Grade 8 - Cape · PDF filethe development of this grade 8 mathematics curriculum guide. ......

Grade 8

ATLANTIC CANADA MATHEMATICS CURRICULUM i

INTRODUCTION

Acknowledgments

The departments of education of New Brunswick, Newfoundland andLabrador, Nova Scotia, and Prince Edward Island gratefully acknowl-edge the contributions of the following groups and individuals towardthe development of this grade 8 mathematics curriculum guide.

• The Regional Mathematics Curriculum Committee; current andpast representatives include the following:

New BrunswickGreta Gilmore, Mathematics Teacher,Belleisle Regional High School

John Hildebrand, Mathematics Consultant,Department of Education

Joan Manuel, Mathematics/Science Supervisor,School District 10

Nova ScotiaRichard MacKinnon, Mathematics Consultant,Department of Education & Culture

Sharon McCready, Mathematics Teacher,Sherwood Park Educational Centre

Newfoundland and LabradorRoy Hodder, Acting Vice Principal,MacPherson Junior High School

Patricia Maxwell, Mathematics Consultant,Department of Education

Prince Edward IslandClayton Coe, Mathematics/Science Consultant,Department of Education

Joan Kennedy, Mathematics Teacher,Stonepark Intermediate School

• The Provincial Curriculum Working Group, comprising teachersand other educators in Newfoundland and Labrador, which servedas lead province in drafting and revising the document.

• The teachers and other educators and stakeholders across AtlanticCanada who contributed to the development of the grade 8 math-ematics curriculum guide.

ACKNOWLEDGEMENTS

ATLANTIC CANADA MATHEMATICS CURRICULUMii

INTRODUCTION

ATLANTIC CANADA MATHEMATICS CURRICULUM iii

INTRODUCTION

I. Background and

Rationale

II. Program Design

and

Components

III. Assessment

and Evaluation

IV. Designing an

Instructional

Plan

V. Curriculum

Outcomes

Specific Curriculum

Outcomes

Table of Contents

TABLE OF CONTENTS

A. Background .................................................................................... 1B. Rationale ........................................................................................ 1

A. Program Organization .................................................................... 3B. Unifying Ideas ................................................................................ 4C. Learning and Teaching Mathematics .............................................. 6D. Meeting the Needs of All Learners ................................................. 6E. Support Resources .......................................................................... 7F. Role of Parents ............................................................................... 7G. Connections Across the Curriculum............................................... 7

A. Assessing Student Learning ............................................................. 9B. Program Assessment ....................................................................... 9

Designing an Instructional Plan ........................................................ 11

Curriculum Outcomes ...................................................................... 13

Number Sense .................................................................................. 8-1Operation Sense ............................................................................. 8-13Patterns and Relations .................................................................... 8-39Measurement ................................................................................. 8-53Geometry....................................................................................... 8-69Data Management ......................................................................... 8-81Probability ..................................................................................... 8-95

ATLANTIC CANADA MATHEMATICS CURRICULUMiv

INTRODUCTIONTABLE OF CONTENTS

ATLANTIC CANADA MATHEMATICS CURRICULUM 8-1

SPECIFIC CURRICULUM OUTCOMES, GRADE 8

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

Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Number Concepts/Number and Relationship

Operations

General Curriculum Outcome A:

Students will demonstrate number sense andapply number-theory concepts.

ATLANTIC CANADA MATHEMATICS CURRICULUM8-2



Elaboration – Instructional Strategies/SuggestionsKSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toi) demonstrate an understanding

of number meanings withrespect to integers and rationaland irrational numbers, andexplore their use in meaningfulsituations

iii) represent numbers in multipleways and apply appropriaterepresentations to solveproblems

SCO: By the end of grade 8,students will be expected toA1 model and link various

representations of squareroot of a number

A2 recognize perfect squaresbetween 1 and 144 andapply patterns related tothem

A1 Students should be able to model perfect squares and square rootsthrough the use of blocks or grid paper. They should make a link betweenthese concrete or pictorial representations of square root and the numericalrepresentations (e.g., 100 - 81 = 19). In the figure below, students should beencouraged to view the area as the perfect square, and either dimension of thesquare as the square root.

6

6

A2 Students should be able to recognize automatically each of the perfectsquares from 1 through 144. They should also be exposed to other perfectsquares up to 400. Some students will be able, at the end of this topic, torecognize the perfect squares up to 400. They should work with patternsrelated to perfect squares of any size. Automatic recognition of perfect squareswill be very useful in later work with algebra and number theory as well as forthe recognition of the reasonableness of results that involve square rootsachieved using a calculator.

It is also valuable to bring out the patterns that emerge from a list of perfectsquares; that is, students should recognize that the differences between theperfect squares increase in a consistent way as shown in the pattern below:

1 4 9 16 25 36 49

3 5 7 9 11 13

These increases can be graphed in conjunction with GCO (C) to observe thepatterns graphically. Another pattern that can be observed includes therecognition that the sum of the square roots of two consecutive perfectsquares is equal to the difference between those two perfect squares. Forexample, 25 + 36 = 6+ 5 =11 and 36 - 25 = 11. Likewise,

100 + 81 =10+ 9 = 19 and 100 - 81 = 19. This pattern can be useful forfinding the next perfect square beyond a known one. For example, if thestudent knows that 122 = 144, then 132 = 144 + 12 + 13 = 169.Such observations allow some students to develop a depth of understandingthat goes beyond the level of the intended outcome. This depth ofunderstanding will assist in retention and connections with other topics.





PerformanceA1.1 Ask students to use grid paper or square tiles to show the square rootof 36, 49, and 81.

A1.2 Ask students to use grid paper or square tiles to show all the perfectsquares less than 150.

Pencil and PaperA2.1 Use square roots to solve each of the following:a) A square has an area of 81 m2. What are the dimensions?b) A cube has a surface area of 294 m2. What are the dimensions?

A2.2 Jackie wrote the following string of consecutive perfect squares: 4, 9,16, 25, 36, 49. When she calculated the differences between the numbers,the following was found: 5, 7, 9, 11, 13. She decided to calculate thedifferences between the numbers once again, and found the following:2, 2, 2, 2. If Jackie started with the set 64, 81, 100, 121, and 144 andfollowed the same procedures, what would she observe?

PortfolioA2.3 John observed that the sum of the square roots of two consecutiveperfect squares was equal to the difference between the perfect squares. Askstudents to select three sets of consecutive perfect squares and use them toconfirm whether or not they think John’s observation is correct.

[For some students, as an extension, this relationship can be proven correctmore generally later in the course, once students have worked with algebratiles. They can use algebra tiles as follows without any formal discussion ofmultiplication of binomials.

x x

1

The dimensions of the squares are x by x and x+1 by x+1.The area of the first figure is x2, and the area of the second figure isx2 + x + x + 1. The difference between the two squares isx2 + x + x + 1 - x2 = x + x + 1, or 2x + 1. Since the dimensions of the squaresrepresent the square roots, the sum of the lengths for the two squares wouldbe x + x + 1 or 2x + 1. This should show that the relationship is true for allpositive values of x.]




Elaboration – Instructional Strategies/Suggestions

A3 It is very important to emphasize the difference between an exact square rootand the decimal approximation. Students should also understand the differencebetween the decimal value for the square root of a non-perfect square and thedecimal value for a rational number, that is, they need to be aware that the squareroot of a non-perfect square has a decimal portion which does not repeat. Thisdistinction will help them to grasp the difference between numbers that arerational and those which are not. It would be appropriate, though not essential,to discuss the term irrational number at this point. Many students will want toattach a name to numbers which are not rational and, if this is the case, thecorrect terminology should be applied. Students can describe a number such as

6 as an irrational number. They should view 6 as the exact value and2.4494489743 as an approximation, regardless of how many places the numberextends beyond the decimal point. Students should be able to model square rootsfor non-perfect squares.

A4 Students will develop a greater intuitive understanding of square root throughpractising estimation skills. For numbers between 1 and 144, it is important tobe able to identify between which two whole numbers the square root will fall. Itis also important to be able to approximate to the point where students canidentify which whole number is closer to the square root. For example, studentsshould know that the square root of 22 is between 4 and 5, and that it is closer tothe 5 than the 4. That is, students should identify that the square root fallsbetween 4.5 and 5.0.

Equally, students should use patterns to determine that since the square root of16 is 4, then the square root of 1600 is 40. Drawing from their experiences withestimation, they should also recognize that the square root of 2200 is between 40and 50, but closer to 50.

Prime factorization is a method used to find the square root of perfect squares.For example, consider 576 .Since 576 = 2 × 288= 2 × 2 × 144= 2 × 2 × 12 × 12= 24 × 24, then 576 = 24.

Some students may continue the factoring as follows before identifying thesquare root:= 2 × 2 × 4 × 3 × 4 × 3= 2 × 2 × 2 × 2 × 3 × 2 × 2 × 3= (2× 2× 2× 3) × (2× 2× 2× 3) group factors in two equal groups= 24 × 24 ∴ 576 = 24.

An alternative approach would be:

400 = (2× 2)× (2× 2)× (5× 5)(organize factors into pairs)

select one factor from each pair to get 2× 2× 5 = 20 ∴ 400 = 20.

Likewise, students may recognize 400 = 4 ×100 = 2×10 = 20.

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toi) demonstrate an understanding



SCO: By the end of grade 8,students will be expected toA3 distinguish between an exact

square root of a number andits decimal approximation

A4 find the square root of anynumber, using anappropriate method





PerformanceA3.1 Jan used grid paper to show that the square root of 20 is not a whole number.She formed a square using 16 blocks. The 4 additional blocks she cut in half andplaced 4 on each of the dimensions of the 4 x 4 square. This produced a figure asshown:

Ask students toa) use the diagram to estimate the square root of 20b) find the square root of 20 on the calculatorc) use the diagram to help justify why there is a difference between the results in part

a) and part b)d) use a similar diagram to estimate the square root of 30

Pencil and PaperA3.2 Julia wanted to find the area of a rectangle with a length of 12 cm. She knew thewidth of the rectangle was the same as the lengths of the sides of an adjacent square.The area of the square was 58 cm2. She used her calculator as follows: 58 = 7.6,therefore, 12 cm× 7.6 cm = 91.2 cm2. Kim solved the same problem as follows:

58 × 12 = 91.4 cm2. Account for the difference in the results.

A.4.1 Find the square root of each of the following, using patterning and/or primefactorization. Justify your answer.a) 6400b) 12 100c) 900d) 676

InterviewA4.2 Ask students to use their knowledge of the square root of 81 to help find thesquare root of 8100. Ask them to explain why it is not as easy to find the square rootof 810 by using information about the square root of 81.

A4.3 Ask students to explain how to find the square root of640 000, using patterns.

A4.4 Ask students why 8 cannot equal a whole number.

PresentationA4.5 Ask students to explaina) why prime factorization cannot be used to find the exact square root of numbers

that are not perfect squaresb) why prime factorization can be used to find the exact square root of numbers that

are perfect squares

Suggested Resources





A5 In grade 7, students worked with scientific notation as it pertained tolarge numbers. Some students may have encountered scientific notation forsmall numbers through using calculators. These very small numbers ofteninvolve powers with negative exponents. (Students should be aware thatsome calculators use an “E” to identify error, while others use “E” to displayexponents.)

Negative exponents are often encountered in the context of place-valuecharts, which often show the place values of tenths, hundredths, andthousandths as 10-1, 10-2, and 10-3. Such experiences can serve as a startingpoint in introducing negative exponents. Generally, negative exponents areintroduced using a pattern such as:

1000 103

100 102

10 101

1 100

0.1 or 110 10-1

0.01 or 1100 or 1

102 10?

0.001 or 11000 or

1103 10?

The main focus should be on work with base ten; however, as an extension,students can also explore other bases (such as base 2 and 3), since these areparticularly useful for work with patterning for exponential growth anddecay. Base-ten blocks were suggested in working with exponents at thegrade 7 level. They should also be considered as a starting point ofinstruction with this outcome.

A6 Some experiences with very small numbers may have occurred throughthe science program. Examples of small numbers include diameter of a cell,diameter of an electron, mass of a hummingbird, and mass of an insect.While the new work related to scientific notation in grade 8 is related tosmall numbers, practice should also include some large numbers as well sothat students make a clear distinction in their minds about how the twodiffer.

In writing numbers in scientific notation, it would be worthwhile to makethe connection to multiplying by 0.1, 0.01, 0.001, ... For example, 0.00621= 6.21× 0.001= 6.21× 10-3 (the link between 0.001 and 10-3 should be clear).

Students should be able to translate numbers like 0.0000361 and651× 10-5 to scientific notation.




SCO: By the end of grade 8,students will be expected toA5 demonstrate and explain the

meaning of negativeexponents for base ten

A6 represent any numberwritten in scientific notationin standard form, and viceversa





Pencil and PaperA5.1 Put 103 ÷ 10 = into your calculator. Keep pressing the equal sign andrecord the list of answers in a table as shown:

103 103 ÷ 10 100 or 102

102 102 ÷ 10 10 or 101

101 101 ÷ 10 1 or 100

100 100 ÷ 10 0.1 or 10-1

10-1 10-1 ÷ 10 0.01 or 10-2

10-2 10-2 ÷ 10 0.001 or 10-3

10-3 10-3 ÷ 10 0.0001 or 10-4

. . .

a) At what point does the calculator convert the numbers to scientificnotation?

b) What does this tell you about the calculator display?c) Rewrite each of the numbers in the table in scientific notation.Try the same activity, but this time enter 10 x 10 in your calculator.Continue to press the = sign and record in a similar manner to that of thetable shown. Repeat a), b), and c) for the new table.

A6.1 Compare 4.2 × 10-3, 42.3 × 10-4, and 0.421 × 10-2, and arrangethem from least to greatest. Justify your ordering.

InterviewA6.2 John read in a magazine article that the mass of a certain insect is2.3 × 10-3 g. Ask students how they would explain to John the meaning ofthe -3.

JournalA6.3 In an article about the solar system, the following is stated: “Mercuryis very close to the sun. The distance from Mercury to the sun is5.8× 10-7 km.” Ask students if this seems reasonable, and to explain why orwhy not.

Research/Mini-ProjectA6.4 Ask students toa) use the Guinness Book of World Records to find examples of very large

and very small things that might be suitably recorded using numbersin scientific notation

b) ask their science teacher to identify some things related to sciencewhich are typically represented using scientific notation, and to rankthem in order of size





A7 Since students have worked with integers, positive fractions, and decimals inprevious grades, they should not have difficulty extending this knowledge toinclude negative decimal and fractional numbers. The issue of placement of thenegative in a fraction should also be addressed. It is important for students tounderstand that -2

3 , - 23 , and 2

-3 are all equivalent fractions. This can be shownparticularly well using problems such as 6

-2 , -62 , and - 6

2 , since the division canbe readily done and students can see that the results are the same, regardless ofwhere the negative sign is placed.

Comparing and ordering numbers draws largely upon the students’ previouslydeveloped repertoire of mental strategies and observations. Many of thestrategies mentioned have been used in previous grades but are now applied toan extended number set. Also, many students can come up with strategiesthemselves. These strategies should also be added to the list. Some possiblestrategies include the following:

• A negative is always less than a positive.

• Positive fractions with common denominators can be compared byexamining numerators (e.g., 3

8 is less than 58 because 3 < 5). Likewise,

when denominators are the same, a larger numerator implies a greaterfraction. These observations are similar for negative fractions (e.g., -

38 is

greater than - 58 because -3 is greater than -5).

• Positive fractions with common numerators can be compared by examiningdenominators (e.g.,

35 is greater than 3

6 because 5 < 6). Likewise, whennumerators are the same, a smaller denominator implies a greater fraction(e.g., -

35 is less than -

36 because -5 is greater than -6). Some students may

have more difficulty with these relationships when they involve negativenumbers and may need to consider a number of cases to be convinced thatthe strategy applies.

• Fractions can be compared to reference points such as 1, 12 , -1, and - 1

2 .This comparison can be done using a number line (e.g., -

13 is greater than

- 58 because it is to the right of - 1

2 , while - 58 is to the left of - 1

2 ).

12

-1 0 1- 12- 5

8- 1

3

• When some numbers are presented in decimal form and others presentedin fractional form, comparison is often aided by changing all numbers to acommon form (often decimal form in order to do place valuecomparisons). However, when fractions are familiar, students should be ableto place them directly. For example, with -

12 and -0.6, students should

recognize, without conversion, which fraction is greater.

The number line will be an essential tool for many students in helping them tovisualize the relative positioning of positive and negative fractions and decimals.



ii) read, write, and order integers,rational numbers, and commonirrational numbers

SCO: By the end of grade 8,students will be expected toA7 compare and order integers

and positive and negativerational numbers (indecimal and fractionalforms)





Pencil and PaperA7.1 Find three numbers which lie between

a) 0 and -1

b) 12 and 1

3

c) - 3.5 and - 3.6

d) - 13 and - 0.4

e) - 23 and - 0.6

A7.2 Replace the with a value to make each sentence true. Explain why yourchoice is correct.

a) -2

> 35

b)37 <

3

c) -0.345 > - 0.34

InterviewA7.3 Ask students to explain their approach in comparing - 4

3 and - 127 .

PortfolioA7.4 Sarah’s class started a stock market club. Each student was given a day ofthe week to be responsible for checking the stock market reports. This morning,Sarah found that there was little change in most of their holdings, except forScotia Silver, which was down a quarter, and Brunswick Copper, which was upan eighth. As it happens, the price of both stocks were previously worth $18.00per share. The group holds 100 shares of each type of stock.a) Ask students if they think the overall value of the stocks is greater or less as a

result of the changes, and to explain their thinking.b) Have students create two other questions which can be answered based on

the information provided, exchange their questions with a partner, andanswer each other’s question.

c) Tell students that, if Sarah had the same stocks on a Canadian exchange, thechange in value would be quoted in decimals. Ask them to represent thestock changes as decimal fractions.

[Note: This problem may require some explanation since many students will notunderstand the significance of stock market changes.]

ProjectA7.5 An interesting project would involve giving each student an imaginary$100 000 to invest in the stock market. Have each student chart once a week thechanges in the value of the stock. This chart can be used to create a graph, aswell as to enhance the study of fractions and decimals. It might be interesting tohave each student choose some stocks from a U.S. exchange and others from aCanadian exchange to ensure a mixture of fractions and decimals. The projectcan extend over several weeks, or an entire term or semester.

Suggested Resources





A8 Students have worked with percent in previous grades. Percents greaterthan 100, however, can be somewhat abstract for many students. Whenstudents hear references to percent, it is often in the context of percentincrease and percent decrease. Other references to percent which can bevery confusing include statements such as, “He gave it a 110% effort.”

In modelling percents greater than hundred, consider the following:

If represents 100%, then represents 200%. If we start with , anincrease of 200% would result in . Likewise, 150% of would berepresented by .

It may be possible to link percents greater than 100 to social studies. Forexample, students may have heard of inflation rates as being on the orderof 200% over a short period of time in certain countries of the world. Theymay also be able to relate percents greater than 100 to change in price overtime. For example, students can compare the cost of an item such as a sodapop in their parents’ youth with the cost today. Such comparisons usuallylead to increases that far exceed 100%. Monthly inflation rates or changesin bank rates on a monthly basis provide good examples of percentages thatare a fractional part of 1%.

A9 Students should understand that a proportion is a statement of equalitybetween two ratios. The emphasis should be on developing proportionalreasoning. Opportunities to apply proportional reasoning occur often inmeasurement, particularly in relation to scale. There are also manyopportunities to apply proportional thinking to money and time, as well asto other aspects of measurement.

A more procedural approach to working with proportions occurs inoutcome B2. In this outcome, students will begin their work withproportion informally, solving problems solely through the use of theirconceptual understanding of equivalent ratios and rates.

Many real-world problems involving proportional thinking can be solvedby recognition of the relationships between numbers.

If a 3-pack of juice costs $1.29, what would 12 juice packs cost?[Students might recognize that since 3× 4 = 12, then $1.29× 4 wouldgive the cost of 12.]

If a 6-pack of pop costs $2.89, what would 9 pop cost? [Studentsmight recognize that since 6× 1

12 = 9, then $2.89× 1.5 would give the

cost of 9 pop.]

Other strategies are outlined in association with B2.




SCO: By the end of grade 8,students will be expected toA8 represent and apply

fractional percents, andpercents greater than 100, infraction or decimal form,and vice versa

A9 solve proportion problemsthat involve equivalent ratiosand rates





PerformanceA8.1 Tell students that a flat from a set of base-ten blocks represents 100% ofsomething. Ask them to use base-ten blocks to representa) 110% c) 200%b) 125% d) 450%

Pencil and PaperA8.2 John’s father said “In my youth I could buy a chocolate bar and a soft drink for20¢.” What would be a typical cost for these items today? Estimate the percentincrease this represents.

A8.3 A pair of figure skates cost $235.00 last year. They are sold this year for$236.00. What percent increase does this represent?

A8.4 Sarah has a savings account that earns 12 % interest monthly. Jane has a savings

account where she earns 534 % annually. Who do you think would have more money

in the bank at the end of one year if they both start with the same amount? Why?

A8.5 When Jane saw the interest rate of 0.9% per annum, she thought that it was areasonable return on her money. Jack advised her that it was a very low rate ofreturn.a) Why do you think Jane thought it a good rate of return?b) Why do you think Jack thought it a bad rate of return?

A9.1 A recipe uses 500 mL of flour for every 125 mL of sugar. How much flourwould be needed when 500 mL of sugar is used?

A9.2 Suzellée found a good deal on pop. She could buy 12 packs for $2.99. Sheneeds 72 pops for her party. Explain how she can calculate the cost.

InterviewA9.3 Tell students that when making lemonade Sue uses 5 scoops of powder for6 cups of water, and Sarah uses 4 scoops of powder for 5 cups of water. Ask studentsthe following:a) Are the situations proportional to each other? Explain why or why not.b) In which situation is it likely the lemonade will be more flavourful? What

assumptions did you make?

Journal EntryA8.6 A certain chemical is dangerous to humans if it is found in the water supply atmore than 275 parts per million.a) Ask students what percent this represents.b) Ask them what presence of the chemical would represent a danger level in a

kilolitre of water.

A9.4 Ask students to discuss in their journals whether or not the following could besolved using a proportion:David is 6 years old and Ellen is 2 years old. How old will Ellen be when David is12 years old?

Suggested Resources



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


Number Concepts/Number and Relationship

Operations

General Curriculum Outcome B:

Students will demonstrate operation sense andapply operation principles and procedures in

both numeric and algebraic situations.





B1 The mathematical properties—commutative (order), associative(grouping), and distributive—have already been explored in grade 7, aswell as in earlier grades. However, they should be revisited so thatstudents can confirm their relevance to the number systems that are newto this grade level. That is, students should understand that, in the sameway that 6× 0 = 0 and -8× 0 = 0, it is also true that -

23 × 0 = 0. Likewise,

in the same way that -5× 4 = 4× (-5), it is also true that4.25× (-0.5) = -0.5× 4.25. Instruction should focus on reinforcing theusefulness of these properties with respect to the extension of thenumber system rather than recognition or matching exercises. Teachersshould refer to the properties by their formal names and encourage theirstudents to use terminology such as commutativity. Discussion shouldinclude why certain properties do not apply to subtraction and division.

When working with properties such as commutativity and associativity,students should be exposed informally to the notion of closure. Closurewas already referenced in grade 7. It is through the discussion of whethersets are closed to particular operations that students realize the need forextension to other systems. For example, in grade 7, discussion about thesolution to the question 2 - 5 led to the realization that the answer is notdefined within the set of whole numbers, which motivated the need forthe set of integers. Likewise, because -6 ÷ 4 is not defined for integers,students begin to appreciate that the set of fractional numbers extends tonegatives as well.

The focus should be on the usefulness of the properties for computation,especially mental computation. Examples:

1) -6× 5.2 + (-6)× 0.8 2) -8.2× 1.2think -6× (5.2 + 0.8) think -8.2× 1 + -8.2× 0.2= -6× 6 = -8.2 + -1.64= -36 = -9.84

3) -9× (-7.2)× 5× (-0.2)think -9× 5 = -45, then -45× (-0.2) = 9, and 9× (-7.2) = -64.8(Order is selected for the ease of the calculation.) This sameproblem might also be solved as follows:5× (-0.2) = -1, then -1× -9 = 9, and 9 × (-7.2) = -64.8

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected tov) apply estimation techniques to

predict, and justify thereasonableness of, results inrelevant problem situationsinvolving rational numbers andintegers

vi) select and use appropriatecomputational techniques ingiven situations and justify thechoice

SCO: By the end of grade 8,students will be expected toB1 demonstrate an

understanding of theproperties of operations withintegers and positive andnegative rational numbers(in decimal and fractionalforms)





Pencil and PaperB1.1 Jan solved these problems as follows:

(14 )× 27 + (

14 )× 73

15 × 43 +

15 × 57

17 × 700 -

17 × 35

= (14 )× (27 + 73) =

15 × (43 +57) =

17 × (700 - 35)

= (14 )× 100 =

15 × (100) =

17 × (665)

= 25 = 20 = 95

Chris solved the same three problems as shown:

(14 )× 27 + (

14 )× 73

15 × 43 +

15 × 57

17 × 700 -

17 × 35

= 6.75 + 18.25 = 8.6 + 11.4 = 100 - 5

= 25 = 20 = 95

a) Are both sets of solutions correct? Explain the thinking used in eachperson’s solutions.

b) If you wished to solve the problem mentally, which method would youchoose, the one used by Chris or by Jan? Explain why.

c) Create two problems, using the same procedures as Jan’s problem, butusing different numbers. One of the problems should be such that useof Jan’s method makes it easier to do mentally, and the other should besuch that use of Jan’s method is not an advantage.

d) Create two problems, using the same procedures as Chris’ problem,but using different numbers. One of the problems should be such thatuse of Chris’ method makes it easier to do mentally, and the othershould be such that use of Chris’ method is not an advantage.

InterviewB1.2 Ask students why the commutative property would be useful infinding this answer mentally: 5

4 × 37 × 45

B1.3 Ask students to find the solution to each of the following mentallyand to explain their strategies.

a) -34 × 17 × 5

24 × 0

b) 5 13 + 4 1

5 + 313 + 7 + 2 1

3





B2 Students should be provided with a number of strategies for solving problemsinvolving proportionality. One strategy is to look for relationships between thevarious terms of a proportion and use these relations to solve for missing values.

Ask students to find how long it would take to produce a 35-page essay ifthey can produce a 5-page essay in 2.2 hours.To solve this problem, studentsmight recognize that since 7× 5 = 35, then 35 pages should take about7 times as long to produce. They may instead create the followingproportion,

2.25 = x

35 . In solving this proportion, students could beencouraged to consider equivalent fractions; that is, we can multiply thefraction

2.25 by a form of one (i.e.,

77 ) to obtain an equivalent fraction with a

denominator of 35; since 15.435 = x

35 , x must equal 15.4. Thus it will take15.4 hours to produce a 35-page essay.

If the problem were changed so that the student was asked how long it wouldtake to produce an 18-page essay, the new proportion would be

2.25 = x

18.Students should recognize now that multiplying 2.2

5 by a form of one, usingintegers, will not produce a proper fraction with a denominator of 18. A differentstrategy might be more appropriate; for example, the solution may be found bymultiplying each side of the proportion by a different form of one, as shownbelow:

2.25 = x

18

2.2 × 185 × 18 = x × 5

18 × 5

39.690 = x × 5

90

Through discussion, students should realize that, since the denominators areequal and both fractions are equal, then the numerators must be equal. Now theproblem is simplified to 39.6 = x× 5. This can be solved by either guess-and-check, re-writing the multiplication sentence as a division sentence (39.6 = x× 5can be written as 39.6÷ 5 = x), or using more formal equation-solving techniquesto find the unknown.

The above problem could also have been solved by determining what number,when multiplied by 5, would equal 18. This number, 3.6, could then bemultiplied by the numerator, 2.2, to solve for x. This is simply using the firststrategy, but now recognizing that decimal numbers could be used as well toobtain the form of one; e.g., 3.6

3.6 .

Another strategy to solve proportions is the unitary method. For example, to findthe cost of 18 bars if 10 bars cost $2.19, the following proportion may be writtento help solve the problem, 2.19

10 = x1 . By solving this, the students will see that

each bar will cost $0.219 or $0.22, so 18 bars will cost 18× $0.22.

Consider how the strategies above could be used to solve proportions if theunknown appears in some other location; e.g., x

3 = 912 or 2

7 = 30x .

Note: The elaboration for B2 is continued on the next 2-page spread.

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toii) model problem situations

involving rational numbersand integers

iii) apply computational procedures(algorithms) in a wide varietyof problem situations involvingfractions, ratios, percents,proportions, integers, andexponents

v) apply estimation techniques topredict, and justify thereasonableness of, results inrelevant problem situationsinvolving rational numbersand integers


SCO: By the end of grade 8,students will be expected toB2 solve problems involving

proportions, using a varietyof methods





Pencil and PaperB2.1 Study each of the proportions and estimate which of a), b), c), or d)represents the largest value. Solve to verify your estimate.

a)37 = a

28 b)b9 = 3

4

c) 5c = 15

33 d) 58 = 23

d

B2.2 Use proportions to find which is the better buy: 1.2 L of orange juicefor $2.50, or 0.75 L of orange juice for $1.40. Explain why it is the betterbuy.

B2.3 Walter and Pat have the same ratio of cats to dogs in their kennels.Walter has three cats for every five dogs.a) In September, Pat had 25 dogs. How many cats did she have?b) In January, Pat had 48 cats and dogs altogether. How many of Pat’s

animals are dogs?

B2.4 What is the scale of a map if 7.2 cm on the map represents a distanceof 1800 km?

B2.5 The ratio of girls to boys in Mr. Gosse’s homeroom is 18 to 12. Whatproportion of the students in Mr. Gosse’s homeroom are boys?

InterviewB2.6 Ask students to explain whether or not, in the ratio 3:8 = 17: ,

can be a whole number.

B2.7 Ask students to explain why 1:20 000 000 is another way to describethe ratio of 1 cm representing 200 km on a map.

PortfolioB2.8 A statue of John Cabot was made from a model. The height of themodel was 25 cm. Ask students to find the height in metres of the statue ifit was made using a scale of 1:15 [scale represents ratio of model to actualheight].

B2.9 On a photograph a man appears 3 cm tall. Ask students what is likelyto be the scale, and to explain their reasoning.





B2 (Cont’d) It is important that students see the usefulness of proportions.The topic is rich in problem-solving opportunities and lends itself to real-world applications; for example, proportions are commonly used for scalemodels, altering a recipe, and comparison shopping. It is when proportionsare embedded in a real-life context that students can more readily relate tothe process used in finding unknown values. The study of scale is animportant application of work with ratio and proportion, and connectswell with the component of transformational geometry related toenlargements and reductions (dilatations) studied at this grade level.

If the scale on a map is 1:50 000, what is the actual distance betweentwo towns, if they are 7 cm apart on the map? Students may recognizethat they need to multiply by 7 and perform this computation directly,or set up and solve a proportion.

150 000 = 7cm

x

x = 350 000cm or 3.5 km

Teachers may wish to study enlargements and reductions at the same timeas ratio, rate, and proportion. Also, outcomes A9 and B2 should beaddressed together.

B3 In the relationship a% of b = c, students should be able to estimate andcalculate a percentage of a given number (find the value of c), thepercentage one number is of another number (find the value of a), and thewhole when a specified percentage is given (find the value of b). This canalso be expressed as

a100 = b(part)

c(whole. All problems related to this topic can beclassified as one of the three types which follow. In each case, practiceshould include percents less than one and greater than 100.

Type 1 - Finding the percentage of a given number.Typical problem: Find 28% of 1200. This problem can be expressed as28% of 1200 = c. Students would normally solve the problem by changingthe percent to a decimal and multiplying, especially if they are solving theproblem using a calculator. However, it should be noted that a solution canbe found by using a proportion

28100 = c

1200. For this problem, studentsshould see that since 1200÷ 100 = 12, then 28× 12 is the answer.

A strategy to estimate the solution for this type of problem is to determinewhat 10% of the given number is, round the given percent to the nearest10, and multiply. In the above example, students should be able to see that10% of 1200 is 120, and, since 28% is close to 30%, therefore, 30% of1200 is 3 times 120. Also, students might round 28% to 30% and find30% of 1200 by multiplying mentally.

Note: The elaboration for B3 is continued on the next two 2-page spreads.






SCO: By the end of grade 8,students will be expected toB2 solve problems involving

proportions, using a varietyof methods

B3 create and solve problemswhich involve finding a, b,or c in the relationshipa% of b = c, usingestimation and calculation





Pencil and PaperB3.1 If 2% of a certain number is 0.46,a) what would 10% of the number be?b) what is the number?

B3.2 A jacket is now selling for $64. The sign above it indicates it wasreduced by 20%. What was the original selling price?

B3.3 McDunphy’s Burger Heaven has a sale on hamburgers. A hamburgeris 1

2 price when you buy a medium drink and a medium fries. The normalprices are as follows: hamburger - $2.30, medium drink - $1.29, andmedium fries - $1.39. What is the actual percentage off the regular pricewhen you take into account what must be purchased to take advantage ofthe sale?

B3.4 A politician was elected with 2145 votes at a convention. If shereceived 58% of the votes cast, about how many votes were cast? [Thesolution mentally might be as follows: 60% of = 2100, guess 3000; 60%of 3000 = 1800, guess 4000; 60% of 4000 = 2400. Since 2100 is exactlyhalf-way between the two guesses, a third guess might be 3500. Since anexact answer is not required, this seems like a reasonable estimate.]

InterviewB3.5 If 30 is close to 80% of a number, ask students what they know aboutthe number.

B3.6 Ask students, if is 60% of , what percent of 2 is 2 ? [This maybe challenging. Students may need to consider giving and values tohelp them think about the question. For example, if 12 is 60% of 20, whatpercent of 2 × 20 is 2 × 12?]

B3.7 Tell students that, in one of the Atlantic provinces, the residents mustcalculate the provincial income tax by finding 60% of the federal incometax. They must also pay a surcharge which is 5% of the federal income tax.If students already know the federal tax, ask them how they can find thetotal tax payable in one single computation.

B3.8 A certain number is between 10 and 100. Ask students to explainwhat can be concluded about 150% of this number.

B3.9 Ask students to explain why 60% is not a good estimate for whatpercentage 30 is of 70.





B3 (Cont’d)

Type 2 - What percentage one number is of another number.Typical problem: What percent of 15 is 12, or, written another way, 12 out of15 is what percent? Using a calculator, students might enter 12÷ 15 and convertthe answer to a percent. They might also write the problem as a% of 15 = 12.Students may translate this into a proportion such as

a100 = 12

15 . Solutions tosome proportions can be found by simplifying the known fraction(denominator and numerator are given) to a commonly used fraction. Forexample, since

1215 = 4

5 , the original proportion now becomes a

100 = 45 . This

proportion should be more straightforward to solve mentally. In instances wherethe known fraction cannot be simplified exactly to a commonly used fraction,an approximation might be appropriate. For example,

1124 might be simplified to

12 for the purpose of approximating an answer.

Type 3 - Finding the whole when a specified percentage is given.Typical problem: 25% of what number is 48? Again, this could be expressed as25% of b = 48, or as a proportion

25100 = 48

b . To estimate, students might thinkthat since 25

100 = 14 and 48 is close to 50, then 1

4 of what number is 50. This samestrategy might also be used to complete a mental calculation. Anotherestimation approach would be to realize that since the second numerator, 48, isnearly twice the first, 25, then the second denominator should be nearly twicethe first denominator, 100. Students can also solve 25% of b = 48, as follows:0.25 b = 48

b = 48÷ 0.25b = 192

Mental strategies can also be used in situations where an exact answer isrequired; e.g., when finding 28% of 1200, consider finding 30% of 1200 and2% of 1200 and subtracting. Another strategy would be to make use of specialcommon fractions; e.g., when finding 26% of 840, first take 25% of 840, or14 of 840 = 210. If an estimate is required, this will suffice. If an exact answer is

needed, find 1% of 840 = 8.40 and add.

In application problems, students will sometimes encounter occasions involvingtwo percentages. Students should realize that, when combining percentages,they cannot add the percentages directly; e.g., if a tennis racket was already onsale for 20% off and the store announced a sale which read, “30% off all itemsin the store, including items already on sale,” students should explore whathappens when the two percentages are combined (e.g., compare calculating a50% discount, versus taking off 20% followed by 30%). However, when anestimate is required, adding these two percentages would provide a roughestimate of the answer. Students should discuss whether the estimate will belarger or smaller than the actual answer, and whether the accuracy of an estimateis affected by the size and order of the discounts. Students should also explorewhen this strategy does not result in a good estimate, such as when 50% off isfollowed by an additional 50% off.







SCO: By the end of grade 8,students will be expected toB3 create and solve problems

which involve finding a, b,or c in the relationshipa% of b = c, usingestimation and calculation





Portfolio EntryB3.10 Elite basketball sneakers, which regularly sell for $185, were markeddown by 25%. To further improve sales, the discount price was reduced byanother 15%.a) Ask students what the final selling price was.b) Ask them what the total percent of discount on the original price was.c) Ask students what the total cost was after 15% tax was added.d) Tell students that Jane decided that, since the second discount was

15% and the tax was 15%, she would save herself a lot of work and letone balance out the other. This way she could calculate the firstdiscount and determine the total cost. Ask students to discuss herreasoning and decide whether it produces an accurate result.

B3.11 Tell students that Sarah found out that the new car she boughtwould depreciate in value by 20% per year. Sarah paid $20,000 for the carand planned to keep it for three years. She wanted to find the car’s value atthe end of three years and asked a friend to help. They decided to do theircalculations independently, and then compare answers. Sarah’s answer was$10,240, but her friend’s answer was $8000.a) Ask students how each of the answers was obtained.b) Ask them who they think is correct, and to explain their choice.





B3 (Cont’d) Students should also be made aware that problems involving adiscount can be solved in more than one way. For example, the discountedprice can be determined either by finding 20% of the original price andsubtracting, or, more efficiently, by finding 80% of the original price.

B4 Problem situations particularly well-suited to this outcome are relatedto finding the percentage markup or markdown in sales situations,Consumer Price Index, and fluctuations in the value of the dollar.

A dress cost $22 to make and was being sold for $40. What was thepercentage markup?

Note: Businesses usually express markup as a percent of the cost,although some businesses calculate it as a percent of the selling price.

percentage markup = (selling price - cost) ÷ cost= ($40 - $22) ÷ $22= 0.818181... or 82%

In general, percentage increase or decrease is found using the following:Percentage increase = increase

original amountPercentage decrease = decrease

original amount






SCO: By the end of grade 8,students will be expected toB3 create and solve problems

which involve finding a, b,or c in the relationshipa% of b = c, usingestimation and calculation

B4 apply percentage increaseand decrease in problemsituations

× 100%

× 100%





Pencil and PaperB4.1 Mr. Jones bought a mining stock at $35. Two weeks later he sold itfor $105. What was the percentage of increase?

B4.2 The Canadian dollar was valued at 70.0¢ U.S. on Friday. OnMonday the opening value was 68.5¢ U.S. What was the percentage ofdecrease?

B4.3 Mack’s Sound Emporium purchased CD players for $129 per unitand is planning to sell them for $195.99. They purchased 150 units.a) What is the percentage of increase (markup) per unit?b) How much can Mack expect to make if he sells all units?c) After 4 weeks, Mack realizes that the CD players are not selling as fast

as he hoped, so he puts them on sale for 20% off. Will there still beprofit? If he sells 56 units for the duration of the sale, how muchmoney will he make on the items sold?

PortfolioB4.4 Ask students to draw a rectangle and a triangle of any dimensions,and to solve the following:a) Find the area and the perimeter of each figure.b) Increase the dimensions of each figure by 30% and find the new

perimeter and area.c) Decrease the dimensions of each figure by 40% and find the new

perimeter and area.d) Find the ratio of new perimeter to original perimeter and the ratio of

new area to original area of each of part a) and b). What do younotice?

B4.5 Every month Statistics Canada calculates the Consumer Price Index(CPI), which is a means of measuring changes in retail prices. Ask studentsto research the CPI to determine how it is calculated and what it is usedfor.





B5 Students worked with addition and subtraction of fractions concretelyand pictorially in previous grades, and this work was continued in grade 7with a focus on estimating sums and differences. A brief review of thefollowing concepts may be necessary: equivalent fractions, lowest terms,and LCM. A number of manipulatives can be used to develop operationswith fractions concretely, including circle models, pattern blocks, tangrams,money, number lines, fraction factory pieces, fraction bars, and otherfraction kits. Students should revisit previous modelling with fractions,such as the following:

When moving towards developing an algorithm for addition andsubtraction of fractions, a major focus should be placed on writing ofequivalent fractions. Once students internalize the fact that fractions can beadded or subtracted symbolically when they reflect equal subdivisions of aquantity, they become less reliant on concrete or pictorial models. At thislevel, students may work with fractions written in improper or mixednumber forms; however, they should be aware that applications of fractionsto algebra in senior high more often involve fractions represented inimproper form.

B6 When any problem involving fractions is presented, it is important thatstudents first attempt to solve it mentally. If it cannot be solved mentally,students should determine whether an estimate is sufficient or if an exactanswer is required. The following are situations where mental computationwould be expected:

• When denominators are the same, or, if the common denominator iseasily determined (e.g., 1

2 + 14, 7

10- 15 ). Such situations occur when

one denominator is a multiple of the other.







SCO: By the end of grade 8,students will be expected toB5 add and subtract fractions

concretely, pictorially, andsymbolically

B6 add and subtract fractionsmentally, when appropriate

Since 112 is the gap,

1112 must be the sum.

14 1

313

56 - 1

3

Start with 56

Represent 13

Place the 13 piece over the 5

6and 3

6 are left.

14 + 1

3 + 13 =

1112

Represent both using the same subdivision of the whole.

+

+

35 + 1

2

Represent both using the same subdivision.912 - 8

12 = 112

34 - 2

3 -

-

610 + 5

10 = 1110 or 1

110

[Combine the two sets toproduce a final answer.]

[Compare the two sets. Thedifference represents the finalanswer.]





PerformanceB5.1 Ask students to show why the following is an incorrect procedurethrough the use of concrete materials or diagrams:

38 - 1

4 = 3 - 18 - 4 = 2

4 = 12

Pencil and PaperB5.2 Create three pairs of fractions whose sum is 1

2 .

B5.3 Create three addition and three subtraction sentences with the sameresult as 6

12 + 312 .

B5.4 A recipe requires 2 13 cups of flour to make 24 muffins. How many

cups of flour would be required to make 60 muffins?

B5.5 What might be the value of if 3 12 -1 2 <2?

InterviewB5.6 Ask students to explain why the following is incorrect through the useof concrete materials: 1

4 + 14 = 2

8

B5.7 Ask students how they would convince someone that the following isincorrect:

56 + 5

8 = 1014

B5.8 Ask studentsa) if an answer can be sixths when you add fourths and thirds, and to

justify their responseb) if an answer can be sevenths when you add fourths and thirds, and to

justify their response

B6.1 Ask students to solve the following mentally. (Mental Computationshould include Oral and/or Timed Response Assessment.)

a)17 + 4

7

b)12 + 5

8

c) 12 - 1

8

d) 5+ 25

e) 4 - 15

f ) 6 - 2 34

PresentationB5.9 Ask students to explain how to find the common denominator of twofractions ifa) one denominator is a multiple of the otherb) two denominators have a factor in common but one is not a multiple

of the other





B6 (Cont’d)

• When a simple fraction is subtracted from or added to a wholenumber (e.g., 2 - 1

3 , 4 - 23 , 3 + 4 2

3 ).

Activities related to mental computation should generally be done forshort periods of time. Five to ten minutes at the beginning of a class isusually sufficient.

B7 Students worked with multiplication of a fraction by a whole numbermentally in grade 7. This topic should start with concrete and pictorialmodels, but develop to the symbolic level in grade 8. Among the simplercombinations to model concretely or pictorially are

• a whole number by a fraction less than one (e.g., 4×13 , uses repeated

addition)

= (43 or 1

13 )

• a fraction less than one by a whole number (e.g., 13 × 6, think 1

3 of 6)

Start with 6 objects Divide into 3 groups

• a fraction less than one by any other fraction, especially when thenumerator is 1

14 of

23

It should be shown that “of” means multiplication. This may be done bycomparing results in examples such as

14 of 8 and

14 × 8.

When work is done at the symbolic level, it should be supported byconcrete or pictorial representations. Grid diagrams should also beconsidered when modelling multiplication. Students should notice that 1

2of 3

4 is represented as 38 . By comparing the question with the result,

students can start to speculate about a possible algorithm.

Modelling should always be related back to the symbols so that studentsmake the connections clearly; otherwise, the use of models may not helpsupport students’ understanding of the algorithms. Students should be ableto work effectively with multiplication of fractions symbolically by the endof grade 8.





SCO: By the end of grade 8,students will be expected toB6 add and subtract fractions

mentally, when appropriateB7 multiply fractions concretely,

pictorially, and symbolically

How many in each group?

14 of each 1

3 togethermakes 2

12 or 16

Divide each 13

into fourths





PerformanceB7.1 Ask students to draw a diagram to show why each of the following istrue:

a)13 × 3 =1

b) 6 × 13 = 2

Pencil and PaperB7.2 What multiplication sentence is illustrated?

0 1 2 3 4 4 5 6 7 823

B7.3 Place the numbers 1, 2, 3, and 4 in the boxes to get the least possibleanswer.

Try the same type of problem, using the numbers 2, 3, 4, and 5. Choose adifferent set of four numbers and repeat the activity.

PortfolioB7.4 Tell students that Frank works 7 1

4 hours a day for a five-day workweek. He works 3 hours on Saturday and is paid at time and a half.Ask students the following:a) For how many hours a week is Frank actually paid?b) What weekly salary would he make at $9.25 per hour?c) If Fred has $95.00 per week taken out of his check for taxes and union

dues, and his father makes him save 35 of his take-home pay for

university, how much spending money would be available to Fred ona weekly basis?

B7.5 Ask students to write two fractional numbers which have a productbetween the given numbers for each of the following:a) 14 and 15b) 1

2 and 13

×





B8 Since division of fractions is an introductory topic, it is important thatsignificant time be spent working with concrete and pictorial models. Initialexamples for modelling should be chosen carefully and worked through by theteacher prior to instruction. These simpler examples should enable students toderive an algorithm. Students should reach the symbolic level with the divisionof fractions at this grade level. Situations which are well suited to modelling withmaterials include

• a simple fraction divided by a whole number (For 12 ÷ 3, divide 1

2 into 3equal parts. What does each part represent?)

• a whole number divided by a simple fraction ( 4 ÷ 12 asks how many 1

2 ’sthere are in 4)

• a simple fraction divided by a simple fraction where the numerator of thedivisor is one and both denominators are the same ( 5

6 ÷ 16 asks how many

16 ’s there are in 5

6 )

• a simple fraction divided by a simple fraction where the numerator of thedivisor is one and the fractions are compatible ( 1

2 ÷ 14 or 3

8 ÷ 14 )

The number line can also provide a useful model for division. For example,suppose it takes 1 1

4 hours to do 3 chores. How long does it take for each chore ifthey all require an equal amount of time? This can be modelled as follows:

There are two common algorithms for division which can be considered. Thecommon-denominator algorithm involves finding a common denominator anddividing the numerators; e.g.,

43 ÷ 1

2 = 86 ÷ 3

6 =8 ÷ 3=223 . This can be modelled

concretely using the process described in bullet #3 above. The more traditionalinvert-and-multiply algorithm involves inverting the divisor and multiplying byit; e.g.,

43 ÷ 1

2 = 43 × 2

1 = 83 = 2 2

3 . It is necessary for students to understand theconcept of reciprocal before they work with division, using the invert-and-multiply algorithm. As a starting point for this algorithm, students can comparethe solution of problems such as 8÷ 1

2 and 8× 2.





SCO: By the end of grade 8,students will be expected toB8 divide fractions concretely,

pictorially, and symbolically

Divide into 3equal parts.

What does eachpart represent?

12

Count the number of halves in 4 objects. Since eachobject has two halves, 4× 2 = 8. By comparing thisto the original question, a connection can be madeto the invert-and-multiply algorithm.

How many 14 ’s

are there in 38 ?

Answer = 1 12

How many 16 ’s are there in 5

6 ?

How many 14 ’s

are there in 12 ?

Answer = 2

Divide each quarter into 3 parts, whichmakes 15 parts in 1 1

4 hours. There willbe 5 parts for each chore, and 12 parts in1 hour, hence, 5

12 hour for each chore.

Answer = 16

Answer = 5

114





PerformanceB8.1a) Ask students to demonstrate, by drawing diagrams, and explain why

each of the following is true:2 ÷ 1

4 = 812 ÷ 2 = 1

4

b) Ask students to compare the solutions in part a) with the solutions to2× 4 and 1

2 × 12 , and to discuss their observations. [Students should

relate this to the invert and multiply algorithm.]

Pencil and PaperB8.2 Write a division sentence for the following:

0 1 2 3 3 4 5 6 7 812

1 2 3 4 5 6 7

B8.3 Complete the following patterns, and extend them for two extra lines.What pattern do you observe?

9÷ 9 =

9÷ 3 =

9÷ 1 =

9÷13 =

9÷19 =

: :

PortfolioB8.4 Caitlin decided to make muffins for the school picnic. Her reciperequires 2 1

4 cups of flour to make 12 muffins. Caitlin found there wasexactly 18 cups of flour in the canister, so she decided to use all of it.

a) Ask students how many muffins Caitlin can expect to get.b) The principal of the school liked Caitlin’s muffins and asked her to

cater the school picnic next year, producing enough muffins for all 400students. Ask students how many cups of flour Caitlin will require.

4÷ 12 =

2÷ 12 =

1÷ 12 =

12 ÷ 1

2 =14 ÷ 1

2 =





B9 Appropriate problems to solve mentally include• for multiplication, i) a fraction by a whole number when the numbers

are compatible (e.g., 15 × 30 and

23 ×12), ii) and any two proper

fractions when the numerators and denominators are relatively simpleto work with (e.g., 2

3 × 34 and 1

3 × 38 ), and iii) a whole number by a

mixed number (e.g., 4× 112 and 3× 2

13 ). In this last example, students

should apply the distributive property to multiply mentally:3× 2 1

3 →3× 2 + 3× 13 →6 + 1 = 7.

• for division, i) a simple fraction divided by a whole number(e.g.,

12 ÷ 4 and

14 ÷ 2), ii) a whole number divided by a fraction

(e.g., 4÷ 13 and 3÷ 3

4 ), and iii) a simple fraction divided by a simplefraction when denominators are the same (e.g.,

34 ÷ 1

4 and 58 ÷ 1

8 ).

Students worked with estimation of sums and differences in grade 7.Estimation should be encouraged prior to computation and be used tocheck reasonableness of results. Rough estimates can usually be achieved byrounding to the nearest whole, and sometimes to the nearest half, inmultiplication and division problems. For example:

3 34 ×5 1

5Ý = 4 ×5 = 20,

49

× 3 57

Ý = 12 × 31

2 = 12 × 3+ 1

2 × 12 =1 1

2 + 14 =13

4 , and78 ÷

13

Ý = 1÷13 = 3.

B10 The order of operations was studied in grade 7 as it pertained to wholeand decimal numbers. Students will need to reconfirm that the same orderis equally relevant for fractional numbers. They should also be providedopportunities to see the ambiguity of results when the order of operations isignored. The order of operations is discussed in some detail in the grade 7curriculum. Students may need to be reminded that the order of operationsis as follows: brackets→exponents→multiplication or division (in theorder they appear)→addition and subtraction (in the order they appear).

B11 It is important that students are able to translate a problem situationinto an arithmetic expression, a picture, or a concrete representation whichcan assist in finding a solution. Much of the assessment of fractionoperations at grade 8 should be embedded in problem situations.

It has become more difficult to find problem situations that typicallyinvolve fractions, as opposed to decimals. Since the measurement system isdecimal-based, such problem situations are better suited to the use ofdecimals. Fractions are still commonly used to describe fractions of anhour, and in non-metric recipes. Many facts about the earth can lead tomeaningful problem contexts (such as 1

9 of an iceberg shows above earth’socean water; water covers 7

10 of the earth’s surface; the Pacific Oceancontains 9

20 of the earth’s ocean water).





SCO: By the end of grade 8,students will be expected toB9 estimate and mentally

compute products andquotients involving fractions

B10 apply the order of operationsto fraction computations,using both pencil and paperand the calculator

B11 model, solve, and createproblems involving fractionsin meaningful contexts





PerformanceB11.1 Ask students to draw a diagram to show that 2

3 × 14 = 1

6 .

Pencil and PaperB10.1 Insert one set of brackets to make the following statements true, andjustify your answer:

a)12 + 1

4 × 23 = 1

2 b)34 × 1

5 + 23 × 5

3 =1 112

B10.2 In order to win a prize, Richard was required to answer correctly thefollowing skill-testing question: 5 - (

12 + 1

2 × 13 )2 ÷ 1

9 . What is the correctanswer?

B11.2 In the gymnasium, 14 of the people present are men,

13 are women,

and the rest are children. If there are 840 people in the gymnasium, howmany are children?

InterviewB9.1 Ask students to estimate each of the following and explain theirthinking:

a) 30 ÷ 2 78 c) 5 1

4 × 8 e) 4 × 838

b) 24÷ 4 14 d) 36 ÷ 31

5 f ) 32 ÷ 7 34

B9.2 Ask students to solve each of the following mentally, and to explaintheir strategy:

a) 30 × 25 c)

34 × 4

5 e)13 ÷ 1

6

b) 56 ÷ 1

6 d) 34 ×16 f ) 4× 2 1

4

PortfolioB11.3 Tell students that Michael ordered 3 extra large pizzas and asked thateach pizza be cut into sixteenths. If each person at Michael’s party is likelyto eat 3 pieces, ask students how many people the pizzas can serve.

B11.4 Tell students that Lisa has 34 of a large candy bar. She gave

13 of

what she had to Shannon.a) Ask students to explain, in at least two different ways, how they know

that Shannon got less than 13 of what would have been a whole bar.

b) Ask them what fraction of a whole candy bar each girl has.





B12 All four of the operations with integers were developed in grade 7.The focus at the grade 7 level was on conceptual understanding, andstudents worked extensively with concrete and pictorial models. All fouroperations were extended to the symbolic level. Once students are able towork confidently with integers, it should be relatively straightforward toextend this to working with the full range of rational numbers. Somerevisiting of the concrete and pictorial representations should take place,but work should progress rapidly to the symbolic level. For details relatedto the development of integers using models, see the elaborations associatedwith outcomes B11, B12, and B13 of grade 7. A number of models shouldbe used in illustrating the operations, including the use of colouredcounters (where one colour represents positive and another representsnegative) and the use of the number line. Students should practise theirestimation skills using rational numbers, and be able to use the +/- key onthe calculator appropriately.

The computational algorithms for operations with decimals may need abrief review. This can be determined through pre-testing. Operations withpositive and negative decimal numbers are new to this grade level. All ofthe following methods should be used: estimation, mental computation,pencil and paper, and calculator. Students at this level often become toodependent on the calculator, and it is timely to emphasize that all fourmethods should be considered when faced with a problem situation. Aswith other number systems, students should always consider whether theproblem can be solved mentally. If it cannot be solved mentally, theyshould determine whether an exact answer or an estimate is required. If anestimate is sufficient, they should estimate the answer. When an exactanswer is necessary, consideration should be given to pencil-and-papermethods, or the calculator. An estimate should still be done to helpdetermine whether the answer is reasonable. Students will work withnegative numbers in fractional form in grade 9.

The order of operations should be revisited in the context of positive andnegative decimal numbers; for example, students should work withexpressions such as -0.2 + 4.5 (-5 + 2.4) - (-6)÷ 0.2.

B13 Students should solve problems related to rational numbers, as well aspose problems for others to solve. They should be engaged in activities suchas the following:

- creating problems which can be solved from a given numberexpression or equation

- creating problems to exchange and solve

- writing questions that can be answered based on a given story orgraphical representation, and then solving them





SCO: By the end of grade 8,students will be expected toB12 add, subtract, multiply, and

divide positive and negativedecimal numbers with andwithout the calculator

B13 solve and create problemsinvolving addition,subtraction, multiplication,and division of positive andnegative decimal numbers





Pencil and PaperB13.1 Pam recorded the daily high temperatures for one week and found theaverage high temperature for the week to be -4.10C. The temperatures fromSunday to Friday were 11.70C, -17.40C, 00C, -23.60C, -13.90C, and 9.10C.Explain how you would estimate the temperature on Saturday. What was theactual temperature on Saturday? Compare the answer and your estimate.

B13.2 John is trying to keep track of his net worth. He owes money to threefriends in the amounts of $4.25, $3.00, and $11.00. On Saturday, he will bepaid $32.50 for his paper route, receive $10.00 in allowance, and get the usual$2.00 from Grandma for shovelling her steps. In addition, his mother gavehim $12 per gift to buy birthday presents for each of the Thomson triplets. Hefound a good deal and purchased the three gifts at a sale of 3 for $22.00. Writethree questions which can be answered from this story, and write a numberexpression for each.

InterviewB12.1 Ask students to explain the key strokes necessary to solve this problem,using a calculator: 3.2 - (-8)× 0.5

Ask students to compare their answer with those of other members of the class.Do all calculators approach this problem in exactly the same way?

JournalB12.2 Tell students that Jared believes that -5.2 - (-3.2) = 2. Ask them to writeto Jared explaining why they agree or disagree, using a diagram to help in theirexplanation.

B13.3 Ask students to write problems which can be solved using the givennumber sentences.a) -2.44 + 3× 7.99b) [-12 +( -7 1

2 )+13 + (-2)]÷ 4

ActivityB12.3 Have students choose a partner. Ask the partners to roll two dice ofdifferent colours, assigning negative to one colour and positive to the othercolour. Ask them to write a number sentence for the sum. Ask partners to rollthe two dice again, find the sum mentally, and add the result to their previousscore. Partners should exchange turns until the first person reaches 20 or -20.Ask, why would it be fair to accept 20 or -20 as the winning score? Tellstudents that this activity can be modified and used with other operations, andby assigning the negative or positive to specific colours after each roll, insteadof maintaining the same designation throughout the game. Ask students if thischange allows them to get to 20 more quickly. Have them consider otherpossible rule changes such as “You can’t go over 20.” Also, a similar game canbe played using a deck of cards, where red represents one sign and black theother sign.

Suggested Resources





B14 Students were exposed to the notion of a variable in grade 7. As partof the early development of algebra, students should be given a variety ofways to relate to the symbolism. Different individuals derive meaning indifferent ways. One connection which may be useful to students relates tomeasurement situations. For example, if we know that a certain distance is3 m and 20 cm, can we say that the distance is 3 + 20? Students have had alot of experience in working with measurement units, and realize that theymust be expressed in the same unit (like terms) before they can be added orsubtracted. When expressed as 3 m + 0.2 m, they can be added to form asingle term, 3.2 m. A parallel can be drawn between this measurementsituation and variable expressions. This parallel provides a goodmeasurement connection. A similar analogy can be drawn with place value.For instance, ask students if we can add 2 tens and 5 ones to get 7 ofsomething.

Another analogy would be the addition of 0.3 + 56 . It is difficult to

combine the two numbers unless both numbers are expressed in a similarformat; i.e., they both need to be expressed as decimals or as fractions.

Students should also be given an opportunity to connect the notion of likeand unlike terms to concrete materials. Algebra tiles can be used to visuallydistinguish between x and y, or x and x2. Students should use thesematerials to develop an understanding that 2x refers to how many x’s,whereas x2 is represented by a square tile with an area of x times x, or x2.

2x is represented as follows:

whereas x2 is represented as

Students should, with the aid of concrete materials, be able to add andsubtract to simplify expressions. They should know which terms can andcannot be combined. When this is done using concrete materials, studentstend to grasp this notion readily. In grade 7, the focus was on termscontaining a single variable, mainly on exponents. However, in grade 8,students should also work with terms containing more than one variableand with any degree. The use of an xy-tile from a set of algebra tiles will beuseful. Students should be able to clearly identify the x2, y2, and xy-tilesand know their names are linked with their dimensions. These tiles areshown below.

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toi) explore and explain, using

physical models, the connectionsbetween arithmetic andalgebraic operations

iv) apply operations to algebraicexpressions to represent andsolve relevant problems

SCO: By the end of grade 8,students will be expected toB14 add and subtract algebraic

terms concretely, pictorially,and symbolically to solvesimple algebraic problems

x2

xy y2





Pencil and PaperB14.1

a) Find the perimeter of the rectangle byi) creating an expression and then substituting a value for the

variableii) creating an expression, simplifying the expression, and then

substituting for the variableb) Discuss the advantages and disadvantages of each method.

B14.2 A rectangular flower garden has a length of 8 cinder blocks and awidth of 9 bricks. Finda) an expression for the perimeter of the flower gardenb) the perimeter of the flower garden when each cinder block is 25 cm

and each brick is15 cm

B14.3 Jane used square tables placed side by side to form a rectangle ofdimensions 2 tables by 8 tables. Each square table has an area of x2 cm2.Write expressions fora) the area of the rectangle formedb) the perimeter of the rectangle

PortfolioB14.4 Four tiles are arranged as shown in the diagram. Each tile has awidth of 2 cm, and the ends are placed 2 cm apart. The length of each tileis x cm.

2 cm

x cm

a) Ask students to find an expression to represent the perimeter.b) Ask students to find the perimeter if x = 6 cm.

6a

4a





B15 While addition of polynomials is often straightforward, for subtractionconsideration should be given to the different representations ofsubtraction, including the following:

• comparison, which refers to comparing and finding the differencebetween two quantities.

• taking away, which refers to starting with a quantity and removing(taking away) a specified amount.

• adding the opposite, which refers to subtracting by first changing thequestion to an addition and then adding the opposite of a quantity. Forexample, instead of subtracting x, one might add -x. Likewise, insteadof subtracting 2x - 1, one might add -(2x - 1), which is the same as -2x+ 1. Students should model 2x - 1 and understand that the opposite(inverse) is found by flipping the tiles.

• missing addend, which asks the question, What would be added to thenumber being subtracted to get the starting amount? For example, for(3x - 2) - (2x + 1), ask: What is added to 2x + 1 to get 3x - 2?

All four of these meanings for subtraction have been developed in previousgrades in the context of number.

B16 Multiplication of a polynomial by a scalar should be developed withconcrete materials and diagrams, using repeated addition. Given a problemsuch as 3 (2x + 1), students should recognize that it is the same as 2x + 1 +2x + 1 + 2x + 1, and therefore model the binomial three times, combine thelike terms, and arrive at an answer, as shown below:

The area model should also be explored in association with the topic, sothat students can relate results achieved through repeated addition withresults achieved using the area model.

3

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toi) explore and explain, using

physical models, the connectionsbetween arithmetic andalgebraic operations

iv) apply operations to algebraicexpressions to represent andsolve relevant problems

SCO: By the end of grade 8,students will be expected toB15 explore addition and

subtraction of polynomialexpressions, concretely andpictorially

B16 demonstrate anunderstanding ofmultiplication of apolynomial by a scalar,concretely, pictorially, andsymbolically

2x + 1 2x + 1 2x + 1 6x + 3

2x - 1 opposite -2x + 1

3(2x + 1)

2x + 1

2x + 1 x - 3 3x - 2

add to get





a) b)

PerformanceB15.1 Ask students to show, through the use of algebra tiles, how the solutions tothe following differ from each other:a) (2x2 + x) + (-4x2 + 5x)b) (2x2 + x) - (-4x2 + 5x)

B16.1 Ask students to record symbolically an expression for each, both as anexpression involving addition and as an expression involving multiplication (shadedpositive, white negative).

a) + +

b) +

B16.2 Ask students to write the dimensions and area for the rectangle shown.

B16.3 Ask students to demonstrate the product for each of the following, usingalgebra tiles or diagrams:a) 2 (x2 + 3) b) 3 (2x - 1) c) 3 (x2 - 2x + 1)

B16.4 Ask students to show the product of 3 and 2x + 4 as the area of a rectangle.

Pencil and PaperB15.2 Write each of the steps in the problems which follow, using symbols, andexplain each step:

+

=

-

+=

=

B15.3 Simplify the following expression and write two symbols which representeach step:

+ + + +

Suggested Resources



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


Patterns and Relations

General Curriculum Outcomes C:

Students will explore, recognize, represent, andapply patterns and relationships,

both informally and formally.





C1 Students should be able to move interchangeably among the variousformats that describe relationships. They should describe in words and useexpressions and equations to represent patterns given in tables, graphs,charts, pictures, and/or by problem situations. Information presented in avariety of formats should be used to derive mathematical expressions andpredict unknown values. While in grade 7, students observed informallyrelationships which produced a variety of shapes; grade 8 will be limited tolinear situations and those that do not produce a regular pattern that canbe readily described by an equation. These generally produce a broken-lineor curved graph.

For example, students might be given information in a table such as

a 1 2 3 4 5b 2 5 8 11 14

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toi) analyse, generalize, and create

patterns and relationships tomodel and solve real-worldand mathematical problemsituations

iii) represent patterns andrelationships in multiple ways(including the use of algebraicexpressons, equations,inequalities, and exponents)

iv) explain the connections amongalgebraic and non-algebraicrepresentations of patterns andrelationships

SCO: By the end of grade 8,students will be expected toC1 represent patterns and

relationships in a variety offormats and use theserepresentations to predictunknown values

C2 interpret graphs thatrepresent linear andnon-linear data

or in a graph such as:

and asked to describe in words or use an expression or equation to describethe pattern they see. Part of the description may be the use of tiles, cubes, orpictures to model what is observed in the table or graph.

Once an algebraic description of a pattern is established, this can be used topredict unknown values. For example, in the table above students mayobserve that the pattern can be described using b = 3a - 1. They can use thisdescription to determine the value of b for any given a value, such as whena = 10 or a = 102. In some situations (e.g., the increase of temperature overtime) the pattern will not be regular. Students will then use interpolationand extrapolation to predict unknown values. Interpolation is theprediction of a value between two known values, and extrapolation is theprediction of a value which goes beyond the data that is given. These wereaddressed in previous grades.

C2 Many relationships produce graphs that are non-linear. Students shouldobserve when looking at tabular data that, when an equal spacing betweenthe values of one variable produces an equal spacing between values of theother variable, the relationship will be linear. Students should also be giventhe graphs of non-smooth situations and asked to interpret them.

The swimming pool below is being filled with water that flows at aconstant rate. Ask students to sketch a graph of the height of the wateragainst volume.

Note: The elaboration for C2 is continued on the next 2-page spread.





Pencil and PaperC1/2.1 For each of the tile patternsa) make a table of values, and describe the patterns in words

b) use the pattern and description to write a mathematical equationidentifying what the variable(s) represent

c) use the equation to help determine the tenth entry in the table

C1/2.2 A certain rectangle has a length which is 12 of the width.

a) Make a table showing the relationship between the width and theperimeter.

b) Describe the relationship between the width and perimeter in words.c) Write a mathematical rule to relate width and perimeter, identifying

what the variable(s) stand for.d) Use the rule to find the perimeter when the width is 99 metres.

PortfolioC1/2.3 Make stacks of rods as shown and find the surface area and volumewhen there is (are) 1, 2, 3, 4, .... 10 rods. Organize this information in atable. Each rod is 6 units long and the ends are 1 unit square. Each rod isplaced 1 cm from the end of the one before it.

a) Ask students to determine a pattern for the surface area and volume forn rods.

b) Ask students to graph the data and discuss the shape of the graph.

Adapted from: Addenda Series - Grades 5-8, “Patterns and Functions,”Investigation #2, p. 46.

i)

ii)





C2 (Cont’d)/C3 Although C3 is stated separately, it will undoubtedly beaddressed at the same time as outcomes C1 and C2. The related quantitiesreferred to in the outcome may be connected to experimental data instatistical probability, measurement, or to investigations pertaining to slope.

Students may be given information in various forms such as tables,diagrams, pictures, graphs, or equations and be expected to use theinformation to describe change in words or a graphical display.

Students will need some practice in both matching a situation to its graphand in sketching graphs for various situations that lead to linear andbroken-line graphs.

The graph at right shows how the height of liquid intwo of the beakers below varies as water is added at aconstant rate. Ask students to discuss and speculateas to what the graph would look like for the othershapes. [For some students, it may be necessary toactually bring in some containers.]

A B C D

E F G

Volume

Hei

ght

A E

Volume

Hei

ght

G

F

Typical Solution:

D C B

The tables below show data for two relationships.

Study the change in y-values. Use the changes to determine which ofthe two represents a linear relation. Justify your choice.

Note: The elaboration for C3 is continued on the next 2-page spread.


patterns and relationships tomodel and solve real-world andmathematical problemsituations

iii) represent patterns andrelationships in multiple ways(including the use of algebraicexpressons, equations,inequalities, and exponents)

iv) explain the connections amongalgebraic and non-algebraicrepresentations of patterns andrelationships

SCO: By the end of grade 8,students will be expected toC2 interpret graphs that

represent linear andnon-linear data

C3 construct and analyse tablesand graphs to describe howchange in one quantityaffects a related quantity

Volume

Hei

ght

B A

A. x 0 1 2 3 4y 0 7 14 21 28

B. x 0 1 2 3 4y 1 4 9 16 25





Pencil and PaperC2.1 The following is a graph showing John running a race. Write aparagraph that explains why the graph has its shape.

Time (Seconds)

Dis

tanc

e

70605040302010

1 2 3 4 5 6 7 8

C2/3.1 When measuring the liquids in irregularly shaped containers, it isimportant to know how the volume depends on the shape of the container.Match the following bottles to the graph that appears to best show how theheight changes with time. There is one graph for which there is nocontainer. Sketch a possible container to match the extra graph.

a)

d)

i)

iv)

vii)

c)

f)

iii)

vi)

b)

e)

ii)

v)

[Answer: a) iv) d) v) diagram for ii]b) iii) e) vii)c) i) f vi)





C3 (Cont’d) The importance of focussing on the different representationsof a relation should be emphasized. For example, when students havedifficulty recognizing a relationship from a table, they should realize thatthe tabular information can be graphed, and that the graph may provide amore visual means of recognizing the relationship.

Because linear relationships are the focus in grade 8, students will do someexploration of patterns associated with parameter changes in a linearequation. The intent is that students will start to make some associationsbetween changes which are made to the equation and how that affects theslant of the graph or its placement on the coordinate plane. Assessmentshould focus on having the students do an additional investigation, asopposed to using the conclusions reached in C3/4.1 and C3/4.2 as toolsfor solving additional problems.

C4 Students should recognize that for linear relationships the ratio ofvertical change to horizontal change is consistent anywhere along the line.This can be discovered by asking students to choose any two points on aline and make a path from point A to point B which involves travellingvertically and horizontally. The vertical change is called the rise, and thehorizontal change is called the run. Determining these two numbers for avariety of points chosen along a line will help students conclude that theratio of vertical change to horizontal change will be the same for any twopoints on the line.

Work with slope should not be limited to graphs. The slope of a staircase,the slope of a roof, and the steepness of roads and inclined planes are allexamples that can be used to relate slope to real-life situations. Determiningthe slope of a line is a good application of ratio. Students should make thelink that, for graphs which rise to the right, the ratio of vertical change tohorizontal change is positive, and that, for graphs that rise to the left, theratio is negative. Likewise, the magnitude of the ratio should be linked withthe steepness of the line. When working with relationships such as

x 2 3 4 5,

y 7 12 17 22

students can observe that, for linear relationships, when x increases byincrements of 1, the difference in the y values corresponds with the slope ofthe line. Therefore, students should be able to conclude from the table thatthe slope is 5. Since as x increases y also increases, the slope must bepositive.


patterns and relationships tomodel and solve real-world andmathematical problemsituations

ii) analyse functional relationshipsto explain how the change inone quantity results in a changein another

SCO: By the end of grade 8,students will be expected toC3 construct and analyse tables

and graphs to describe howchange in one quantityaffects a related quantity

C4 link visual characteristics ofslope with its numericalvalue by comparing verticalchange with horizontalchange





Pencil and PaperC3/4.1a) Graph each of the following equations, using a table of values, computer software, or a

graphing calculator.i) y = 2x + 1ii) y = 2x + 3iii) y = 2x + 5

b) How are the graphs alike?c) How are the graphs different?d) How are the equations alike?e) How are the equations different?f) What conclusions can you draw?g) How would the graph of y = 2x + 2 compare with the graphs of the above three? [Possible

answer: The graph would be parallel to the other graphs and lie between graph i) and ii).]

C3/4.2a) Graph each of the following equations, using a table of values, computer software, or a

graphing calculator.i) y =

12 x + 2

ii) y = -2x + 2iii) y = 4x + 2iv) y = -

14 x + 2

b) How are the graphs alike?c) How are the graphs different?d) How are the equations alike?e) How are the equations different?f) What conclusions can you draw?g) How would the graph of y = 5x + 2 compare with the graphs of the above four? [Possible

answer: The graph passes through the intersection point of the previous graphs. It has aslant that is similar to graph iii).]

C3/4.3 According to the graph below, who ran fastest? Explain.

C3/4.4 Assuming that cost is proportional to time, and that all calls were made at the sametime of day, which graph is likely to represent each of the following:a) Dave phoned his mother who lives 50 km away and chatted a long time.b) Sarah phoned a friend 200 km away and talked a long time.c) Bill phoned his sister in Turkey and had a brief conversation.

Suggested

Resources

Dis

tanc

e

Time(s)

JoeJim

Jack

Cos

t

Time(min)

III

III





C5 Students have graphed linear equations in previous grades. To find theintersection of two graphs, students can start with equations, tables ofvalues, or verbal descriptions of two linear situations and consider wherethe two graphs intersect. There are three strategies which should beconsidered:

• Students can generate tables of values and look for points in the tablesthat are the same. That is, if they identify the same point in both tablesof values, they should be able to conclude that this is the intersectionpoint.

• Students can use information in tables of values to generate a series ofordered pairs for each equation. Each series of ordered pairs produces agraph. Students can then look for the coordinates of the point ofintersection of the two graphs.

• Students can graph both equations directly (without generating a tableof values), using computer software or the graphing calculator. Thegraphing calculator can find the point of intersection directly or by thetrace function.

• As an extension, some students may, as a result of the work done inC4, use the slope-intercept form to produce the graphs (slope-interceptform is not a core topic at the grade 8 level, but some students mayarrive at the slope-intercept form as a result of exploration in C3/4.1and C3/4.2 ).

As a starting point, students in small groups or as a whole class mightconsider problems such as the following:

Suppose the school were planning a skating party. The president of thestudent council calls Memorial Stadium and is told it will cost a flatrate of $120 plus $1 per person. She calls Glacier Stadium and isquoted a price of $2.25 per person but with no flat-rate charge.

a) Ask students to create a table of values for each situation, showingthe costs for 20, 40, 60, 80, ... students.

b) Ask students to graph both situations and determine at whatnumber of students both costs will be the same.

c) Ask students to determine which rink would be most economical,if each of the following numbers of students sign up for the skatingparty: 55, 85, 115, 145, and 175.

d) Tell students that a third rink charges a set rate of $250 with nolimit on the number of students. Ask them how this rink comparesto the first two for each of the numbers of students indicated inpart c).

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toiii) represent patterns and

relationships in multiple ways(including the use of algebraicexpressions, equations,inequalities, and exponents)

SCO: By the end of grade 8,students will be expected toC5 solve problems involving the

intersection of two lines on agraph




Worthwhile Tasks for Instruction and/or Assessment Suggested ResourcesSuggested Resources

Pencil and PaperC5.1 Sarah started a summer business, making notepaper. She got a grant from theDevelopment Bank to get her business going. Her costs and sales are shown below.Cost - To produce p packages, the cost is $2 per package plus the initial design costof $180. Total cost C is given by C = 180 + 2n.Total Sales - Sarah sells the packages of notepaper for $8 a package. If n packagesare sold, the amount received A is given by A = 8n.

C=180+2n

C n

A=8n

A n

a) Complete the tables of values and graph the two relations on the samecoordinate plane.

b) Find the point of intersection of the two graphs.c) What does the point of intersection represent in this situation?d) What was the cost per package of notepaper?e) How many packages does she have to sell to break even?f) What would her summer profit be if she sold 50 packages of notepaper? 1000

packages?

C3/4/5.1 Fred and his sister ran a race. Fred runs at an average speed of 3 metresper second, and his sister runs at an average speed of 4 metres per second. In a500m race, Fred gets a 100 metre head start.a) Make a table of values and draw a graph of the race.b) Determine the slope of each line, using the vertical change and horizontal

change. What does the slope correspond with?c) Who won the race?

PortfolioC3/4/5.2 A tennis club opened in Toni’s neighbourhood. There are two methodsof payment offered, and Toni is trying to figure out which one would be mosteconomical for her needs. She can either pay a yearly membership fee of $75 plus$2 for each time she uses the courts, or she can sign up for an associate membershipwhere the fee is only $10, but she must pay $4 every time she uses the courts. Askstudents the following:a) How many times will Toni need to use the tennis courts for both costs to be

the same?b) If Toni usually plays tennis about 20 times a year, which would be the most

economical method of payment?c) If Toni plays tennis twice a week, which would be the most economical

method of payment? [It may be necessary to discuss this problem with somestudents before it is assigned. For example, students may need to be led to theequations and then assigned the problem for completion. This will depend onthe nature of individual students, or perhaps the class as a whole.]





C6 Students have done significant work with concrete materials anddiagrams in solving linear equations in grade 7. In grade 8, instructionshould start with concrete materials and pictorial models, but move quicklyto the symbolic, with the ultimate goal that students can solve one- andtwo-step equations with or without concrete or pictorial support by theend of grade 8. The focus of the work should be on whole and integercoefficients, as well as some extension to include fractions and decimals.The cover-up method and the balance method can be used to develop thesymbolic algorithm. These are elaborated on in the grade 7 curriculum.

For example, using the cover-up method to solve -3m + 4 = -20, ask thequestion, “What added to 4 equals -20?” Since -24 + 4 = -20, the answer is-24. Then ask the question, “What multiplied by -3 equals -24?” Since -3times 8 equals -24, the answer is 8, therefore m = 8.

An example using the balance method:

Solving equations using algebra tiles:

The focus in instruction should be on equations such as the following:

x + 3 = -7 x4 = 12

-5x = 30x3 + 4 = -2

12 x - 5 = 10

x15 =

23

Students should consider in advance what might be a reasonable solution,and be aware that once they acquire a solution, it can be checked foraccuracy by substitution into the original equation.

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected tov) apply algebraic methods to solve

linear equations andinequalities, and investigatenon-linear equations

SCO: By the end of grade 8,students will be expected toC6 solve and verify simple linear

equations algebraically

unknownpositivenegative→ 2x + 3 = -7

→ 2x + 3+ -3=-7+ (-3)

→ -2x - 4=6

→ -2x - 4+4=6+4

→ -2x =10

→ - x

→ - 1

→ 2x = -10

→ x = -5

→ 1

→ x

→ -x =5

→ x = -5





PerformanceC6.1 Ask students to solve the equation shown, using tiles, and to write thesymbolism for each step involved in the solution.

C6.2 Ask students toa) solve the equation shown in the balanceb) sketch diagrams to show each step in the solutionc) write symbols to describe the steps involved in the solution

C6.3a) Ask students to solve the following:

i) x2 +1 = 5

ii) x + 2 = 10iii) 4x + 8 = 40

b) Ask students what they notice about the answers for each of theequations in part a)

c) Ask them to analyse the three equations in a) to determine why theanswers all turned out the way they did.

Pencil and PaperC6.4 Which of the following produces the smallest value for p?

a) -2p + 4.5 = 12.9 b) 6 + 2.2p = 14.8

c)p5 = 11.2 d) 7

2 + p = 6

C6.5 Write the symbolism to match each step of the solution shown andexplain what is happening in each step:

Step 1

Step 2

Step 3Step 4

=





C7 It is often difficult to create problem situations which require algebra tosolve them. Most such problems can be solved using methods such asguess-and-check and systematic trial. It may therefore be necessary tospecify the strategy in some instances to ensure that problem solving usingalgebra is done at this time. Also, when the numbers used are large, it canbe more easily illustrated that algebra is a tool which can readily solveproblems that might otherwise be very tedious to solve using methods suchas guess-and-check. Instruction can capitalize, however, on studentexperience with guess-and-check and use this strategy in establishing thevariable.

Ticket sales for a concert were $2790. There were 50 tickets for $15seats and 70 for $12 seats. How many tickets were sold for $8 seats?

$8 $15 $12 Totalseats seats seats sales

Guess 50 50 70 50 × $8 + 50 × $15 + 70 × $12 = $1990

Guess 100 50 70 100 × $8 + 50 × $15 + 70 × $12 = $2390

Guess 200 50 70 200 × $8 + 50 × $15 + 70 × $12 = $3190

Guess p 50 70 p × $8 + 50 × $15 + 70 × $12 = $2790

∴ 8p + 750 + 840 = 2790

8p + 1590 = 2790

8p = 2790 - 1590

8p = 1200

p = 1200 ÷ 8

p = 150

Students should also be given situations such as the following and be askedwhat relevant questions can be posed. They can then exchange questionswith a partner to solve.

An equilateral triangle has the same side lengths as a square, but theperimeter of the square is 14 cm longer than the perimeter of thetriangle. [Questions that may be created include the following: What isthe length of each side? What is the perimeter of the equilateraltriangle? What is the perimeter of the square? What is the area of thesquare?]

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 and willalso be expected tov) apply algebraic methods to solve

linear equations andinequalities, and investigatenon-linear equations

SCO: By the end of grade 8,students will be expected toC7 create and solve

problems, using linearequations





Pencil and PaperC7.1 Some cows and some chickens live on a farm. If the total number oflegs is 38, and the total number of heads is 16, use algebra to find howmany cows and how many chickens live on the farm. (Hint: If there are xcows, there are 16 - x chickens.)

C7.2 A taxicab company charges a basic rate of $2.00 plus $1.50 for everykilometre driven. If the total bill was $21.50, use algebra to find how farthe cab ride was.

C7.3 Write a question that can be answered based on the informationgiven. Exchange questions with a partner to solve.a) Ricki had 15 hockey cards. When he went shopping on Saturday, he

bought 4 packages with his allowance. Each package cost $1.25. Henow has 47 hockey cards.

b) For a party, Jack ordered 3 extra large pizzas. Each pizza was cut into16 pieces. He invited 16 friends, and all but three of them ate exactlythe same number of pizza slices. Sue ate 1 less then the others, Bill ate2 less than the others, while Cara didn’t eat any.

PortfolioC7.4 Ask students to create story problems that could be solved using analgebraic equation involvinga) addition and multiplicationb) addition and divisionc) subtraction and multiplicationd) subtraction and divisionHave students exchange the problems with a partner and solve.

C7.5 Ask students to create a story problem that could be solved using analgebraic equation involvinga) addition and multiplication where the answer is 12b) addition and division where the answer is 4c) subtraction and multiplication where the answer is 18d) subtraction and division where the answer is 6Have students exchange problems with a partner and solve, using algebrato confirm that the answer is as intended. Tell them to revise the problem ifnecessary.



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


Geometry

General Curriculum Outcome E:

Students will demonstrate spatial senseand apply geometric concepts, properties,

and relationships.





E1 Students’ mathematical experience with 3-D is often drawn from two-dimensional pictures. It is important that students be able to interpretinformation from 2-D pictures of the world, as well as to represent real-world information in 2-D. Because of their flexibility, interlocking cubesare most often used as the basic building blocks for 3-D objects.

Students can be given a set of plans, such as the following, and be asked toconstruct, using cubes, a building that adheres to the plans. Such plans areoften referred to as orthographic plans or drawings.

TopFront Right

When interpreting orthographic drawings, it is sometimes the case that notall students will produce exactly the same building structure. For example,with the set of plans above, each of the following mat plans would apply.

2 1

1

1 2

2 2

1

2 2

Likewise, each of the following isometric drawings satisfies theorthographic plans at the top.

Students can compare structures and come to realize that there is morethan one structure that fulfils the information in the set of plans. They canthen explore such questions as:- Suppose we start with one of the mat plans. Would there be a number

of ways a set of orthographic drawings or an isometric drawing couldhave been produced?

From there, they can explore such questions as the following:- What is the minimum number of cubes which can be used to fulfil the

plans provided?- What is the maximum number of cubes which can be used to fulfil the

plans provided?- How many different objects can be built to fulfil the plans?

Note: The elaboration for E1 is continued on the next 2-page spread.

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toi) construct and analyse 2- and

3-D models, using a variety ofmaterials and tools

iv) represent and solve abstractand real-world problems interms of 2- and 3-D geometricmodels

SCO: By the end of grade 8,students will be expected toE1 demonstrate whether a set

of orthographic views, a matplan, and an isometricdrawing can represent morethan one 3-D shape





PerformanceE1.1 Ask students to use the following plans to

Top

Front Right

a) find as many different shapes as possible that satisfy the plans and drawa single mat plan for each

b) find the minimum number of cubes needed to construct the structurec) find the maximum number of cubes needed to construct the structure

E1.2 Ask students to use the isometric drawing shown to

a) find as many different shapes as possible that satisfy the isometricdrawing shown


Pencil and PaperE1.3 Using a collection of interlocking cubes and the set of plans shown,complete the following:

Top

Front Right

a) Make a mat-plan record of every possible building which can beconstructed. [You may work in pairs to produce buildings and thencome together to consider whether all possible mat plans have beencreated.]

b) What is the maximum number of cubes used for a building?c) What is the mimimum number of cubes used for a building?d) What do all of the diagrams have in common?





E1 (Cont’d) This outcome can also be explored using isometric drawings.Students can discover that, when they are given only one view of anisometric drawing, they often cannot see all the cubes because some arehidden. Students can be given an isometric drawing, such as the one shownbelow, and be asked to create a building from it.

Generally, not all students make the same structure, and they realize thatone drawing can lead to more than one 3-D shape. They can again explorethe maximum, minimum, and variety of structures which can support agiven drawing.

Note: Depending on the experience students bring to making isometricdrawings, some background work may be necessary. To initiate such work,have students copy an isometric drawing, a simple figure requiring onlytwo or three interlocking cubes, and build from there. Some work has beendone with these drawings in previous years, including grade 7.

E2 Naturally, if more than one 3-D shape can be built from a singleisometric drawing, students need to explore how the plans can be mademore precise. This can be done by examining an isometric drawing andbuilding a solid which it appears to represent. Students can then constructa second isometric drawing from the same 3-D shape. In some cases, theywill find that even with two isometric drawings of an object, theinformation is still insufficient to ensure uniqueness.

The two isometric drawings shown below are both of the same 3-Dshape. Discuss with students why the drawings do not providesufficient information so that every person would necessarily build thesame object. Discuss where there may be hidden cubes.

Front left view Front right view



iv) represent and solve abstract andreal-world problems in terms of2- and 3-D geometric models

SCO: By the end of grade 8,students will be expected toE1 demonstrate whether a set of

orthographic views, a matplan, and an isometricdrawing can represent morethan one 3-D shape

E2 examine and drawrepresentations of 3-Dshapes to determine what isnecessary to produce uniqueshapes





PerformanceE2.1 Ask students to use the following mat plan to

Top

Front Right

a) draw two different isometric views and discuss whether this providesenough information to define a unique object

b) determine how many isometric views would be necessary to ensurethat the information defines a unique object

E.2.2 Ask students, if enough information were provided using a mat plan,whether there would be more than one way of constructing the object or itsisometric drawing. Have them test their answers by starting with thefollowing mat plan and determining whether the figure produced isunique.

2 1 1

1 2 1

1

3

E1/2.1 Give students a supply of interlocking cubes and ask them to builda shape, using a specified number of cubes. Ask students to

a) make a sketch of their shape, on isometric dot paperb) exchange sketches with a partner and build the 3-D shape from their

partner’s sketch [The two students can compare objects as a means ofchecking their work. If the figures are not identical, both studentsshould review the process to determine if an error was made or if bothobjects actually met the specifications.]

c) repeat the process, this time exchanging the models and producingsketches of the partner’s model on isometric dot paper [They can thencompare the two sketches and discuss the sources of any differencesobserved.]

E1/2.2 Ask students to find out how many different shapes can beconstructed using 2 interlocking cubes, 3 interlocking cubes, and4 interlocking cubes.





E3 Visualization of the movement of three-dimensional objects is animportant life skill, as anyone who has attempted to move furniture aroundin a living room or move a sofa through a door will attest. The purpose ofthis topic is to provide students with some experiences in visualizing andrecording the movement of 3-D shapes.

Students should be given a 3-D object, asked to sketch it from a particularpoint of view, perform the indicated transformation, and resketch theobject, all on isometric paper. For example, have students build the 3-Dobject from the set of plans below.

Top Front Right

With the front of the object facing the students, have them turnit 45° clockwise and sketch the object. Now have them turn it another90° clockwise and sketch it again. Have them turn it one more time90° clockwise and produce the third sketch. Have them continue rotatingat 90° intervals until the sketch looks identical to one already drawn.Repeat the above activity, but this time use a different 3-D object and havethe students make sketches from three different points of view. [Somestudents may be ready to try to sketch without physically rotating theobject first, but this will be very difficult for most students.]

Students can also explore making a 3-D object and sketching the object onisometric dot paper, reflecting the object about a horizontal or vertical lineto produce an image, and finally creating the object which is the reflectionof the original object. Students can start with a shape, such as the oneshown below, and sketch reflections to see how many different possiblepositions for a given shape can be drawn.

Students can also work with translations, but this should be a much morestraightforward activity. Translations simply involve reproducing theidentical shape at some other point along the grid. Because it is sostraightforward, it might make a good starting point for instruction relativeto this outcome.



iii) develop and analyse theproperties of transformationsand use them to identifyrelationships involvinggeometric figures


SCO: By the end of grade 8,students will be expected toE3 draw, describe, and apply

transformations of 3-Dshapes





PerformanceE3.1 Have students start with a shape, such as the one shown, and rotatethe shape, using 90-degree rotations clockwise and counterclockwise todetermine how many distinct drawings can be created. [If needed, allowstudents the opportunity to use cubes to make the shape.]

E3.2 Have students start with a shape, such as the one shown, and througha series of reflections, determine how many distinct drawings can becreated.

E3.3 Give students a sketch of a room and provide the dimensions. Askthem how many different ways a given corner-shaped sectional sofa can bearranged in the room, if the back of the sectional must be against at leastone wall, and the back of the other side of the sectional must touch anadjacent wall. Ask students how many of the arrangements which arepossible actually make sense, in terms of making best use of the sectionalwithin the room. Students will have to take doorways, fireplaces, etc. intoconsideration.

Pencil and PaperE3.4 Start with a shape such as the one shown and shift ita) 3 units to the left and 4 units downwardb) 5 units to the right and 3 units upward





E4 When studying polygons, students can use a table, such as the oneshown, to organize information. By extending the table, they can observepatterns and generalize about the sum of the measures of the interior anglesof various polygons, as well as about the measure of each interior angle of aregular polygon. The sum of the measures of the interior angles of apolygon is found by dividing the polygon into triangles as shown.

Since the sum of the measures of the angles in a triangle is 180° , the sumof the interior angles in the pentagon shown is 5× 180° , subtract the360° at the centre: 5× 180 - 2× 180 = (5 - 2)× 180.

Regular Polygons# of sides sum of interior∠ s measure of each ∠

3 180(1 × 180) 604 360(2 × 180) 905 540(3 × 180) 108. [always multiplying 180. by 2 less than the #. of sides]n (n - 2) × 180

(n - 2) × 180n

This is a good opportunity to reinforce work with graphing. Students cangraph the sum of the interior angles against the number of sides, or themeasure of each angle against the number of sides, to determine if therelationships are linear.

Similarly, patterns related to diagonals can be explored.Polygons

# of sides # of diagonals total # ofat one vertex diagonals

3 0 04 1 25 2 56 3 97 4 14..n n - 3

n × (n- 3)2

Again, the relationships between the number of sides and number ofdiagonals at one vertex, and number of diagonals in all, can be graphed todetermine if they are linear.

Symmetry of the various regular polygons can be explored to determine ifthe number of lines of symmetry is related to the number of sides.

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toii) compare and classify geometric

figures, understand and applygeometric properties andrelationships, and representgeometric figures viacoordinates

v) draw inferences, deduceproperties, and make logicaldeductions in synthetic[Euclidean] andtransformational geometricsituations

SCO: By the end of grade 8,students will be expected toE4 analyse polygons to

determine their propertiesand interrelationships





PerformanceE4.1 Ask students to sketch any triangle and, through a series of reflections,determine if a regular polygon can be produced. Ask students to considerwhat characteristics would be necessary in a triangle so that, through aseries of reflections, a regular polygon is produced. [Possible answer: Whenthe triangle is isosceles and the reflections are being done using the equalsides a regular polygon is produced, as long as the product of the numberof sides and the measure of each vertex angle is 360 degrees.]

E4.2 Ask students to sketch a series of regular polygons with from 3 to 10sides.a) Have students work in pairs to find the number of lines of symmetry

which can be drawn for each polygon. Ask them if any patternsemerge, and to explain their answers.

b) Have students find the centre of each polygon by finding theintersection point of the right bisectors of the sides of the polygon.When each vertex of the polygon is joined to the centre, the anglesformed are called the central angles. Ask students to record themeasures of the central angles for the series of regular polygons. Askthem if any patterns emerge, and to explain their answers.

E4.3 Ask students to speculate about the sum of the measures of theexterior angles of polygons, and then ask them to set up an investigation toexplore and determine if there is a relationship between the number of sidesand the sum of the measures of the exterior angles. [An exterior angle isdefined as an angle formed when one side of a polygon is extended at anyvertex.]

exterior angle

Pencil and PaperE4.4 Using the relationship established between the number of sides andthe angle measures in a polygon, finda) the number of sides if the sum of the interior angles is 16200

b) the number of sides if the measure of each interior angle of a regularpolygon is 1440

c) the sum of the interior angles if there are 15 sidesd) the measure of each interior angle of a regular 12-sided polygon

E4.5 Using the relationship established between the number of sides andthe number of diagonals which can be drawn, finda) the total number of diagonals which can be drawn in a figure with 20

sidesb) the number of sides a polygon has if 14 diagonals can be drawn from

one vertexc) the number of sides a polygon has if a total of 27 diagonals can be

drawn





E5 The use of computer software can allow a great deal of flexibility for theinvestigation of enlargements and reductions. It should be noted that,when a ratio is used to represent an enlargement or a reduction, the formatof the ratio is New:Original. A ratio of 2:1 means the new figure is anenlargement to twice the original. Likewise, a ratio of 1:3 means the newfigure is a reduction to

13 of the original, or the original is three times the

new figure.

As well as considering the scale, the centre of a dilatation must be identifiedin order to locate the position of the dilatation image. It is when the issueof dilatation centre is considered that an enlargement or reduction can bedescribed as a dilatation. Students should work with centres which arelocated inside the original figure, on a side of the original figure, andexternal to the original figure. Both opaque projectors and overheadprojectors are useful tools in working with dilatations.

When students construct a dilatation image, the following steps areimportant:• Use a straightedge to draw a faint ray from the dilatation centre to each

vertex of the original figure (in the case of enlargements, extend theline).

• Measure the distance from each vertex of the original figure to thedilatation centre.

• Use the scale ratio to calculate distances for the image. For example, ifthe scale ratio is 1:3, representing a reduction of 1

3 , each distance fromthe dilatation centre to the image vertex will be calculated bymultiplying by

13 .

• The image vertices will be on the line joining the vertex of the originalfigure to the centre, but will be

13 of the distance between the two

points.

This is shown below:

CÕ

C

D

A

B

BÕ

DÕ

AÕ0

9.7 cm

7.6 cm

7.9 cm

9.2 cm

As a means of reinforcing other transformations, students should also workwith combinations of transformations that include dilatations, such as anenlargement followed by a reflection.

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toiii) develop and analyse the

properties of transformationsand use them to identifyrelationships involvinggeometric figures

SCO: By the end of grade 8,students will be expected toE5 represent, analyse, describe,

and apply dilatations

OC = 9.2 cm OC’ = 9.2 ÷ 3 = 3.1 cmOB = 7.9 cm OB’ = 7.9 ÷ 3 = 2.6 cmOA = 7.6 cm OA’ = 7.6 ÷ 3 = 2.5 cmOD = 9.7 cm OD’ = 9.7 ÷ 3 = 3.2 cm





a)

b)

PerformanceE3/5.1 Give pairs of students a bucket of interlocking cubes.a) Show students a shape. Ask them to construct a shape which

represents an enlargement by a factor of 3.b) Ask students to each make a shape, using exactly 6 cubes, exchange

shapes with a partner, and ask their partner to create an enlargement ofthe shape by a factor of 2. [Note: Many of the 3-D objects explored inassociation with E1-E3 can be used in this context as well.]

c) Ask students whether this activity can be represented by a dilatation,and ask them to show how the cubes would need to be placed to makea dilatation. [The use of a dilatation centre differentiates a dilatationfrom a simple enlargement or reduction.]

Pencil and PaperE5.2 Write about whether congruence, area, length, and orientation arepreserved as they pertain to enlargements and reductions. Use diagrams tohelp support your writing.

E5.3 Determine the scale factor for each of the following:

AÕ

A

0

BÕ

B

CÕC

E5.4 A photograph is 5 cm by 7 cm. It is enlarged by a scale factor of 2.5.What are the new dimensions?

E5.5 In the diagram below, using the origin as the centre of dilatation,a) enlarge the given figure by a scale factor of 2b) reduce the given figure by a scale factor of 1

2c) record the vertices for each image

0 1 2 3 4 5 6

2

1

BÕ

AÕ

CÕ0

A

BC





Geometry

General Curriculum Outcome E:

Students will demonstrate spatial senseand apply geometric concepts, properties,

and relationships.





E1 Students’ mathematical experience with 3-D is often drawn from two-dimensional pictures. It is important that students be able to interpretinformation from 2-D pictures of the world, as well as to represent real-world information in 2-D. Because of their flexibility, interlocking cubesare most often used as the basic building blocks for 3-D objects.

Students can be given a set of plans, such as the following, and be asked toconstruct, using cubes, a building that adheres to the plans. Such plans areoften referred to as orthographic plans or drawings.

TopFront Right

When interpreting orthographic drawings, it is sometimes the case that notall students will produce exactly the same building structure. For example,with the set of plans above, each of the following mat plans would apply.

2 1

1

1 2

2 2

1

2 2

Likewise, each of the following isometric drawings satisfies theorthographic plans at the top.

Students can compare structures and come to realize that there is morethan one structure that fulfils the information in the set of plans. They canthen explore such questions as:- Suppose we start with one of the mat plans. Would there be a number

of ways a set of orthographic drawings or an isometric drawing couldhave been produced?

From there, they can explore such questions as the following:- What is the minimum number of cubes which can be used to fulfil the

plans provided?- What is the maximum number of cubes which can be used to fulfil the

plans provided?- How many different objects can be built to fulfil the plans?

Note: The elaboration for E1 is continued on the next 2-page spread.



iv) represent and solve abstractand real-world problems interms of 2- and 3-D geometricmodels

SCO: By the end of grade 8,students will be expected toE1 demonstrate whether a set

of orthographic views, a matplan, and an isometricdrawing can represent morethan one 3-D shape





PerformanceE1.1 Ask students to use the following plans to

Top

Front Right

a) find as many different shapes as possible that satisfy the plans and drawa single mat plan for each


E1.2 Ask students to use the isometric drawing shown to

a) find as many different shapes as possible that satisfy the isometricdrawing shown


Pencil and PaperE1.3 Using a collection of interlocking cubes and the set of plans shown,complete the following:

Top

Front Right

a) Make a mat-plan record of every possible building which can beconstructed. [You may work in pairs to produce buildings and thencome together to consider whether all possible mat plans have beencreated.]

b) What is the maximum number of cubes used for a building?c) What is the mimimum number of cubes used for a building?d) What do all of the diagrams have in common?





E1 (Cont’d) This outcome can also be explored using isometric drawings.Students can discover that, when they are given only one view of anisometric drawing, they often cannot see all the cubes because some arehidden. Students can be given an isometric drawing, such as the one shownbelow, and be asked to create a building from it.

Generally, not all students make the same structure, and they realize thatone drawing can lead to more than one 3-D shape. They can again explorethe maximum, minimum, and variety of structures which can support agiven drawing.

Note: Depending on the experience students bring to making isometricdrawings, some background work may be necessary. To initiate such work,have students copy an isometric drawing, a simple figure requiring onlytwo or three interlocking cubes, and build from there. Some work has beendone with these drawings in previous years, including grade 7.

E2 Naturally, if more than one 3-D shape can be built from a singleisometric drawing, students need to explore how the plans can be mademore precise. This can be done by examining an isometric drawing andbuilding a solid which it appears to represent. Students can then constructa second isometric drawing from the same 3-D shape. In some cases, theywill find that even with two isometric drawings of an object, theinformation is still insufficient to ensure uniqueness.

The two isometric drawings shown below are both of the same 3-Dshape. Discuss with students why the drawings do not providesufficient information so that every person would necessarily build thesame object. Discuss where there may be hidden cubes.

Front left view Front right view




SCO: By the end of grade 8,students will be expected toE1 demonstrate whether a set of

orthographic views, a matplan, and an isometricdrawing can represent morethan one 3-D shape

E2 examine and drawrepresentations of 3-Dshapes to determine what isnecessary to produce uniqueshapes





PerformanceE2.1 Ask students to use the following mat plan to

Top

Front Right

a) draw two different isometric views and discuss whether this providesenough information to define a unique object

b) determine how many isometric views would be necessary to ensurethat the information defines a unique object

E.2.2 Ask students, if enough information were provided using a mat plan,whether there would be more than one way of constructing the object or itsisometric drawing. Have them test their answers by starting with thefollowing mat plan and determining whether the figure produced isunique.

2 1 1

1 2 1

1

3

E1/2.1 Give students a supply of interlocking cubes and ask them to builda shape, using a specified number of cubes. Ask students to

a) make a sketch of their shape, on isometric dot paperb) exchange sketches with a partner and build the 3-D shape from their

partner’s sketch [The two students can compare objects as a means ofchecking their work. If the figures are not identical, both studentsshould review the process to determine if an error was made or if bothobjects actually met the specifications.]

c) repeat the process, this time exchanging the models and producingsketches of the partner’s model on isometric dot paper [They can thencompare the two sketches and discuss the sources of any differencesobserved.]

E1/2.2 Ask students to find out how many different shapes can beconstructed using 2 interlocking cubes, 3 interlocking cubes, and4 interlocking cubes.





E3 Visualization of the movement of three-dimensional objects is animportant life skill, as anyone who has attempted to move furniture aroundin a living room or move a sofa through a door will attest. The purpose ofthis topic is to provide students with some experiences in visualizing andrecording the movement of 3-D shapes.

Students should be given a 3-D object, asked to sketch it from a particularpoint of view, perform the indicated transformation, and resketch theobject, all on isometric paper. For example, have students build the 3-Dobject from the set of plans below.

Top Front Right

With the front of the object facing the students, have them turnit 45° clockwise and sketch the object. Now have them turn it another90° clockwise and sketch it again. Have them turn it one more time90° clockwise and produce the third sketch. Have them continue rotatingat 90° intervals until the sketch looks identical to one already drawn.Repeat the above activity, but this time use a different 3-D object and havethe students make sketches from three different points of view. [Somestudents may be ready to try to sketch without physically rotating theobject first, but this will be very difficult for most students.]

Students can also explore making a 3-D object and sketching the object onisometric dot paper, reflecting the object about a horizontal or vertical lineto produce an image, and finally creating the object which is the reflectionof the original object. Students can start with a shape, such as the oneshown below, and sketch reflections to see how many different possiblepositions for a given shape can be drawn.

Students can also work with translations, but this should be a much morestraightforward activity. Translations simply involve reproducing theidentical shape at some other point along the grid. Because it is sostraightforward, it might make a good starting point for instruction relativeto this outcome.



iii) develop and analyse theproperties of transformationsand use them to identifyrelationships involvinggeometric figures


SCO: By the end of grade 8,students will be expected toE3 draw, describe, and apply

transformations of 3-Dshapes





PerformanceE3.1 Have students start with a shape, such as the one shown, and rotatethe shape, using 90-degree rotations clockwise and counterclockwise todetermine how many distinct drawings can be created. [If needed, allowstudents the opportunity to use cubes to make the shape.]

E3.2 Have students start with a shape, such as the one shown, and througha series of reflections, determine how many distinct drawings can becreated.

E3.3 Give students a sketch of a room and provide the dimensions. Askthem how many different ways a given corner-shaped sectional sofa can bearranged in the room, if the back of the sectional must be against at leastone wall, and the back of the other side of the sectional must touch anadjacent wall. Ask students how many of the arrangements which arepossible actually make sense, in terms of making best use of the sectionalwithin the room. Students will have to take doorways, fireplaces, etc. intoconsideration.

Pencil and PaperE3.4 Start with a shape such as the one shown and shift ita) 3 units to the left and 4 units downwardb) 5 units to the right and 3 units upward





E4 When studying polygons, students can use a table, such as the oneshown, to organize information. By extending the table, they can observepatterns and generalize about the sum of the measures of the interior anglesof various polygons, as well as about the measure of each interior angle of aregular polygon. The sum of the measures of the interior angles of apolygon is found by dividing the polygon into triangles as shown.

Since the sum of the measures of the angles in a triangle is 180° , the sumof the interior angles in the pentagon shown is 5× 180° , subtract the360° at the centre: 5× 180 - 2× 180 = (5 - 2)× 180.

Regular Polygons# of sides sum of interior∠ s measure of each ∠

3 180(1 × 180) 604 360(2 × 180) 905 540(3 × 180) 108. [always multiplying 180. by 2 less than the #. of sides]n (n - 2) × 180

(n - 2) × 180n

This is a good opportunity to reinforce work with graphing. Students cangraph the sum of the interior angles against the number of sides, or themeasure of each angle against the number of sides, to determine if therelationships are linear.

Similarly, patterns related to diagonals can be explored.Polygons

# of sides # of diagonals total # ofat one vertex diagonals

3 0 04 1 25 2 56 3 97 4 14..n n - 3

n × (n- 3)2

Again, the relationships between the number of sides and number ofdiagonals at one vertex, and number of diagonals in all, can be graphed todetermine if they are linear.

Symmetry of the various regular polygons can be explored to determine ifthe number of lines of symmetry is related to the number of sides.

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toii) compare and classify geometric

figures, understand and applygeometric properties andrelationships, and representgeometric figures viacoordinates

v) draw inferences, deduceproperties, and make logicaldeductions in synthetic[Euclidean] andtransformational geometricsituations

SCO: By the end of grade 8,students will be expected toE4 analyse polygons to

determine their propertiesand interrelationships





PerformanceE4.1 Ask students to sketch any triangle and, through a series of reflections,determine if a regular polygon can be produced. Ask students to considerwhat characteristics would be necessary in a triangle so that, through aseries of reflections, a regular polygon is produced. [Possible answer: Whenthe triangle is isosceles and the reflections are being done using the equalsides a regular polygon is produced, as long as the product of the numberof sides and the measure of each vertex angle is 360 degrees.]

E4.2 Ask students to sketch a series of regular polygons with from 3 to 10sides.a) Have students work in pairs to find the number of lines of symmetry

which can be drawn for each polygon. Ask them if any patternsemerge, and to explain their answers.

b) Have students find the centre of each polygon by finding theintersection point of the right bisectors of the sides of the polygon.When each vertex of the polygon is joined to the centre, the anglesformed are called the central angles. Ask students to record themeasures of the central angles for the series of regular polygons. Askthem if any patterns emerge, and to explain their answers.

E4.3 Ask students to speculate about the sum of the measures of theexterior angles of polygons, and then ask them to set up an investigation toexplore and determine if there is a relationship between the number of sidesand the sum of the measures of the exterior angles. [An exterior angle isdefined as an angle formed when one side of a polygon is extended at anyvertex.]

exterior angle

Pencil and PaperE4.4 Using the relationship established between the number of sides andthe angle measures in a polygon, finda) the number of sides if the sum of the interior angles is 16200

b) the number of sides if the measure of each interior angle of a regularpolygon is 1440

c) the sum of the interior angles if there are 15 sidesd) the measure of each interior angle of a regular 12-sided polygon

E4.5 Using the relationship established between the number of sides andthe number of diagonals which can be drawn, finda) the total number of diagonals which can be drawn in a figure with 20

sidesb) the number of sides a polygon has if 14 diagonals can be drawn from

one vertexc) the number of sides a polygon has if a total of 27 diagonals can be

drawn





E5 The use of computer software can allow a great deal of flexibility for theinvestigation of enlargements and reductions. It should be noted that,when a ratio is used to represent an enlargement or a reduction, the formatof the ratio is New:Original. A ratio of 2:1 means the new figure is anenlargement to twice the original. Likewise, a ratio of 1:3 means the newfigure is a reduction to

13 of the original, or the original is three times the

new figure.

As well as considering the scale, the centre of a dilatation must be identifiedin order to locate the position of the dilatation image. It is when the issueof dilatation centre is considered that an enlargement or reduction can bedescribed as a dilatation. Students should work with centres which arelocated inside the original figure, on a side of the original figure, andexternal to the original figure. Both opaque projectors and overheadprojectors are useful tools in working with dilatations.

When students construct a dilatation image, the following steps areimportant:• Use a straightedge to draw a faint ray from the dilatation centre to each

vertex of the original figure (in the case of enlargements, extend theline).

• Measure the distance from each vertex of the original figure to thedilatation centre.

• Use the scale ratio to calculate distances for the image. For example, ifthe scale ratio is 1:3, representing a reduction of 1

3 , each distance fromthe dilatation centre to the image vertex will be calculated bymultiplying by

13 .

• The image vertices will be on the line joining the vertex of the originalfigure to the centre, but will be

13 of the distance between the two

points.

This is shown below:

CÕ

C

D

A

B

BÕ

DÕ

AÕ0

9.7 cm

7.6 cm

7.9 cm

9.2 cm

As a means of reinforcing other transformations, students should also workwith combinations of transformations that include dilatations, such as anenlargement followed by a reflection.

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toiii) develop and analyse the

properties of transformationsand use them to identifyrelationships involvinggeometric figures

SCO: By the end of grade 8,students will be expected toE5 represent, analyse, describe,

and apply dilatations

OC = 9.2 cm OC’ = 9.2 ÷ 3 = 3.1 cmOB = 7.9 cm OB’ = 7.9 ÷ 3 = 2.6 cmOA = 7.6 cm OA’ = 7.6 ÷ 3 = 2.5 cmOD = 9.7 cm OD’ = 9.7 ÷ 3 = 3.2 cm





a)

b)

PerformanceE3/5.1 Give pairs of students a bucket of interlocking cubes.a) Show students a shape. Ask them to construct a shape which

represents an enlargement by a factor of 3.b) Ask students to each make a shape, using exactly 6 cubes, exchange

shapes with a partner, and ask their partner to create an enlargement ofthe shape by a factor of 2. [Note: Many of the 3-D objects explored inassociation with E1-E3 can be used in this context as well.]

c) Ask students whether this activity can be represented by a dilatation,and ask them to show how the cubes would need to be placed to makea dilatation. [The use of a dilatation centre differentiates a dilatationfrom a simple enlargement or reduction.]

Pencil and PaperE5.2 Write about whether congruence, area, length, and orientation arepreserved as they pertain to enlargements and reductions. Use diagrams tohelp support your writing.

E5.3 Determine the scale factor for each of the following:

AÕ

A

0

BÕ

B

CÕC

E5.4 A photograph is 5 cm by 7 cm. It is enlarged by a scale factor of 2.5.What are the new dimensions?

E5.5 In the diagram below, using the origin as the centre of dilatation,a) enlarge the given figure by a scale factor of 2b) reduce the given figure by a scale factor of 1

2c) record the vertices for each image

0 1 2 3 4 5 6

2

1

BÕ

AÕ

CÕ0

A

BC



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


Probability

General Curriculum Outcome G:

Students will represent and solve problemsinvolving uncertainty.





G1 The need to find the probability that something will not occur presentsitself perhaps as often as the need to find the probability that somethingwill occur. If an event is described as E, the probability of the event can bedescribed as P(E). The probability that the event will not occur can then bedescribed as not P(E). Not P(E) is said to be the complement of P(E).

If the probability of getting a two when you roll a die is 1 out of 6,what is the probability of not getting two on the same roll?

The sum of the probabilities of an event and its complement is alwaysequal to 1. That is, P(E) + not P(E) = 1.

In situations for which the probability of various events occurring is notequally likely, experimentally is often the only method of determiningprobability.

One experiment often conducted is that of tossing a paper cup to see ifit lands upright, upside down, or on its side. The probability of a papercup landing upright is not

13 , and the only means of determining the

probability is experimental.

A similar experiment involves determining the probability that athumb tack will land on its head. Again, the probability of a tacklanding on its head is not

12 . To find the probability in this situation

would again require experimentation.

G2 Students already learned how to calculate the theoretical probability ingrade 7, using

# of favorable outcomesP(E) =

Total # of possible outcomes

This can only be used when dealing with equally likely outcomes or eventsand, therefore, is not applicable to the paper-cup or the thumb-tackexamples above. To find the probability of a complementary event, theformula 1 - P(E) can be used. If, for example, the probability of an eventoccurring is

14 , then the probability of it not occurring is 1 -

14 =

34 .

G3 Once students have worked with probability experiments and derivedtheoretical probability, they should be able to compare the results obtainedfrom each method. Discuss with students when they can be comfortablethat the experimental probability is a close approximation of the theoreticalprobability, and what can be done to increase their confidence inexperimental results. Discussion here should focus on the influence ofincreasing sample size.

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 andwill also be expected toi) make predictions regarding,

and design and carry out,probability experiments andsimulations in relation to avariety of real-world situations

ii) derive theoretical probabilities,using a range of informal andformal techniques

iii) determine and compareexperimental and theoreticalresults

SCO: By the end of grade 8,students will be expected toG1 conduct experiments and

simulations to findprobabilities of single andcomplementary events

G2 determine theoreticalprobabilities of single andcomplementary events

G3 compare experimental andtheoretical probabilities





PerformanceG1/2/3.1a) Ask students to work with a partner with one person acting as recorder

and the other tossing a standard six-faced die. Ask them to roll the die50 times, record the results, and use the results to estimate theprobability ofi) tossing a 5ii) not tossing a 5iii) tossing a number less than 4iv) not tossing a number less than 4v) tossing a number greater than or equal to 3vi) tossing a number not greater than or equal to 3

b) Ask what they notice about the sum of the results for a) and b), c) andd), and e) and f ).

c) Ask students to use tree diagrams, or the formula for theoreticalprobability, to determine the probabilities for i) - vi).

d) Ask students to compare theoretical and experimental probabilities.

G1/2/3.2 Prior to class, put 2 counters of one colour (black), 3 of anothercolour (red), and 3 of a third colour (yellow) in a bag. Ask students to workwith a partner, with one person acting as recorder and the other drawingobjects from the bag.a) Ask them to draw a counter from the bag, repeating this activity 50

times, and recording the results. Ask them to use the results to estimatethe probability of getting a black, the probability of getting a red, andthe probability of not getting a red.

b) Ask students to find the theoretical probability and compare withresults found through experiments.

Pencil and PaperG2.1 A survey was conducted to answer the question, What is yourfavourite sport? The results are shown below.

Sport Percent

Baseball 18Basketball 15Tennis 12Swimming 28Skating 11Soccer 16

a) What is the probability that a friend’s favourite sport is not baseball?b) What is the probability that a friend’s favourite sport is not played on a

field?





G4 In the survey work which has been done in relation to GCO (F), therewere many opportunities for students to draw conclusions aboutpopulations on the basis of experimental results. To give students a betterunderstanding of the use of probability, invite a person who usesprobability data on a regular basis as part of his/her work to talk with theclass. For example, an insurance agent could talk about mortality tables andhow they are developed and used to establish rates. They can also talkabout how data on car accidents and the age of drivers is used to establishdifferent rates for younger drivers, and for male and female drivers. Ameteorologist could explain how the probability of rainfall is determined.Similarly, a person who works for a polling company could explain theaccuracy of a survey as it is typically reported (e.g., media often uses phrasessuch as 95% accurate, 19 times out of 20).

Ask students to work in small groups to brainstorm about thedecisions people make and how statistics and probability increase theirknowledge base for decision making. Ask them them to discuss thefollowing question: How does probability affect decisions people makein the following areas?- smoking/health- speeding/accidents- drugs/side effects- allergies/side effects- wearing seat belts/injuries- education/earning potential

KSCO: By the end of grade 9,students will have achieved theoutcomes for entry-grade 6 and willalso be expected toiv) relate a variety of numerical

expressions (ratios, fractions,decimals, percents) to thecorresponding experimental orsimulation situation

SCO: By the end of grade 8,students will be expected toG4 demonstrate an

understanding of how data isused to establish broadprobability patterns





Pencil and PaperG4.1 Working in pairs, draw an irregular area on a 6 × 6 grid. This figureshould not be seen by your partner. Ask your partner to pick thecoordinates of 25 points at random, and record the selections as hitting ormissing the irregular figure. Use the ratio of hits to tries to predict thepercentage of the grid which the irregular figure covers. [Note: This can bedone using repeated samples, and students can look intuitively at thevariability of repeated samples.]

G4.2 Of 1000 people surveyed in Lightnerville, 750 were found to beblond.a) What would you conclude about the people of Lightnerville?b) Billy-Bob is on a basketball team from Lightnerville High, and there is

only one blond on that team. Is this possible? Explain.

PresentationG4.3 Ask students to select one of the following, do some research, andprepare a brief presentation to the class.a) How are batting averages used in baseball in decision making?b) How is the probability of getting lung cancer increased if you smoke?c) How is the probability of avoiding major injury increased when you

wear a seatbelt?d) How is the probability of avoiding major injury increased when air

bags are used? Why is this not necessarily true for young children?e) How is the probability of getting a good job affected by education

levels?

PortfolioG4.4 Ask students to find out the percentage of the population which isexpected to be of the various blood types, and to use the information todetermine the chance of finding a person in their school who is of each ofthe blood types. Ask them to express each result as a ratio and as a fraction.Ask students to discuss the possibility that no one at their school is A+ orAB-.

Grade 8 - Cape · PDF filethe development of this grade 8 mathematics curriculum guide. ......

Documents

Transcript of Grade 8 - Cape · PDF filethe development of this grade 8 mathematics curriculum guide. ......