Walle 1994, 399)

F7 Students should understand that the average, or mean, is asummary statistic that gives a general sense of a set of data. It may beused to give a sense of such things as how much a car costs, how wella baseball player performs, or the size of families in a certain geo-graphical area, e.g., If the mean of a set of test scores is 90%, while itmay not tell how many students achieved above or below 90%, itdoes tell that the class, as a whole, did very well.

Students could learn to calculate means in the course of learningabout division. They should learn to determine a mean visually byrepresenting the data with counters or on a bar graph and seeking toeven out the bars or counters, e.g., Suppose one student has saved $3,another $5, and another $10, and they want to know the averageamount saved. The mean of 3, 5, and 10 is 6 because they can movethe objects below into 3 equal lines of 6, as shown.

To practise the skill of finding means, students can use data aboutthemselves, e.g., leg and arm lengths, heights, number of siblings.

• Suggest that students state how many first cousins they have. Aclass mean can then be determined using a calculator. It may bethat the mean is not a whole number. Have students discuss howthis should be handled. Ask students then to explore what themean would have been if

- each student had had one more cousin- half the students had each had one fewer cousin- one student had had one more cousin than reported and another one fewer.





PerformanceF7.1 Have students create lines of cubes that are 14, 18, and 31cubes long. Ask them to show how to manoeuvre the cubes physi-cally to calculate the mean.

Paper and PencilF7.2 Ask students to create three different sets of data for which themean is 6.

F7.3 Ask students to create a set of data with the same mean as thatfor 36, 48, 52, and 67.

InterviewF7.4 Have students determine the mean of the following test scores:49, 49, 49, 50, 51, 52. Then ask, Is it possible for the mean score ona test to be greater than 50 if more than half of the students havemarks less than 50? Have students explain their responses.

F7.5 Ask students to describe situations in which it might be usefulto find a mean.

F7.6 Ask students, When will someone divide by 5 to find a mean?

F7.7 Ask students why someone might find the mean of 51, 58, and59 by finding the mean of 1, 8, and 9, and adding it to 50. Wouldthis work? Have the student give another example of this strategy.

PortfolioF7.8 Have students interview some adults to determine when theywould find it helpful to know the mean. Ask them to write a reporton their findings.





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v) formulate and solve simpleproblems (both real-worldand from other academicdisciplines) that involve thecollection, display andanalysis of data and explainconclusions which may bedrawn

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F8 explore real-world issues ofinterest to students and forwhich data collection isnecessary to determine ananswer

One of the most important rules tofollow in conducting graphing andstatistics activities is to let students gathertheir own data. (Van de Walle 1994,391)

F8 When investigating real-world issues students will need to makedecisions with respect to questions such as

- What questions should be asked?- Should we give a choice?- How will we group the responses?- What if there is a tie?- Will everyone be surveyed or will we randomly select a few students?

• Students might explore questions such as- How can we decide what colour we should paint the

classroom?- How can we determine how much to charge for a concert ticket in order to make a profit?- How can we find what pizza toppings we should order for our class party?

• If students were to explore the question of pizza toppings for the classparty, they might need to find out

- what toppings are available- how much it costs for different numbers of toppings- whether fractional parts of pizzas can be topped differently- whether only some possibilities should be presented- etc.





InterviewF8.1 Ask students how they might determine not only the mostpopular type of sport among 9-year-olds, but also possible reasons forthat result.

PresentationF8.2 Invite groups of students to determine a question to whichthey would like an answer. They should then plan how to collect,organize, and display the information.

PortfolioF8.3 Ask students how they would collect, organize, and displaydata to show the mean number of students absent each day during a2-week period.

F8.4 Measure the heights of everyone in the class and record thedata on paper as you gather it. Provide students with copies of thisdata. Ask them to prepare a stem-and-leaf plot for this data, deter-mine the mean height for the class, and to prepare at least threequestions that could be answered by examining their plots.

Data Management

and Probability:Probability




GCO G: Students will represent and solve problems involving uncertainty.

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i) explore, interpret and makeconjectures about everydayprobability situations byestimating probabilities,conducting experiments,beginning to construct andconduct simulations, andanalysing claims which theysee and hear

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G1 predict probabilities aseither close to 0, near 1, ornear

1

2

G2 cite examples of everydayevents with very high orvery low probabilities

Whenever possible ... we should try touse an experimental approach in theclassroom. (Van de Walle 1994, 384)

G1 Students should recognize that a probability close to 0 meansthat an event rarely occurs, and that a probability near 1 means thatan event that almost always occurs. Students should explore themiddle of this range by thinking of as many situations as they canfor which the probability is about

1

2, e.g.,

- A coin flip results in heads.- A die is rolled and the result is an odd number.- A child is born and is a boy.

• Provide students with hundreds charts. Have them describesituations that would have low, high, and middle-range prob-abilities, e.g., The probability that a dropped chip will land onan even number on the hundreds chart is

1

2, that it will land on a

number greater than 95 is almost 0, and that it will land on anumber less than 120 is 1.

G2 Students should be able to give examples of real-life ormathematical events with very low or very high probabilities, e.g.,The probability is close to 0 that an elephant will ever be in theschool, but the probability is close to 1 that a fly will be.

You should take every opportunity to ask about probabilities whenstudents are studying other mathematics topics, other subjects, andparticipating in other activities, e.g., When they are estimatingproducts such as 34 x 56, you could ask, What is the probabilitythat the product of this will be greater than 1500? greater than2400?





InterviewG1.1 Ask students to think of an event related to eating that has aprobability very close to 0.

G1.2 Ask students, Why might the probability of seeing a polar bearbe 0 for some people, but close to 1 for others?

G1.3 Ask students to describe situations for which a probabilitymight be very high in one case, but very low in another.

G2.1 Ask students, Why might it be silly to talk about the probabil-ity of something that has already happened?

G2.2 Ask students, What does the forecaster mean if he/she says theprobability of rain is almost 100%?

PortfolioG1.4 Ask students to invent or describe a game in which theprobability of a certain event might be very close to 1. Have themdiscuss why one might get different numbers of points for differentevents in that game.

G2.3 Ask students to think of some words that people use every daythat might be considered probability words, e.g., surely, not likely.





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i) explore, interpret and makeconjectures about everydayprobability situations byestimating probabilities,conducting experiments,beginning to construct andconduct simulations, andanalysing claims which theysee and hear

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G3 predict whether one simpleoutcome is more or lesslikely than another

G4 use fractions to describeexperimental probabilities

G3 Using spinners, dice, coloured cubes, or number possibilities,students should be able to predict whether one outcome is more orless likely than another, e.g.,

(a) From a bag with 8 red cubes and 4 yellowcubes, they are more likely to pull out a red cube,in fact twice as likely.

(b) If they use this spinner, they are less likely tospin an even number than an odd one.

(c) When they roll two dice, they are more likely to get numbers thatsum to 7 than numbers that sum to 11. (They should investigatewhy the number 7 is referred to as “lucky.”)

You should encourage students to predict probabilities and thencheck them by carrying out experiments, gathering data for them-selves and pooling data with their classmates.

Introduce students to games of chance some of which they discoverare fair and some not fair. Discuss how the rules might be changed tomake unfair games fair. Students could then invent games related tosums and products using dice, spinners, and cards, e.g.,You get apoint if the sum of the two cards you pick is highest. Then, studentswould try to decide if their games are fair.

G4 Students should use common fractions to describe simple experi-mental results, e.g., If students toss coins 10 times and gets heads 4times, they would use the fraction to describe the experimentalprobability of getting a head. It would be advisable, at this stage, torestrict the total number of tries to 12 or less, so that the fractionsthey get are more meaningful to them.

Students should learn from experience that if an experiment is re-peated or more data is gathered, the results could be different. Theyshould compare their results with the results of others to see that,indeed, the values are often different.

Other simple probability situations include (a) spinning spinnerswith differently coloured or marked sections, (b) choosing colours ofcubes out of a bag, and (c) counting the number of reds when dou-ble-sided counters are tossed.

1 2

53





PerformanceG3.1 Have students design experiments for which a certain outcomehas a probability near 0, but not 0.

G3.2 Have students design experiments with three possible out-comes in which one of the outcomes has a probability near

1

2.

G4.1 Ask students to choose novels. Have them open their novel toany page and note whether or not the first letter is a “t.” Have themcheck 10 pages in this manner. Ask them then to describe the prob-ability that “t” is the first letter on a page of a book.

G4.2 Have students station themselves within or near the schoolwhere they can see passing cars. Have them record the colours of thefirst 10 cars they see. Then ask them to describe the probability thata passing car will be blue. Have them discuss why they might get adifferent probability the next time they perform the experiment.Then check to see if they do.

PresentationG3.3 Have students create lists of the names of students in the class.Ask them to generate some observations about the names, e.g., Morethan half start with a letter between A-M or less than half of the lastnames have fewer than 5 letters. Ask them to produce a “quiz” dealingwith the probability that various statements about the names aretrue, e.g., Two types of quiz items:

Type 1: ______ is more likely than _______ .Type 2: Is the probability of ________ closer to 0,

1

2, or 1?

PortfolioG3.4 Provide an example of a spinnergame in which one person wins if thesum of the numbers on two spinners iseven, and another wins if this sum isodd. Have students discuss the fairness of the game and then inventtheir own fair and unfair spinner games.

G4.3 Ask students to perform the following experiment: Turn on aradio. Note whether the first voice you hear is a female or a malevoice. Turn off the radio. Change the station. Try again 15 minuteslater. Repeat the experiment five times. Describe the probability ofhearing a male voice.

Grade 5

Number Concepts/



Number Concepts


GCO A: Students will demonstrate number sense and apply number theory concepts.



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i) demonstrate an understand-ing of number meaningswith respect to wholenumbers, fractions, anddecimals

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A1 represent whole numbers tothe millions

A1 Students will continue to use whole numbers as they performcomputations or measurements and as they read and interpret data. Tohave a better understanding of large numbers, such as a million, studentsneed opportunities to investigate problems involving these numbers.

• Have students investigate the following problems:- How many $100 bills would it take to make $1 million?- How long would a line of one million centicubes be?- How many garbage bags would be needed to hold one million 2-litre pop bottles?- How much grid paper would be needed to show one million squarecentimetres?

There are many children’s books that are good sources of explorations ofnumber meanings.

• To help students visualize one million, show them a large cube in thebase-10 blocks and ask, How many cubic centimetres does this show?Place 10 of these large cubes in a row to make a large base-10 “rod.”Ask, How many cubic centimetres in this rod? Highlight the connec-tion between this “rod” representing 10 thousand and the small rodrepresenting 10. Ask, If we had enough large cubes to make ten rowsof 10 thousand, what would we have? Highlight the connectionbetween this large “flat” representing 100 thousand and the flatrepresenting 100. Ask, If we were to stack ten of these large“flats,”what would we have? Highlight the connection between thisnew large “cube” representing 1 million and the cubes representing 1thousand and 1. As well, make the connection between this cuberepresenting 1 million cubic centimetres and representing 1 cubicmetre. Have students use metre sticks to make a cubic metre andplace the large base-10 cube and the rod made up of ten large cubesinside to help in their visualization of 1 million cubes.





PerformanceA1.1 Ask students to predict whether the gymnasium could hold amillion cereal boxes and to make sufficient measurements to check theirpredictions.

InterviewA1.2 Ask students to explain how they know that 1 345 121 is greaterthan 1000 thousands and to suggest what this number might be used torepresent. Ask how a number such as this might be written in a newspa-per.

A1.3 Ask students to decide if they have lived 1 000 000 hours yet andto explain the reasoning for their decision.

A1.4 Ask students, How does a million compare to a thousand? to tenthousand?

A1.5 Ask students to explain how they know that 2 345 121 is greaterthan 2000 thousands and to tell where they might see such a numberused.

PortfolioA1.6 Ask students to use newspapers and/or catalogues to find items tobuy that would total $1million. Limit the number of any one item theycan purchase to 5. You could have them follow this up by conductinginterviews with senior citizens to find out what could have been pur-chased with $1million fifty years ago. Ask students to write a report ontheir findings.

A1.7 Students could work in pairs to create 2-page spreads for a classbook about a million. Each spread could begin, If you had a million__________, it would be ___________.





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By the end of grade 6, studentswill have achieved the outcomesfor entry–grade and will also beexpected to


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A2 interpret and modeldecimal tenths, hundredths,and thousandths

A2 Students should continue to use materials to represent or modeldecimals. In this way, they can better see the relationship betweenhundredths and thousandths, e.g., Studentsmight use thousandths grids of the samesize as hundredths grids to model decimalsto the thousandths.

Base-10 blocks should be used to illus-trate decimals. Within a given context,the large block could represent 1, thenthe flat would represent 0.1, the rod0.01, and the small cube 0.001. Themodel for 3.231 would be shown as:

You should vary which block represents 1 so students will developflexibility in thinking about decimal fractions. Students seem to find iteasier to adjust their thinking about the blocks if appropriate contextsare used, e.g., imagining the large cube as a block of 1L of ice cream, theflat as 1 slice of bread, or the rod as 1 package of candy.

Since 1mm = 0.001m, students can also represent thousandths usinglength measurements, e.g., 0.423 m can be represented as 423 mm, 42.3cm, and 4.23 dm.

Students should be encouraged to represent the symbols for decimals ina variety of ways, e.g., 0.452 is 452

1000,

45

100 +

2

1000,

4

10 +

52

1000.

To help students develop number sense, encourage them to usereference points, e.g., 0.452 m is a little less than half a metre. Somestudents may recognize that it is only 0.048 m less than half a metre.

• Provide opportunities for students to find and share how largenumbers are represented in newspapers, magazines, and almanacs.

Students should recognize that decimals can represent something quitesmall in some situations and something very large in other situations,e.g., 0.025 m is only 2.5 cm; however, 0.025 of the population ofCanada refers to 25 out of every thousand people, 250 of every 10 000people, 2500 of every 100 000 people, 25 000 of every million people,and if we were to round Canada's population to 30 million, 30 x 25 000or 750 000 people. Discussions such as these help students developnumber sense.





PerformanceA2.1 Have students model 0.025 using decimal squares. Ask, Howdoes this model differ from the model for 25 hundredths? Ask them tomodel both amounts using base-10 blocks. Ask, What block did you useto represent the ones place? Why?

Paper and PencilA2.2 Have students identify the decimal represented by the shadedportion of the diagram if the hundred grid represents 1 whole. Ask,How much more is required to make a whole?

InterviewA2.3 Show students cards on which decimals have been written, e.g.,0.75 m and 0.265 m. Ask them to place the cards appropriately on ametre stick.

A2.4 Ask students to express 0.135 in at least three different ways.

A2.5 Ask students to identify situations in which 0.25 represents asmall amount and situations in which it represents a very large amount.

A2.6 Tell students that a new bakery slices its loaves of bread into 10equal pieces, makes bread sticks by cutting each slice of bread into 10equal pieces, and makes croutons by cutting each bread stick into 10equal pieces. Ask them to model this using base-10 blocks. Ask ques-tions such as, What part of the loaf is 1 slice? 3 slices? 1 stick? 5 sticks?1crouton? 9 croutons? 3 slices and 2 sticks? What part of 1 slice is 4sticks? 6 croutons? 2 sticks and 3 croutons? Then have students use theblocks to show quantities such as 0.2 loaf, 0.14 loaf, 1.5 loaves, 0.5 slice,0.25 slice, 0.7 stick, and 0.3 stick.





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A3 interpret, model, andrename fractions

A3 Developing number sense with fractions takes time and is bestsupported with a conceptual approach using materials. Using a varietyof manipulatives helps students understand properties of fractions andrealize that the relationship between the two numbers in a fraction isthe focus. Possible manipulatives are pattern blocks, Fraction Factory,geoboards, colour tiles, counters, and egg cartons.

Provide students with a variety of activities that include threeinterpretations of fractions: (1) part of a whole, e.g.,

1

3 of a chocolate

bar; (2) part of a set, e.g., 2

5 of a bag of marbles; and (3) part of a

linear measurement, e.g.,3

4 of a race track.

It is important that students areable to visualize equivalentfractions as the naming of thesame region partitioned indifferent ways as shown here.

Some manipulatives that illustrate equivalent fractions well include

-fraction circles or squares -pattern blocks

11

2 =

3

2 1

1

3 =

8

6 =

4

3

- geoboards/geopaper - egg carton

2

5 =

4

10

9

12 =

3

4

More than one egg carton can be used to show mixednumbers and fraction equivalents, e.g., 1

2

3 =

5

3

You should avoid introducing a rule about multiply-ing numerators and denominators to form equivalents. You couldconfirm such as rule if students observe it; however, the explanationshould be connected to manipulatives and pictures.

• Ask students to use Fraction Factory or coloured strips to findexamples of equivalent fractions.

• Show the green triangle in the pattern blocks and tell students itrepresents

1

3. Ask them to show 1 using pattern blocks.

1234567123456712345671234567

1234567123456712345671234567

12345678123456781234567812345678

1234567123456712345671234567

24

36

48

= ==12





PerformanceA3.1 Ask students to use their fingers and hands to show that

1

2and

5

10 are equivalent fractions. Alternatively, you could ask students

to choose a manipulative of their own choice to show this or someother equivalence.

A3.2 Ask students to use Fraction Factory or coloured strips to show1

2

3

5

3= and 1

3

4

7

4= .

A3.3 Ask students to use the circle below to find 2

3 of 18.

A3.4 Tell students that you have a pan of squares representedby a geoboard. Ask them to use geoboards to explain the equivalenceof

1

2,

2

4, and

4

8 and to make the connection to the pan of squares.

A3.5 Have pairs of students use coloured tiles to show equivalencesas written on selected cards, e.g.,

3

4 and

6

8,

3

9 and

1

3,

2

3 and

8

12.

Paper and PencilA3.6 Show students a picture that represents 6

8.

Explain to them that to rename this 6

8 as 3

4,

they can think of “clumping” the 8 sections ofthe whole into 2s. There are then four groups of2 sections. Out of these four groups, three areshaded.

Have students make a diagram and identify the “clump size” thatshould be used to show that

10

15

2

3= . Ask how they might predict the

“clump size” without drawing the diagram.

PortfolioA3.7 Have students prepare posters showing all the equivalent fractionsthey can find using a set of no more than 30 pattern blocks.





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A4 demonstrate an understand-ing of the relationshipbetween fractions anddivision

A4 Students should understand the relationship between fractionsand division to help them interpret fractions of sets and later toconvert fractions to decimals.

When looking at a division situation, such as 16 ÷ 3,students can visualize it as

1

3 of 16—the share on

1

3 of a

mat when the 16 is shared equally among the 3 parts ofthis mat (

16

3 or 5

1

3 ).

When they look at a fraction, students can think of it as an alternativeway of expressing division, e.g.,

2

3 is the amount each person would

get when 3 people share 2 items.Each person gets

1

3 of each item.

1

3 +

1

3 =

2

3

Alternatively, 8

3 tells how many groups of 3 in 8.

2

3 of a group of 3

• Remind students that the blue rhombus in the pattern blocksrepresents

1

3 of a hexagon. Ask them to put together 14 of these

rhombuses—3 at a time—to make hexagon shapes. When thetask is complete, ask students to discuss why another name for

14

3is 4

2

3 , and why they could think of it as 14 ÷ 3.

A problem that would be represented in this way would be: If eachperson at a party of 14 could eat 1

3of a large pizza, how many

pizzas would you have to buy?

As students work through a number of these types of problems, theywill see the pattern that dividing the numerator by the denominatorwill change an improper fraction to a mixed number. It would beinappropriate to just tell students to divide the denominator into thenumerator to change an improper fraction to a mixed number with-out developing the conceptual understanding for such a procedure.Ideally, you will have students who discover this pattern and share itwith their peers.

✔ ✔

1 group of 3 2nd group of 3





PerformanceA4.1 Tell students that there were three pizzas left from the class party.Everyone agreed that the organizing committee would share what wasleft. Ask, What part of the pizzas would each of the four membersreceive if the pizzas were divided equally?

A4.2 Ask students to use pattern blocks to explain how they know17

3 is the same as 5

1

3.

Paper and PencilA4.3 Tell students that you divided one number by another and theresult was 2

1

2. Ask them what the two numbers might have been.

A4.4 Have students draw two different pictures of squares to show4

5—one picture to represent part of a whole and one to show sharing.

Ask them to write possible stories that each picture could represent.

InterviewA4.5 Show tstudents the following and tell them that one person said itrepresented

5

4 and another said it was

5

8. Ask them which person they

think was correct and to give reasons for their answers.

A4.6 Ask students how many buckets of water they would need towater nine plants if each plant needs

1

2 bucket.

A4.7 Ask students to explain why they would divide 16 by 3 to find themixed number name for 16

3. Invite them to use drawings or models in

their explanations.

PresentationA4.8 Give a group of students a fraction. Ask them to act it out in adivision “skit.” The rest of the class has to guess the fraction beingacted out, e.g., For

13

4, students might pretend to make families of

four using thirteen classmates to see how many families there are.





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ii) explore integers, ratios andpercents in common,meaningful situations

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A5 explore the concepts ofratio and rate informally

A5 Ratio is a multiplicative comparison of two numbers or measures orquantities, e.g., 10:1 is the ratio of the value of a dime as compared tothat of penny; 3:2 is the ratio of the number of boys to the number ofgirls in a group of 3 boys and 2 girls. You are using ratio when stating,“She is running twice as fast now as last year.” (2:1) or “She has done 3times as much today as she did yesterday.” (3:1). You would read ratiossuch as 10:1 as “ten is to 1” or “10 to 1.”

In the past, students have often compared two quantities in a subtractiveor additive way. For ratio, emphasis needs to be given to the multiplica-tive comparison, e.g., If you ask them to compare a dime to a penny,students may say that the dime is 9 cents more than the penny but this isnot a ratio. Help them view the comparison as 10 cents to 1 cent or thevalue of a dime as 10 times the value of a penny.

A special ratio is a rate. It is a multiplicative comparison of two quanti-ties with the quantities described in different units, e.g., 2 cans for 98¢ isa price rate for a product and 20 km/hour is a rate of speed.

Students should relate ratios and common fractions, e.g.,The ratio of shaded parts to the total number of parts inthe circle is 1:4 or

1

4. (This is also the fraction of shaded

parts.) The ratio of shaded parts to unshaded parts is 1:3or

1

3 which conceptually is not a common fraction. It is suggested that if

a fraction such as 1

3 is used to represent a ratio, you describe it as “1 is

to 3” rather than as “one third.”

• Provide students with geometric situations such as the ratio of thenumber of sides in a hexagon to the number of sides in a square(6:4), the ratio of the number of vertices to the number of edges in arectangular prism (8:12), and the ratio of the number of vertices in ahexagon to the number of sides (6:6).

• Provide students with numerical situations such as the ratio compar-ing the value of a quarter to that of a dime (25:10), the rate of pay fora job ($5/hour), and the ratio comparing the number of multiples of2 to the multiples of 4 for numbers from 1 to 100 (2:1 or 50:25).

• Provide students with measurement situations such as the ratedescribing the “crowdedness” of a classroom (25 people/60m2), theratio of perimeter to side length of a square (4:1), the ratio of describ-ing the enlargement factor on a Xerox copy (3:2), and the ratecomparing centimetres to decimetres (10:1).





PerformanceA5.1 Ask students to model situations in which there are six of onething for every two of another thing.

Paper and PencilA5.2 Ask students to fill in the blanks in as many ways as possible tocreate a true statement about mathematical situations, e.g., For every ______ ______ , there are _______ _______ .

(Sample response: For every one dozen, there are 12 items.)

A5.3 Ask students to make one drawing that would show both ofthe following:

For every one pencil, there are three pieces of paper.For every three pencils, there are nine pieces of paper.

Ask the students what they would say for every five pencils.

InterviewA5.4 Ask students to give a number of ratios that relate to sports,e.g., For every 5 players on the starting lineup in basketball, there are9 players in baseball. Have them share their answers.

A5.5 Ask students to place two different pattern blocks side by sideand state their ratio in two ways, e.g., The yellow block is three timesthe size of the blue block or the blue block is one-third the size of theyellow block.

A5.6 Ask, How can you give the length of an object in centimetres ifyou know the length in millimetres? Would it be possible to give thelength in centimetres if you knew the length in metres? Which doyou find easier? Why?

A5.7 Ask students to give as many ratios as theycan using a set of buttons or a picture of buttons.

A5.8 Tell students that the ratio of pens to pencils in a boxis 4:3. Ask them to fill in the blanks: The number of pens is ___times the number of pencils; the number of pencils is ___ times thenumber of pens.

A5.9 Ask students to find sets of two rectangles in the FractionFactory that are in the ratio 1:2, in the ratio 3:1, and in the ratio 3:2.

PortfolioA5.10 Ask students to describe five situations that depict a rate of3:1





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iii) read and write wholenumbers and decimals, anddemonstrate an understand-ing of place value (tomillions and to thou-sandths)

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A6 read and represent numbersto millions

Two important ideas developed for three-digit numbers should be carefullyextended to larger numbers. First, thegrouping idea should be generalized.That is, ten in any position makes asingle thing in the next position, andvice-versa. Second, the oral and writtenpatterns for numbers in three digits isduplicated in a clever way for every threedigits to the left. (Van de Walle 1994,173)

A6 Students should be aware of the pattern in the place-valuesystem—each group of three digits is read as a number (up to 999)with the appropriate unit, e.g., 42 135 456 is read as 42 million,135 thousand, 456.

A place-value chart blockedinto sets of 3 digits is useful todemonstrate this idea.

Students should be exposed to reading whole numbers in a variety ofways, e.g., 6 200 000 as “six milliion, two hundred thousand” or as “6and 2 tenths million” since of a million is 100 000; 2 153 456 as “2million, one hundred fifty-three thousand, four hundred fifty-six” or as“two thousand one hundred fifty-three thousand, four hundred fifty-six.”

• Provide students with practice placing counters or digits on place-value charts to represent numbers that you state orally. Have themwrite the numerical form of the numbers once the charts are filled in.Ask students to read back the numbers formed.

• When students have had sufficient practice with place-value charts,have them write only the numerical form of numbers that you readaloud to them. Vary the difficulty by including numbers that includeseveral zeros. Ask students to read the numbers back to the class.Whenever an activity such as this is designed for a class, it is recom-mended that more be done with the numbers than simply writingthem properly, e.g., discussing situations where the numbers mightbe used, placing them on number lines, and asking for numbers thatare related such as one thousand more or one hundred less. Thesetypes of activities will help students develop a sense of large numbers.

• As an extension to reading and writing numbers, ask students topractise telling how many more must be added to make a particularnumber, e.g., Have students write the numerical form of “ninehundred eighty thousand, four” and ask how many more wouldbe needed to make a million. Students should find ways thatmakes sense to them to find the difference, e.g., Some may recog-nize that twenty thousand less four would make a million, ornineteen thousand, nine hundred ninety-six. Learning experiencessuch as this provide practice with mental math strategies.





PerformanceA6.1 Have students use all the digits 2, 0, 4, 0, 5, and 3 three timesto record three different whole numbers. Ask them to read eachnumber. As an extension, ask students to determine how manydifferent numbers can be made using these digits.

A6.2 Ask students to make the largest possible number on theircalculators. Have them read the number.(They will probably pointout the fact that numbers without spaces are more difficult to read.)Ask, If you were to subtract 98 765 432 from the number displayed,predict what the calculator will show when you press the equal sign.Check it.

Paper and PencilA6.3 Ask students to write in numerals the populations of BritishColumbia which is “three million, two hundred eighty thousand,”and of Quebec, which is “six and nine-tenths million.”

InterviewA6.4 Ask students to explain everything they would know about anumber if they were told that it has eight digits.

PortfolioA6.5 Have students write reports on the different ways numbers arewritten in newspapers and magazines. A possible approach might beto report on “Estimation in the Media.”

A6.6 Ask students to make a list of whole numbers that take threewords to say, e.g., 9 000 080, 600 000, and 403.

A6.7 Tell students that a company has 1.45million paperbacks. Ask,How many boxes, and what size, would be needed to hold thesebooks? Could the students in the school read this many books and, ifso, how long would it take them? Have them determine how manylibrary shelves this many paperbacks would fill.

PresentationA6.8 Have students rearrange the cards below in a variety of ways. Askthem to write the numerical form of the number represented by each oftheir arrangements.

five hundred two million three thousand four





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A7 read and represent decimalsto thousandths

A7 It is important that students recognize that the third place afterthe decimal represents thousandths, the second place representshundredths, and the first place represents tenths. Students should beable to read decimal numbers in print and record the numerical formof decimals upon hearing them orally or seeing them written out.

Students should state the fractional value of the decimal digits whenreading decimal numbers using the word “and” only for the decimalpoint, e.g., 16.8 is read “sixteen and eight tenths,” 0.57 is read “fifty-seven hundredths,”and 2.091 is read “two and ninety-one thou-sandths.”

Students should also be able to place decimal numbers on number lines,e.g., Given a segment with end points labelled 2 and 4, students shouldmark where they think each of the following numbers would be anddefend their decisions: 2.3, 2.51, 2.999, 3.01, 3.75, 3.409 and 3.490.

Reading decimal numbers in context, such as with kilograms ofhamburger or with litres of juice, is useful when making the connectionbetween fractions and decimals. 6.25 L is six and twenty-five hundredthslitres, thought of as 6

25

100, which some might recognize as 6

1

4 .

Students should also be able to interpret large whole numbers written indecimal format, e.g., 5.1million as 5 100 000).

18.5 can be read “18 and 5 tenths” but is often read “18 and a half.”Have students practise estimating and reading numbers in this way, e.g.,6.497 may be read as “about 6 and a half,” 48.73 as “about 48 and 3quarters,” and 12.254 as “about 12 and a quarter.”

• Have the students discuss when common fractions rather thandecimal numbers are likely to be used and when decimal representa-tions would be more appropriate.





PerformanceA7.1 Give pairs of students three dice and ask them to take turnstossing all three. Ask them to make the largest number they canusing the digits to represent tenths, hundredths, and thousandths.Have them represent the number using base-10 blocks. Ask themhow much would need to be added to their number to make onewhole, e.g., If they were to toss 3, 6, and 2, the largest possiblenumber would be 0.632 represented by 6 flats, 3 rods, and 2 smallcubes if the large cube is a whole; 0.368 would need to be added tomake one whole.

A7.2 At a centre, place five different displays of combinations ofbase-10 blocks. Ask students to visit the centre and record the fivedecimals displayed if they are told that the large cube represents 1.

Paper and PencilA7.3 Tell students that gasoline is priced at 56.9¢ per litre. Ask,What part of a dollar is this?

A7.4 Ask students to write the decimal numbers for "two hundredfifty-six thousandths” and “two hundred and fifty-six thousandths.”Ask them to explain why watching and listening for “and” is impor-tant when interpreting numbers.

InterviewA7.5 Ask students to explain why newspapers might record anumber as 2.5 million instead of 2 500 000. Ask them to discusswhether or not this is a good idea.

A7.6 Show students a written request that Samuel’s teacher gavehim: “Please cut 3.25 m of ribbon for me.” Ask students to read thenote and explain how many centimetres of ribbon this would be.

A7.7 Tell students that you drank 0.485 L of juice. Ask, About howmuch more would I have to drink to equal 0.5 L?

PortfolioA7.8 Ask students to write reports on the use of 0.5 and

1

2 . Have

them survey adults and check newspapers and magazines to findwhen each is used.

A7.9 Ask students to write ten different decimal numbers that havetenths, hundredths, and/or thousandths. Have them make base-10block pictures that would represent their numbers.





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iv) order whole numbers,fractions, and decimals, andrepresent them in multipleways

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A8 compare and order largenumbers

A8 Students should be able to compare two whole numbers when - both numbers are written out in full, e.g.,

34 256 876 > 34 255 996 - both numbers are written in decimal form, e.g.,

34.25 < 34.3 - one number is written in full and the other in decimal form, e.g.,

34 256 876 < 35.2 million - numbers are written using different groupings, e.g.,

3423 thousand 453 > 3 325 146The latter two formats tend to be more challenging for students. It isimportant, therefore, that they have practise renaming numbers indifferent ways. Once students recognize that there are 1000 thousands ina million, they realize that 6 million and 6000 thousands are the same.Students need to have a sense of the size of large numbers—simply beingable to point out the place value of digits is not sufficient.

• Provide students with problems in which the numbers to be com-pared are in context. (a) Mrs. McKinnon won $2 435 752 in thelottery. She already had $2.5 million saved. Has she doubled hermoney? Explain. (b) Order the populations of the following metro-politan cities from least to greatest:New York—17.95 million Paris—8 720 000Tokyo—28.4 million London, England— 7 000 000Ask students to make comparative statements about the populations,e.g., The population of Tokyo is about four times that of London.Paris is about half the population of New York City. The populationsof New York and Paris together are less than the population of Tokyo.

You will probably need to give attention to rounding to the nearestmillion or hundred thousand for the purpose of comparing numberseasily.





PerformanceA8.1 Ask student to order the following number cards from least togreatest.

A8.2 Tell students that Jason spent $980 every day for 5 years, Suespent $854 every day for 6 years, and Keri spent $1156 every day for4 years. Ask them to predict who spent the most and the leastmoney. Have them use calculators to check their predictions.

A8.3 Have students predict which three cities in the world theythink have the most people. Then provide them with data (or havethem find the data on the Internet) on the populations of ten of themost populated countries in the world. Ask them to order the data toverify their predictions.

InterviewA8.4 Tell students that Mary compared 3 425 630 and 3 524 013by explaining that 34 hundred thousand is less than 35 hundredthousand. Ask them to explain Mary’s reasoning and to identify otherapproaches that could be used to make the comparison.

PortfolioA8.5 Have students design games for their classmates that requirethem to compare and order large numbers.





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A9 compare and order decimals

A9 Students should be able to determine which of two decimal num-bers is greater by comparing the whole number parts first and then theamounts to the right of the decimal points. It is important that studentsunderstand decimal numbers do not need the same number of placesafter the decimal point to be compared. They should also understandthat a number having more places after the decimal point than anotherdoes not mean it is smaller nor does it mean it is larger—these arecommon misconceptions. That is, some students think that 0.101 islarger than 0.11 because 101 is larger than 11; other students think it issmaller just because it has thousandths while the other number has onlyhundredths. (These same students would also say 0.101 is smaller than0.1 because it has thousandths while 0.1 has only tenths.) These miscon-ceptions are best dealt with by having students make base-10 blockrepresentations of numbers that are being compared.

Encourage students to use benchmarks such as one-fourth, one-half, andthree-fourths, e.g., They can quickly conclude that 0.8 > 0.423 becausethe former is much more than half and the latter is less than half.

Encourage students to visualize the base-10 block representations ofnumbers they are comparing. Have them first model such comparisonsusing the actual blocks to see that it is a powerful strategy.

Students should recognize that thousandths may be small in comparisonwith some other numbers, e.g., 0.003 is small compared to 3; therefore,thousandths may make little difference when two numbers are comparedunless, of course, both numbers are very small as in the case of 0.014 mand 0.009 m. There are times, however, when thousandths are not small,e.g., If a population of 3.124 million is to be compared with otherpoplations, the 4000 represented by the thousandths place would not beconsidered small. The size is relative to the context.

Encourage students to round decimals being compared to the nearesttenth or hundredth to get a relative sense of their sizes.

Measurement provides valuable contexts for decimal numbers becauseany measurement can be written in an equivalent unit that wouldrequire decimals, e.g., 345mL is 0.345 L.

• Give students eight blank cards and ask them to write a decimal oneach. Have them challenge partners to order the number cards.





PerformanceA9.1 Provide sets of “digit cards.” Ask students to place cards in themissing spaces, in as many ways as possible, to make the sentencebelow true.

9. 8 < . 2 0

A9.2 Have students copy a template of the type shown.

Roll a die. As you call out the number, students should fill in one blankon their papers. Roll the die 18 times. The students who end up withthree true number sentences win a point. Repeat the process.

A9.3 Give student partners six cards with different base-10 blockpictures on them. Ask them to order the numbers represented andread the decimals to one another. Explain that the flat equals 1 forthis activity.

InterviewA9.4 Ask students to explain why they cannot compare two decimalnumbers by simply counting the number of digits in each number.

A9.5 Give students the number cards 9.023, 10.9, 9.05, 10.11, and9.8. Ask them which decimal they think is closest to 10. Have themexplain how they made their decisions.

PortfolioA9.6 Provide examples of some of the best javelin throw distancesthat have occured in past Olympics, e.g.,

1972—90.48 m1980—91.20 m1988—84.28 m1992—89.66 m

Ask students to arrange the distances in order and determine whetherrecords always improve. Have students follow up by choosing recordsfrom other Olympic events to order and include in their portfolios.





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A10 compare and order fractionsusing conceptual methods

An understanding of fraction conceptsand order and equivalence relations is aprerequisite for success in computationwith fractions. (NCTM 1989a, 160-61)

A10 Students should continue to use conceptual methods to com-pare two fractions. These methods include (i) comparing both to abenchmark such as one-half, (ii) comparing the two numerators whenthey have the same denominator, and (iii) comparing the two de-nominators when they have the same numerators.

With the third method, students often make a common error becauseof their experience comparing whole numbers, i.e. In comparingfractions such as

10

7 and

10

6, theythink

10

7 is greater than

10

6 because

of the relationship between 7 and 6 as whole numbers. You will needto spend considerable time on activities and discussion to developnumber sense for fractions. Contexts such as the “pizza model” workwell, e.g., Ask students, Which would you rather have, a piece ofpizza divided into 6 equal parts or a piece of the same pizza dividedinto 7 equal parts?

Pose questions such as - Which is greater,

3

10 or

3

8 ? A possible answer, “I know

3

8 >

3

10because eighths are larger than tenths.”

- Which is greater, 3

8 or

7


7

10 is

greater than 3

8 because

3

8 is less than half and

7

10 is greater than

half.”

Students should be encouraged to compare fractions greater than oneby considering them as mixed numbers, e.g., Which is greater,

10

8 or7


7

5 is greater because 7

5 is 1 2

5 , 10

8is 1

2

8, and

2

5 is greater than

2

8.”

• Invite students to create patches (made of paper) fora class patchwork quilt in which the colours on theirpatches show a particular comparison.

Some students may be ready to compare pairs offractions by finding equivalents that share the same numerator orsame denominator, e.g.,

(i) When comparing 3

5 and

7

10,

3

5 can be renamed as

6

10.

(ii) To compare 10

7 and

5

3 , rename

5

3 as

10

6.

A focus on a procedure for finding equivalent fractions is not importantat this grade level.





PerformanceA10.1 Provide students with pattern blocks. Ask them to arrange theblocks to model two different fractions, one being less than the other.Have them record the number sentence that describes the model.

Paper and PencilA10.2 Ask students to select, from the digits 1 to 9, those that canfill the boxes to a make true statements. Ask them for three differentpossibilities.

InterviewA10.3 Ask students, How do you know that

1

3

3

4< ?

A10.4 Ask students, If you know that 2 2

7G> , what do you know

about the value of ? Explain.

A10.5 Give students cards on which the following fractions arewritten:

2

5

1

4

6

5

7

8

5

10

Ask them to order the fractions from least to greatest and to give reasonsfor the order. This task could be modified to include some decimals,particularly tenths, and common fractions, e.g., 5

1 0could be written

as 0.5.

PresentationA10.6 Have students conduct an experiment rolling a pair of coloureddice. The number on the red die is used as the numerator of a fractionand the number on the blue die is used as the denominator. Have thempredict whether or not the fraction will usually be less than half. Allowstudents to conduct the experiment to verify their predictions andpresent their findings to the class.

Paired DiscussionA10.7 Ask pairs of students to find ways of showing which of

7

8 and

5

6 is greater. Ask them to provide explanations that are easily under-

stood without the use of materials. Have them list pairs of fractions thatthey find more difficult than others to compare and to explain why.





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v) apply number theoryconcepts (e.g., primenumbers, factors) in rel-evant situations with respectto whole numbers,fractions, and decimals

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A11 recognize and find factorsof numbers

A11 Students should already be familiar with the term “factor.” Nowthey should also recognize that - the factors of a number are never greater than the number - the number is always a multiple of any of its factors - division can be used to find factors - the greatest factor is always the number itself - the least factor is always 1 - the second greatest factor is always

1

2 the number or less

To find factors, students should use a variety of strategies such as(a) creating rectangles of a particular area using square tiles or grid paper.The lengths and widths of these rectangles are factors of the numberrepresenting the area and when they find all possible rectangles of thatarea, they will have found all the factors.

(b) dividing by each number that is less than or equal to the givennumber and in that way may find all the possible factors.

(c) using knowledge of special properties of multiples of specific factors,e.g., There are only even digits in the ones place of multiples of 2. Theones digit of multiples of 5 is either 5 or 0.

Some students confuse the terms “factor” and “multiple.” You shouldmodel this language consistently for students making statements such as“2 is a factor of 4” or “4 is a multiple of 2.” Also, provide a variety ofexperiences that will require students both orally and in their writing touse the words factor, multiple, and product, e.g., Show students arectangle with its dimensions and area marked and ask them as partnersto take turns making as many different statements as possible about thefactors, multiples, and product shown by the rectangle.

• Ask students to tell what they know about multiplication sentencessuch as 22 x 12 = 264. (It is expected that they will know that 22 and12 are factors of 264, that 264 is the product of 22 and 12, and that264 is a multiple of 22 and 12. Some students may recognize that 22x 12 is the same thing as 11 x 24, consequently deducting that 11and 24 are factors of 264.)





PerformanceA11.1 Ask students to use 24 colour tiles to find all the factors of 24.Have them then use more or fewer tiles to try to find another numberthat has the same number of factors. Ask them to record their results.

A11.2 Ask students to use colour tiles to determine which of 24, 36,and 45 has the greatest number of factors and to keep a record of howthey arrived at their answers.

Paper and PencilA11.3 Tell students that a certain number has 2, 3, and 4 as factors.Ask, What might the number be? Could there be more than one answer?

A11.4 Have students compare the factors of a number and of itsdouble, e.g., 12 and 24, and describe what they notice.

A11.5 Ask students to express 36 as the product of two factors in asmany ways as possible.

InterviewA11.6 Ask students to explain how they know, without dividing, that 2cannot be a factor of 47.

A11.7 Ask students to describe how they would go about finding thefactors of a number.

A11.8 Ask students why they would know without finding allfactors of 42 that the two greatest ones are 21 and 42.

PortfolioA11.9 Ask students to explain, in a few sentences, why every wholenumber greater than one has at least two factors.

A11.10 Ask students to use 12 square tiles to make rectangles of varioussizes, writing about their findings using the words “factor,” “product,”and “multiple” in their writings.

A11.11 Tell students that a marching band has 120 members. Ask themto explore the many different ways the band could be arranged in equalrows for marching.

Number Concepts/



Number Operations





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i) model problem situationsinvolving whole numbers,decimals, and fractions byselecting appropriateoperations and procedures

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B1 find sums and differencesinvolving decimals tothousandths

B1 You should have students use materials, e.g., thousandths andhundredth grids and base-10 blocks, to model solutions to addition andsubtraction questions involving tenths, hundredths and thousandths,and to make connections with the paper-and-pencil algorithms.

When using base-10 blocks,if the large cube is used torepresent the whole, 2.4 +1.235 is modelled as:

Addition and subtractionquestions should be presented both horizontally and vertically toencourage alternative computational strategies, e.g., For 1.234 +1.990, students might calculate 1.234 + 2 = 3.234 and then 3.234 -0.01 = 3.224.

Students have to make choices when doing computations. First, theyhave to determine whether the answer must be exact or if an approxima-tion is sufficient. If an exact answer is required, other decisions must bemade after they estimate, such as “Can I compute this mentally?” Ifnot, “Will I use a paper-and-pencil method or a calculator?” At first,students will not automatically make these decisions; they must beencouraged to do so. It is important to remember, however, thatbecause students must become proficient with paper-and-pencilmethods, the use of calculators should not always be an option. As ageneral rule, students should use mental and paper-and-pencilstrategies for those computations that are outcomes for their gradelevel or below unless the computations are involved in problems thatrequire many repetitions and/or many computational steps—for thesethey should use calculators. As well, they should use calculators toexplore computations normally beyond their grade level.

You should allow students to use algorithms of choice when theycalculate with pencil-and-paper methods. While it is important thatyou respect algorithms developed by students, if they are inefficient,you should guide them toward more appropriate ones.

Estimation is an important first step in any paper-pencil or calculatorcomputation in order to be alert to the reasonableness of solutionsobtained. You should model this estimation behaviour yourself.





PerformanceB1.1 Provide base-10 blocks or thousandths grids. Have studentschoose addition or subtraction questions involving decimal numbersand demonstrate them with the models.

B1.2 Model 4.23 and 1.359 with base-10 blocks. Ask students to findthe difference between these two numbers using base-10 blocks.

B1.3 Provide students with a variety of labels from grocery itemsshowing either their contents in millilitres or their mass in grams. Askstudents to find three items with a total capacity or total mass greaterthan a specified number of litres or grams.

Paper and PencilB1.4 Provide students with the batting averages of some baseballplayers. Have them calculate the spread between the player with thehighest average and the one with the lowest. Have students createproblems using the averages on the list.

B1.5 Ask students to fill in the boxes sothat the answer for each question is 0.4with the stipulation that the digit 0 cannotbe used.

B1.6 Request that students provide examples of questions in which twodecimal numbers are added and the answers are whole numbers.

InterviewB1.7 Tell students that you have added three numbers, each less than 1,and the result is 2.4. Ask them if all three decimal numbers could be lessthan one half and explain why or why not. Once students realize thenumbers cannot all be less than one half, ask them how many of themcould be less than one half.

B1.8 Present the following question in which Jane made an error whenshe subtracted. Ask students what they might say to help Jane under-stand why the answer is incorrect. 5.23

-1.4533.783

PresentationB1.9 Have students conduct research to find situations where decimalsare added and subtracted in everyday life and present their findings in avideo, as an oral report, or as a written report.





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B2 multiply 2-, 3-, and 4-digitnumbers by 1-digit numbers

B3 find the product of two2-digit numbers

B2 Have students continue to use the language of models to explainmultiplication algorithms, e.g., “4 sets of 2 units is 8 units. 4sets of 5 rods is 20 rods which is traded for 2 flats, leaving 0rods. 4 sets of 4 flats is 16 flats. The 2 additional flats make18 flats which is traded for a 1 thousand cube and 8 flats. 4sets of 3 thousand cubes is 12 thousand cubes; the additional1 thousand cube makes 13 thousand.”

Encourage mental math strategies, e.g., By rearranging factors, 5 x 66is the same as 10 x 33 and 44 x 25 is the same as 11 x 100. Usingthe front-end strategy, 3 x 213 becomes 3 x 200 + 3 x 10 + 3 x 3.

B3 Similarly, materials should be used to modelmultiplication of 2-digit by 2-digit numbers.Efficient models for such products are areas ofrectangles with dimensions equal to the two num-bers. This can be shown using base-10 blocks.Students should be able to relate the model to thealgorithm by explaining each symbolic step in relationto the appropriate part of the rectangular display.

When students understand the area model, they may choose to use agrid-paper drawing as an explanation. It is suggested that studentsinitially use an algorithm that shows the four products such as theone used in the above example. (An alternative would be to show thefour products starting with the smallest.) You could present thestandard algorithm; however, it is important that you provide anexplanation with models not just give procedural rules. That is, inthe above example, students should understand that the standardalgorithm involves finding the area of the bottom portion of therectangle (4 x 31) in one step and finding the area of the top portionof the rectangle (20 x 31) in another step.

You should give students the choice of using the standard algorithmor an alternative one. If, however, students are using inefficientalgorithms, you should guide them to select more appropriate ones.And, as with all computational questions, students should estimatebefore calculating.

Immediate recall of basic multiplication facts is a necessary prerequi-site for paper-and-pencil algorithmic procedures, for estimation, andfor further mental computation.





PerformanceB2.1 Ask students to use models to explain how three theatres, eachhaving 243 seats, could hold 729 people.

B3.1 Have students use models to show how to find the amount ofmoney collected for photos if 13 students each bring in $23.

Paper and PencilB2.2 Have students compare the results of 36 x 4 and 46 x 3 and theresults of 74 x 5 and 54 x 7. Ask them to make general statements aboutthe switching of the tens digit in the 2-digit factor with the single-digitfactor.

B3.2 Ask students to explain why the product of two different 2-digitnumbers is always greater than 100.

InterviewB2.3 Tell students that a number is multiplied by 8 and the product is11 384. Ask them how they would know that the number was greaterthan 1000 and had a 3 or an 8 in the ones place.

B2.4 Tell students that to find 7 x 513, Anne began with 3500. Askwhat she would do next to complete the product.

B2.5 Remind students that 24 x 5 = 120. Ask them how this could beused to find the product of 34 and 5.

PortfolioB3.3 Ask students to explore the pattern in these products: 15 x 15,25 x 25, 35 x 35, ... Have them describe the pattern and explain howthis pattern could be used to predict 85 x 85 and 135 x 135. Ask themto check their predictions using calculators. You could also have studentsexplore the pattern in these products: 19 x 21, 29 x 31, 39 x 41, ... anduse it to make predictions for 79 x 81 and 109 x 111.

B3.4 Ask students to explore the following:(a) 24 + 35 is the same as 25 + 34. Is 24 x 35 the same as 25 x 34?Explain.(b) 19 + 32 is the same as 20 + 31. Is 19 x 32 the same as 20 x 31?Explain.





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B4 divide 2-, 3-and 4-digitnumbers by single-digitdivisors and investigatedivision by 2-digit divisors

B4 Division questions can be viewed either as sharing or as findinghow many groups, e.g.,

3 4532

300015321500

−

−

Using base-10 blocks, students should model division-as-sharingstory problems and describe orally and in writing the steps they tookto find the solutions. This will lead naturally to the tradi-tional long-division algorithm. In the algorithm to theright, students should realize the “4” represents 4 thou-sands and that if three share these, each gets 1 thousandwith 1 thousand left. This traded and put with the 5hundreds gives 15 hundreds. If these are shared among 3, each gets 5hundreds with 0 left over, and so on until students get a remainder of2; they should eventually learn how to continue the division processto get decimal solutions.

Models help students understand why the number of “leftover” unitsafter the sharing must be less than the divisor. Models also help clarify acommon mistake of writing a remainder as a decimal no matter what thedivisor, such as writing the remainder of 2 as .2 in the question above.

Students should also have opportunities to investigate division by 2-digitdivisors. Begin by having students divide by 10, 20, 30, ... Given thequestion 869 20÷ , encourage students to think of 20 sharing 869. Theyshould then think that each gets 40 because 20 x 40 = 800 with 69 left.Each could get 3 more because 20 x 3 = 60 and 9 are left as a remainder.Follow these explorations with estimation activities so students becomeproficient at estimating quotients with divisors of 10, 20, 30, ... Finally,explore questions such as 2713 31÷ , where students find an estimateby rounding it to 2700 30÷ , or 9118 ÷ 16, where students find anestimate by dividing near compatibles: 9000 15÷ . They should havesufficient experience with 2-digit divisors to understand the process.

Because calculators will be used for many questions that requiredivision by more than one digit, it is important that students betaught to interpret decimal remainders, e.g.,when the remainder canbe ignored or rounded up, and when it forms an important part ofthe solution.

“Make 1000 sets of 3, using3000; 1532 left.” Make 500 setsof 3 using 1500; 32 left...” Thenumbers of sets at each stagetends to be a multiple of 1000,100 or 10 to make it easier.

OR

3 453231515

03

15

−

−

“4 thousands shared among 3,each gets 1 thousands using3000: 1532 left. Trade 1thousand for 10 hundreds toshare now 15 hundreds, etc.”





PerformanceB4.1 Ask students to use base-10 blocks to model how to divide 489 by7.

Paper and PencilB4.2 Ask students to fill in the boxes so that nodigit is used more than once, there is no remain-der, and the resulting division sentence is correct.

B4.3 Ask students to fill in the boxes to makethe division sentence true.

InterviewB4.4 Ask students to tell what division question is being modelled inthis picture. Ask them to provide word problems that would be solvedby this model.

B4.5 Tell students that 612 students in a school are being bussed to amuseum. The law states that a maximum of 45 students is allowed oneach bus. Ask them to estimate and then calculate how many buses willbe needed.

PortfolioB4.6 Ask students to write word problems involving division by a 2-digit number for each of the following situations: (a) the remainder isignored, (b) the remainder is rounded up, and (c) the remainder is partof the answer. The following are sample word problems for these situa-tions:(a) A person has 78¢. Pens cost 19¢ each. How many pens can theperson buy?(b) 126 people need to be transported. Twelve passengers can travel ineach van. How many vans are needed?(c) 75 metres of ribbon are to be shared among 10 students to makecrafts. How much ribbon will each student get?





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B5 find simple products ofwhole numbers anddecimals

B5 Students should use materials to model the multiplicationproblems involving whole numbers and decimals, e.g., Each costumefor the class play requires 4.35 square metres offabric. How much fabricwill need to be purchasedto make 3 of these cos-tumes? 3 x 4.35 = 13.05may be modelled using(a) base-10 blocks

(b) decimal grids (c) money

Students should also model solutions to problems that involve taking adecimal part of a whole, e.g., It was agreed that Elaine’s share of anypackages of candy found will be 0.4. If 12 packages of candy are found,how many packages will Elaine get? The solution to 0.4 × 12 could bemodelled with base-10 blocks by first showing the 12 with rods. Thenshow 4-tenths of each rod as 4 small cubes. The total number of cubeswill be 48—4 rods and 8 cubes or 4.8 packages. If the 12 packages wereshown with flats, then 4-tenths of each flat would be 4 rods for a total of48 rods—4 flats and 8 rods or 4.8 packages.

Before modelling procedures or calculating using paper-and-pencil,students should estimate the products.

You should point out that, for some questions, the first strategy thatshould be used is a mental strategy such as the front-end method,e.g., 4 × 20.12 and 8 × 4.5

• Provide students with examples of multiplication questions. Askthem to identify the questions that lend themselves to the front-end strategy. 23.31 67.9 2 × 435.24

× 3 × 7 8 × 35.48





PerformanceB5.1 Request that students draw models to show 8 x 2.03.

B5.2 Ask students to explain the difference between 3 x 0.6 and 0.6x 3 using base-10 blocks as models. Then have them create two storyproblems that would illustrate the difference.

Paper and PencilB5.3 Ask students to fill in the boxes with digits to make the com-putation true.

InterviewB5.4 Ask students to explain how and why the answers to 435 x 7and 43.5 x 7 are alike and how they are different.

B5.5 Tell students that a flat represents onewhole. Ask them what multiplication ques-tion is represented by this base-10 blockdisplay. Have them do the pencil-and-papercomputation that is represented, explainingthe relationship between each step in theprocess and this display.

B5.6 Tell students that to find 2.25 x 8,Jane said, “16 + 2 = 18.” Ask them to explain Jane’s thinking.

B5.7 Ask students to find the answers to 3 x $2.13 and 5 x $4.25using the front-end method.

PortfolioB5.8 Ask students to make a list of ten questions that require multi-plication of decimal numbers by whole numbers. Have them includesome examples that their classmates could solve using the front-endstrategy. These lists could be exchanged in class.

4. 8

× 6.2 8





B6 With the support of models, such as base-10 blocks, studentswill come to see that the process of dividing decimal numbers bywhole numbers as identical to that involving the division of wholenumbers, e.g.,

Students need to be reminded of the importance of estimating beforemodelling a question or using paper-and-pencil procedures.

It is also important to discuss the nature of remainderswhen dividing decimal numbers, e.g., In the divisionquestion shown at right, the remainder is 0.1, not 1. Ifdesired, this leftover could be traded for 10 hundredths andthe division continued to make the quotient 115.13, and so on.

• Using store flyers, prepare a list of specific items such as: 5 items for$4.65, 8 items for $16.88. Provide students with the flyers and lists.Ask them to determine what the items must be.

• Have students research winning relay times in Olympic events. Askthem to compute the average time taken by each of the 4 competitorsin these events. (The Canadian Almanac and other references providedetailed Olympic data.)

Explain to students how to round numbers when determining the priceof single items, e.g., If you want to buy one can of peas that is priced at2/99¢, your price per can would be 50¢. If grapefruit are 3/$1, individu-ally they cost 34¢.

• Ask the students to calculate the cost of:2 cans of peas, priced 6/$4.504 pkg. tissue, priced 10/$7.951 kg ground beef @ 3 kg/$101 can ginger ale @ 12/$4.99

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B6 divide decimal numbers bysingle-digit whole numbers

“4 tens shared among 4 uses 40;5.4 left to share; each gets 1whole using 4: 1.4 is left to share;14 tenths shared among, eachgets 3 tenths; 2 tenths left or 20hundredths; each gets 5 hun-dredths.”





PerformanceB6.1 Tell students that 4 people are sharing a pizza which costs $14.40.Ask them what each person’s share of the cost should be. Have themprove that their answers are correct.

Paper and PencilB6.2 Ask students to fill in the box so that the amount of money willbe shared equally with none remaining.

B6.3 Ask students for some possible values for the dividend if the resultis 5.2 when you divide by a single-digit divisor.

B6.4 Have students fill in the boxes below in more than one way,making certain not to use the same digit twice within one divisionquestion.

B6.5 Tell students that the flat represents one whole in the divisionquestion pictured below. Ask them to express the division questionsymbolically and to create an accompanying word problem.

B6.6 Tell students that 5 boxes have to be wrapped using 4.36 m ofribbon. Have them calculate how much ribbon should be cut off foreach box. Ask them to explain what the remainder is and what could bedone with it.





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B7 determine whether anopen sentence is always,sometimes, or never true

B7 Generally, students’ previous experiences have been to interpret anopen sentence as a statement that is true when the one right number isfound. In fact, if the frame is on the right-hand side of the equal sign,they often interpret it to mean “and the answer is.... .”

They need to be exposed to a number of open sentences that arealways true and ones that are never true. They will need help torealize that they cannot immediately determine the type of sentence,e.g.,“523 + is even” is a sentence that is sometimes true—there are manypossible numbers to make it so.“523 + is greater than 500” is a sentence that is always true if is a positive number.“523 + is a fraction” is a sentence that is never true if is awhole number.

You should have your students write examples of open sentences thatare always, sometimes, and never true. Encourage them to make opensentences spanning all four operations.

Give examples of problem situations to the class and have students writeopen sentences for each. It is useful to include some examples of prob-lems that require more than one step. They may wish to use a box, atriangle, or a letter to represent the unknown number(s). The intent isfor students to practise writing open sentences, not necessarily solvethem. Some students will write an all inclusive sentence, while othersmay need to use more than one open sentence, e.g., Consider thisproblem situation: “Jake bought a poster and 6 books for a total of$27.12. The books were priced at $3.69 each. How much was theposter?” One student might give + 6 x $3.69 = $27.12 as the sen-tence, while another might give 6 x $3.69 = and + = $27.12

Provide students with open sentences and ask them to create problemsituations to match each one, e.g.,2 x + $0.49 = $7.07 2 x + 13 = 43 4 x + 7 =

Once they have created the problems, have them share them, anddiscuss whether there is only one solution or more than one.





Paper and PencilB7.1 Ask students, If the missing number in each of the open sentencesbelow is a whole number, can you tell whether the open sentence issometimes true, always true, or never true? Explain your thinking.

3 x is even3 x is a multiple of 33 x is greater than 5003 x is 0.

B7.2 Have studentss create open sentences that are always true,others that are never true, and still others that are only true for oneparticular value replacing the open frames.

B7.3 Tell students that some children share 43 candies equally andthere is one candy left over. Have them write open sentences thatwould show ways to find how many children shared the candy.Discuss their open sentences, one of which could be 43 ÷ = r 1.

InterviewB7.4 Ask students whether they see a difference between the meaningsof the in these two sentences:

5+5= 5+ =15

B7.5 Show students open sentences such as those shown below and askthem which ones can be solved if the box must represent a wholenumber. Have them create word problems to match those that theyselected.

$7.45 + = $9.2245 ÷ = 1876 + + 27 = 100216 - = 44

x 0 = 49





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B8 solve and create additionand subtraction problemsinvolving whole numbersand/or decimals

B9 solve and createmultiplication and divisionproblems involving wholenumbers and/or decimals

B8 Students should continue to use addition and subtraction to solveand create relevant problems.

An example of a mathematical problem is the cryptogram, in which eachletter stands for a different digit. These problems require students toapply knowledge about the concepts of addition and subtraction and toperform numerous calculations. Students can create and sharecryptograms. It is helpful to repeat some letters/numbers, e.g.,

ABC (123) S E N D ( 9567)CBD (328) M O R E (1085)EFA (451) M O N E Y (10652)

Students should be solving multi-step word problems involving combi-nations of the four operations as well as creating their own. Requiringstudents to create their own problems provides opportunities for themto explore the operations in depth. It is a complex skill requiring solidconceptual understandings and must be part of the student’s problem-solving experiences.

B9 Students should also create and solve problems that use multiplica-tion and division procedures. You should encourage them to solve andcreate problems that focus on the various meanings of the two opera-tions. For multiplication, these would include equal-group problems thatare modelled using sets or arrays, rate problems that are often modelledusing number lines, combination problems that are modelled with arrowsbetween sets or in tables, comparison problems that are often modelledusing two sets, and product-of-measures problems such as areas of rectan-gles. For all of these multiplication problems there are correspondingdivision problems. Particular emphasis should be placed on equla-groupdivision problems that require finding “how many sets” and “the size ofsets,”e.g., How many packages of 9 marbles can be made with 172marbles? How many marbles will each of 6 children get if they share 172marbles? You will probably have to model problems that illustrate someof these meanings and get students to create problems like the ones youhave modelled.

It is important that, among the problems presented or created by stu-dents, some lend themselves to mental computation, others requirepaper-and-pencil computation, and still others call for the use of calcula-tors.





Paper and PencilB9.1 Ask students to create word problems, incorporating the numbers64.2 and 3, in which division will be required to find solutions.

B8.1 Have students create word problems that will involve subtrac-tion to find solutions and for which the answers are 1359.2.

B9.2 Ask students to make problems that would require someone tocompute 1000 - 385.

B8/9.1 Present students with a variety of different shapes, clearlyindicating their dimensions some of which are decimals. Ask them tofind the perimeters of these shapes.

B9.2 Ask students to create and solve problems that involve the divisionof a decimal by a whole number.

InterviewB9.3 Ask students to explain how they could use multiplication to helpthem estimate the number of words in a book.

PresentationB8/9.2 Provide store flyers and ask students to create a series of prob-lems based on the information in these flyers. Have them present theseproblems to the class.

B8/9.3 Provide data about product sales in Atlantic Canada in variouscategories. Request that the students produce a series of meaningfulword problems that relate to the data. (Canadian Global Almanac orInternet are possible sources for this information.)

PortfolioB8.3 Present the following challenge and ask students to describe, inwriting, their thinking processes: Find the result of adding all the oddnumbers from 1 to 101 and then subtracting all the even numbers from2 to 100 without doing all the compuations.

B8/9.4 Ask students to create money word problems that involve morethan one step and more than one type of operation.

B8.4 Have students create addition problems that avoid using cues suchas “altogether” or “in all.”

B8.5 Have students create subtraction problems that involve decimalsand ask that they avoid cues such as “more than” or “less than.”





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B10 estimate sums anddifferences involvingdecimals to thousandths

Children with number sense usenumbers flexibly and choose the mostappropriate representation of a numberfor a given circumstance. When solvingproblems, they are able to select fromvarious strategies and tools - they knowwhen to estimate, when to use paper andpencil, and when to use a calculator.They predict with some accuracy theresult of an operation and describe therelationships between various forms ofnumbers. This 'friendliness withnumbers' goes far beyond mere memori-zation of algorithms and number factsand implies an ability to recognize whenoperations are required and when theyhave been correctly performed. (NCTM1992b, 8)

(a)

B10 Students should estimate automatically whenever faced with acalculation whether, or not, one is required. With continued practise ofthe basic facts and of a variety of mental math and estimation strategies,they will gain the confidence to undertake such computations.

Rounding is a commonly used strategy for estimation, e.g., To find$2.99 + $7.98 + $4.98, you can round to $3 + $8 + $5 for a total of$16; to find 2794 - 1616, you can round to 28 hundred - 16 hundred toget 1200. Students should see that estimation is a flexible process toprovide as accurate an answer as possible. Simply obeying some rules forrounding will not always produce the best estimate, e.g., The bestestimate for 147 + 142 would be found by rounding up 147 to 200 androunding down 142 to 100; the best estimate for 174 + 280 + 451would be found by rounding up 174 and 280 and rounding down 451to get 200 + 300 + 400 or 900.

Compatible numbers involveslooking for number combina-tions that go together to makeapproximately 10, 100, or1000, e.g.,

Front-end involves estimating using digits on the left (highest placevalue) and making an adjustment for the rest of the digits. This adjust-ment may include “clustering,” e.g., 24 and 73 together to make100, or further front-end strategies, e.g.,

For = 70 and

2 + 3+ 2 = 7 for a total of 77; then 87 hundredths is about 1,29 hundredths and 83 hundredths is about another 1; so, 79 is a good estimate of this sum.

52 87 329 22 8350 20. . . ,+ +

+think





Paper and PencilB10.1 Tell students that 2.89 + 7.17 is estimated to be 48. Askthem to provide three sets of digits that could go in the boxes to makethis estimate possible.

B10.2 Ask students to provide examples of addition questions forwhich it would be appropriate to ignore the three digits after the decimalpoint when estimating. Ask for examples for which the digits should beconsidered when estimating the sums.

InterviewB10.3 Have students estimate the difference: 13.240 - 1.972. Ask themto describe two other subtraction questions for which one would suggestthe same estimate for the difference.

B10.4 Provide the following calculations and ask students to explainhow they would estimate the answers:

24.3 + 39.16 + 75.03 + 62.2 998.201 - 249.6

PortfolioB10.5 Have students use grocery store flyers to select at least eight itemsthat would total close to, but less than, $20. Ask that they list the items,provide estimates for the totals, and explanations for their estimates.

B10.6 Tell students that Hank always uses the “rounding rule.” Askthem to write to Hank to try and convince him that he may want toreconsider always using the same strategy.

B10.7 Ask students to estimate totals of their purchases, or theirparents’ grocery orders. Have them report on the strategies they usedand their progress with estimating.





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B11 estimate products andquotients of two wholenumbers

B12 estimate products andquotients of decimals bysingle-digit whole numbers

B11/B12 Students should continue to use a variety of estimationstrategies for multiplication and division.

Rounding There are a number of things to consider when rounding formultiplication and division estimates. (a) If one of the factors is a singledigit, consider rounding the other factor, e.g., For 8 x 693, rounding693 to 700 and multiplying by 8 is a much closer estimate than round-ing 8 to 10 and multiplying 10 by 693. (b) If the two factors are 2-digitwith the ones digits 5 or more, consider rounding the smaller factor upand the larger factor down, e.g., For 76 x 36, rounding to 70 x 40produces a closer estimate than rounding to 80 x 40 or to 80 x 30.(c) If rounding for a division estimate, look for compatible numbers,e.g., For 4719 ÷ 6, use 4800 ÷ 6; for 3308 ÷ 78, use 3200 ÷ 80.

Front-end involves beginning at the largest place value, e.g., An estimateof 8 x 823.24 would be 8 x 800 or 6400. If a more accurate estimate iswanted, use additional place values, e.g., For 8 x 823.24, 8 x 20 or 8 x25 could be added to 6400 for an estimated product of 6560 or 6600.When using front-end estimation for divisions by 2-digit divisors, youmight try to convert the question to have a single-digit divisor, e.g., For7843 ÷ 30, think of it as 750 tens ÷ 3 tens to get 250.

Doubling for division involves rounding and doubling both the dividendand divisor. This does not change the solution but can produce “nicer”divisors, e.g., 2223.89 ÷ 5 can be thought of as 4448 ÷ 10, or about 445.It is important that the students understand why this works; they mightthink that since twice as many people are sharing twice as much, eachstill gets the same share.

• Have students play the Range Game for all the operations, e.g., Formultiplication, have students enter a “start” number into theircalculators and press “x.” Give them a range for the product. Theythen enter their estimated second factors into their calculators, andpress “=” to see if they get a product within the target range. A pointsystem could be devised by the players.

Start Number Target 12 550 - 630 48 2500 - 2700126 1000 - 1100





Paper and PencilB12.1 Ask students to provide estimates in metres for the lengths of aladybug and a caterpillar. Have them use their estimates to calculatehow much longer 10 caterpillars would be than 10 ladybugs.

InterviewB12.2 Ask students to give estimates for the cost of six packages ofcheese at $3.49 each.

B12.3 Tell students that Jacques estimated the cost of 3 packages ofgum at $0.79 each and 4 packages of potato chips at $0.79 each to beabout $7. Hilary gave her estimate to be $5.60. Ask students how theythink Jacques and Hilary arrived at their estimates and to indicate whichestimate was closer to the actual cost.

B11.1 Tell students that 834 ÷ 6 is about 300. Ask them to decidewhat should go in the box.

B11.2 Tell students that you have multiplied a 3-digit number by a 1-digit number and the answer is about 1000. Ask the student to describethree possible multiplication questions that you might have had.

B11.3 Ask students for estimates if they are told that a number between300 and 400 is divided by a number between 60 and 70.

B11.4 Tell students that a bus holds 72 students. Ask them how theywould estimate how many buses are required to transport 3000 students.

PortfolioB11.5 Present students with Susan’s approach to estimate 4598 ÷ 36.She says, “You just replace all the digits except the first ones with 0.Since 4000 ÷ 30 is about 130, the answer is about 130.” Ask students tocomment on Susan’s approach and to provide examples to back up theirconclusions.

B11/12.1 Direct students to write to a classmate who has beenabsent because of an operation and has missed the classwork onestimation. Ask them to write explanations for the absent student onwhat was missed.





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B13 perform appropriate mentalmultiplications with facility

B13 Mental math strategies are strategies that enable students tocalculate “in their heads” actual solutions to computations, notmerely estimates of them. While students should already possess avariety of strategies, it is important to recognize that these strategiesdevelop and improve over the years with regular practice. This meansthat mental mathematics must be part of instruction at all gradelevels. It is suggested that you plan a 3–5 minute daily mental mathor estimation activity, where a strategy is highlighted or rehearsed.Sharing computational strategies within the context of problem-solving situations is also an essential part of the students’ strategiesdevelopment.

Students should regularly perform the following types of multiplica-tions mentally and discuss the strategies they use:(a) multiplication of two single-digit numbers—the multiplication facts

(b) multiplication by 10, 100, and 1000

(c) multiplication of single-digit multiples of powers of ten, e.g.,70 x 30, 30 x 400, and 20 x 3000

(d) multiplications that can have the factors rearranged to make simplerproducts, e.g., 25 x 40 can be arranged as 25 x 4 x 10 to make 100 x 10,16 x 4 can be rearranged as 8 x 2 x 4 to make 8 x 8.

(e) multiplications that lend themselves easily to the front-end strategy,e.g., 3 x 2320 can be thought of as: 3 x 2000 + 3 x 300 + 3 x 20 = 6960;4 x 1.21 can be thought of as: 4 x 1 + 4 x 0.2 + 4 x 0.01 = 4.84. Thisstrategy is often used with some form of adjustment, e.g., 3 x 3260 canbe thought of as: 3 x 3000 + 3 x 250 + 3 x 10 = 3750 + 30 = 9780.

(f ) multiplications that involve a factor that ends in a 9 where thecompensating strategy can be used—multiply by the near multiple of tenand subtract the one extra set to find the actual product, e.g., 39 x 7, canbe thought of as: 40 x 7 less the one extra set of 7 to get 273; 79 x 9 canbe thought of as: 80 x 9 less the extra set of 9 to get 711. This samestrategy could be when a factor ends in an 8, e.g., 68 x 7 can be thoughtof as 60 x 7 less two extra sets of 7 to get 420 - 14 = 406.When presented with a computation to do mentally, students should beencouraged to use different strategies or combinations of strategies. Youshould only insist that students use specific strategies when they arebeing introduced and initially practised. Afterwards, they should selectthe strategy that makes sense to them and is efficient.





InterviewB13.1 Tell students that when asked to multiply 36 by 11, Kellysaid, “I think 360 + 36 = 396.” Ask students to explain Kelly'sthinking.

B13.2 Tell students that to mentally multiply 25 by 84, Stephanieexplained that it was the same as 100 x 21. Ask students if this wouldprovide the correct answer and, if so, to explain how it works. Havethem provide other examples of questions for which this strategy couldbe used.

B13.3 Ask students, How might you easily multiply 2 x 57 x 5mentally?

B13.4 Ask students to explain why it is easy to calculate mentallythe following questions: 48 x 20

50 x 86242 ÷ 2

B13.5 Tell students that Sue bought 24 cans of pop for a party. The saleprice was 2 for $0.89 with a 5-cent deposit on each can. The cashier toldSue that the total was $35.90. Ask students, Does this sound reasonable?Why not? Can you mentally calculate the correct cost? How?

B13.6 Ask students why Lynn multiplied 11 x 30 to find 22 x 15.

PortfolioB13.7 Ask students to keep track of when they use their mental mathstrategies in and out of school and to write about the situations whenthey do.

B13.8 Ask students to list the mental math strategies that they useregularly.

B13.9 Tell students that Mark said he would prefer to use the front-endmental method for finding the answers to 2 x 244 and 3 x 325 ratherthan use a calculator or a paper-and-pencil method. Ask students toexplain how Mark might have found these products.

B13.10 Ask students to provide explanations for how they couldmultiply a 1-digit number by 99 mentally. Have them apply theirstrategies to specific questions.





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B14 divide numbers mentallywhen appropriate

B15 multiply whole numbersmentally by 0.1, 0.01, 0.001

B14 Students should regularly perform the following types of divisionmentally:(a) division of 2-digit numbers by single-digit divisors, e.g., 56 ÷ 8,29 ÷ 3, and 75÷ 3.

(b) division by 10, 100, or 1000

(c) division where the divisor is a multiple of 10 and the dividend is amulitple of the divisor, e.g., 400 ÷ 20, 900 ÷ 30, 6000 ÷ 30, 1000 ÷ 50.

Division by a power of ten should be understood to result in a uniform“shrinking” of hundreds, tens, units, ... which could be demonstratedand visualized with base-10 blocks, e.g., Division by 10 involves each1000 (large cube) becoming a 100 (flat), each 100 (flat) becoming a 10(rod), and each 10 (rod) becoming a 1(unit).

The think multiplication strategy is a convenient one to use when divid-ing mentally, e.g., When dividing 60 by 12, think, “What times 12is 60?” This could also be used in combination with other strategies,e.g., For 920 ÷ 40, think, “20 groups of 40 would be 800, leaving120, which is 3 more groups of 40 for a total of 23 groups.”

B15 Students should relate multiplication by 0.1, 0.01, 0.001 to divi-sion by 10, 100, or 1000, respectively.

To facilitate student understanding, it may be useful to remind themthat multiplication indicates the number of groups of something.Therefore, just as 2 x 40 is two sets of 40, 0.1 x 40 is one tenth of 40. If40 is shown with 4 base-10 block rods, 0.1 x 40 could be shown astaking one-tenth of each rod—getting one unit—for a total of 4 units.For 40 ÷ 10, each rod would be divided by 10—getting one unit—for atotal of 4 units.

Another approach is based on following this pattern:

1000 x 4 = 4000 As the first number is divided successively 100 x 4 = 400 by 10, so is the product. 10 x 4 = 40 1 x 4 = 4 Therefore, it is only reasonable that

0.1 x 4 = 0.40.01 x 4 = 0.04

0.001 x 4 = 0.004





PerformanceB15.1 Tell students to start with 3452 displayed on their calculatorsand predict how many times they would need to press “x,” “0.1,” and“=” to get a number less than 1. Ask them to confirm their predictions.

Paper and PencilB14.1 Have students fill in the box to produce a true statement.

345 ÷ 10 = ÷ 100

InterviewB15.2 Tell students that 178 is multiplied by 0.01. Ask, What digit willbe in the tenths place? Why?

B14.2 Tell students that when a certain number was divided by 10, theresult was 45.95. Ask them what the result would have been if thedivision had been by 100 and if it had been by 1000.

B14.3 Ask students for a quick way to divide 60 by 2 and then todivide the result by 5. Have them give other examples of questionswhere the same quick way could be used.

B15.3 Ask students to explain whether or not this statement is true:“When you multiply a number by 0.01, the product is always less thanthe number.”

B15.4 Explain to students that Jacob was told that one-tenth of thestudent body bring their lunches. Jacob said that he was going to divideto find how many students bring lunches. Sammy disagreed and saidthat he should multiply. Mary said that they were both right. Askstudents if they agree with Jacob, Sammy, or Mary, and to justify theirpositions.

PresentationB14.4 Ask students to prepare lists that describe strategies to calcu-late mentally various types of division questions.

Patterns and

Relations




GCO C: Students will explore, recognize, represent and apply patterns andrelationships, both informally and formally.

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By the end of grade 6, studentswill have achieved the outcomesfor entry–grade 3 and will all beexpected to

i) describe, extend, and createa wide variety of patternsand relationships to modeland solve problems involv-ing real world situations andmathematical concepts

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C1 use place value patterns toextend understanding of therepresentation of numbersto millions

C2 recognize and explain thepatterns in dividing by 10,100, and 1000 and/or inmultiplying by 0.1, 0.01,0.001

C3 solve problems usingpatterns

Earlier work with patterns has enrichedeach student's basic understanding ofmathematics. Investigating additionalpatterns develops and refines theirmathematical abilities and enables themto describe, extend, create, analyze, andpredict knowledgeably. (NCTM 1992b,1)

C1 Students have learned that numbers are represented in groups of 3,where there are ones, tens, and hundreds and then ones, tens, andhundreds of thousands. Students will extend this understanding to thenext group of millions, i.e., ones, tens, and hundreds of millions.

C2 Students should recognize the pattern in either dividing by 10,100, and 1000 respectively or multiplying by 0.1, 0.01, and 0.001.When initially presenting these multiplication and division opera-tions, it would be inappropriate to teach the “rule” of moving thedecimal point to the right or left by counting spaces. Actually, thepattern which will emerge shows that, depending on the division ormultiplication, the digits move to appropriate positions. For example,dividing by 100 would force the digit in the hundreds place to moveto the ones place, with all other digits moving along with it. It isimportant that reasons for the patterns be understood. Have studentsexplain these patterns and the meanings of the operations. Studentsshould recognize that the answer to 2341 x 0.001 is one thousandthof 2341, and can be thought of as a division question.

2341 ÷ 10 = 234.1 2341 x 0.1 = 234.12341 ÷ 100 = 23.41 2341 x 0.01 = 23.412341 ÷ 1000 = 2.341 2341 x 0.001 = 2.341

C3 Many problems solved easily through the use of patterns are appro-priate for grade five students. Examples are:

• Use the following pattern to figure out what 9 x 999 would be.2 x 9993 x 9994 x 999

• If you keep dividing the square as shown, how many sections willthere be in the tenth picture?

etc.





PerformanceC3.1 Tell students that when a piece of paper is folded once, you gettwo sections. When it is folded twice, you get four sections. Ask them toinvestigate the number of sections you would get with 3 folds and with4 folds and have them predict the number of sections with 5 folds. Havethem check their predictions and explain how you would predict thenumber of sections for 8 folds, if it were possible to do it.

Paper and PencilC3.2 Have students predict the numbers in the next two rows of

11 1

1 2 1

1 3 3 1

C3.3 Ask students to find the products below, to look for a pattern, andto use this pattern to predict 1111 x 2222.

1 x 2 =11 x 22 =111 x 222 =

Ask them if there is a similar pattern for these products.1 x 311 x 33111 x 333

InterviewC1.1 Ask students to create place-value mats for numbers as high ashundred millions. Have them explain what is meant by the places beinggrouped in threes.

C2.1 Tell students that a number is divided by 100 and the result is427.4. Ask how they know that the original number could not have hada 3 as one of its digits.

C2.2 Ask students to tell what they know about the following withoutgiving the answers: 4567 x 0.1

4567 x 0.014567 x 0.0001

C2.3 Tell students that Jake said, “You always get a larger answerwhen you multiply.” Ask them to respond to Jake's observation.





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ii) explore how a change in onequantity in a relationshipaffects another

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C4 rearrange factors to makemultiplication simpler

C5 recognize and explain how achange in one factor affectsa product or quotient

C6 predict how a change inunit affects an SI measure-ment

C7 manipulate the dimensionsof a rectangle so that thearea remains the same

C4/C5 Students should recognize that a rearrangement of factorscan often make a multiplication much simpler, e.g., Rearrange 28 x 250to 7 x 1000 by noticing that 4 is a factor of 28, with 7 as the otherfactor, and that 4 x 250 is 1000. 7 x 1000 is certainly easier to find than28 x 250! Some students might view this as taking one-fourth of 28 andmultiplying 250 by 4. It works because we are taking one-fourth of thegroups—7 groups instead of 28—but we are making each group 4 timesas great—1000 instead of 250. You might find other students whothink, “28 x 1000 would be 28 000, but 250 is only one-fourth of 1000,so my answer must be one-fourth of 28 000—7 000.”

This strategy is particularly useful when numbers are halves orfourths of 10, 100 and 1000, i.e., 2.5, 5, 25, 50, 250, and 500, andthe products are to be found mentally.

Rearranging factors can also be useful in finding products by pencil-paper methods, e.g.,

15 x 12 = (3 x 5) x (2 x 6) = (3 x 6) x (5 x 2) =18 x 10 = 180

C6 Students should generalize that using a smaller measurement unitwill increase the number of those units and that using a larger unit willdecrease the number of those units. This would enable students to thinkthat 28 cm must be 0.28 m, not 2800 m, since metres are larger thancentimetres or that 352 m must be 352 000 mm, not 0.352 mm, sincemillimetres are smaller than metres.

C7 Students should be aware that many rectangles can have the samearea and that if this is to be true, a longer length must accompany ashorter width. In fact, students may recognize that if one dimension ismultiplied by any factor, the other dimension must be divided by thatfactor to retain the same area. This generalization may be made if youplan a guided investigation, asking students to draw specific rectangles,to find the areas, to change the dimensions in specific ways, and tocompare the new areas to the areas of the original rectangles and the newdimensions to the original dimensions.

• Tell pairs of students to find as many different rectangles and squaresas they can that have an area of 64 m2. Ask them to display theirfindings on grid paper and to identify the one that would representthe best “room” dimensions for a bedroom and to explain why.





PerformanceC7.1 Ask students to use square tiles to show that if the length of arectangle is halved and the width doubled, the area remains the same.Ask them how this is related to the half-and-double strategy in mentalmath that could be used for questions such as 56 x 50.

Paper and PencilC5.1 For each of the following, ask students to tell how many times asgreat the second product is than the first:

44 x 25 compared to 44 x 10075 x 20 compared to 75 x 10010 x 70 compared to 90 x 703 x 100 compared to 12 x 250

InterviewC6.1 Ask students whether each of the following numbers will increaseor decrease as the measurement unit is changed as indicated:

0.04m to centimetres3.02cm to metres0.002L to millilitres2.005km to metres

C4.1 Ask students to explain why the result of 320 x 500 has to be halfof 320 000.

C5.2 Ask students to explain whether 5600 ÷ 5 is double or half of5600 ÷ 10 and to explain why.

C4.2 Ask students, How does multiplying 44 by 100 help to compute44 x 25? What must be done next in order to find the answer? Can youexplain another way to mentally compute 44 x 25?

PortfolioC4.3 Ask students to explore the calculator exercise below for a pattern.Ask them to find explanations for why this exercise works the way itdoes. Have them write about the explorations.

Enter a one-digit numberMultiply it by 7Multiply the answer by 143





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C8 demonstrate anunderstanding that themultiplicative relationshipbetween numerators anddenominators is constantfor equivalent fractions

C8 Students should have experience forming equivalents for varioussimple fractions using materials such as Fraction Factory. Encouragethem to display these fractions in a table and to look for patterns,e.g., If the equivalents to

1

2 are

displayed in a table, studentsshould observe that the numeratoris always one half the denominator. Ask, If you double the numeratorand denominator of any fraction in the table, do you get anotherfraction in the table? What if you triple instead of double? Can youmultiply the numerator and denominator by any number and getanother fraction that would be in this table? Can you predict thenumerator in the table if the denominator is 48? Will there be adenominator of 37 in this table? Can you predict the denominator inthis table if the numerator is 16? Will there be a numerator of 37 inthis table? These and other similar questions should provoke classdiscussion about equiavlent fractions.

• Have students fold sheets of paper into thirds and unfold thesheets. Ask them to shade two-thirds. Have them fold the sheetsin half in the other direction and unfold. Ask, How many sectionsare now shown on your sheets? How does this number of sectionscompare to the 3 you had initially? How many of these 6 sectionsare shaded? How does this number of shaded sections compare tothe 2 you had initially? What do you notice about the two com-parisons? Have them fold the sheets in half and in half again, andthen unfold. Ask the same questions. Revisit each of these foldsrecording the symbols that describe the fraction shaded. Havestudents connect the multiplicative comparison between the totalnumber of sections and of those shaded as the papers are folded tothe multiplicative comparison between the numerators anddenominators of the equivalent fractions that describe the frac-tional parts which are shaded.

Through a variety of concrete and pictorial activities with equivalentfractions and pattern observation, students should notice that betweentwo equivalent fractions, the denominators have the same multiplicativecomparison as the numerators.

Numerator 1 2 3 4 5 6 7

Denominator 2 4 6 8 10 12 14





PerformanceC8.1 Ask students to make a table of equivalent fractions for 3-fourths.

C8.2 Ask students to use paper folding to convince you that 2-eighths is equivalent to 1-fourth.

Paper and PencilC8.3 Ask students to fill in the missing numbers to create a table ofequivalent fractions.

C8.4 Ask students to put the numbers shown below these two tables ofequivalent fractions in the correct spots.

InterviewC8.5 Ask students why

21

25

is another name for 1

2 and

2

4.

C8.6 Present this table of equivalent fractions.

Ask students for the multiplicative relationship between the numera-tors and denominators of pairs of fractions in the table, e.g.,

8

20and

4

1 0 .

Portfolio

C8.7 Provide students with sheets containing pictures of eightcongruent rectangles. Ask them to shade 3 fourths of all eight rectan-gles in the same way. Have them leave one rectangle showing 3fourths but have them subdivide the other seven rectangles in differ-ent ways to show seven different fractions that are equivalent to 3fourths.





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C9 represent measurementrelationships using tablesand two-dimensional graphs

C9 As students explore perimeters, areas, and volumes, they should beencouraged to display some of the data in tables and graphs. Thesedisplays will assist students in making inferences about the data.

Suppose students examine the perimeters of regular hexagons withvarious side lengths. They could record the data in a table and draw agraph as shown below.

From either of these displays, they can easily see, for example, why theperimeter of a regular hexagon with a side length of 5.5cm is 33cm.

Other types of measurement graphs could include- areas of squares for different side lengths- volumes of cubes for different side lengths- areas of rectangles with length of 10cm for different widths

Students might compare the perimeter of rectangles with area 24 fordifferent widths and notice how the perimeter shrinks and grows.

Alternately, students might compare the areas of rectangles with perim-eter 24 and notice how the area grows and shrinks.

Perimeter(cm)

Side Length (cm)1 2 3 4 5

10

20

30

Side Length (in cm) 1 2 3 4 5 6 7

Perimeter (in cm) 6 12 18 24 30 36 42





PerformanceC9.1 Provide students with multilink cubes. Have them build cubes ofdifferent sizes. In each case, ask them to create tables to show the variousside lengths and volumes and create graphs to display the information.Then ask them to predict the volume of a cube with a side length of 2.5units.

C9.2 Provide square tiles. Ask students to build rectangles 2 tiles widebut of different lengths. Ask them to record the perimeter and area ofeach rectangle in a table and to look for patterns in their tables.

Paper and PencilC9.3 Ask students to create graphs showing the relationship betweenside length in centimetres and area in square centimetres. Ask them touse the graphs to determine the area of a square with side length 5.5cm.

C9.4 Ask students to explore the relationship between areas of rectan-gles of different sizes and the areas of the rectangles created by doublingboth dimensions, e.g.,

InterviewC9.5 Ask students to describe what happens to the area of a rectangle ifits width stays the same, but its length increases by one unit.

C9.6 Ask students to discuss the relationship between the shapesdescribed by these measurements.

C9.7 Provide the following table and ask students to discuss the shapesdescribed by the table.





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i) extend understanding ofmeasurement concepts andattributes to includevolume, temperature,perimeter, and angle

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D1 solve simple problemsinvolving the perimeters ofpolygons

D2 calculate areas of irregularshapes

D1 Students should conceptualize perimeter as the total distancearound an object or figure. They might observe that, for certain figures,the perimeter is easy to compute, e.g.,(a) For an equilateral triangle, the perimeter is 3 times the side length.(b) For a square, the perimeter is 4 times the side length.(c) For a rectangle, the perimeter is double the sum of length and

width.

• Provide students with loops of string of fixed length such as 30cm.Have them form the loops into various polygons on top of a pieces ofgrid paper and estimate the areas of these polygons. Ask, What do allyour shapes have in common? For this perimeter, which shape seemsto have the most area?

D2 Students should use transparent grids and geoboards to help themcalculate the areas of a variety of shapes in their environment, e.g.,shapes of hands, feet, and leaves.

• Have students create shapes on geoboards and challenge otherstudents to find the areas of these shapes.

Areas involving square metres can be calculated by making newsprintmodels of 1 square metre and using them to fill in areas of large surfaces.Students could model larger areas involving square metres or squarekilometres using scale drawings in which, forexample, 1 cm might represent 1 m or 1 km.

• Challenge students to find as many shapes aspossible on a geoboard with an area of 5. Somepossible shapes are illustrated.

When determining the area of the polygon on the dot paper to theright, students may think of it as the 2 obvioussquare units and 5 obvious half squares (2

1

2) for a

sum of 41

2square units. All that is remaining to

find is the upper right triangle which is formedby the diagonal of two squares. The area of thetriangle would represent half of this, or 1moresquare unit, for a total of 5

1

2 square units. Ask

students to find other ways to determine thisarea.





PerformanceD2.1 Give pairs of students 24 colour tiles. Ask them to find differentshapes, all with an area of 24, but with different perimeters. Ask them tofind a way to keep track of their shapes and perimeters. They shouldfind what shape has the largest perimeter and what shape has the small-est perimeter. Before they start, have students reach a concensus on therules for the shape formation, e.g., Can there be shapes other thanrectangles? Must tiles have full sides abutting each other?

Paper and PencilD1.1 Ask students to draw three different polygons with the sameperimeters.

D2.2 Have students use dot paper to compare the areas of rectangles ofthe following dimensions:

2 cm x 3 cm, 4 cm x 3 cm, 6 cm x 3 cmAsk them what pattern they observed and to give another set of dimen-sions that would follow the pattern.

InterviewD1.2 Tell students that the longest side of a triangle is 10cm and that astudent claims that its perimeter must be greater than 20cm. Ask, Doyou agree with this student’s claim? Explain your position.

D1.3 Ask students to explain why the perimeter of rectangles withwhole-number side lengths is always even.

D2.3 Show students a shape on a geoboard or dot paper. Ask them todetermine its area and explain how they calculated it, e.g.,





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i) extend understanding ofmeasurement concepts andattributes to includevolume, temperature,perimeter, and angle

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D3 determine the measures ofright angles, acute angles,and obtuse angles

D3 Students often find the measurement of angles a difficult taskbecause the degree units are so very small coupled with the many linesand the double numbering running clockwise and counterclockwise onmost protractors. Before they begin to use a standard protractor, it isuseful for them to move from the non-standard unit wedges, as pre-sented in grade 4, to an intermediary protractor. The problems encoun-tered with angle measurement will be eased considerably by involvingstudents in making their own protractors.

• Provide the students with semicircular shapes cut from tracing paper,or construction paper. (The tracing paper would allow students to seethe angle vertex and follow the arms in order to measure more easily.)Have them fold the semicircle in half, forming a right angle or squarecorner. Explain that angles are measured in degrees and that a rightangle is made up of 90 of these degrees. Have this fold named 90 °.Ask them to fold once again and have them determine, and name,the size of the new angles generated by thefolds. One further fold would provideangles halfway between 0° and 45° and 45°and 90°. Discuss the measurement of thesefolds with the class and how they can assistwith estimation of angle sizes.

Once students have had practise estimating (see SCO, D7)and measur-ing angles using their homemade protractors, they can be introduced tothe standard protractor. You should choose these protractors carefullybecause some are easier for students to use than others. Before they makeany measurement with a protractor, they should be able to estimate theresult within 5–10 degrees. This ability will make protractor readingmuch easier.

Students have previously classified angles as right, acute, and obtuse bytheir overall appearances. Their understanding now should include - a right angle has a measure of 90° - a straight line, made by two right angles, has a measure of 180° - an acute angle has a measure less than 90° - an obtuse angle has a measure between 90° and 180°The purpose of an interview is to

uncover how students think aboutmathematics, so provide opportunitiesfor contradictions in students’ beliefsabout mathematical concepts to emerge.(NCTM 1991, 29) obtuse acute right





PerformanceD3.1 Have students estimate and draw acute angles that are less thanhalf the size of a right angle. Ask them to check their angles by measur-ing them.

D3.2 Ask students to put three green pattern blocks together to make atrapezoid. Have them examine where the three blocks share a commonpoint. Ask, How many degrees are in the straight line formed by theblocks? What must the measure of each angle in the green blocks be?Have students similarly try to find the measures of the angles in theother pattern blocks without using a protractor.

Paper and PencilD3.3 Ask students to draw shapes with two obtuse angles and threeacute angles. Then have them measure the angles.

InterviewD3.4 Ask students to find examples of obtuse angles in the classroom.Ask them to check by finding the approximate measures in degrees.

D3.5 Show students this diagram and ask them to identify and measurethe marked angles. Have them explain how the answers were deter-mined.

D3.6 Ask students, Why do you think right angles are more commonin our world than acute or obtuse angles?

PortfolioD3.7 Ask students to design rooms of furniture, all of which have noright angles. Have them write a report listing advantages and disadvan-tages of their designs.

orHave students write advertisements for their designs detailing why theyare superior to conventional furniture designs.





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ii) communicate using stand-ard units, understand therelationship among com-monly used SI units (e.g.,mm, cm, m, km) and selectappropriate units in givensituations

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D4 demonstrate an understand-ing of the relationshipsamong particular SI units

D4 The particular relationships that students should understandinclude (a) 1 metre is 10 dm, 100 cm or 1000 mm; (b) 1 kilometre is1000 m; (c) 1 litre is 1000 mL; (d) 1 gram is 1000 mg; and(e) 1 kilo-gram is 1000 g.Students should gain recognition and undertanding of these relation-ships through a variety of direct experiences. Many students will alreadyknow some of these from their work in previous grades in mathematicsand science. If they have developed good estimation skills, they shouldbe able to visualize particular units, e.g., If they hear the word decimetre,some might think right away about the length of the rods in the base-10blocks and use this as a referent for judging the lengths of other things.Then they know from direct experience that these rods are 10cm long;hence, the relationship between decimetres and centimetres is obviousto them. They should use these relationships to rename measurementswhen comparing them.

• Share short paragraphs describing the measurements of a variety ofclassroom items. Ask students to insert the appropriate units, e.g.,The table was 1524 ___ long. On it was a pencil that was 0.17 __long.

• Discuss with students the scale on a map of Canada. Invite them toinvestigate how the map would differ if the scale were changed, e.g.,from 1cm representing 100 km to 1 mm representing 100 km or1dm representing 100 km.

• Ask students to investigate individual portions of pop, juice, andmilk and then to predict and test the portion of a 1-litre containereach would fill.

It is important to help the students develop mental images of variousmeasurement units—these personal referents are necessary for successfulestimation. As referents for units of length, students might use thingssuch as their rulers for 30 cm or 300 mm, metre sticks or the height of adoor knob for 1m or 100 cm, and base-10 blocks for 1 cm, 1 dm or 10cm. If asked about particular lengths, they can visualize them in terms oftheir referents, e.g., They might think about 62 cm as approximately tworulers or as a little more than 6 base-10 rods.

• Have students match objects to cards on which mass measurementsare written. Use different names for the same object such as 1.5 kgand 1500 g. Encourage students to estimate measurements beforeactually using the scales.





PerformanceD4.1 Ask students to show, with fingers or arms, the length of:

550 mm, 68 cm, 0.02 m, 4 dmAsk them to tell another way of expressing each of the lengths usingdifferent units.

Paper and PencilD4.2 Ask students to determine the length of time it might take to walk1 000 000 millimetres.

D4.3 Have students use metric units to fill in these blanks in as manyways as possible: 1000_____ = 1 ____

D4.4 Ask students to list some ratios involving metric units that wouldbe 10:1 and some that would be 1:100, e.g., 1 dm:1 cm would be 10:1and 1cm:1m would be 1:100.

InterviewD4.5 Ask students if it would be possible to walk 0.001 km in a minuteand to explain their thinking.

D4.6 Ask students, If you change metres to centimetres, will the nu-merical values become greater or less? Why?

D4.7 Show students a variety of containers and ask them to identify theone probably designed to hold 500 mL. Ask them for reasons for theirchoices.

D4.8 Ask students to describe the size of the cage they might need tocomfortably hold a 50 000 mg animal.

D4.9 Have a variety of containers lined up and ask students to matchthese measurements on cards with the containers that would hold them:

3000 millilitres 2 litres500 millilitres 0.45 litres





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iv) develop and apply rules andprocedures for determiningmeasures (using concreteand graphic models)

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D5 develop formulas for areasand perimeters of squaresand rectangles

D5 Students need to have many opportunities to experiment withdeveloping their own formulas for calculating the perimeterss and areaof squares and rectangles.

You should try to elicit from the students ways to find the areas ofsquares and rectangles by working with a variety of materials. Youshould not simply provide the formula—multiplying the length by thewidth or A=l x w.

• Provide students with worksheets containing 6–8 pictures of rectan-gles all of which have whole number dimensions and with acetatecentimetre graph paper. First ask them to order by sight these rectan-gles from largest to smallest area. Have them check their predictionsby using the acetate grids to find the areas in square centimetres. Thisshould be followed by a discussion emphasizing that rectangles canbe filled with square centimetres in rows and columns and theconnection to arrays in multiplication. Then provide students with1 cm x 12 cm grid strips on acetate and ask them to use only thesestrips to find the areas of other rectangles. They should see that thesestrips could be used to find in each rectangle the number of squarecentimetres in each row and the number of rows. Finally have themuse rulers to find how many square centimetres would occupy thefirst row in a rectangle—indicated by its length in centimetres—andhow many rows of these square centimetres there would be—indi-cated by its width in centimetres. Multiplying these two valuesshould then be apparent.

When students investigate the distance around a rectangle, they willproduce their own formulas for perimeter that might include: l + w + l + w, 2(l+w), and 2l + 2w.

• Have students work in pairs to solve the following problem: You needto determine the amount of fencing required to build a dog pen thathas a length twice as long as its width. What are the dimensions ofthe pen? How can the perimeter of the pen be found without addingevery length?You want to cover the floor of the pen with square paving stones.How many will you need? Find a way to calculate this withoutcounting each stone.

The development of area formulas is afantastic opportunity to follow the spiritof the NCTM Standards. First, aproblem-solving approach can meaning-fully involve students and help them seethat mathematics is a sense-makingendeavor. Second, the connectedness ofmathematics can be clearly seen. (Van deWalle 1994, 313)





PerformanceD5.1 Provide students with centimetre graph paper. Have them cut outrectangles that have these dimensions: 4 cm x 7 cm, 5 cm x 8 cm, and 7 cm x 8 cm. Ask them to explain the relationship between the dimen-sions of each rectangle and the number of square centimetres shown ineach. Have them investigate whether this relationship holds for rectan-gles with decimal dimensions, e.g., 4 cm x 6.5 cm, by cutting out a fewexamples.

D5.2 Tell students that in your desk drawer you have a rectangle thathas a perimeter of 42 cm. Ask students to draw what it might look like.Have them share their drawings before you bring out your rectangle tocompare with theirs. Ask, If I tell you the perimeter of a rectangle, canyou be sure of its dimensions?

Paper and PencilD5.3 Tell students that the perimeter of a square is 38 cm. Ask them todraw it on graph paper and predict whether everyone’s square will bethe same.

InterviewD5.4 Tell students that the length of a rectangle is increased by one unitand its width decreased by one unit. Ask, What happens to its area andits perimeter?

D5.5 Ask students, Can the number describing the perimeter of apolygon be less than the number describing its area?

D5.6 Tell students that you measured the large table top in squarecentimetres and square metres. Ask, Which number will be greater andwhy? What do you think would be reasonable numbers for this area?

D5.7 Ask students how many base-10 flats could fit in a square metre.Ask , What is the relationship then between dm2 and m2?

D5.8 Ask students to name things with an area of 4 dm2 in the room.

D5.9 Tell students that the area of a classroom is 600 m2 and its perim-eter is 100 m. Ask, What are the dimensions of the classroom?

PortfolioD5.10 Have students calculate how much carpet would be needed tocover the floor of a room in their home. Ask them to include a floorplan showing where furniture is located. Ask, How much of the floorspace is taken up by furniture?





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D6 solve simple problemsinvolving volume andcapacity

D6 Students should recognize the difference between volume—theamount of space occupied by a three dimensional object—and capac-ity—the amount a container is capable of holding. The volume unitswhich they will generally encounter are cubic centimetres (cm3), cubicdecimetres (dm3), and cubic metres (m3) while capacity units will bemillilitres (mL) and litres (L). Capacity units are usually associated withmeasures of liquid, e.g., litres of milk, juice, fuel oil, and gasoline.

They should develop personal referents for these units. The use ofpersonal referents will help students establish the relationships betweenthe units, e.g., The small cube in the base-10 blocks is 1 cm3 and wouldhold 1mL and the large cube is 1dm3 and would hold1L. If they explore the size of a million using visuali-zation of base-10 blocks (see SCO, A6) and build acubic metre with metre sticks, they will have a goodmental image of 1m3.

Students should have a sense of which unit is the most appropriatevolume or capacity unit to use in any circumstance.

• List a variety of situations on the board, e.g., taking cough syrup,buying gasoline, determining the amount of air in a room, andgetting a flu shot. Ask students to choose the unit of measurementthat would be used for each. Have them compare their answers anddefend their choices.

• Have students measure the volume of small rectangular prisms bycounting the number of cubes it takes to build a duplicate of it.

• Invite groups of students to investigate the capacities of variousbeverage containers to determine which size container is found mostoften. They might record their findings in a graph or table andpresent it to the class.





D6.1 Ask students to calculate the volume of each type of base-10block, i.e., the volumes of

Paper and PencilD6.2 Tell students that a container holds 1.5 L. Ask if it is large enoughto make a jug of orange juice if the concentrate is 355 ml and you haveto use the concentrate can to add three full cans of water.

InterviewD6.3 Ask students how they could use a 1-L milk carton to estimate750mL of water.

D6.4 Ask students to estimate the number of cubic metres in theclassroom and to give an explanation for how the estimate was deter-mined. Follow this with actual measurement and have students use thisas a referent for estimating the volume of another differently sized room.

D6.5 Tell students that you need a box with a volume of 4000 cubiccentimetres to hold a gift you have purchased. Ask them to describewhat such a box might look like and what that gift might be.





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D7 estimate angle size indegrees

D7 Students should be able to estimate, within a reasonable range, themeasurements of angles in degrees, e.g., Astudent should be able to say which of theangles at right is nearer to 30o and be ableto provide an estimate for the other.

To estimate well, students need to have a sense for the sizes of a few anglessuch as

• Provide students with pipe cleaners that they can bend to form angles.Ask them to make an angle is about 30o . Have them compare theirestimates with neighbours. Draw an angle with a measure of 30o onthe overhead and allow students to compare their pipe-cleaner anglesto the projected image. Continue by asking them to make other anglesgiving them the degree measures.

Geo-strips and straws are other sources of materials that can be used tomake or show angles. With many different experiences over time, studentswill develop good estimation skills. The goal is to be able to estimate themeasure of angles to within 5–10 degrees of their actual sizes.

• After students are quite capable of estimating angle size, have themeach write the numbers 1–10 in a column in their scribblers. Showthem ten angles, one at a time, and ask them to estimate each and torecord their estimates. Be sure to show the angles in a variety ofpositions and with arms of varying length. Afterwards go over theirsolutions and ask students to share the strategies they used. Repeat thisactivity a few days later and note any improvements students havemade.

• If a computer is available with the Logo language, students can play agame in which a circle target is randomly placed. The turtle is aimed atthe target by indicating an angle at which to shoot and a distance.Students could get a number of turns to try to shoot the target.





PerformanceD7.1 Have students estimate and then use their homemade protractorsto approximate the measures of a variety of angles provided onworksheets.

D7.2 Ask students to examine several printed capital letters that aremade with only segments, e.g., K, L, M, and V. Ask them to classify theangles as right, acute, or obtuse, to estimate their sizes in degrees, andthen to measure them.

D7.3 Have pairs of students work together. Ask one student to make anangle and the other to estimate its size in degrees with the first studentthen checking the estimate with a protractor. Have them change roles.

Paper and PencilD7.4 Ask students to draw angles of approximately 45° and 135°.

D7.5 Show students a 60° angle. Tell them that it measures 60° and askthem to draw one that is about

1

2 the size. Repeat using other angles.

InterviewD7.6 Show students the diagram below and ask why is is easy to tellthat the segment makes a 45°angle.

D7.7 Show students an angle of 135°and tell them that someone saidthat it was 45°. Ask students to explain how they think such an errorcould be made.

D7.8 Ask, Which angle sizes do you find easiest to estimate? Why?

D7.9 Show students angle wedges and ask them to estimate their sizes.





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D8 determine which unit isappropriate in a givensituation and solve prob-lems involving length andarea

D8 Students should be familiar with four standard area units—thesquare centimetre (cm2), the square decimetre (dm2), the square metre(m2), and the square kilometre (km2).

The establishment of personal referents for these units is critical forthe development of good estimation skills. The base-10 blocks wouldprovide a good referent for square centimetres since the area of one faceof the small cube is 1 cm2 and of one face of the large cube is 100 cm2 .As well, the large faces of the flats or one face of the large cube is1dm2. It would be helpful for students to make a square metre onthe floor using masking tape and to keep it for a while as a referentfor estimation activities. They might discover that other things in theroom are almost 1 m2 , e.g., the area of the door from height of thedoor knob to the floor.

Students benefit from observing how many ofone area unit it takes to create another, e.g.,100 dm2 are required to make a square metresince it takes 10 dm to make a metre.Students could use base-10 flats inside themasking tape square metre to observe this.

Students should have a sense of which unit would be most appropri-ate to measure certain areas, e.g., a postage stamp, a farm field, and aclassroom.

Students should be able to write area measurements in terms of thestandard units, using decimals where necessary, e.g., The area of apaperback is 348.5 cm2.

Students can solve and create problems involving a variety of measure-ments in their everyday experience or using their imaginations. Ideally,some problems will focus on particular measurements, e.g. area orlength, and others will combine measurements, e.g., If the perimeter of arectangle is 18 cm and its area is 20 cm2, what are the dimensions?

Present students with an interesting way to determine the length of awoundup piece of wire. Compare its mass to the mass of a known lengthof wire, e.g., If a ball of wire has a mass of 36 g, and a 10 cm strip of thesame wire has a mass of 3.4 g, then the ball is probably a bit more than 1m long. Students come to appreciate that there are often indirect ways todetermine measurements found by using problem-solving strategies.





Paper and PencilD8.1 Have students use centimetre grid paper to design floor plans fora home. Have each square centimetre represent a square metre and askthem to calculate the area and perimeter of each room. Ask students todiscuss how using formulas helped with these calculations.

D8.2 Tell students that the grade five class has decided to sell fudge thathas been cut into 3 cm x 3 cm pieces. Ask them to write explaining howthey might determine what different size cardboard could be cut to holda layer of a dozen pieces of fudge.

InterviewD8.3 Tell students that Keri says the way to find the perimeter of arectangle is to use the formula: (l+w) x 2, but Ted says you must use(l x 2) + (w x 2). Ask, Who is correct? Why? Is there any other way tofind the perimeter of a rectangle?

D8.4 Tell students that Maryann said she could figure out the length ofa rectangle if she were given the width and the area. Ask, Is this possible?Explain.

D8.5 Ask students to tell you everything they can about a rectangle thathas a length of 12 cm and an area of 144 cm2 .

D8.6 Tell students that the area of a shape is 24 square units. Thelength is 6 units. Ask, How could you determine the width?

PresentationD8.7 Ask pairs of students to work together to develop a strategy todetermine the cost of carpeting the classroom floor. Invite them topresent their strategies to the rest of the class.

D8.8 Ask students to solve and present this problem: A rope 1.2 mlong is wound around. In what area might it fit?

Shape and Space:Geometry




GCO E: Students will demonstrate spatial sense and apply geometric concepts,properties and relationships.

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i) identify, draw, and buildphysical models of geomet-ric figures/shapes

iv) solve problems usinggeometric relationships andspatial reasoning.

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E1 draw a variety of nets forvarious prisms andpyramids

E2 identify, describe, andrepresent the various cross-sections of cubes andrectangular prisms

Building solid or three-dimensionalshapes presents a little more difficultythan two-dimensional shapes but isperhaps even more important. Building amodel of a three-dimensional shape is aninformal way to get to know andunderstand the shape intuitively in termsof its component parts. (Van de Walle1997, 355)

E1 Students will have cut out and assembled prepared nets forprisms and pyramids in previous grades. They should now investigateand draw a variety of possible nets for these shapes. These nets shouldbe drawn by “rolling and tracing”1 faces, cutting out, and wrappingactual 3-D solids, e.g., For a square pyramid, these are some possiblenets

Note: A net is not a different net if it is a reflection or rotation of oneyou already have.

E2 A cross-section is the 2-D shape of the face produced when aplane cut is made through a solid, e.g., A cube could be cut toproduce these shapes (among others)

Students should investigate cross-sections of prisms made by plane cuts.Some of these cuts should be parallel to faces and others oblique; theyshould start at a vertex and at different points along the edges of theprisms. Possible sources for these prisms are plasticine, styrofoam,cheese, brownies, and rice crispie squares. Students could make cuts withpiano wires or wire cheese cutters; you could use a knife.

Students should try to visualize shapes made by cuts and then checktheir predictions. If you wrap a shape with an elastic band where the cutwill be made, it will be helpful to some students to visualize.

If geoblocks are available, cubes, square prisms, and rectangular prismscan be built in a variety of ways; thereby, some cross-sections of theseprisms can be demonstrated without having to cut.

1This means placing the 3-D shape on a sheet of paper; tracing aroundthe face of the shape with a pencil; rolling the shape over so that anotherface is on the paper; tracing that face, being careful that it is attached tothe first face; and so on until all faces have been drawn.


Worthwhile Tasks for Instruction and/or Assessment Suggested Resources



PerformanceE1.1 Provide each group of four students with a triangular prism, atriangular pyramid, a square prism, and a square pyramid. Each studentin the group should use one of these shapes to trace a net for it then cutthe net out. The group should discuss other possible nets for each shapeand assign responsibility for creating some of these alternative nets.

E1.2 Have students draw all the possible nets for a triangularpyramid that has all faces equilateral triangles. Repeat for one with anequilateral base and three isosceles triangular faces. Ask, Did you getmore nets for one of them? Why do you think this happened?

E1.3 Provide students with a prism or pyramid and wrapping paper.Ask them to trace a net for the shape, cut it out, and actually wrap theshape to check it. Ask them to unwrap the shape, cut off one face, andfind the possible places this face could be re-attached to produce othernets. Advise them to use tape to re-attach the face and check. Extension:If centimetre graph paper were used for this activity, a good connectionto surface area could be made.

E1.4 Provide students with net-like pictures of a hexagon with 6triangles attached that are actually not nets for a hexagonal pyramid. Askstudents to predict if they are nets, check their predictions by cuttingthem out, and make any changes needed to create a true net.

E2.1 Cut off the tops of 1-L milk cartons to make open square prisms.Fill them half full of water. Have the students tip the carton in differentways and examine the shapes made by the surface of the water. Havethem draw the shapes they find and discuss these as cross-sections of asquare prism. (You could also use clear plastic cubes.)

InterviewE1.5 Show students nets for 3-D shapes. Ask them to point anddescribe how the nets would fold up to make shapes, and to name theshapes.

E2.2 Ask students to find three different cross-sections of a Playdohcube. Have them try to visualize to predict the shapes of the cross-sections before the actual cuts are made.

E2.3 Ask students to draw the shape they would see if the corner of arectangular prism were cut off.





E3 For these drawings, you will need isometric dot paper. Thispaper should be positioned as shown below.

It is interesting to note that the angle made above is 120o but incontext of a drawing of a cube (above), it will appear as 90o . This isanother example of perceptual constancy in spatial visualization.

Isometric drawings are always done from a front-right, front-left, back-right, or back-left view, not by looking straight on.

• Start with a simple shape like A (below). Have students make thiswith three cubes and replicate this drawing of it. Start by drawing theforemost (front) cube and then draw the other two attached to it.This is the front-left view. Have students turn their shape so they canlook at it from the front right (B) and draw it. Again, start by draw-ing the foremost (front) cube, then the one attached to it, and finallythe 3rd. (Be careful because you only see 1

1

2 faces of this one!)

It might be helpful to have the students use a paper mat (below) sothey can place their structures on it and move it to the desiredposition, e.g., front right, as they draw.

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E3 make and interpretisometric drawings ofshapes made from cubes





PerformanceE3.1 Have students make three-cube towers and isometric drawings ofthese towers. Have them topple these towers and make isometric draw-ings from the front left and the front right.

E3.2 Have students use five cubes to make T-shapes and four differ-ent isometric drawings of them.

E3.3 Have students construct with cubes each shape shown below.These shapes are drawn from the front left. Ask students to makeisometric drawings from the front right.

E3.4 Ask students if there could be any hidden cubes in the drawingsin E3.3. If so, ask them where they could be.

E3.5 Have students make a structure using eight cubes and draw anisometric view of it. Have them exchange drawings with classmates andeach of them build the other’s structure from its drawing.

E3.6 Sep up a learning centre using 8–10 of the structures andcorresponding pictures the students completed in the task describedin E3.5. Ask students to match the drawings and structures.

PortfolioE3.7 Have students find all the different shapes that can be made fromfour cubes. For each one, have them make an isometric drawing torecord it.





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E4 explore relationshipsbetween area and perimeterof squares and rectangles

E5 predict and constructfigures made by combiningtwo triangles

A teacher’s questioning techniques andlanguage in directing students’ thinkingare critical to the students’ developmentof an understanding of geometricrelationships. Students should bechallenged to analyze their thoughtprocesses and explanations. (NCTM1989b, 112)

E4 Through structured exploration activities, students shouldconclude that all squares with the same area have the same perimeterand all squares with the same perimeter have the same area. However,rectangles with the same area can have different perimeters, and rectan-gles with the same perimeter can have different areas. The generalizationabout squares is often over-generalized causing a common misconcep-tion about the relationship between area and perimeter in rectangles andin other polygons.

E5 To promote spatial sense and to develop visualization skills,students should work through a series of activities from simple tocomplex that involve constructing polygons from two triangles.

• Plan activities that will have students investigating the variouspolygons they could make using each of the following pairs oftriangles:- two congruent equilateral triangles

(possible source: pattern blocks)- two congruent isosceles right triangles

(possible source: tangram pieces)- two congruent isosceles triangles

(possible source: a rectangle cut along both diagonals)- two congruent right triangles

(possible source: a rectangle cut along one diagonal)- two congruent acute/obtuse triangles

(possible source: a parallelogram cut along one diagonal)- two different isosceles triangles with equal bases

(possible source: a kite cut on the diagonal joining equal angles)

The emphasis should be on visualization—seeing in their minds. Afterstudents have done the investigations by manipulating both triangles,you should show only one triangle on the overhead and ask students todraw the various results of combining it with another congruent trian-gle.





PerformanceE4.1 Provide students with grid paper. Have them each draw a squarethat has sides of two units and ask them to find its perimeter and area.After sharing their results, have them repeat the tasks with squares thathave other side measurements. Ask students if they see a relationshipbetween the side length of a square and its perimeter.

E4.2 Ask students to construct a square on their geoboards that has aperimeter of 12 units and to compare their squares. Ask them if every-one constructed congruent squares? Repeat this task but this time askthem to make a rectangle with a perimeter of 12 units.

E4.3 Have students use square tiles to find all the possible rectanglesthat can be made from 12 tiles. Have them record their findings on gridpaper and to note the areas and perimeters of each rectangle. Ask themwhat they notice about the appearance of the rectangle with the greatestperimeter and with the least perimeter. Have them repeat using 24 tiles.

E5.1 Have students cut rectangles along one diagonal and investigatethe other polygons that can be made using these two triangles. After thestudents have found the possible polygons, you could have them placethe two triangles to make the rectangle, keep one triangle in place andslide the other triangle to make a parallelogram. Now have them flip thistriangle to make a larger triangle. Ask them to return the triangle tomake a rectangle and repeat the sliding and flipping in another direc-tion. Finally, have them return the triangle to make a rectangle and askthem to describe the motion of the triangle that would result in a kite.

Paper and PencilE4.4 Tell students that a farmer has 100m of fencing to make a pen forhis pigs. He decides a square or a rectangle would be the best shape. Askthem for some possible sizes of pens he could make. Ask them how theareas of each compare and what size they would recommend and why.

E5.2 Have students describe the types of possible pairs of congruenttriangles that would combine to make (i) a square, (ii) a rectangle, (iii) akite, and (iv) a parallelogram.





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ii) describe, model, andcompare 2-D figures and 3-D shapes, explore theirproperties, and classify themin alternative ways.

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E6 recognize, name, describe,and represent perpendicularlines/segments, bisectors ofangles and segments, andperpendicular-bisectors ofsegments

E7 recognize, name, describe,and construct right, obtuse,and acute triangles

Definitions should evolve from experi-ences in constructing, visualizing,drawing, and measuring two- and three-dimensional figures, relating propertiesto figures, and contrasting and classifyingfigures according to their properties.Students who are asked to memorize adefinition and a textbook example or twoare unlikely to remember the term or itsapplication. (NCTM 1989b,112)

E6• Show students how to use a mira to draw perpendicular lines: Draw a

line segment and place the mira across it, moving the mira until theimage of the part of the segment on one side of the mira falls on thesegment on the other side. The line on which the mira sits is perpen-dicular to the original line; therefore, place a pencil against thebevelled edge of the mira and draw in this perpendicular. When theimage of the endpoint of the segment falls on the other endpoint, themira is bisecting the segment; therefore, the line on which the mirasits would be the perpendicular-bisector of the segment.

• Have students use two straws or two toothpicks to make arrange-ments in different configurations (estimate first and check), e.g.,

- parallel to one another- intersecting- perpendicular at an end point of one straw- perpendicular at endpoints of each straw- one straw perpendicular to the other straw at its midpoint- one straw bisecting the other straw but not perpendicular- each straw bisecting the other straw but not perpendicular- one straw bisected by the other straw and perpendicular- each straw bisecting the other straw and perpendicular

• Have students bisect angles by folding one arm onto the other, byusing a mira to find where one arm would reflect onto the otherwhen it is placed through the vertex, and by measuring.

E7 Give students 12 cards with examples of right, acute, and obtusetriangles on them. Ask them to sort them into three groups by thenature of their angles and share how they were sorted. (This can be doneeven if the names are not known.) Attach the names for these classifica-tions to the student sort. Look for examples in the world of each type;examine familiar materials in the classroom, e.g., pattern blocks,tangrams. Have students choose straws of different lengths or geostripsto make examples of each type.

This angle classification should be connected to the side classification—equilateral, isosceles, scalene—which was studied in grade 4.

mira





PerformanceE6.1 Have students each draw a right angle, an acute angle, and anobtuse angle. Ask them to use only a mira to draw in the bisector of eachangle. Have them check their results by paper folding and by measuringthe angles with a protractor.

E6.2 Have students make the capital letters of the alphabet that useonly segments. Have them find examples of bisectors of segments,perpendicular segments, and perpendicular-bisectors. Have them sharetheir findings.

E7.1 Have students construct specific triangles on their geoboards andrecord them on geopaper, e.g., an acute triangle that has one side usingfive pins, a right triangle that is also isosceles, and an obtuse triangle thathas one side using five pins.

Paper and PencilE6.3 Have students draw, without measuring, a segment that is ap-proximately 10cm long. Have them construct a perpendicular bisectorof this segment, using only a mira.

E7.2 Have students draw three examples of each type of triangle.

InterviewE6.4 Have students fold sheets of paper in half and then in halfagain. Open them up to reveal the folds and draw in segmentsalong these folds. Have them describe the relationship betweenthese two segments.

PresentationE7.3 Provide pairs of students with two 6cm straws, two 8cm straws,and two 10cm straws. Have them investigate the triangles they can makeusing 3 straws at a time. Complete a table of results.

Straws Used Type of Triangle

PortfolioE7.4 Have students investigate how many different isosceles trianglescan be made on 5-pin x 5-pin geoboards. For this activity, differentmeans the side lengths are different, not the position on the geoboard.Ask them to record the triangles on geopaper and classify them as acute,obtuse, or right.





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ii) describe, model, andcompare 2-D figures and 3-D shapes, explore theirproperties, and classify themin alternative ways.

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E8 make generalizations aboutthe diagonal properties ofsquares and rectangles andapply these properties

E8 You should plan guided investigations using paper folding, miras,and direct/indirect measurements of lengths and angles, so that studentswill describe the patterns regarding diagonals in squares and rectan-gles—these patterns will be the diagonal properties of these shapes.

• Provide each student in a group of 4 with a different square. Askthem how they think the lengths of the two diagonals of their squarescompare. Check by measuring them and sharing their findings. Askthem how the diagonals appear to intersect. Check their predictionsby measuring and sharing their findings. Ask them what the diago-nals appear to do to each vertex angle. Ask them what they think themeasure of each of the angles at a vertex is. Check by measuring andsharing their findings. Have them cut out the four triangles made bythe two diagonals. Ask them to describe and compare these triangles.Ask them to describe everything they learned about squares in thisinvestigation. Ask them to write about all the properties of squaresthat they now know. (See SCO, E8 in grade 4 for other properties.)

From investigations like this, students should conclude that the diago-nals of a square (a) are equal in length; (b) bisect each other; (c) intersectto form four right angles and combined with (b) means they are perpen-dicular-bisectors of each other; (d) are bisectors of the vertex angles ofthe square, thus forming 45o angles; and (e) form four congruent isosce-les right triangles.

Similarly, by investigating rectangles, they should conclude that thediagonals of a rectangle (a) are equal in length, (b) bisect each other,(c) form two pairs of equal opposite angles at the point of intersection,(d) form two angles at each vertex of the rectangle that sum to 90o andhave the same measures as the two angles at the other vertices, and (e)form two pairs of congruent isosceles triangles.

You should engage students in a variety of activities that require them toapply these diagonal properties.

(See SCO, E11 for rotational symmetry properties which could bedeveloped at the same time through these guided investigations.)

These properties of squares and rectangles should be added to the side,angle, and reflective symmetry properties established in grade 4, SCO,E8.





PerformanceE8.1 Have students draw squares that have diagonals of length 8cm.Ask them what properties of a square they used to do this? Did everyonedraw the same square?

E8.2 Have students cut a square along both diagonals. Have theminvestigate the different shapes that they can make (i) using two of thetriangles formed if equal sides must abut, (ii) using three of the triangles,and (iii) using all four triangles.

E8.3 Have students draw rectangles that have diagonals that intersectto form 60o angles. Ask, Did everyone in the class draw the same one?How do all the rectangles compare?

E8.4 Ask students, When the diagonals are drawn in a rectangle, howdo you know that each triangle formed is 1

4 of the rectangle?

Paper and PencilE8.5 Have students draw an isosceles right triangle. Have them use amira to draw the square for which the triangle is one-quarter.

E8.6 Have them draw a segment 12cm long. Have them, using only amira, construct the square that has this as a diagonal.

E8.7 Explain that all triangles are rigid while rectangles are not. One orboth diagonals are often used in the real world to make a rectangularshape rigid. Have students explain what this means and to give a real-world examples.

InterviewE8.8 Have students draw isosceles triangles. Have them explain howthey could use their triangles to construct rectangles that would have theisosceles triangles represent one-quarter of their areas.

PresentationE8.9 Explain that a family of rectangles has a perimeter of 38cm and allof their sides are whole numbers. Have students draw this family ofrectangles on graph paper. Ask, Which family member has the greatestarea? the longest diagonal?

E8.10 Have students draw rectangles and show the two diagonals.Have them measure one angle at a vertex and one angle at the centre.Ask them to find the measures of all the other angles in the figure usingonly these two angle measurements and their knowledge of properties.





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iii) investigate and predict theresults of transformationsand begin to use them tocompare shapes and explainconcepts.

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E9 make generalizationsabout the properties of

translations andreflections and applythese properties

Good geometry activities almost alwayshave a spirit of inquiry or problemsolving. Many of the goals of problemsolving are also the goals of geometry.(Van de Walle 1997, 344)

E9 You should introduce these transformations by planning activitiesthat allow students to make and recognize translations and reflections ofshapes.

• Have students draw or trace shapes, and using a mira, draw mirrorlines and the reflected images.

- Have them compare the shapes and their reflected images usingtracing paper or by folding over and looking through the paper at alight source. They should conclude that they are congruent.

- Ask them to label the shapes with A, B, C, D ..., the correspondingvertices of the reflected image with A’, B’, C’, D’ ..., and to nameboth shapes clockwise starting at A and A’ (see below). They shouldconclude that the shapes and images are of opposite orientation.

- Have them make segments by joining corresponding vertices andexamine the angles made by the mirror lines with these segments.They should conclude that the mirror lines are perpendicular toall segments joining corresponding image points.

- Have them make a direct comparison of (or measure) the distancefrom corresponding vertices to the mirror line. They should concludethat corresponding points are equidistant from the mirror lines. Inshort, the mirror line is the perpendicular-bisector of all segmentsjoining corresponding points.

These properties should be seen to hold true for a variety of shapesreflected in different mirror lines. After the students write summariesof the properties of a reflection, they should apply these properties ina variety of ways.

Similarly, through guided investigations of shapes under translations,students should conclude that a shape and its image (a) are congru-ent, b) have the same orientation, (c) have corresponding sidesparallel, and (d) that all segments joining corresponding points areequal and parallel to one another.





PerformanceE9.1 Have groups of three students place three geoboards in a row. Onthe middle one, have them construct a shape. Using the edges of thisgeoboard as mirror lines, have them construct the reflected images onthe other two geoboards. Have them convince one another that thefigures are of opposite orientation and that corresponding points areequidistant from the mirror lines.

Paper and PencilE9.2 Have students use properties to help them draw the image undereach transformation.

a. LMNO reflected in b. PQR under a translationline l where P → P'

E9.3 Have students find the reflected image of∆ABC in the given mirror line using only a sheet ofpaper as a measuring tool. Check using a mira. Askthem what properties of reflections they used whileperforming this task.

E9.4 Explain to students that these twotriangles are reflections of one another.Have them use rulers to find the mirrorline. Check using miras.

E9.5 Explain to students that Jeri started to trans-late ABCD. He located the image of A and markedit. Ask them to finish the translated image for Jeri.





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E10 explore rotations ofone-quarter, one-half, andthree-quarter turns using avariety of centres

E10 Rotations are the most perceptually challenging of thetransformations. Students need many first-hand experiences makingrotations and examining the results before they will be able to identifyrotations. With both drawing and identifying rotational images, theemphasis should be on rotations of one-quarter, one-half, and three-quarter turns with centres on lines that are extensions of sides of theshape.

Students’ prior experiences with rotations would have been quarterand half turns of triangles and quadrilaterals with their vertices ascentres of rotation. Using this as a starting point, you should havestudents move on to explore quarter (90o) and half (180o) turns ofother shapes with their vertices as centres. Then, they could rotateshapes with centres located along lines formed by extending the sidesof the shape. Finally, they could undertake some work involvingthree-quarter turns.

• Have students use square dot paper to rotate rectangle ABCD 90o

clockwise about point P.

Note: P is on the line containing DC . One way to do the rotation is forstudents to use tracing paper, trace ABCD, hold a pencil firmly at pointP, rotate the tracing paper 90o clockwise, and locate the rotated images ofA, B, C, and D.





PerformanceE10.1 Have each student trace a pattern block and choose one of itsvertices as a centre of rotation. Have them extend one of the sides of theblock through this vertex so that they have a straight line (180o angle).Have them rotate the pattern block 180o from its original positionaround the chosen vertex and trace the pattern block again. Have themexamine these half-turn images. Repeat using different vertices, otherpattern blocks, and 90o rotations, i.e., make 90o angles at the chosenvertex.

E10.2 Have students fold plain sheets of paper into four quarters andlabel them (see below).

A B

C D

Have them arrange four or five pattern blocks along the horizontalsegment in section A and arrange the same blocks in the same orderalong the left vertical segment in section B. Ask,What is the relationshipbetween these two arrangements? Have them arrange the same blocks insection D so it will be a half-turn of the arrangement in section A. Havethem arrange the same blocks in section C so it will be a half-turn of thearrangement in B. Ask, What is the relationship between the arrange-ments in A and C?

E10.3 Have students investigate to see if there is any difference betweenthe images made by 180o clockwise rotations and 180o counter-clockwiserotations.





E11• Ask each student to use a square from the pattern blocks, mark one

of its vertices, and trace the block to make a square on a sheet ofpaper. With the square block placed inside its picture, rotate itclockwise with the centre of rotation being the centre of the square(intersection point of its two diagonals) until it perfectly matches itspicture again. Notice that the marked vertex is at the next corner.Repeat this rotation. Because the square can appear in four identicalpositions during one complete 360o rotation (see below), it is said tohave rotational symmetry of order 4.

Similarly, it could be shown that a rectangle has rotational symmetry oforder two.

These rotational symmetry properties should be combined with otherproperties of squares and rectangles. (See SCO, E8.)

E12 A 2-D figure is said to tessellate if an arrangement of replicationsof it can cover a surface without gaps or overlapping, e.g., If a number oftriangles in the pattern blocks were used, you would be able to use themto cover a surface; therefore, this triangle is said to tessellate. Investiga-tions should include some shapes like pentagons and octagons that willnot tessellate. The octagon is the shape often used in flooring and tileswhere squares fill the gaps because octagonal tiles won’t tessellate.

E13 Students need hands-on experiences cutting polygons and trans-forming them to develop spatial visualization skills, e.g., changing atriangle to a parallelogram by cutting off the triangle formed by joiningthe midpoints of two sides and rotating it about either of those mid-points.

This dissection process can be helpful in developing area formulas.

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E11 make generalizations aboutthe rotational symmetryproperty of squares andrectangles and apply them

E12 recognize, name, andrepresent figures thattessellate

E13 explore how figures can bedissected and transformedinto other figures

A tessellation is a tiling of the planeusing one or more shapes in a repeatedpattern with no holes or gaps . . . Moststudents will benefit from using actualtiles with which to create patterns. (Vande Walle 1994, 339)





PerformanceE11.1 Have students investigate to find out if a square is the onlyquadrilateral with rotational symmetry of order 4.

E11.2 Have students investigate what other quadrilaterals besidesrectangles have rotational symmetry of order 2 and which ones also havetwo lines of reflective symmetry.

E12.1 Have students investigate all of the different pattern blocks fortheir ability to tessellate.

E13.1 Have students cut out a parallelogram. From any point on oneside, have them draw a perpendicular segment to the opposite side. Havethem cut along this segment and translate (to the left) the piece on theright until two sides match. Ask them what shape they get.

(1) (2) (3)

E13.2 Have students cut out a trapezoid;fold it into two parts so that one parallelsides matches the other parallel side;unfold and draw a perpendicular any-where in the top part (see diagram); cut itinto three parts; rotate parts 1 and 2 asshown. Ask them what new shape was produced from this action.

PresentationE12.2 Have students fold a sheet of paper in half again and again untilthey have 8 sections. With it completely folded, have them draw anytriangle on the exposed surface and cut it out going through all 8sections. Using the 8 congruent triangles, have them test to see if thistriangle tessellates. Have them share their observations. Ask, Did every-one’s triangle tessellate? Did we have different triangles—acute, obtuse,right, isosceles, scalene? What conclusion might we make about thetessellating ability of any triangle?

E12.3 Using any one of the pattern blocks and half of a sheet of plainpaper, have students trace the block to completely cover this paper. Havethem color one block in the centre blue and color the rest of the shapesobeying these rules: if two shapes share a common side, they must be adifferent color (sharing a common vertex is OK); use as few colors asyou can.

Data Management





GCO F: Students will solve problems involving the collection, display and analysis of data.

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ii) construct a variety of datadisplays, including tables,charts, and graphs, andconsider their relativeappropriateness and

iii) read, interpret, and make,and modify predictionsfrom, displays of relevantdata

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F1 use double bar graphs todisplay data

F2 use pictographs and bargraphs to display andinterpret data

Certainly graphs, pictures, and charts canbe used to show data to an audience oncethe data have been collected, summa-rized, and analyzed. A pictorial represen-tation is an effective way to make apoint. However, a real user of statisticsemploys pictures and graphs as tools tounderstand the data during the processof analysis ... Representing data in apicture, table, or graph is a way todiscover the features of the data, to arraythe data so that their shape and relation-ships among aspects of the data can beseen. (NCTM 1989a, 142)

F1 Students should be aware that sometimes two pieces of data about acertain population are collected and need to be displayed simultane-ously, e.g., Census data often shows male and female data separately fordifferent years in a double bar graph—a graph where sets of two bars,each representing a different yet related item, abut and are differentiatedin some way (see examples below).

As an example, present the data below and explain to students that youhave asked five students in the class how many brothers and sisters theyhave. Brothers SistersStudent 1 1 1Student 2 2 0Student 3 1 2Student 4 0 1Student 5 2 1Display the data both ways:

Discuss with students how this type of graph allows them not only tocompare students in terms of how many brothers they have or howmany sisters they have, but also to compare the number of brothersversus the number of sisters.

F2 While students have been making and interpreting bar graphs andpictographs in previous grades, they should continue with these datadisplays. The scales of the bar graphs and the numbers represented bythe icons in pictographs should reflect the sizes of the numbers thatgrade 5 students should use, e.g., a pictograph showing the annual milkproduction of three provinces might use a cow for 100 000 litres.

Brothers

SistersStudent

1

#1

#2

#3

#4

#5

2





PerformanceF1.1 Provide students with the mean marks of boys and girls on threequizzes. Ask them to draw double bar graphs to illustrate this data.

F1.2 Ask students each to roll a die 25 times, to record the results in afrequency table. Have them work as partners to create double bar graphsto show the comparison of their frequencies of outcomes.

F1.3 Ask students to draw double bar graphs comparing how manynumbers between 1 and 100 are multiples of 2 versus multiples of 4,multiples of 3 versus multiples of 6, and multiples of 4 versus multiplesof 8. Ask students what conclusions might be drawn from their displays.

InterviewF1.4/2.1 Ask students to describe some data which would be appropri-ate to display using a bar graph, a double bar graph, and a pictograph.

F2.2 Show students an unlabelled bar graph of the populations of theCanadian provinces. Ask them to decide which bar they think goes withwhich province. Afterwards, have them draw pictographs to display thepopulations of each province.

PresentationF1.5/2.3 Have students collect information on various animal lengthsand masses. Ask them each to display the lengths in a bar graph, themasses in a bar graph, and the lengths and masses in a double bar graph.Ask what conclusions they might draw from the double bar graph thatthey might not have from the separate bar graphs.

F1.6 Have students find male and female Olympic track records forvarious events. Ask them to draw double bar graphs to display thisinformation. Have them share what conclustion they came to afterexamining their graphs.

PortfolioF2.4 Give students the graph below and ask them to find the errormade in gathering the data. Have them include their work.





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F3 use coordinate graphs todisplay data

F4 create and interpret linegraphs

F3 Coordinate graphing is a very useful tool for students to develop.While they should continue to use coordinates for the purposes oflocating points, they should also begin to consider what sort of datacould be displayed using coordinate graphs, e.g., If they consider thepairs of numbers that sum to ten, theycould use these pairs as coordinates andplot these points. They should notice thatthey will get a straight line if they join thepoints. Have them estimate the coordinatesof some other points on this line. Ask, Dothese coordinates also sum to 10? If one ofthe coordinates is 3.5, what is the othercoordinate?

• To reinforce plotting points, have students create “join-the dots”pictures on coordinate grids. Explain that after they draw theirpictures, they should list the coordinates of points in order of con-nection. Have them share their lists with other students for them tofigure out what the pictures are.

F4 As shown above, coordinate points can be joined to create a linegraph. The purpose of a line graph is to focus on trends implicit in thedata, e.g., If students measure the temperature outside every hour duringa school day, they could create a graph where the ordered pair is(hour number, temperature). By connecting the points with lines, theysee the trend in the temperatures throughout the day.

Temperature

Hour Number





PerformanceF3.1 Ask students to plot the points (2, 5) and (3, 7.5). Have themconnect the points and give the coordinates of 3 other points on theline.

F3.2 Ask different groups of students to create and plot ordered pairsthat satisfy these relationships: - the two numbers in the pair add to 8 - the two numbers in the pair add to 12 - the two numbers in the pair add to 15 - the two numbers in the pair add to 17Have students compare their graphs to observe the similarities anddifferences.

F4.1 Have students collect information about the number of studentsin the school in grades 1, 2, 3, 4, and 5. Ask them to draw a line plot toshow this data. Ask, Are there “bulges” in the numbers in certaingrades? (Note: You will probably have to remind students that thevertical scale should go up by 5s, 10s, 20s, or whatever number isappropriate for the data collected.)

F4.2 Ask students to look up the hockey scores for a favourite team overthe course of 10 games. Have them create line graphs with the orderedpairs being (game number, number of goals scored by favourite team).Ask them to create a second graph with the ordered pairs being(game number, goals scored by opposing team). Have them compare thetwo graphs.

Paper and PencilF3.3 Ask students to give the coordinates of all the points that are twounits from (3,3) if distance is measured only along the grid lines.

PortfolioF3.4 Ask students to compare the pictures formed by connecting thesetwo sets of ordered pairs, in order, joining the last point to the first asthe last step.Set 1: (2,1), (3,1), (4,1), (5,1), and (4,3)Set 2: (1,2), (1,3), (1,4), (1,5), and (3,4)Then have students design their own diagrams with flip images and listthe two sets of points.





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i) collect, organize, anddescribe relevant data inmultiple ways



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F5 group data appropriatelyand use stem-and-leaf plotsto describe the data

F5 Stem-and-leaf plots provide a convenient organization for recordinggrouped numerical data, e.g., If students in the class list their heights incentimetres as 140, 135, 127, 128, 131, 130, 121, 119, 124, 127, 130,131, 139, 142, 143, 118, 129, you notice that the data ranges from 118to 143. It would be convenient to group it into the 110s, 120s, 130s,and 140s and this is what a stem-and-leaf plot does.

Heights of Students (in cm)

11 8 912 1 4 7 7 8 913 0 0 1 1 5 914 0 2 3

The stems are the number of tens in the data—11, 12, 13, and 14. Theleaves represent the ones digits in the data, arranged from smallest tolargest and repeated as needed. The least values appear at the top and thegreatest at the bottom of the plot. Within each grouping, the dataappears in order from left to right. It is suggested that students use gridpaper to help them line up the data, putting one leaf in each square.

Students should be encouraged to notice the shape of the data, e.g.,There are more values in the middle than at the top or bottom. Theycould find the median of the data easily because the data is arrangedfrom least to greatest. In the example above, since there are 17 pieces ofdata, the median is the 9th piece of data, which is 130.

Students will need to make decisions about what to use as the stem, e.g.,If data ranges from 100 to 1000, the stem might be the number ofhundreds in the number and two-digit numbers would represent theleaves. If data involves decimal amounts, they might use the wholenumbers as the stem and the part of the number after the decimal as theleaves.

• Have students gather data about any of the following, make stem-and-leaf plots, and pose questions for other students to answer fromtheir plots:- the number of marbles different students own- the last three digits of the phone numbers of different students- the number of pages in favourite novels





Paper and Pencil

F5.1 Ask students to find the median, maximum, minimum, andrange of the data displayed in the stem-and-leaf plot below. It dis-plays the results of rolling a die, that has 0, 1, 1, 2, 3, and 4 on itsfaces, twice and forming the 2-digit number from the results of theserolls in order of appearance. Two-digit Numbers

1 1 1 1 2 32 2 3 43 3 54 0

Ask, What is the experimental probability that 11 will be the 2-digitnumber rolled?

F5.2 Tell students that you have collected data on the population ofCanada for each year since 1867. Ask them to suggest how the datacould be grouped for presentation.

InterviewF5.3 Ask students to describe the characteristics of a stem-and-leaf plot.

PortfolioF5.4 Ask students to gather information about the number of phonecalls that come into their home over the course of the week. Groups ofstudents could collect their data and create stem-and-leaf plots. Theyshould draw conclusions from the information gathered.

F5.5 Ask students to gather information on Olympic swimmingrecords over the course of the last 30 years for whichever event theywish. Have them create stem-and-leaf plots to show the information.

F5.6 Have students research the cost of at least 30 different cars,rounding these costs to the nearest hundred dollar and recording thecosts as thousands, e.g., $16 790 would be rounded to $16 800 and thecost recorded as $16.8 thousand. Have them display this data as stem-and-leaf plots. Ask students to examine their plots and write a fewsentences that describe the cost of cars in general.

F5.7 Have groups of students research populations for ten locationswithin their province, ten others in the country, and ten others in theworld. Ask students to display the data, explaining why they wouldorganize the populations in different groupings for the three displays.





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iv) develop and apply measuresof central tendency (mean,median, and mode)

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F6 recognize and explain theeffect of certain changes indata on the mean of thatdata

Technology provides teachers with aneffective method of examining the meanwith their classes. Spreadsheets areunbelievably easy to use. . . With thistool it is easy to . . . change the data inany way at all to observe the effect on themean. (Van de Walle 1994, 400)

F6 Students should develop a sense of the mean of a set of data as its“middle”—the number for which the total of the differences from it tothe data below it balances the total of the differences from it to the dataabove it. Consider the data: 6, 9, 10, 12, and 13; the mean is 10 sincethe distances from 10 to 9 and from 10 to 6—5 in total—balances thedistances from 10 to 12 and from 10 to 13—5 in total. This conceptualview of the mean should be connected to the view of the mean as thevalue of each data point should all the data be redistributed equally. Inthis case, if 2 from the 12 and 3 from the 13 were redistributed bygiving 1 to the 9 and 4 to the 6, the data points would be 10, 10, 10, 10,and 10. This could be illustrated by using cubes to make towers withheights equal to the original data and redistributing the cubes.

Through directed activities in which students work with actual data,they should come to the generalizations that the mean of a set of data - increases if any piece of data increases - decreases if any piece of data decreases - increases if a piece of data below the existing mean is removed - decreases if a piece of data above the existing mean is removed

• Provide a set of ten salaries of office workers in a certain company,e.g., 5 salespersons @ $25 000

3 secretaries @ $20 0002 clerks @ $17 500.

Ask students to determine the mean salary. Then have them predictand verify whether this mean salary goes up or down (a) if a secretaryresigns, (b) if a clerk resigns, and (c) if two more salespeople arehired. Ask them to describe at least three ways that the mean salary ofthe office workers could increase by $5000.

Note: In common usage, the word “average” is used as a synonym forthis “mean”; however, in mathematics, there are other averages such as“median” and “mode.” It is suggested that you use the word “mean” evenif you use the word “average” in conjuction with it.





PerformanceF6.1 Have students use counters to show visually why the mean of 8, 10,and 15 has to be 2 more than the mean of 6, 8, and 13.

Paper and PencilF6.2 Ask students to create sets of measurements that would maintainthe same average or mean even if two pieces of the data were removed.

F6.3 Ask students, Which two numbers could be removed from thisset of amounts of students’ savings and the mean would remain thesame?

$37, $40, $43, $20, $60, $40Have them predict the effect on the mean if everyone added $4 to theirsavings. Ask them to check their predictions. Have them predict the effecton the mean if the student with the least amount saved withdrew all hermoney. Ask them to check their predictions.

InterviewF6.4 Present students with the data: 9, 6, 8, 4, 7, 10, 5, 5, 8, 3. Ask themto explain the effect on the mean if the 7 were removed from the set.

F6.5 Ask students why the average or mean of this set of class sizeswould not change much even if the 30 were removed from the set.

20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 30

F6.6 Ask students, Why is it easy to tell that the mean of the data belowis 45?

43, 45, 47, 42, 48, 48, 49

F6.7 Tell students that Bing was asked to find the mean of 70, 75, 80,84, and 91. He said, “That’s easy! It’s 80 because 10 plus 5 is 15 and 4plus 11is 15.” Ask students to explain Bing’s method.





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v) formulate and solve simpleproblems (both real-worldand from other academicdisciplines) that involve thecollection, display, andanalysis of data and explainconclusions which may bedrawn

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F7 explore relevant issues forwhich data collection assistsin reaching conclusions

Real data are either collected by thestudents or obtained from real-worldsources. Real data are, by their verynature, “messy.” More data might beavailable than are needed to solve theproblem being considered. Unusualcharacteristics of the data mightnecessitate many attempts at selection,sorting, and representation in an effort tomake sense of them. (NCTM 1989a, 35)

F7 Students have had previous experience in collecting data to explorerelevant issues of interest to them. You should now focus on choosingthe best way to display that data, e.g., Suppose students have collectedinformation about the amount of fat and protein in various types ofsnack foods and have to decide whether to display that information inbar graphs, pictographs, or double bar graphs. If they choose bar graphs,they would have to consider whether the bars should be organized to gofrom least to greatest, greatest to least, or to use some other organization.They would also have to consider the scale they will use for the heightsof the bars, whether wide bars or narrow bars will be used, and why thesame width should be used for all bars. On the other hand, if theychoose pictographs, students would have to consider what icons shouldbe used for fat and protein and how many grams each icon shouldrepresent. Finally, if they choose double bar graphs, they will have toconsider how to differentiate the fat and protein bars as well as the order,scale, and width of the bars.

You should have students examine pictographs and bar graphs fromdifferent sources, e.g. web pages, newspapers, and magazines, to see anddiscuss what decisions must have been made to display the data.

You will need to emphasize the analysis of various displays of data inways that will cause students to do more than merely read informationfrom them. They should be starting to analyse data displays to drawconclusions, to make decisions, or to stimulate other questions, e.g., Ifon their bar graphs for fat and protein in foods you notice that the barsfor fat are always higher than the bars for protein, ask students, What doyou notice about the bars for fat? Can you conclude from this that snackfoods are bad for you? What questions might you want to ask a nutri-tionist? If you want a snack high in protein but with the lowest amountof fat intake, which food would you choose?





PerformanceF7.1 Have students collect, display, and analyse information on thenutritional value of various cereals. The information is usually foundon the sides of cereal boxes.

PresentationF7.2 Have groups of students determine questions to which they wouldlike an answer. Have them (a) determine how to collect the informationthey would like to have, (b) collect it, and (c) display it in the mannerthey feel is the most appropriate. Some possible questions include - What clothing and shoe sizes are most common for ten year olds? - About how many minutes a day is a household phone generally in use? - Do more people in Canada live in city limits, suburbs, or in the country?

PortfolioF7.3 Have students collect data on the prices of lettuce at differentstores in a particular week. Ask them to display the data and describewhether this information would help a shopper decide on the best placeto shop. Ask them to record other questions they might have after theyexamine the data displays.

F7.4 Ask groups of students to devise a way to determine how muchtaller grade 5 students are, on average, than grade 4 students. Have thestudents collect data, display it, and analyse it.

Data Management






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i) explore, interpret, and makeconjectures about everydayprobability situations byestimating probabilities,conducting experiments,beginning to construct andconduct simulations, andanalysing claims which theysee and hear

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G1 conduct simple experimentsto determine probabilities


G1 Students should continue to perform simple experiments and usefractions to indicate the experimental probabilities that result. Althoughdice, coins, and spinners are typically used for these experiments, thereare other materials which can be used and contexts that also providepractise in computational skills as illustrated by the following activities.

• Have students conduct simple experiments on hundreds charts. Havethem begin at a designated number and roll a die to determinewhere to go next:

1— down 1 and right 12— down 2 and right 23— down 3 and left 14— down 4 and left 25— down 56— up 1

They then determine the probability that after 5 rolls they will landin some designated range of numbers, or on a certain type ofnumber, such as an even number or a multiple of 3.

• Have the students use the random number function on calculators orspreadsheets to generate two 2-digit numbers and to add the num-bers. This is repeated a number of times to determine the probabilitythat the sum of the numbers will be greater than 100.

• Have pairs of students take turns rolling two dice. One number isused as the numerator of a fraction and the other as the denominator.Have them determine the probability that the fraction generated is inlowest terms, i.e., has no equivalent using smaller numbers.

Students should use decimals to describe experimental results, e.g., If anevent occurs 9 times out of 100, students could report the probabilityas

9

100 or 0.09. Students should recognize that the more times they

repeat experiments the more reliable the probability will be, e.g., Ifthey were trying to determine the probability of getting a 3 when adie is rolled and only rolled the die 6 times, the result would verylikely be less reliable than if they had rolled the die 36 times.

If your students have access to the technology, they should conductsimulations of these types of experiements to determine experimentalprobabilities.This would allow them to see the effect on probabilities ofincreasing the number of trials to several hundreds or thousands.





PerformanceG1.1 Have pairs of students roll a die 4 times. Ask them to create two2-digit numbers and subtract them. Have them repeat the experiment20 times. Have them calculate the probability that the difference theyget is less than 10. Have them repeat the experiment another 20 timesand compare the probability for 20 rolls compared to 40 rolls.

G1.2 Provide spinners marked in 10 equal sections, labelled 0,1,2, ...9.Ask the students to spin the spinners five times and total the numbersspun. Have them repeat this 10 times. Ask, What is the probability thatthe sum of the five numbers is greater than 25? Have the studentscompare their findings. Combine all their results for a class value.

G1.3 Provide pairs of students with bags containing 20 green cubesand 5 red cubes. Ask that one student pull out a cube, state the colour,and return it to the bag while the other student records the color. Havethem repeat this 20 times. Have students report their probabilities that agreen cube was chosen.

G1.4 Ask students to place a counter on the number 50 on hundredscharts and to flip a coin. Explain that if the result is heads, they movedown one space and if the flip is tails, they move up one space. They willflip the coin 10 times to complete one experiment. They should repeatthe experiment 20 times in total. The objective is to calculate theprobability that they will move their counters off the board during thecourse of the experiment. Have them share the experimental probabili-ties they got and combine their results to calculate a class value.

Paper and PencilG1.5 Tell students that you rolled a pair of dice 25 times and the sumof the numbers was 8 on 4 of the rolls. Ask, What is the experimentalprobability that the sum is 8? Does this seem reasonable?

InterviewG1.6 Ask students to describe an experiment to determine the prob-ability that the difference between the numbers on a pair of dice is 1.

G1.7 Tell students that two people performed an experiment where acoin was tossed and the probability of tossing a head was calculated.One person got a probability of

3

5. The other person got a probability of

47

100. Ask students which person they think got the more reliable result

and why.





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ii) determine theoreticalprobabilities using simplecounting techniques

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G2 determine simple theoreticalprobabilities and usefractions to describe them

... Theoretical probability is based on alogical analysis of the experiment, not onexperimental results. (Van de Walle 1994,383)

G2 Experimental probabilities are calculated by performing experi-ments. Theoretical probabilities are calculated by determining all thepossible outcomes of an event and comparing how many times a par-ticular outcome occurs to the total outcomes. If a die is rolled 60 times,the number 4 might come up 15 times giving an experimental probabil-ity of

15

60. However, the theoretical probability is

1

6, because there are

six equally likely possible outcomes (1, 2, 3, 4, 5, 6) when a die isrolled and one of these outcomes is the number 4—1 is compared to6 to get the ratio

1

6 .

Students need experiences to show them that the more times that anexperiment is conducted, the closer the experimental probability gets tothe theoretical value.

Another consideration in determining theoretical probabilities is thelikelihood of an outcome. In the case of the rolling of a die, all sixnumbers have an equal chance of being rolled—we say that all outcomesare “equally likely.” On the other hand, consider the spinner shownbelow.

Even though there are three outcomes, they are not equally likely. Thetheoretical probability of spinning a 1 is

1

2, not

1

3. This can be

determined by calculating the fractional part of the spinner coveredby 1.

Students should describe theoretical probabilities using both frac-tions and decimals, e.g., Students should be able to determine thatthe theoretical probability of:

• spinning a B is 1

3 or 0.333... • rolling a 5 is

1

6

• rolling an even number is 1

2 or 0.5





PerformanceG2.1 Ask students to put coloured cubes in a bag so that the theoreticalprobability of choosing a red one is

1

2 and choosing a green one is

1

4.

Ask, Why is there more than one way to model this situation?

G2.2 Ask students to list the first 20 multiples of 3 and to deter-mine the probability that a multiple of 3 is also a multiple of 6 and isalso a multiple of 9.

Paper and PencilG2.3 Provide hundreds charts. Ask students to determine the theoreti-cal probability that a number randomly chosen on the chart (a) ends in a5, (b) is even, (c) is less than 50, and (d) is on the diagonal.

G2.4 Provide students with array forms of the multiplication tables.Ask them to determine the theoretical probability that two 1-digitnumbers chosen randomly would have a product that is (a) even, (b) lessthan 5, (c) greater than 70, and (d) 24.

InterviewG2.5 Ask students why the probability that the sum of the numbers ona pair of dice will be 3 is not the same as the probability that the sumwill be 7.

G2.6 Ask students to explain how to determine the theoreticalprobability of rolling a 3 on a regular die and how this would changeif the die contained the numbers 1, 3, 3, 3, 5, and 6.

Grade 6

Number Concepts/



Number Concepts





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By the end of grade 6, thestudents will have achieved theoutcomes for entry–grade 3 andwill also be expected to


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A1 represent large numbers in avariety of forms

A1 Students will still need further experiences reading and recordingvery large numbers which may include fractions or decimals. An empha-sis should be placed on representing large whole numbers using round-ing and decimal notation, e.g., Rather than writing 34 518, studentsshould be encouraged to approximate it as 34.5 thousand; 12 340 000might be recorded as 12.34 million. In this case, the “unit” would bemillions, although the number could be written as 123.4 hundredthousand especially if comparisons were being made to other numbers inthe hundred thousands.

Have students practise recording numbers that are presented to themorally and rounding each one to the nearest tenth or hundredth of amillion, e.g., One hundred eight million, ninety-three thousand, forty-six might be rounded to 108.1 million or 108.09 million.

Students should also be able to recognize and represent fractional partsof large numbers.

43 489 784 is about 431

2 million

247 986 is about 1

4 million

8 762 154 is about 83

4 million

• Use real data when possible, e.g., The population of Atlantic Canada,as recorded in the 1991 census, is two million, three hundred sev-enty-eight thousand, two hundred ninety-seven. The populations ofthe Atlantic Provinces are: Newfoundland - 568 474, Prince EdwardIsland - 129 765, Nova Scotia - 920 000, and New Brunswick -760 058.

• Ask students to find examples of large numbers used in newspapersand magazines. Encourage discussion on the need for accuracy inreporting and the appropriate use of rounded numbers.

• Have students prepare reports on populations of Canadian cities. Askthat the report include a graph of the five most populous cities,comparing them to the five most populous cities of another country.





PerformanceA1.1 Prepare and shuffle 5 sets of number cards (0–9 for each set).Have students select nine cards and ask them to arrange the cards tomake the greatest possible and least possible whole number. Havestudents read each of the numbers. Extend this activity by asking stu-dents to determine (a) at least ten different whole numbers they couldmake using the nine digits selected and (b) the number of $1000 billsthey would get if they were given the greatest and least numbers asamounts of money.

InterviewA1.2 Discuss the idea of counting to 100 by 10s. Ask, How manynumbers are named? Then ask, How many numbers are named whencounting to 1 000 000 by thousands? by hundreds? by tens? Finally ask,How many numbers are named when counting to 100 000 000 byhundred thousands? by thousands? by hundreds?

A1.3 Tell students that light from a star takes 7000 centuries to reachthe earth. Ask, How many years is that?

A1.4 Present the following library information to students:Metropolitan Toronto Library 3 068 078 booksBibliotheque de Montreal 2 911 764 booksNorth York Public Library 2 431 655 books

Ask them to rewrite the numbers in a format such as . millionor . million books. Then ask them to make comparison state-ments about the number of books.

PortfolioA1.5 Have students collect newspaper and magazine clippings in whichlarge numbers are used. Discuss the type of situations in which they aremost likely to encounter large numbers and why that might be.





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By the end of grade 6, thestudents will have achieved theoutcomes for entry–grade 3 andwill also be expected to


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A2 represent fractions anddecimals

A2 Mixed number notation and improper fractions seem to be moreproblematic for students than proper fractions. Students should easilymove between mixed number and improper fraction formats of anynumber. Rather than only applying a rule to move from one format tothe other, students should be encouraged to focus on the meaning, e.g.,

Since 14

3 means 14-thirds and it takes 3 thirds to make 1whole,

14

3represents 4 wholes and another 2-thirds or 4

2

3.

Often it is easier for students to grasp the size of mixed numbers than

improper fractions, e.g., Students might know that 41

3 is a bit more

than 4; however, they might have little sense of the size of 13

3.

Students should be familiar with base-10 block representations ofdecimals, e.g., Since the large cube is 1 litre, it follows that

= 1, = 0.1, = 0.01, and = 0.001.

These base-10 block representations help students visualize the relativesizes of decimals.

Measurement contexts continue to be natural ones for decimal situa-tions, e.g., Students might consider how many kilograms of ground beefone would need for four hamburgers, or how many kilometres onecould walk in a minute.

• As a class activity, have students use a place-value chart that is dividedinto six sections representing numbers to the thousandths. Selectnumber cards, one at a time, and have each student decide in whichsection to place the digit as it is selected, in order to try to make thegreatest possible number. After six digits are called, have the studentscompare their numbers. This activity encourages students to takerisks and to think about probability. Extensions might includeestimating how far each student's number is from a target number ordetermining how the number would be rounded for a newspaperarticle and what it might represent. Alternatively, you might havethem try to make the least possible number.





PerformanceA2.1 Ask students to use pattern blocks to show why 3

1

3 =

10

3. Observe

which block they use to define the whole.

Paper and PencilA2.2 Present students with contexts that use large numbers such asastronomical data and demographic data, and contexts that use decimalthousandths such as sports data and SI measurements. As well, havestudents consider decimals in situations where decimals are not normallyencountered, e.g., Ask them to complete the following sentences:

In 0.1 years, I could...In 0.01 years, I could...In 0.0001 years, I could...

A2.3 Ask students to determine the number of whole numbers between2.03 million and 2.35 million.

A2.4 Provide students with thousandths grids. Ask them to shade thegrids, one at a time, to show the following decimals:

0.0040.2030.0231.799

Ask them to tell which was easiest to do and why.

PortfolioA2.5 Ask students to write reports on what they have learned aboutdecimals and what questions they now have concerning the topic.





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ii) explore integers, ratios, andpercents in common,meaningful situations

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A3 write and interpret ratios,comparing part-to-part andpart-to-whole

Both part-to-whole and part-to-partratios are comparisons of two measure-ments of the same type of thing. Themeasuring unit is the same for eachvalue. (Van de Walle 1994, 275)

A3 Use the students themselves, counters, or other models to review theconcept of ratio as a comparison between two numbers, e.g., In a groupof 3 boys and 2 girls,

3:2 tells the ratio of boys to girls3:5 tells the ratio of boys to the total group2:5 tells the ratio of girls to the total group2:3 tells the ratio of girls to boys.

Students should read “3:2” as “3 to 2” or “3 __ for every 2 __.”

A ratio is a multiplicative comparison between two quantities. When theratio of boys to girls is written as 3:2, that comparison is another way ofsaying that the number of boys is 1

1

2 times the number of girls or the

number of girls is 2

3 the number of boys.

• Have students use pattern blocks and geo-blocks to find ratiosbetween the sizes of the various blocks, e.g., Using pattern blocks, theratio of the blue blocks to the red blocks is 2:3, the ratio of the yellowblocks to the green blocks is 6:1, and the ratio of the green blocks tothe red blocks is 1:3.

Exploring ratios can take place in everyday situations, e.g., The ratio ofwater to concentrate to make orange juice is 3:1 or “3 to 1.” Ratiosshould also be explored in other topics in mathematics, e.g., Finding theratio of the perimeter of a square to its side length, the ratio of thelength of the diagonal of a square to its side length, and the ratio of thecorresponding sides of similar shapes.

Students should be reminded that all fractions are ratios. A fractioncompares a part to a whole, e.g., If

3

5 of a rectangle is shaded, the ratio

of shaded parts to the whole is 3:5. However, not all ratios are fractions.A ratio can be the comparison of a part to a part, e.g., In the rectanglewith 3 of the 5 parts shaded, the ratio of shaded parts to unshaded partsis 3:2. This is particularly problematic because this ratio can be writtenusing fraction symbols as

3

2. It is recommended that ratios written with

fraction symbols be read using ratio language, i.e., as “three is to two”rather than as “three halves.”

Students will also encounter ratios that involve the comparison of twoquantities that have different units—these ratios are called rates. Theywill use rates when they shop, e.g., oranges at 2/$1.09, when they dealwith speed, e.g., 100 km in 2 hours, and when they convert betweenunits of measurement, e.g., 1000 m in 1 km.





PerformanceA3.1 Ask students to model two situations that could each be describedby the ratio 3:4. Specify that the second situation must involve a differ-ent total number of items from the first.

A3.2 Ask students to find the following body ratios, comparing theirresults with those of others:(a) wrist size: ankle size(b) wrist size: neck size(c) head height: full height

A3.3 Ask students to select 20 tiles of four different colours so thatpairs of colours show the following ratios: 4 to 3, 2:1, and

1

3.

Paper and PencilA3.4 Provide students with information such as that below. Ask themto write ratio comparisons between/among the items.

4 cats 3 goldfish 2 hamsters

InterviewA3.5 Ask students whether they believe that the ratio of the populationof any city in Canada to the total population of Canada could be 1:2.Have them explain their response.

A3.6 Ask students, Why might you describe the ratio below as 4:1? as1:4? Are there other ratios to describe the boys and girls?

B B B B G B= boy G=girl

PresentationA3.7 Have students investigate the number of deaths in Canada causedby various diseases and to examine the ratios involved. You could alsohave them investigate the funding for research in the study of thesevarious diseases to see if the ratio of money spent is similar to the ratioof deaths caused by the diseases.

PortfolioA3.8 Have students report on ratios found in the classroom. Theycould include such ratios as boys:girls, teacher:pupils, desks:students,tables:students, pencils:students, and m2 of classroom:student.





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A4 demonstrate an understand-ing of equivalent ratios

A4 Students should understand why, forexample, the ratios 2:3 and 4:6 representthe same relationship. They should see thatif for every 2 of one item, there are 3 ofanother, then for 4 of the first item, therewould be 6 of the second.

Many students will recognize the similarity between the concept ofequivalent ratios and the concept of equivalent fractions, e.g., In thediagram above,

2

5 of the counters in the top row are white, but

4

10 of the

counters in total are white, so 2

5 =

4

10 .

• Have students use pattern blocks and geo-blocks to find pairs ofblocks that have equivalent comparisons in size, e.g., Using thepattern blocks, they would find that the ratio of the red blocks to theyellow blocks is the same as the ratio of the green blocks to the blueblocks—they are both 1:2.

Students should use the concept of equivalent ratios to make interpreta-tion of situations easier, e.g., In a large bag of marbles, the ratio of bluemarbles to the total number of marbles is 4:10, e.g., 4 out of every 10marbles are blue. Then to answer, “How many blue marbles would youexpect in 100 selections?”, intuitively students would likely use theequivalent ratio 40:100.

• Have students work in pairs or small groups to discuss equivalentratios if Sue received 36 votes and Sam received 9 votes.36:9 or 4:1 (Sue received 4 votes for every 1 vote Sam received.)9:36 or 1:4 (Sam received 1 vote for every 4 Sue received.)

36:45 or 4:5 (Sue received 4 votes for every 5 votes cast.)

9:45 or 1:5 (Sam received 1 vote for every 5 cast.)

• Ask students to work in pairs writing situations for which classmateswould practise dealing with equivalent ratios.

• Have students work in groups of 4. Provide them with multi-linkcubes in 2 colours. Ask that one student in each group make twotowers, each in a different colour, that would represent a ratio of theirchoice. Then ask the other three students each to make two towers,using different numbers of cubes, that would be in the same ratio asthe towers of the first student in the group. Have a class sharing ofresults with each group explaining the relationship among the towers.

2:3

4:6





PerformanceA4.1 Ask students to use 2-colour counters to set up displays that showthe ratio 5:6. Have them set up other displays to show equivalent ratiosand to explain these equivalencies.

Paper and PencilA4.2 Present students with the diagram below.

x x x o

x x x o

x x x oAsk them, What equivalent ratios does this diagram show?

A4.3 For each of the following, ask students to find an equivalent ratioin which one of the terms is 20.

(a) 4:6 (b) 10:30 (c) 3:5 (d) 4:5 (e) 3:6

A4.4 Ask students to list all the ratios that are equivalent to 1:2 inwhich the second term is less than 50.

InterviewA4.5 Tell students that in a class of 30 students, there are 20 girls. Askthem to explain why the ratio of boys to girls is 1:2 and not 1:3.

A4.6 Ask students to explain how a place-value chart could be used togenerate equivalent ratios.

A4.7 Ask students, Why do you get an equivalent ratio by multiplyingboth terms of a ratio by 3?

A4.8 Ask students, Can the ratio 4:5 be equivalent to any other ratiothat has two terms that differ by only 1? Why or why not?

PresentationA4.9 Tell students that 758 people were surveyed and 248 respondedthat they used Brighto detergent. Ask students, working in pairs, toestimate the ratio that would best describe the number of people whouse Brighto. Have them make up similar situations for their classmatesto solve.





A5 Students should view percent as a special ratio, a ratio for which thesecond term is 100. They should not be computing with percentages atthis time and need not work with percentages greater than 100.

They should explore situations in which percent is commonly used. Thiscould be accomplished by looking for percents in newspapers, maga-zines, sale flyers, and other advertisements.

They should make and interpret diagrams that represent various per-centages, e.g., 2% and 35% represented in hundreds grids.

Students should recognize and express the relationship between thepercent and decimal names of these special ratios, e.g., 48% and 0.48.Using a metre stick as a model for percent is helpful for students tounderstand that 48 cm is 48% or 0.48 or

48

100 of the stick.

Students should know the percent equivalents for common ratios suchas 1:4 (

1

4), 1:2(

1

2), 3:4 (

3

4), and 1:1.

Students should understand that finding a percentage is the same asfinding an equivalent ratio out of 100.

• Have students explore a variety of geographic or social studies dataexpressed in terms of percentages, e.g., About 70% of the earth iswater; about 68% of Canadian households own microwaves; andover 80% of car passengers wear seat belts.

• Have students cut sheets of paper and lengths of string to showpercents such as 50%, 10%, and 25%.

• Have the students predict percentage results, explain their predictionstrategies, and check their predictions, e.g., Ask students to estimatethe percentage of (a) red counters that will be displayed when fifty 2-coloured counters are shaken and spilled; (b) each colour of Bingochip when you show a total of 100 blue, red, and green chips on anoverhead for 10 seconds; and (c) a hundreds grid that is shaded inwhen you display a picture made by shading in squares on such agrid.

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A5 demonstrate an understand-ing of the concept ofpercent as a ratio





PerformanceA5.1 Have the student draw a design in a hundredths grid (or partiallycover a flat) and describe the percentage of the grid covered. Ask furtherquestions such as: How many more squares would you have to shade in(or cover) to cover

1

2 (

1

4, 0.68, 80%) of the squares?

A5.2 Ask the student to shade in hundredths grids to showparticular percentages.

Paper and PencilA5.3 Describe for students a family of five that includes three children.Ask them to indicate the percentage of the family that the childrenrepresent and the percentage that each child represents. Have themdescribe families of different sizes with the same percentage of childrenas the one you have described.

InterviewA5.4 Ask students, Which of the following is least? most? Explain youranswers.

1

2020% 0.020

A5.5 Ask students, What percent of a metre stick is 37 cm?

A5.6 Ask students to name percents that indicate (a) almost all ofsomething, (b) very little of something, and (c) a little less than half ofsomething.

A5.7 Ask students, Why might teachers use percentages to indicatemarks on tests rather than just indicating the number correct? Why is itnot necessary to have 100 marks on a test to use percent?

PortfolioA5.8 Have students create crayon quilts made of patches of variouscolours. Ask them to describe the percentages of each colour within thepatch (as approximations if necessary). Have them also estimate thepercent of the total quilt represented by each colour.

A5.9 Ask students to write letters to a friend/relative/teacher tellingwhat they have learned about ratio and percent.

A5.10 Ask students to use newspapers, flyers, and magazines to collectexamples of situations in which percent is used and have them make acollage for a class display.





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A6 demonstrate an understand-ing of the meaning of anegative integer

Negative numbers are an important setof numbers. They can and should beexplored before they are encountered inalgebra. In fact, students almost everyday either have some interaction withnegative numbers or experience aphenomenon that negative numbers canmodel. (Van de Walle 1994, 411)

A6 Students will have previously encountered negative integers infor-mally, as in dealing with winter temperatures. To build on this informalunderstanding, it might be useful to start with a vertical number linewhich resembles a thermometer.

The main ideas students should understand are:(a) each negative integer is the mirror image of a positive integer withrespect to the 0 mark,(b) 0 is neither positive nor negative, and(c) negative integers are all less than any positive integer.

Students should read -5 as “negative 5.” Saying “minus 5” should bediscouraged because it describes the operation of subtraction. This use ofthe same sign for two different concepts is rather confusing; however,students need to be reminded of the difference and to use the appropri-ate language for each concept.

Other useful contexts for considering negative integers are(a) elevators which go both above and below ground, so floors can be given both positive and negative labels(b) golf scores above and below par(c) money situations involving debits and credits(d) height above and below sea level.

Addition and subtraction situations involving integers should only bedealt with informally, if at all.





PerformanceA6.1 Have students place the numbers -4, -3, and 2 at the appropriateplaces on the number line.

Paper and PencilA6.2 Ask students, if possible, to name all the negative integers that aregreater than -7?

A6.3 Tell students that a number is 12 jumps away from its opposite ona number line. Ask, What is the number?

InterviewA6.4 Ask students to explain why -4 and +4 are closer to each otherthan -5 and +5.

A6.5 Ask students, In what situations might you encounter negativenumbers?

A6.6 Have students explain ways in which -4 is like +4.

A6.7 Ask students, Why can an integer never be 11 away from itsopposite on a number line?

PortfolioA6.8 Ask students to design simple games for which positive andnegative points might be awarded. Have them play and keep track oftheir total scores.





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A7 read and write wholenumbers in a variety offorms

A8 demonstrate an understand-ing of the place-valuesystem

A7 You will need to review reading and writing numbers in the mil-lions. Such numbers should be written out fully, e.g., 345 321 400, readas 345 million, 321 thousand, 400 and should be rounded to usedecimal notation, e.g., 345.3 million.

Students should explore writing numbers using exponential notation,e.g., Write 102 to mean 10 x 10 or 100. Write 103 to mean 10 x 10 x 10or 1000. Therefore, 3422 may be written as 3 x 103 + 4 x 102 + 2 x 10 +2.

Opportunities should be provided for students to record the numericalform of a number that is either spoken or written out in word form.Conversely, students should have the opportunity to both write out andsay the word form of a number expressed in its numerical form. Specialattention should be given to numbers for which the numerical expres-sions include a number of internal zeros, since these tend to cause themost difficulty for students, e.g., nine hundred two million, thirtythousand, three.

A8 Students should understand that the place-value system follows apattern such that

• each position represents 10 times as much as the position to its right• each position represents

1

10 as much as the position to its left

• positions are grouped in 3s for purposes of reading them, both beforeand after the decimal point.

All students should be aware that numbers extend to the left at least intothe billions group and to the right into the ten thousandths, hundredthousandths and millionths places. If students inquire about theseextensions, a discussion is in order.

Although “billions” refers to numbers rarely found in students' experi-ences, they may be interested in investigating numbers of this magnitudeas they relate to national debt, personal fortunes, populations, pieces oftrivia, e.g., How long is a billion millimetres?

Base-10 blocks can be used to model larger numbers and these patterns,e.g., A long rod of ten large cubes representing 10 000 parallels the rodrepresenting 10); visualizing a large flat of 100 large cubes arranged in a10 x 10 rectangle representing 100 000 parallels the flat representing100; visualizing an even larger cube made by stacking 10 large flatsrepresenting 1 000 000 parallels the cubes representing 1 and 1000.





PerformanceA7.1 Ask students to read the following numbers:

105 020 00364 203 006920 000 029

A7.2 Ask students to arrange the cards shown below in at least threeways and to record the numeral for each.

A7.3 Provide students with “pretend” cheques for which the dollaramounts have been listed. Ask them to write out the word form of eachamount.

Paper and PencilA8.1 Ask students to use only the digits 2, 3, and 4 to create threenumbers with values between 42 million and 43 million if each digit canbe used more than once.

A7.4 Ask students to write the number 3 thousand as millions.

InterviewA8.2 Ask students to explain the difference(s) in how the followingnumbers are written:

two thousandthstwo thousandtwenty thousandtwenty thousandths

PortfolioA7/8.1 Have each student prepare a “lesson plan” to teach a grade 5student about a billion. You could have them actually do the teachingand report on their experiences.





A9 Some students may already know the decimal equivalents of some

simple fractions, e.g., 1

2 = 0.5,

1

4 = 0.25,

1

5 = 0.2 and any fraction with

a denominator of 10, 100, or 1000. They may use such relationshipswhen they have to locate points on number lines, e.g., To locate 8.75 ona number line, many students think of 0.75 as being three quarters ofthe way from 8 to 9.

Many students, however, believe that the only fractions which can bedescribed by decimals are those with denominators that are a power of10 such as 10, 100, and 1000.

By building on the connection between fractions and division, studentsshould be able to represent any fraction in decimal form, using thecalculator as an aid, e.g.,

2

3 means 2 wholes shared among 3, so

2

3 = 2 ÷ 3. The calculator display would show 0.6666666. Physically,

this could be modelled as 2 pizzas shared by 3 people.

or 2

3 of one whole

Students should recognize when a decimal repeats, but need not dealwith the symbolism for handling repeating decimals at this time.

Base-10 blocks can be used to explain the decimal equivalents to frac-tions, even when these decimals repeat, e.g., 1 ÷ 3 could be modelled asfollows: The large cube represents one whole. It must be shared by 3people. Trade the cube for 10 tenths—flats. Each of the 3 people gets 3tenths, so the decimal begins with 0.3. Trade the leftover tenth for 10hundredths—rods. Each of the 3 people gets 3 hundredths, so the nextdigit in the decimal is 3, i.e., the decimal begins with 0.33. Continue theprocess with thousandths—cubes. Then students will have to visualizeten thousandths and further subdivisions.

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A9 relate fractional and decimalforms of numbers

To connect the two numeration systems,fractions and decimals, students shouldmake concept-oriented translations fromone system to another . . . The calculatorcan also play a significant role in decimalconcept development. (Van de Walle1994, 262)





Performance

A9.1 Have students use calculators to find decimal forms of 4

6 and

1

6and then subtract these decimals. Ask, How might you have predictedthe difference?

Paper and PencilA9.2 Ask students to explain how they should know that the decimalform of

4

15 cannot begin with 0.6.

A9.3 Have students suggest common fractions that are a bit less than0.4 and to justify their suggestions. Ask, Can you name another fractionthat is between 0.4 and your suggestion?

InterviewA9.4 Ask students to identify how the decimal forms of

1

8 and

1

4 are

related and to explain what this tells about fractions. Have them provideanother pair of fractions with the same relationship.

A9.5 Ask students how the diagram below shows that

0.625 = 5

8 =

1

2 +

1

2 of

1

4

Ask, What fraction/decimal equivalence would be shown if twice asmuch were shaded?

A9.6 Ask students, How does knowing that 1

4 = 0.25 help you find the

decimal form of 3

4? of

5

4?

PortfolioA9.7 Ask students to use calculators to find decimal forms for a groupof fractions and to make as many observations as possible about thedecimals obtained, e.g.,

1

8

2

8

3

8

4

8

5

8

6

8

7

8

8

8.

A9.8 Ask students to respond in writing to the following question:How are fractions and decimals alike and how are they different?





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v) apply number theoryconcepts (e.g., primenumbers, factors) inrelevant situations withrespect to whole numbers,fractions, and decimals

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A10 determine factors andcommon factors

Number theory is the study of relation-ships found among the natural numbers.At the elementary level, number theoryincludes the concepts of prime number,odd and even numbers, and the relatednotions of factor, multiple, and divisibil-ity. (Van de Walle 1994, 404)

A10 Most students should have little difficulty with the concept of onenumber being a factor of another. These can be found by dividing bysmaller numbers and looking for remainders of zero. This conceptextends directly from previous work in multiplication and division. Theconcept of common factors, however, will be new for most students. Itmay be useful to ensure that students understand that “common” is usedin the sense of “joint,” rather than “ordinary.” This is a typical misun-derstanding on the part of students because of the use of the word inother contexts such as skipping is a common activity, chocolate is acommon flavour of ice cream, and jeans are common pants for studentsto wear. You will need to have them consider situations such as whatfriends they have in common with their brother/sister and what traitsthey have in common with their friends.

To introduce the concept of common factors, have students compare thefactors of two numbers, and note any factors which are factors of bothnumbers, e.g., For 16 and 18, they would find the factors of 16 to be1, 2, 4, 8, 16 and the factors of 18 to be 1, 2, 3, 6, 9, 18. Examiningthese lists, students will see that 1 and 2 are the only factors common toboth 16 and 18; therefore, they would say, “1 and 2 are the commonfactors of 16 and 18.”

Students should soon conclude that 1 is always a common factor of anytwo numbers.

Another way to find common factors of a pair of numbers is to begin bycreating a rectangle, using the two numbers as the length and width. Acommon factor is the side length of any square which can be used tocover the rectangle exactly, e.g., For a 20 x 30 rectangle, 2, 5, and 10 arecommon factors since only the following tilings are possible:





PerformanceA10.1 Ask students to draw one or more rectangles to show that 8 is acommon factor of 16 and 24.

A10.2 Provide students with hundreds charts and differently colouredcubes. Have them skip count by 2s, putting a red cube on each number;skip count by 3s, putting a blue cube on each number; skip count by 5s,putting a yellow cube on each number. Ask, What numbers have both ared and blue cube? What does this tell you about these numbers? Whatnumbers have cubes of all three colours? What does this tell you aboutthese numbers?

Paper and PencilA10.3 Have students find numbers that have 4, 7, 28 and 12 as factors.Ask, Is there a smaller number than yours that will meet the conditions?Explain why or why not.

A10.4 Ask students to find numbers with six factors.

A10.5 Ask students, If 3 and 4 are common factors of a pair of num-bers, what are three possibilities for the pair of numbers?

InterviewA10.6 Tell students that the common factors of a particular pair ofnumbers includes 10. Ask them to explain what other common factorsthis pair must have.

A10.7 Ask students, Why is it not possible for a common factor of 38and 90 to be greater than 20?

PortfolioA10.8 Have students design tests they think could be used to determineanother student’s understanding of common factors.





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v) apply number theoryconcepts (e.g., primenumbers, factors) inrelevant situations withrespect to whole numbers,fractions, and decimals

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A11 distinguish between primeand composite numbers

Prime numbers can be viewed asfundamental building blocks of the othernatural numbers. (Van de Walle 1994,405)

A11 A prime number is a number that has only two different wholenumber factors—1 and itself—such as 29 which has factors of 1 and 29.28 which has factors of 1, 2, 4, 7, 14 and 28 would not be prime—it is acomposite number.

Students might find it helpful to think of aprime number as a number for which thatnumber of squares can be arranged in a rectan-gle in only one way, e.g., 7 can only bearranged as a rectangle in one waywhile 8 which is not prime can bearranged as a rectangle in two ways.

• Have the class use up to 36 coloured tiles to explore the differentrectangles that can be made for each number through 36. Pairs ofstudents may be assigned 2 or 3 numbers each, e.g., one pair mightexplore the numbers 21, 22, and 23. Have the numbers (1-36)written horizontally across the front of the room or on the board. Askeach pair to cut out of squared paper all the rectangles that can bemade for each of their numbers and to display them under thenumbers.

Students should recognize that the concept of prime numbers appliesonly to whole numbers. Although students should have strategies fordetermining whether or not a number is prime, it is not essential forthem to be able to quickly recognize whether or not a number is prime.Exceptions would be that students should be able to identify right awaythat all even numbers, except 2, and all numbers ending in 5 or 0, except5, are not primes.Many students do not realize that 1 is not a prime number. There aremany explanations for this, but it is sufficient for students to realize that1 has only one factor and not two different factors.Students should be introduced to the term “composite” to describenumbers with more than two factors. They should be encouraged to uselanguage such as multiple, common multiple, factor, common factor,prime, and composite. As well, they should explore numbers andbecome familiar with their composition.

• Ask students to write about the number 36, describing it using“factor language” in as many ways as they can.





PerformanceA11.1 Ask students to draw diagrams to show why 10 is not a primenumber.

A11.2 Ask students to express even numbers greater than 2 in terms ofsums of prime numbers, e.g.,

4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, ...,48 = 43 + 5, 50 = 47 + 3, ...

A11.3 Have students identify the prime numbers to 100 by exploringthe sieve of Eratosthenes using hundreds charts. Before they start,remind students that 1 is neither prime nor composite—it is in a cat-egory by itself. Begin by having them circle 2 (the first prime number)and cross out every second number (the multiples of 2) which must becomposites. Next, have them circle 3 (the next prime number) and crossout every third number (the multiples of 3), some of which have alreadybeen crossed off. Then ask them to proceed to 5—the next number notcrossed out—circling it and crossing out its multiples. Explain that theyshould continue this process of circling the next number not crossed outand crossing out all its multiples. When they repeat this process untilthey circle the last number on their hundreds harts that is not crossedout, the circled numbers will be the primes up to 100.

Paper and PencilA11.4 Ask students, Are there more prime numbers between 50 and 60than between 60 and 70?

A11.5 Ask students to find three pairs of prime numbers that differ bytwo, e.g., 5 and 7.

InterviewA11.6 Ask students, Why is it easy to know that certain large numbers,e.g., 4 283 495, are not prime without factoring them?

A11.7 Remind students that the numbers 2 and 3 are consecutivenumbers and both are prime numbers. Ask, Why can there be no otherexamples of consecutive prime numbers?

PortfolioA11.8 Have students use computers or calculators to help them deter-mine the prime numbers up to 100. Ask them to prepare a reportdescribing as many features of their list as they can.

Number Concepts/



Number Operations





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i) model problem situationsinvolving whole numbers,decimals and fractions byselecting appropriateoperations and procedures

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B1 compute products of wholenumbers and decimals

B1 Students should be able to compute products of whole numbersusing paper-pencil algorithms; however, they should also know when itis appropriate to use mental procedures or calculators. Students requirepractice estimating products as well—this should be done before anyprocedure is undertaken. It is essential that students have quick recall ofthe multiplication facts.

They should continue to use base-10 blocks and money models to makesense of multiplication algorithms involving decimals. It is not enoughto tell students to multiply, estimate, and decide where to put thedecimal point; they need to see why the procedure works. Base-10blocks are effective to model products of whole numbers and decimals,e.g., If the flat represents one, each rod is a tenth; therefore,

3 x 0.5 = 15 tenths (1.5)

To model 0.2 x 3, students would first have to show 3 which they coulddo by using 3 flats. Then 2-tenths of this set of 3 would be shown bytaking 2 rods from each flat for a total of 6 rods or 0.6. If, on the otherhand, they used a rod to represent one, they would show 3 with 3 rodsand take 2-tenths of each rod—2 small cubes—for a total of 6 cubes or0.6.

One way students might do multiplication as a paper-pencil task is torewrite it, e.g., 5.4 rewritten 54 tenths

x 2 as x 2 108 tenths or 10.8.

Similarly, base-10 blocks can be used to represent the multiplication ofhundredths by a whole number. If the flat represents one, each rodrepresents 0.1 and each small cube represents 0.01.

7 x 0.05 = 7 sets of 5-hundredths





PerformanceB1.1 Ask students to draw or build models to illustrate 4 x 3.453.

B1.2 Have students describe how to calculate 3 x 4.23 using moneymodels.

Paper and PencilB1.3 Ask students to determine how much more five cans of juice costat $1.29 each than six cans at $0.99 each.

B1.4 Have students identify a decimal which when multiplied by 500will produce a result of 200. Have them check using a calculator.

B1.5 Ask students to find the missing digits: 5. 3

x 3 .58

Ask them to “think about their thinking” and be prepared to explainwhat steps they took, and why, in finding the missing digits.

InterviewB1.6 Ask students to respond to the following: Jane said 3.45 x 4 mustbe 1.380. There is only one digit before the decimal place in 3.45, sothere must be one digit before the decimal place in the product.

B1.7 Ask students if the result of multiplying a decimal by a wholenumber can be a whole number.

B1.8 Present the arrangement of blocks shown .Ask, What multiplication is displayed if the flatis assumed to represent one whole?





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B2 model and calculate theproducts of two decimals

B2 Patterns can also be used to help students understand the placementof the decimal in the product of two decimal amounts, e.g., Havestudents use calculators to explore a series of related products such as2 x 4, 0.2 x 4, 2 x 0.4, and 0.2 x 0.4 and discuss any patterns theydiscover.

Base-10 blocks can effectively model the product of two decimals, e.g.,For 0.4 x 0.6, students will find it helpful to describe this as “4-tenths ofa set of 6-tenths.” If they use the flat to represent one, then 6-tenthswould be represented by 6 rods. Taking 4-tenths of these 6 rods wouldinvolve taking 4 small cubes from each rod for a total of 24 cubes. These24 cubes represent 24-hundredths of the flat; so, 0.4 x 0.6 = 0.24.

A picture of this same product could be drawn using a 10 x 10 square ongrid paper. If this square represents one, 6-tenths could be shown by

marking off 6 columns of 10. Then 4-tenths of these 6columns could be shown by shading in 4 small squares ineach of the 6 columns. The result—the 24 shadedsquares—represents the product, 24-hundredths of the10 x 10 square.

It is helpful to think of a product in terms of the two dimensions of arectangle used to find its area, e.g., 2.2 x 5.6 as the width and length of arectangle as shown. 5 0.6

2

0.2

Area = 10(2 x 5)+ 1.2(2 x 0.6)+ 1.0(0.2 x 5)+ 0. 12(0.2 x 0.6) = 12. 32

Rather than providing a rule about “counting decimal places”—a rulewhich students often mix up—it is better if students understand therelationship between whole number and decimal products. Encouragethem to think first about the product if there were no decimals; then totake into account the affect on this product of decimal(s) in the factors.

Students should estimate products before calculating, e.g., Round eachof the decimals in 2.86 x 8.153 for an estimate of 24 (3 x 8). For 0.6 x34.5, think that 6-tenths of 34.5 is a little more than half of 34.5.Estimation also helps students place the decimal point in a solution.





Paper and PencilB2.1 Tell students that two decimals are multiplied. The product is0.48. Ask, What might they have been? Give two other pairs.

B2.2 Ask students to work in pairs sharing strategies for estimating andsolving such calculations as

- cost of 6.15 m of material at $4.95 a metre- the area of a rectangular plot of land 24.78 m x 9.2 m- 0.5 of a length of rope that is 20.6 m long.

Have them write similar questions to share with their classmates.

InterviewB2.3 Ask students, Why is the answer to 0.6 x 0.4 a whole number ofhundredths?

B2.4 Ask students, Is it possible to multiply 2 decimals and get thesame result as if you had multiplied 2 whole numbers?

B2.5 Ask students, When you multiply two decimals, how does theresult compare in size to the numbers you multiplied?

PresentationB2.6 Ask students to find their own heights in metres. Have themresearch some animal sizes and prepare a report using statements such as

An animal that is about 0.1 of my height is ______.An animal that is about 0.2 of my height is ______.

PortfolioB2.7 Present students with the following problem: A multiplication oftwo decimal numbers was modelled using exactly 13 base-10 blocks.What numbers might have been multiplied?





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B3 compute quotients of wholenumbers and decimals

B3 Students should continue to use manipulative materials to modeldivision of a decimal by a whole number, e.g., 3.44 ÷ 3 would bemodelled as shown.

If = 1

A common context in which this calculation would emerge, and towhich the students would relate, is the sharing of money or unit pricing.Other possible contexts are sharing metres of ribbons, or litres of juice,or kilograms of meat.

Students should be expected to estimate quotients, e.g., 4.28 ÷ 3 will bea bit more than 1, but 4.28 ÷ 5 will be close to

4

5 or 0.8. Some students

may think of 4.28 ÷ 5 as 428 hundredths divided by 5, or about 85-hundredths, or as 42.8-tenths ÷ 5 or about 8-tenths.

Students should understand that the “remainder” when they perform thedivision of a decimal number is different than with whole numbers, e.g.,When dividing 3.4 by 3, the remainder “1” at the bottom is really “0.1,”not “1.” They should recognize that for more accuracy, they couldcontinue to find how many hundredths.

Many students will be ready to use “short” division when dividing by asingle digit number with few or no trades needed. For this, they wouldshow no work below since it is being done mentally, e.g.,

3 3 403310

91

1 13..

−

−

4 2 480 62..

or 3 4.261.42





PerformanceB3.1 Have students use base-10 blocks to show how to find 5.28 ÷ 4and draw corresponding pictures.

Paper and PencilB3.2 Tell students that a can of pop holds 0.355 L. Ask them to deter-mine how many such cans they would have to buy if they wanted 5 L ofpop.

InterviewB3.3 Present the display shown at right andask students to explain what division questionis being modelled assuming the flat represents1.

B3.4 Ask students, How are the results of 423 ÷ 3 and 42.3 ÷ 3 related?

B3.5 Ask students, Why is the remainder not 1 when you divide 2.1 by4?

B3.6 Ask students to complete the calculation shown.Observe how they treat the remainder. Ask them to createstory problems that would be solved by this computation.Have them explain what to do with the remainders in theirproblems.

PresentationB3.7 Ask students to use store flyers to find items that are sold in twos,threes, or other groupings, and to provide unit prices for the variousitems. Have them compare these unit prices to prices of the same orsimilar products in other stores’ flyers.

4 34 6

32

2

8..

−





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B4 model and calculate thequotients of two decimals

B4 As with multiplication, students should relate the division of adecimal by a decimal to the corresponding division of a decimal by awhole. They could consider the unit, e.g., If they think of 43.2 ÷ 0.5 as432-tenths ÷ 5-tenths, then determining how many 5-tenths are in 432-tenths becomes simply 432 ÷ 5. If they think of 43.25 ÷ 0.5 as 432.5-tenths ÷ 5-tenths, then determining how many 5-tenths are in 432.5tenths becomes 432.5 ÷ 5.

Through activities considering the unit, students may notice that since43.2 ÷ 0.4 is equivalent to 432 ÷ 4, they could simply change thedivision before doing any calculation, e.g.,

Students might find it productive to use a money model, e.g.,43.2 ÷ 0.4might be interpreted as determining the number of sets of 4 dimes in$43.20. Since 10 sets of 4 dimes (40¢) is $4, 100 sets of 4 dimes is $40.Eight forties would make the extra $3.20. Therefore, 43.2 ÷ 0.4 is 108.

Base-10 blocks could be used to model divisions of relatively smallnumbers, e.g., For 3.8 ÷ 0.4, if the flat is one, 3 flats and 8 rods wouldrepresent 3.8. To show how many sets of 4 rods there are in 3.8, the 3flats would be traded for 38 rods and sets of 4 rods formed. There are 9full sets of 4 rods and half another set (2 rods out of 4); therefore,3.8 ÷ 0.4 = 9.5.

As with all operations, division should be practised in contexts to givethem meaning, e.g., Have students compete in paper airplane races.Each student flies his/her plane 3 times, measuring each distancetravelled in metres. Each student gets a score equal to the average dis-tance. That is, if a student’s flights are 2.43 m, 1.89 m, and 2.24 m, theaverage distance = 6.56 m ÷ 3 = 2.186 m.





PerformanceB4.1 Ask students to use a number line and base-10 blocks to showwhy 4.2 ÷ 0.2 is the same as 42 ÷ 2. Have them explain why thisrelationship is helpful to remember.

B4.2 Ask students to use a metre stick and decimetre/centimetre unitsto explain why 3.4 ÷ 0.2 is the same as 34 ÷ 2.

Paper and PencilB4.3 Ask students which question below has an answer different fromthe others and how they would know without performing the opera-tions.

42.5 ÷ 0.5 425 ÷5 85 ÷ 1 0.425 ÷ 0.05

B4.4 Ask students to fill in the boxes with digits to make a true divisionsentence: 4. ÷ 0. = 14.

B4.5 Have students explain the following division sentence by referringto coins: 2.40 ÷ 0.1 = 24

InterviewB4.6 Ask students to explain how the diagram below illustrates that1.8 ÷ 0.3 = 6.

B4.7 Ask students to explain why someone might find it easier to divide8.8 by 0.2 than 1.1 by 0.3?





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ii) model problem situationsinvolving the addition andsubtraction of simplefractions

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B5 add and subtract simplefractions using models

When area models are used for additionand subtraction, common denominatorsare frequently not involved at all. (Van deWalle 1994, 244)

B5 It is important to continue to provide concrete experiences to helpstudents build an understanding of simple fraction operations usingmaterials such as Fraction Factory, pattern blocks, fraction strips, andfraction circles. The intent of this outcomes is to find solutions toaddition and subtraction questions using models such as these and notto use paper-pencil algorithms, e.g., Using pattern blocks to model4

6 +

1

3 and fraction circles to model

1

4 +

3

6, the displays might be:

Hexagon = 1

Students can find the solutions to these computations by examining therepresentations. For the pattern blocks, they will see clearly that 4triangles (each

1

6) and 1 rhombus (

1

3) make 1 whole hexagon. For the

fraction circles, it is clear that the sum is 3

4 of a circle.

Students generally have no difficulty adding or subtracting fractionswith like denominators when these are shown with materials, e.g., IfFraction Factory pieces are used to model

3

5 +

2

5, it is easy for them to

see that 3 blue pieces and 2 blue pieces are 5 blue pieces. You shouldemphasize that “fifths” is the family name for these fractions and theycan easily add or subtract fractions in the same family. Students familiarwith Fraction Factory pieces can easily make the connection of addingone-half and one-fourth to trading pieces so they have fractions in thesame family, i.e., The orange piece (one-half ) would be traded for 2purple pieces (2-fourths) so it can be easily added to the 1 purple piece.

• Have students find many sums for 1 by covering the hexagon patternblock in as many different ways as possible and recording eachcoverage using the appropriate common fractions, e.g.,

1 = 2

3 +

2

6 (2 blue + 2 green)

= 1

2 +

3

6 (1 red + 3 yellow)

= 4

6 +

1

3 (4 green + 1 blue)





PerformanceB5.1 Ask students to use pattern blocks to model

2

3

1

2+ .

Paper and PencilB5.2 Ask students to state the fraction addition that is modelled on dotpaper, assuming the portion of the paper shown is the whole.

B5.3 Have students explain how thediagram at right shows that

4

12

5

12

3

121+ + =

B5.4 Have students list a group of three fractions that add to 1

2.

InterviewB5.5 Ask students to explain why

2

3

4

5+ has to be greater than 1.

B5.6 Tell students that Jane said that 4

8

2

8

6

16+ = . Ask, Is she right or

wrong? Why?

B5.7 Tell students that you have subtracted two fractions and the resultis less than

1

2. Ask, Can both fractions be greater than

1

2? less than

1

2?

PortfolioB5.8 Have students use triangular grid paper to copy the models theycreate to show the different ways that pattern block hexagons can becovered. Ask them to write the fraction addition sentence that woulddescribe each of their models.





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B7 demonstrate an understand-ing of the function nature ofinput-output situations

B7 To introduce the concept of a func-tion, the “function machine” works well.Show students examples of how inputsare acted upon by functions to producean output. They should understand thatthe machine does the same operation(s)to every input value.

Then challenge them to decide what the function machine does if theyare given a series of inputs and outputs, e.g. If 4→ 12, 6 → 16, and10 → 24, what did the machine do? Answer: it doubled the input andadded 4.

Number tricks provide an enjoyable context for students to practise theidea of function, e.g., Consider the “trick:” Explanation

Choose a number.Add 8. Γ + 8Multiply by 2. 2 ⋅ Γ + 16Subtract 14. 2 ⋅ Γ + 2Divide by 2. Γ + 1Subtract 1. Γ

Students recognize that the result depends on the original numberchosen. After discussing the “trick” and seeing why it works, studentswill enjoy making up their own, thereby practising the notion offunction.

It is recommended that you help students make the transition fromusing “x” to show multiplication, such as 3 x Γ , to using the impliedmultiplication, such as 3Γ , by getting them used to the raised dotsymbol, such as 3 ⋅ Γ . Students have experienced using letters withnumbers in measurement, such as 3 L and 6 m, and using boxes formissing digits in numerical forms of numbers, such as 3Γ 7. They donot associate either of these with multiplication; therefore, the impliedmultiplication will take time for them to associate with this operation.





Paper and PencilB6.1 Tell students that you intend to create number “tricks” in whichyou will perform a sequence of calculations. For each trick, ask studentsto describe the one calculation that could be substituted for the sequenceof calculations.(a) add 5, add 20, subtract 10(b) multiply by 5, multiply by 20, divide by 10(c) add 4, double, add 6, divide by 2

InterviewB6.2 Tell students that you put 25 into a function machine and 77came out. Ask them to state four possible functions that would do this.Then tell them that the output for 30 was 92 and ask if one of their fourfunctions would also do this.

B6.3 Ask pairs of students to perform the following sequence of opera-tions twice and to observe what happens each time. Have them provideexplanations for this “trick.”

Pick a number between 0 and 10Add 7 to the number.Double the new number.Add 11.Subtract 25Divide by 2.

PresentationB6.4 Ask students to prepare a set of five “number tricks”—the trickierthe better. Each trick should be based on doing various computationswith an initial number that the other person will choose. Studentsshould share their tricks to create books of tricks with explanations forthe users.





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iv) apply computational factsand procedures (algorithms)in a wide variety of problemsituations involve wholenumbers and decimals

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B7 solve and create relevantaddition, subtraction,multiplication, and divisionproblems involving wholenumbers

B8 Students should continue to use the four operations to solve math-ematical and real world problems. They should also have the opportu-nity to create problems for others to solve.

Students should also be encouraged to estimate answers to test forreasonableness and whenever the calculation can be done mentally,children should do so.

There are many interesting sources of data, both on the Internet and inreference books. Some of the most useful print resources include theCanadian Global Almanac, the Guiness Book of World Records and the TopTen of Everything.

Internet searches can be done for data relating to any topic of studentinterest, whether sports, populations, or food.

You will find it helpful to refer to Van de Walle 1994 or 1997 for thechapter “Developing Meanings for the Operations” as a reminder of thevarious structures of problems for the four operations.

You should present students with problems that require many computa-tional steps and ask them to create such problems, e.g., You went to thestore to buy 3 books that cost $4.50 and a scribbler that costs $2.98. Ifthe store has all taxes included in their prices, how much change willyou get from a $20 bill?





Paper and PencilB7.1 Provide students with appropriate data and ask them to determinehow much farther away Jupiter is from Earth than from Mars. Ask thatthey express this as a ratio as well as an actual distance.B7.2 Ask students to determine all possible sums using any combina-tion of the numbers 389, 243, 301, 332 and 91.

InterviewB7.3 Ask students, When could you multiply to find the perimeter of ashape?

PresentationB7/B8.1 Have students pretend to work in a post office. Explain thatone of the rules states that the total length, height, and width of apackage has to be less than 100 cm to mail it. Provide them with severalpackages for them to determine whether these packages will be mailed.In addition, ask them to create a list of package sizes that would justmake the requirement.

PortfolioB7.4 Have students plan a trip with various stops. Explain to them thatthey should use distance charts or maps, find the distances between thestops, and determine the length of the entire round trip. Alternatively,you might ask them to plan a trip of a specified length such as between1200 and 1500 km. You could extend this task to include the calcula-tion of the total cost of the trip.

B7/B8.2 Ask students to create a variety of problems that involve tasksthat would actually not likely be done, e.g., How many toothbrushes arerequired to make a line that is 2 km long? How many pennies must belined up to make a kilometre?

B7.5 Have students create problems based on information provided,e.g., Urban/Rural Population 1991

RURAL URBAN

Canada 6 389 724 20 909 135NFLD 264 023 304 451NS 418 434 481 508PEI 77 952 51 813NB 378 686 345 214





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iv) apply computational factsand procedures (algorithms)in a wide variety of problemsituations involve wholenumbers and decimals

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B8 solve and create relevantaddition, subtraction,multiplication and divisionproblems involving deci-mals

B9 Provide problems involving decimals using contexts such as moneyand measurement.

Demographic data can be a good source of material for problem solvingand creation. Provide students with information about the populationsof various places in Canada. Have them create authentic problems basedon data provided in decimal form, e.g., How many people in BC areliving in urban areas? How many more people live in Edmonton thanSaskatoon? How many more people live in Halifax than Saint John?How many people/km2 in Canada’s largest cities? How would you graphthe populations of Canada’s capital cities?

Some problems can be strictly mathematical challenges, e.g. Ask stu-dents to find as many combinations of digits as they can to fill in thetemplate below so that the sum or differenceis 10.0.

Pieces of children’s literature often provide interesting contexts forproblems, e.g., Code Red at the Supermall by Eric Wilson and Countingon Frank by Rod Clement.

You should provide students with multi-step problems and ask them tocreate such problems.





PerformanceB8.1 Have students draw quadrilaterals with perimeters of 16.3 cm.

Paper and PencilB8.2 Ask students to create measurement problems that involve theaddition and/or subtraction of hundredths of metres.

B8.3 Ask students to create addition, subtraction, multiplication, anddivision word problems that will all have a solution of 4.2.

B8.4 Provide students with word problems that will require two ormore steps and two or more different operations to solve, e.g., Jason has24.5 m of ribbon. He uses it to make six 2.75 m bows for the largeChristmas wreaths. How much ribbon will he have left?

InterviewB8.5 Tell students that you have bought 1.362 kg of cheese and 0.485kg of ham. Ask them to estimate how many ham and cheese sandwichescould be prepared with these amounts.

B8.6 Ask students to create story problems that would be solved byadding the result of 12.5 ÷ 0.25 to 6.

PortfolioB7/8.1 Have students each create a “Problem Sheet of the Week” basedon a predetermined topic or holiday. Select one or more to assign to theclass for homework. You may wish to have them publish a class problembooklet.

B8.7 Have students create problems involving given data, e.g.,

Population by Official Language 1991 (%)

English French Bilingual

NF 96.5 0.04 3.3NS 91.1 0.02 8.6PE 89.6 0.2 10.1NB 57.9 12.5 29.5





B9 Estimation should precede all calculations so results can be testedfor reasonableness.

When considering multiplication by a decimal, students should recog-nize that, for example, 0.9 of something will be almost that amount, butnot quite, and 2.4 multiplied by an amount will be double the amountwith almost another half of it added on.It is important for students to realize estimation is a useful skill in theirlives; therefore, regular emphasis on real-life contexts should be pro-vided. On-going practice in computational estimation is a key to devel-oping understanding of number and number operations and increasingmental process skills. Although rounding is often the only estimationstrategy used, there are others, many of which provide more accurateanswers. These other strategies should be part of a student's repertoire.

Rounding in Multiplication 688 x 79 is easily rounded to 700 x 80 to geta good estimate of 56 000. Consider, however, 653 x 45. If these wererounded according to a “rounding rule,” the estimate of 35 000 wouldnot be close to the actual answer of 29 385. Multiplying 600 by 50would give a closer estimate. It is important that students explorepossible rounding combinations with their calculators and discuss thereasons for the variances.

Rounding in Division For 789.6 ÷ 89, think, “90 multiplied by whatnumber would give an answer close to 800?”

Front-end Multiplication 6.1 x 23.4 might be thought of as 6 x 20 (120)plus 6 x 3 (18) and a little more for an estimate of 140 or might bethought of as 6 x 25 for an estimate of 150.Front-end Division Pencil-paper division involves front-end estimation.The first step is to determine in which place value the first digit of thequotient belongs. For 8)424.53, the first digit is a 5 and is in the tensplace. The front-end estimate is therefore 5 tens or 50.

• Have students estimate each of the following and tell which of theirestimates is closer and how they know: 9.7 kg of beef at $4.59/kg and4.38 kg of fish at $12.59/kg.

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B9 estimate products andquotients involving wholenumbers only, wholenumbers and decimals, anddecimals only

Number sense is a critical component ofour students’ education. Encouragingstudents to estimate and check answers asan integral part of any numericalexercise, discussing common measure-ment situations with them, and askingthem to justify their mathematicalchoices will help students develop thiscrucial ability. (NCTM 1992c, 10)





Paper and PencilB9.1 Have students compute the approximate number of hours in10 000 seconds and 100 000 seconds.

B9.2 Tell students that you have estimated the product of a decimal anda whole number to be 5.5. Ask, What might be the numbers that weremultiplied?

B9.3 Provide students with supermarket check-out slips on which thetotal has been removed. Have students estimate the total amounts.

B9.4 Tell students that it takes about 0.08 kg of beef to make a ham-burger patty. Sue checks the label on the package and finds she has 2.456kg of beef. Ask, About how many patties can she make?

InterviewB9.5 Tell students that the product of two numbers is about 40 000.Ask, What might the numbers be?

B9.6 Ask students, Which of the following would give the best estimatefor 37 x 94? Why?

30 x 90 40 x 100 40 x 95 40 x 90

B9.7 Ask students why someone might estimate 516 x 0.48 by takinghalf of 500.

B9.8 Tell students that the cashier told Samantha that her total for 3 kgof grapes at $3.39/kg was $11.97. Ask how Samantha knew right awaythe cashier made a mistake.

B9.9 Tell students that Sandy’s marks for English, Math, Science, andFrench were all about the same. When she added them up, she had atotal of 319. Ask students to estimate Sandy’s average mark.

B9.10 Ask students for an estimate of the cost of 25 pens which are$0.79 each. Ask what estimating strategies they used and if there areother ways to estimate the answer.





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vi) select and use appropriatecomputational techniques(including mental, paper-and-pencil and technologi-cal) in given situations

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B10 divide numbers by 0.1,0.01, and 0.001 mentally

In fact, there are relatively few workabledivisions that can be done mentallycompared with the other three opera-tions . . . That does not mean thatdivision is less important as a mentalcomputation skill. However, mentaldivision is more of a tool for estimation.(Van de Walle 1994, 209)

B10 Students have mentally multiplied and divided numbers by 10,100, and 1000, and multiplied numbers mentally by 0.1, 0.01, and0.001. Students should add division by 0.1, 0.01, and 0.001 to theirmental repertoires.

Since students generally expect the division process to result in a quo-tient which is smaller than the dividend, it is important that studentsunderstand why that is not the case here. One way to illustrate this is byreminding them of the concept, e.g., Students will understand that oneway to illustrate 12 ÷ 3 is to consider how many groups of 3 there are in12. Obviously, there are 4. Similarly, to illustrate 2.6 ÷ 0.1, considerhow many one-tenths there are in 2.6. Clearly, there are 10 in each unitand another 6 in 0.6, for a total of 26 groups of one-tenth.

Contexts often lead naturally to multiplying by 10 when diving by 0.01,e.g., If students are asked how many one-tenth metre pieces of ribbonthey can cut from 2 m, they think, “That would be 10 pieces in 1 m or20 pieces in 2 m.”

Ultimately, students should see that dividing by 0.1 (one-tenth) increasesthe number of parts—the answer—by a factor of 10. Similarly, theyshould understand that dividing by 0.01one-hundredth results inincreasing the answer by a factor of 100, and dividing by 0.001 increasesthe answer by a factor of 1000.

Students should be able to describe these changes interms of place value, e.g., They should be able to explainthat, when dividing by 0.01, each hundredth becomes aunit, each tenth becomes a ten, each unit becomes ahundred, each ten becomes a thousand, and so on. Thiscan readily be illustrated using a place -value chart.

Include dividing by 0.1, 0.01 and 0.001 as part of regular mental mathactivities. As students become comfortable with questions of this type,include a mix of questions involving multiplication and division by1000, 100, 10, 0.1, 0.01, and 0.001.





Performance

B10.1 Have students divide 0.0034 by 0.1. Repeat the division by 0.1three more times. Have students tell whether the result is greater than orless than 1. Ask, Will repeatedly dividing by 0.1 always lead to a numbergreater than 1? Why or why not?

Paper and Pencil

B10.2 Present students with the following:4[]6.[] ÷ 0.1 = []5.[]3 ÷ 0.01

Ask, What digits belong in the boxes?

B10.3 Ask students, Which answer will have a 3 in the tens place?

42.345 ÷ 0.1 42.345 ÷ 0.01 42.345 ÷ 0.001

Interview

B10.4 Ask students, What digit would be in the tens place after divid-ing 453.2 by 0.01? Why?

B10.5 Tell students that you divided a decimal number by 0.001 andthe answer is a decimal number also. Ask, What do you know about theoriginal decimal number?





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v) select and use appropriatecomputational techniques(including mental, paper-and-pencil and technologi-cal) in given situations

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B11 calculate sums anddifferences in relevantcontexts by using the mostappropriate method

B11 Students should recognize the need for different approaches tocomputation depending on the problem situation. Estimation must beused with all computations, but when an exact answer is required,students need to decide whether it is more appropriate to use a mentalstrategy, a pencil-and-paper calculation, or some form of technology,most often the calculator.

Students should practise mental math strategies on an on-going basis,not as a self-contained unit of work. You should devote 3–5 minutes aday to some form of mental math or estimation activity. One objective isfor them to use it in their daily lives, not just in math class. It is alsoimportant to point out that students benefit immeasurably from explor-ing number and number patterns while developing mental math strate-gies.

Students should explore strategies in problem-solving contexts, e.g.,They should solve the following problem using mental strategies:

Mason bought 32.5 m of fencing that was on sale for $3 ametre. How much change did he receive from a $100 bill?

Students should perform mental computations with facility usingstrategies that were outcomes in previous grades, e.g.,Front-end24 345 Beginning at the front, or ten thousand place in this 3 214 case, this sum might be read thirty-seven thousand10 116 (24 + 10 + 3 ), six hundred (3+2+1), sixty (4+1+1), no,

that’s seventy (6 + 4 is another 10), five.

(a) 28 164 (b) 15 347 (a) can easily be solved with the front-end -12 052 -9 579 strategy. A quick glance at the digits in (b)

would lead to considering paper-pencilinstead of mental math.

CompensationFor $25.95 + 3.99 + 12.98, it is helpful to think of $26 + $4 + $12 andto subtract $0.08; $42 subtract $0.08 is $41.92.For the following subtraction, students might use a combination offront-end and compensation:

7683 24 hundred, forty, less 6 (9 - 3) is 24 hundred thirty- - 5249 four or 2434.





InterviewB11.1 Ask students to describe ways to compute 3000 - 2898 mentally.

B11.2 Ask students to explain how to solve each of the following usingmental strategies:

$75 + $12.25 + $5.75 = 470 + 1068 + 30 =

2435.7 4579.25 304.1 -2134.141050.2

B11.3 Remind students that pairs of numbers such as 75 and 25, 45and 55, 340 and 660 are called “compatible” numbers. Ask them to statethe number that would be compatible with each of the following:

40 49 700 21 7 490 880 910

B11.4 Provide students with a set of computations such as the onebelow. Ask them which ones they could do mentally and which onesthey would use a paper-and-pencil method to solve. Have them find thesolutions.(a) 2 x 22.3 (b) 3.1 x $2.38 (c) 100 - 12(d) $4.63 + 11.2 (e) 24.8 x 0.5 (f ) $126.48 - $14.20(g) 1097.2 - 39.8 (h) $1.99 + $2.99 + $5.98 + $0.99

PortfolioB11.5 Have students make up practice sheets of a range of computa-tion questions that would employ the mental strategies discussed inclass. You could use some of these student-generated sheets in futureclasses.





B12 Students should know multiplication and division concepts andhave mastered the multiplication facts and mental strategies for multi-plying and dividing whole and decimal numbers by multiples of ten.

Front-end Multiplication3 x 325.15 Using the front-end strategy, one would say, 9

hundred (3 x 300) seventy-five (3 x 25) and45 (3 x 15) hundredths.

Students very often proceed to find solutions using a paper/pencilalgorithm without first checking to see if a mental strategy could beemployed. It is important that there is a mix of mental and paper-pencilopportunities when problem situations are presented.Front-end DivisionUnlike other operations, we traditionally proceed from left to right indivision pencil-paper algorithms. Students should be able to decide byexamining the division question if it can be found quickly by a mentalfront-end approach or if a paper-pencil algorithm would be necessary.Even then they should use a“short” method rather than always dealingwith each place value separately. To provide practice, give students a mixof division questions, e.g.,

3)120.96 5)176.28 12)2400 4)248.04

CompensationIn multiplication, students should be able to recognize opportunities touse this strategy, e.g., For 9 x $4.95, 9 x $5.00, or $45, less $0.45 (9 x 5)gives the total of $44.55.

In division, students should recognize that dividing by 5 can often beeasier if the dividend is doubled and they divide by 10, e.g., 1632 ÷ 5 isthe same as 3264 ÷ 10 which is easier to do mentally.

Rearranging FactorsIf students are to multiply 25 x 16, 25 x 12, or 25 x 44, where thesecond number has 4 as a factor, they should rearrange the factors to get100 x 4, 100 x 3, or 100 x 11 all of which are easier to do mentally. Thisis also helpful in situations such as 22 x 15 where 2 is a factor of 22 and15 x 2 makes 30—a “nice” number—so 11 x 30 is easier to do mentally.

In problem-solving situations involving numbers of many place valuesor repetitions of operations that would get tedious, students should usecalculators. These should not be used, however, for operations that arepaper-pencil and mental outcomes for the current or previous grades.

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vi) select and use appropriatecomputational techniques(including mental, paper-and-pencil and technologi-cal) in given situations

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B12 calculate products andquotients in relevantcontexts by using the mostappropriate method





Paper and PencilB12.1 Have students give written explanations for how and why thedouble/halve strategy works.

InterviewB12.2 Ask students how to calculate 14 000 ÷ 50 mentally.

B12.3 Have students provide two numbers greater than 100 that wouldbe easy to multiply mentally and to explain their choices.

B12.4 Provide students with a list of computation questions and askwhich ones they could easily compute mentally and which ones theywould choose to calculate using paper-and-pencil algorithms. Havethem give an estimate for each one they do not do mentally.

2 x 315.2 35 x 40 99 x 85 47 x 58

B12.5 Ask students how to use a calculator to help find the solution to999 999 x 343 343.

B12.6 Have students provide two numbers greater than 100 that wouldbe easy to divide mentally. Ask them to defend their choices.

B12.7 Explain to students that you need to know approximately howmany 750 mL bottles of pop to buy to be sure of having a total of 50 L.Ask them to help you find out and to explain their methods.

PresentationB12.8 Ask pairs of student to look for a pattern when squaring 2-digitnumbers ending in 5, e.g., 552 is 3025, 452 is 1225. Have them describethe pattern, apply it to 952 , and to explain why it works.

Patterns and

Relations





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C1 solve problems involvingpatterns

... Students can explore patterns thatinvolve a progression from step to step.In technical terms these are called“sequences”. We will simply call them“growing patterns”. With these patterns,students not only extend patterns butlook for a generalization or algebraicrelationship that will tell them what thepattern will be at any point along theway. (Van de Walle 194, 376)

C1 Students should continue to use patterns to help them solve math-ematical problems, e.g., Ask the student to use a calculator to computethe following set of sums:

1

21

21

21

2

+ 1

4

+ 1

4 +

1

8

+ 1

4 +

1

8 +

1

16

What do you think + 1

4 +

1

8 +

1

16 +

1

32 +

1

64 +

1

128 will be?

• Have students con-tinue this pattern tofind the next threehexagonal numbers.

• Ask students to find the number of factors of 10, of 20, of 40, and of80. Ask, Can you predict the number of factors of 640?

• Explain to students that two grade 6 students started an environmen-tal club. They agreed to each get a new member every month. Eachnew member would recruit a new member by the end of their firstmonth, and every month thereafter. How many members will therebe at the end of one year?

Good sources of number patterns are tables of function values. Studentslook for patterns to predict other values in the table, e.g., Present stu-dents with the table below and ask them to fill in the missing values.

Input Output 0 4 1 7 2 10 3 Γ 4 Γ Γ 25

6 15 28





PerformanceC1.1 Have students explore the potential difficulty in using a calculatorto multiply very large numbers where the product will be greater thanthe display capability, e.g., 999 999 999 x 999 999 999. For productssuch as this, students might look for a pattern and use the pattern tofind the large product, e.g.,

99 x 99 =999 x 999 =

9999 x 9999 =

Paper and PencilC1.2 Have pairs of students calculate the following products and lookfor a pattern: 38 x 32

36 x 3437 x 33

Ask, What do you notice? Predict what 58 x 52 will be. Explain yourprediction and verify it.

C1.3 Ask students to find the sum of the first 30 even numbers withoutactually listing them all and adding all 30 numbers.

C1.4 Ask students to find the sum of the first two odd numbers, thesum of the first three odd numbers, the sum of the first four odd num-bers, and the sum of the first five odd numbers. Have them examinethese sums to try to detect a pattern. Ask them to predict the sum of thefirst eight odd numbers and to verify the prediction. Ask, What is thesum of the first 12 odd numbers?





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C2 use patterns to exploredivision by 0.1, 0.01, and0.001

C2 Students should have had experiences mentally multiplying anddividing numbers by 10, 100 and 1000; mentally multiplying numbersby 0.1, 0.01, 0.001; and examining the patterns connected with theseoperations. This C2 outcome should be connected to SCO, B10 thatexpects students to mentally divide numbers by 0.1, 0.01, and 0.001.Students should recognize that the pattern of changes produced bydividing by 0.1, 0.01, and 0.001 is the same as the pattern of changesproduced by multiplying by 10, 100, and 1000, e.g.,

4.71 × 10 = 47.1 → 4.71 ÷ 0.1 = 47.1

4.71 × 100 = 471 → 4.71 ÷ 0.01 = 471

4.71 × 1000 = 4710→ 4.71 ÷ 0.001 = 4710

You could provide students with similar questions and ask them to usecalculators to get the answers. Encourage them to look for a pattern inthe solutions. Hopefully, it will be the students who conclude thatdividing by tenths is the same as multiplying by 10.

It is important that students be able to explain these patterns withrespect to place-value changes, not just in terms of a rule involvingmoving the decimal point.

Contexts that focus on the conceptual understanding of division bydecimals help many students understand the connection between thetwo operations, e.g., If 5.6 kg of gold is to be put in packages of0.001 kg, how many such packages will there be? Most students wouldthink, “If there is 0.001 kg in each package, there would be 1000packages made from 1 kg. Since there are 5.6 kg, then there would be5.6 x 1000 or 5600 packages.” In fact, many students do not realize thatthis is a division story problem. Therefore, you will have to help themassociate stories such as this one with the concept of division by adecimal.





InterviewC2.1 Tell students that Frank divided 42.8 by 0.1 and got an answerbetween three and four hundred. Ask the students to explain how theyknow Frank’s answer must be incorrect.

C2.2 Ask students to explain why dividing a number by 0.01 results ina greater number than he/she originally started with.

C2.3 Ask students to predict (in terms of change in place value) whatthe effect would be of dividing by 0.00001.





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ii) explore how a change in onequantity in a relationshipaffects another

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C3 recognize and explain howchanges in base or heightwill affect areas of rectan-gles, parallelograms, ortriangles

C4 recognize and explain howan increase in height, width,or length of a rectangularprism changes its volume

C5 recognize and explain howthe change in one term of aratio affects the other term

C3 As students develop the formulas for areas of rectangles, parallelo-grams, and triangles, they should understand the contributions of themeasures of lengths, widths, and heights to the measures of areas, e.g.,With the formula for area of a parallelogram (A = bh), students shouldunderstand a variety of relationships such as: if b is doubled, then so isA; if b and h are doubled, then A is quadrupled; if b is doubled but h ishalved, A remains the same. Students should explore these relationshipswith actual paralllelograms, e.g.,

C4 Students should explore the volumes of a set of rectangular prismsthat have their lengths, widths, and heights related in different ways.They should investigate the affect on the volume if one, or more,dimensions are multiplied by a specific factor, e.g., the length is dou-bled, the width is tripled, or the height is halved.

C5 Just as with equivalent fractions, students should be aware that asone term of a ratio is multiplied or divided by a particular quantity, so isthe other. Through explorations of a variety of ratios, students shouldbecome aware that, when a particular quantity is added to or subtractedfrom each term of a ratio, the resulting ratio is not equivalent—believingthat they are equivalent is a common student misconception.





PerformanceC3.1 Provide students with multilink cubes. Have them constructrectangular prisms with the dimensions: 3 x 5 x 2 and 6 x 5 x 2. Askthem to find the volume of each prism and the ratio of these volumes.Ask, How could you have anticipated that the volume of the largerprism would be twice the volume of the smaller? How do you think thevolume of a 6 x 5 x 4 prism would compare to the 3 x 5 x 2 prism?

C3.1 Ask students to draw three different triangles on grid paper, all ofwhich have a base of 4 units and an area of 8 square units. Have themchoose one of these triangles and change its base so that its area will be16 square units. Ask them to explain how the height, rather than thebase, of the original triangle could be changed to make an area of 16square units.

Paper and PencilC3.2 Ask students each to draw on grid paper a 2 cm x 2 cm square.Then ask them to draw rectangles with areas exactly three times the areaof the square and to mark the dimensions of these rectangles. Finally, askthem to compare the dimensions of their rectangles to those of thesquare and explain how they could have predicted the area relationship.

C5.1 Tell students that you have two ratios that are equivalent with thefirst term of one ratio being 10 and the first term of the other ratio being25. Ask them to explain how the second terms of the ratios are relatedand to suggest some possibilities for these second terms.

C3.3 Tell students that one parallelogram has the same height asanother and its base is three times as long. Ask, How are the areas ofthese two parallelograms related? Why?

InterviewC5.2 Ask students to explain why halving one term of a ratio necessi-tates halving the other term if the ratio is to be preserved.

PortfolioC4.2 Ask students to design a number of different rectangular boxes forfudge. Explain that each box must have a volume of 1200 cm3. Havethem record their designs by making isometric or orthographic drawingsand marking the dimensions on the drawings. Have them write reportson their favourite designs providing reasons for their choices.





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iii) represent mathematicalpatterns and relationships ina variety of ways (includingrules, tables and one- andtwo-dimensional graphs)

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C6 represent equivalent ratiosusing tables and graphs

C5 By representing equivalent ratios in tables and graphs, students willclearly see both the relationship between the two elements of the ratioand the relationship between the two equivalent pairs, e.g., The ratio 2:3is equivalent to:

This can be graphedas shown:

Student can see from the table that, in each ratio, the first term is 2

3 of

the second term. Also, for each equivalent ratio, the same factor is usedto multiply both 2 and 3 to create the new ratio. Students could alsoapproximate from the graph that a ratio such as 6.66:10 is also equiva-lent to 2:3.

• Have students investigate ratios using pattern blocks. Ask them toplace two different blocks beside each other and to state the ratio thatwould describe how they compare, e.g., If they used a trapezoid and atriangle, the ratio would be 3:1. Have them triple the number of eachblock, placing the blocks of the same type together. Ask them if thetwo regions covered by these blocks are still in the same ratio, e.g., Ifthey look at the region covered by three trapezoids compared to theregion covered by three triangles—which make a trapezoid—the ratiowould still be 3:1. The comparison could be made by comparing thenumbers of triangles needed to cover both regions—this would be9:3. Have them now place one block of the same type onto the tworegions and ask if the two regions covered by the blocks are still inthe same ratio, e.g., If they add a triangle to both the regions coveredby three trapezoids and three triangles and compare them, the ratio isnot 3:1—it is 10:4 if triangles needed to cover both regions arecompared or 5:2 if rhombuses needed to cover both regions are used.





PerformanceC6.1 Have students draw graphs to show the equivalents of the ratio4:5.

Paper and PencilC6.2 Provide students with the table below. Ask them to state thelowest term ratio of seniors to children under 4 and to fill in the missingterms.

InterviewC6.3 Tell students that a certain ratio is listed as 11:32. Ask, Whatsimpler ratio would be a good estimate for this? How do you know?





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iii) represent mathematicalpatterns and relationships ina variety of ways (includingrules, tables and one- andtwo-dimensional graphs)

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C7 represent square andtriangular numbersconcretely, pictorially,and symbolically

C7 Students may have had informal exposure to square and triangularnumbers. These special numbers have both geometric and numericalsignificance.

Have students make squares using square or colour tiles and ask themhow many tiles it took to make each square. From this investigationstudents should get examples of square numbers such as

x x x x x x x x x x xx x x x x x x x x

x x x x x x x x x x1 4 9 16

Square numbers are numbers that can be shown in square arrays. Theyare the products of numbers multiplied by themselves, e.g., 4 = 2 x 2.

Similarly, provide students with counters and show them the number 3made in triangular form using 3 counters. Challenge them to find othernumbers that would form triangles with counters. They should findexamples of triangular numbers such as

x x xx x x x x x

x x x x x xx x x x

3 6 10

1 is also a triangular number. You will probably have to tell students this!

If students observe that each triangular number is half the number in anarray with dimensions that are one unit apart, it becomes easier for themto determine the numbers, e.g., 6, the third triangular number, is halfthe number in a 3 x 4 array.

x - - -x x - -x x x -

Likewise 10, the fourth triangular number, is half the number in a4 x 5 array. x - - - -

x x - - - x x x - - x x x x -

Students should also notice that triangular numbers are the result ofadding the consecutive numbers beginning at 1, e.g., 6 = 1 + 2 + 3.





PerformanceC7.1 Ask students to draw pictures of the 8th triangular number andstate what it is.

C7.2 Ask students to draw a number pattern where the numbers mightbe called “doublesquares.” Ask them how to test whether a number isdoublesquare.

Paper and PencilC7.3 Ask students to investigate whether two triangular numbers addedtogether could ever make a square number. Also, have them investigatewhether two square numbers could be added together to make a triangu-lar number.

C7.4 Ask students to investigate whether the sum of two square num-bers is ever a square number and whether the sum of two triangularnumbers is ever a triangular number.

InterviewC7.5 Tell students that Sara said 100 is not a square number because ifyou draw a 25 x 4 array, it’s not square. Ask, Do you agree with Sara?Explain.

C7.6 Ask students, Why is 8 not a square number?

C7.7 Ask students whether 144 is a square number and to give reasonsfor their responses.

PortfolioC7.7 Ask students whether they think there are hexagonal numbers andto investigate which numbers they think they would be. Have themresearch these numbers including interviewing older students, math-ematics teachers, and others.





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iii) solve linear equations usinginformal, non-algebraicmethods

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C8 solve simple linearequations using open frames

C9 demonstrate understandingof the use of letters toreplace open frames

C8 Students should be able to solve simple linear open-frame sentences.The formal use of letters to represent variables should be introducedonly after students have had success using boxes, triangles, and otheropen frames.

Introduce these sentences by having them represent various contexts,e.g., Suppose in a class of 23, 8 students are working independently andthe others are in groups of 3. How many groups? An equation thatwould represent this situation is 3 ⋅ Γ + 8 = 23. The number of groups,5, is the missing number.

Model the language used for the meaning of equations. The aboveexample might be read, “Three times a number and 8 more is 23.” Havestudents read equations for meaning and write equations when given themeanings.

C9 Students can be shown the parallel between sentences such as5 + Γ = 8 and5 + n = 8

They must realize that the particular letter used is irrelevant.

Students should understand that this use of a letter is simply a conven-tion and no more or less meaningful than the use of the open frame.They must also be taught that we often do not write the multiplicationsign when a letter is used, e.g., instead of writing 3 x n, we write 3n.Many students misinterpret 3n to mean a number in the thirties, so it isvery important that this convention be made clear. See SCO, B6.

There are many patterns students could investigate that would lead tothe use of variables, e.g. How many people can be seated at “n” tables?

... n tables

4 people 6 people 8 people 2n + 2 at 1 table at 2 tables at 3 tables people at

n tables





Paper and PencilC8.1 Provide a few simple linear equations using open frames forstudents to solve, e.g., Γ + 7 = 10, 3 x Γ = 21, 2 x Γ + 3 = 16.

C8.2 Have students create three different open sentences for which thesolution is 5.

C9.1 Ask students to write open sentences using letters or figures foreach of the following:(a) twelve less than a number is twenty(b) the quotient of a number and five is two and five-tenths(c) the product of seven and a number is seventeen and a half(d) the difference between a number and ten is greater than twoHave them find solutions to their open sentences.

InterviewC8.3 Tell students that Γ represents one digit. Ask, Why would youknow from examining 2 ⋅ Γ +8 = 35 and 2 ⋅ Γ + 4 = 31 that theirsolutions must be the same?

C9.2 Remind students that open frame expressions can be translatedinto English phrases, e.g., 3 ⋅ Γ might represent “three times as much.”Ask them to state phrases that could be associated with each of thefollowing: (a) 5 + n

(b) 3 - n(c) 2 x n(d) n ÷ 2

C9.3 Ask students which would have the larger value, “n” or “y,” and toexplain their choice.

2 ⋅ n + 16 = 32 2 ⋅ y + 16 = 36





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ii) communicate using standardunits, understand the relation-ship among commonly usedSI units (e.g., mm, cm, m,and km) and select appropri-ate units in given situations

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D1 use the relationships amongparticular SI units tocompare objects

D2 describe mass measurementsin tonnes

D1 Students should know some of the relationships between metricunits from direct experiences and from visualization of the actual units,e.g., From visualizing the large cube in the base-10 blocks, they wouldknow that 1000 millilitres is equivalent to 1 litre or that 1mL = 0.001 L;from visualizing a metre stick, they would know that 100 cm = 1 m orthat 1 cm = 0.01 m; from visualizing the length of a rod in the base-10blocks, they would know that 10 cm = 1 dm or that 1 cm = 0.01 dm.With activities involving such associations, students should review themeaning of the metric prefixes, i.e.,

milli (1

10of ) centi (

1

100of )

deci (1

1000of ) kilo (1000 of )

Using these relationships, students should learn to compare amounts,e.g., - Which is greater, 3.45 L or 345 mL?

- How many milligrams make a kilogram?- How many metres is 45.2 cm?

Students should realize that the relationships between linear SI units isnot the same as the relationships between corresponding SI area andvolume units, e.g., While 100 cm = 1 m, 100 cm2 ≠ 1 m2 and 100 cm3 ≠1 m3. Believing that these relationships are the same is a commonstudent misconception. You should plan investigations that would verifywhat the actual relationships are, e.g., Have students make 1 m2 on thefloor using masking tape. Ask them to fill the space with base-10 blockflats, each of which has a top face area of 100 cm2. They should see that10 000 cm2= 1 m2.

D2 Students should be introduced to the “tonne” as being equivalent to1000 kg and associated with objects of large mass. You will probablyhave to help students distinguish this tonne from the ton that is used inthe United States to represent 2000 pounds. Help them develop aconcept of 1 tonne by exploring how many common objects, such astext books or students, it would take to make 1 tonne. Have themresearch the masses of familiar objects that would have tonne measures,such as cars and large animals.

Students should be aware of the types of items which commonly wouldhave masses measured in tonnes. They should relate the tonne to othermass units, e.g., 456 kg = 0.456 tonnes, and solve problems involvingtonnes.

InterviewThe purpose of an interview is touncover how students think aboutmathematics, so provide opportunitiesfor contradictions in students’ beliefsabout mathematical concepts to emerge.(NCTM 1991, 29)





PerformanceD1.1 Have students place a 20 cm3 structure into a full container ofwater. Ask, How many millilitres of water have spilled out?

Paper and PencilD1.2 Ask students to explain how someone can determine the numberof cubic decimetres in a cubic metre.

PresentationD2.1 Ask students to work in pairs and decide whether or not itemswith the following masses would be easy to lift:0.001 tonnes 0.001 kg10 000 tonnes 10 000 gHave them share their conclusions and reasoning with the class.

Paper and PencilD1.3 Describe for students an object as being 0.003 dm long. Ask themwhether this object would be a whole number of centimetres long and awhole number of millimetres long.

D1.4 Tell students that the area of a rug is 9000 cm2 and ask how manysquare metres that is.

InterviewD1.5 Ask students, What do you think “kilosecond” means? Can youthink of situations where this unit would be used?

D1.6 Tell students Sue claims that since 10 dm = 1 m, 10 dm2 = 1m2.Ask, Do you agree with Sue? Why or why not?

D1.7 Tell students that the area of a rectangular rug is 10 000 cm2. Ask,What might the dimensions of this rug be?

D2.2 Prepare a set of 10 cards, 5 with pictures of items that would havelarge but varying masses and 5 with their actual masses in tonnes. Askstudents to match the items and masses.

D2.3 Tell students that 1 mL of water has a mass of 1 gram. Ask themto determine how many millilitres of water would they have to have tohave a mass of 1 tonne.

D2.4 Have students investigate the number of grade 5 children itwould take to balance an elephant.





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ii) communicate using stand-ard units, understand therelationship among com-monly used SI units (e.g.,mm, cm, m, and km) andselect appropriate units ingiven situations

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D3 demonstrate an understand-ing of the relationshipbetween capacity andvolume

D3 It is important to have students explore the relationships betweenthe cubic units of volume and the units of capacity. Through exploringproblems involving both capacity and volume , students should come tounderstand the following relationships between the two:

1 cm3 = 1 mL 1 dm3 = 1 L 1 m3 = 1 kL

Students could make these connections by using base -10 blocks. Thesmall cubes (1 cm x 1 cm x 1 cm) have a volume of 1 cubic centimetreand would displace 1 mL of liquid in a container; the large cubes(10 cm x 10 cm x 10 cm) have a volume of 1000 cubic centimetres or 1cubic decimetre and would displace 1 L of liquid. This could be ex-tended by making a large rod of 10 large cubes (10 000 cm3) within askeletal model of a cubic metre made with metre sticks. Students couldeasily visualize that 10 of these rows (100 000 cm3) would make a largeflat within the cubic metre and that 10 of these large flats (1 000 000cm3) would fill the cubic metre. Since this visualized filling of 1 m3

would use 1000 large cubes to construct and it is known that 1 largecube displaces 1 L, it follows that 1000L (1 kilolitre) is what 1 m3

would displace.

While capacity and volume are both measures of the size of a 3-D regionof space, capacity units (mL, L, kL) are usually associated with measur-ing liquids and/or the containers which hold fluids. Volume units (cm3 ,dm3,m3) are more commonly associated with solid objects.

1 cm3 = 1 mL 1000 cm3 = 1 L• Have students measure in centimetres the three inside dimensions of

a variety of cardboard boxes and explain how these dimensions couldhelp them determine how many cubic centimetres would fill theboxes. Have them repeat the process measuring the outside dimen-sions of the boxes. Ask them to explain which results would be thevolumes of the boxes and which results would be the capacities of theboxes. Then have them report their results using the appropriateunits.





PerformanceD3.1 Have students use base-10 blocks to make a structure with avolume of 1256 cm3. Ask, If your structure were placed in water, howmany litres of water would it displace?

Paper and PencilD3.2 Ask students to decide whether they would use volume units orcapacity units to describe: - the amount of water in a pool - the amount of wheat in a barrel - the living space in a house

InterviewD3.3 Ask students to visualize the large cube in the base-10 blocks. Ask,If you had a box that just held this block, what would be the capacity ofthe box? Although the unit decilitre is not commonly used, what base-10 block would represent 1 dL? How many millilitres would be equiva-lent to 1 dL?

D3.4 Provide students with a strong box or a picture of a strong box.Explain that its volume was reported in cm3 and its capacity was re-ported in mL but the numbers were different. Ask students to explainwhy this might be so and which number would likely be greater.

PortfolioD3.5 Ask students to design lesson plans for a grade four class in whichthey will address the following: - What does volume mean? - What does capacity mean? - How are they similar? different?Invite them to teach their lessons to small groups of children and writereports on the experience.





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D4 estimate and measure anglesusing a protractor

D5 draw angles of given sizes

D4 Students should learn how to use standard protractors to measureangles reasonably accurately. Since theseprotractors have double scales,students will need to deter-mine which number to usein a given situation. This isbest accomplished byhaving students firstestimate the size of theangle and use this estimate to decide which reading makes the mostsense.

• Have students find the measures of each of the angles in severalquadrilaterals and to find the totals of these measurements. Ask themif they see a pattern in the totals they get. Similarly, you could havethem find the sums of the angles in other types of polygons.

D5 Students need to learn how to use a protractor to draw angles.Initially, concentrate on having them draw angles between 0o and 90o

and then on angles between 120o and 180o. For most students, this isprobably their first experience viewing a straight line as an angle with ameasure of 180o.

In order to produce an accurate drawing, students should be aware ofthe importance of positioning the 0o line on the protractor so that itcoincides with the first arm of the angle and the centre point on theprotractor so it coincides with the vertex of the angle.

• If the computer programme, Logo, is available, have students exam-ine the effect of various angle turns. Students will note that, althoughthe angle in the triangle is 50o, the size of the required turn is 130o.This will help them experience first hand the connection between thetwo angle measures on their protractors.

The protractor is one of the most poorlyunderstood measuring instruments foundin schools ... By making a protractorwith a large unit angle, all of [the]mysterious features can be understood.Then, a careful comparison with astandard protractor will permit thatinstrument to be used with understand-ing. (Van de Walle 1994, 305)





PerformanceD4.1 Ask students to measure the angles found in various printedcapital letters of the alphabet.

D4.2 Ask students to show with their hands estimates for a 120° angle.

D4.3 Provide each student with two geo-strips and a fastener. Askstudents to make and display their estimates for a variety of angles whosemeasures you give orally one at a time. Have them as partners comparetheir estimates. Display your angles (checked with a protractor) and havethem compare theirs with yours.

D5.1 Ask students to fold paper to make a 135° angle without using aprotractor.

Pencil and PaperD5.2 Provide students with pictures of angles, e.g., 60o. Ask them todraw angles that are 90° greater without using protractors.

D5.3 Ask students to draw angles they think might measure 150o. Askthem to check their drawn angles with protractors and comment on howclose they were.

InterviewD4.4 Tell students that Jeff measured this angle and said it measured50o. Ask, Do you agree? Why or why not?

D4.5 Tell students that Marc said he could make an angle bigger byextending both angle arms. Ask them what they think of Marc’s plan.





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D6 continue to solve measure-ment problems involvinglength, capacity, area,volume, mass and time

D6 Students should solve problems involving a variety of types ofmeasurement. These problem-solving experiences should occur not onlywhile studying measurement, but also while studying other strands inthe mathematics curriculum such as number, operations, and geometry.Many problems can and should be coordinated with the teaching ofother curricular areas such as social studies and science.

Time problems should include the use of a variety of time units, explo-ration of the recording of time using the 24-hour clock, and explorationof the idea of time zones. This provides opportunities for students toplan “worldwide” trips using travel schedules and taking time zones intoaccount. Students also need experiences adding and subtracting timemeasures because of the 60-minute-for-1-hour substitution rather thanthe more familar trade of 100 for 1 hundred, e.g., How much time haselapsed between 6:40 pm and 9:10 pm?

The following are some possible sources of problems to connect units ofmeasurement: - time and length measures to determine speed - area and number measures to calculate population densities - area and length measures to find ratios in similar figures - mass and capacity measures to conclude that the mass of 1L of water is 1kg.





PerformanceD6.1 Ask students to find the area of this trapezoid and to explain theirmethods.

D6.2 Provide students with airline schedules and ask that they work inpairs to discuss the times of flight arrivals and departures. Have themplan a cross-Canada trip visiting at least 6 major cities. Also, ask themquestions such as: What is the quickest way to go to Victoria, B.C.?

InterviewD6.3 Ask students how they might estimate the volume of a beachball.

PortfolioD6.4 Have students plan a trip from St. John, NB to St. John’s, NF bycar and ferries only. Ask them to start on a Monday morning at SaintJohn, NB and to assume that they will average 90 km/h on land. Ask,When would you expect to arrive in St. John’s, Newfoundland? Ask thatthey include all their routes, times, and schedules.

D6.5 Ask pairs of students to design pens for 6 gorillas in a zoo. Ex-plain that the animals need an exercise area and a watering hole. Encour-age them to be creative in their designs. Ask them also to calculate thecost of tiling the floors of their pens.





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D7 demonstrate an understand-ing of the relationshipsamong the base, height andarea of a parallelogram

D7 Students should recognize that the area of a parallelogram is thesame as the area of a related rectangle, i.e., one with the same base andheight. They should experience this relationship by cutting out parallelo-grams, drawing perpendiculars as shown in the diagram below, cuttingoff the triangles, and translating the triangles to the opposite sides of theparallelograms to form rectangles.

• Have students work as partners with two geoboards. Ask one partnerto make a parallelogram on a geoboard that has an area of 4 squareunits. Then ask the other partner to make a rectangle on his/hergeoboard that has the same base and area as the partner’s parallelo-gram. Ask, What else is the same about the two shapes on thegeoboards? (Repeat this using other areas and with partners exchang-ing roles.)

Once students have a clear visual image of the relationship between theheight of a parallelogram and the width of its related rectangle, theyshould be able to relate the formula for the area of a rectangle (A = l x w)to the formula for a parallelogram (A = l x h). They should be givenpictures or cutouts of parallelograms and asked to measure what theyneed to determine the areas of these shapes. Then, if students are giventhe measures of two of base, height, and area, they should be able to findthe measure of the third.

Students should recognize that a variety of parallelograms can have thesame area, e.g.,

• Make a flexible rectangle using geostrips or cardboardstrips and brads. Begin to tilt the rectangle. Ask studentswhether the area has changed. Keep tilting the shape. Discuss howwith each tilt, a new parallelogram is created. Ask, What is changingthat is causing the area to get smaller?





PerformanceD7.1 Ask students to draw parallelograms with an area of 24 cm2 ongrid paper. Then ask them to draw three different parallelograms thathave the same base lengths and areas as the first ones they drew. Askthem to draw another parallelogram that has the same area but a differ-ent base length than the ones they have drawn already.

Paper and PencilD7.2 Ask students, If two parallelograms have the same areas, do theyhave to be similar?

D7.3 Tell students that a parallelogram has an area of 42 and a heightof 6 cm. Ask them to find the length of its base and to draw on gridpaper examples of such a parallelogram.

InterviewD7.4 Ask students, Do these parallelograms have the same area? Howdo you know?

D7.5 Ask students to determine which of the two shapes below has thegreater area and to provide explanations.





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D8 demonstrate an understand-ing of the relationshipbetween the area of atriangle and the area of arelated parallelogram

D9 demonstrate understandingof the relationships betweenthe three dimensions of arectangular prism and itsvolume and its surface area

D8 Plan activities that will enable students to recognize that any trian-gle is one half of a parallelogram. Thus, students should see that the areaof the triangle is just one-half of the area of the related parallelogram.

is 1

2 of

is 1

2 of

Students can use this relationship to find areas of simple triangles.Students should understand that, as long as the base andheight are the same, the areas of visually-different trian-gles are the same.

• Have students locate on geoboards as many trianglesas possible which have an area of 2 square units.

D9 To determine volumes and surface areas students should buildstructures with centimetre cubes, always estimating before calculating.

Although students need not commit formulas to memory, their experi-ences should indicate to them that each of the three dimensions of aprism—the height, depth, and width—affects its volume and its surfacearea.

• Have students build a 3 cm x 6 cm x 2 cm boxes with centicubes. Adiscussion afterwards should bring out that a layer of 18 centicubes(3 x 6 ) would make one-half the box and 2 such layers or 2 x 18 =36 cm3 must be its volume. (Students should have sufficient experi-ences so they visualize the relationship between the three lineardimensions and the placement of cubes to build the rectangularprism.) Continue this activity by having them determine the surfacearea of their boxes and discuss the relationships between the dimen-sions and the areas of the rectangles they needed to find.

By building rectangular prisms from the large cubes in the base-10blocks with each block having a volume of 1 dm3 or 0.001 m3 and eachface a surface area of 1 dm2 or 0.01m2, students can extend their workwith volume and surface to units other than cm, cm3, and cm2 .





PerformanceD8.1 Ask students each to draw a triangle, such as the one below, ongrid paper. Then ask them to draw a parallelogram with twice the area oftheir triangle.

Extension: If they draw this triangle on grid paper, trace it on tracingpaper placed over it, rotate the tracing paper 180o about the midpoint ofany one of the three sides, and trace the rotated triangle on the gridpaper, they will have a parallelogram that is twice the area of the triangle.

D8.2 Ask students to draw five different triangles on grid paper all ofwhich have an area of 6.5 square units. Have them present their draw-ings and explain how they know that the triangles all have the same area.

D9.1 Ask students to build rectangular prisms using one large cube andfour flats from the base-10 blocks. Have them make either isometric ororthographic drawings of their prisms. Ask them to record on theirdrawings (a) the dimensions of their prisms in centimetres and indecimetres, (b) the volume of their prisms in cubic centimetres and incubic decimetres, and (c) the surface areas of their prisms in squarecentimetres and in square decimetres.

Paper and PencilD9.2 Ask students to write explanations for why a prism with a basethat is 5cm x 3 cm and a height of 4 cm must have a volume of 60 cm3.

D9.3 Ask students, Which has the greater effect on the volume of aprism—doubling the area of its base or doubling its height?

D8.3 Ask students to find the height of a triangle that has an area of 18square metres and a base of 4.5 metres.

PortfolioD8.4 Present the following scenario to students: The area of a firsttriangle is found. The area of a second triangle is found to be 2 squareunits less than the first. The height of the second triangle is 1 unit lessthan the height of the first triangle. Ask the students to explain whatthey know about the bases of these two triangles.

Shape and Space:

Geometry

ATLANTIC CANADA MATHEMATICS CURRICULUM: GRADES 4–6 6-78


GCO E: Students will demonstrate spatial sense and apply geometric concepts, proper-ties, and relationships.


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i) identify, draw, and buildphysical models of geomet-ric figures

iv) solve problems usinggeometric relationships andspatial reasoning

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E1 describe and represent thevarious cross-sections ofcones, cylinders, pyramids,and prisms

E1 A cross-section is the 2-D face produced when a plane cut is madethrough a 3-D shape. For example, consider a right-circular cone.(a) If it is cut in any plane parallel to its base, the face produced is acircle.

(b) If it is cut down through its vertex, the exposed face is a triangle.

(c) If it is cut in a plane parallel to a plane of symmetry, the shapebelow is produced.

(d) If the cone is cut obliquely—not parallel to its base—the faceproduced is an oval (ellipse).

Cross-sections should be investigated by actually cutting shapes or byobserving water surfaces in models of shapes. Students should have hadprior experiences with cross-sections of cubes, square prisms, andrectangular prisms.

It is not intended at this stage for students to remember the variouscross-sections for each 3-D shape without the presence of the shape;however, they should try to draw their predictions of the cross-sectionsbefore they cut the shape. Using elastic bands on the shapes to representcuts is a way to help students visualize.

Students should be challenged to analysetheir thought processes and explanations.They should be allowed sufficient timeto discuss the quality of their answers andto ponder such questions as, Could it beanother way? What would happen if ... ?(NCTM 1989b, 113)





PerformanceE1.1 Have students make some triangular prisms with Plasticine. Askthem to predict and draw the polygonal cross-section(s) that wouldresult with each of the following cuts and to check their predictions bymaking the appropriate cuts:(a) cut parallel to its bases(b) cut parallel to one of its rectangular faces(c) cut obliquely (slanting) towards its bases(d) cut obliquely to a rectangular face

E1.2 Ask students to explain and demonstrate how a square pyramidcould be cut to produce each of the following cross-sections:(a) a circle (b) a rectangle (c) a trapezoid

PresentationE1.3 Have students stack four hexagonal pattern blocks to makehexagonal prisms. Ask them to discuss as partners some ways theseprisms could be cut and what shapes would be produced. Have thempresent their ideas to the class, including their pictures of the differentcross-sections.

InterviewE1.4 Provide a variety of 3-D shapes for students to examine. Explainto them that you have a mystery 3-D shape that has been cut to make atriangular cross-section. Ask them to think of four possibilities for thismystery shape. Have them describe for each shape the cut that wouldhave been made to produce this cross-section

E1.5 Ask students to describe how a cylinder could be cut to produceeach of the following cross-sections:(a) a circle (b) a rectangle (c) an oval





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i) identify, draw, and buildphysical models of geomet-ric figures

iv) solve problems usinggeometric relationships andspatial reasoning

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E2 make and interpretorthographic drawingsof 3-D shapes madewith cubes

E2 Orthographic drawings are a series of 2-D views of a 3-D shapedrawn by looking at the shape straight down (to get a top view) andstraight on (to get front, back, right, and left views). The figure at rightcould be interpreted in a mat plan that shows its base with numbersindicating the number of cubes high.

Students should build this shape with cubes, place their shapes on matplans, and draw the various orthographic views on square dot paper.The following are the views for this shape:

Top View Front View Left View Right View Back View

The use of mats often helps students with these drawings. A square ofplain paper appropriately marked with directionswould be a simple mat for this purpose. The stu-dents could then place a 3-D shape on the mat andmove the mat to make the drawing of each view.Note: Left and right are always relative to the front.

Some students might find it helpful to close one eyeand place themselves so that they are at eye levelwith the shape; they should then see only one face of the 3-D shape.

• Provide students with directional mats and 3-D shapes made of eightcubes. Have them draw mat plans and make top, front, and rightorthographic views of these shapes.

• Provide students with top, front, and left orthographic views ofvarious 3-D shapes. Have them use cubes to build the shapes withthese views and to draw mat plans and two isometric views.





PerformanceE2.1 Have students use cubes to construct shapes that have this matplan.

Ask them to place them on mats and to draw the various orthographicviews, labelling each view.

E2.2 Ask students to use cubes to construct the shape shown at right.Ask them to draw mat plans for this shapeand the top, front, and left orthographicviews, labelling each. Ask, Would the backand right orthographic views be needed forsomeone to be able to build this shape orwould the three views you have be sufficient?

E2.3 Ask students to use cubes to construct buildings that would havethese orthographic views.

TOP FRONT RIGHT

Ask them to draw mat plans for this shape.

Paper and PencilE2.4 Provide students with this picture of a building drawn from itsfront-right corner. Ask them which one of A–E is the right orthographicview of the same building.





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ii) describe, model, and com-pare 2- and 3-D figures andshapes, explore their proper-ties and classify them inalternative ways

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E3 make and apply generaliza-tions about the sum of theangles in triangles andquadrilaterals

E4 make and apply generaliza-tions about the diagonalproperties of trapezoids,kites, parallelograms, andrhombi

E5 sort the members of thequadrilateral “family” underproperty headings

E3 Students should investigate these generalizations in a variety of ways.For example, if each student cuts out a triangle, tears off its three angles,and places them together, the three angles will form a straight line(180o); if each student cuts out three copies of a triangle, labels the threeangles, and starts a tessellation, they can see that the sum of the angles is180o;

if students measure the three angles in a variety of triangles and addthem, the 180o relationship might be revealed, although measurementoften produces only approximate results.

Similarly, students could be lead to generalize the 360o sum of the anglesof any quadrilateral.

E4 Generalizations about diagonal properties should result from guidedinvestigations.

(a) For a rhombus, its diagonals are perpendicular bisectors of eachother, form four congruent right triangles, bisect the angles of therhombus, and are the two lines of reflective symmetry of the rhombus.

(b) For a parallelogram, the diagonals bisect each other and form twopairs of congruent triangles.

(c) For a kite, the diagonals are perpendicular and form two pairs ofcongruent right triangles; one of the diagonals is bisected, and the otherdiagonal is a line of reflective symmetry and it bisects two oppositeangles of the kite.

(d) For a trapezoid, there are no special properties of its diagonals.

These properties should be developed for each figure, applied in avariety of ways, compared to the others, and combined with the side andangle properties of the figures.

E5 Students should be able to sort pictures or cutouts of the variousquadrilaterals into sets according to one or more properties. Theseproperties include: diagonals bisect each other, opposite sides congruent,opposite sides parallel, has four right angles, diagonals perpendicular toeach other, opposite angles congruent, has reflective symmetry, anddiagonals form two pair of congruent triangles.

Modelling, mapping, and engaging inactivities and spatial experiencesorganized around physical models canhelp students discover, visualize, andrepresent concepts and properties ofgeometric figures in the physical world.(NCTM 1992d, 1)





PerformanceE3.1 Tell students that you heard someone say, “Since any quadrilateralcan be divided into two triangles and a triangle has a sum of angles of180o, it is obvious that the sum of the angles of a quadrilateral is 360o.”Have students draw pictures of various quadrilaterals to verify the truthof this statement. Have them extend this thinking to find the sum of theangles of a pentagon.

E4.1 Have students draw scalene right triangles. Ask them to use mirasto make pentagons by drawing two reflection images of the trianglesusing both arms of the right angles as mirror lines. Have them join twovertices of the pentagons to produce quadrilaterals that will have theirtwo diagonals showing. Ask them to name the type of quadrilateral. Askif everyone in the class got the same type of quadrilateral. Have them listthe properties of this quadrilateral that they can confirm from the waythey drew it.

E5.1 Make a set of shape cards with a variety of pictures of differentmembers of the quadrilateral family. Distribute them to the students.Choose an attribute card, e.g., opposite sides parallel. Have students puttheir shape cards under this attribute, if appropriate, and to discuss whyor why not each card got placed. Choose another attribute card, e.g.,diagonals bisect each other, and place it with the first card. Have stu-dents discuss which shape cards should now be removed and why.

Paper and PencilE3.2 Ask students to find the size of the missing angle(s) in each of thefollowing triangles and to draw the triangle : (a) two of its angles are 70o

and 45o, (b) two of its angles are each 75o, (c) it is a right triangle with a60o angle, and (d) it is an isosceles triangle with an angle of 102o.

E3.3 Ask students to draw parallelograms on square dot paper. Havethem measure only one of the angles of their parallelograms and deter-mine the measures of the other three angles by using known relation-ships.

E4.2 Ask students to list the properties of a rhombus that are the sameas those of a kite and the properties that are different.

E4.3 Ask students each to draw a segment that is 6–10cm long. Havethem use only miras to draw rhombuses that have this segment as one oftheir diagonals. Ask them to explain the processes they used.





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ii) describe, model, and com-pare 2-D and 3-D figuresand shapes, explore theirproperties and classify themin alternative ways

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E6 recognize, name, describe,and represent similar figures

E6 Students have an intuitive sense of similar shapes—shapes that areenlargements or reductions of each other. Students’ experiences withnegatives of photographs that can be developed in different sizes, withmaps or pictures that are drawn to scale, and with images produced bymagnifying glasses provide natural connections for this concept. Over-head projectors, photocopiers, and film projectors are other sources ofreal-world contexts to relate to similar figures.

Students should discuss what the word similar might mean to them ineveryday contexts. Compare these meanings to the specific meaning ofthe word in mathematics, i.e., corresponding angles are equal and pairsof corresponding sides have the same ratio.

• Prepare pairs of shapes some of which are similar and some which arenot. Tape the larger onto the board and place the smaller on theoverhead projector. Have a student move the projector until theimage coincides, or does not, with the one taped on the board. Theyare similar if a match can be made.

• Place a red pattern block on the overhead projector. Have studentscompare the projected image to the actual block, asking them what isthe same and what is different. Have a student place the block in thecorresponding angles of the projected image. This should emphasizethe role angles play in making shapes similar. Informally compare thelengths of corresponding sides of the block and of the projectedimage, seeing approximately how many times longer the image sidesare than the sides of the block. This will be easier if you move theprojector in advance so the sides will be a whole number times largerrather than a fractional times larger.

All dilatation images of a shape (see SCO E9) are similar; however, notall similar figures are merely dilatation images of one another—similarfigures can be on different planes and/or be the result of a dilatation incombination with other transformations.

Students are likely to recognize the similarity of different sizes of regularpolygons, e.g., equilateral triangles and squares. Also, because all trian-gles with equal angles are similar, they are more likely to recognizesimilar triangles than most similar quadrilaterals.





PerformanceE6.1 Have students examine the three different-sized triangles in atangram set for similarity. Ask them if they think these triangles aresimilar. Have them compare the angles and the lengths of thecorresponding sides of these triangles to confirm or refute their answers.

E6.2 Provide students with worksheets of rectangles, most of which aresimilar. Ask them to cut out the rectangles they think are similar and todraw in their diagonals. Ask them to lay the similar rectangles on top ofone another, starting with the largest and in such a way that they share avertex. Ask them what they notice about the diagonals of the similarrectangles. Have them place on their piles one of the rectangles, with itsdiagonals drawn, that is not similar. Ask them what they observe aboutthe diagonal of this rectangle. Have them investigate whether otherquadrilaterals could use this “diagonal test” for similarity.

E6.3 Have students each make a triangle on geoboards using thebottom left peg and the pegs directly above and to the right of it. Askthem to make four different triangles, all similar to this one.

E6.4 Using a flashlight and a shape in a dimmed classroom, move theflashlight to cast different shadows on the wall. Ask students to identifywhen the shadow is/is not similar to the shape. Repeat using othershapes. Ask, Where does the flashlight need to be held to produce asimilar shape?

E6.5 Have students use only the triangles in the pattern blocks to makeother larger triangles. Ask, Are these larger triangles similar to the greenpattern block? Have them hold green blocks close to one of their eyesand stand over one of the larger triangles staring down at them usingtheir eyes with the blocks in front of them. Ask them to move the blocksuntil they coincide with the larger triangles. (This is another informalway to test for similarity.) Have them compare the sizes of the angles inthe small and large triangles as well as the lengths of the correspondingsides. Repeat these tasks using the other pattern blocks.

Paper and PencilE6.6 Have students draw scalene triangles, cut them out, and use thesetriangles to draw smaller and larger triangles that are similar to them.

E6.7 Provide students with pictures drawn on one-sized grid paper.Have them use a different-sized grid paper to draw similar pictures.





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iii) investigate and predict theresults of transformationsand begin to use them tocompare shapes and explaingeometric concepts

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E7 make generalizations aboutthe planes of symmetry of3-D shapes

E7 In the same way that some 2-D shapes have lines of reflectivesymmetry, some 3-D shapes have planes of reflective symmetry, i.e.,planes that bisect 3-D shapes such that all points in one-half are mirrorimages of the corresponding points in the other half. These planes ofsymmetry should be connected to cross-sections as examples of specialcuts. A cube, for example, has nine different planes of symmetry asshown in the figure below. Although faces B and C could also be cut byperpendicular planes as shown on face A below, only one cut on face Bwould produce a different result.

Students should investigate planes of symmetry of right triangular,square, rectangular, pentagonal, and hexagonal pyramids. Studentsshould discover the pattern that the number of planes of reflectivesymmetry for these pyramids is equal to the number of lines of reflectivesymmetry of their bases, e.g., A right square pyramid has 4 planes ofsymmetry and its square base has 4 lines of symmetry.

• Provide students with right triangular, rectangular, square, pentago-nal, and hexagonal prisms. Have them investigate the number oflines of reflective symmetry of the bases of these prisms. Ask, Willplanes through these lines of symmetry be planes of symmetry ofthese prisms? Do these prisms have other planes of symmetry? Havestudents explain how to find the number of planes of symmetry of aprism.

When students examine cones and cylinders for planes of symmetry, theconcept of infinite number will likely arise as students notice that thereare “a whole bunch” of planes that would cut through the centre of thecircular bases to be planes of symmetry. Also, spheres will be found tohave infinitely-many planes of symmetry.

Note: The cones, cylinders, prisms, and pyramids used should be right,such as the ones typically found in basic sets of solids.





PerformanceE7.1 By stacking pattern blocks, construct a 3-D configuration that hastwo planes of symmetry. Ask students to show where its two planes ofsymmetry are.

E7.2 Ask students to use sets of pyramids that have regular polygonalbases to help complete this table.

______Pyramid___ No. of Planes of Symmetry

triangular (one equilateral face) squarepentagonalhexagonal

Ask, Do you see a pattern that you could use to predict how manyplanes of symmetry an octagonal pyramid has?

E7.3 Have students examine real-world objects of a variety of shapes,e.g., boxes, containers, toys, candies, and candles, for planes of reflectivesymmetry.

E7.4 Ask students to use 12 multi-link cubes to build shapes that havetwo planes of reflective symmetry.

InterviewE7.5 Provide pictures of houses, garden sheds, gazebos, and otherstructures. Ask students to choose which structures have plane(s) ofsymmetry and describe the location(s) of these planes.

E7.6 Ask students what is meant by this statement: “The number ofplanes of symmetry that a cone has is related to the number of diagonalsa circle has.”

E7.7 Ask students to compare the number of planes of symmetry of asquare prism—not a cube—and a rectangular prism. Have them explainwhy the square prism has more planes of symmetry than the rectangularprism. Have them explain why they both have fewer planes of symmetrythan a cube.

E7.8 Show students a water glass that has an interestingly shapedbottom. Ask them to describe any plane(s) of symmetry that this glassmight have.





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E8 make generalizations aboutthe rotational symmetryproperty of all members ofthe quadrilateral “family”and of regular polygons

E8 If a shape can be turned about a point so that it exactly coincideswith its original position at least once in less than a complete rotation, itis said to have rotational symmetry. The number of times it appears inthe identical position during one complete rotation is the order ofrotational symmetry, e.g., If an equilateral triangle is turned clockwise120o about its centre point, the image is identical; if it is turned another120o, again the image is identical. It is said to have rotational symmetryof order 3.

If a shape has to be rotated a full 360 degrees before it fits its tracedimage, it does not have rotational symmetry.

• Have students use a blue block from the pattern blocks, tracing it onpaper. Have them place the block on its traced image and lightly puta pencil mark in the upper left corner of the block. Ask them to turnthe block within its traced image to the right until it fits its tracedimage again. Have them notice where the mark on the block is now.Have them continue turning the block to the right until it fits itstraced image again, noticing where the mark is on the block.Through this activity, students should conclude that a rhombus hasrotational symmetry of order 2.

Through activities such as the one above with other members of thequadrilateral family of shapes, students should generalize that a squarehas rotational symmetry of order 4; a rhombus, parallelogram, andrectangle each have rotational symmetry of order 2; a kite and a trap-ezoid do not have rotational symmetry.

There are many handy contexts for exploring rotational symmetry, e.g.,Consider the common toddler toy which involves fitting blocks throughopenings. In how many ways can the hexagonal block be fitted throughthe hexagonal opening?Symmetry in two and three dimensions

provides rich opportunities for studentsto see geometry in the world of art,nature, construction, and so on.(NCTM 1989b, 115)





PerformanceE8.1 Have students trace hexagonal pattern blocks on paper andmark a dot at one vertex on the block. Have them rotate the blocksclockwise inside their traced images until they fit the traced hexagonsexactly.

Have them continue rotating the blocks until the marked vertexreturns to its original position. Ask them how many times the blocksappeared in identical positions. Have them describe the rotationalsymmetry of a regular hexagon.

E8.2 Repeat task E8.1 for a square, rhombus, and trapezoid usingpattern blocks. Provide cutouts to repeat task E8.1 for a rectangle,parallelogram, and kite.

E8.3 Provide students with pictures of designs and quilt patternssuch as the ones below. Ask them to predict whether they haverotational symmetry. Have them use tracing paper to confirm theirpredictions by tracing the patterns and rotating the tracing paper ontop of the pictures. Have them also check for reflective symmetry.

E8.4 Ask students to use geoboards to make shapes that haverotational symmetry of order 2.

InterviewE8.5 Ask students to explain how someone would know the order ofrotational symmetry for any regular polygon, e.g., a regular octagon,without needing to use tracing paper to test for it.

PortfolioE8.6 Have students examine newspapers and magazines for picturesand logos that have rotational symmetry, e.g., car hood ornaments.Ask them to select four of their favourite ones, paste them on paper,and write short descriptions of their symmetry, including commentson their reflective symmetry if they have it. Invite them to designtheir own logos with rotational/reflective symmetry.





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E9 recognize and representdilatation images of2-D figures and makeconnections to similarfigures

E9 Introduce dilatations by having students participate in activitiessuch as the one below.

• Have each student draw a scalene triangle, labelled JKL, on a sheet ofpaper and select a point C outside this triangle. Ask them to measurethe distance from C to J, and triple this distance for CP; measure thedistance from C to K, and triple this distance for CQ; and measurethe distance from C to L, and triple this distance for CR. Have themdraw a triangle by joining P, Q, and R and ask them to compare thetwo triangles.

Explain to students that if lines through all corresponding vertices oftwo shapes on a plane converge at a single point as they do in theactivity above—these shapes are dilatation images of one another. Thepoint of convergence is the centre of dilatation, and the two shapes aresimilar.

∆ ′ ′ ′A B C is the dilatation imageof∆ABC with T as the centre ofdilatation; ′ ′ ′Q R is thedilatation image of∆PQR withP as the centre of dilatation.Comparing the distances fromP to each of the corresponding points shows that they are all twice as farout; comparing the distances from T to each of the correspondingpoints shows that they are all half as far.

The centre of dilatation could be connected to the concept of vanishingpoint if students have done perspective drawing in art.

Students should be encouraged to compare dilatation images in order tofind relationships between them. Some students might notice thatcorresponding sides are parallel and that each pair of correspondingsides have the same ratio.

∆P





PerformanceE9.1 Have students each trace the large triangle from tangram sets onlined paper so that its longest side lies along a line (see below). Havethem also trace one of the smaller triangles from the sets so that itslongest side lies along a different line. Ask them to draw lines betweencorresponding vertices of the two triangles and extend them. Ask, Whatis the point where the three lines intersect called? Have them investigatewhether lined paper could be used in the same way to set up any twosimilar figures to be dilatation images.

E9.2 Have students try to visualize the existence of a centre of dilata-tion for each of the following pairs as a means of predicting whetherthey are dilatation images of one another. Have them check by actuallylocating the centre using a straight edge.

a) b) c)

E9.3 Have students each trace a red pattern block on a sheet of plainpaper. Ask them to draw dilatation images of this block using centres oftheir choice. Ask them to compare the angles and the lengths of thesides of their two trapezoids.

E9.4 Have students draw 9 cm x 12 cm rectangles on lined paper, witha 12cm length lying along one of the lines on the paper. Have themdraw a 3cm x 4cm rectangle somewhere else on their papers with a 4cmlength lying along a line. Ask, Are the two rectangles dilatation imagesof one another? How do you know?





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E10 predict and represent theresult of combiningtransformations

E10 Students should understand that two congruent shapes on thesame plane are images of one another under a translation, reflection,rotation, or any combination of these three transformations. If twosimilar shapes are on the same plane, they are dilatation images ordilatations in combination with translations, reflections, or rotations.Students should investigate a variety of combinations, each time tryingto visualize the result to make a prediction before actually carrying outthe transformations. These combinations could include a reflectionfollowed by a translation, two translations, two reflections, a translationfollowed by a rotation, two rotations, and a dilatation followed by atranslation.

• Place three geoboards side by side. Have one student make a scalenetriangle on the first geoboard. Ask another student to construct onthe second geoboard the image of this triangle if the right side of thefirst geoboard is used as a mirror line. Ask another student to con-struct on the third geoboard the image of the triangle on the secondgeoboard under a 90 degree counterclockwise rotation. Repeat thisactivity using other shapes.

• Provide each student with grid paper marked with a coordinatesystem and three pattern blocks of the same type. Ask students toplace one block on the system so that one of its vertices is at (-5, 3).Ask them to place a second block so that it would be the image of thefirst block under a horizontal translation of 10 units. Then ask themto place the third block so that it is the image of the second blockunder a reflection in the x-axis. Have them compare the first andthird blocks. Repeat this activity using two other transformations.Extension: Have each student carry out two transformations of theirchoice on the coordinate system and leave only the first and thirdblocks in place. Have them exchange coordinate systems with apartner and have them try to predict the two transformations thattook place. Share their predictions and actual transformations.

• Have students investigate such questions as (a) If a shape undergoes 2translations, does it matter in which order they take place? and (b)Does a reflection followed by a translation produce the same result asthe translation followed by the reflection?Computer software allows students to

construct two- and three- dimensionalshapes on a screen and then flip, turn, orslide them to view them from a newperspective. (NCTM 1989b, 114)





PerformanceE10.1 Have students locate the image of ∆ABC after a reflection inline 1 followed by a reflection in line 2. Ask them what singletransformation of∆ABC would produce the same resultant image.

E10.2 Ask students to draw isosceles triangles on dot paper and totranslate them 4 units horizontally. Ask them to describe the reflectionthat would produce the same resultant images.

E10.3 Present students with the pictures on grid paper of two congru-ent shapes—the initial shape and the resultant image after two transfor-mations were performed. Ask students to predict what two transforma-tions were performed. Ask, Could this have been done in more than oneway? Could this have been done by a single transformation?

Paper and PencilE10.4 Have students each draw a 4 x 4 square on grid paper and shadesome of the small squares inside to create a design. Have them drawanother 4 x 4 square attached under the first square. Ask them to drawin this square the image of their design under a reflection in the bottomside of the first square. In another 4 x 4 square attached under thesecond square, have them draw the half-turn rotation image of thedesign in the second square. Ask them to compare the designs in thefirst and third squares.

Data Management






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i) collect, organize, anddescribe relevant data inmultiple ways

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F1 choose and evaluateappropriate samples fordata collection

F2 identify various types ofdata sources

F1 To gather information about a large population, when it isimpossible to check every person involved, samples are used. Usinginformation gathered from such samples, generalizations are madeabout the entire population. However, it is recognized that conclu-sions drawn from samples may not always be true for the population,but in order to minimize the degree of error, care is taken in theselection of samples.

Students should consider both how to choose samples and how safe itis to generalize to the full populations, e.g., Suppose they wanted todetermine people’s favourite take-out food, they should realize that itwould not be wise to choose a sample of patrons of Pizza Palacebecause that sample could be biased in favour of pizza.

In choosing a sample, students should carefully consider the informa-tion being sought and how people answering questions could bebiased, e.g., If students want to find what radio station is mostpopular, they should probably consider the mix of ages within thesample, the male-female distribution within the sample, the availabil-ity of a variety of stations to those sampled, and the time of day.

A sample should be constructed to deal with potential biases.

F2 Students will realize that although some data is collected first-hand by interviewing or observing, much of the data to which theyare exposed is second-hand data. Students should explore, throughdiscussion, how second-hand data might be collected and howreliable they feel it is, e.g., If students read that 30% of children inCanada are not physically fit, they might then wonder about the datasource: Was a sample used? Were children tested directly or was datacollected by asking doctors or teachers?

Since students should realize they must be careful about drawingconclusions from reported data, they should become familiar withsources for different types of data.





InterviewF1.1 Ask students to describe situations in which they believe thatsamples might be biased.

F1/2.1 Ask students what sample/data source they would use todetermine the amount of water an average Canadian uses in a day.

F1.2 Tell students that to judge the popularity of the prime minis-ter, news reporters talked to a number of visitors to Parliament Hill.Explain that the reporters felt their sample was unbiased since peopleof all age groups were there. Ask the students if they agree and toexplain why.

F1.3 Ask students, How would you choose a sample to interview inorder to predict a provincial election result?

F2.1 Ask students where they might go to find data about thenumber of school-aged children in their province.

F1.4 Ask students why a sample of 5 year-olds might not be the bestone to find out what playground equipment a school should have.

PresentationF1/2.2 Have students locate media stories which present statisticalinformation about Canadians. Ask them to tell how they believe thedata was collected and whether they believe the public can be reason-ably certain that the data is reliable.





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ii) construct a variety of datadisplays (including tables,charts, and graphs) andconsider their relativeappropriateness

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F3 plot coordinates in fourquadrants

F3 Students should plot data points in all four quadrants.

This could be motivated by using linking geoboards. If one board hasall the pegs labelled with coordinates and another board is attachedto the left, the new pegs must be labelled—this creates a need for away to label the other pegs.

Students should recognize that(a) a negative number for thefirst coordinate indicatesthat the point is to the left of thevertical axis(b) a negative number for the secondcoordinate indicates that the point isbelow the horizontal axis(c) the point at which the axesintersect has coordinates (0,0)and is known as the origin.• Students might create drawings using all four quadrants of the

coordinate grid. They could then provide other students with alist of thevertices in order, for each drawing created. The otherstudents would subsequently re-create the drawings.

Situations which might be modelled using 4-quadrant graphs includehigh vs low temperatures for different days; mathematical relation-ships, e.g., a number vs its double; and locations such as blocksnorth, south, east, and west from the town centre.

• Show a map (graph) like the one at right.Ask, How many blocks north of towncentre Mary lives? How manyblocks east? Write Sue’s locationas an ordered pair.

(-1, 2)

(2, 3)

(2,-1)(0,0)

(-3, -3)





PerformanceF3.1 Ask students to investigate what happens to shapes plotted oncoordinate planes if all the first coordinates of points on them areswitched with the second coordinates, e.g., (3, -2) becomes (-2, 3).

Paper and PencilF3.2 Ask students to describe the shape formed by the points(-4, 2), (-2, 1), (0, 2) and (0, 4).

F3.3 Ask students to describe where each of these points would belocated following a half-turn about the origin: (-3, -5), (3, 6), and(-2, 4).

F3.4 Ask students to plot 10 points in quadrant one for which thedifference between their first and second coordinates is 3. Ask themto identify other points along the continuation of this line whichhave coordinates with negative values and to list three pairs of suchcoordinates.

F3.5 Ask students to name five coordinate pairs that together willmake a shape in the top left quadrant of a graph.

F3.6 Give the coordinates of a triangle, e.g., (1, 2), (3, 5) and (4, 0).Ask students to reflect the triangle in the horizontal axis and label thecoordinates of the image. Ask them to reflect it in the vertical axis aswell. Ask them to examine the corresponding points to see if they candetect patterns for the reflections in the axes, state these patterns forreflections, and apply their pattern rule to some other points.

InterviewF3.7 Ask students to plot 10 points for which the first coordinate isthe opposite of the second, e.g., (5,-5), (-1, 1). Have them describethe pattern they see and explain why they might have expected thatpattern.

F3.8 Ask students to plot the scores on the various holes in thismock golf game and then explain how the graph depicts theperformance of the player.





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F4 use bar graphs, double bargraphs, and stem-and-leafplots to display data

F4 Students should regularly use bar graphs, double bar graphs, andstem-and-leaf plots to display and organize data. Data can be col-lected in surveys, through experiments or through research. Topicsmay include areas of mathematics, other curricular areas and real-lifesituations.

• Have students might gather information about the ages of theirgrandparents and display it in either a double bar graph or stem-and-leaf plot.

4 5 6 65 0 0 1 1 1 2 3 5 5 5 6 6 7 7 76 1 1 1 2 3 5 5 8 97 0 0 1 2 5

Num

ber

of G

rand

pare

nts

5

10

1234123412341234 grandfather

grandmother

123412341234123412341234123412341234123412341234123412341234123412341234123412341234123412341234

123412341234123412341234123412341234123412341234123412341234123412341234

1234123412341234

1234123412341234

Age (by decade)

40s 50s 60s 70s

Student’s Grandparents

Student’s Grandparents





PerformanceF4.1 Ask students to create graphs which illustrate both the first andsecond choices of favourite sports of class members.

F4.2 Ask students to draw a bar graph to compare the number ofcalories used by an adult female in one hour for each activity listedbelow:

Sleeping 55Walking 180Walking uphill 360Running 420

InterviewF4.3 Ask students, What scale would you use to graph the followingdata?

Category A- 25Category B- 1000Category C -5000.

F4.4 Ask students to survey their classmates to find out the arm spanlengths and leg lengths of members of the class. Have a group ofstudents create a stem-and-leaf plot for each female arm span and leglength, and each male arm span and leg length. The various graphscould then be compared.





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F5 use circle graphs torepresent dataproportionally

F5 Students should realize that circle graphs are used to describehow a whole is distributed into its componentparts. Students should be able to estimate per-cents when shown a circle graph, e.g., Theyshould estimate that A is about 50% and B isabout 30%.

There are many easy ways toconstruct a circle graph such as

(a) circle mats divided up intotenths and hundredths (black-linemaster in Van de Walle 1994)

(b) a strip of equal-sized squares shaded by categories, taped togetherto make a circle, with lines drawn from the centre of the circle to thepositions where the categories change on the strip

(c) fraction circle pieces

It is important that students understand a circle graph describesrelative size, not actual size. If, for example, two circle graphs werecreated to show the age distributions of people in Nova Scotia and inPrince Edward Island, it would not be apparent from looking at thesetwo graphs that there are more people in Nova Scotia than in PEI.

A “circle” or “pie” graph is used when atotal amount has been partitioned intoparts and interest is in the ratio of eachpart to the whole and not so much in theparticular quantities. (Van de Walle 1994,395-96)





PerformanceF5.1 Ask students to research the percentage of people speakingvarious languages in a given province in Canada. Using estimates ofthe percentages, ask students to create a circle graph showing thedistribution.

F5.2 Ask students to draw circle graphs they predict will show thedistribution of favourite types of music of their classmates if theclassmates choose among country, rock, and classical. Then let themgather data, draw the actual circle graph, and compare it to theirpredictions.

F5.3 Ask students to draw a circlegraph to show the country of originof baseball or hockey players from afavourite team.

F5.4 Have students convert theinformation on the hundredths chartsto a circle graph.

Paper and PencilF5.5 Ask students to estimate the percent for eachsection of this graph. Ask them to describesituations for which this circle graph would be agood display and to label the sections appropri-ately.

InterviewF5.6 Ask students, Why do you think circle graphs are often used toshow how a government is spending its budget?

F5.7 Ask students to describe situations for which they might usecircle graphs instead of a bar graph.

F5.8 Explain to students that Jesse claimed that in his circle graphhe had sections labelled 35%, 25%, 30%, and 15%. Ask students ifthey think Jesse is mistaken and to explain why.

Yellow

RedBlueWhite





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F6 interpret data representedin scatterplots

F6 Scatterplots are used to show the relationship between twoquantities. The plot is made up of “scattered” points which areordered pairs. Each ordered pair tells the simultaneous value of thetwo quantities, e.g., If the plot shows the height for people of differ-ent masses, (mass, height of a person with that mass) would be whatis represented by the coordinates on this plot. If the plot mightshows height for people of different ages,each ordered pair would be(age, height for a person with that age).

Other topics for scatterplots might be temperature at different timesof day, plotting hour against temperature; and tree height for differ-ent aged trees, plotting height against age.

Students should observe that fairly clear relationships are often showneven though a few pieces of data may not fit the relationship well.They should be able to describe these relationships.

Characterizing the shape of the data, as ifyou were going to sculpt this shape withclay, is a prerequisite to summarizing andtheory building. Paying attention to theshape of the data may be the mostimportant idea we can communicate tostudents about data analysis. When youlook at a table or graph, what strikes youabout the data? Where are the dataclumped? Is there more than one clump?Are there bumps of data in surprisingplaces? Are there holes that contain nodata? (NCTM 1989a, 138-39)





Paper and PencilF6.1 Ask students, What conclusionmight be drawn from this scatterplot?

InterviewF6.2 Ask students, How is a scatterplot like other graphs and how isit different?

F6.3 Show students a scatterplot of provincial populations forvarious years. Ask, What conclusions can be drawn from this graph?





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iii) read, interpret, and makeand modify predictionsfrom displays of relevant

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F7 make inferences from datadisplays

F7 Students are usually intrigued by unusual graphs, e.g., The graphbelow displays the level of water in a bathtub as someone takes abath.

In this type of situation, students should each tell a “story” about thegraph by describing what they think happened to lead to the shapeof the graph.

Students should also be drawing inferences from conventional graphsand tables. These could include, among other things, predictions ofvalues not actually gathered but in intervals between values that weregathered and predictions of values in intervals before and/or aftervalues that were gathered. If, for example, a line graph displays thenumber of millilitres of rainfall between 12 noon and 4 pm for actualmeasurements made every 30 minutes, students might be asked tofind the rainfall at 2:45 pm and to find the rainfall at 4:30 pm. Theyshould be aware of the assumption(s) being made when they findthese values.

If students are presented with a bar graph showing ages, such as intask F7.2, and asked to create a stem-and-leaf plot that would berepresented by this bar graph, they would have to make many infer-ences about the actual ages in the intervals.

Time

WaterLevel





Paper and Pencil

F7.1 Show students this circle graph depicting ages of Canadians.Ask them to explain all the information they can get from the graph?

F7.2 Present students with the graph below. Explain that it showsthe age distribution of members of a local legion branch. Ask them toprepare questions that other students could answer by getting infor-mation from this graph.

PortfolioF7.3 Ask students to collect a variety of graphs from the media andorganize them by topic. Have them create brief reports on theinformation provided by the visuals.





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iv) develop and apply meas-ures of central tendency(mean, median and mode)

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F8 demonstrate an under-standing of the differencesamong mean, median, andmode

F8 Students have previously encountered the concept of mean, i.e.,the average calculated by taking the total amount and sharing itequally. They have also alreadyseen that the total amount abovethe mean balances the totalamount below.

The median is another type of average. It tells the middle number ina set of data. Students should recognize that the mean and medianmay be the same or may be different as shown below.

The mode is yet another type of average, in some ways the easiest todetermine. It is the piece of data that appears most often, e.g.,Consider the following data:

5 5 5 5 10 3 3The mode is 5 (which appears 4 times). In this case, the median isalso 5, but the mean is not.

Students might explore the “stability” of the mean and median, e.g.,Ask them to compute both statistics for 3, 10, 15, 22, 45, and alsofor 3, 10, 15, 22, 100. They will see that an “odd piece of data” hasmuch more impact on the mean than on the median.

Students might find situations for which averages are described andtry to decide whether it was a mean, median, or mode that was beingreported in each case, e.g., A baseball player’s batting average is amean. If the average price of a house in a particular subdivision isreported, however, it might more likely be a median or a mode. Themode is the usually the average used to describe shoe size.





Paper and PencilF8.1 Ask students to create sets of three numbers for which themean is a lot less than the median.

F8.2 Ask students to change just one piece of the following data toincrease the median:

2, 3, 4, 5, 6.

F8.3 Ask students to create two different sets of data, both with amode of 3, one with a mean the same as the median, and one with amean different from the median.

F8.4 Ask students, Are the mean and median of this data the same?30, 35, 37, 39, 49

InterviewF8.5 Ask students to identify data situations for which the means areusually greater than the medians.

F8.6 Ask students for examples of situations in which it might bedifficult to determine a mode.

F8.7 Ask, Do you believe that the mean or the median is the mostappropriate average to use to describe scores on a test? Why?

F8.8 Tell students that an average amount of TV viewing time wasreported as 20 hours a week. Ask, Which average do you think isbeing used? Why?

F8.9 Tell students that a shoe salesman said the average men’s shoesize is 10. Ask students what average they think he is referring to andwhy. Ask them why a shoe store is not likely to find the mean shoesize very meaningful.





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v) formulate and solve simpleproblems (both real-worldand from other academicdisciplines) that involvethe collection, display, andanalysis of data and explainconclusions which may bedrawn

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F9 explore relevant issues forwhich data collectionassists in reachingconclusions

There are many opportunities to includestatistics in the sixth-grade curriculum.In so doing, students review manymathematical ideas, relate mathematicsto the real world, and extend their ideasabout statistics. (NCTM 1992c, 16)

F9 Students should continue to think about how to collect anddisplay data; however, they should be concentrating on the analysis ofdata. In particular, students should consider how to interpolate orextrapolate from data provided.

Interpolating involves describing data between existing pieces ofinformation while extrapolating involves describing data extendedbeyond the presented information, e.g., Suppose students have thefollowing information from a survey in a certain school district:

Students should be able to describe what the data tells us—olderstudents generally do more homework but there is not necessarily anincrease each grade. They might also consider whether they could usethe data to determine what would be an average amount of home-work at grade 8 and what the dangers might if they extrapolated.

• Provide the following data:

Ask students to draw a scatterplot and ask whether they couldpredict a math test score of students who do 20 minutes of home-work per day. Have the students explain their reasoning.





PerformanceF9.1 The following data describes the percentage of the populationof Canada that was rural in different years:

Ask students to draw a scatterplot and analyse the data. Ask them topredict what the value was in 1996 and in 1982, and to comment onthe degree of confidence they have in their predictions.

PresentationF9.2 Ask students to find an Internet site that displays data aboutthe attendance at sports events for a particular team over a period ofyears. Ask them to display the data. Ask, Could you use the informa-tion to predict the attendance in future years?

F9.3 Have students collect and display information about thechange in the cost of postage stamps over the last 50 years. Then askthem to predict the cost to mail a letter in the year 2020, based onthe data. Ask them to be able to justify their predictions.

F9.4 Have groups of students select a question to which they wouldlike an answer. Ask them to collect and display the appropriate datathat will help answer their question. Have them discuss their dataand conclusions. Three examples of questions that might be selectedare: What is the level of physical exercise of eleven-year-olds in ourprovince? In what ways does one school differ from another? What arethe proportions of various problems that a local doctor or hospitaltreats?

Data Management


ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4– 65-106


Elaboration— Instructional Strategies/Suggestions


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i) explore, interpret, and makeconjectures about everydayprobability situations byestimating probabilities,conducting experiments,beginning to construct andconduct simulations, andanalysing claims which theysee and hear

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G1 conduct simple experimentsto determine probabilities


G1 Students should continue to perform simple experiments and usefractions to indicate the experimental probabilities that result. Althoughdice, coins, and spinners are typically used for these experiments, thereare other materials which can be used and contexts that also providepractise in computational skills as illustrated by the following activities.

• Have students conduct simple experiments on hundreds charts. Havethem begin at a designated number and roll a die to determinewhere to go next:

1— down 1 and right 12— down 2 and right 23— down 3 and left 14— down 4 and left 25— down 56— up 1

They then determine the probability that after 5 rolls they will landin some designated range of numbers, or on a certain type ofnumber, such as an even number or a multiple of 3.

• Have the students use the random number function on calculators orspreadsheets to generate two 2-digit numbers and to add the num-bers. This is repeated a number of times to determine the probabilitythat the sum of the numbers will be greater than 100.

• Have pairs of students take turns rolling two dice. One number isused as the numerator of a fraction and the other as the denominator.Have them determine the probability that the fraction generated is inlowest terms, i.e., has no equivalent using smaller numbers.

Students should use decimals to describe experimental results, e.g., If anevent occurs 9 times out of 100, students could report the probabilityas

9

100 or 0.09. Students should recognize that the more times they

repeat experiments the more reliable the probability will be, e.g., Ifthey were trying to determine the probability of getting a 3 when adie is rolled and only rolled the die 6 times, the result would verylikely be less reliable than if they had rolled the die 36 times.

If your students have access to the technology, they should conductsimulations of these types of experiements to determine experimentalprobabilities.This would allow them to see the effect on probabilities ofincreasing the number of trials to several hundreds or thousands.

ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4– 6 5-107




PerformanceG1.1 Have pairs of students roll a die 4 times. Ask them to create two2-digit numbers and subtract them. Have them repeat the experiment20 times. Have them calculate the probability that the difference theyget is less than 10. Have them repeat the experiment another 20 timesand compare the probability for 20 rolls compared to 40 rolls.

G1.2 Provide spinners marked in 10 equal sections, labelled 0,1,2, ...9.Ask the students to spin the spinners five times and total the numbersspun. Have them repeat this 10 times. Ask, What is the probability thatthe sum of the five numbers is greater than 25? Have the studentscompare their findings. Combine all their results for a class value.

G1.3 Provide pairs of students with bags containing 20 green cubesand 5 red cubes. Ask that one student pull out a cube, state the colour,and return it to the bag while the other student records the color. Havethem repeat this 20 times. Have students report their probabilities that agreen cube was chosen.

G1.4 Ask students to place a counter on the number 50 on hundredscharts and to flip a coin. Explain that if the result is heads, they movedown one space and if the flip is tails, they move up one space. They willflip the coin 10 times to complete one experiment. They should repeatthe experiment 20 times in total. The objective is to calculate theprobability that they will move their counters off the board during thecourse of the experiment. Have them share the experimental probabili-ties they got and combine their results to calculate a class value.

Paper and PencilG1.5 Tell students that you rolled a pair of dice 25 times and the sumof the numbers was 8 on 4 of the rolls. Ask, What is the experimentalprobability that the sum is 8? Does this seem reasonable?

InterviewG1.6 Ask students to describe an experiment to determine the prob-ability that the difference between the numbers on a pair of dice is 1.

G1.7 Tell students that two people performed an experiment where acoin was tossed and the probability of tossing a head was calculated.One person got a probability of

3

5. The other person got a probability of

47

100. Ask students which person they think got the more reliable result

and why.

ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4– 65-108


Elaboration— Instructional Strategies/Suggestions


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ii) determine theoreticalprobabilities using simplecounting techniques

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G2 determine simple theoreticalprobabilities and usefractions to describe them

... Theoretical probability is based on alogical analysis of the experiment, not onexperimental results. (Van de Walle 1994,383)

G2 Experimental probabilities are calculated by performing experi-ments. Theoretical probabilities are calculated by determining all thepossible outcomes of an event and comparing how many times a par-ticular outcome occurs to the total outcomes. If a die is rolled 60 times,the number 4 might come up 15 times giving an experimental probabil-ity of

15

60. However, the theoretical probability is

1

6, because there are

six equally likely possible outcomes (1, 2, 3, 4, 5, 6) when a die isrolled and one of these outcomes is the number 4—1 is compared to6 to get the ratio

1

6 .

Students need experiences to show them that the more times that anexperiment is conducted, the closer the experimental probability gets tothe theoretical value.

Another consideration in determining theoretical probabilities is thelikelihood of an outcome. In the case of the rolling of a die, all sixnumbers have an equal chance of being rolled—we say that all outcomesare “equally likely.” On the other hand, consider the spinner shownbelow.

Even though there are three outcomes, they are not equally likely. Thetheoretical probability of spinning a 1 is

1

2, not

1

3. This can be

determined by calculating the fractional part of the spinner coveredby 1.

Students should describe theoretical probabilities using both frac-tions and decimals, e.g., Students should be able to determine thatthe theoretical probability of:

• spinning a B is 1

3 or 0.333... • rolling a 5 is

1

6

• rolling an even number is 1

2 or 0.5

ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE: GRADES 4– 6 5-109




PerformanceG2.1 Ask students to put coloured cubes in a bag so that the theoreticalprobability of choosing a red one is

1

2 and choosing a green one is

1

4.

Ask, Why is there more than one way to model this situation?

G2.2 Ask students to list the first 20 multiples of 3 and to deter-mine the probability that a multiple of 3 is also a multiple of 6 and isalso a multiple of 9.

Paper and PencilG2.3 Provide hundreds charts. Ask students to determine the theoreti-cal probability that a number randomly chosen on the chart (a) ends in a5, (b) is even, (c) is less than 50, and (d) is on the diagonal.

G2.4 Provide students with array forms of the multiplication tables.Ask them to determine the theoretical probability that two 1-digitnumbers chosen randomly would have a product that is (a) even, (b) lessthan 5, (c) greater than 70, and (d) 24.

InterviewG2.5 Ask students why the probability that the sum of the numbers ona pair of dice will be 3 is not the same as the probability that the sumwill be 7.

G2.6 Ask students to explain how to determine the theoreticalprobability of rolling a 3 on a regular die and how this would changeif the die contained the numbers 1, 3, 3, 3, 5, and 6.

atlantic canada mathematics curriculum guide: grades...

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