Let's Do Mathematics 5 – Worktext B by Regal Education

Workt ext

for learners 10 - 11 years old

Aligned to the US Common Core State Standards

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Workt ext

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for learners 10 - 11 years old

Let’s Do Mathematics

Let’s Do Mathematics is a series covering levels K-6 and is fully aligned to the United States Common Core State Standards (USCCSS). Each level consists of two books (Book A and Book B) and combines textbook-style presentation of concepts as well as workbook practice.

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Central to the USCCSS is the promotion of problem-solving skills and reasoning. Let’s Do Mathematics achieves this by teaching and presenting concepts through a problem-solving based pedagogy and using the concrete-pictorial-abstract (CPA) approach. Learners acquire knowledge and understanding of concepts through a guided progression beginning with concrete examples and experiences which then flow into pictorial representations and finally mastery at the abstract and symbolic level. This approach ensures that learners develop a fundamental understanding of concepts rather than answering questions by learned procedures and algorithms. Key features of the series include:

Anchor Task

Angles of Triangles Anchor Task

Open-ended activities serve as the starting point for understanding new concepts. Learners engage in activities and discussions to form concrete experiences before the concept is formalized.

$0.75 each

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$5.25 per pack

$1.45 each

Multiplying

Let’s Learn

Concepts are presented in a clear and colorful manner. Worked problems provide learners with guided step-by-step progression through examples. Series mascots provide guidance through helpful comments and observations when new concepts are introduced.

Operations on Decima ls

Anchor Task

Let’s Learn

by 1-digit Nu

mbers

Step 3

A superma rket is sellin g pistachio will 3 kg of nuts for $21.3 pistachio nuts 0 per kilogram cost? . How much We need to multiply 21.3 by 3 to find help find the out. Let’s use answer. a place value chart to Tens Ones Tenths . Each row represents the . cost of 1 kg of pista chio . nuts.

Multiply the

chio nuts costs

3 x 0.3 = 0.9

tenths.

1 . 3

Tens

Multiply the

Tens

Ones

3 . 9

6.83 x 4 = 27.32

3x1=3

ones.

2 1 . 3 x

. .

Tenths

4 . 4

4 using the

od.

6 . 18 3

Tenths

$63.90.

7 27 . 4

77.4 x 6 = 464.4

Tenths

. .

Step 2

column meth

. 4

Find 6.83 x

Ones

. 9

6 using the

7 27 . 4

Ones .

21.3 x 3 = 63.9 So, 3 kg of pista

$21.30 Step 1

Tens

3 6 3 . 9

Find 77.4 x

Multiply the

tens.

2 1 . 3 x

7 27 . 4

6 4 6 4 . 4

column meth

od.

6 . 18 3 4 . 3 2

6 . 18 3

4 2 7 . 3 2

We can use rounding and estimation to check our answers.

Let’s Practice

Fill in the blanks.

(a)

Let’s Practice

Ones

Tens

Hundreds

Ten Thousands Thousands

Hundred Thousands

Millions

Fill in the blanks.

(a)

Learners demonstrate their understanding of concepts through a range of exercises and problems to be completed in a classroom environment. Questions provide a varying degree of guidance and scaffolding as learners progress to mastery of the concepts.

556,795

536,795

516,795

576,795

100,000 more

100,000 less

ds place

Look at the ten thousan

(b)

increases by The ten thousands digit The numbers increase

Tens

Hundreds

Ones

in each step. 125,000 more

125,000 less

in each step.

The next number in

the pattern is

(b)

2,824,575

1,574,575

The numbers increase

(c)

Millions

4,074,575

(d)

Hundred Thousands

Millions

the pattern is

Ten Thousands Thousands

Hundreds

Ones

Tens

1,500,000 more

1,500,000 less

in each step.

+ The next number in

Hundred Thousands

324,575

Hundred Thousands

Millions

Ten Thousands Thousands

Hundreds

Ones

Tens

10,000 more

10,000 less

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At Home

Classify each triangle .

(a)

Classify each triangle . Choose one classific ation per triangle . (b)

(a)

Right-angled

At Home

Scalene

Isosceles

(b)

Further practice designed to be completed without the guidance of a teacher. Exercises and problems in this section follow on from those completed under Let’s Practice.

Right-angled

(c)

Scalene

(d)

Isosceles

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(c)

Right-angled

Scalene

Isosceles

(e)

(f)

(d)

Right-angled

Scalene

Isosceles

Hands On

ps of 4-5. in your it number Work in grou n. write a 7-dig n and 6 millio As a group, een 5 millio that is betw notebook square. on the start a counter e Plac the 2. ter forward e your coun dice and mov n on your dice. 3. Roll the spaces show number of ber plete the num p must com in the grou 4. Everyone order to move forward. pattern in till nal number with the origi steps 3 to 4 at Repe 5. the finish. you reach

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Learners are encouraged to ‘learn by doing’ through the use of group activities and the use of mathematical manipulatives.

Solve It!

Solve It! (a) OPQR is a parallelo gram. SP is a straight line. Find OPQ O

P 20º

R S

(b) MNOP is a trapezo id. NP is a straight line. Find t. M

38º

47º

(c)

GHIJ is a parallelogram. HJ is a straight line. Find G

56º H

m J

44º

120

2. Use the ordered pairs to plot the points

Looking Back 1. The line plot shows the distances the school fun run.

Looking Back

Consolidated practice where learners demonstrate their understanding on a range of concepts taught within a unit.

118º

Activities that require learners to apply logical reasoning and problem-solving. Problems are often posed which do not have a routine strategy for solving them. Learners are encouraged to think creatively and apply a range of problem-solving heuristics.

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students in Grade 5 ran during the

Fun Run Distances

on the coordinate grid.

(a) A (1, 2)

(b) F (4, 4)

(c)

J (3, 7)

(d) W (3, 2)

(e) C (9, 9)

(f)

H (9, 6)

(g) E (4, 8)

(h) R (8, 4)

(i)

O (6, 5)

10 9 3 4

1 4

1 2

3 4

1 4

Miles

(a) How many students ran 2 miles? than (b) How many students ran further (c)

6 1

What is the combined distance ran by 1 mile of less?

1 miles? 2

the students who ran 4

mi ran by (d) What is the combined distance or further?

3 the students who ran 1 4 miles

3 2 1 0

239 238

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Contents

2 3 14 22 29 36

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5 Operations on Decimals Adding and Subtracting Decimals Multiplying by 10s, 100s and 1,000s Multiplying by 1-digit Numbers Dividing by 1-digit Numbers Word Problems

Geometry Types of Triangles Angles of Triangles Angles of Quadrilaterals

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6 Ratio Finding Ratio Ratio and Measurement Equivalent Ratios and Simplest Form Word Problems

84 85 98 108

8 Measurement Converting Measurement Units Word Problems

124 125 133

9 Volume Volume and Unit Cubes Volume of Rectangular Prims Volume and Capacity Word Problems

150 150 168 181 186

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48 48 58 65 74

206 206 224

11 Problem Solving Act It Out Draw a Model Guess-and-Check Make a List Look for Patterns Work Backwards Simplify the Problem Solve Part of the Problem Before-After Concept Make Suppositions

242 242 248 254 259 264 270 276 282 287 293

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10 Data and Graphs Line Plots Graphing Equations

Operations on Decimals

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Anchor Task

$1.45 each

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$5.25 per pack

Adding and Subtracting Decimals Let’s Learn

32.5

bag

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Ethan weighs 32.5 kg. His schoolbag weighs 5.8 kg. Find the total weight of Ethan and his schoolbag.

5.8

To find the total weight, we add. Step 1

Step 2

Add the tenths. 2 . 5

5 . 8

. 3

Tens

8 . 3

Ones

Add the tens.

2 . 5

5 . 8

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Add the ones.

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Step 3

2 . 5

5 . 8

+ 3

8 . 3

We can regroup 13 tenths into 1 one and 3 tenths.

Tenths

. .

Find the sum of 148.27 and 61.58. Step 1

We can regroup 15 hundredths into 1 tenth and 5 hundredths.

8 . 12

1 . 5

Hundreds

Tens

Ones

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Tenths

Hundredths

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Add the hundredths.

. .

Step 2

Step 3

Add the tenths.

8 . 12

1 . 5

. 8

Tens

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Hundreds

8 . 12

1 . 5

9 . 8

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Add the ones.

Ones

. . .

Tenths

Hundredths

Step 4

We can regroup 10 tens into 1 hundred.

8 . 12

1 . 5

9 . 8

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Add the tens.

Step 5 1

8 . 12

1 . 5

9 . 8

Ones

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Hundreds

Tens

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Add the hundreds.

Tenths

Hundredths

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148.27 + 61.58 = 208.85

Add 27.93 to 83.12. 1

7 . 9

3 . 1

1 . 0

We can use rounding and estimation to check our answers.

27.93 + 83.12 = 111.05

Sophie had a piece of ribbon 68.4 cm in length. She cut 45.7 cm of the ribbon to tie around a gift. What length of ribbon does she have left?

ribbon left

45.7

Step 1

Subtract the tenths. 6

8 . 144 5 . 7

Tens

Ones

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–

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To find the length of ribbon left, we subtract.

. 7

Step 2

Step 3

Subtract the ones. 7

–

Subtract the tens.

8 . 144

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5 . 7 2 . 7

–

8 . 144

5 . 7

2 . 7

68.4 – 45.7 = 22.7 Sophie has 22.7 cm of ribbon left.

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ribbon cut

68.4

We can regroup 1 one into 10 tenths.

Tenths

Subtract 7.49 from 22.36. Ones

Tenths

Hundredths

Tens

– 1

2 . 123

7 . 4

4 . 8

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22.36 – 7.49 = 14.87

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Find the sum and difference of 53.18 and 62.57.

To find the sum, we add. To find the difference, we subtract.

3 . 11

2 . 5

5 . 7

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–

2 . 45

3 . 1

9 . 3

The sum of 53.18 and 62.57 is 115.75. The difference of 53.18 and 62.57 is 9.39.

Let’s Practice (a) (b) 3 1 4 . 1 5 9

2 . 6

6 . 5 .

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9 . 3

5 . 1

1 . 7

3 . 8

3 . 6

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(g) (h) 3 9 5 . 8 3

3 . 8 .

1 . 7

(e) (f) 5 8 3 . 6 8 +

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5 . 8

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1. Add.

3 +

3 . 5

8 . 3 .

2. Subtract. (a) (b) 9 7 . 5 –

7 . 9

–

5 . 3

3 . 7 .

9 . 8

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–

(e) (f) 4 6 . 8 5 –

5 . 7

–

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6 . 3

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–

5 . 9

2 . 5

3 . 1

4 . 8

8 . 4

(i) (j) 8 8 . 4 3 –

(g) (h) 8 6 . 4 6 –

7 . 9

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–

8 . 4 .

–

5 . 3

3. Use the column method to add or subtract. (b) 13.56 – 3.59 =

(d) 33.65 – 23.59 =

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(a) 4.69 + 13.59 =

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(e) 43.35 + 136.94 =

(f) 87.48 – 13.34 =

(b) Home At

(a) (b) 5 7 . 7 4 . 9

8 . 8 .

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8 . 4

5 . 6

4 . 5

4 5 6 . 7 3 (e) (f) +

5 . 8

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5 . 7

7 . 8

(h) 48.39 + 212.48 =

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(g) 4.69 + 4.69 =

1. Add.

2. Subtract.

–

6 . 2

–

4 . 3

5 . 7 .

4 . 6 .

–

3 . 6

–

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(g) 45.94 – 24.69 =

6 . 4

6 . 8

(e) (f) 7 6 . 4 2 –

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–

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(a) (b) 8 9 . 6

5 . 3

4 . 3

(h) 94.05 – 45.39 =

3. Find the sum and difference of each pair of numbers.

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sum =

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(a) 67.58 and 45.38

difference =

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(b) 35.48 and 125.94

sum =

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sum =

difference =

Multiplying by 10s, 100s and 1,000s

Hundreds

Tens

Ones

Tenths

Hundredths

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Thousands

Anchor Task

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(a) 2.1

(b) 3.06

(d) 9.49

2.1 x 10

3.06 x 10

1.28 x 10

9.49 x 10

2.1 x 100

3.06 x 100

1.28 x 100

9.49 x 100

2.1 x 1,000

3.06 x 1,000

1.28 x 1,000

9.49 x 1,000

Let’s Learn Let’s use place value disks to help multiply numbers by 10.

0.1

100 100

x 10

24.3

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Find 24.3 x 10.

243

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24.3 x 10 = 243

Let’s use place value disks to help multiply 3.52 by 10. 1

0.1

0.01 0.01

3.52

x 10

0.1

35.2

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3.52 x 10 = 35.2

Let’s use a place value chart to help multiply numbers by 10. Find 29.47 x 10.

Tens

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Hundreds

Ones

Tenths

Hundredths

. .

29.47 x 10 = 294.7

Multiply 7 . 4

5 7 . 0

3 . 8

1 8

0. 4

(b) 3.81 x 80 = 3.81 x 8 x 10 = 30.48 x 10 = 304.8

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(a) 7.4 x 50 = 7.4 x 5 x 10 = 37 x 10 = 370

Find 3.14 x 100. Hundreds

Tens

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Let’s use a place value chart to help multiply numbers by 100.

Ones

Tenths

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3.14 x 100 = 314 Multiply.

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(a) 8.3 x 600 = 8.3 x 6 x 100 = 49.8 x 100 = 4,980

(b) 2.35 x 400 = 2.35 x 4 x 100 = 9.4 x 100 = 940

8 . 3

6 4

9 . 8

2 . 23

5 4

9. 4

Hundredths

Find 0.45 x 1,000. Hundreds

Tens

Ones

Tenths

Hundredths

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Multiply.

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0.45 x 1,000 = 450

(a) 5.6 x 3,000 = 5.6 x 3 x 1,000 = 16.8 x 1,000 = 16,800

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(b) 2.91 x 3,000 = 2.91 x 3 x 1,000 = 8.73 x 1,000 = 8,730

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5 . 6

6 . 8

2 . 9

8 . 7

1 . 7

4 6 . 8

Let’s Practice 1. Multiply by 10, 100 and 1,000.

1.2 x 200 =

1.2 x 2,000 =

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10.11 x 30 =

10.11 x 300 =

10.11 x 3,000 =

(d) 9.87 x 10 =

9.87 x 100 =

9.87 x 1,000 =

(b) 0.8 x 8 =

0.8 x 80 =

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1.2 x 20 =

4.5 x 1,000 =

3.1 x 1,000 =

4.5 x 100 =

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3.1 x 100 =

(b) 4.5 x 10 =

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(a) 3.1 x 10 =

0.8 x 800 =

0.8 x 8,000 =

(d) 0.01 x 5 = 0.01 x 50 = 0.01 x 500 = 0.01 x 5,000 =

Multiply. (b) 7.4 x 100 =

(f) 43.42 x 200 =

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(e) 2.42 x 5 =

(d) 5.56 x 4 =

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(a) 3.2 x 10 =

(g) 11.13 x 2,000 =

(h) 24.34 x 3,000 =

(b) Home At 1. Fill in the blanks. (a)

0.1

(b) 100

101 x (c)

1,000

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2.1 x

100

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1,000

100 10

1,000

100 10

1,000

100 10

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(d)

100

1,000

100 10

1,000

100 10

1,000

100 10

1,000

100 10

2. Find the products.

Multiply.

3.05 x 1,000 =

(d) 0.28 x 4 =

0.28 x 40 =

0.28 x 400 =

0.28 x 4,000 =

(b) 12.4 x 100 =

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(a) 8.1 x 10 =

3.05 x 100 =

0.42 x 1,000 =

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0.42 x 100 =

(b) 3.05 x 10 =

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(a) 0.42 x 10 =

(d) 7.72 x 3 =

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Multiplying by 1-digit Numbers Let’s Learn

A supermarket is selling pistachio nuts for $21.30 per kilogram. How much will 3 kg of pistachio nuts cost?

$21.30

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We need to multiply 21.3 by 3 to find out. Let’s use a place value chart to help find the answer. Each row Tens Ones Tenths represents the cost . of 1 kg of pistachio . nuts. .

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3 x 0.3 = 0.9

Step 1

Multiply the tenths. 2

1 . 3

Tens

Ones

Tenths

. 9

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. 3x1=3

Step 2

Multiply the ones. 2 1 . 3

3 . 9

Tens

Ones

. . . .

Tenths

Step 3

Multiply the tens. Tens

2 1 . 3

Ones

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6 3 . 9 21.3 x 3 = 63.9 So, 3 kg of pistachio nuts costs $63.90.

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Find 77.4 x 6 using the column method. 7 27 . 4

7 27 . 4

. 4

7 27 . 4

4 . 4

77.4 x 6 = 464.4

Tenths

4 6 4 . 4

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Find 6.83 x 4 using the column method. 6 . 18 3

4 2

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6 . 18 3

. 3 2

6 . 18 3 4

2 7 . 3 2

6.83 x 4 = 27.32

We can use rounding and estimation to check our answers.

Let’s Practice 1. Multiply. (a) (b) 2 5 . 6 x

2 9 . 2 4

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9 .

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6 .

(i) (j) 7 3 3 . 2 4 5

3 0 7 . 4 5

(g) (h) 9 3 . 0 7 x

7 4 . 1

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(e) (f) 8 3 . 5 5 x

2 7 . 8

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1 0 8 8 . 4 6 x

8 .

2. Use the column method to multiply. (b) 135.3 x 5 =

(d) 582.44 x 7 =

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(a) 38.2 x 4 =

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(e) 787.68 x 4 =

(f) 3,206.53 x 6 =

Solve It! How do you make an octopus laugh?

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To find the answer, multiply the numbers. Write the matching letters in the boxes according to their order.

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17.32

5.84 x 3

5.66 x 3

3.58 x 4

1.47 x 7

2.7 x 8

17.52

21.6

17.52

0.76 x 5

0.82 x 9

R eg k

4.33 x 4

2.04 x 8

1.05 x 8

3.8 –

17.52

16.32

10.29

16.98

14.32

7.38

16.32

8.4

(b) At Home 1. Multiply. (a) (b) 2 0 . 8 x

2 0 6 . 2 5

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5 0 5 . 0 5 x

(i) (j) 8 1 3 . 1 7 x

6 .

(g) (h) 5 8 . 2 8

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(e) (f) 4 4 . 1 3 x

8 . 6 5

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9 .

1 3 . 6

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2 3 4 . 5 6

7 .

2. Use the column method to multiply. (b) 105.7 x 3 =

(d) 851.28 x 8 =

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(a) 19.3 x 6 =

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(e) 992.64 x 7 =

(f) 4,083.26 x 4 =

Dividing by 1-digit Numbers Let’s Learn

Let’s use place value disks to divide 9.6 by 4.

0.1

0.1 0.1

0.1

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0.1

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Regroup 1 one into 10 tenths. Now we can make equal groups!

0.1

There are 4 equal groups of 2.4. 9.6 ÷ 4 = 2.4

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Michelle has a piece of string 1.56 m in length. She cuts it into 6 pieces of equal length. Find the length of each piece of string.

0 . 2 6 1 . 5 6 1 2 3 Step 2

6 6

Bring down the 6 hundredths. Now there are 36 hundredths.

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0 . 2 6 1 . 5 1 2 3 3

Divide 1 one by 6. Regroup 1 one into 10 tenths. Add the 5 tenths and divide. 15 tenths ÷ 6 = 2 tenths remainder 3 tenths. 15 tenths – 12 tenths = 3 tenths.

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Step 1

6 6 0

36 hundredths ÷ 6 = 6 hundredths.

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1.56 ÷ 6 = 0.26 Each piece of string has a length of 0.26 m

0.26 m = 26 cm. So each piece of string has a length of 26 cm.

Find 84.35 ÷ 5. 1 5 8 5 3 3

84.35 ÷ 5 = 16.87 Divide 145.18 by 7.

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4 8

4 0 4

1 5 8 5 3 3

6 . 8 4 . 3 5 4 0 4 4

1 5 8 5 3 3

6 . 8 7 4 . 3 5 4 0 4 4

3 0 3

3 0 3 5 3 5 0

Use rounding and estimation to check the answer.

8 8 0

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2 0 . 7 1 4 5 . 1 1 4 0 5 1 4 9 2 2

6 . 4 . 3 5

Step 4

1 . 5 8 4 . 3 5 5 3

Step 3

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Step 2

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Step 1

145.18 ÷ 7 = 20.74

Let’s Practice 1. Divide.

5 2 2 4 . 5

6 8

1 . 2 4

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3 6 7 . 4 4

(a) (b) (c)

(d) (e) 1 5 9 . 8 8

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8 2 3 6 2 . 5 6

2. Complete the following. Show your working. (b) 71.19 ÷ 9 =

(d) 223.14 ÷ 6 =

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(a) 82.08 ÷ 6 =

(f) 5,599.26 ÷ 9 =

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(e) 3,456.78 ÷ 6 =

At Home 1. Divide.

5 3 7 5 . 5

3 5 9 . 0 7

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7 6 7 . 2 7

(a) (b) (c)

(d) (e)

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5 4 4 4 . 4 5

6 7 0 . 3 2

2. Complete the following. Show your working. (b) 42.21 ÷ 9 =

(d) 158.48 ÷ 8 =

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(a) 29.61 ÷ 7 =

(f) 5,308.35 ÷ 5 =

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(e) 1,009.56 ÷ 4 =

Word Problems Let’s Learn

$1.38

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At the school bookshop, a pen costs $1.38 and a notebook costs $3.76. Find the total cost of 3 pens and 1 notebook.

$3.76

pen

notebook

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Find the cost of the pens by multiplying $1.38 by 3. 1

1 . 23 8

4 . 1

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The cost of 3 pens is $4.14. Add the cost of the notebook to find the total cost. 4 . 11

+ 3

7 6

7 . 9 0

The total cost of 3 pens and a notebook is $7.90.

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12 l

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Halle made 12 liters of lemonade. She poured an equal volume of lemonade into 6 bottles. She had 4.32 liters remaining. Find the amount of lemonade in each bottle.

bottle bottle bottle bottle bottle bottle

remaining 4.32 l

Find the total volume of lemonade poured into the 6 bottles by subtracting the remaining amount from the original amount. 0

2 . 90 100 4

3 2

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–

7 . 6 8

There is 7.68 liters of lemonade in 6 bottles. To find the volume of orange juice in each bottle, we divide. 8 8

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1 . 2 6 7 . 6 6 1 6 1 2 4 4

8 8 0

Each bottle contains 1.28 liters of lemonade.

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Wyatt and Jordan have combined savings of $120. Wyatt has $10.74 more than twice of Jordan's savings. How much does Jordan have?

Wyatt's savings Jordan's savings

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$10.74

–

2 9 0 . 9 0 100 1 0 . 7 4

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3 units = $120 – $10.74 = $109.26

1 0 9 . 2 6

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1 unit = 109.26 ÷ 3 = 36.42

Jordan has $36.42

3 3 1 0 9 1 1

6 . 4 2 9 . 2 6 9 8 1 1

2 2 6 6 0

$120

A jar of jam has a mass of 123.6 grams. A tub of butter has a mass of 314.6 grams. Sophie buys 6 jars of jam and 4 tubs of butter. Find the total mass of the items. 123.6 g

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jam

butter

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314.6 g

Find the mass of the jars of jam. Multiply the mass of 1 jar by 6. 1

2 33 . 6

6 7 4

1 . 6

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The jars of jam have a mass of 741.6 grams. Now find the mass of the tubs of butter. Multiply the mass of 1 tub by 4. 3 11

4 . 6

2 5 8 . 4

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The tubs of butter have a mass of 1,258.4 grams. Add the masses to find the total mass of the items. 1

7 14

1 . 6

2 5 8

Can you express the mass in kilograms?

2 0 0 0 . 0

The total mass of the items is 2,000 grams.

Let’s Practice

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1. Riley made 20 liters of fruit punch. She poured an equal volume of punch into 4 bowls. She had 1.04 liters remaining. Find the volume of punch in each bowl.

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2. Keira and her baby sister have a combined mass of 56 kilograms. Keira's mass is 4.63 kilograms less than twice the mass of her baby sister. Find Keira's mass.

R eg

al E

du ca

tio

3. At the bakery, a chocolate eclair costs $2.40 and a scone costs $1.25. (a) Dominic buys 3 chocolate eclairs and 6 scones. How much did he spend in total? (b) Ethan has $10 and he buys 6 scones. Does he have enough money left over to buy a chocolate eclair?

At Home

du ca

tio

1. A phone company charges $1.22 to connect calls. It then charges $0.64 per minute. How much does it cost to make an 18-minute call in total?

R eg

al E

2. Mr. Romero's car uses 8.36 liters of gas per hour. Mr. Romero filled his car with gas and then drove for 5 hours. There was 8.2 liters of gas remaining in his tank. How much gas was in his car before the journey?

du ca

tio

3. Wyatt bought 30 kilograms of flour. He poured an equal mass of flour into 8 containers and had 3.12 kilograms remaining. Find the mass of flour in each container.

R eg

al E

4. A brick fence has a height of 8 bricks. A flag pole of height 2.28 meters is erected on top of the fence to give a total height of 5 meters. Find the height of 1 brick.

Looking Back 1. Find the sum and difference of each pair of numbers.

sum =

difference =

al E

(b) 12.29 and 10.37

du ca

tio

(a) 14.6 and 23.8

sum =

difference =

R eg

sum =

2. Use the column method to multiply. (b) 231.7 x 5 =

(d) 342.84 x 6 =

al E

du ca

tio

(a) 25.3 x 4 =

R eg

(e) 503.39 x 9 =

(f) 4,207.47 x 4 =

3. Complete the following. Show your working. (b) 61.35 ÷ 5 =

(d) 349.92 ÷ 8 =

al E

du ca

tio

(a) 17.15 ÷ 7 =

R eg

(e) 1,448.48 ÷ 4 =

(f) 9,898.28 ÷ 7 =

R eg

al E

du ca

tio

4. At the cinema, movie tickets are $15.92 for adults and half price for 1 children. A tub of popcorn is $4.86 and juice boxes are the price of 3 popcorn. A family of 2 adults and 3 children buy tickets to the movie as well as 4 tubs of popcorn and 3 juice boxes. How much did they spend in all?

Ratio

Finding Ratio

R eg

al E

du ca

tio

Anchor Task

Let’s Learn We can use ratio to compare two or more quantities.

tio

Sophie has 3 yellow shirts and 2 pink shirts.

du ca

The ratio of the number of yellow shirts to the number of pink shirts is 3 : 2. The ratio of the number of pink shirts to the number of yellow shirts is 2 : 3. We read the ratio of 3 : 2 as 3 to 2.

al E

Each bunch has the same number of fruits.

R eg

The ratio of the number of tomatoes to the number of bananas is 4 : 2. The ratio of the number of bananas to the number of tomatoes is 2 : 4. Notice that the ratio compares the bunches of fruit, not the total number of individual fruits.

tio

We can use ratio to compare a quantity of a part with a quantity of the total.

du ca

The ratio of the number of chocolate ice creams to the number of strawberry ice creams is 3 : 5.

The ratio of the number of strawberry ice creams to the number of chocolate ice creams is 5 : 3.

al E

There are 3 chocolate ice creams. There are 8 ice creams in all. The ratio of the number of chocolate ice creams to the total number of ice creams is 3 : 8.

R eg

What is the ratio of the number of red roses to the total number of roses?

There are 6 red roses. There are 9 roses in all. The ratio of the number of red roses to the total number of roses is 6 : 9. 50

Let’s Practice 1. Write the ratio.

tio

(a)

al E

(b)

du ca

The ratio of the number of pencil cases to the number of bags

The ratio of the number of shoes to the number of socks is :

R eg

(c)

The ratio of rectangles to triangles is

. 51

2. Write the ratio.

du ca

tio

(a)

The ratio of the number of eggs to the number of donuts is

R eg

al E

(b)

The ratio of pancakes to cookies is 52

tio

3. Find the ratio of basketballs compared to the total number of balls.

du ca

The ratio of the number of basketballs to the total number of balls .

R eg

al E

4. Find the ratio of milk cartons to the total number of cartons.

The ratio of the number of milk cartons to the total number of cartons is

Hands On 1. Place a paperclip in the center of the circle as shown.

du ca

4. Write the ratio to compare the number of times the paperclip landed on each color.

tio

3. Repeat Step 2 for a total of 20 spins.

2. Flick the paperclip and record the color it lands on in a tally.

Color

Tally

red

yellow

R eg

al E

green blue

Color red to blue yellow to red blue to green blue to green and yellow yellow to red and blue

Ratio

At Home 1. Write the ratio.

tio

(a)

al E

(b)

du ca

The ratio of the number of beef burgers to chicken burgers

R eg

The ratio of the number of lemons to the number of oranges is

(c)

The ratio of circles to squares is

. 55

2. Write the ratio.

du ca

tio

(a)

The ratio of the number of raspberries to the number of cherries is

R eg

al E

(b)

The ratio of bananas to apples is

du ca

tio

3. Find the ratio of the number of noodle boxes to the total number of food boxes.

The ratio of the number of noodle boxes to the total number of food boxes is

al E

R eg

4. Find the ratio of the number of kittens compared to the total number of pets.

The ratio of the number of kittens to the total number of pets is

. 57

Ratio and Measurement Let’s Learn

al E

0 4 kg

3 kg

1 kg

2 kg

tio

du ca

Compare the masses of the fruits.

We can use ratio to compare measurements such as length, mass and volume. We can also use ratio to compare the amount of money or time. When we use ratio to compare measurements, we do not include the units of measurement.

0 4 kg

3 kg

1 kg 2 kg

R eg

The mass of the watermelon is 3 kilograms. The mass of the pineapple is 2 kilograms. The ratio of the mass of the watermelon to the mass of the pineapple is 3 : 2. The ratio of the mass of the pineapple to the mass of the watermelon is 2 : 3. The ratio of the mass of the watermelon compared to the total mass of the fruits is 3 : 5.

The yellow crayon is 7 centimeters in length. The blue crayon is 5 centimeters in length.

tio

Compare the length of the crayons.

du ca

The ratio of the length of the blue crayon to the length of the yellow crayon is 5 : 7.

The ratio of the length of the yellow crayon to the length of the blue crayon is 7 : 5. Compare the volume of liquid in the beakers. B

R eg

al E

The volume of the liquid in Beaker A is 400 milliliters. The volume of the liquid in Beaker B is 300 milliliters. The ratio of the volume of the liquid in Beaker A to the volume of the liquid in Beaker B is 400 : 300. The ratio of the volume of the liquid in Beaker B compared to the total volume of the liquid in both beakers is 300 : 700. 59

Let’s Practice Use ratio to compare measurements. 1.

du ca

tio

The ratio of the volume of liquid in Beaker B to the volume of liquid in Beaker A is

al E

400 g

R eg

300 g

100 g

200 g

400 g 300 g

100 g

200 g

The ratio of the mass of the salmon to the mass of the beef is

tio

The ratio of the length of the nail to the length of the :

du ca

screw is

al E

The ratio of the length of the toothpaste to the length of the

R eg

toothbrush is

Hands On

1. Empty the contents of your pencil case onto your desk. 2. Use a ruler to measure the lengths of 6 objects. Have a classmate measure the same items from their pencil case. 3. Use ratios to compare the lengths of your objects with your classmate's objects. 61

At Home Use ratio to compare measurements.

du ca

tio

The ratio of the length of the fork to the length of the spoon is

al E

4 kg

R eg

3 kg

4 kg

1 kg

3 kg

1 kg

2 kg

The ratio of the mass of the apples compared to the total mass of oranges and apples is

Hands On At home, use a ruler to find the length of each object. Then, use ratio to compare the lengths.

tio

1. The ratio of the length of my shoe to the length of my father's shoe

2. The ratio of the length of a spatula to the length of a fork :

du ca

3. The ratio of the length of my television to the height of my television is

al E

4. The ratio of the length of my mathematics book to the breadth of my mathematics book is

5. The ratio of the height of my schoolbag to the height of my chair :

R eg

6. The ratio of my height to the height of my sibling is

Solve It!

du ca

Chelsea has $

tio

1. The ratio of Sophie's savings to Chelsea's savings is 2 : 1. If Sophie has $100, how much does Chelsea have? Show your working.

al E

2. On Mr. Taylor's farm, the ratio of the number of ducks to chickens is 1 : 5. If there are 10 ducks, how many chickens are there on Mr. Taylor's farm? Show your working.

On Mr. Taylor's farm there are

chickens.

R eg

3. Mrs. Taylor bought 7 kilograms of apples and oranges. The ratio of the mass of the apples to the total mass of the apples and oranges is 3 : 7. What is the mass of the oranges? Show your working.

The mass of the oranges is

kilograms.

Equivalent Ratios and Simplest Form Let’s Learn

Riley has 4 oranges and 8 apples. The ratio of the number of oranges to the number of apples is 4 : 8.

1 unit oranges apples

du ca

tio

We can represent the ratio using a bar model.

R eg

al E

Riley puts the fruit on plates in groups of 2.

There are 2 plates of oranges and 4 plates of apples. The ratio of the number of oranges to the number of apples is 2 : 4. 1 unit

oranges apples

The ratio of oranges to apples has not changed. 4 : 8 and 2 : 4 are equivalent ratios. 65

du ca

tio

Riley puts the fruit on plates in groups of 4.

There is 1 plate of oranges and there are 2 plates of apples. The ratio of the number of oranges to the number of apples is 1 : 2. 1 unit oranges apples

al E

4 : 8, 2 : 4 and 1 : 2 are equivalent ratios. 1 : 2 is the ratio of oranges to apples in its simplest form.

We can find the simplest form of a ratio by dividing each term in the ratio by the greatest common factor.

R eg

Let's find the simplest form of the ratio 12 : 4. The ratio 12 : 4 has two terms, 12 and 4. The lowest common factor of 12 and 4 is 4. ÷4

12 : 4

= 3 : 1

÷4

A ratio is in its simplest form when the only common factor of the terms is 1.

The ratio 12 : 4 in its simplest form is 3 : 1.

We can use multiplication to find missing terms in equivalent ratios. Let's find the missing term in the equivalent ratio 4 : 5 = 12 : ?

4 : 5

= 12 : 15

du ca

The missing term is 15. 4 : 5 and 12 : 15 are equivalent ratios.

tio

In the equivalent ratio, the first term has been multiplied by 3. To find the missing term, multiply the second term by the same multiplying factor.

We can also use division to find missing terms in equivalent ratios. Let's find the missing term in the equivalent ratio 18 : 6 = 3 : ?

In the equivalent ratio, the first term has been divided by 6. To find the missing term, divide the second term by the same factor. 18 : 6

al E

÷6

= 3 : 1

The missing term is 1. 18 : 6 and 3 : 1 are equivalent ratios.

R eg

In the ratio 3 : 1, the only common factor of the terms is 1. So, 3 : 1 is in its simplest form.

Let’s Practice 1. Find the equivalent ratios.

tio

(a)

al E

(b)

du ca

The ratio of the number of red cans to number of blue cans is : .

R eg

The ratio of the number of red cans to number of blue cans is : . (c)

The ratio of the number of red cans to number of blue cans : . is 68

tio

(d)

du ca

The ratio of the number of red cans to number of blue cans is : . 2. Use the bar models to write the equivalent ratios.

al E

(a) 1 unit

Ratio = 1 unit

R eg

(b)

Ratio = (c)

1 unit

Ratio =

: 69

3. Express each ratio in its simplest form. 24 : 6 25 : 10 (a) (b) (c) :

÷5

(d) (e) (f) 50 : 40 16 : 24 ÷ 10

÷8

÷6

÷3

÷6

du ca

÷3

(g) (h) (i) 30 : 24 27 : 3 ÷6

4. Fill in the blanks. Express each ratio in its simplest form. (a) (b) 8 : 24 ÷

al E

R eg

12 : 4

16 : 40

5. Write each ratio in its simplest form. (a) 30 : 3 =

(b) 3 : 12 =

(d) 32 : 16 =

(e) 4 : 24 =

(f) 9 : 6 =

(g) 100 : 20 =

(h) 15 : 40 =

(e) (f) 11 : 33 =

10 : 30

12 : 6

÷6

: 9 : 15

÷3

tio

÷ 10

÷6

18 : 42 :

÷6

6. Fill in the missing numbers. Show your working. (b)

: 5 = 5 : 25

: 9 = 21 : 27 (d) 32 : 8 = 4 :

(f)

: 4 = 28 : 16

al E

(e) 3 : 1 = 9 :

du ca

(c)

tio

(a) 15 : 6 = 5 :

: 5 = 27 : 45 (h) 36 : 12 = 72 :

R eg

(g)

(i) 5 : 6 =

: 48 (j) 9 : 4 = 90 :

At Home 1. Express each ratio in its simplest form. 12 : 10 25 : 15 (a) (b) (c) =

(d) (e) (f) 48 : 8 32 : 24 ÷8

÷8

(g) (h) (i) 27 : 18 18 : 48 ÷9

÷9

÷ 10

du ca

÷9

÷6

÷ 16

2. Fill in the blanks. Express each ratio in its simplest form. (a) (b) 7 : 21 ÷

al E

R eg

28 : 14

30 : 45

10 : 12

3. Write each ratio in its simplest form. (a) 30 : 6 =

(e) 4 : 24 = (g) 90 : 60 = 72

: :

(b) 20 : 16 =

(d) 64 : 24 =

(f) 21 : 6 = :

(h) 18 : 63 =

(e) (f) 16 : 2

27 : 9

÷9

÷5

tio

÷2

10 : 30

÷ 10

32 : 16 :

÷ 16

4. Fill in the missing numbers. Show your working. (b)

: 21 = 21 : 7

: 16 = 22 : 32 (d) 9 : 6 = 36 :

(f)

: 1 = 27 : 9

al E

(e) 12 : 1 = 48 :

du ca

(c)

tio

(a) 27 : 18 = 54 :

: 25 = 20 : 100 (h) 18 : 12 = 54 :

R eg

(g)

(i) 16 : 4 =

: 1 (j) 49 : 14 = 7 :

Word Problems Let’s Learn

Number of boys = 60 – number of girls = 60 – 35 = 25

tio

At Mermaid College, there are 60 students in Grade 5. 35 students are girls. Find the ratio of the number of boys to the number of girls in Grade 5. Express the ratio in its simplest form.

1 unit Boys

du ca

The ratio of the number of boys to the number of girls is 25 : 35.

60 students

Girls

÷5

25 : 35

÷5

al E

= 5 : 7 In its simplest form, the ratio of the number of boys to the number of girls in Grade 5 at Mermaid College is 5 : 7.

R eg

On the weekend, Riley and Halle made $63 selling lemonade. Riley sold more lemonade, so they split the money at a ratio of 4 : 3. How much money did each child make? 1 unit

Riley

$63

Halle

Riley received 4 units. 4 units 4 x $9 = $36

$63 $63 ÷ 7 = $9

Halle received 3 units. 3 units 3 x $9 = $27

Riley made $36 and Halle made $27. 74

7 units 1 unit

The ratio of the mass of Ethan's suitcase to the mass of Blake's suitcase is 7 : 3. The mass of Blake's suitcase is 24 kg. Find the total mass of both suitcases.

24 kg

24 kg 24 kg ÷ 3 = 8 kg

Ethan's suitcase = 7 units. 7 units 7 x 8 kg = 56 kg

du ca

3 units 1 unit

tio

Blake

Ethan

Total mass of the suitcases = 24 kg + 56 kg = 80 kg The total mass of both suitcases is 80 kg.

Keira has a piece of rope that is 48 meters in length. She cuts the rope into two pieces in the ratio 9 : 7. Find the length of each piece of rope.

al E

48 m

48 m 48 m ÷ 16 = 3 m

R eg

16 units 1 unit

Length of longer piece of rope = 9 units. 9 units 9 x 3 m = 27 m Length of shorter piece of rope = 7 units. 7 units 7 x 3 m = 21 m The length of the longer piece of rope is 27 m. The length of the shorter piece of rope is 21 m. 75

Let’s Practice

du ca

tio

1. Joe's Aquarium sells goldfish and turtles. On Saturday, the ratio of the number of goldfish sold to the number of turtles sold was 5 : 7. If 35 goldfish were sold on Saturday, how many turtles we sold?

On Saturday, Joe's Aquarium sold

turtles.

R eg

al E

2. Wyatt spent a total of 55 minutes doing his mathematics and science homework. He spent the first 35 minutes doing his mathematics homework and the rest of the time doing his science homework. Find the ratio of time spent doing science homework to time spent doing mathematics homework. Write the ratio in its simplest form.

In its simplest form, the ratio of time Wyatt spent doing his science homework to the time spent doing his mathematics homework : . was 76

du ca

tio

3. A farmer picks apples and oranges from his orchard. The ratio of the mass of the apples to the mass of the oranges is 5 : 2. The mass of the apples is 55 kilograms. Find the mass of the oranges.

The mass of the oranges is

kilograms.

R eg

al E

4. On a field trip, Chelsea spotted 49 butterflies. She spotted 35 orange butterflies. The rest of the butterflies she spotted were blue. Find the ratio of the number of orange butterflies to the number of blue butterflies Chelsea spotted. Express the ratio in its simplest form.

In its simplest form, the ratio of the number of orange butterflies to the : . number of blue butterflies Chelsea spotted was

At Home

There are

du ca

tio

1. The ratio of the number of boys in the schoolyard to the number of girls in the schoolyard is 4 : 7. There are 63 girls in the schoolyard. How many boys are in the schoolyard?

boys in the schoolyard.

R eg

al E

2. Sophie is using pink and purple beads to make necklaces for her friends. The ratio of the number of pink beads to the number of purple beads is 8 : 5. On Friday, Sophie used 24 more pink beads that purple beads. How many purple beads did Sophie use on Friday?

On Friday, Sophie used

purple beads.

Mr. Whyte spent

du ca

tio

3. On a weekend away, Mr. Whyte spent $320 on flights and accommodation at a ratio of 8 : 2. How much did Mr. Whyte spend on accommodation only?

on accommodation only.

R eg

al E

4. A carpenter cuts a plank of wood into two pieces in the ratio 9 : 4. The longer piece of wood is 99 centimeters in length. What is the length of the shorter piece of wood?

The shorter piece of wood is

in length.

Solve It!

du ca

tio

1. The ratio of Michelle's savings to Riley's savings is 3 : 5. In all, they have $96. Michelle received another $14 from her grandfather. What is the ratio of Michelle's savings to Riley's savings now? Write the ratio in its simplest form. Show your working.

R eg

al E

2. Blake has 60 more marbles than Keira. The ratio of the number of Blake's marbles to the number of Keira's marbles is 7 : 3. If Blake gives Keira 5 marbles, what will the ratio of Blake's marbles to Keira's marble be? Write the ratio in its simplest form. Show your working.

Looking Back 1. Write the ratio.

tio

(a)

al E

(b)

du ca

The ratio of the number of yellow roses to red roses is : .

R eg

The ratio of the number of hamsters to the total number of pets : . is (c)

The ratio of the length of the nail to the length of the screw is

. 81

2. Use the bar models to write the equivalent ratios. (a)

Ratio =

tio

(b)

1 unit

du ca

1 unit

Ratio =

3. Express each ratio in its simplest form.

(a) (b) (c) 2 : 8 35 : 45 ÷2

÷2

al E

÷5

36 : 27

÷9

4. Fill in the blanks. Express each ratio in its simplest form. (a) (b) 35 : 21 ÷

R eg

12 : 48

33 : 22

5. Write each ratio in its simplest form. (a) 40 : 10 =

(b) 25 : 15 =

(d) 54 : 9 =

(e) 24 : 60 =

(f) 21 : 9 =

: :

÷9

6. Fill in the missing numbers. Show your working. (b)

: 3 = 20 : 60

: 8 = 36 : 48 (d) 7 : 9 = 49 :

du ca

(c)

tio

(a) 27 : 81 = 3 :

al E

7. A fisherman caught 300 kilograms of prawns and crabs at a ratio of 8 : 7. How many kilograms of crabs did the fisherman catch? Show your working.

The fisherman caught

kilograms of crabs.

R eg

8. The ratio of the number of teachers to students at Broadbeach College is 2 : 7. There are 125 more students than teachers. How many teachers are at Broadbeach College? Show your working.

There are

teachers at Broadbeach College. 83

Geometry

R eg

al E

du ca

tio

Anchor Task

Types of Triangles Let’s Learn

This triangle is called an equilateral triangle. A

tio

Recall that triangles are 3-sided figures. We can group triangles based on their sides and internal angles. There are 6 main types of triangles. Some triangles can belong to more than 1 group. Let’s look at each group.

du ca

It has the following properties:

• All sides are the same length. We write: AB = BC = CA

• All internal angles are the same. We write: x = y = z

R eg

al E

The markings show sides of equal length. On this triangle, all sides are equal!

These figures are all equilateral triangles. E

T N

This triangle is called a right-angled triangle. Right-angled triangles have one right angle (90º). We mark right angles with 2 straight lines. PRQ is a right angle.

du ca

tio

These figures are all right-angled triangles.

R eg

ABC is a right angle. We can also see that BA = BC.

al E

Are the other angles in a right-angled triangle bigger or smaller than 90º?

A triangle with 2 or more equal sides is called an isosceles triangle. L

LMN =

LNM

tio

du ca

The markings show LM = LN. Triangle LMN is an isosceles triangle!

These figures are all isosceles triangles.

al E

Triangle ABC is both an isosceles and an equilateral triangle.

R eg

Triangle TUV is both a right-angled triangle and an isosceles triangle!

A triangle where all 3 sides are of different lengths is called a scalene triangle.

We use different markings on each side to show they are not equal.

tio

These figures are scalene triangles. Q

du ca

12 in

14 m

17 m

18 in

20 m

al E

8 in

R eg

3 cm

This scalene triangle is also a right-angled triangle!

5 cm

4 cm

A triangle where all the angles are less than 90º is called an acute triangle. K 79º

57º

tio

What is the sum of the angles? 44º L

du ca

These figures are acute triangles. Q

81º

11 cm

12 cm

47º

52º

al E

15 cm

This acute triangle is also an isosceles triangle!

R eg

This acute triangle is also a scalene triangle!

60º

38º

71º 71º

This acute triangle is also an equilateral triangle!

A triangle where one of the angles is greater than 90º is called an obtuse triangle. X

Which angle is greater than 90º?

du ca

These figures are obtuse triangles.

tio

33º

51º

96º

This obtuse triangle is also a scalene triangle!

30º 131º

al E

19º

R eg

98º

41º

Triangle ABC is also an isosceles triangle!

Let’s Practice

(a)

tio

Equilateral

66º

1. Use a ruler to measure the sides of the triangles. Check to classify. You may check more than 1 box.

Isosceles

(b)

Scalene

du ca

48º

66º

Equilateral

45º

Isosceles

90º

Scalene

al E

45º

(c)

60º

R eg

60º

Equilateral Isosceles Scalene

(d)

Equilateral

68º

53º

Isosceles 59º

Scalene

2. Use a protractor to measure the internal angles of the triangles. Check to classify. You may check more than 1 box. (a)

Equilateral Right-angled Acute

tio C

Obtuse

du ca

(b)

Equilateral Right-angled

Acute

al E

Obtuse

(c)

R eg

31º

(d)

Right-angled

22º

Acute Obtuse

127º M

Equilateral

Equilateral Right-angled Q

Acute Obtuse

R eg

al E

du ca

tio

(a) Right-angled triangle (b) Scalene triangle (c) Acute triangle (d) Isosceles triangle (e) Obtuse triangle

3. Use a ruler and pencil to draw and label the following figures on the dot paper.

Hands On

Work in pairs and classify each triangle.

41º

75º

73º

tio

109º

32º

62º

75º

D 59º

du ca

30º

al E

59º

30º

64º

R eg

49º

67º

75º

91º

59º

30º

45º

Triangle A B

n tio

45º

Type(s)

al E

90º

du ca

D E F

R eg

G H I

K L

At Home 1. Classify each triangle. (a)

Right-angled

tio

Scalene

(b)

du ca

Isosceles

Right-angled Scalene

al E

Isosceles

R eg

(c)

(d)

Right-angled Scalene Isosceles

2. Classify each triangle. Choose one classification per triangle.

du ca

tio

(a) (b)

al E

R eg

(e) (f)

Angles of Triangles

R eg

al E

du ca

tio

Anchor Task

Let’s Learn Recall that the sum of angles on a straight line is 180º. Line ST is a straight line. Add SVU and UVT.

The sum of the angles on a straight line is 180o.

du ca

tio

180o

126o

54o

al E

SVU + UVT = 126o + 54o = 180o

You have learned that the angles of a triangle always combine to form a straight line. The sum of the internal angles of a triangle is 180º

R eg

Find the unknown angle of each triangle. (a)

23º

ABC = 180º – 23º – 31º = 126º

31º

23o

31o

180o ABC ?

(b)

34º P 180º 34o

PQR

du ca

43o

tio

43º

PQR = 180º – 43º – 34º = 103º

al E

(c)

66º

R eg

180º

90o

c = 180º – 90º – 66º = 24º

100

66o

c ?

(d)

This is an equilateral triangle. All of the angles are the same! c

tio

Each angle is 60º.

al E

(e)

du ca

a + b + c = 180º a = b = c = 180º ÷ 3 = 60º

32º

This is an isosceles triangle. Two of the angles are the same!

R eg

180º – 32º = 148º ZXY +

XZY = 148º

ZXY = XZY = 148º ÷ 2 = 74º

1 01

Let’s Practice 1. Find the unknown angle. Show your working. (a)

tio

75º

37º D

EDF =

(b)

du ca

90º

al E

31º

(c)

30º

R eg

73º

ABC =

(d)

1 02

(e) 70º

(f)

tio

du ca

m 68º

al E

(g)

R eg

42º

(h)

TVU =

103

2. Read the clues to find the unknown angles. Show your working. (a) a is twice the size of

tio

a= b=

du ca

(b) Triangle ABC is an isosceles triangle. w is 4 times larger than y. A

al E

5 the size of 6

R eg

104

114º

p= q=

Hands On

3 cm

5 cm

R eg

al E

4 cm

du ca

tio

Work with your partner to draw these Pythagorean triples. (5, 12, 13), (6, 8, 10) and (8, 15, 17) Can you find more?

This right-angled triangle is special because the length of its sides are all whole numbers. A right-angled triangle that has whole number side lengths is called a Pythagorean triple. We can write this Pythagorean triple as (3, 4, 5) to show the side lengths are 3 cm, 4 cm and 5 cm.

105

At Home Find the unknown angle. Show your working.

(a)

tio

41º

du ca

(b)

(c)

al E

R eg

62º

39º

ABC =

(d)

31º

106

z 31º

Solve It! Find the unknown angles. (a) LMO is a triangle. LN is a straight line.

du ca

tio

67º

49º M

(b) ABC is an equilateral triangle. AD is a straight line A

al E

39º

60º

R eg

50º

40º S

PSQ =

107

Angles of Quadrilaterals

R eg

al E

du ca

tio

Anchor Task

108

du ca

The sum of the angles of a triangle is 180º. The sum of 2 triangles must be 360º!

tio

Recall that quadrilaterals are 4-sided shapes. Quadrilaterals can be divided into 2 triangles by cutting a straight line between opposite vertices.

al E

The sum of the internal angles of a quadrilateral is 360º

R eg

A square is a quadrilateral with 4 right angles. 4 x 90º = 360º!

109

We can group quadrilaterals as shown. We can use line markings to show sides of equal length and arrow markings to show sides that are parallel. quadrilaterals

• at least 1 pair of

• 2 pairs of parallel sides • opposite sides are equal

n • no parallel sides

du ca

parallel sides

parallelograms

tio

trapezoids

• opposite angles are equal

rhombi

al E

rectangles

• all angles are equal

• all sides are equal

R eg

squares

110

• all sides and angles are equal

Figure ABCD is a quadrilateral. Let’s find the missing angle. A

360o 77º

72o

77o

99o

ABC

tio

99 º

72º

= 112º

du ca

ABC = 360º – 72º – 77º– 99º

Figure LMNO is a parallelogram. Let’s find the missing angles. L

38º

38o

Opposite angles are equal. So, y = 38º

R eg

x + z = 360º – 38º – 38º = 284º x = z = 284º ÷ 2 = 142º x = 142º,

360o

al E

y = 38º,

From the model is equal to half of 360º. Can you solve the problem another way?

z = 142º

111

Figure ABCD is a trapezoid. Find

DAB. B

tio

33º D

du ca

DAB = 360º – 33º – (2 x 90º) = 327º – 180º = 147º

Figure HIJK is a parallelogram. Let’s find the missing angles. I

al E

43º

R eg

In a parallelogram, opposite angles are equal. IJK = KHI = 43º HIJ + HKJ = 360º – (2 x 43º) = 274º HIJ = HKJ = 274º ÷ 2 = 137º

112

Let’s Practice Find the unknown angle. Show your working. L 61º

(a)

tio

(b)

LMN =

73º

115º

al E

110º U

RST =

91º

R eg

(c)

du ca

110º

EFG = FGH = GHE =

113

(d)

tio

92º

BCD =

(e)

du ca

78º

al E

GHI =

(f)

R eg

60º

O PMN = MNO = NOP =

114

(g)

155º

du ca

tio

BCD =

DAB =

al E

51º S

R eg

(h)

CDA =

QRS =

STQ =

TQR =

115

Hands On

al E

du ca

tio

Work in pairs and classify each quadrilateral.

R eg

116

Quadrilateral

Type(s)

al E

du ca

tio

R eg

E F

G H I

117

At Home Find the unknown angle. Show your working. W

(a)

72º

tio

92º

104º

(b)

A 72º

du ca

WZY =

al E

112º

77º

R eg

(c)

ADC =

68º

F DEF =

118

120º

z C

(d)

tio

du ca

y= z=

(e)

al E

55º

R eg

LMN = MNO = NOL =

119

Solve It! (a) OPQR is a parallelogram. SP is a straight line. Find OPQ

118º P 20º

du ca

(b) GHIJ is a parallelogram. HJ is a straight line. Find m.

al E

56º

44º

R eg

120

tio

Looking Back 1. Find the unknown angle. Show your working. (a)

68º

tio

37º

DEF =

du ca

(b)

59º

al E

(c)

77º

30º

R eg

BCA =

(d)

121

2. Classify each figure. Choose one classification per figure.

du ca

tio

(a) (b)

al E

R eg

(e) (f)

1 22

3. Find the unknown angle. Show your working. A

B 73º

62º

(a)

tio

115º C

(b)

du ca

ADC =

44º

al E

(c)

R eg

120º

Z XYZ = YZW = ZWX =

123

Measurement

du ca

1,200 g = 1.2 kg

tio

Anchor Task

2 km 600 m = 2.6 km

R eg

al E

5 weeks =

124

Converting Measurement Units Let’s Learn

Use the chart below to help you answer the word problems.

tio

Length Metric

Customary

1 foot (ft) = 12 inches (in)

1 decimeter (dm) = 10 centimeters

1 yard (yd) = 3 feet

1 meter (m) = 100 centimeters

1 mile (mi) = 1,760 yards (yd)

du ca

1 centimeter (cm) = 10 millimeters (mm)

1 kilometer (km) = 1,000 meters

Mass

Metric

Customary

1 pound (lb) = 16 ounces (oz)

1 kilogram (kg) = 1,000 grams (g)

al E

1 ton (T) = 2,000 pounds

Volume

Metric

Customary 1 cup (c) = 8 fluid ounces (fl oz)

R eg

1 liter (l) = 1,000 milliliters (ml)

1 pint (pt) = 2 cups 1 quart (qt) = 2 pints 1 gallon (g) = 8 pints

Time 1 minute = 60 seconds 1 hour = 60 minutes 1 day = 24 hours 1 week = 7 days 1 year = 52 weeks

125

du ca

tio

Sophie and Riley made 4 gallons of lemonade to sell at a school fundraiser. They plan to sell the lemonade for $3 per pint.

What is the volume of lemonade Sophie and Riley made in pints? How much money will they raise if they sell all of their lemonade? 4 gallons = 4 x 8 pt = 32 pt

So, Sophie and Riley made 32 pints of lemonade.

al E

32 x $3 = $96

Sophie and Riley will raise $96 if they sell all of their lemonade.

R eg

Jordan's newborn kitten weighed 275 grams. It now weighs 12 times as much as it did as a newborn. How many kilograms does Jordan's kitten weigh now? 275 g x 12 = 3,300 g

3,300 g ÷ 1,000 = 3.3 kg Jordan's kitten weighs 3.3 kilograms now.

126

Mrs. Jenkins has 10 pounds of 2 cooked rice. She needs 4 ounces of rice to make a serving of Thai green

1 lb x 16 = 168 oz 2

du ca

tio

Thai green curry, how many ounces of rice will Mrs. Jenkins have left?

curry. If she makes 40 servings of

Mrs. Jenkins has 168 ounces of rice. 40 x 4 oz = 160 oz 168 oz – 160 oz = 8 oz

al E

Mrs Jenkins will have 8 ounces of rice left.

Wyatt has 1.5 liters of water in his drink bottle. After a run he drinks 525 milliliters of water. How much water is left in his drink bottle?

R eg

1.5 l = 1.5 x 1,000 ml = 1,500 ml

Before the run, there was 1,500 ml of water in Wyatt's drink bottle. 1,500 ml – 525 ml = 975 ml After his run, there was 975 ml of water in Wyatt's drink bottle.

127

Let’s Practice 1. Convert the customary units of length. Show your working. 1 ft = 2

in (b) 66 in =

yd (f) 2 yd =

al E

(e) 2 4 mi =

yd (d) 366 ft =

du ca

tio

(a) 2

2. Convert the metric units of length. Show your working.

R eg

(a) 2.6 km =

m (b) 12,345 m =

m (d) 390 cm =

(e) 6.32 m =

mm (f) 296 mm =

1 28

3. Convert the customary units of mass. Show your working.

(e) 9,000 lb =

T (d) 122 oz =

tio

oz (b) 256 oz =

du ca

(a) 17 lb =

T (f) 122 oz =

4. Convert the metric units of mass. Show your working. g (b) 3,250 g =

al E

(a) 17 kg =

R eg

(e) 10.11 kg =

g (d) 13,400 g =

g (f) 1,270 g =

129

5. Convert the customary units of volume. Show your working.

gal (d) 12 gal =

(e) 1.5 qt =

fl oz (f) 66 cu =

qt (b) 32 fl oz =

du ca

tio

(a) 15 pt =

al E

6. Convert the metric units of volume. Show your working. (a) 1,200 ml =

l (b) 620 ml =

ml (d) 9,900 ml =

(d) 0.22 l =

ml (e) 3.11 l =

R eg

130

At Home 1. Convert the customary units of measurement. Show your working. oz (b) 96 oz =

(e) 192 in =

T (d) 6.2 T =

du ca

tio

(a) 4.5 lb =

ft (f) 72 ft =

ft (h) 2.5 mi =

(i) 13 pt =

fl oz (j) 24 qt =

R eg

al E

(g) 51 yd =

(k) 10.5 gal =

pt (l) 12 fl oz =

gal

131

2. Convert the metric units of measurement. Show your working.

(e) 22,200 ml =

l (f) 1.22 l =

ml (h) 950 ml =

al E

(g) 0.13 l =

R eg

(i) 91 dm =

(k) 12.8 km =

1 32

kg (d) 15,300 g =

tio

g (b) 0.02 kg =

du ca

(a) 12.1 kg =

ml (j) 92.5 cm =

m (l) 0.41 m =

Word Problems

al E

du ca

tio

1. Ethan fills a watering can with 3 gallons of water. After watering his garden, there is 2 quarts of water remaining in the watering can. How many quarts of water did Ethan use?

Let’s Practice

R eg

2. A bottle contains 32 fluid ounces of ketchup. How many pints of ketchup are in 4 bottles?

133

al E

du ca

tio

3. Mr. Rogers has 10 meters of wire. He cuts the wire into 8 pieces of equal length to repair his fence. He has 3.6 meters of wire left. Find the length of each piece of wire he cut in centimeters.

4. Sophie ran 9 laps of the athletics track at school. She ran a total distance of 4

1 kilometers. 2

R eg

How many meters is 1 lap of the track?

1 34

du ca

tio

5. Jordan and Wyatt make 12 liters of lemonade to sell at the local market. Each cup of lemonade has 300 milliliters of lemonade. At the end of the day, they have 1.2 liters of lemonade left. How many cups of lemonade did they sell?

R eg

al E

6. A farmer has 24 pounds of strawberries. He packs the strawberries into punnets weighing 12 ounces each. How many punnets of strawberries can the farmer make?

135

du ca

tio

7. A bag of potatoes weighs 64 ounces. Mr. Whyte buys 4 bags of potatoes. How many pounds of potatoes did Mr. Whyte buy?

R eg

al E

8. Wyatt is giving away one balloon to each friend who attends his birthday party. On each balloon he ties 2.5 feet of string. How many yards of string will Wyatt need if 12 friends attend his birthday party?

136

du ca

tio

9. The distance from the campsite to the waterfall is 3 miles. Halle leaves the campsite and hikes for 2.5 miles in the direction of the waterfall. How many more yards does Halle have to hike to reach the waterfall?

R eg

al E

10. Railway workers install 58 meters of train track per day. It takes them 47 days to install a track between Beach Station and Sunnybank Station. Find the distance between the stations in kilometres and meters.

137

du ca

tio

11. Michelle takes 6 minutes and 39 seconds to swim 6 laps of a swimming pool. How long did Michelle swim in seconds?

R eg

al E

12. Halle buys ribbon to tie bows on some gifts. She uses 8 lengths of ribbon that are each 36 cm in length and has 22 cm of ribbon left. Find the total length of ribbon she bought in meters.

138

du ca

tio

13. Chelsea is on a flight from Beijing to Dubai. The total flight time is 7 hours and 20 minutes. The plane has been flying for 5 hours and 25 minutes. How many more minutes does the plane need to fly before Chelsea arrives in Dubai?

R eg

al E

14. Sophie and Halle made 4.5 liters of fruit punch to sell at the town fair. They plan to sell the fruit punch for $2.5 per pint. How much money will Sophie and Halle make if they sell all of the fruit punch?

139

R eg

al E

tio

du ca

1. The desks in an exam hall are to be spaced 4 decimeters apart. Each desk has a width of 7 decimeters. The exam hall is a rectangular-shaped room with a width of 14 meters. How many desks can be placed across the room? Note the desks on the sides can touch the wall.

Solve It!

140

R eg

al E

du ca

tio

2. A bathtub contains 120 liters of water. Ethan pulls the plug and the water drains from the bathtub at a rate of 250 ml per second. If Ethan pulled the plug at 6:00 p.m., what at what time will the bathtub drain completely?

1 41

du ca

tio

1. Keira is building a square picture frame with a side length of 15 inches. She cuts the sides of the frame from a piece of wood that is 6 feet long. What is the length of the wood leftover? Express your answer in inches.

At Home

R eg

al E

2. Danny the bricklayer loads 9,000 pounds of bricks onto his truck. He uses 2,000 pounds of bricks to build a retaining wall. What is the mass of the bricks left on his truck? Express your answer in tons.

142

du ca

tio

3. A dripping tap leaks 1 milliliter of water every second. Find the volume of water leaked in 2 hours in liters and milliliters.

R eg

al E

4. Blake is on summer break for 6 weeks. On the first day of summer break, he goes on a diving trip with his father for 12 days. He then stays at his grandmother's house for 2 weeks. How long before Blake goes back to school? Express your answer in weeks and days.

143

du ca

tio

5. A paint store buys a large, 44-gallon drum of paint. It sells the paint in 2-pint tins. In 1 day, the store sells 170 tins of paint. Find the volume of paint left in the drum in pints.

R eg

al E

6. There is a dance performance at the local department store. There are 3 shows of the same duration with a 15-minute break between each show. The first show starts at midday. The last show finished at 4:00 p.m. What is the length of 1 show in hours and minutes?

144

7. It takes Riley 3 minutes to walk 250 meters. How far does Riley walk in 1

1 hours? 2

du ca

tio

Express your answer in kilometers.

R eg

al E

8. A painter needs 4 gallons of paint to complete painting a living room. He finds three old tins of paint. The first tin contains 2 quarts of paint. The second tin contains 5 pints of paint. The third tin contains 1 gallon of paint. Does the painter have enough paint to finish painting the living room? If no, how much more paint will he need? If yes, how much paint will be left over? Express you answer in pints.

145

Solve It! A computer store sells data cables in 2 price plans. Coils of cable please Natcha

4 km of data cable. 5

R eg

al E

du ca

1. Find the cheaper plan for buying

tio

Plan 2: 40¢ per meter for any length.

Plan 1: 50¢ per meter for the first 500 meters, then 20¢ per meter thereafter.

2. How much money is saved on the cheaper plan?

146

du ca

tio

3. At what length are both Plan 1 and Plan 2 the same price?

R eg

al E

4. The store is running a sale for Plan 2 – 50% off for purchases over 5 km. Which is the cheaper plan for buying a 5-kilometer cable?

147

Looking Back 1. Convert the units of measurement. Show your working. m (b) 12,345 m =

mm (f) 56 fl oz =

al E

(e) 8.23 m =

pt (d) 6.5 T =

du ca

tio

(a) 12.5 km =

2. A baker bakes 20 baguettes that are each 1 foot and 1 inch in length. He cuts each baguette into slices that are 3

R eg

baguettes slices can the baker make?

1 48

1 inch in length. How many 4

du ca

tio

3. Jordan is jogging around the school running track to train for a fun run. The distance around the track is 600 meters. Jordon completes 4 laps of the running track. How many more laps does he need to run to cover a total distance of 6 kilometers?

R eg

al E

4. The school football coach prepared 3 gallons of sports drink for his team of 14 players. After the game, each player drank 3 cups of sports drink. How many pints of sports drink are remaining?

149

Volume

Volume and Unit Cubes

R eg

al E

du ca

tio

Anchor Task

150

Let’s Learn Compare the tennis ball and the soccer ball. Which takes up more space?

du ca

tio

The soccer ball is bigger and takes up more space.

The soccer ball takes up more space than the tennis ball. The amount of space an object takes up is its volume. The volume of the soccer ball is greater than the volume of the tennis ball.

al E

Compare the volume of the boxes.

Box A

Box B

R eg

The volume of Box A is greater than the volume of Box B. The volume of Box B is smaller than the volume of Box A. We compared the volumes of the boxes. How can we measure the volume?

1 51

tio

The solids below are rectangular prisms. Which prism has the greater volume?

Prism A

Prism B

du ca

To find out which prism has the greater volume, we can divide them into unit cubes. Each edge in a unit cube is 1 unit in length. A unit cube has a volume of 1 cubic unit. All sides of the cube have the same length.

Prism B can be divided into 6 unit cubes. It has a volume of 6 cubic units.

R eg

al E

Prism A can be divided into 4 unit cubes. It has a volume of 4 cubic units.

The volume of Prism B is greater than the volume of Prism A.

152

Let's find the volume of the prism.

The prism has 3 rows of 4 unit cubes.

4 unit cubes

tio

4 unit cubes

The prism is made up of 12 unit cubes. Its volume is 12 cubic units.

du ca

Let's find the volume of this Solid A. 1 unit

2 unit cubes

Divide the solid into layers and count the unit cubes in each layer.

6 unit cubes

al E

Solid A

Solid A is made up of 9 unit cubes. Its volume is 9 cubic units. Let's find the volume of this Solid B.

R eg

1 unit cubes

Solid B

Can you divide the solid into layers another way?

3 unit cubes 5 unit cubes

9 unit cubes

Solid B is made up of 18 unit cubes. Its volume is 18 cubic units.

1 53

Solid E

Solid F

al E

R eg

Solid G is made up of 9 unit cubes. Its volume is 9 cubic units.

Solid I

Solid I is made up of 14 unit cubes. Its volume is 14 cubic units.

154

Solid F is made up of 27 unit cubes. Its volume is 27 cubic units.

du ca

Solid E is made up of 15 unit cubes. Its volume is 15 cubic units.

Solid G

tio

Let's find the volume of these solids.

Solid H

Solid H is made up of 14 unit cubes. Its volume is 14 cubic units.

Solid J

Solid J is made up of 31 unit cubes. Its volume is 31 cubic units.

Let’s Practice 1. Compare the volumes of the objects. Check the object with the greater volume.

al E

(b)

du ca

tio

(a)

R eg

(c)

(d)

155

2. Compare the volumes of the rectangular prisms. Fill in the blanks. Circle the prism with the greater volume.

Prism B

tio

Prism A

cubic units.

Prism B has a volume of

cubic units.

al E

Prism C

du ca

Prism A has a volume of

(b)

Prism D

Prism C has a volume of

cubic units.

Prism D has a volume of

cubic units.

R eg

(c)

Prism E

Prism F

Prism E has a volume of

cubic units.

Prism F has a volume of

cubic units.

156

(a)

Prism G

tio

(d)

Prism H

cubic units.

du ca

Prism G has a volume of Prism H has a volume of

al E

(e)

cubic units.

Prism I

R eg

Prism I has a volume of

Prism J has a volume of

Prism J

cubic units. cubic units.

1 57

Hands On

tio

1. With a classmate, use unit cubes to build each rectangular prism. Record the volume of each prism in the table.

Prism P

Prism R

du ca

Prism Q

Prism T

Prism S

Volume (cubic units)

al E

Prism P

Q R

R eg

Prism U

2. Take the cuboid apart and build another cuboid of the same volume. Did the volume of the cuboid change when you re-built it? Explain your answer.

158

Let’s Practice 1. Compare the volumes of the solids. Fill in the blanks. Circle the solid with the smaller volume.

Solid B

du ca

Solid A

tio

(a)

Solid A has a volume of

cubic units.

Solid B has a volume of

cubic units.

al E

(b)

Solid C

Solid D

Solid C has a volume of

cubic units.

Solid D has a volume of

cubic units.

R eg

(c)

Solid E

Solid F

Solid E has a volume of

cubic units.

Solid F has a volume of

cubic units.

159

Solid H

tio

Solid G

cubic units.

Solid H has a volume of

cubic units.

al E

Solid I

du ca

Solid G has a volume of

(e)

Solid I has a volume of

Solid J has a volume of

Solid J

cubic units. cubic units.

R eg

(f)

Solid K

Solid L

Solid K has a volume of

cubic units.

Solid L has a volume of

cubic units.

160

(d)

R eg

al E

du ca

tio

2. Match the solids with the same volume.

161

Hands On

Solid V

Solid X

Solid Y

Solid Z

Volume (cubic units)

al E

Solid

Solid W

du ca

Solid U

tio

1. With a classmate, use unit cubes to build each solid. Record the volume of each solid in the table below.

R eg

X Y Z

2. Arrange the solids in order from the least volume to the greatest volume.

least volume

1 62

greatest volume

At Home 1. Circle the object with the smaller volume.

al E

(b)

du ca

tio

(a)

R eg

2. Arrange the balls in order from the greatest volume to the smallest volume.

bowling ball

greatest volume

beach ball

basketball

smallest volume

163

3. Find the volume of each prism. Circle the prism with the greater volume. Circle both solids if they have the same volume.

cubic units

du ca

cubic units

tio

(a)

al E

(b)

cubic units

R eg

(c)

cubic units

164

cubic units

4. Find the volume of each solid. Circle the solid with the smaller volume. Circle both solids if they have the same volume.

cubic units

du ca

cubic units

tio

(a)

al E

(b)

cubic units

R eg

(c)

cubic units

165

Solve It!

Solid W

du ca

tio

1. Compare the solids.

Solid X

(a) What is the volume of Solid W?

(b) How many unit cubes need to be added to Solid W to make Solid X?

R eg

al E

2. Compare the solids.

Solid Y

Solid Z

(a) What is the volume of Solid Y? (b) How many unit cubes need to be taken away from Solid Y to make Solid Z?

166

tio

3. Halle is using unit cubes to build stairs.

du ca

(a) What is the volume of the stairs Halle has built so far?

(b) Halle wants to build the stairs to a height of 6 steps. Complete the table to show how many unit cubes Halle will need to add for each step. Volume (cubic units)

al E

Steps 1

3 4

R eg

5 6

(d) What would be the volume of a set of stairs with 10 steps?

1 67

Volume of Rectangular Prisms

R eg

al E

du ca

tio

Anchor Task

168

Let’s Learn Riley makes a rectangular prism from 1-centimeter cubes.

1 cm

du ca

2 units = 2 cm

tio

1 cm

Each cube has a side length of 1 cm. The volume of each cube is 1 cubic cm. We write: 1 cm3 We say: 1 cubic centimeter

3 units = 3 cm

al E

4 units = 4 cm

2 units = 2 cm

4 x 3 x 2 = 24. The rectangular prism has a volume of 24 cm3.

3 units = 3 cm

R eg

4 units = 4 cm

Volume of a rectangular prism = length x width x height = 4 cm x 3 cm x 2 cm = 24 cm3

V=lxwxh

We can use this formula to find the volume of any rectangular prism!

169

3 cm

Length = 12 cm Width = 8 cm Height = 3 cm

tio

8 cm 12 cm

6 in

Length = 6 in Width = 6 in Height = 6 in

al E

6 in

A rectangular prism where the sides are of equal length is called cube.

du ca

6 in

8 ft

R eg

8 ft

20 ft

12 m

170

Let's identify the length, width and height of these rectangular prisms. Make sure to include the correct units.

Length = 20 ft Width = 8 ft Height = 8 ft

4m 9m

Length = 12 m Width = 9 m Height = 4 m

Find the volume of the rectangular prisms. (a)

Volume = l x w x h = 12 x 5 x 4 = 60 x 4 = 240 ft 3

4 ft

5 ft

tio

12 ft

(b) 6 cm

du ca

Volume = l x w x h = 8 x 7 x 6 = 56 x 6 = 336 cm3

7 cm 8 cm

al E

(c)

5 6 6 3 3 6

R eg

Volume = l x w x h = 8 x 8 x 8 = 64 x 8 = 512 m3

6 4 x 8 5 1 2

(d)

23 in

7 in

Volume = l x w x h 9 in = 23 x 9 x 7 = 1,449 in3

171

Let’s Practice 1. Find the length, width and height of the rectangular prisms. (a)

Width =

tio

Height =

du ca

(b)

Length =

Width = Height =

(c)

al E

Length =

Width = Height =

R eg

(d)

Length = Width = Height =

172

Length =

(e) 11 cm

Length = Width =

9 cm

Height =

tio

14 cm

du ca

(f) 9 in

Length = Width =

5 in

13 in

al E

(g)

Height =

12 m

10 m

R eg

28 m

Length = Width = Height =

(h)

15 ft

Length = Width =

12 ft 32 ft

Height =

173

2. Find the volume of the rectangular prisms. Show your working. (a)

tio

14 cm

12 cm 16 cm

du ca

Volume =

(b)

al E

25 cm

12 cm

Volume =

7 cm

R eg

(c)

10 in

12 in

19 in

Volume =

174

(d)

tio

16 m

Volume =

du ca

16 m

(e)

9 cm

al E

14 cm

11 cm

Volume =

R eg

(f)

9 ft

7 ft

Volume =

3 ft

175

At Home 1. Complete the table. Show your working. 1 cm

Prism B 1 cm

tio

6 cm

du ca

11 cm

21 cm

Prism C

Prism A

6 in

al E

Prism D

10 in

17 in

R eg

22 ft

1 in 20 cm

1 in

10 ft

1 ft

Prism E

7 cm 4 cm Prism F

176

n tio du ca al E

Prism

Length

Width

Height

Volume

R eg

A B

D E F

177

2. Find the volume of the rectangular prisms. Show your working. (a)

tio

du ca

Volume =

(b)

al E

16 in

Volume =

R eg

(c)

10 ft

8 ft

9 ft

Volume =

178

(d)

10 m

tio

du ca

Volume = (e)

7 cm

8 cm

al E

20 cm

Volume =

R eg

(f)

10 m

Volume =

179

Solve It! Wyatt cut some wood to make a rectangular shaped block.

tio

• The volume of the block is 384 cm3. • The length of the block is 3 times the height. • The width of the block is 2 times the height. • All sides have whole number lengths in centimeters.

Width

al E

Length

du ca

(a) Read the clues to find the dimensions of the block.

Height

R eg

(b) Wyatt makes a second block where the length and width are double that of his original block. Find the volume of the second block. Write the volume as a multiple of the first block's volume.

180

Volume and Capacity Let’s Learn

du ca

tio

A cube-shaped container of side length 10 cm is completely filled with water. Let's find the volume of the water.

10 cm

10 cm 10 cm

Recall that 1 cm3 = 1 ml 1,000 cm3 = 1 l The container has 1 liter of water.

al E

Volume = 10 x 10 x 10 = 100 x 10 = 1,000 cm3

R eg

The amount of liquid a container can hold when filled is called capacity.

Capacity: 1,000 cm3 Volume: 0 cm3

The container has a capacity of 1 liter. When it is empty, its volume is 0.

Capacity: 1,000 cm3 Volume: 1,000 cm3

181

The water is poured into a rectangular container that has a length of 10 cm, width of 10 cm and height of 20 cm.

How much more water do we need to add to fill this container?

10 cm

10 cm 10 cm

du ca

10 cm

tio

20 cm

The volume of liquid in the container is 1,000 cm3. The capacity of the container is 2,000 cm3.

R eg

al E

A tank contains water to a height of 4 cm. Find the volume and capacity of the tank in liters and milliliters.

12 cm

4 cm

15 cm

Capacity = 15 x 12 x 10 = 1,800 cm3 = 1,000 cm3 + 800 cm3 = 1 l 800 ml

182

10 cm

Volume = 15 x 12 x 4 = 720 cm3 = 720 ml

Let’s Practice Find the capacity and volume of each rectangular tank in liters and milliliters.

(a)

5 cm

du ca

2 cm

tio

5 cm

13 cm

R eg

(b)

Volume =

al E

Capacity =

6 cm

Capacity =

9 cm

20 cm

30 cm

Volume =

183

(c)

9 cm

Capacity =

Volume =

al E

(d)

du ca

36 cm

4 cm

R eg

45 cm

Capacity =

184

tio

13 cm

6 cm

Volume =

8 cm

15 cm

At Home Find the capacity and volume of each rectangular tank.

(a)

tio

12 cm

14 cm

du ca

7 cm 40 cm

4 cm

R eg

(b)

Volume =

al E

Capacity =

9 cm

10 cm

45 cm

Volume =

185

Word Problems

Volume = 5,6003

du ca

tio

The base of a rectangular prism measures 20 cm by 14 cm. If the volume of the prism is 5,600 cm3, find its height.

Let’s Learn

14 cm 20 cm

al E

Use the formula V = l x w x h and work backwards!

R eg

Volume = l x w x h 5,600 = 20 x 14 x height height = 5,600 ÷ (20 x 14) = 5,600 ÷ 280 = 560 ÷ 28 = 20 cm

The height of the rectangular prism is 20 cm.

186

Both numbers are multiples of 10. We can simplify! 5,600 ÷ 280

A rectangular tank measuring 30 cm by 30 cm by 12 cm is filled

1 with 3

1 3

30 cm

du ca

30 cm

tio

12 cm

water. Find the volume of the water. Give your answer in liters.

First, find the height of the water. 1 of 12 = 12 ÷ 3 3 = 4

al E

The height of the water is 4 cm. Now, let's find the volume. Volume = l x w x h

= 30 x 30 x 4

600 ÷ 1,000 = 0.6 3 liters + 0.6 liters = 3.6 liters

R eg

= 3,600 cm 3

3,600 cm3 = 3.6 l

1 87

Container B

tio

Container A

28 cm

40 cm

du ca

40 cm 40 cm

Two identical rectangular containers have a square base with a side length 4 of 40 cm. They are filled with water to a height of 28 cm. A jug is used to 5 move water from Container A to an identical Container B.

40 cm

Jordan filled the jug with water and poured it into Container B five times.

Container B

R eg

al E

Container A

6 cm

The water level of the Container B is now 6 cm from the brim. Find the capacity of the jug. First, let's find the volume of water that is originally in Container A.

Volume = 40 x 40 x 28 = 44,800 cm3

Container A originally contained 44,800 cm3 of water.

1 88

Next, find the height of the container.

du ca

Find the volume of water in Container B.

tio

5 of the height = 5 x 7 = 35 cm 5

4 of the height = 28 cm 5 1 of the height = 28 ÷ 4 5 = 7 cm

Height of water in Container B is 35 cm – 6 cm = 29 cm Volume = 40 x 40 x 29 = 46,400 cm3

al E

Find the volume of water added. 46,400 – 44,800 = 1,600 cm3

The jug was filled 5 times. Divide by 5 to find the capacity of the jug. 1,600 ÷ 5 = 320

R eg

The capacity of the jug is 320 cm3.

Can you find the answer another way?

189

Let’s Practice

du ca

tio

1. The volume of a cuboid is 2,025 cm3. Its height is 25 cm. What is the area of its base? Use the space provided to draw a diagram and show your working.

R eg

al E

2. The base of a rectangular prism measures 20 cm by 25 cm. If the volume of the cuboid is 7,500 cm3, find its height. Use the space provided to draw a diagram and show your working.

190

du ca

tio

3. Ethan has two identical coolers. Each one measures 50 cm long, 40 cm wide and 60 cm high. What is the total volume of the two coolers in liters? Use the space provided to draw a diagram and show your working.

R eg

al E

4. Peter has two identical bookshelves stacked one on top of the other. The two identical bookshelves hold 72,000 cm3. If the area of the base is 1,200 cm2, how tall is each bookshelf? Use the space provided to draw a diagram and show your working.

191

R eg

al E

du ca

tio

5. A rectangular tank, 40 cm long and 30 cm wide, was filled with water to a depth of 15 cm. When Halle poured out some water from the tank, the water level dropped to 9 cm. How many liters of water did Halle pour out? Use the space provided to draw a diagram and show your working.

192

R eg

al E

du ca

tio

6. A rectangular container is 16 cm wide and 30 cm in length. It is completely filled with water. Half of the water in the container is poured out. How much water is left in the container? Use the space provided to draw a diagram and show your working.

193

R eg

al E

du ca

tio

7. A fish tank measures 45 cm by 35 cm by 30 cm and holds 25 l of 2 water. Sophie wants to fill the tank to of its height. How much 3 more water is needed? Give your answer in liters. Use the space provided to draw a diagram and show your working.

194

At Home

du ca

tio

1. The volume of a box is 3,600 cm3. Its width is 10 cm. Its length is twice its width. Find its height. Use the space provided to draw a diagram and show your working.

R eg

al E

2. Riley built a house for her two hamsters out of two boxes. One box measures 16 cm by 10 cm by 12 cm and the other measures 8 cm by 12 cm by 6 cm. What is the total volume of the hamsters' house? Use the space provided to draw a diagram and show your working.

195

R eg

al E

du ca

tio

3. A rectangular tank has a base of 80 cm by 50 cm and a height of 30 cm. It contains 100 liters of water. Find the height of the water level in the tank. Use the space provided to draw a diagram and show your working.

196

4. The figure shows a solid that is made up of 4 cubes of edge 2 cm.

R eg

al E

du ca

tio

(a) Find the volume of the solid. (b) If the solid is painted white, find the total area which is painted white. Use the space provided to draw a diagram and show your working.

197

5. The figure shows a cuboid consisting of 12 cubes. The area of the shaded face is 100 cm2.

R eg

al E

du ca

tio

(a) Find the volume of each cube. (b) Find the volume of the cuboid. Use the space provided to draw a diagram and show your working.

1 98

Looking Back 1. Find the volume of each prism. Circle the prism with the greater volume. Circle both solids if they have the same volume.

du ca

tio

(a)

cubic units

al E

(b)

cubic units

R eg

(c)

cubic units

199

2. Find the volume of each solid. Circle the solid with the smaller volume. Circle both solids if they have the same volume.

cubic units

du ca

cubic units

tio

(a)

al E

(b)

cubic units

R eg

(c)

cubic units

2 00

cubic units

3. Find the volume of the rectangular prisms. Show your working. (a)

tio

12 cm

10 cm

(b)

Volume =

du ca

10 cm

al E

6 ft

7 ft

8 ft

R eg

(c)

Volume =

5 in 10 in

20 in

Volume =

2 01

(d)

tio

12 m

Volume =

du ca

12 m

(e)

9 cm

al E

10 cm

13 cm

Volume =

R eg

(f)

12 m

8m 5m

202

Volume =

4. Find the capacity and volume of each rectangular tank in liters and milliliters.

4 cm

5 cm

Volume =

al E

Capacity =

du ca

10 cm

tio

8 cm

(a)

R eg

(b)

2 cm

Capacity =

6 cm

4 cm 20 cm

Volume =

203

(c)

6 cm

tio

12 cm

Volume =

al E

(d)

du ca

15 cm

Capacity =

R eg

10 cm

Capacity =

2 04

8 cm

15 cm

25 cm

Volume =

du ca

tio

5. The volume of a rectangular container is 512 cm3. It width is two times its height. Its length is two times its width. Find its height. Use the space provided to draw a diagram and show your working.

al E

6. X and Y are two rectangular containers. The base area of X is 40 cm2 and the base Y has dimensions as shown. X contained 1,000 cm3 of water and Y was empty.

20 cm Y

8 cm

R eg

(a) What was the height of the water level at X? (b) Blake poured some water from X into Y without spilling. After that, the height of the water at level of X was the same as that level of Y. How much water did Blake pour into Y?

205

10 Data and Graphs Line Plots

tio

Anchor Task Hours Spent Reading Per Night (min) 30

R eg

al E

du ca

206

Let’s Learn

A line plot, sometimes called a dot plot, is a way to show how frequently data occurs along a number line. An 'X' or a dot is placed above a number each time it occurs in the data set. Riley emptied her pencil case on her desk and measured the length of each

tio

1 object to the nearest inch. She recorded the data in the table below. 4 Length of Pencil Case Objects (in) 1 2

3 4

1 4

1 2

du ca

3 4

Let's represent the data in a line plot and interpret the data.

al E

Length of Pencil Case Objects

R eg

7 7

7 7 7 7 7 7

1 2

3 4

7 7

4 inches

1 4

7 7 7 4

1 2

7 7 4

3 4

207

From the line plot, we can see that most objects are 3

3 inches in length. 4

We can see that 8 objects are shorter than 4 inches. 1 inches in length. 2 3 The longest objects are 4 inches in length. 4

tio

The shortest objects are 3

3 1 3 2 4 - 3 = 4 - 3 4 2 4 4 1 = 1 4

du ca

What is the difference between the shortest and longest object?

1 The difference between the shortest and longest object is 1 inches. 4

al E

Find the total length of the objects that are 4

1 inches in length. 2

1 1 1 1 + 4 + 4 = 13 2 2 2 2

R eg

The total length of the 4

208

1 1 inch objects is 13 inches. 2 2 What are some other ways we can interpret the data in the line plot?

Let’s Practice 1. The line plot shows the hours of exercise each student did in 1 week.

1 2

du ca

tio

Hours of Exercise

Hours

1 (a) How many students exercised for 3 hours? 2

(b) How many students exercised for less than 3 hours?

al E

R eg

1 3 hours exercising? 2

(e) What was the total time the students spent exercising?

209

2. The line plot shows the amount of fruit Mr. Whyte sold at his grocery store over a 15-day period.

1 2

7 7 23

1 2

tio

7 7

7 7 7

du ca

7 7 7

7 7 7 7

Fruit Sold

Pounds

1 2

(a) On how many days did Mr. Whyte sell 24 pounds of fruit? (b) What was the least amount of fruit sold on any day?

al E

(d) On how many days did Mr. Whyte sell more than 24 pounds of fruit?

R eg

(e) On how many days did Mr. Whyte sell more than 22 pounds of fruit?

(f) What is the total mass of fruit Mr. Whyte sold over the 15-day period?

Mr. Whyte sold

21 0

pounds of fruit over the 15-day period.

3. The line plot shows the lengths of fishing hooks in Ethan's tackle box.

1 2

3 4

7 1

1 4

Inches

7 7

tio

7 7

7 7 7

1 2

du ca

7 7 7

7 7 7 7

Fishing Hook Sizes

3 4

(a) How many hooks are 1 inch in length?

(b) How many hooks are longer than 1 inch?

al E

1 (c) What is the total length of the -inch hooks? 2

R eg

(d) What is the total length of the hooks shorter than 1 inch?

(e) What is the difference in the length of the longest and shortest hooks?

211

4. The line plot shows the amount of sports drink consumed by the players in a football team during a match. Represent the data in a line plot. Sports Drink Consumed (pints) 1

1 4

1 2

3 4

du ca

tio

1 2

al E

5. The line plot shows the length of nails and screws in Riley's dad's toolbox. Represent the data in a line plot. Length of Nails and Screws (in.)

3 4

R eg

212

1 4

1 2

3 4

6. The students in class 5A measured different amounts of water into 12 identical beakers. The amount of water in each beaker is shown below.

3 cup 4

1 cup 2

3 cup 4

1 cup 2

3 cup 4

1 cup 4

3 cup 4

tio

1 cup 2

1 cup 4

du ca

1 cup 2

R eg

al E

(a) Represent the amount of water in each beaker in a line plot.

1 (b) How many beakers contain cup of water? 2 1 (c) How many beakers contain more than cup of water? 4 1 (d) What is the total volume of water in the beakers with cup of water? 2

cups

213

1 (e) What is the total volume of water in the beakers with cup of water? 4

cups

du ca

tio

3 (f) What is the total volume of water in the beakers with cup of water? 4

cups

(g) What is the total volume of water in the beakers with more than

al E

1 cup of water? 4

cups

R eg

(h) Find the amount of water each beaker would contain if the total amount in all the beakers were redistributed equally.

If redistributed equally, each beaker would contain of water.

21 4

cups

1 1 3 1 1 1 1 1 in., in., in., in., in., in., in., in., 4 8 4 2 2 2 8 4 1 1 1 3 1 1 1 3 in., in., in., in., in., in., in., in., 4 4 8 4 2 2 2 8

du ca

tio

(a) Represent the lengths of strings in a line plot.

7. Sophie uses strings of different lengths to make bracelets for her friends. The lengths of the strings are shown below.

1 inch long? 2

al E

(b) How many strings are

3 inch long? 4 1 (d) How many strings are longer than inch? 4 (c) How many strings are shorter than

R eg

(e) What is the total length of the strings?

The total length of the strings is

215

Hands On

du ca

tio

1. Place your hand on a sheet of paper. Stretch your fingers apart as far as you can. Use a pencil to mark the tip of your thumb and the tip of your little finger. Use a ruler to measure your hand span to the nearest one eighth of an inch.

2. Share and collect data on the hand spans of your classmates. Record the data in the table below. Hand Span Lengths

Hand Span (in.)

R eg

al E

Name

21 6

Name

Hand Span (in.)

du ca

tio

3. In the space below, create a line plot to represent the data you collected.

(a) What is the most common hand span width? (b) What is the least common hand span width?

R eg

al E

in.

(d) What is the combined length of the two longest hand spans?

in.

217

At Home 1. The line plot shows the amount of sugar in coffees served at the Espresso Express Cafe on Tuesday.

1 4

1 2

du ca

tio

Sugar in Coffees

3 4

1 4

1 2

3 4

Teaspoons

(a) What is the most amount of sugar in a coffee?

al E

(b) What is the difference between the least and most amount of sugar in a coffee?

tsp.

R eg

tsp.

218

2. The table shows the daily amount of flour used at Mrs. William's bakery for 2 weeks.

7 7

1 2

du ca

Kilograms

7 7

7 7 7

tio

7 7 7 7

Fruit Sold

(a) On how many days was more than 9 kilograms of flour used?

al E

(b) On how many days was less than 10 kilograms of flour used?

R eg

(d) What is the total mass of the flour used?

219

3 1 1 1 1 7 3 3 in., in., in., in., in., in., in., in., 4 4 4 2 2 8 8 4 3 1 3 5 5 1 1 3 in., in., in., in., in., in., in., in., 8 2 8 8 8 2 2 8

du ca

tio

(a) Represent the lengths of the insects in a line plot.

3. Blake measured the lengths of different insects he spotted in his garden. The length of each insect is shown below.

1 inch long? 2

al E

(b) How many insects are

1 inch? 2

R eg

(d) What is the total length of all insects longer than

(e) What is the total length of all insects shorter than

1 inch? 2

in. 1 inch? 2

in.

22 0

3 7 1 1 3 5 1 3 1 3 lb, lb, lb, lb, lb, lb, lb, lb, lb, lb 4 8 2 8 4 8 2 8 8 4 7 1 1 3 1 5 1 7 3 1 lb, lb, lb, lb, lb, lb, lb, lb, lb, lb 8 8 2 8 4 8 2 8 8 8

du ca

tio

(a) Represent the patties sold in a line plot.

4. Bob's Burger Shack sells burgers with patties of different weights. The weights of the patties sold during lunch time are shown below.

1 pound or heavier? 2 3 (c) How many patties are lighter than pound? 4 1 (d) What is the total weight of the pound patties? lb 2 (e) If Bob's Burger Shack used the same amount of meat to make half pound patties, how many patties could they make?

R eg

al E

(b) How many patties are

221

Hands On 1. Collect 16 objects from around your home that are between 1 and 2 inches in length. Write the names of the objects in the table below.

Objects at Home Length (in.)

Object

Length (in.)

R eg

al E

du ca

Object

tio

2. Use a ruler to measure the length of each object to the nearest inch. Write the length of each object in the table.

22 2

du ca

tio

3. In the space below, create a line plot to represent the data you collected.

(a) What is the most common object length? (b) What is the least common object length?

R eg

al E

in.

(d) What is the combined length of the four longest objects?

in.

223

Graphing Equations

tio

Anchor Task

du ca

11 10 9 8

al E

7 6

R eg

3 2 1

224

Let’s Learn

A coordinate grid can be used to locate points on a plane. A coordinate grid is made up of a horizontal number line, called the x-axis and a vertical line called the y-axis.

tio

Each location on a coordinate grid can be described using an ordered pair of numbers. The first number in an ordered pair is the x-coordinate and is located on the x-axis. The second number in an ordered pair is the y-coordinate and is located on the y-axis.

du ca

Two points are marked on the coordinated grid below. The ordered pair for A is (2, 8). The ordered pair for B is (7, 4). y 10 9

A (2, 8)

al E

7 6

B (7, 4)

R eg

2 1

Mark two more points on the grid. Use ordered pairs to describe the location to a friend.

225

You can use ordered pairs and a coordinate grid to describe locations. 10

tio

8 7

du ca

6 5 4

2 1

al E

R eg

22 6

(1, 8)

(8, 9)

(4, 6)

(2, 2)

(7, 4)

(7, 1)

You can use a coordinate plane to graph an equation.

Kiera wants to buy cupcakes from her local bakery. The cupcakes cost $2 each. The table below shows the total cost of buying different amounts of cupcakes. Cost of Cupcakes 1

Cost ($)

tio

Number of Cupcakes

du ca

The data in the table can be plotted on a coordinate plane using the ordered pairs (1, 2), (2, 4), (3, 6), (4, 8), (5, 10).

Drawing a line through the plots forms a straight line graph. The straight line graph represents an equation that shows how the x-axis and y-axis are connected. The equation is y = 2x. 11

al E

10 9 8

R eg

Cost ($)

5 4

3 2 1

3 4 5 6 7 8 Number of Cupcakes

227

A hardware store sells wood in 3-meter lengths for $12 each. The table shows the total cost of buying a different number of lengths of wood. Cost of Wood 3

Cost ($)

Length of Wood (m)

tio

Let's use the equation y = 4x to draw a straight line graph on a coordinate plane.

du ca

132 120 108 96

al E

Cost ($)

60 48

R eg

9 12 15 18 21 Length of Wood (m)

Blake's father needs 21 meters of wood to build a new fence. Using the straight line graph, we can see that the total cost of 21 meters of wood is $84.

228

24 27 30 How much does 27 m of wood cost?

Let’s Practice se the ordered pairs to draw and color circles at the correct place on U the coordinate plane.

tio

9 8

du ca

7 6 5

al E

4 3 2 1

R eg

(1, 5)

(2, 9)

(6, 6)

(8, 8)

(9, 5)

(8, 1)

229

2. The coordinate plane shows the location of each child's house. Name the ordered pair for each house.

tio

9 8

du ca

7 6 5 4

al E

3 2 1

R eg

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

230

3. A supermarket sells watermelons for $6 each. The cost of watermelons is plotted on the coordinate plan using the equation y = 6x. Watermelons Purchased 66

tio

54 48

du ca

Cost ($)

42 36 30 24 18 12

al E

4 5 6 7 Watermelons

R eg

(a) How much does 2 watermelons cost? (b) How much does 5 watermelons cost? (c) How much does 9 watermelons cost? (d) Riley spent $36 on watermelons. How many did she buy? (e) Blake has $54. How many watermelons can he buy?

How much change will he get?

231

square has 4 sides of equal length. So, we can calculate the A perimeter of a square by multiplying the side length by 4.

Perimeter of a Square (cm) 1

Perimeter (P)

3 16

tio

Side Length (s)

(a) Complete the table.

(b) Use the equation P = 4s to create a straight line graph. Perimeter of a Square

du ca

40 36

28 24 20

al E

Perimeter (cm)

16 12 8

R eg

3 4 5 6 7 Side Length (cm)

(d) A square has a side length of 8 cm. What is its perimeter? (e) A square has a perimeter 8 cm. What is its side length? (f) A square has a perimeter 16 cm. What is its side length?

232

5. A painter needs 27 liters of paint to paint a house. Paint is sold in cans of 3 liters for $9 each. (a) Complete the table.

Cost ($)

12 27

15 45

tio

Paint (l)

Cost of Paint

(b) Use the equation y = 3x to create a straight line graph. Cost of Paint

du ca

90 81 72

54 45

al E

Cost ($)

36 27 18

R eg

12 15 18 Paint (l)

24 27 30

(d) How much will 9 liters of paint cost? (e) The painter has $20. How much more money does he need to buy the paint he needs?

233

At Home 1.

Use the ordered pairs to plot the points on the coordinate grid.

tio

(d) W (2, 7) (e) C (6, 9) (f) H (3, 6)

(a) T (3, 3) (b) A (1, 9) (c) Z (3, 9)

(g) S (7, 7) (h) B (5, 8) (i) O (1, 5)

du ca

10 9 8 7

al E

6 5 4

R eg

3 2 1

23 4

2. The coordinate plane shows the location of the cars. 10

tio

8 7

du ca

6 5 4

al E

3 2 1

R eg

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

235

hardware store sells metal chain for $6 per meter. A The minimum length is 4 meters.

(a) Create a straight line graph to show the cost of chain using the equation y = 6x. Use the data in the graph to answer the questions. Cost of Metal Chain

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60 54 48

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Cost ($)

42 36 30 24 18

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12 6

3 4 5 6 7 Length of Chain (m)

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(b) What is the cost of 4 meters of metal chain?

(c) What is the cost of 5 meters of metal chain? (d) What is the cost of 8 meters of metal chain? (e) What length of chain can you buy with $48? (f) What length of chain can you buy with $54?

236

4. Joe's Subs sell submarine sandwiches for $2 per inch. The shortest submarine sandwich is 5 inches.

(a) Create a straight line graph to show the cost of sandwiches using the equation y = 2x. Use the data in the graph to answer the questions. Cost of Submarine Sandwich

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28 26 24

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Cost ($)

22 20 18 16 14

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12 10

5 6 7 8 9 10 11 12 13 Length of Submarine Sandwich (in.)

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(b) What is the cost of the shortest submarine sandwich? (c) What is the cost of a 13-inch submarine sandwich? (d) Halle has $23. What is the longest submarine sandwich she can buy?

(e) Chelsea has half as much money as Halle. What is the longest submarine sandwich she can buy?

237

Looking Back 1. The line plot shows the distances the students in Grade 5 ran during the school fun run.

3 4

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Fun Run Distances

1 4

1 2

Miles

3 4

1 4

(a) How many students ran 2 miles?

(b) How many students ran further than 1

1 miles? 2

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3 (d) What is the combined distance ran by the students who ran 1 4 miles or further?

23 8

Use the ordered pairs to plot the points on the coordinate grid.

(a) A (1, 2) (b) F (4, 4) (c) J (3, 7)

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(g) E (4, 8) (h) R (8, 4) (i) O (6, 5)

(d) W (3, 2) (e) C (9, 9) (f) H (9, 6)

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9 8 7 6

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5 4

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2 1

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3. Mrs. Taylor needs 12 pounds of flour to bake bread for the school fundraiser. Flour is sold in packets of 2 pounds for $4 each. (a) Complete the table.

Cost ($)

8 12

10 20

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Flour (lb)

Cost of Flour

(b) Use the equation y = 2x to create a straight line graph. Cost of Flour

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36 32

24 20 16

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Cost ($)

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8 10 12 Flour (lb)

(d) How much will 14 pounds of flour cost? (e) Mrs. Taylor has $50. How much money will she have left when she buys the flour she needs?

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n equilateral triangle has 3 sides of equal length. We can calculate the A perimeter of an equilateral triangle by multiplying the side length by 3.

(a) Complete the table.

Perimeter (P)

5 12

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Side Length (s)

Perimeter of an Equilateral Triangle (in.)

(b) Use the equation P = 3s to create a straight line graph.

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Perimeter of a Equilateral Triangle

21 18 15

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Perimeter (cm)

12 9

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3 4 5 6 7 Side Length (cm)

(d) An equilateral triangle has a perimeter of 21 cm. What is the length of each side?

2 41

Problem Solving

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Example 1 Wyatt has 2 empty water bottles. Wyatt wants to give 5 liters of water to his friend, Halle. How can he measure out 5 liters of water using the bottles?

Act It Out

Step 1 Fill the 4-liter bottle and pour it into the 7-liter bottle.

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Step 2 Fill the 4-liter bottle again and pour it into the 7-liter bottle. Wyatt now has 1 liter left in the 4-liter bottle.

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Step 3 Empty the 7-liter bottle and pour the 1 liter into the 7-liter bottle.

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Step 4 Fill the 4-liter bottle once more and pour the water into the 7-liter bottle. 4L

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Wyatt now has 5 liters of water to give to Halle.

24 3

Example 2 How many different triangles can you make by joining the dots?

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Look at the different ways that three points can join together.

ABC ABE

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ABF

ACD ACF ADE ADF AEF BCD BCE BDF CDE CDF CEF DEF

15 different triangles can be made by joining the dots. 244

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1. 17 army soldiers need to cross a river using a small boat. The boat can only carry 3 army soldiers at a time. How many times did the boat cross the river?

24 5

2. During a pandemic period, it is advised that work meetings should be held at a circular table, with a maximum of 5 people. How many ways can meeting members sit with the boss remaining in the same spot?

Boss

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246

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3. In a mathematics lesson, the teacher wants to arrange 8 students to form 4 straight lines. The teacher wants 3 students in each line by intersecting some of the lines. How can the teacher arrange this?

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The teacher now wants to arrange 10 students to form 5 straight lines, with 4 students in each line. How can the teacher arrange this?

24 7

Draw a Model

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Look at the figure on the following page. In round 1, there will be 8 matches played. In round 2, there will be 4 matches played. In round 3, there will be 2 matches played. In round 4, there will be 1 match played.

Example There are 16 teams in a football league. When 2 teams compete, there can be only 1 winner. The winner will move on to the next match. How many matches will be played in total after the final match?

To find the total number of matches played, add the number of matches played in each round. 8 + 4 + 2 + 1 = 15

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Therefore, there will be 15 matches played throughout the tournament.

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Draw a diagram when you are given complex information.

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Diagrams are a great way to communicate and present ideas.

Round 1 Team 1

Round 2 Round 3

Team 2

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Team 3

Round 4

Team 5 Team 6 Team 7 Team 8

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Team 9

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Team 4

Team 10 Team 11

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Team 12

Team 13 Team 14

Team 15 Team 16

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1. The cost of a present was shared among Ethan, Halle and Sophie. Ethan paid one fifth the cost of the present. Sophie paid $15 more than one third of the remaining amount. Halle paid $30 more than Sophie. What was the price of the present?

Halle

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Sophie

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Ethan

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2. There are 20 students in a class. 8 students each received 3 red stars. 5 students each received 2 blue stars and 3 red stars. The rest of the students each received 1 yellow star. Express as a fraction the number of blue stars awarded over the total number of stars awarded.

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3. Three quarters of the number of zebras in a nature reserve is equal to half the number of deer. One third of the number of deer is equal to three fifths the number of lions. If there were 15 more zebras than lions, how many deer were there?

2 52

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4. A parachute is dropped from the peak of a 200 m high mountain on Planet X. The parachute drops at a speed of 56 m/s. Every second after that, the speed at which the parachute falls is halved until it is 89.75 m from the ground. After which, the parachute falls at a steady rate of 1.25 m/s. How long would it take for the parachute to reach the ground?

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Guess-and-Check

Wyatt's amount

Blake's amount

$40

40 - 23 = 17

$41

41 - 23 = 18

$42

42 - 23 = 19

$43

43 - 23 = 20

$44

44 - 23 = 21

Total

Left over

Check

40 - 18 = 22

17 + 22 = 39

39 - 40 = -1

41 - 18 = 23

18 + 23 = 41

41 - 41 = 0

42 - 18 = 24

19 + 24 = 43

43 - 42 = 1

43 - 18 = 25

20 + 25 = 45 45 - 43 = 2

44 - 18 = 26

21 + 26 = 47

yes

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Cost of the gift

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Example Wyatt and Blake want to buy a birthday present for their teacher. Wyatt is $23 short of the amount needed to buy the gift while Blake is $18 short of the amount needed to buy the gift. They decided to combine their money and buy the gift together. After buying the gift, they will have an extra of $3 left over. How much is the gift?

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Therefore, the gift costs $44.

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Keep guessing and checking until you arrive at the correct answer

47 - 44 = 3

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1. Packs of decorative lights come in the form of star or moon shapes. A pack of star-shaped lights are 7 meters long and a pack of moonshaped lights are 5 meters long. During a parade, exactly 174 meters of lights were used for decoration. How many packs of star-shaped and moon-shaped lights were used?

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2. At a manufacturing company, all employees work an equal number of hours each day. The total combined number of work hours of all employees for a day is 231. If each employee works at least 5 hours a day, how many employees are there in the company?

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3. Farmer Joe has 22 chickens and sheep in his yard. The chickens and the sheep have a total of 74 legs. How many chickens and how many sheep are in Farmer Joe's yard?

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4. Find two numbers with a product of 90 and sum of 21.

2 58

Make a List

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Let's make a list to keep track of all the different combinations

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Example Halle has a blue pencil, a black pencil and a red pencil. She also has a blue sharpener, a black sharpener and a red sharpener. In how many different ways can she match a pencil with a sharpener?

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List key blue pencil (B) blue sharpener (b) black pencil (K) black sharpener (k) red pencil (R) red sharpener (r) Bb Kb Rb Bk Kk Rk Br Kr Rr

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Therefore, there are 9 possible combinations.

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1. How many different 3-digit numbers can be formed from the numbers: 7, 4, 1 and 3? Each number can only be used once.

2 60

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2. There are 5 colored beads on a table. The beads are colored: red, blue, orange, yellow and green. Keira is to choose 3 beads. In how many ways can she choose 3 beads?

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3. The total area of 3 squares is 89 cm2. Each has a side length that is a whole number. If the side length of the smallest square is more than 1 cm and the side length of the biggest square is less than 10 cm, what is the perimeter of the 3 squares?

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4. A rectangle has an area that is equal to its perimeter. If the length of the rectangle is longer than the breadth, what is the length and breadth of the rectangle? Both the length and breadth are whole numbers.

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Look for Patterns

Example A group of students were at a farewell exchange ceremony. Each student exchanged their gift with every other student who was present at the ceremony.

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(a) If there were 4 students, how many gifts were exchanged?

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To find the total number of gifts exchanged we can look at the number of exchanges between the students. In this example there are 4 students.

3 + 2 + 1 = 6

There are 6 exchanges between the 4 students. In each interaction, 2 gifts are exchanged. So the total number of gifts is double the amount of exchanges. 6 x 2 = 12 gifts 12 gifts were exchanged between the 4 students.

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To find the number of exchanges, find the sum of the whole numbers from 1 to 1 fewer than the total number of students.

(b) If there were 13 students, how many gifts were exchanged?

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12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 78 There were 78 exchanges. 78 x 2 = 156 gifts Therefore, 156 gifts were exchanged between 13 students.

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Divide by 2 to find the number of exchanges. 210 ÷ 2 = 105

There were 105 exchanges. Let's try 14 students. 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 91 This is too low.

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105 – 91 = 14 105 = 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 So, if 210 gifts were exchanged, there were 15 students at the ceremony.

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2 rows

3 rows

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1 row

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1. You arrange table tennis balls in triangular shapes as shown. How many balls will there be in a triangle that has 10 rows?

2 66

2. Look at the pattern below. M A T H E M A T I C S M A T H E M A T I C S M A T H E M A T I C S . . .

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(a) What is the 34th letter?

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(b) What is the 125th letter?

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3. Dominic wrote all the numbers between 1 and 85 on the whiteboard as seen below. How many digits did he write in total?

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1, 2, 3, 4, 5, 6, 7, 8, 9 . . . . . . . . . 83, 84, 85

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4. What are the next two numbers in the pattern? , 2, 6, 12, 20, 30, 42, 56, 72, 90,

269

Work Backwards

? Interchange

– 1 4 +3 12 passengers

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– 1 2 +2

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Example A bus left an interchange with some passengers on it. At the first bus stop, half of the passengers got off and 2 people got on. At the second stop, a quarter of the people got off and 3 people got on the bus. When the bus left the second stop, there were 12 people on it. How many people were on the bus when it left the interchange?

First Stop

Second Stop

Working backwards, the last event that occured was 3 people got on the bus. We take away 3 from the final number of people on the bus. 12 - 3 = 9.

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We know that 9 people were on the bus before the last 3 people hopped on.

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Interchange

– 1 2 +2

– 1 4 9 passengers

First Stop

Second Stop

Moving back, a quarter of the people got off the bus. This means that three quarters would remain on the bus. We multiply the number of people on the bus by four thirds. 4 36 9 x = 3 3 = 12

27 0

Interchange

First Stop

Second Stop

– 1 2 +2 12 passengers

12 - 2 = 10.

? Interchange

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– 1 2

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Working back we see 2 people got on at the first stop. To work backwards, we take away 2 people.

10 passengers First Stop

Second Stop

Finally, the first event was half the people got off, leaving half the people on board. To work backwards, we multiply the number of people by 2.

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10 x 2 = 20.

Therefore, there were 20 people on the bus at the interchange.

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If you know the final answer but not the starting point, then you should work backwards!

271

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1. Sophie has some savings in her bank account. During the day she spends $74.50 on shoes, spends $80 on food and deposits $535 into her account. At the end of the day she sees that her balance is $851.80. How much money did she have in the bank at the beginning of the day?

272

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2. Riley has a certain number of masks. She gave half of the masks to Ethan and then took 5 masks back. Riley then gave a quarter of her remaining masks to Halle and then took 6 masks back. Finally, Riley gave half of the remaining masks to Blake. Riley had 30 masks left. How many masks did she have at first?

273

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3. Wyatt has 69 stickers on Wednesday. On Tuesday, he bought 12 stickers and then gave away a quarter of the stickers he had. On Monday, he shared all his stickers equally with his 4 friends. How many stickers did Wyatt have at the start of Monday?

27 4

4. I am thinking of a 7-digit number. The millions digit is the difference between the ten thousands digit and the ones digit.

The hundred thousands digit and ten thousands digit combine to form a 2-digit number that can be divided by 6 with a remainder of 5. When the same number is divided by 5, the remainder will be 0.

tio

The thousands digit is an odd number, a multiple of 3 and a factor of 36 but is not 3.

The ones digit is 3.

du ca

The hundreds digit is the sum of the one and ten digits minus 5. The tens digit is double the ones digit plus 3.

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What is the number that I am thinking of?

275

Simplify the Problem

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Example The figure below shows 1 unit of a repeating pattern. The unit is made up of 1 smaller equilateral triangle, 2 identical squares and 1 larger equilateral triangle. The sum of the perimeter of the small equilateral triangle and 1 square is double the perimeter of the larger equilateral triangle. If the total perimeter of 28 unit patterns is 4,872 cm, find the length of one side of the larger equilateral triangle.

Let's find the perimeter of 1 unit of the pattern

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4,872 ÷ 28 = 174

So, one unit pattern has a perimeter of 174 cm. The smaller triangle and the 2 squares all have the same side length as denoted by the markings.

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Perimeter of smaller triangle and 1 square = 7 units The larger triangle has a perimeter that is double of the perimeter of the smaller triangle and 1 square. 1

Perimeter larger triangle = 2 of 7 units 1 = 3 2 units

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Total perimeter = 3 units + 4 units + 3 2 units + 4 units 1

= 14 2 units 1

Perimeter of larger triangle = 3 2 units 1

= 3 2 x 12

du ca

= 42 cm

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1 unit = 174 ÷ 14 2 = 12 cm

14 2 units = 174 cm

Divide by 3 to find the length of 1 side. 42 ÷ 3 = 14

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The larger triangle has a side length of 14 cm.

R eg

That was a lot easier than working with 28 unit patterns!

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1. The diagram below is formed by 3 squares – A, B and C. The side length of square B is twice that of square A The side length of square C is twice that of square B. The perimeter of the figure is 75 cm. Find the area of square C.

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2. What is the maximum number of 4 cm cubes that can be cut from a wooden cuboid measuring 30 cm by 18 cm by 5 cm?

279

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3. A carpenter is cutting cubes from a large block of wood. The carpenter takes 2 minutes to cut one cube from the large block. He then arranged the cubes to form the given figure. After arranging the cubes, he paints all of the exposed faces including the underside of the figure. Each face takes 1 minutes to paint. How long did he take to complete the whole process in hours and minutes?

2 80

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4. I am thinking of a number. I add 4 to my number. I then multiply by 3. The final value is 7 times my original number. What is my original number?

28 1

Solve Part of the Problem

Thursday

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Tuesday

Wednesday

This diagram will help us to solve the different parts of this problem

tio

Monday

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4 parts of Thursday = $128 1 part = 128 ÷ 4 = 32

4 parts of Wednesday = 3 parts of Thursday 3 x 32 = 96 96 ÷ 4 = 24 1 part of Wednesday = 24

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4 parts of Tuesday = 3 parts of Wednesday 3 x 24 = 72 72 ÷ 4 = 18 1 part of Tuesday = 18 Monday = 3 parts of Tuesday 3 x 18 = 54 So, Ethan saved $54 on Monday

28 2

Example Ethan saves a sum of money on Monday. Every subsequent day, he saves one third more than he did the day before. By Thursday, he had saved $128. How much did he save on Monday?

21.8 m

West

East

16.1 m

South

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South

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Junction

North

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North

1. The diagram shows part of a 3-tunnel system. 2 parts of the tunnel system runs North to South while another part runs East to West. Two tunnels meet at a traffic light junction. The width of each tunnel is 1.9 m. Find the area of road covered by the tunnel system shown in the diagram.

28 3

2. The diagram below shows a regular pentagon, a circle and an isosceles triangle. Find the value of angle y.

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3. The diagram below is made up of 15 identical rectangles. Using the dimensions given in the diagram, find the area of all the rectangles used to make up the figure.

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15 cm

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26 cm

28 5

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4. The diagram shows 1 large circle surrounded by 6 identical smaller circles. Straight lines are extended from the center of the larger circle onto the sides of the smaller circles. Straight lines are also extended from the center of each of the 6 smaller circles onto the side of the larger circle. They form a right angle with the lines that extended from the large circle. Find the sum of the angles. Note that all 6 angles are equal.

286

Before-After Concept

Example The ratio of the number of sweets Riley had to the number of sweets Wyatt had was 5 : 3. When Riley gave 40 sweets to Wyatt, the ratio became 7 : 17. How many sweets did Riley have at first?

Riley Wyatt

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Before

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Let's look at the ratio before the exchange and after the exchange.

Before the exchange: 5 : 3 – which is a total of 8 units. After

Riley Wyatt

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After the exchange: 7 : 17 – which is a total of 24 units.

To make the two ratios have an equal amount of units, we can multiply the before ratio by three.

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Before

Riley

Wyatt

Before the exchange: 15 : 9 – which is a total of 24 units. After

Riley

Wyatt

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The before and after ratios, add up to the same amount of units. This allows us to compare them.

Before: 15 : 9 After: 7 : 17

Before

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When Riley gave away 40 sweets, her ratio decreased from 15 units to 7 units. There is a difference of 8 units. Riley

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Wyatt 8 units

After

Riley Wyatt

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So, 40 sweets is equal to 8 units. To find the number of sweets in one unit, we divide 40 by 8. 40 ÷ 8 = 5

1 unit = 5 sweets

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We can use this to find out how many sweets Riley had at first by using the ratio. Riley had 15 units at first. 5 x 15 = 75 sweets.

So, Riley had 75 sweets at first.

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1. Halle and her sister had $100 altogether. If their father gives Halle another $30, Halle will have as much money as her sister. How much money does Halle have?

28 9

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2. Chelsea has a total of 40 oranges and pears. If she exchanges every pear for 2 oranges, she will have 56 oranges. How many oranges and how many pears does she have?

29 0

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3. There are some coins in the 2 boxes, labeled A and B. Box A has 2 more coins than Box B. If we move 1 coin from Box B to Box A, Box A will have twice as many coins as Box B. How many coins are there in Box A at first?

29 1

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4. Dominic, Ethan and Sophie had some arcade tickets. The ratio of Dominic’s tickets to Ethan’s was 9:5. The ratio of Ethan’s tickets to Sophie’s was 4:3. After Dominic used 55 of his tickets, the number of tickets Ethan had was four fifths that of Dominic’s. How many more tickets does Dominic have than Sophie now?

29 2

Make Suppositions

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Let's suppose that all I sold were large mangoes.

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Example Blake sold a total of 40 large and small mangoes in a particular day. Large mangoes sell for $5 and small mangoes sell for $3. At the end of the day, he collected a total of $168. How many large mangoes did Blake sell?

40 x 5 = $200 at the end of the day

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This supposition was incorrect. The difference between our supposition outcome and the actual value was: $200 - $168 = $32

The difference in price between one large mango and one small mango is: $5 - $3 = $2

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So, when we change our supposition and replace a large mango with a small mango, the outcome will decrease by $2. We need to find the number of small mangoes that will decrease our supposition outcome by $32. 32 ÷ 2 = 16

Exchanging 16 large mangoes with small mangoes, we will reach $168. 40 - 16 = 24.

Therefore, Blake sold 24 large mangoes. 29 3

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1. Sophie had 24 chickens and goats on her farm. The total number of legs on her animals is 68. How many chickens and goats are there on her farm?

29 4

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2. There are some zebras, goats and ducks in a zoo. There are 2 times as many ducks as there are goats. There are a total of 42 animals and a total of 120 legs in the zoo. How many zebras are there in the zoo?

29 5

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3. During a homecoming school event, there are 100 cupcakes to share among 100 people. Each adults eats 2 cupcakes. Every 3 children share 1 cupcake. How many children are there?

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4. A bicycle shop had a total of 37 bicycles and tricycles. Each bicycle sold for $189 while each tricycle sold for $99. If there were a total of 90 wheels in all the bicycles, how much would the shopkeeper have earned if he sold all of the bicycles and tricycles?

29 7

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© Blue Ring Media Pty Ltd ACN 161 590 496 2013 - 2021. This publication would not have been possible without the tireless effort of our production team. Special thanks to: Daniel Cole, Matthew Cole, Wang Hui Guan, Kevin Mahoney, Winston Goh, Jesse Singer, Joseph Anderson, Halle Taylor-Pritchard, Sophie Taylor-Pritchard, Tejal Thakur, Natchanuch Nakapat,Varasinun Mathanattapat, Kanungnit Pookwanmuang, Saijit Lueangsrisuk Original Illustrations: Natchanuch Nakapat, GraphicsRF, Blue Ring Media and Interact Images Royalty-free images: Shutterstock, Adobe Stock