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Elementary – Grade 5

Nathalie FortierAnnie Leblanc

WORKBOOK

M A T H E M A T I C S

B

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Registration of copyright – Bibliothèque et Archives nationales du Québec, 2014 Registration of copyright – Library and Archives Canada, 2014

Printed in Canada 7890 HLN 22 21 20 ISBN 978-2-7613-6133-0 13325 ABCD OF10

© ÉDITIONS DU RENOUVEAU PÉDAGOGIQUE INC., 2014 Member of Pearson Education since 1989

1611 Crémazie Boulevard East, 10th Floor Montréal, Québec H2M 2P2 Canada Telephone: 514 334-2690 Fax: 514 334-4720 [email protected] pearsonerpi.com

ENGLISH VERSION

Project Editor and TranslatorAmy Paradis

ProofreaderBrian Parsons

Art DirectorHélène Cousineau

Graphic Design CoordinatorSylvie Piotte

Cover Frédérique Bouvier

Electronic PublishingCatherine Boily

ORIGINAL VERSION

Managing EditorMonique Daigle

Project Editor and Linguistic Reviewer Lina Binet

Project Editor and Photo Research Marie-Claude Rioux

ProofreaderLucie Lefebvre

Coordinator, Rights and PermissionsPierre Richard Bernier

Art DirectorHélène Cousineau

Graphic Design CoordinatorSylvie Piotte

Graphic Design and CoverFrédérique Bouvier

Electronic PublishingInterscript

IllustratorMichel Rouleaup. 43

ENGLISH VERSION

Pedagogical ReviewerPaul Lamarche

Pedagogical ConsultantBob Butler, Teacher, Chelsea Elementary School,

Western Québec School Board

ORIGINAL VERSION

Science Content ReviewerPhilippe Bazinet, Mathematics Pedagogical Consultant,

Western Québec School Board

ConsultantsAnn-France Couture, Teacher, École Auclair,

commission scolaire des Trois-LacsCaroline Geoffroy, Teacher, École Saint-Donat,

commission scolaire de MontréalJulie Germain, Teacher, École Saint-Benoît,

commission scolaire de MontréalAntoine Leblanc, Teacher, École Sainte-Anne,

commission scolaire des Hautes-Rivières

Stock Illustrationsshutterstock

Meaning of Pictograms

Student constructs knowledge with teacher guidance.

Student applies knowledge by the end of the school year.

Problem-solving steps are presented on the inside cover of the workbook.

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B-III

TABLE OF CONTENTS

THEME

Showtime! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

4.1 Arithmetic Following the Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4.2 Arithmetic Associative, Commutative and Distributive Properties . . . . . . . . . . . . . . . . . . . . . 5

4.3 Arithmetic Multiplying Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.4 Arithmetic Dividing Natural Numbers with a Decimal Remainder . . . . . . . . . . . . . . . . . . . . . . 12Dividing by 10, 100 and 1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.5 Statistics Creating Questions for a Survey, Collecting and Organizing Data . . . . . . . . . . . 16

4.6 Statistics Understanding and Calculating the Arithmetic Mean . . . . . . . . . . . . . . . . . . . . . . . 19

4.7 Arithmetic Dividing a Decimal by a Natural Number Less than 11 . . . . . . . . . . . . . . . . . . . . 23

4.8 Measurement Estimating and Measuring Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

MAKING CHOICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

PROBLEM SOLVING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

THEME

Let’s Move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1 Arithmetic Reading and Writing Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Locating Integers on a Number Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Comparing Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Measurement Estimating and Measuring Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Geometry Locating Points in a Cartesian Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4 Geometry Observing and Producing Frieze Patterns Using Translations . . . . . . . . . . . . . . . 53Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.5 Measurement Estimating and Measuring Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Establishing Relationships Between Units of Measure . . . . . . . . . . . . . . . . . . . . . 57

5.6 Measurement Establishing Relationships Between Units of Measure for Time . . . . . . . . . . 60

III


B-IV

5.7 Measurement Estimating and Measuring Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.8 Measurement Estimating and Measuring Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Establishing Relationships Between Units of Measure . . . . . . . . . . . . . . . . . . . . . 67

MAKING CHOICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

PROBLEM SOLVING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

THEME Adventures in Nature . . . . . . . . . . . . . . . . . . . . . . . 79

6.1 Probability Enumerating Possible Outcomes of a Random Experiment Using a Table and a Tree Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2 Probability Comparing the Outcomes of a Random Experiment with Known Theoretical Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3 Geometry Describing and Classifying Prisms and Pyramids Using Faces, Vertices and Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.4 Geometry Nets of Solids or Convex Polyhedrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.5 Geometry Testing Euler’s Theorem on Convex Polyhedrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

MAKING CHOICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

PROBLEM SOLVING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Final Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121


Oftentimes, contest winners are asked to answer a skill-testing question containing a chain of operations. They have to answer this question correctly to claim the prize!

A Winning Chain of Operations

4.1 Following the Order of Operations

4.2 Associative, Commutative and Distributive Properties

4.3 Multiplying Decimals

4.4 Dividing Natural Numbers with a Decimal RemainderDividing by 10, 100 and 1000

4.5 Creating Questions for a Survey, Collecting and Organizing Data

4.6 Understanding and Calculating the Arithmetic Mean

4.7 Dividing a Decimal by a Natural Number Less than 11

4.8 Estimating and Measuring Surface Area

WHAT I’LL LEARN

Showtime!

To estimate the cost of a renovation, a contractor must first determine the total surface area that needs renovating. With this information, the contractor can then set the costs for supplies and labour.

Price Per Square Metre

TH

EM

E

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LEARN ABOUT IT

Following the Order of Operations The order of operations is the order you must follow when making calculations in a chain of operations.

To avoid getting an incorrect result from a chain of operations, it is important to follow the order of operations.

Here is the order to follow.

1. Operations in brackets.

2. Exponentiation (exponents).

3. Multiplication and division, from left to right.

4. Addition and subtraction, from left to right.

Here are the steps for solving the following chain of operations: 5 × 22 − 6 + 35 ÷ (2 + 3) = ?

Steps Example

1. Do all operations in brackets. 5 × 22 − 6 + 35 ÷ (2 + 3) = ?5 × 22 − 6 + 35 ÷ 5 = ?

2. Do the exponentiation (exponents). 5 × 22 − 6 + 35 ÷ 5 = ?5 × 4 − 6 + 35 ÷ 5 = ?

3. Do the multiplication and division, from left to right.

5 × 4 − 6 + 35 ÷ 5 = ? 20 − 6 + 7 = ?

4. Do the addition and subtraction, from left to right.

20 − 6 + 7 = ? 14 + 7 = 21

5 × 22 − 6 + 35 ÷ (2 + 3) = 21

ARITHMETIC Chain of Operations SECTION 4.1

WORK IT OUT

1 Underline the operation that has to be done first in each chain of operations.

a) 35 + 20 × 12 = b) 4 + (15 − 12) + 3 = c) 7 + 4 − 5 + 3 × 4 =

d) 32 ÷ 8 + 4 = e) 42 + 3 × 6 = f) (4 + 3) × 4 =

g) 38 − 20 + 52 = h) 50 ÷ 10 ÷ 5 = i) 23 × 52 + (27 − 11) =

A series of mathematical

operations is called a “chain

of operations.”

B-2 Section 4.1


LEARN ABOUT IT2 Calculate these chains of operations

by following the order of operations.

example 4 + 5 × 3 + 8 − 6 = a) 30 − 4 × 6 + 12 × 6 =

4 + 15 + 8 − 6 = 21

b) 2 × 12 ÷ 4 + 5 = c) (4 + 8) + 15 − (2 × 3) =

d) 5 × 7 − (4 + 8 − 3) = e) (12 − 5) + 3 × 7 =

3 Match each chain of operations with its result.

a) 3 × 9 − 2 + 2 × 6 • • 3

b) 3 × (9 − 2) + 2 × 6 • • 37

c) 3 × 9 − (2 + 2) × 6 • • 33

4 Circle the operation that will solve the following problem.

At the improv show, 15 teams of 4 girls, 7 teams of 4 boys and 8 mixed teams of 4 people are competing. How many participants are there in all?

a) 15 + 7 + 8 = 30 b) (15 × 4) + (7 × 4) + (8 × 4) = 120

My Calculations

When writing out a chain of operations, align the mathematical symbols on the next line to make sure you don’t forget any.

Think about it!

Theme 4 B-3


USE REASONING

1 At Melody’s ballet recital, there are 10 rows of 8 spectators to the left of the stage, 10 rows of 6 spectators to the right of the stage, 15 rows of 12 spectators in the middle and 6 rows of 20 spectators on the balcony. How many spectators are there at the recital? Use a chain of operations to help you calculate.

Solution:

2 Isaac is buying some souvenirs at his favourite singer’s rock concert. He chooses 2 posters for $7.50 each, a shirt for $23.75 and 3 pins for $5.25 each. He has $60 in his pocket. How much money does he have left after he makes his purchases? Use a chain of operations to help you calculate.

Solution:

B-4 Section 4.1


WORK IT OUT

LEARN ABOUT IT

Associative, Commutative and Distributive Properties Associative, commutative and distributive properties make calculating mathematical operations easier.

• The associative property applies to addition and multiplication. It groups the numbers in an equation in different ways without changing the result.

Examples:

• The commutative property also applies to addition and multiplication. It moves around the numbers in an equation in different ways without changing the result.

Examples:

• The distributive property applies to multiplication. It distributes multiplication over addition or subtraction.

Examples:

Associative Property of Addition Associative Property of Multiplication

(4 + 18) + 2 + 5 = 4 + (18 + 2) + 5

22 + 2 + 5 = 4 + 20 + 5

29 = 29

(3 × 7) × 5 × 2 = 3 × (7 × 5) × 2

21 × 5 × 2 = 3 × 35 × 2

105 × 2 = 105 × 2

210 = 210

Commutative Property of Addition Commutative Property of Multiplication

28 + 13 + 6 = 13 + 6 + 28

41 + 6 = 19 + 28

47 = 47

8 × 3 × 5 = 3 × 5 × 8 24 × 5 = 15 × 8 120 = 120

Distributive Property of Addition Distributive Property of Subtraction

1. 5 × (9 + 3) = (5 × 9) + (5 × 3)

2. = 45 + 15

= 60

1. Distribute 5 over 9 and 3.

2. Add the 2 products.

1. 3 × (11 − 5) = (3 × 11) − (3 × 5)

2. = 33 − 15

= 18

1. Distribute 3 over 11 and 5.

2. Subtract the 2 products.

ARITHMETIC Determining Numerical Equivalencies Using

Relationships Between Operations SECTION 4.2

Theme 4 B-5


WORK IT OUT

1 Use the associative property to do these operations.

a) 13 + 29 + 6 =

b) 10 × 16 × 3 × 4 =

c) 124 + 52 + 14 + 36 =

d) 5 × 111 × 2 × 7 =

2 True or false? True False

a) 15 + 30 + 22 = 15 + (30 + 22)

b) (46 − 16) − 6 = 46 − (16 − 6)

c) 144 ÷ (12 ÷ 4) = (144 ÷ 12) ÷ 4

d) (32 × 10) × 14 = 32 × (10 × 14)

3 Find 2 equivalent chains of operations by applying the commutative property to each equation. Then, calculate the result.

a) 11 + 9 + 30 =

b) 2 × 6 × 11 =

c) 40 + 14 + 60 + 27 =

d) 6 × 9 × 5 × 10 =

My Calculations

B-6 Section 4.2


4 Create 2 chains of operations to demonstrate the commutative property.

a) Commutative property of addition:

b) Commutative property of multiplication:

5 Use the distributive property to do these operations.

a) 6 × (10 + 7) =

b) (30 − 12) × 5 =

c) (14 + 22) × 4 =

6 Angela finds 3 boxes, each containing 15 hats and 12 scarves, in a dressing room. She also finds 4 boxes, each containing 10 pairs of glasses and 8 purses.

a) Use the distributive property to write the chains of operations needed to calculate the number of accessories that Angela has found.

Hats and scarves:

Glasses and purses:

b) Calculate the total number of accessories that Angela has found.

7 Indicate the property applied in each chain of operations.

a) (87 + 38) + 23 = 87 + (38 + 23)

b) 66 × 3 × 7 = 3 × 7 × 66

c) 9 × (25 − 10) = (9 × 25) − (9 × 10)

My Calculations

Theme 4 B-7


LEARN ABOUT IT

ARITHMETIC Multiplying Decimals SECTION 4.3

Multiplying Decimals Decimals are multiplied in the same way that natural numbers are multiplied. The decimal point is then added based on the number of decimal places in the 2 factors.

Here are the steps for multiplying one decimal by another decimal.

Steps Example: 15.4 × 6.3

1. Align the 2 numbers to be multiplied into columns.

1 5 . 4 × 6 . 3

2. Multiply the 2 numbers. Ignore the decimals points.

3 2 1 1

1 5 4 × 6 3 1

4 6 2 + 9 2 4 0

9 7 0 2

3. Count the total number of digits after the decimal point in both factors. (In this case, there are 2.)

15.4 There is 1 digit after the decimal point.

6.3 There is 1 digit after the decimal point.

4. Add a decimal point to the product before the last 2 digits.

1 5 .4 × 6 .3

9 7 . 0 2 There are 2 digits after the decimal point.

1 5 . 4 × 6 . 3

4 6 2 + 9 2 4 0

9 7.0 2

1 5 × 6

9 0

3 2 1 1 3

1

The product is therefore close to 90, not 900.

To make sure the decimal point is placed correctly, round

the 2 factors to make an approximation of the product.

Then, multiply.

Think about it!

When multiplying the 1st factor by the digit in the tens position in the 2nd factor, start by placing a 0.

B-8 Section 4.3


LEARN ABOUT IT

WORK IT OUT

1 Make an approximation of the product, then calculate it.

Approximation Approximationa) 3 .7 × 2 . 4

b) 1 8 . 2 × 4 . 6

2 Do these multiplications. Then, match each product with a letter in the secret code to discover Elsa’s favourite type of dance.

b) 3 9 . 6 × 8 . 5

e) 6 3 . 5× 1 4.8

c) 8 9 .3 × 6 .7

f) 8 6 . 3× 3 2.2

a) 1 4 . 7 × 4 . 9

d) 5 7 . 8× 2 5.3

Secret Code

A = 720.31 B = 5983.1 C = 2448.48 D = 598.31 E = 1462.34 F = 2448.44

G = 16 721.5 H = 9390.86 I = 1815.22 J = 16 717.05 K = 335.55 L = 2778.68

M = 72.03 N = 2778.86 O = 336.60 P = 16 717.5 Q = 2449.44 R = 939.80

S = 16 717 T = 939.89 U = 1815.20 V = 7203 W = 3366 X = 1462.38

Elsa’s favourite type of dance:

Theme 4 B-9


3 Solve these problems.

a) William buys a bouquet of flowers for $14.70 and a bunch of balloons that is 1.3 times more expensive. How much does the bunch of balloons cost?

b) Max’s grandfather says that a theatre ticket used to cost $2.30 when he was young. Max mentions that his ticket to see the play Clever Cats costs 8.6 times more. How much does Max’s ticket cost?

c) A circus horse gallops 2.3 km to complete one lap of the track. How many kilometres does the horse run after 8.5 laps?

d) Jessie performs 2 dance routines. Before the first routine, she drinks 1.2 L of lemon water. She drinks 2 times that amount before the second routine. Each litre of water contains 4.7 g of natural sugar. How many grams of sugar does Jessie drink?

My Calculation

My Calculation

My Calculation

My Calculation

B-10 Section 4.3


USE REASONING

1 The width of a theatre stage is 4.5 m. Its length is 2.7 times greater. The width of the giant screen on the wall behind it is 2.2 m. Its length is 2.4 times greater. The decorator installs a rope light all around the stage and another around the screen. How long is each of the 2 rope lights?

Solution:

2 Natasha buys 17.5 m of cotton and 8.2 m of silk to make superhero costumes. The cotton costs $14.50 per metre and the silk costs $18.30 per metre. Natasha has a budget of $500.00 to make the costumes. Does she have enough money to also buy 3.4 m of linen at $15.70 per metre? If yes, how much money does she have left?

Solution:

Theme 4 B-11


LEARN ABOUT IT

Dividing Natural Numbers with a Decimal RemainderDivision is an operation for finding how many times a divisor goes into a dividend. If the quotient is not an integer (whole number), the remainder is expressed with decimals.

Here are the steps for expressing the remainder of a division with decimals.

Steps Example: 25 3361. The divisor 25 does not go into 3 hundreds,

the 1st digit of the dividend. So, use 33 tens.• 25 goes into 33 one time.

Write the number 1 above 3 tens.• Multiply 1 × 25 = 25 and write 25

below 33 tens. • Subtract 33 − 25 = 8.

2. Bring down the 6 units. • 25 goes into 86 three times.

Write the number 3 above 6 units.• Multiply 3 × 25 = 75 and write 75 below 86 units.

Subtract 86 – 75 = 11.

3. Since the difference is not equal to 0, there is a remainder. • Add a decimal point and 2 zeroes to the dividend

and a decimal point to the quotient.

4. Bring down the 1st 0. • 25 goes into 110 four times.

Write the number 4 above 0 tenths.• Multiply 4 × 25 = 100 and write 100 below 110. • Subtract 110 − 100 = 10.

5. Bring down the 2nd 0. • 25 goes into 100 four times.

Write the number 4 above 0 hundredths.• Multiply 4 × 25 = 100 and write 100 below 100. • Subtract 100 – 100 = 0.

Do the inverse operation to check your answer: 13.44 × 25 = 336

ARITHMETIC Dividing Natural Numbers with a Decimal Remainder

Dividing by 10, 100 and 1000 SECTION 4.4

Divisor Dividend Quotient

5 36 = 7.20

Decimal

25

1 3 H T U

3 3 6 − 2 5 0 8 6 − 7 5 1 1

21

25

H T U

3 3 6 . 0 0 − 2 5 0 8 6 − 7 5 1 1 0 − 1 0 0 1 0 0 − 1 0 0 0

21

1 3 . 4 4

B-12 Section 4.4


LEARN ABOUT IT

WORK IT OUT

1 Calculate the quotient of each division.

a) b) c)

d) e) f)

g) h) i)

Do the inverse operation to check your answers.

Think about it!

6 9 012

25

32 16 50

20 22

30 209 6 3 1 7 2 6

4 5 6 2 8 4 4 5 5 5 5

3 2 4 0 2 4 2 8 4 8 3 2

Theme 4 B-13


WORK IT OUT

Calculate the quotient of each division. Use mental calculation strategies.

a) 480 ÷ 100 = b) 2589 ÷ 10 =

c) 57.2 ÷ 100 = d) 1643.7 ÷ 10 =

e) 241.8 ÷ 10 = f) 7809.5 ÷ 100 =

Dividing by 10, 100 and 1000 Look at these examples for dividing integers.

57 ÷ 10 = 5.7 57 ÷ 100 = 0.57 57 ÷ 1000 = 0.057

325 ÷ 10 = 32.5 325 ÷ 100 = 3.25 325 ÷ 1000 = 0.325

1426 ÷ 10 = 142.6 1426 ÷ 100 = 14.26 1426 ÷ 1000 = 1.426

When dividing an integer by a multiple of 10 (10, 100, 1000, etc.), move the decimal point one or more places to the left. Add a decimal point and one or more zeroes if needed.

325 ÷ 10 = 32.5 325 ÷ 100 = 3.25 325 ÷ 1000 = 0.325

Look at these examples for dividing decimals.

58.9 ÷ 10 = 5.89 58.9 ÷ 100 = 0.589

314.2 ÷ 10 = 31.42 314.2 ÷ 100 = 3.142

1458.2 ÷ 10 = 145.82 1458.2 ÷ 100 = 14.582

When dividing a decimal by a multiple of 10 (10, 100, 1000, etc.), move the decimal point one or more places to the left. Add one or more zeroes if needed.

For example, to calculate 314.2 ÷ 100, move the decimal point 2 places to the left to get 3.142.

To divide by 10, move the decimal point one place to the left. To divide by 100, move the decimal point two places to the left. To divide by 1000, move the decimal point three places to the left.

LEARN ABOUT IT

B-14 Section 4.4


USE REASONING

1 Tickets sold during 4 comedy shows brought in $6328.00. Each show attracted 100 spectators. Tickets sold during 5 circus performances brought in $143 500.00. Each show attracted 1000 spectators. What is the cost of 1 ticket to the comedy show? What is the cost of 1 ticket to the circus?

Solution:

2 At the puppet theatre, 391 students are signed up for a puppet-making workshop and 245 students are signed up for a puppeteering workshop. Organizers break up the students into groups of 46 for the puppet-making workshop and groups of 25 students for the puppeteering workshop. How many of each type of workshop must the organizers give if all of the students want to participate?

Solution:

Theme 4 B-15


LEARN ABOUT IT

Creating Questions for a Survey, Collecting and Organizing DataAn investigation or survey is a means to gather information for obtaining statistics.

• Survey questions are designed to collect data. Here are 3 characteristics of an effective survey question:

Examples1. A clear question linked to the subject

of the survey.• In which season do the students

in the class celebrate their birthdays?

2. A question, often multiple choice, that is easy to read and simple to answer.

• Which of these colours do you like the least: blue, red or green?

3. A question that leads to an answer that is easy to process.

• Which of these is your favourite subject: English, math or science?

• A data table can help you to organize survey data and compile your answers.

Example:

A data table has 4 elements.

1. A survey title.

2. Answer categories.

3. A tally of answers (each answer is represented by ).

4. Results (the total number of answers in each category).

1 Seasons Students in Class Celebrate Birthdays

2 Spring Summer Fall Winter

3 4 4 6 9 6

STATISTICS Creating Questions for a Survey, Collecting,

Describing and Organizing Data SECTION 4.5

1 Circle the questions that qualify as effective survey questions.

a) What did you do yesterday? b) Do you prefer dancing or singing?

c) In what year were you born? d) What is your brother’s name?

e) In what month were you born? f) Who is your favourite actor?

WORK IT OUT

B-16 Section 4.5


2 Use the data gathered from Tara’s survey to fill in the table below.

Musical Instruments Played by Students in the School Band

Flora: trombone Alex: trombone Mylie: flute Wendy: clarinet

Lucas: drums Mia: trumpet Lily: drums Evie: flute

Theo: clarinet Laurie: flute Angie: trumpet Luke: trumpet

Carl: flute Victor: trombone Sarah: clarinet Leo: trumpet

Emma: trumpet Alison: flute Felix: clarinet Noah: trombone

Data Table

Instrument

Tally

Results

3 Use the following data to fill in the table below.

• More than 1

3 of the students prefer hip-hop.

• Classical music is the least popular type of music.

• Pop music is 2 times more popular than rock music.

• 30 students answered the survey.

Pop

2 10

Theme 4 B-17


4 Matthew asks the students in the auditorium what they liked best about the school play. Here are the results of his survey.

Part of School Play Most Liked by Students

Singing Dancing Acting Comedy

11 16 8 5

Answer these questions:

a) What part of the school play did the students like the least?

b) How many students saw the play?

c) Which part was 2 times more popular than the acting?

d) Which part did the students like best?

5 Look at the graphs below. Form a survey question based on the data in each graph.

a) b)

Survey question: Survey question:

Types of Performers

Circus Performers

Num

ber

of P

eopl

e

20181614121086420

AnimalsClowns

Acrobats

Lion tamers

Types of Movies

Comedy

Drama

Science �ction

Horror

Suspense

35%

25%

25%

10%

5%

B-18 Section 4.5


LEARN ABOUT IT

Understanding and Calculating the Arithmetic MeanThe arithmetic mean is the sum of a set of numbers divided by the total number of parts in the set.

In statistics, it is used for analyzing a group of data and for finding an average value.

To calculate the arithmetic mean:

1) add all of the data within a group;

2) divide the sum by the total number of data values.

Example 1: Example 2:Calculate the average price of a ticket to the circus:• Flying Circus: $15.25• Magical Circus: $18.75• Modern Circus: $14.00• Circus Explosion: $16.35• Bravo Circus: $12.65• Musical Circus: $22.001. Add 15.25 + 18.75 +

14 + 16.35 + 12.65 + 22 = 99

2. Divide by the number of pieces of data (6): 99 ÷ 6 = 16.5

The average price of a ticket to the circus is $16.50.

Calculate the average number of songs sung by each student in the school musical:• Julianne: 5 • Christopher: 6• Stella: 3

• Dimitri: 2

1. Add 5 + 6 + 3 + 2 = 16

2. Divide by the number of pieces of data (4): 16 ÷ 4 = 4

Each student sings an average of 4 songs.

STATISTICS Understanding and Calculating the Arithmetic Mean SECTION 4.6

If the quotient is not a natural number,

round it to the nearest tenth.

Theme 4 B-19


1 Calculate the arithmetic mean of each group of data.

a) 87 59 66 68

Arithmetic mean:

b) 119 121 118 123 114 125

Arithmetic mean:

c) 324 319 299

Arithmetic mean:

d) 68 63 67 68 64 66

Arithmetic mean:

2 Six students sold tickets to a rap concert. What is the average number of tickets sold per student?

Student Number of Tickets Sold

Elisabeth 8

Joannie 13

Riley 5

William 11

Nigel 10

Emily 7

My Calculation

My Calculation

My Calculation

My Calculation

My Calculation

WORK IT OUT

B-20 Section 4.6


3 Last month, Brad the actor tallied the hours he spent rehearsing his lines each week.

This month, Brad’s average weekly rehearsal time was one and a half hours less than the month before. What is this month’s weekly average?

4 This bar graph shows the number of students in each Grade 5 and Grade 6 class who signed up for the after-school hip-hop program.

What is the average number of students per class signed up for the program?

My Calculation

My Calculation

Class

Number of Students Per Class Signed Up for Hip-hop Program

Num

ber

of S

tude

nts

20181614121086420

5A 5B 5C 6A 6B 6C

Last Month’s Rehearsal Hours

Week 1 6 hours

Week 2 5 hours

Week 3 5 hours

Week 4 4 hours

Theme 4 B-21


USE REASONING

1 A magician is analyzing data on the age of his audience members to help him prepare for his shows.

Age GroupNumber of Audience Members Per Day

Wednesday Thursday Friday Saturday Sunday

Children 33 39 48 73 56

Teenagers 54 57 59 47 67

Adults 63 54 43 30 27

What is the average number of audience members for each age group?

Age Group Average Number of Audience Members

Children

Teenagers

Adults

Solution:

2 Here is data on the length of time that a juggler and a clown spend on stage during 2 acts at a circus show.

Performer 1st Act 2nd Act Encore Average Time Spent on Stage

Jingles the Juggler 17 minutes 13 minutes 14 minutes

Boppy the Clown 22 minutes 18 minutes 16 minutes

How much time does each performer spend onstage during the encore?

Solution:

B-22 Section 4.6


WORK IT OUT

LEARN ABOUT IT

Dividing a Decimal by a Natural Number Less than 11Division calculates the number of times a divisor fits into a dividend. The dividend is not always an integer.

Decimals are divided in the same way that natural numbers are divided. Simply add one more step: the decimal point.

Steps Example: 4 236.81. Write the numbers in the division by placing

the divisor to the left of the bracket.

2. Divide the whole part as you would normally do. • The divisor 4 does not go into 2 hundreds.

Use 23 tens. 4 goes into 23 five times: 4 × 5 = 20. There are 3 tens left.

• Bring down the 6 units. There are now 36 units to divide. 4 goes into 36 nine times: 4 × 9 = 36. There are 0 units left.

3. When you have finished dividing the whole part, move on to the decimal part. • Add a decimal point to the quotient

before bringing down the tenths.• Bring down the 8 tenths.

4 goes into 8 two times: 4 × 2 = 8.

ARITHMETIC Dividing a Decimal by a Natural

Number Less than 11 SECTION 4.7

5 9 . 24

H T U

2 3 6 . 8 − 2 0 0 3 6 − 3 6 0 8 − 8 0

Do the inverse operation to check your answer:

3

5 9 . 2 × 4

2 3 6 . 8

5 94

H T U

2 3 6 . 8 − 2 0 0 3 6 − 3 6 0

Theme 4 B-23


1 Add a decimal point to each quotient.

a) 275.94 ÷ 7 = 3 9 4 2 b) 3647.7 ÷ 9 = 4 0 5 3

c) 242.25 ÷ 5 = 4 8 4 5 d) 869.7 ÷ 3 = 2 8 9 9

e) 317.48 ÷ 4 = 7 9 3 7 f) 557.52 ÷ 6 = 9 2 9 2


a) b) c)

d) e) f)

WORK IT OUT

2 0 8.2 57

3 5 6

4 83 3 7.84 6 0 0.48

2 4 1 4.1 7 3 4.5 3 5 8.8

B-24 Section 4.7


USE REASONING

1 There are 9 members in the school’s hip-hop dance troupe. One parent gives them 191.7 m of fabric to make costumes for their recital. Each dancer gets an equal amount of fabric. How many metres of fabric does each student get for a costume?

Solution:

2 A group of volunteers is putting on a benefit concert to send 8 choir singers to New York for a music competition. They sell 365 tickets for $10.50 each, and they make $331.90 from selling refreshments during intermission. If the money raised is divided equally, how much money does each musician get for the trip to New York?

Solution:

Theme 4 B-25


LEARN ABOUT IT

Estimating and Measuring Surface Area Area is the measure of a figure’s surface.

Measure the area to calculate the surface of a stage, a screen, a wall, a floor, etc.

Use the international system of units (SI) to measure surface area. The most common units of measure are square kilometres (km2), square metres (m2) and square centimetres (cm2).

Common Units of Measure:km2 area of a city or a countrym2 area of a stage or a tabletopcm2 area of a playing card or a poster

To calculate the area of a figure, count the number of units that cover a surface. For example, the area of this figure is 4 square units or 8 triangular units.

Reminder

The area of a square or a rectangle can also be calculated with the help of a mathematical formula.

Area of a Square Area of a Rectangle

Multiply the measurement of one side by itself.

Formula: Area = side × side = (side)2

Area = 3 cm × 3 cm

Area = 9 cm2, which reads, “9 square centimetres.”

Multiply the measurement of its length by its width.

Formula: Area = length × width

Area = 4 cm × 2 cm

Area = 8 cm2, which reads, “8 square centimetres.”

3 cm

3 cm

3 cm 3 cm

2 cm

2 cm

4 cm 4 cm

MEASUREMENT Estimating and Measuring Surface Area SECTION 4.8

B-26 Section 4.8


1 Which unit of measure is best for calculating the area of each surface?

a) The floor of a concert hall.

b) The fabric needed to make a scarf.

c) The size of Canada.

d) A computer screen.

e) A schoolyard.

cm2 m2 km2

2 Estimate the area of each object in square centimetres.

a) The cover of your Decimal workbook.

b) A sheet of loose-leaf paper. c) A stamp.

d) The top of your desk. e) The seat of your chair.

3 Estimate the area of each surface in square centimetres. Then, use your ruler to calculate the exact area.

a) b)

Estimate: Estimate:

Area: Area:

WORK IT OUT

Theme 4 B-27


4 The length of a rectangle is 3 cm. Its width is equivalent to double its length. What is the area of the rectangle? Draw the rectangle in the space below.

5 A rectangle is 8 cm wide and 2 cm long. What is the measurement of one side of a square with the same area as the rectangle?

6 Calculate the area of each figure in square decimeters.

a) 8 m

3 m

1 m

7 m

10 m

b) 6 dm

25 cm25 cm

600 mm

10 cm

Aera: Area:

Think about it!

Divide the figure into squares or rectangles.

My Calculation

B-28 Section 4.8


USE REASONING

1 The workmen from the Mirror, Mirror company install mirrors on 2 walls in a dance studio. The 1st wall is 6 m wide and 3 m high. The 2nd wall is 4 m wide and 3 m high. Each mirror is 2 m wide and 1.5 m long, and costs $52.50. How many mirrors are needed to completely cover both walls? What is the total cost of the mirrors?

Solution:

2 A magician uses a table measuring 2 m wide and 0.7 m long. On it, he puts down a box with sides each measuring 4.5 dm. What area of the table is still bare?

Solution:

Theme 4 B-29


MAKING CHOICES

Circle the correct answer(s) to each question below.

Show your calculations.

1 What is the result of this chain of operations?

(48 ÷ 6) × (50 − 30) + 27 = ?

a) 376 b) 187

c) 160 d) 167

2 What is the product of 62.9 × 21.6?

a) 1962.3 b) 13 586.4

c) 225.84 d) 1358.64

3 What is the quotient of 25 256?

a) 1.24 b) 100.24

c) 10.24 d) 10.42

4 Here is the number of theatre tickets bought by 4 separate groups: 7 tickets, 4 tickets, 12 tickets, 5 tickets. What is the average number of tickets purchased?

a) 5.6 tickets b) 7.4 tickets

c) 8 tickets d) 7 tickets

5 If the area of a rectangle is 32 cm2 and it is 4 cm long, how wide is it?

a) 8 cm b) 16 cm

c) 8 cm2 d) 12 cm

My Calculation

My Calculation

My Calculation

My Calculation

B-30 Making Choices


TH

EM

E

ARITHMETIC

1 Calculate these chains of operations by following the order of operations.

a) (25 + 14) − (15 − 7) = b) 120 ÷ (24 + 8 × 2) =

c) 245 − 9 × 20 + 46 = d) 18 − 32 + 7 × 4 =

2 Match each equation to the correct property.

Property

a) 6 × (10 − 4) = (6 × 10) − (6 × 4) = 36 • • Associative property of multiplication

b) (125 × 5) × 3 × 7 = 125 × 5 × (3 × 7) = 13 125 • • Commutative property of multiplication

c) 35 + 18 + 15 = 35 + 15 + 18 = 68 • • Distributive property of multiplication

d) 9 × 7 × 10 = 10 × 9 × 7 = 630 • • Commutative property of addition

3 Calculate the product of each multiplication.

a) b) c) d)

REVIEW SECTIONS 4.1 TO 4.8

4 .8 × 2 . 9

2 7 . 3 × 3 . 6

6 8 . 5 × 2 1 . 3

2 4 5 . 3 × 3 3 . 9

Theme 4 B-31



a) b) c)

d) e) f)

5 Calculate the quotient of each division. Use mental calculation strategies.

a) 69.8 ÷ 10 = b) 2457.3 ÷ 100 =

c) 485 ÷ 1000 = d) 287.3 ÷ 10 =

e) 8478.5 ÷ 100 = f) 2698 ÷ 1000 =

9 9 012

25 3220

30 157 2 9 1 5 3 9

8 2 4 2 9 4 3 5 9 6 0

B-32 Review



a) b) c)

d) e) f)


a) Tickets to a piano concert each cost $49.85. Four people sell a total of 1000 tickets. If they equally share the money from the ticket sales, how much does each person get?

5

3 4 9

671 2 7.8 5 3 2 3.4 5 1 0.36

2 2 5 9.6 1 8 2 5.28 1 1 7 2.25

My Calculation

Theme 4 B-33


b) A costume designer makes 8 identical costumes with 126 m of fabric. How many metres of fabric does she use for each costume?

c) A magician packs his props in 3 boxes. The width of the first box is 2.7 times greater than the width of the second box. The width of the second box is 10 times smaller than the width of the third box. The width of the third box is 3 m. What is the width of the first box?

d) The height of a movie theatre screen is 40 dm and its width is triple its height. What is the area of the screen in square metres?

STATISTICS

8 Look at the data table, then write 2 statements that describe the results.

Grade 5 Students’ Favourite Dance Styles

Hip-hop Jazz Break Dancing Ballet

8 4 5 2

a)

b)

My Calculation

My Calculation

My Calculation

B-34 Review


ARITHMETIC9 Calculate the arithmetic mean of each group of data.

a) 14 19 11 15 21 16

Arithmetic mean:

b) 111 114 100 115

Arithmetic mean:

10 Here are the scores earned by a dance troupe at a regional competition.

8.3

10

9.5

10

9.4

10

8.8

10

9

10

What is the average score?

MEASUREMENT

11 Calculate the area of each figure.

a) 52.4 cm b) 50 cm

Area: Area:

120 cm

40 cm

30 cm

My Calculation

My Calculation

My Calculation

Theme 4 B-35


PROBLEM SOLVING

What I’m Looking ForWhat I Know

A New AuditoriumThe school principal plans to renovate the auditorium. Her budget is between $4000.00 and $4800.00. Here is a list of the work she wants to accomplish, the dimensions and the cost of the materials.

Work Dimensions Cost of Materials

Paint 2 walls. 1st wall: 22 m wide by 3.5 m high.2nd wall: 17 m wide by 3.5 m high.

One 4-litre paint can covering 10 m2: $35.25.

Change the flooring.

Floor: 20.25 m wide by 14 m long.

4 tiles cover 1 m2.1 tile: $2.75.

Make drapes for 3 windows.

3 windows on the 1st wall measuring 1 m wide by 2.5 m high.

1 m of opaque fabric: $15.00.Add 1 m of extra fabric per drape for hemming the top and bottom.

The principal would also like to install a new sound system and better lighting.

What is the total cost of the renovation? What is the maximum amount of money that the principal can put towards a new sound system and lights?

PROBLEM SOLVING

B-36 Problem Solving


My Solution

Validation

Solution:

Theme 4 B-37


Fill in the grid. Write 1 digit per box.

Cross Number Puzzle

A B C D E F

1

2

3

4

5

6

1 Arithmetic mean of the following group: 235, 238, 231, 240. Result of 5 × 42 + (2 × 9) – 7.

2 Result of 2 × (11 – 9). Product of 558.4 and 2.5.

3 ? ÷ 10 = 45.3

4 Measurement of the sides of a square with an area of 81 cm2. 25.18 ÷ ? = 2.518

5 9956 ÷ ? = 99.56 Result of 136 ÷ (2 × 4) × 31.

6 Missing number in a group to get an arithmetic mean of 7. Area of a rectangle measuring 45 cm long by 36 cm wide.

Across Down

A Width of a rectangle with an area of 216 cm2 and a length of 9 cm. 9150 ÷ 10

B Arithmetic mean of the following group: 2.7, 3.3, 2.9, 3.1. Result of 15 ÷ 5 + 9 × 3 – 30.

C 0.61 x 100 Product of 8.08 and 12.5.

D ? ÷ 25 = 13.6 Area of a rectangle measuring 3 m long and 2 m wide.

E Product of 79.6 and 12.5. Result of 4 × (7 + 6).

F ? ÷ 20 = 8.15 269.9 ÷ ? = 26.99

B-38 Game Time


Let’s MoveT

HE

ME

As Time Goes By

It takes 24 hours for Earth to make one complete rotation on its axis. This sets the precise time frame of one full day on Earth. Our planet spins like a top, moving little by little around the Sun and making one complete revolution in 365 days on average. This trajectory determines the length of one full year on Earth.

Negative integers are essential for describing many scenarios. Without them, how could we express sub-zero temperatures, geographical coordinates below sea level or historical events that occurred before our era?

Very Real Numbers5.1 Reading and Writing Integers Locating Integers on a

Number Line Comparing Integers

5.2 Estimating and Measuring Temperature

5.3 Locating Points in a Cartesian Plane

5.4 Observing and Producing Frieze Patterns Using Translations

Tessellations

5.5 Estimating and Measuring Mass Establishing Relationships

Between Units of Measure

5.6 Establishing Relationships Between Units of Measure for Time

5.7 Estimating and Measuring Volume

5.8 Estimating and Measuring Capacity

Establishing Relationships Between Units of Measure

WHAT I’LL LEARN

13325_decimale_5b_th05_an.indd 39 2015-07-09 3:12 PM

LEARN ABOUT IT

ARITHMETIC Reading and Writing Integers Locating Integers on a Number Line Comparing Integers

SECTION 5.1

Reading and Writing IntegersIntegers are whole numbers that are part of a set {…, –3, –2, –1, 0, 1, 2, 3, …} represented by the symbol . This set contains positive integers (greater than 0) and negative integers (less than 0).

–4 –3 –2 –1 0 1 2 3 4

Negative integers Positive integers

Positive and negative integers are used every day to express such things as:

• temperature (–5 °C, 12 °C);

• money (– $10 is a loss, while $10 is a gain).

Locating Integers on a Number LineIntegers can be represented on a horizontal axis or a vertical axis.

• On a horizontal axis, negative integers are to the left of 0 and positive integers are to the right of 0.

−7 −6 −5 −4 2−3 3−2 4−1 50 61 7−8−9−10 8 9 10

Negative integers Positive integers

• On a vertical axis, negative integers are below 0 and positive integers are above 0.

Positive integers

Negative integers

−7−6−5−4

2

−3

3

−2

4

−1

5

8

0

6

9

1

7

10

−10−9−8

The number 0 is both a positive

integer and a negative integer.

B-40 Section 5.1


LEARN ABOUT IT

Calculate the difference between 2 integers by counting the spaces that separate them on a number line.

−7 −6 −5 −4 2−3 3−2 4−1 50 61 7

On this number line, there are 4 spaces between –3 and 1: 3 spaces on the negative integer side and 1 space on the positive integer side. Therefore, there is a difference of 4 between –3 and 1.

WORK IT OUT

1 Arrange these integers correctly on the number line.

a) −3 8 −6 5 −1

0 3−2

b) 12 −18 15 −9 −3

0 6−6

c) 60 −40 −50 40 −10

0 20−30

2 Indicate the value of each letter on the number line.

D C

0

AB

8−12

A : B : C : D :

3 Use the number line from Exercise 2 to calculate the difference between these letters.

a) A and B : b) A and C : c) C and D : d) B and D :

Pay attention to the intervals on a number line.

Think about it!

Theme 5 B-41


Comparing IntegersIntegers can be easily compared on a number line. Numbers are arranged in increasing order from left to right.

As the number line goes further to the left, the numbers are lesser in value.

Look at the numbers in red on the number line.

−7−10 −6−9 −5−8 −4 2−3 3−2 4−1 5 80 6 91 7 10

We see that: −7 < −4 −4 < 0

0 < 3 3 < 8

These numbers are arranged this way in increasing order: −7, −4, 0, 3, 8

As a vertical number line goes down, the numbers are lesser in value.

We see that: −5 < −2 −2 < 0

0 < 3 3 < 4

These numbers are arranged this way in increasing order: −5, −2, 0, 3, 4

−5−4

2

−3

3

−2

4

−1

5

01

WORK IT OUT

1 Compare these numbers using the < or > symbol.

a) 8 −9 b) 15 6 c) −6 −2

d) −11 −12 e) −25 −28 f) −20 25

g) −5 −4 h) 9 −10 i) −1 1

LEARN ABOUT IT

B-42 Section 5.1


2 Arrange these numbers in increasing order.

a) 15 7 −15 9 −45 −3

b) −34 −43 −90 −12 −5

c) −26 76 34 −2 −48

3 Indicate the integer that corresponds to each floor in the building.

0

4 Use the image from Exercise 3 to determine the number of floors that Theo must take.

a) Theo parks his car in the 2nd parking lot, then goes to the meeting room.

b) Theo leaves his office to buy a coffee at the restaurant.

c) After watching a presentation in the auditorium, Theo has lunch at the restaurant, then goes back to his car in the 2nd parking lot.

Restaurant

Auditorium

Offices

Meeting room

Lobby

Theme 5 B-43


USE REASONING

1 Jody has $20. This weekend, she would like to repay her brother the $10 that he lent her, buy a $15 T-shirt, earn $5 babysitting the little girl next door and ask her grandmother for the $15 that she was promised. Then, she would like to give her friend a gift worth $20.

Represent each action with a positive or negative integer. Does Jody have enough money to buy a gift?

Solution:

2 A submarine travels 25 m deep to explore the ocean. It goes up to 12 m below the surface, then back down to 30 m before resurfacing. What is the total vertical distance travelled?

Solution:

Action Integer

Starting amount

Money repaid to brother

T-shirt purchase

Babysitting

Money from grandmother

Gift purchase

−40

−30

−20

−10

40

30

20

10

0

B-44 Section 5.1


LEARN ABOUT IT

MEASUREMENT Measuring and Estimating Temperature SECTION 5.2

Estimating and Measuring Temperature A thermometer is used to measure temperature.

The most common unit of measure is degree Celsius (°C).

A thermometer has a vertical scale.

• The temperature gets hotter as the numbers get closer to the top of the thermometer.

• The temperature gets colder as the numbers get closer to the bottom of the thermometer.

Examples:12 °C is warmer than 5 °C.−7 ºC is colder than −3 ºC.

Just like the numbers on a number line, all numbers below 0 °C represent negative temperatures, such as −15 °C.

All numbers above 0 °C represent positive temperatures, such as 20 °C.

As with integers, you can calculate the difference between 2 temperatures.

Example: If it is −4 °C in the morning and 3 °C in the afternoon, the difference between the 2 temperatures is 7 degrees: 4 degrees between −4 and 0, and 3 degrees between 0 and 3.

Here are some important temperatures:

• Water freezes at a temperature of 0 °C.

• Water boils at a temperature of 100 °C.

• The normal body temperature of a human is 37 °C.

−30

−20

−40

−10

0

10

20

30

40

50

60

°C

0 °C

Glass tube

Positivetemperature

scale

ReservoirDegree

Celsius

Negative temperature

scale

Morning: –4 °C

Difference

Afternoon: 3 °C

4

3+

−20

−30

−10

0

10

20

30

°C

Theme 5 B-45


WORK IT OUT

1 Estimate the temperature indicated on each thermometer.

a)

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

b)

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

c)

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

2 What is missing on the thermometers in Exercise 1 to help you tell the exact temperature?

3 Find the temperature indicated on each thermometer.

a)

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

b)

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

c)

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

−30

−20

−10

0

10

20

30

°C

B-46 Section 5.2


4 Here are the cities that Marjorie is visiting by train. Look at the map, then answer the questions.

Montréal4 °C

Québec1 °C

Gaspé–5 °C

Gatineau3 °C

Mont-Tremblant

–9 °C Sherbrooke6 °C

Val-d’Or–11 °C

Saguenay2 °C

La Tuque–1 °C

Rivière-du-Loup–4 °C

a) Which city is the coldest?

b) Which city is the warmest?

c) What is the difference between the coldest and warmest temperatures?

d) What is the difference between the temperature in Gaspé and the temperature in Québec?

e) What is the average temperature registered in Montréal, Québec, Sherbrooke and Gatineau?

My Calculations

5 This morning, the temperature was 2 °C. It doubled by noon, then fell 5 degrees in the afternoon. In the evening, the temperature fell 6 degrees. What was the temperature by the end of the evening?

−7−10 −6−9 −5−8 −4 2−3 3−2 4−1 5 80 6 91 7 10

Theme 5 B-47


LEARN ABOUT IT

SECTION 5.3 GEOMETRY Locating Points in a Cartesian Plane

Locating Points in a Cartesian Plane

You are familiar with the one-quadrant Cartesian plane containing positive coordinates, like this one. You can easily locate the ordered pair (3, 4).

Reminder

Horizontal axis

Vert

ical

axi

s

1 2 3 4 5

y

x

5

4

3

2

1

0

A 4-quadrant Cartesian plane is made up of 2 perpendicular number lines.

• The horizontal number line is called the x-axis.

• The vertical number line is called the y-axis.

• The point at which these 2 axes meet in the Cartesian plane is called the origin.

• A Cartesian plane is separated into 4 quadrants, which correspond to the 4 regions bordered by the axes.

• Each number line has a scale of negative and positive numbers. The negative numbers are to the left of 0 on the horizontal axis and below 0 on the vertical axis.

X-Axis

Origin

Y-Axis

2nd quadrant(−x, +y)

1st quadrant(+x, +y)

3rd quadrant(−x, −y)

4th quadrant(+x, −y)

0 1 2 3 4 5

y

x

5

4

3

2

1

−1

−2

−3

−4

−5

−5 −4 −3 −2 −1

• A point’s position is indicated by a pair of numbers known as coordinates. These coordinates are shown between brackets and are separated by a comma, such as (3, 4) and (3, −4).

• The 1st number in a pair indicates the position of a point on the horizontal axis (x), while the 2nd number shows its position on the vertical axis (y).

B-48 Section 5.3


WORK IT OUT

1 Find the coordinates of the following points.

0−1

−1 1 2 3 4 5 6 7−2−3−4−5−6−7

−2

−3

−4

−5

−6

−7

A

7

6

5

4

3

2

1

F

B

C

E

D

y

x

2 Cora says that these are the coordinates of the triangle below:

A: (0, 4)

B: (0, −5)

C: (−2, −3)

Is she correct?

0−1

−1 1 2 3 4 5 6−2−3−4−5−6

−2

−3

−4

−5

−6

A

y

x

5

4

3

2

1

6

B

C

A:

B:

C:

D:

E:

F:

Theme 5 B-49


3 Represent the coordinates.

a) Plot the points in the Cartesian plane. Write the letters next to the points.

example A: (1, 2) B: (3, 5) C: (−2, 5) D: (−4, 2)

E: (−3, 1) F: (−3, −5) G: (2, −5) H: (4, −3)

I: (2, −1) J: (2, 1) K: (5, 1) L: (5, 5)

M: (5, 0) N: (7, 0) O: (7, –5) P: (5, –5)

0−1

−1 1 2 3 4 5 6−2−3−4−5−6

−2

−3

−4

−5

−6

A

6

5

4

3

2

1

7 8−7−8

7

8

−7

−8

y

x

b) Connect each set of points in order. Then, write the name of the polygon that they form.

• Points A, B, C and D form

• Points E, F, G, H and I form

• Points J, K and L form

• Points M, N, O and P form

B-50 Section 5.3


4 Plot the following points in the Cartesian plane, then connect them in order.

A: (2, 4)

B: (5, 0)

C: (6, −4)

D: (2, −6)

0−1

−1 1 2 3 4 5 6 7 8−2−3−4−5−6−7−8

−2

−3

−4

−5

−6

−7

−8

8

7

6

5

4

3

2

1

y

x

a) Using the vertical axis as a line of reflection, draw a reflection of the figure.

b) Indicate the new coordinates of this figure.

A : B : C : D :

5 Add a point to the Cartesian plane to form a trapezoid. Draw the figure, then write the coordinates of the point added.

0−1

−1 1 2 3 4 5−2−3−4−5

−2

−3

−4

−5

5

4

3

2

1

y

x

Reflection is a transformation that produces an inverted image in relation to the line of reflection. The figure and its image are equidistant from the line of reflection.

Think about it!

Theme 5 B-51


USE REASONING

Thomas is making deliveries in the neighbourhood. Use the directions below to map out his route.

• Thomas moves only horizontally and vertically.

• He cannot go past the library due to roadwork.

• The pharmacist, the dentist and the grocer are expecting deliveries.

• Thomas cannot take the same route twice.

• Thomas must stop at the gas station, the stationery store and the pet shop, but not at the bakery.

• He must take the shortest route possible.

• Each square is equal to 1.5 km.

Indicate the coordinates corresponding to each of Thomas’ stops. Then, calculate the number of kilometres travelled.

0−1

−1 1 2 3 4 5 6 7 8−2−3−4−5−6−7−8

−2

−3

−4

−5

−6

−7

−8

8

7

6

5

4

3

2

1

Library

Start/Finish(−5, 2)

y

x

Grocery store

Gas station

Bakery

Bank

Stationery store

Pet shop

Dentist

Restaurant Pharmacy

Coordinates of each stop:

Solution:

B-52 Section 5.3


LEARN ABOUT IT

SECTION 5.4 GEOMETRY Observing and Producing Frieze Patterns and

Tessellations Using Translations

Observing and Producing Frieze Patterns Using TranslationsA frieze pattern is a rectangular strip with a repeating pattern.

Use reflection or translation to create a frieze pattern from a starting pattern.

Figure 1

Example of a frieze pattern created by reflecting a starting patternLine of reflection

Figure 2

Example of a frieze pattern created by translating a starting patternTranslation arrow

Translation is a geometric transformation that moves all of a figure’s points in the same direction and along the same distance. The figure keeps its shape, its orientation and its dimensions; it slides.

Translation is represented by a translation arrow. This arrow shows the translation’s direction and the length that it must move.

In Figure 2, the translation arrow indicates that the pattern must move 3 squares to the right.

In Figure 3, the translation arrow shows that the pattern must move 4 squares to the right and 1 square down.

Figure 3

A B

CD

A B

CD

Theme 5 B-53


WORK IT OUT

1 Complete the frieze pattern by following the line of reflection.

2 Complete the frieze patterns by following the translation arrows.

a)

b)

c)

B-54 Section 5.4


LEARN ABOUT IT

TessellationsA tessellation is a collection of geometric figures that cover a surface.

• There is no empty space between the figures.

• The figures never overlap.

Reflection or translation, or a combination of both, can be used to create a tessellation from a starting pattern.

Example of a tessellation

WORK IT OUT

1 Look at these 2 patterns.

Pattern A Pattern B

Which pattern is not a tessellation?

Why?

Theme 5 B-55


2 Create a tessellation by adding 6 more figures.

3 Carry out the translation of the figure below, then create a tessellation by adding 8 more figures.

4 The figure below was created using different translations. Colour in all the triangles in each translation the same colour.

How many different colours are there?

B-56 Section 5.4


LEARN ABOUT IT

MEASUREMENT Estimating and Measuring Mass

Establishing Relationships Between Units of Measure SECTION 5.5

Estimating and Measuring MassMass is the measure of the amount of matter in an object or living thing.

The international system of units (SI) is used to measure mass. Grams (g) and kilograms (kg) are the most common units of measure for mass.

1 kilogram (kg) = 1000 grams (g)

A scale is used for finding the mass of an object.

A piece of candy has a mass of approximately 1 g while a bag of flour has a mass of approximately 1 kg.

Establishing Relationships Between Units of MeasureThe following table compares units of measure for mass.

Unit of measure Kilogram Hectogram Decagram Gram Decigram Centigram Milligram

Symbol kg hg dag g dg cg mg

1st number 4

Equivalency 4 0 0 0

2nd number 2 6 3 4

Equivalency 2 6 3 4

3rd number 0 7 2 5

Equivalency 0 7 2 5

Note: Hectograms, decagrams, decigrams, centigrams and milligrams are not studied at the primary level, but it is useful to know their place in the table of units of measure.

The table demonstrates that:

1) 4 kg = 4000 g 2) 2634 g = 2.634 kg 3) 0.725 kg = 725 g (4 × 1000) (2634 ÷ 1000) (0.725 × 1000)

.

.

Theme 5 B-57


WORK IT OUT

1 Indicate the unit of measure (g or kg) that best describes the mass of each object.

a) Tire b) Windshield wiper blade

c) Train ticket d) Boat

e) Helicopter blade f) Cushion

2 Convert these measurements.

a) 2 kg = g b) 546 g = kg

c) 2.426 kg = g d) 6400 g = kg

e) 8.31 kg = g f) 99 g = kg

3 Compare these numbers using the <, > or = symbol.

a) 5700 g 57 kg b) 8 kg 8000 g

c) 345 g 3.45 kg d) 0.3 kg 299 g

e) 200 g 0.2 kg f) 650 g 6.5 kg

4 Jeffrey’s suitcase weighs 958 g. What is

the mass of clothing that he must remove

from his suitcase for it to weigh 1

2 kg ?

My Calculation

B-58 Section 5.5


USE REASONING

1 A truck is loaded with 46 boxes each weighing 45.5 kg and 32 boxes each weighing 3520 g. What mass must be added to the truck for it to weigh 3000 kg?

Solution:

2 Here is a list of rates that an airline company charges customers for checking in their baggage.

Type of Baggage Rate

Baggage 21 kg or less Free

Baggage 22 kg to 32 kg $65.25

Sporting equipment 15 kg or less $36.95

Sporting equipment 16 kg to 35 kg $58.80

Joannie and her family bring 2 suitcases each weighing 20 100 g, a suitcase weighing 30 220 g, a bag weighing 24 325 g, 2 golf bags each weighing 14 553 g and scuba diving equipment weighing 34 890 g. How much will it cost Joannie and her family to check in their baggage?

Solution:

Theme 5 B-59


LEARN ABOUT IT

MEASUREMENT Establishing Relationships

Between Units of Measure for Time

SECTION 5.6

Establishing Relationships Between Units of Measure for TimeTime is measured in years, months, days (d), hours (h), minutes (min) and seconds (s).

• One year = 12 months. • One day = 24 hours.

• One year = 52 weeks. • One hour = 60 minutes.

• One year = 365 days. • One minute = 60 seconds.

• One leap year = 366 days.

• One month = 30 or 31 days, except February which has 28 days, 29 in a leap year.

Use base 60 to convert units of measure for hours, minutes and seconds.

Examples:

Convert 3 h 35 min to Minutes Convert 123 Minutes to Hours

1. 1 hour = 60 minutes, so 3 hours = 3 × 60 min = 180 min

2. 180 min + 35 min = 215 min

3. So, 3 h 35 min = 215 min

1. 60 minutes = 1 hour, so 60 min × ? ≈ 123 min 60 × 2 = 120

2. 2 h + 3 min = 2 h 3 min

3. So, 123 min = 2 h 3 min

Note: The “≈” symbol means, “is more or less equal to.”

WORK IT OUT

1 How many minutes are there in each length of time?

a) 1 h 54 min =

b) 5 h 16 min =

c) 10 h 48 min =

d) 17 h 25 min =

e) 20 h 32 min =

My Calculations

B-60 Section 5.6


2 Convert these measurements.

a) 168 hours = days.

b) 36 months = years.

c) 300 seconds = minutes.

d) 540 minutes = hours.

e) 1 h 45 min = minutes.

My Calculations

3 Match the equivalent lengths of time.


a) Eva takes the train at 10:45 a.m. and arrives at her destination at 11:30 a.m. How many minutes does her journey last?

1

2

3

4

567

8

9

10

111224

23

22

21

2019 18 17

16

13

15

14

My Calculation

b) Ben’s bus route lasts 58 min in the morning and 1 h 9 min in the evening. In all, how many minutes does he travel?

My Calculation

My Calculationsa) 156 weeks • • 168 months

b) 2160 minutes • • 2880 minutes

c) 14 years • • 3 years

d) 2 days • • 1620 minutes

e) 27 hours • • 36 hours

Theme 5 B-61


USE REASONING

1 Here is a list of the departure and arrival times of some trains, along with the duration of their routes. Use the available data to fill in the table.

Destination Departure Arrival Duration

Ottawa 6:20 p.m. 5 h 35 min

Montréal 7:30 a.m. 11:45 a.m.

Toronto 7:00 p.m. 6 h 20 min

Gaspé 5:30 a.m. 14 h

1

2

3

4

567

8

9

10

111224

23

22

21

2019 18 17

16

13

15

14

2 Paresh is going on a trip to the Bahamas. His plane leaves Montréal at 3:30 p.m. He makes a stopover in Toronto at 4:50 p.m. He takes off at 5:45 p.m. and arrives in the Bahamas at 9:15 p.m. How many minutes does Paresh spend in the air before arriving at his destination?

1

2

3

4

567

8

9

10

111224

23

22

21

2019 18 17

16

13

15

14

Solution:

B-62 Section 5.6


LEARN ABOUT IT

SECTION 5.7 MEASUREMENT Estimating and Measuring Volume

Estimating and Measuring VolumeVolume is the measure of space occupied by a solid. Space has 3 dimensions: length, width and height.

Volume is measured in cubic units. The most common units of measure are cubic centimetres (cm3), cubic decimetres (dm3) and cubic metres (m3).

To calculate the volume of a solid, count the number of cubic units.

Example: The volume of this solid is 10 cubic units.

Reminder

The volume of a prism can be calculated with the following mathematical formula:

Volume = length (l) × width (w) × height (h)

Multiply the measurement of the length by the width and by the height.

Examples:

Volume of a Cube

Volume: 3 cm × 3 cm × 3 cm

Volume: 27 cm3, which reads, “27 cubic centimetres.”

Height

Length

3 cm

Width

Volume of a Rectangular Prism

Volume: 6 cm × 2 cm × 4 cm

Volume: 48 cm3, which reads, “48 cubic centimetres.”

4 cm

6 cm

2 cm

Most Common Units of Measure:

cm3 volume of a wallet

dm3 volume of a travel bag

m3 volume of a train car

The exponent 3 means that the unit

of measure is multiplied by itself

3 times.

Theme 5 B-63


WORK IT OUT

1 What is the most suitable unit of measure to calculate the volume of each object?

cm3 dm3 m3

a) Hot-air balloon basket

b) Toy car

c) Hull of a boat

d) Bicycle seat

e) Bus rearview mirror

2 Find the volume of each structure in cubic units.

a) b)

Volume: Volume:

c) d)

Volume: Volume:

B-64 Section 5.7


3 Calculate the volume of each prism.

a)

5 cm

5 cm

5 cm

b)

27 dm

10 dm5 dm

Volume:

Volume:

c)

9 m

7 m

5 m

d)

10 cm

5.3 cm

2.7 cm

Volume:

Volume:

4 A box containing a remote-controlled

helicopter is 5 dm wide. Its length

is triple its width, while its height

is 1

5 of its width. What is the volume

of the prism?

My Calculation

Theme 5 B-65


USE REASONING

1 Julia and her family have packed 4 suitcases, 3 backpacks and a cooler. They wonder whether to use the family car or Julia’s grandmother’s car. Here are the dimensions of both car trunks.

Trunk Length Width Height

Family car 1.2 m 1 m1

2 the length

Grandmother’s car 1.5 m 1 m 5.6 dm

Which car has the largest storage volume? How many cubic metres greater is the larger car trunk?

Solution:

2 A freight car measures 14 m long by 3 m wide by 4.5 m high. It is transporting boxes measuring 1.5 m long by 1 m wide by 2 m high. What is the maximum number of boxes that the freight car can transport?

Solution:

1.5 m1 m

2 m

B-66 Section 5.7


LEARN ABOUT IT

SECTION 5.8 MEASUREMENT Estimating and Measuring Capacity

Establishing Relationships Between Units of Measure

Estimating and Measuring CapacityCapacity is the volume of matter, often liquid, contained in an object.

The international system of units (SI) is used to measure capacity. Litres (L) and millilitres (ml) are the most common units of measure. These measurements are often used in cooking.

1 L

500 ml

15 ml5 ml

2 ml

1 ml

Establishing Relationships Between Units of MeasureThe following table compares units of measure for capacity.

Unit of measure Kilolitre Hectolitre Decalitre Litre Decilitre Centilitre Millilitre

Symbol kl hl dal L dl cl ml

1st number 1 5

Equivalency 1 5 0 0 0

2nd number 2 2 5

Equivalency 0 2 2 5

3rd number 4 6

Equivalency 4 6 0 0

Note: Kilolitres, hectolitres, decalitres, deciliters and centilitres are not studied at the primary level, but it is useful to know their place in the table of units of measure.

The table demonstrates that:

1) 15 L = 15 000 ml 2) 225 ml = 0.225 L 3) 4.6 L = 4600 ml (15 × 1000) (225 ÷ 1000) (4.6 × 1000)

.

.

1 L = 1000 ml1

2 L = 500 ml

Theme 5 B-67


WORK IT OUT

1 Which unit of measure (L or ml) is best for calculating the capacity of each container?

a) Perfume bottle b) Aquarium

c) Cup of tea d) Swimming pool

2 Convert each capacity into litres.

a) 22 000 ml =

b) 431 ml =

c) 1700 ml =

d) 56 ml =

3 Arrange these capacities in increasing order.

0.255 L 5500 ml 5 L 250 ml 0.5 L

4 Compare these capacities using the <, > or = symbol.

a) 2 L 2200 ml b) 699 ml 0.699 L c) 45 L 45 000 ml

d) 10.2 L 1020 ml e) 1

4 L 400 ml f) 555 ml

1

2 L

5 Complete these equivalencies.

a) 6501 ml = 2 L +

b) 3.35 L = 3035 ml +

c) 8.05 L + 805 ml = 10 000 ml –

My Calculations

My Calculations

B-68 Section 5.8


USE REASONING

1 Mr. Adams needs 24 L of paint to repaint his boat. The hardware store sells 10 L cans for $85.25 each, 20 000 ml cans for $155.95 each and 4 L cans for $45.99 each. Which is the least expensive combination of paint cans?

Solution:

2 At the Clean & Go car wash, it takes 33 L of water and 65 ml of soap to wash one car. The car wash is open 12 hours a day, and an average of 6 cars are washed per hour. How many litres of water and how many litres of soap are needed over a period of 7 days?

Solution:

Theme 5 B-69




1 In which set are the integers arranged in decreasing order?

a) –12, –8, –4, 2, 7, 15 b) 17, 7, 2, –4, 0, –5

c) 20, 10, –15, –10, –5, 0 d) 20, 15, 5, 0, −5, −8

2 At 7:00 a.m., the temperature is −12 °C. By 1:00 p.m., it warms up to 2 °C. What is the temperature difference between these 2 times of day?

a) 14 ºC b) 10 ºC

c) 12 ºC d) 8 ºC

My Calculation

3 Which figure represents a translation of the first helicopter?

A B C D

4 How many grams are there in 3.890 kg?

a) 389 g b) 38.9 g c) 3890 g d) 3908 g

5 How many minutes are there in 4 hours and 44 minutes?

a) 284 minutes b) 444 minutes c) 244 minutes d) 280 minutes

6 What is the volume of a suitcase that is 50 cm long by 2.5 dm wide by 20 cm high?

a) 2500 cm3 b) 250 cm3

c) 250 dm3 d) 25 000 cm3

My Calculation

MAKING CHOICES

B-70 Making Choices


TH

EM

E


ARITHMETIC

1 Write the missing integers on each number line.

a) 6–4 0

b) –15 205

2 Add 3 integers to each series while following the pattern.

a) 10, 6, 2, , ,

b) –14, –17, –20, , ,

c) 40, 30, 20, , ,

3 Compare these numbers using the < or > symbol.

a) 4 –3 b) 17 –17 c) –6 –1

d) –36 6 e) –1 0 f) –125 –152

g) –10 1 h) 72 27 i) –15 –5

4 Noah dives into a swimming pool from a diving board that is 5 metres high. He goes 3 metres underwater before swimming up to the surface. What is his total vertical distance travelled?

My Calculation

Theme 5 B-71


5 Max parks his car in an underground parking garage. He rides the elevator to the 13th floor. The elevator shows that he has gone up 15 floors. On what floor is Max’s car parked?

My Calculation

MEASUREMENT

6 Fill in the table by calculating the temperature differences between noon and midnight.

Day Temperature at Noon (°C)

Temperature at Midnight (°C)

Temperature Difference (°C)

Friday −2 −24

Saturday 4 −15

Sunday 0 −9

a) Which night is the warmest?

b) Which afternoon is the coldest?

7 The thermometer inside Margot’s house shows that it is 20 °C. The thermometer outside her house shows that it is −24 °C. What is the temperature difference between these 2 areas?

8 Transform each mass into grams.

a) 2.255 kg = g b) 20.5 kg = g

c) 25.105 kg = g d) 2.51 kg = g

e) 21 kg = g f) 5.02 kg = g

−30

−20

−10

0

10

20

30

°C

B-72 Review


9 A delivery truck is transporting 76.6 kg of sand. It delivers 67 670 g to a client. How many kilograms of sand are left?

My Calculation

10 Convert these measures of capacity.

a) 2534 ml = L b) 1.69 L = ml

c) 89 ml = L d) 0.78 L = ml

11 How many millilitres are there:

a) in a 3

4 litre bottle?

b) in a 1

2 litre glass?

c) in a 1

4 litre cup?

12 Matt’s bottle has 1.5 L of water in it. He wants to pour the entire contents of his bottle into small 50 ml cups. How many cups can Matt fill?

My Calculation

13 Arrange these measures of time in increasing order.

a) 366 seconds 6 minutes 1200 minutes 65 seconds 2 hours

b) 14 months 4 years 60 months 200 weeks 12 weeks

Theme 5 B-73


14 Ranya’s airplane ride lasts two and a half hours each way. What is the total length of her trip in minutes?

My Calculation

15 Noah flies a helicopter 1 h 30 min per day from Monday to Thursday. On Friday, he flies for 2 h 45 min, then on Saturday for 3 h 35 min. How much time does Noah spend flying the helicopter in one week?

My Calculation


a)

6 dm6 dm

6 dm

b)

3 dm5 cm

2 dm

Volume:

Volume:

17 The volume of a box containing a miniature airplane is 90 dm3. It is 2 dm high and 5 dm wide. What is the length of the box?

My Calculation

B-74 Review


GEOMETRY

18 Zack and Emily are playing a game of Battleship. Look at the position of Zack’s ships in the Cartesian plane.

70−1

−1 1 2 3 4 5 6−2−3−4−5−6

−2

−3

−4

−5

−6

6

5

4

3

2

1

7

8−7−8

y

x

−7

−8

E F

G

Ship 2

Ship 1

H

8

CD

A B

b) Draw Emily’s 2 ships in the Cartesian plane by placing a point at each coordinate. Then, connect the dots in order.

19 Complete the frieze pattern by translating the pattern with the help of the translation arrow.

a) Give the coordinates of Zack’s 2 ships.

Zack’s Ship 1

A:

B:

C:

D:

Zack’s Ship 2

E:

F:

G:

H:

Emily’s Ship 1 Emily’s Ship 2

I: (4, –1)

J: (6, –1)

K: (7, –3)

L: (3, –3)

M: (–2, 1)

N: (–2, 3)

O: (–6, 3)

Theme 5 B-75


What I Know What I’m Looking For

PROBLEM SOLVING

A Long Voyage“Time difference” is literally the difference in local time between 2 locations. For example, the time difference between Paris and Montréal is 6 hours. Therefore, if it is 9:00 a.m. in Montréal, it is 3:00 p.m. in Paris.

Jada and her family are going to Paris to visit her grandparents. They board the plane at 8:20 a.m. The airplane ride lasts 7 hours and 30 minutes.

When they arrive in Paris, they have a snack that lasts 1

2 an hour.

Then, they take a 40-minute ride on a high-speed train.

At the train station, they get in her grandparents’ car.

The car ride lasts 3

4 of an hour.

How much time, in hours and minutes, do Jada and her family spend travelling on different modes of transportation?

At what time (local Paris time) do they arrive at her grandparents’ house?

PROBLEM SOLVING



My Solution

Validation

Solution:

Theme 5 B-77


Hot-Air Balloon Ride

Colour in the hot-air balloon patches that contain the measurements equivalent to the measurements below. Use the colour code.

Measures of Time

measures = to 4 hours

measures = to 4 days

measures = to 420 minutes

Measures of Mass

measures = to 1

4 kg

measures = to 1

2 kg

measures = to 480 g

Measures of Capacity

measures = to 3

4 L

measures = to 1

2 L

measures = to 3650 ml

2500 g

500 g0.5 kg

0.48 kg345 600 s

750 ml

4.8 kg

0.480 kg3.650 L

0.50 L500 ml25 200 s 3.65 L

0.75 L36.5 L

7 h

17 h

5 kg

90 h

7.5 L

14 400 s 250 g

240 min 0.25 kg 96 h 5.5 L

B-78 Game Time


Adventures in NatureT

he

me

Platonic Solids

Tetrahedrons, cubes, octahedrons and icosahedrons are also known as “Platonic solids,” named after the ancient philosopher Plato. Greeks associated these solids with elements found in nature, such as fire, earth, air and water.

A soccer ball is made up of 20 white hexagons and 12 black pentagons. The polyhedron formed by these figures is called a “truncated icosahedron” or “Archimedean solid,” named after the famous Greek mathematician Archimedes. Euler’s theorem can be applied to a soccer ball: 32 faces + 60 vertices – 2 = 90 edges.

6.1 Enumerating Possible Outcomes of a Random Experiment Using a Table and a Tree Diagram

6.2 Comparing the Outcomes of a Random Experiment with Known Theoretical Probabilities

6.3 Describing and Classifying Prisms and Pyramids Using Faces, Vertices and Edges

6.4 Nets of Solids or Convex Polyhedrons

6.5 Testing Euler’s Theorem on Convex Polyhedrons

WhAT I’LL LeARN

An Archimedean Ball

13326_decimale_5b_th06_an.indd 79 14-02-25 16:24

LEARN ABOUT IT

Probability Enumerating Possible Outcomes of a Random

Experiment Using a Table and a Tree Diagram SecTIoN 6.1

Enumerating Possible Outcomes of a Random Experiment Using a Table and a Tree DiagramThe outcome of a random experiment is based on chance. It is therefore impossible to know the outcome in advance. For instance, it is impossible to know:

• on which number a dice will land;

• which card will be randomly drawn from a deck of cards;

• whether a coin will land on “heads” or “tails.”

A tree diagram or table can be used to represent all the possible outcomes of a random experiment.

tree Diagram representation

Here are all the possible outcomes of drawing 2 leaves from a bag containing 3.

All PossibleOutcomes

2nd Draw1st Draw

If you take into account the order in which the leaves are drawn, there are 6 possible combinations.

There are also 3 possible pairs of colours: green and red, green and yellow, and red and yellow.

B-80 Section 6.1


LEARN ABOUT IT

table representation

1st Draw 2nd Draw All Possible Outcomes

There are 6 possible outcomes.

Here are 2 probabilities derived from the results.

• The probability of drawing a green leaf and a red leaf is 2

6 or

1

3.

• The probability of drawing a yellow leaf is 4

6 or

2

3.

WoRK IT oUT

1 Nathan is going to camp. Here are the contents of his suitcase: 3 pairs of pants (one grey, one tan and one black), 4 shirts (one khaki, one orange, one red and one yellow) and 2 pairs of socks (one navy and one white).

a) Nathan randomly chooses his clothing. Indicate all the possible combinations.

Pants Shirts Socks Possible Combinations

b) What is the number of possible combinations?

Theme 6 B-81


2 The camp’s adventure trail has 3 starting points. Nathan can choose from different routes.

a) Use a tree diagram to indicate all the possible routes beginning with each starting arrow.

3 C

2 b

1 a D

E

F

Start 1st Choice 2nd Choice Possible routes

b) What is the number of possible routes?

1st ChoiceStart 2nd Choice

B-82 Section 6.1


LEARN ABOUT IT

SecTIoN 6.2 Probability Comparing the Outcomes of a Random

Experiment with Known Theoretical Probabilities

Comparing the Outcomes of a Random Experiment with Known Theoretical Probabilities

A random experiment is based on chance. It is impossible to know

the outcome in advance. However, it is possible to determine

the probability that an event will happen.

theoretical probability relies on mathematics.

For example, it can be determined that the probability of randomly

drawing a red marble from this bag is 3

6 or

1

2, therefore 50%.

In theory, if you were to repeat this experiment 10 times,

you should draw the red marble 5 times since 1

2 is equal to

5

10.

In reality, this is not always the case.

Look at the results obtained by 3 students who conducted

the marble experiment.

Draws

Students 1 2 3 4 5 6 7 8 9 10

Adele

Leo

Mila

Outcome After 10 Draws

Outcome Compared to the Theoretical Probability of

1

2

Adele drew 6 red marbles out of 10 or 6

10. Greater than the theoretical probability.

Leo drew 5 red marbles out of 10 or 5

10. Equal to the theoretical probability.

Mila drew 2 red marbles out of 10 or 2

10. Less than the theoretical probability.

As an experiment is repeated more often, the results get closer to the theoretical probability. For this reason, several people’s results can be tallied.

Theme 6 B-83


WoRK IT oUT

1 Determine the probability that the following results will occur.

a) Getting an even number with a throw of one dice.

b) Getting a 5 or a 6 with a throw of one dice.

2 Determine the probability that the following results will occur after 30 throws.

a) Getting an even number with a throw of one dice.

b) Getting a 5 or a 6 with a throw of one dice.

3 Check the probabilities indicated in Exercises 1 and 2. Roll a dice 30 times, then tally your results in the table by placing a checkmark in the correct column each time.

Getting an Even Number Getting a 5 or a 6 Other Results (1 or 3)

Result: Result: Result:

4 Once you have completed 30 throws, compare your results from Exercise 3 to the theoretical probability determined in Exercises 1 and 2.

Less than the Theoretical Probability

Greater than the Theoretical Probability

Equal to the Theoretical Probability

Getting an Even Number with a Throw of One Dice

Getting a 5 or 6 with a Throw of One Dice

Think about it!

If you roll a 6, place a checkmark in 2 columns since it is an even number too.

B-84 Section 6.2


5 Determine the probability of randomly drawing the following cards from a standard deck of 52 playing cards.

One Draw 60 Draws

a) A diamond.

b) A black card.

6 Draw a card at random from a standard deck of 52 playing cards. Write a checkmark if it is a black card, then put it back in the deck. Repeat the experiment 15 times.

Drawing a black Card

a) Write your outcome out of 15 in the table. Add 3 other students’ outcomes.

My outcome Student 1 Student 2 Student 3 total

/15 /15 /15 /15 /60

b) Is your total outcome less than, greater than or equal to the theoretical probability of drawing a black card determined in Exercise 5 b)? Explain your answer.

7 In this spinning wheel game, players win 2 points if the arrow stops on a yellow or green section. Jessica gets a total of 90 points after spinning the wheel 100 times. Is her outcome less than, greater than or equal to the theoretical probability that the arrow will stop on yellow or green? Explain your answer.

Theme 6 B-85


LEARN ABOUT IT

GEoMEtriC Describing and Classifying Prisms and Pyramids SecTIoN 6.3

Describing and Classifying Prisms and Pyramids Using Faces, Vertices and Edges

A solid is a 3-dimensional geometric figure formed by at least one closed surface. There are 2 categories of solids: curved bodies and polyhedrons.

Reminder

Solids

Prisms have 2 congruent and parallel faces as bases. The remaining faces are rectangles. Generally, prisms are named according to the shape of their bases.

Pyramids have only one face as a base. The remaining faces are triangles that meet at a single vertex.

Polyhedrons are characterized by the number of faces, vertices and edges.

An edge is the segment where 2 faces meet.

A vertex is the intersecting point of at least 2 edges.

Polyhedrons

These solids are formed by polygons (plane surfaces).

PrismsThese have 2 congruent and parallel polygons as bases.

Triangular Prism

Pentagonal Prism

Rectangular Prism

Octagonal Prism

PyramidsThese have a single polygon as a base.

Rectangular Pyramid

Hexagonal Pyramid

Triangular Pyramid

Pentagonal Pyramid

Curved bodies

These solids have at least one curved surface.

Cone Cylinder Sphere

VertexBase

Edge

Face

B-86 Section 6.3


WoRK IT oUT

1 Look at the polyhedrons, then fill in the table.

Name of Polyhedron

Number of Faces

Number of Vertices

Number of Edges

a)

b)

c)

d)

e)

f)

Theme 6 B-87


2 Sam builds a castle tower with 2 polyhedrons. What are they?

3 Which polyhedron am I?

a) I have 6 identical square faces.

b) I have 4 triangles and a square base.

c) I have 7 faces: 2 pentagons and 5 rectangles.

d) I only have 4 triangles.

e) I have 6 faces, 12 edges and 8 vertices. All of my faces are rectangles.

4 Draw each face of a hexagonal pyramid. Use a ruler.

5 This is a model of one of the cabins at summer camp. How many faces, edges and vertices are there in all?

Number of faces:

Number of edges:

Number of vertices:

B-88 Section 6.3


LEARN ABOUT IT

GEoMEtry Nets of Solids or Convex Polyhedrons SecTIoN 6.4

Nets of Solids or Convex Polyhedrons A convex polyhedron contains all the line segments connecting any two vertices (points) within it. Every face of a convex polyhedron can be laid on a flat surface.

A nonconvex, or concave, polyhedron has at least one line segment connecting two vertices (points) outside it.

The net of a polyhedron is the 2-dimensional plane figure obtained when its surfaces are laid flat as if it were unfolded.

Polyhedron Net

Theme 6 B-89


WoRK IT oUT

1 Circle the convex polyhedrons and mark an X on the nonconvex polyhedrons.

2 Lucas has to put together these 4 boxes during a treasure hunt at summer camp. What polyhedrons do they form?

a) b)

c) d)

B-90 Section 6.4


3 Complete the net of each solid. Use a ruler.

a)

b)

c)

d)

Theme 6 B-91


LEARN ABOUT IT

GEoMEtry Testing Euler’s Theorem on Convex Polyhedrons SecTIoN 6.5

Testing Euler’s Theorem on Convex Polyhedrons In the 18th century, the great mathematician Leonhard Euler discovered a formula for easily calculating the relationship between the number of faces (F), vertices (V) and edges (E) of a convex polyhedron. This formula is called “Euler’s theorem.”

Euler’s theorem: F + V – 2 = E

• Add the number of faces and vertices.

• Subtract 2 from this number.

• The result is the number of edges.

Example:

Polyhedron Euler’s Theorem: F + V – 2 Number of Edges

Square prism

6 faces + 8 vertices = 14

14 – 2 = 12 12

Pentagonal prism


12 – 2 = 1010

Triangular prism


11 – 2 = 99

B-92 Section 6.5


WoRK IT oUT

1 Use Euler’s theorem to help you determine the number of edges for each polyhedron.

Polyhedron Euler’s Theorem Number of Edges

a)

b)

c)

d)

2 Does Euler’s theorem apply to these polyhedrons? Explain your answer.

a) b) c)

Theme 6 B-93


3 Calculate the number of faces and vertices for each pyramid. Then, use Euler’s theorem to help you determine the number of edges.

Pyramid Number of Faces

Number of Vertices Number of Edges

a)

b)

c)

d)

4 What do you notice about the number of faces and vertices for the pyramids in Exercise 3?

5 Calculate the number of edges for the following polyhedrons.

a) A pyramid with 10 faces. b) A pyramid with 12 faces.

B-94 Section 6.5


USe ReASoNINg

At summer camp, 2 teams of campers are spending the night in the forest. Each team must use the following materials to build a shelter in the shape of a convex polyhedron: branches at each edge to support the structure, 2 m2 of canvas to cover each face of the structure and 1 metre of rope per vertex to stabilize the shelter.

• Lorenzo’s team makes a shelter in the shape of a square prism.

• Marika’s team makes a shelter in the shape of a pyramid with 15 faces.

How many branches, metres of rope and square metres of canvas does each team need?

lorenzo’s team Marika’s team

(square prism) (pyramid with 15 faces)

Solution:

Theme 6 B-95




mAKINg choIceS

1 Mary Ann is making sandwiches. She has 2 types of bread, 3 types of meat and 3 types of condiments. How many different sandwiches can she make if each sandwich includes one of each type?

a) 16 b) 27

c) 18 d) 36

My Calculation

2 What is the theoretical probability of randomly drawing a red face card from a standard deck of 52 playing cards?

a) 2

13 b)

3

26

c) 7

52 d)

3

13

My Calculation

3 What plane figures does Aaron need to build a hexagonal pyramid?

a) 5 triangles and 1 hexagon b) 6 triangles and 2 hexagons

c) 7 triangles and 1 hexagon d) 6 triangles and 1 hexagon

4 Which polyhedron can be formed from this net?

a) Rectangular pyramid

b) Triangular prism

c) Rectangular prism

d) Triangular pyramid

5 Which mathematical formula, known as Euler’s theorem, helps to calculate the number of edges in a convex polyhedron?

a) Number of faces + Number of vertices – 2 = Number of edges

b) Number of vertices + 2 – Number of faces = Number of edges

c) Number of faces × 2 – Number of vertices = Number of edges

d) Number of vertices + Number of faces + 2 = Number of edges

B-96 Making Choices


Th

Em

E


PRobAbILITy

1 At sports camp, the children invent a game with the objects they find in 2 boxes.

• Contents of box 1: a hoop, a basketball, a rope and a bowling pin.

• Contents of box 2: a scarf, a cone and a tennis ball.

a) Use this table to help you count all the possible combinations of randomly picking 1 object from each box.

1st Object Picked 2nd Object Picked Combination of Possible Objects

b) How many possible combinations of objects can the children make?

c) How many combinations include a bowling pin?

d) How many combinations include a cone?

e) What are the chances of getting a combination that includes a rope?

f) What are the chances of getting a combination that does not include a basketball?

Theme 6 B-97


2 Naomi and her group of campers randomly choose 2 colours for their flag. The background can be red, orange or white, while the rhombus can be blue, green, purple or yellow.

a) Use the tree diagram to help you count the number of possible colour combinations.

background Colour

rhombus Colour Possible Colour Combinations

b) How many possible colour combinations are there?

c) What is the probability that the flag will have a purple rhombus?

d) What is the probability that the flag will have an orange background?

3 Julian is drawing a card from a standard deck of 52 playing cards. What is the theoretical probability that he will draw one of the following cards?

a) A Queen. b) A club.

c) A face card. d) A black card after repeating the experiment 20 times.

B-98 Review


4 The campers are playing the Animal World game. They spin 2 wheels that are each divided into 3 equal sections. On each section is a different animal: an owl, a squirrel and a wolf.

a) Use this table to help you count all the possible animal combinations.

1st Wheel 2nd Wheel Possible Combinations

b) How many possible animal combinations are there?

c) What is the theoretical probability of getting 2 identical animals?

d) What is the theoretical probability of getting an owl and a squirrel, regardless of the order in which they are picked?

5 Kai spins the 2 wheels from Exercise 4 a total of 27 times. Here are his results.

O, S S, W O, O O, W S, O O, W W, O S, O W, O Legend

O, O W, S O, S W, W W, O S, S O, O W, W O, O O: OwlS: SquirrelW: WolfW, S O, S W, S W, O W, W S, O O, S W, O S, S

Indicate whether each outcome is less than, greater than or equal to the theoretical probability.

a) Getting 2 identical animals.

b) Getting an owl and a squirrel, regardless of the order in which they are picked.

Theme 6 B-99


geomeTRy

6 Name each polyhedron. Draw all of its faces. Then, count the number of edges and vertices.

Name of Polyhedron Faces Number

of EdgesNumber

of Vertices

a)

b)

c)

7 Circle the nonconvex polyhedrons.

a)

b)

c)

d)

e)

B-100 Review


8 Circle the correct net of each polyhedron.

a)

b)

c)

9 Jasmine is building a tower with an octagonal prism and an octagonal pyramid. She has 27 rods and 27 magnetic marbles to build it. How many rods is she missing? Use Euler’s theorem.

a) Octagonal prism b) Octagonal pyramid

Number of vertices: Number of vertices:

Number of faces: Number of faces:

Euler’s theorem: Euler’s theorem:

Rods needed: Rods needed:

Missing rods:

Theme 6 B-101


PRobLem SoLvINg

What I’m Looking ForWhat I Know

Unique Tents At Camp Spark, Justin is helping to create a diorama of a campsite. He is asked to build 3 different tents shaped like convex polyhedrons. He must use as many of these polygons as possible:

• 11 • 2 • 4

For each of the 3 tents:

• draw the net of the polyhedron that matches its shape;

• write the name of the polyhedron;

• indicate the number of faces, vertices and edges.

PRobLem SoLvINg



My Solution

Validation

tent 1

Net:

Name: Faces: Vertices: Edges:

tent 2

Net:


tent 3

Net:


Theme 6 B-103


A Mobile Made of Solids

Use the hints to help you write the names of these 7 solids in the correct polygons.

Star Pyramid

Hexagonal Prism

Hexagonal Pyramid

Cylinder Star Dodecahedron

Octagonal Prism

Pentagonal Pyramid

Hints

• There is a curved body in the square.

• Each rectangle contains a nonconvex polyhedron.

• The convex polyhedron in the pentagon has fewer vertices than the convex polyhedron in the trapezoid.

• The convex polyhedron in the hexagon has more edges than the convex polyhedron in the rhombus.

B-104 Game Time


Final Review

Theme

Theme

Theme

Theme

Theme

Theme

13229_decimale_5b_grande-revision_an.indd 105 14-02-25 16:07

FINAL revIew

ArIthmetIc

1 Look at these numbers and answer the questions.

261 238 26 382 454 699 549 563 49 568

a) In which number is the digit 6 worth 6000?

b) Which number has 495 hundreds?

c) Subtract 25 thousands from the smallest number. What number do you get?

d) Arrange these numbers in increasing order.

2 Decompose these numbers in 2 different ways.

a) 232 982

b) 380 709

3 Write the number that corresponds to each equivalent form.

a) 256 T + 14 U + 130 H =

b) 89 Th + 18 T + 4 TTh + 12 H + 12 U =

c) 50 T + 45 TTh + 7 H =

My Calculations

B-106 Final Review


4 Complete the number lines.

a)

25 682 25 782

b)

200 6200

c)

132 420 133 190

d)

245 555 247 555

5 Mark an X on the numbers that become 270 000 when rounded to the nearest ten thousand.

264 448 265 803

269 418

270 556

280 496

275 589

266 628


a) b) c)

2 6 0 × 4 9

1 2 8 × 3 5

5 4 3 × 2 6

Final Review B-107



a) The Olympic organizing committee orders 102 boxes each containing 12 medals. How many medals are there in all?

My Calculation

b) At a soccer tournament, the teams have access to 3 boxes of soccer balls. Each box has 10 bags of 10 balls. How many balls are there in all?

My Calculation

c) Seven groups of 25 people bought tickets to a hockey game. The arena can accommodate 305 people. How many groups of 26 people can also buy tickets to the game?

My Calculation

8 True or false?

True False

a) The cube of 5 is 25.

b) 2 to the power of 5 is 10.

c) In the expression 83 = 512, 3 is the exponent.

d) The square of 7 is 343.

My Calculations

B-108 Final Review


9 Look at these numbers.

125 890 41 505348 633284 380 758 040

Which numbers are not divisible:

a) by 2? b) by 3?

c) by 5? d) by 10?

10 Decompose each number into prime factors. Use exponential notation to express your answer.

a)

40

b)

56

40 = 56 =


a) b) c)

4 8 1 2 2 9 9 2 4 0 013 19 25

Final Review B-109


12 In a group of 12 students, 6 are playing tennis, 4 are playing badminton and 2 are playing ping-pong. Answer each question with 2 equivalent fractions.

a) What fraction of the group is playing badminton?

b) What fraction of the group is playing ping-pong?

13 Circle the fraction in each set that is reduced to its simplest form.

a)

7

49

8

15

3

39

2

52

8

32

b)

3

36

6

18

2

25

4

48

12

15

c)

11

48

9

36

11

99

9

48

3

54

d)

7

21

6

15

7

56

6

57

5

41

14 Arrange the fractions in increasing order.

a)

5

8

5

13

5

18

5

3

5

21

b)

1

2

2

3

1

6

5

12

5

6

c)

7

20

3

10

7

5

1

4

2

5

My Calculations

15 Calculate the result of each operation. Reduce the result to its simplest form when possible.

a) 2

3 +

1

6 = b)

4

21 +

6

7 =

c) 11

18 –

2

9 = d)

1

4 –

1

16 =

e) 5 × 1

8 = f) 3 ×

6

9 =

B-110 Final Review


16 Place the numbers on the number line.

a) 42.19 42.26 42.48 42.33 42.08

42.0 42.1 42.2 42.3 42.4 42.5

b) 7.269 7.282 7.299 7.258 7.290

7.25 7.26 7.27 7.28 7.29 7.30

17 Write the number that corresponds to each equivalent form.

a) (5 × 1

10) + (8 × 102) + (6 × 100) + (9 × 1

100) + (2 × 1

1000) =

b) (2 × 1000) + (2 × 10) + (7 × 0.1) + (3 × 0.001) + (6 × 100) =

My Calculations

18 Use these expressions to complete the equivalencies.

78 tenths 0.075 75.01 0.078 7.81

a) 781 hundredths = b) 7.8 =

c) 75 thousandths = d) 75 + 10

1000 =

19 Find four 6-digit decimals that become 436.7 when rounded to the nearest tenth.

Final Review B-111



a) b) c)

21 Write the missing multiple of 10 (× 10, × 100 or × 1000).

a) 27.85 × = 2785 b) 89.70 × = 897

c) 5.061 × = 5061 d) 751.51 × = 7515.1

22 Rewrite each fraction as a decimal and as a percentage.

Fraction Decimal Percentage

a) 3

10

b) 3

25

c) 3

20

d) 3

4

23 Calculate these chains of operations by following the order of operations.

a) 8 + 10 × 3 + 17 – 9 = b) 5 × 12 ÷ 6 + 82 =

231.89× 7

487.05× 23

652.94× 35

B-112 Final Review


24 Use the distributive property to do these operations.

a) 5 × (4 + 8) =

b) 12 × (10 – 4) =


a) b) c)


a) b) c)

45.6× 7.3

32.6× 12.8

96.2× 33.7

2 8 6 4 2 1 2 4 1 5 0 320 16 15

Final Review B-113



a) Cayden would like to buy headphones for $29.65. If he has $9.25 saved up and gets $15.75 for his birthday, how much money does he need to buy the headphones?

My Calculation

b) One group of 23 students and another group of 26 students, along with 4 adult chaperones, are going to the Museum of Technology. Tickets to the museum cost $5.35 per child and $9.85 per adult. How much will the museum visit cost in all?

My Calculation

c) Damian is updating software on

3 computers. After 15 minutes, the

1st computer has completed 35% of its

updates, the 2nd has completed 2

5 and

the 3rd has completed 45 hundredths.

Which computer has completed

the greatest portion of its updates?

My Calculation


a) b) c)

6 7 41 6 9.5 2 4 6.4 1 4 0.88

B-114 Final Review


29 Calculate the quotient of each division. Use mental calculation strategies.

a) 1047 ÷ 1000 = b) 72.1 ÷ 100 =

c) 456.2 ÷ 100 = d) 599.02 ÷ 10 =

30 Write the missing integers on each number line.

a) −6 0 9

b) −14 350

31 Arrange the integers in decreasing order.

a) –12 14 –6 18 6

b) 25 0 –15 –10 –1 5

c) –42 24 –4 –12 2 21

Geometry

32 True or false?

a) An isosceles triangle has 3 congruent sides.

b) A right triangle can have an obtuse angle.

c) An equilateral triangle has 3 congruent angles.

d) A scalene triangle has 3 sides of different length.

e) An isosceles triangle has 2 angles of the same measure.

True False

Final Review B-115


33 Look at the circle, then answer the questions.

a) What is segment AC called?

b) What is the measurement of segment AC?

c) What is segment BD called?

d) What is the measurement of segment BD?

e) What is the measurement of the central angle ACB?

34 Plot these points in the Cartesian plane. Then, connect them in order.

A: (–3, 5) B: (−1, 5)

C: (0, 2) D: (1, 5)

E: (3, 5) F: (3, −4)

G: (1, −4) H: (1, 1)

I: (0, 0) J: (–1, 1)

K: (–1, –4) L: (–3, –4)

What figure do you get?

0−1

−1 1 2 3 4 5 6 7 8−2−3−4−5−6−7−8

−2

−3

−4

−5

−6

−7

−8

8

7

6

5

4

3

2

1

y

x

B

D

A

C

B-116 Final Review


35 Carry out the translation of this pattern with the help of the arrow.

B

C

D

A

H

G

EF

36 Write the name of each polyhedron. Then, indicate the number of faces and vertices. Use Euler’s theorem to calculate the number of edges.

Name of Polyhedron Number of Faces

Number of Vertices Number of Edges

a)

b)

meAsuremeNt

37 Indicate the type of angle. Then, measure it with your protractor.

a)

Type:

Measurement:

b)

Type:

Measurement:

c)

Type:

Measurement:

Final Review B-117


38 Compare these units of measure for length using the <, > or = symbol.

a) 1.5 m 251 dm b) 35 cm 3.5 dm

c) 245 mm 2.45 m d) 186 cm 1.68 m

e) 5.3 km 5030 m f) 4.805 km 4805 m

39 Calculate the area of each figure.

a)

2.5 cm

b)

155 cm

33 dm

Area: Area:

40 Calculate the temperature differences between morning and noon.

Morning Noon Difference

–25 oC –12 oC

0 oC –6 oC

41 Convert these units of measure.

a) 2000 g = kg b) 1010 g = kg

c) 0.21 kg = g d) 11 kg = g

42 Complete these equivalencies.

a) 8963 ml = 4 L +

b) 4.85 L = 4005 ml +

c) 26 L = 2600 ml +

My Calculations

B-118 Final Review



a) Ludovic does judo 40 minutes each day from Monday to Thursday. How much time does he train each week? Express your answer in hours and minutes.

My Calculation

b) Samantha can make a necklace in 18 minutes. How many necklaces can she make in 1 hour and 30 minutes?

My Calculation

c) Amanda leaves school at 3:50 p.m. and arrives at her house at 4:35 p.m. How much time does it take for her to get home?

My Calculation


a)

6 cm4 cm

2 cm

b)

4.5 m1 m

25 dm

Volume:

Volume:

Final Review B-119


stAtIstIcs

45 Look at the circle graph, then answer the questions.

Favourite Types of NovelsAmong Grade 5 Students

Fantasy novels

Adventure novels

Detective novels

Science �ction novels 40%

30%

12%

Favourite Types of NovelsAmong Grade 5 Students

Fantasy novels

Adventure novels

Detective novels

Science �ction novels 40%

30%

12%

a) What fraction of students prefer fantasy novels?

b) True or false? More than one quarter of students prefer adventure novels.

c) What percentage of students prefer science fiction novels?

46 Calculate the arithmetic mean of each group of data.

a) 14 16 25 18 12 14

Arithmetic mean:

b) 125 132 141 123

Arithmetic mean:

My Calculations

ProbAbILIty

47 Lucy puts 5 pencils, 7 stickers, 9 rubber balls and 4 key chains in a draw box. Indicate the probability of winning each prize as a fraction over 100, a decimal and a percentage.

Fraction Over 100 Decimal Percentage

a) Pencil

b) Sticker

c) Rubber ball

B-120 Final Review


GLOSSARY

A

Area (p. 26)Measurement of a figure’s surface.

Arithmetic mean (p. 19)Sum of a set of numbers divided by the total number of parts in the set.

Associative property (p. 5)Property applied to addition and multiplication that groups the numbers in an equation in different ways without changing the result.

C

Capacity (p. 67)Volume of matter, often liquid, contained in an object.

Cartesian plane (p. 48)Plane formed of 2 perpendicular lines: the horizontal axis (x) and the vertical axis (y).

Commutative property (p. 5)Property applied to addition and multiplication that moves around the numbers in an equation in different ways without changing the result.

Convex polyhedron (p. 89)Polyhedron containing all the line segments connecting any two vertices (points) within it.

Coordinates (x, y) (p. 48)Pair of numbers that indicate the position of a point in a Cartesian plane. The 1st number corresponds to its position on the horizontal axis (x) and the 2nd number corresponds to its position on the vertical axis (y).

Cubic centimetre (cm3) (p. 63)Unit of measure equal to the volume of a cube with 1 cm sides.

Cubic decimetre (dm3) (p. 63)Unit of measure equal to the volume of a cube with 1 dm sides.

Cubic metre (m3) (p. 63)Unit of measure equal to the volume of a cube with 1 m sides.

Curved body (p. 86)Solid with at least one curved surface.

D

Decimal (p. 8, 12, 14, 23)Number comprised of 2 parts separated by a decimal point: a whole part and a decimal part.

Decimal notation (p. 8, 12, 14, 23)Representation of a base 10 number made up of 2 parts (a whole part and a decimal part) separated by a decimal point.

Distributive property (p. 5)Property applied to multiplication that distributes multiplication over addition or subtraction.

Dividend (p. 12, 23)In a division, the number that is divided.

Division (p. 12, 23)Operation that shares a quantity into a number of equal parts.

Divisor (p. 12, 23)In a division, the number that divides.

Glossary B-121

13326_decimale_5b_gloss_an.indd 121 14-02-25 11:57

E

Edge (p. 86)Segment of a solid where 2 faces meet.

Euler’s theorem (p. 92)Formula designed to easily calculate the relationship between the number of faces (F), vertices (V) and edges (E) of a convex polyhedron. The formula is: F + V – 2 = E.

F

Factor (p. 8)Multiplication term.

Frieze pattern (p. 53)Rectangular strip with a regularly repeating pattern.

G

Gram (g) (p. 57)Unit of measure for mass 1000 times smaller than a kilogram (0.001 kg).

H

Hexagon Six-sided polygon.

Hour (h) (p. 60)Unit of measure for time lasting 60 minutes or 3600 seconds. There are 24 hours in a day.

Hundredth (p. 12, 14)In the decimal notation of a number, it is the 2nd digit to the right of the decimal point.

One hundredth = 1

100 or 0.01.

I

Integer (p. 40)Number belonging to a group {… –3 , –2, –1, 0, 1, 2, 3, …}. This group includes positive integers (greater than 0) and negative integers (less than 0).

K

Kilogram (kg) (p. 57)Unit of measure for mass 1000 times greater than a gram (1000 g).

L

Litre (L) (p. 67)Unit of measure for capacity 1000 times greater than a millilitre (1000 ml).

M

Mass (p. 57)Quantity of matter in an object or living thing.

Millilitre (ml) (p. 67)Unit of measure for capacity 1000 times smaller than a litre (0.001 L).

Minute (min) (p. 60)Unit of measure for time lasting 60 seconds. There are 60 minutes in an hour.

Multiplication (p. 8)Operation that finds the product of 2 or more factors.

N

Natural number Integer greater than or equal to 0.

B-122 Glossary


Nonconvex polyhedron (p. 89)Concave polyhedron with at least one line segment connecting two vertices (points) outside it.

O

Octagon Eight-sided polygon.

Origin (p. 48)Point at which 2 axes meet in a Cartesian plane with the coordinates (0, 0).

P

PentagonFive-sided polygon.

Polygon Plane figure formed by closed straight lines. A polygon can be convex or nonconvex.

Polyhedron (p. 86)Solids formed only by polygons (plane surfaces).

Prism (p. 86)Polyhedron with 2 congruent and parallel polygons as bases, and rectangles as its remaining faces.

Probability (p. 83)Possibility that a result will occur. Probability can be given a value between 0 and 1. Zero indicates the impossibility that a result will occur, while 1 indicates the certainty that it will happen. An event can be more or less likely than another, or just as likely as another.

Product (p. 8)Result of a multiplication.

Pyramid (p. 86)Polyhedron with a single polygon as a base, and triangles as its remaining faces.

Q

Quadrant (p. 48)Region of a Cartesian plane bordered by its axes.

QuadrilateralFour-sided polygon.

Quotient (p. 12, 23)Result of a division.

R

Random experiment (p. 80)Experiment with an outcome that is based entirely on chance.

S

Second (s) (p. 60)Base unit of measure for time for the international system of units. There are 60 seconds in a minute and 3600 seconds in an hour.

Solid (p. 86)Three-dimensional geometric figure formed by at least one closed surface.

Square centimetre (cm2) (p. 26)Unit of measure equal to the area of a square with 1 cm sides.

Square kilometre (km2) (p. 26)Unit of measure equal to the area of a square with 1 km sides.

Glossary B-123


Square metre (m2) (p. 26)Unit of measure equal to the area of a square with 1 m sides.

Surface (p. 26, 86)Part of a plane figure or a collection of faces of a solid.

Survey (p. 16)Research meant to gather information on a precise subject to obtain statistics.

TTenth (p. 12, 14, 23)In the decimal notation of a number, it is the 1st digit to the right of the decimal point.

One tenth = 1

10 or 0.1.

Tessellation (p. 55)Collection of geometric figures that cover up an entire surface, with no empty space or overlap.

Thousandth (p. 14)In the decimal notation of a number, it is the 3rd digit to the right of the decimal point.

One thousandth = 1

1000 or 0.001.

Translation (p. 53)Geometric transformation that moves all of a figure’s points in the same direction and the same distance. The figure keeps its shape, its orientation and its dimensions.

Translation arrow (p. 53)Arrow showing the direction and length that a translation must move.

Tree diagram (p. 80)Diagram representing all of the possible results of a random experiment.

V

Vertex (p. 86)Intersecting point of at least 2 edges of a solid.

Volume (p. 63)Measure of space occupied by a solid. Space has 3 dimensions: length (l), width (w) and height (h).

B-124 Glossary


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