Let's Do Mathematics 5 – Worktext B

n at

Wo ork ktex t e xt

5B

ffor o r lle e arners a r n e r s 10 - 11 year yea r s o l d

Let’s Do Mathematics

Re ga le du ca tio n

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

Central to the USCCSS is the promotion of problem-solving skills ls and reasoning.

Let’s Do Mathematics achieves this by teaching and presenting ng g concepts through throu a problem-solving based pedagogy and using the concrete-pictorial-abstract pictorial-abstract torial-abstract ((CPA) approach. Learners acquire knowledge and understanding a ng off concepts through thr th guided progression beginning with concrete examples and experiences which then w flow into pictorial representations and finally mastery at the and symbolic level. e abstract an a This approach ensures that learners develop a fundamental understanding of concepts damental ental understa underst rather than answering questions by learned procedures edures es and algorithms. algorit

Key features of the series include:

Anchor Task

5

Angles of Triangles h r Task hor Anchor

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

$5.25 per pack

$1.45 each

$0.75 each

98

Let’s Learn n

Concepts are presented sented in a clear and a colorful manner.. Worked rked problems problem provide learners step-by-step ers with ith guided step-by ste progression through Series gh examples. Ser mascots provide rovide de guidance through throug throu helpful comments observations omments ments and observa when new concepts oncepts are introduced. intro

Operations on Decima ls

Anchor Task

2

Multiplyi l ing

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 cost?

Multiply the

kilogram. 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 rrepresents the . cost oof 1 kg of pista chio . nuts.

tens.

2 1 . 3

x

Tens

Find 77.4 x

6 using the

od.

x

Multiply the 2

tenths.

1 . 3

x

Tens

3

.

Tenths

.

4 using the

6 . 18 3

x

4

.

.

.

Step 2

Multiply the

ones.

2 1 . 3

x

3

Tens

Ones

.

3 . 9

.

.

22

6.83 x 4 = 27.32

3x1=3

Tenths

2

7 27 . 4

x

4 . 4

77.4 x 6 = 464.4

Find 6.83 x

Ones

. 9

4

6

. 4

3 x 0.3 = 0.9

$63.90.

7 27 . 4

6

$21.30

Tenths

column meth

7 27 . 4

x

.

.

21.3 x 3 = 63.9 So, 3 kg of pistachio nuts costs

?

Step 1

Ones

3 6 3 . 9

6 4 6 4 . 4

column meth

od.

3

6 . 18 3

x

4

. 3 2

3

6 . 18 3

x

4 2 7 . 3 2

We can use rounding and estimation to check our answers.

.

23

ii

Let’s Practice

Fill in the blanks.

2.

Let’s Practice

(a)

Ones

Tens

dre Hundreds

Ten Thousands Thousands

Hundred Thousands

Millions

Fill in the blanks.

1.

(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) Millions

7

5

3

n Ten Thousands usands Thousa

Tens

eds Hundreds

Ones

Re ga le du ca tio n

1

Hundred Thousands

The ten thousands digit The numbers increase

=

(b)

2,824,575

1,574,575

The numbers increase

(c)

.

is number in the pattern

324,575

more 125,000 m

less 1 125,000

in each step.

by

+

The next

in each step.

increases by

ons Million

?

4,074,575

Ten Thousands Thousands

eds Hundreds

Ones

Tens

1,500,000 more

500,000 less 1,500,000

in each step.

by

(d)

=

+

The next number in

Hundred Thousands

Hundred Thousands Thousand

Millions

Ten Thousands Thousands

Hundredss

Ones

Tens

.

the pattern is

10,000 more

less 10,000 le

47

At Home

1.

Classify each triangle .

2.

Classify each triangle e.. Choose one cl classification per triangle. (b)

(a)

At Home

(a)

Right-angled

Scalene

Isoscele osceless

(b)

Right-angled Rig

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.

(c)

Scalene Sca

(d)

Isosceles Isosce

(c)

Right-angled

Scalene

Isosceles

(e)

(f)

(d)

Right-angled d

Scalene

Isoscele sceless

96

97

Hands On

Hands On

ps of 4-5. er in your ber umb it num Work in grou mill n. w write a 7-dig n and 6 millio llion As a group, een 5 millio th that is betw notebook

1.

t square. on the start forward the your counter . and move dice y Roll the dice Ro hown on your spaces show number of ber plete the num p must com oup grou the in fo ard.

4. Everyone order to move forw pattern in ber till origi o nal num the with 4 steps 3 to 5. Repeat the finish. you reach

2.

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

ter Place a coun

3.

Solve It!

(a) OPQR is a parallelo gram.

SP is a straight line. Find OPQ O

Solve It!

118º

P

20º

R

S

50

Q

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

Find t.

Activities that require learners earners ers to ng and nd problem-solvin problem-s apply logical reasoning problem-solving. Problems hich do o not have a rou are often posed which routine strategy rners are encouraged encoura enc for solving them. Learners to think y a range of probl p creatively and apply problem-solving heuristics.

M

N

38º

t

47º

P

O

(c)

GHIJ is a parallelogram.

HJ is a straight line. Find G

m.

56º

H

m

J

44º

I

120

pairs to plot the points 2. Use the ordered p

Looking Back

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

Looking Back

Consolidated nsolidated practice whe where learners demonstrate on a emonstrate their understanding under u range ange of concepts taught taug within a unit.

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

1

1 4

1

1 2

1

3 4

2

2

1 4

8

Miles

7

(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

5

the students who ran

4

3

mi

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

3 the students who ran 1 4 miles

2 1

0

1

2

3

4

5

6

7

8

9

10

mi

239

238

iii

Contents 2 3 14 22 29 36

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

48 48 58 65 74

Re ga le du ca tio n 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

7

iv

Geometry es Types of Triangles Angles of Triangles an Quadrila uadri als Angles of Quadrilaterals

84 85 98 108

8 Measurement ent erting ng Measurem Measu Converting Measurement Units rd Problems roblems Word

124 125 133

9 Volume ume Volume an and U Unit Cubes Volume of R Rectangular Prims and Capacity Volume a Word Pr Wor Problems

150 150 168 181 186

206 206 2 224

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 Conceptt Make Suppositions

242 242 248 254 259 264 270 276 282 287 293

du ca tio n

10 Data and Graphs Line Plots Graphing Equations

v

Operations on Decimals

Anchor Task

on

5

$5.25 25 per pack $1.45 each

$0.75 each

2

Adding and Subtracting Decimalss

on

Let’s Learn Ethan weighs 32.5 kg. His schoolbag weighs 5.8 kg.

Find the total weight of Ethan and his schoolbag. oolbag.

? bag

32.5

5.8

Re ga le du c

Ethan

To find the total weight, we add.

Step 1

Step 2

Add the tenths.

3

+

1

Step 3

Add the ones. ne

2 . 5

3

5 . 8

1

2 . 5

5 . 8

+

. 3

Add the tens.

8 . 3

3

1

2 . 5 5 . 8

+

3

8 . 3

We ca can regroup t 13 tenths into 1 one and 3 tenths. on

Tens

Ones O

.

Tenths

. .

3

Find the sum of 148.27 and 61.58.

Step 1

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

Re ga le du ca tio n

Add the hundredths.

1

+

4

8 . 12

7

6

1 . 5

8

.

5

Hundreds

Tens

Ones

.

Tenths enths

Hundredths

. .

Step 2

Step 3

Add the tenths.

1

+

Add the he ones.

4

8 . 12

7

6

1 . 5

8

. 8

5

Hundreds

Tens en

1

+

4

8 . 12

7

6

1 . 5

8

9 . 8

5

Ones

. . .

4

Tenths

Hundredths

Step 4

We can regroup 10 tens into 1 hundred.

Add the tens.

1

4

8 . 12

7

6

1 . 5

8

0

9 . 8

5

Re ga le du ca tio n

1

+

Step 5

Add the hundreds.

1

1

+

2

4

8 . 12

7

6

1 . 5

8

0

9 . 8

5

Hundreds

Tens

Ones ne

.

Tenths

Hundredths

. .

148.27 + 61.58 = 209.85 9.85

Add 27.93 to 83.12.

2.

1

+

1

2

1

7 . 9

3

8

3 . 1

2

1

1 . 0

5

We can use rounding and estimation to check our answers.

27.93 + 83.12 = 111.05

5

n

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?

68.4 ribbon left

45.7

?

Re ga le du ca

ribbon cut

To find the length of ribbon left, we subtract.

We ccan We regroup regro 1 one into 10 tenths.

Step 1

Subtract the tenths.

6

–

7

8 . 144

Tens

Ones O

.

5 . 7

4

. 7

.

Step 2

Step 3

Subtract the ones.

6

–

4

7

Subtract the tens.

Subtra

8 . 144 5 . 7 2 . 7

6

–

8 . 144

4

5 . 7

2

2 . 7

68.4 8.4 – 45.7 = 22.7 Sophie phie h has 22.7 cm of ribbon left.

6

7

Tenths

Subtract 7.49 from 22.36.

Ones

.

Tenths

Hundredths

Re ga le du ca tio n

Tens

.

1

2

–

1

11

2 . 123

16

6

7 . 4

9

4 . 8

7

22.36 – 7.49 = 14.87

Find the sum and difference of 53.18 8 and 62.57.

62.5 62.

To find the sum, we add.

+

1

5

3 . 11

8

6

2 . 5

7

1

5 . 7

5

To find the difference, we subtract.

5

6

–

5

12

2 . 45

17

7

3 . 1

8

9 . 3

9

The sum of 53.18 3.18 and 62.57 is 115.75.

The difference fference of 53.18 53 and a 62.57 is 9.39.

7

Let’s Practice Add.

Re ga le du ca tio n

1.

(a)

3

+

1

4 . 1

5

9

2 . 6

5

(b)

5

+

3

5 . 8

1

2

1 . 7

7

.

(c)

+

.

3

1

0 . 2

9

1

7

6 . 5

2

(d)

+

2

3

5 . 1

1

1

3

1 . 7

1

.

(e)

+

.

5

8

3 . 6

8

1

4

9 . 3

8

(f) (f

+

2

8

3 . 8

4

2

8

3 . 6

2

.

(g)

+

3

9

5 . 8

2

6

3 . 8 .

8

.

3

(h)

3

+

9

3 . 5

3

8 . 3 .

3

2.

Subtract.

9

7 . 5

2

7 . 9

(b)

7

5 . 3

4

3 . 7

Re ga le du ca tio n

(a) –

–

.

(c)

–

3

7 . 5

2

9 . 8

.

(d)

–

4

7 . 9

3

5 . 9

.

(e)

–

4

6 . 8

4

5 . 7

.

5

(f) f)

–

6

2 . 5

7

5

3 . 1

8

.

(g)

–

.

8

6 . 4

6

3

6 . 3

8

(h)

–

8

4 . 8

4

3

8 . 4

9

.

(i)

–

.

8

8 . 4

3

2

8 . 4

3

.

(j)

–

9

5 . 3

6

4

5 . 3

2

.

9

3.

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

Re ga le du ca tio n

(a) 4.69 + 13.59 =

(c)

29.46 + 9.21 =

(e) 43.35 + 136.94 6.94 =

10

(d) 33.65 33 65 – 23.59 =

(f)

87.48 – 13.34 =

(b) Home At Add.

Re ga le du ca tio n

1.

(a)

5

(b)

7 . 7

4 . 9

+

+

4

6 . 3

2

5 . 8

.

(c)

+

.

2

5

7 . 4

7

5

6

8 . 8

5

(d)

+

4

6

5 . 6

4

5

7

4 . 5

7

.

(e)

+

.

4

5

6 . 7

3

2

7

8 . 4

5

(f)

+

7

4

5 . 7

2

1

4

7 . 8

5

.

(g) 4.69 + 4.69 9=

.

(h) 48.39 + 212.48 =

11

2.

Subtract.

8

9 . 6

3

6 . 2

(b)

8

4 . 3

2

5 . 7

Re ga le du ca tio n

(a) –

–

.

(c)

–

.

8

5 . 4

2

2

4 . 6

8

(d)

–

8

6 . 4

3

2

6 . 8

5

.

(e)

–

7

6 . 4

2

6

3 . 6

2

.

(g) 45.94 – 24.69 4.69 =

12

.

(f) (f

–

8

5 . 3

2

7

4 . 3

2

.

(h) 94.05 – 45.39 =

3.

Find the sum and difference of each pair of numbers.

Re ga le du ca tio n

(a) 67.58 and 45.38

sum =

difference ifference =

(b) 35.48 and 125.94

sum =

(c)

difference =

146.59 and 256.59 56.59

su = sum

difference =

13

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

ga le du ca tio n

Anchor Task Thousands

Hundreds

Tens

Ones

.

Tenths

Hun Hundredths

.

.

.

(a) 2.1

14

(b) 3.06

(c)

1.28

(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 1

3.06 x 1,000

1.28 x 1,000

9.49 x 1,000

Let’s Learn

Re ga le du ca tio n

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

Find 24.3 x 10.

10

10

1

1

1

1

0.1

0.1

0.1

100 100

x 10

10

1 10

10

1

1

1

10 0

243

24.3

24.3 x 10 = 243

ly 3.52 2 by b 10.

Let’s use place value disks to help multiply 1

1

1

0.1

0.1

0.1

0.1

0.1

x 10 0

0.01 0.01

3.52

10 0

110

10

1

1

1

1

1

0.1

0.1

35.2

3.52 x 10 = 35.2

Let’s use a place value e chart cha to help p multiply m numbers by 10.

Find 29.47 x 10.

Hundreds

Tens Ten

Ones

.

Tenths

Hundredths

. .

29.47 x 10 = 294.7

15

Multiply 2

7 . 4

Re ga le du ca tio n

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

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

x

5

7 . 0

3

6

3 . 8

x

1

8

3

0. 4

8

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

10 Find 3.14 x 100.

Hundreds

Tens

Ones

Tenths Te

. . .

3.14 x 100 = 314

Multiply.

(a) 8.3 x 600 = 8.3 x 6 x 100 49 x 100 = 49.8 4,980 = 4,980

(b) 2.35 x 400 = 2.35 2 x 4 x 100 = 9.4 x 100 = 940 94

16

1

8 . 3

x

6

4

1

9 . 8

2 . 23

x

5 4

9. 4

0

Hundredths

Find 0.45 x 1,000.

Tens

Ones

Tenths

.

Hundredths dredths

ed uc ati on

Hundreds

. .

0.45 x 1,000 = 450 Multiply.

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

(b) 2.91 x 3,000 = 2.91 x 3 x 1,000 000 = 8.73 x 1,000 00 = 8,730

(c)

.72 x 4 x 1,000 1.72 x 4,000 = 1.72 = 6.88 8 x 1,000 = 6,880 880

1

5 . 6

x

3

1

2

6 . 8

2 . 9

x

1

3

2

8 . 7

3

1 . 7

2

x

4 6 . 8

8

17

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

Re ga le du ca tio n

1.

(a) 3.1 x 10 =

(c)

2.

3.1 x 100 =

4.5 x 100 =

3.1 x 1,000 =

4.5 x 1,000 ,000 00 =

8.34 x 10 =

(d) 9.87 x 10 =

8.34 x 100 =

9.87 .87 x 100 =

8.34 x 1,000 =

9.87 x 1,000 1,00 =

Find the products. (a) 1.2 x 2 =

(c)

18

(b) 4.5 x 10 =

(b)) 0.8 0 x8=

1.2 x 20 =

0.8 x 80 =

1.2 x 200 =

0.8 x 800 =

1.2 x 2,000 0=

0.8 x 8,000 =

10.11 x 3 =

(d) 0.01 x 5 =

10.11 0.11 x 30 0=

0.01 x 50 =

10.11 .11 x 300 =

0.01 x 500 =

10.11 x 3,000 =

0.01 x 5,000 =

3.

Multiply. (b) 7.4 x 100 =

Re ga le du ca tio n

(a) 3.2 x 10 =

(c)

5.7 x 1,000 =

(d) d) 5.56 6x4=

(e) 2.42 x 5 =

(f)

43.42 x 200 =

(g) 11.13 3 x 2,000 =

(h) 24.34 x 3,000 =

19

(b) Home At Fill in the blanks.

Re ga le du ca tio n

1.

(a)

1

10

x

1

10

0.1

1

2.1 x

=

(b)

100

x

1

101 x

1,000

10

0 1,000

10

=

(c)

100

10

10

x

x

1,000 1 000 1,

100 10

100 10

11,000

100 10

100 10

1,000

100 10

100 10

=

(d)

1

10

x

20

x

=

100

10

100

10

x

1,000

100 10

1,000

100 10

1,000

100 10

1,000

100 10

2.

Find the products. (b) 3.05 x 10 =

Re ga le du ca tio n

(a) 0.42 x 10 =

(c)

3.

0.42 x 100 =

3.05 x 100 =

0.42 x 1,000 =

3.05 x 1,000 000 =

7.3 x 3 =

(d) 0.28 8x4=

7.3 x 30 =

0.28 8 x 40 =

7.3 x 300 =

0.28 x 400 =

7.3 x 3,000 =

0.28 0 x 4,000 4,0 4 =

Multiply.

(a) 8.1 x 10 =

(b) b) 12 12.4 x 100 =

(c)

(d) 7.72 x 3 =

9.38 x 1,000 00 =

21

Multiplying by 1-digit Numbers

Re ga le du ca tio n

Let’s Learn

A supermarket is selling pistachio nuts for $21.30 per kilogram. ram. m.

How much will 3 kg of pistachio nuts cost?

We need to multiply 21.3 by 3 to find out. er.

Let’s use a place value chart to help find the answer.

Tens

Ones

.

Ea row Each rrepr represents the cost oof 1 kg of pistachio nuts.

Tenths

. .

?

$21.30

3 x 0.3 = 0.9

Step 1

Multiply the tenths.

Tens

2

x

Ones

1 . 3

.

Tenths

.

3

.

. 9

.

3x1=3

Step 2

Multiply tiply the ones.

2 1 . 3

x

3

3 . 9

22

Tens

Ones

. . . .

Tenths

Step 3

Multiply the tens.

Tens

Ones

Tenths

.

Re ga le du ca tio n

2 1 . 3

x

3

.

6 3 . 9

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

Find 77.4 x 6 using the column method.

7 27 . 4

x

6

7 27 . 4

x

6

4

x

4 . 4

. 4

7 27 . 4

6

4 6 4 . 4

77.4 x 6 = 464.4

Find 6.83 x 4 using the column mn method.

6 . 18 3

x

4

.

2

3

6 . 18 3

x

4

. 3 2

3

6 . 18 3

x

4

2 7 . 3 2

6.83 x 4 = 27.32 32

We can use rounding and estimation to check our answers.

23

Let’s Practice Multiply.

Re ga le du ca tio n

1.

(a)

(b)

2 5 . 6

x

6

1

x

4

.

(c)

.

(d)

7 6 . 4

x

2 9 . 2 4

x

9

7

.

(e)

.

f) (f)

8 3 . 5 5

x

4

1

(h)

8

x

5

3 0 7 . 4 5

x

.

6

.

7 3 3 . 2 4

x

5

.

24

8

.

9 3 . 0 7

(i)

7 4 . 1

x

.

(g)

2 7 . 8

(j)

1 0 8 8 . 4 6

x

8

.

2.

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

Re ga le du ca tio n

(a) 38.2 x 4 =

(c)

1,672.6 x 3 =

(e) 787.68 8x4=

(d) 582.44 582 44 x 7 =

(f)

3,206.53 x 6 =

25

Solve It!

uc ati on

How do you make an octopus laugh?

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

e

w

4.33 x 4

Re ga l

h

k

17.32

l

5.84 .84 x 3

c

5.66 x 3

s

3.58 x 4

n

1.47 x 7

i

2.7 x 8

17.52

21.6

17.52

0.76 x 5

0.82 x 9

t

21.6

2.04 x 8

1.05 x 8

3.8

–

17.52

26

16.32

10.29

16.98

14.32

7.38

16.32

8.4

(b) At Home Multiply.

Re ga le du ca tio n

1.

(a)

(b)

2 0 . 8

x

4

2

1 3 . 6

x

3

.

(c)

.

(d)

5 2 . 7

x

1

x

9

5

.

(e)

.

f) (f)

4 4 . 1 3

x

7

2 0 6 . 2 5

x

4

.

(g)

.

(h)

5 8 . 2 8 8

x

5 0 5 . 0 5

x

9

.

(i)

8

8 . 6 5

.

1 3 . 1

x

7

6

.

(j)

1

2 3 4 . 5 6

x

7

.

27

2.

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

Re ga le du ca tio n

(a) 19.3 x 6 =

(c)

474.7 x 5 =

(e) 992.64 4x7=

28

(d) 851.28 851 28 x 8 =

(f)

4,083.26 x 4 =

Dividing by 1-digit Numbers

Re ga le du ca tio n

Let’s Learn

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

Regroup oup 1 one 0 tenths. Now into 10 n make equal we can groups!

1

1

0.1

0.1 0

0.1 0 0.

0.1

1

1

0.1

0.1

0.1

1

0.1

1

1

0.1

0.1 0

0.1

1

0.1

1

0.1

0.1

0.1

0.1

1

0.1 0

0.1

0.1

1

1

0.1

1

1

1 1 1

0.1

0.1

1

0.1

1

0.1

1

0.1

1

0.1

1

0.1

1

0.1

1

0.1

1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

There of 2.4. here are 4 equal equa groups gro 9.6 .6 ÷ 4 = 2.4 2

29

Re ga le du ca tio n

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.

Step 1

0 . 2 6 1 . 5 6 1 2 3

Divide 1 one by 6.

Regroup 1 one into 10 0 tenths. ths Add the 5 tenths and nd divide. hs remainder remain remainde 3 tenths.

15 tenths ÷ 6 = 2 tenths enths = 3 tenths. tenths 15 tenths – 12 tenths

Step 2

0 . 2 6 1 . 5 1 2 3 3

6 6

6 6 0

Bring down wn the 6 hundredths. hundre Now there ere are 36 hundredths. hund

36 hundredths h dths ÷ 6 = 6 hundredths.

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

0.26 m = 26 cm.

So each piece of string has a length of 26 cm.

30

Find 84.35 ÷ 5.

Step 2

1 .

5 8 4 . 3 5 5 3

1 5 8 5 3 3

Step 3

6 .

4 . 3 5

1 5 8 5 3 3

Step 4

6 . 8 4 . 3 5

1 5 8 5 3 3

6 . 8 7 4 . 3 5

Re ga le du ca tio n

Step 1

4 0 4

4 0 4 4

3 0 3

4 0 4 4

3 0 3 5 3 5 0

84.35 ÷ 5 = 16.87

Divide 145.18 by 7.

7

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

4 8

Use rounding and estimation to check the answer.

8 8 0

145.18 ÷ 7 = 20.74

31

Let’s Practice Divide.

Re ga le du ca tio n

1.

(a)

(b)

5 2 2 4 . 5

(d)

3 6 7 . 4 4

6 8

(e)

7

32

(c)

1 5 9 . 8 8

8 2 3 6 2 . 5 6

1 . 2 4

2.

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

Re ga le du ca tio n

(a) 82.08 ÷ 6 =

(c)

161.56 ÷ 7 =

(e) 3,456.78 78 ÷ 6 =

(d) 223.14 ÷ 6 =

(f)

5,599.26 ÷ 9 =

33

At Home Divide.

Re ga le du ca tio n

1.

(a)

(b)

5 3 7 5 . 5

(d)

7 6 7 . 2 7

3 5 9 . 0 7

(e)

5 4 4 4 . 4 5

34

(c)

8

1

6 7 0 . 3 2

2.

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

Re ga le du ca tio n

(a) 29.61 ÷ 7 =

(c)

319.2 ÷ 6 =

(e) 1,009.56 56 ÷ 4 =

(d) 158.48 ÷ 8 =

(f)

5,308.35 ÷ 5 =

35

Word Problems

Re ga le du ca tio n

Let’s Learn

At the school bookshop, a pen costs $1.38 and a notebook k costs sts $3.76.

Find the total cost of 3 pens and 1 notebook.

$1.38

$3.76 3.76

pen

pen

pen

notebook

?

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

1

1 . 23 8

x

3

4 . 1

4

The cost of 3 pens ens is $4.14.

Add the cost of the he notebook to find the total cost.

4 . 11

4

+ 3 . 7 6

7 . 9 0

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

36

Re ga le du ca tio n

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 emon each bottle.

12 l

bottle bottle bottle bottle bottle bottle

remaining ma 4.32 l

?

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

–

1

11

2 . 90 100

4 . 3 2 7 . 6 8

There is 7.68 liters off lemonade nade in 6 bottles.

To find the volume bottle, we divide. e of lemonade in each e 1 . 2 6 7 . 6 6 1 6 1 2 4 4

8 8

8 8 0

Each bottle 1.28 liters of lemonade. le contains con

37

Re ga le du ca tio n

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

$10.74

Wyatt's savings

$120

Jordan's savings

?

3 units = $120 – $10.74 = $109.26

1

–

1

2 9 0 . 9 0 1100 1 0 . 7 4

1 0 9 . 2 6

1 unit = 109.26 ÷ 3 = 36.42

3 3 1 0 9 1 1

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

2 2

6 6 0

Jordan has $36.42 $

38

Re ga le du ca tio

A jar of jam has a mass of 123.6 grams.

A tub of butter has a mass of 314.6 grams. s.

Sophie buys 6 jars of jam and 4 tubs of butter. Find the total mass of the items.

123.6 g

jam

?

butter

314.6 g

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

1

2

1

2 33 . 6

x

6

7 4

1 . 6

The jars of jam have a mass ss off 741.6 grams. gra gram Now find the mass of the tubss of butter. butter Multiply the mass of 1 tub by 4.

3 11

2

x

4 . 6 4

2 5 8 . 4

1

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

1

+

1

7 14

1

1 . 6

2 5 8 . 4

Can you express the mass in kilograms?

2 0 0 0 . 0

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

39

Let’s Practice Riley made 20 liters of fruit punch. She poured an equal volume me of punch into 4 bowls. She had 1.04 liters remaining. Find the he volume olume of punch in each bowl.

2.

Keira and her baby sister mass of 56 kilograms. ter have a combined com co Keira's mass is 4.63 k kilograms ms less tthan twice the mass of her baby sister. Find Keira's mass. mas

Re ga le du ca tio n

1.

40

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 d he spend in total? (b) Ethan has $10 and he buys 6 scones. Does he have enough ugh money left over to buy a chocolate eclair?

Re ga le du ca tio n

3.

41

At Home A phone company charges $1.22 to connect calls. It then charges rges $0.64 $0.6 per minute. How much does it cost to make an 18-minute ute call? all?

2.

Mr. Romero's car uses 8.36 liters of gas g ga per hour. Mr. Romero filled his car with gas and then There was 8.2 liters of gas he drove ve e for 5 hours. h remaining in his tank. much ank. nk. How H ch gas was in his car before the journey?

Re ga le du ca tio n

1.

42

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

Re ga le du ca tio n

3.

4.

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

43

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

Re ga le du ca tio n

1.

(a) 14.6 and 23.8

sum =

difference =

(b) 12.29 and 10.37

sum =

(c)

123.177 and nd 318.26

sum =

44

difference =

difference =

2.

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

Re ga le du ca tio n

(a) 25.3 x 4 =

(c)

105.72 x 3 =

(e) 503.39 9x9=

(d) 342.84 342 84 x 6 =

(f)

4,207.47 x 4 =

45

3.

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

Re ga le du ca tio n

(a) 17.15 ÷ 7 =

(c)

105.66 ÷ 6 =

(e) 1,448.48 48 ÷ 4 =

46

(d) 349.92 ÷ 8 =

(f)

9,898.28 ÷ 7 =

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 he price o of 3 popcorn. A family of 2 adults and 3 children buy tickets to the he movie mo as well as 4 tubs of popcorn and 3 juice boxes. How much ch did they spend in all?

Re ga le du ca tio n

4.

47

Ratio

Finding Ratio Anchor Task

48

n

6

Let’s Learn

Re ga le du ca tio n

We can use ratio to compare two or more quantities.

Sophie has 3 yellow shirts and 2 pink shirts.

e number mber of pink pin shirts is 3 : 2.

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

Each bunch has the same number er of fruits.

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

49

Re ga le du ca tio n

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

ms to o the number numb num The ratio of the number of chocolate ice creams of strawberry ice creams is 3 : 5.

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

There are 3 chocolate ice creams. There are 8 ice creams in all.

The ratio of the number of chocolate olate e ice creams to the total number of ice creams is 3 : 8.

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

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

50

Let’s Practice Write the ratio.

Re ga le du ca tio n

1.

(a)

The ratio of the number of pencil cases ases to o the number num numb of bags

is

:

.

(b)

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

.

(c)

The ratio of rectangles to triangles is

:

.

51

2.

Write the ratio.

du ca tio n

(a)

The ratio of the number of eggs ggs to o the number numb of donuts is

Re ga l

(b)

:

.

Th ratio ra of pancakes to cookies is The

52

:

.

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

du ca tio n

3.

num The ratio of the number of basketballss to the total number of balls is

.

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

Re ga

4.

:

The of milk cartons to the total number of cartons he ratio tio of the number numb num

is

:

.

53

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

Re ga le du ca tio n

1.

2.

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

3.

Repeat Step 2 for a total of 20 spins.

4.

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

Color

Tally

red

yellow green blue

Color

red to blue

yellow to red

blue to green

blue to green and yellow yellow to red and blue

54

Ratio

At Home Write the ratio.

Re ga le du ca tio n

1.

(a)

The ratio of the number of beef burgers chicken burgers ers to chicke

is

:

.

(b)

The ratio io of the number numbe of lemons to the number of oranges is :

.

(c)

The ratio of circles to squares is

:

.

55

2.

Write the ratio.

du ca tio n

(a)

The ratio of the number of raspberries berries es to the nu number of cherries is

Re ga

(b)

:

of bananas to apples is The ratio o

56

.

:

.

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

Re ga le du ca tio n

3.

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

boxes is

4.

:

.

e nu kitt Find the ratio of the numberr of kittens compared to the total number of pets.

The he ratio rat of the t number of kittens to the total number of pets

is

:

.

57

Let’s Learn

tio n

Ratio and Measurement 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.

Re ga le du ca

Compare the masses of the fruits.

0 4 kg

3 kg

0 4 kg

1 kg g

g 2 kg

3 kg

1 kg

2 kg

The mass off the e watermelon watermelo is 3 kilograms. The mass of the pineapple is 2 kilograms. lograms.

The ratio io off the mass of the th watermelon to the mass of the pineapple apple is 3 : 2.

2 The he ratio of the m mass of the pineapple to the mass of the watermelon atermelon is 2 : 3. aterme 3

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

58

ed uc ati on

Compare the length of the crayons.

The yellow crayon is 7 centimeters in length.

The blue crayon is 5 centimeters in length.

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

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

Compare the volume of liquid in the e beakers.

B

Re g

A

The volume lume e of the liquid in Beaker A is 400 milliliters. The volume of the liquid quid in Beaker B is 300 3 milliliters. volume of the liquid in Beaker A to the volume of the The ratio of the v volum liquid B is 400 : 300. quid in Beaker B 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

ed uc ati on

Use ratio to compare measurements.

1.

A

B

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

:

2.

.

Re ga

400 00 g

300 00 g

100 g

200 g

400 g 300 g

100 g

200 g

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

is

60

:

.

uc ati on

3.

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

:

.

Re ga l

4.

The ratio of the to the length of the he length ength of the toothpaste too

toothbrush h is

:

.

Hands On

1.

Empty mpty the contents of your pencil case onto your desk.

2.

Use a ruler to measure the lengths of 6 objects. Have a mea m classmate the same items from their pencil case. ssmate measure m meas

3.

Use ratios rat to compare the lengths of your objects with your classmate's objects. sm smate' 61

At Home

Re ga le du ca tio n

Use ratio to compare measurements.

1.

The ratio of the length of the fork k to the e length of o the spoon

is

:

.

2.

4 kg g

3 kg

4 kg

1 kg

3 kg

1 kg

2 kg

2 kg

The ratio of o the mass of the apples compared to the total mass of nges and nge an apples is oranges

62

:

.

Hands On

Re ga le du ca tio n

At home, use a ruler to find the length of each object.

Then, use ratio to compare the lengths.

1.

The ratio of the length of my shoe to the length of my father's ather's shoe

is

2.

:

.

vision to the height heig of my television The ratio of the length of my television

is

4.

.

ength th of a fork The ratio of the length of a spatula to the length

is

3.

:

:

.

The ratio of the length of my book to the breadth of my y mathematics mathema

mathematics book is

5.

.

The ratio of the e height eight of my sschoolbag to the height of my chair

is

6.

:

:

.

atio of my height to the height of my sibling The ratio

is

:

.

63

Solve It! The ratio of Sophie's savings to Chelsea's savings is 2 : 1. Iff Sophie phie has $100, how much does Chelsea have? Show your working. ng.

Re ga le du ca tio n

1.

Chelsea has $

2.

.

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

On Mr. Taylor's farm arm m there ther are

3.

Mrs. Taylor bought ught 7 kilogr 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. th tot What iss the mass of the oranges?

Show your working. t o r

The he ma mass of the oranges is

64

chickens.

kilograms.

Equivalent Ratios and Simplest Form or

Re ga le du ca tio n

Let’s Learn

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

We can represent r the ratio using a bar model.

1 unit

oranges apples

Riley puts the fruit on plates in groups roups ps of 2.

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

oranges ang anges apples

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

65

Re ga le du ca tio n

Riley puts the fruit on plates in groups of 4.

tes of apples. apple The ratio of the There is 1 plate of oranges and there are 2 plates number of oranges to the number of apples pples iss 1 : 2.

1 unit

oranges apples

4 : 8, 2 : 4 and 1 : 2 are equivalent valent nt ratios.

1 : 2 is the ratio of oranges to o apples pples in its it simplest form.

We can find the simplest mplest st form of o a ratio by dividing each term in the ratio by the greatest factor. est common fac factor Let's find the simplest lest form of the ratio 12 : 4.

The ratio 12 : 4 has as two terms, term 12 and 4. The greatest 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 1 : 4 in its it simplest form is 3 : 1.

66

We can use multiplication to find missing terms in equivalent ratios.

Re ga le du ca tio n

Let's find the missing term in the equivalent ratio 4 : 5 = 12 : ? y 3. To find the In the equivalent ratio, the first term has been multiplied by missing term, multiply the second term by the same factor. tor. x3

4 : 5

x3

= 12 : 15

The missing term is 15.

4 : 5 and 12 : 15 are equivalent ratios.

g terms ms in equiv We can also use division to find missing equivalent ratios.

Let's find the missing term in the equivalent quivalent ratio 18 : 6 = 3 : ?

d In the equivalent ratio, the first term has been divided by 6. de the e second te tterm by the same factor.

To find the missing term, divide 18 : 6

÷6

÷6

= 3 : 1

The missing term iss 1. uivalent ratios. ratio 18 : 6 and 3 : 1 are equivalent

In the ratio 3 : 1, the factor of the terms is 1. he only common com So, the ratio form. o 3 : 1 is in its simplest simples sim

67

Let’s Practice Find the equivalent ratios.

Re ga le du ca tio n

1.

(a)

The ratio of the number of red cans ns to the number numbe of blue cans

is

:

.

(b)

The ratio of the he number numbe of o red cans to the number of blue cans

is

:

.

(c)

T ratio of the number of red cans to the number of blue cans The

is 68

:

.

Re ga le du ca tio n

(d)

The ratio of the number of red cans to the e number of o blue cans

is

2.

:

.

Use the bar models to write the equivalent quivalent alent ratios.

(a) 1 unit

Ratio =

(b)

1 unit

Ratio atio =

(c)

:

:

1 unit

Ra = Ratio

:

69

3.

Express each ratio in its simplest form. (a)

24 : 6

(b)

÷6

25 : 10 ÷5

(c)

÷5

12 : 6 ÷6

Re ga le du ca tio n

÷6

:

=

50 : 40

(d)

÷ 10

÷ 10

(i)

÷3

:

18 : 42

=

(d)

=

(f)

÷

16 : 40

÷

:

:

=

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

:

(b) 3 : 12 =

(c)

18 : 16 =

:

(d) 32 : 16 =

(e) 4 : 24 =

:

(f)

(g)) 100 : 20 =

:

9:6=

(h) 15 : 40 =

÷

:

=

11 : 33

÷

12 : 4

÷

:

=

(e)

÷

÷

:

=

6 : 30

÷

10 : 30

÷

:

=

(c)

÷

:

:

:

:

÷

÷3

:

÷6

(b)

8 : 24

÷

70

9 : 15

Fill in the blanks. Express each ratio tio in itss simplest simples form.

(a)

5.

=

:

÷3

=

÷3

:

=

(f (f)

÷8

:

27 : 3

(h)

÷6

÷6

4.

÷8

=

30 : 24

(g)

=

16 : 24

(e)

:

=

:

=

÷6

:

÷6

6.

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

: 5 = 5 : 25

Re ga le du ca tio n

(a) 15 : 6 = 5 :

(c)

: 9 = 21 : 27

(d) 32 : 8 = 4 :

(e) 3 : 1 = 9 :

(f)

(g)

(h) 36 : 12 = 72 :

(i)

: 5 = 27 : 45

5:6=

: 48

(j)

: 4 = 28 : 16

9 : 4 = 90 :

71

At Home Express each ratio in its simplest form.

Re ga le du ca tio n

1.

(a)

12 : 10

48 : 8

:

=

2.

Fill in the blanks. Express each form. h ratio atio in its simplest simp

(a)

7 : 21

÷

(c)

(e)

=

(f)

÷

10 : 12

÷

:

:

=

(c)

:

32 : 48 =

(e)) 4 : 24 =

(g) 90 : 60 =

:

(b) 20 : 16 =

:

(d) 64 : 24 =

:

(f)

:

:

21 : 6 =

(h) 18 : 63 =

÷

:

Write rite each ratio in its ssimplest form. (a) 30 : 6 =

72

30 : 45

÷

÷

:

=

16 : 2

÷

(d)

÷

:

=

28 : 14

÷

=

32 : 48

÷

(b) b)

÷

:

=

3.

32 : 16

=

:

:

÷

÷ 10

:

÷ 16

:

=

110 0 : 30

(i)

÷6

÷6

÷9

:

÷ 10

=

18 : 48

(h)

÷9

÷9

(f)

÷8

:

=

27 : 18

(g)

÷8

:

27 : 9

÷9

=

32 : 24

(e)

÷8

(c)

÷5

:

=

÷8

=

25 : 15

÷5

:

=

(d)

(b)

÷2

÷2

:

÷ 16

4.

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

: 21 = 21 : 7

Re ga le du ca tio n

(a) 27 : 18 = 54 :

(c)

: 16 = 22 : 32

(d) 9 : 6 = 36 :

(e) 12 : 1 = 48 :

(f)

(g)

(h) 18 : 12 = 54 :

(i)

: 25 = 20 0 : 100

16 : 4 =

:1

(j)

: 1 = 27 : 9

49 : 14 = 7 :

73

Word Problems

Re ga le du ca tio n

Let’s Learn

At Mermaid College, there are 60 students in Grade 5. 35 students dents are girls. gir g Find the ratio of the number of boys to the number of girls in n Grade 5.

5 Express the ratio in its simplest form.

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

ber of girls is 25 : 35.

The ratio of the number of boys to the number 1 unit

Boys

60 sstudents

Girls

25 : 35

÷5

÷5

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

y and Halle ma On the weekend, Riley made $63 selling lemonade. Riley sold hey split the th money mo more lemonade,, so they at a ratio of 4 : 3.

How much money make? oney y did each child c m 1 unit

Riley

$63

7 units 1 unit

$63 $63 ÷ 7 = $9

Halle

Riley received 4 units units.

4 units 4 x $9 = $36

Halle received 3 units.

3 units 3 x $9 = $27

Riley made $36 $3 and Halle made $27.

74

Re ga le du ca tio n

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.

Ethan

?

Blake

24 kg

3 units 1 unit

24 kg 24 kg ÷ 3 = 8 kg

Ethan's suitcase tcase se = 7 units.

7 units 7 x 8 kg = 5 56 kg

Total mass of the suitcases = 24 kg + 56 kg g = 80 0 kg k The total mass of both suitcases is 80 kg. g.

Keira has a piece of rope that is 48 length.

8 meters in lengt leng She cuts the rope into two piecess in the ratio 9 : 77. e.

Find the length of each piece of rope. ?m

48 m

?m

16 units 1 unit

48 m 48 m ÷ 16 = 3 m

Length off longer rope = 9 units. nger piece of rop ro 9 units 9 x 3 m = 27 m

Length ngth of shorter p piece of rope = 7 units.

7 units uni 7 x 3 m = 21 m

The length ength of the longer piece of rope is 27 m.

The length the shorter piece of rope is 21 m. h of th

75

Let’s Practice Joe's Aquarium sells goldfish and turtles. On Saturday, the e ratio tio 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? d?

Re ga le du ca tio n

1.

On Saturday, Joe's Aquarium sold old

2.

turtles. tur

5 minutes inutes doing his mathematics and science Wyatt spent a total of 55 he first rst 35 minutes minu mi homework. He spent the doing his mathematics homework and the rest of the time doing his science homework. Find the ratio of the time doing homework to the time spent me spent ssp oing science s ticss homework. hom homew doing mathematics Write the ratio in its simplest form.

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

76

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.

Re ga le du ca tio n

3.

The mass of the oranges is

4.

kilograms. ograms.

ted 49 butterflies. butterflies She spotted 35 orange On a field trip, Chelsea spotted rflies she sspotted were blue. Find butterflies. The rest of the butterflies range butterflies butte the ratio of the number of orange to the number of blue otted. d. Express the ratio in its simplest form. butterflies Chelsea spotted.

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

77

At Home The ratio of the number of boys in the schoolyard to the number mber of girls in the schoolyard is 4 : 7. There are 63 girls in the schoolyard. hoolyard. yard. How many boys are in the schoolyard?

Re ga le du ca tio n

1.

There are

2.

Sophie is using pink and purple rple beads to make necklaces for her friends. The ratio of the e number mber of pink pin beads to the number of purple beads is 8 : 5. Sophie used beads than purple beads. How ed 24 more pink p many purple beads ds did Sophie hie use? use

Sop Sophie used

78

boys in the schoolyard. choolyard.

purple beads.

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

Re ga le du ca tio n

3.

Mr. Whyte spent

4.

on accommodation commodation o only.

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

shorter piece of wood is The sshor ter pie

in length.

79

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

Re ga le du ca tio n

1.

2.

80

Blake has 60 more marbles rbles es than Keira.

Keira Keir The ratio of the number of Blake's marbles to the number marbles is 7 : 3. If Blake gives mber of Keira's Ke K Keira 5 marbles, what ha will the he ratio of Blake's marbles to Keira's marbles be? Write the ratio io o in its simplest s st form. Show your working.

Looking Back Write the ratio.

Re ga le du ca tio n

1.

(a)

The ratio of the number of yellow roses oses to o red roses

is

:

.

(b)

The ratio off the numbe number hamsters to the numb b off h h totall number b off pets

is

:

.

(c)

ratio of the length of the nail to the length of the Th rat The

screw is

:

.

81

2.

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

Re ga le du ca tio n

1 unit

Ratio =

(b)

:

1 unit

Ratio =

3.

:

Express each ratio in its simplest est st form.

(a)

2 : 8

÷2

36 : 27

÷9

:

:

=

(b)

35 : 21

÷

(c)

÷

=

(d)

÷

33 : 22

÷

:

:

=

Write each ratio in its it simplest form. (a) (a 4 40 : 10 =

:

(b) 25 : 15 =

(c)

48 : 36 =

:

(d) 54 : 9 =

(e) 24 : 6 60 =

:

(f)

21 : 9 =

÷

:

=

72 2 : 63

÷

12 : 48

÷

:

=

82

=

(c)

÷5

Fill in the blanks. Express xpre each xp ch ratio in its simplest form.

(a)

5.

35 : 45

÷5

:

=

4.

(b))

÷2

:

:

:

÷

÷9

6.

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

: 3 = 20 : 60

Re ga le du ca tio n

(a) 27 : 81 = 3 :

(c)

7.

: 8 = 36 : 48

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

man caught aught The fisherman

8.

(d) 7 : 9 = 49 :

kilograms of crabs. k

The ratio to students at Broadbeach College o of the number of teachers t is 2 : 7. There students than teachers. How many teachers e are 125 more m roadbeach College?

Colle Col are at Broadbeach Show your working.

There are

teachers at Broadbeach College.

83

Geometry

Re ga le

Anchor Task

84

on

7

Types of Triangles

Re ga le du ca tio n

Let’s Learn

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

This triangle is called an equilateral triangle.

A

It has the following owing properties: prope

x

• Allll sides length. des are are the same s We write: AB = BC = CA

y

• All internal a angles are the same.

We write: write x = y = z

z

C

B

The markings marki w sides of equal show le length.

On this triangle, all sides are equal! s

These figures triangles. res are re all equilateral equilatera equilat E

M

D

O

T

F

N

85

This triangle is called a right-angled triangle.

Right-angled triangles have one right angle (90º).

Re ga le du ca tio n

We mark right angles with 2 straight lines. PRQ is a g ang right angle.

P

Q

R

gles.

These figures are all right-angled triangles.

Y

B

A

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

C

X

M

Z

N

O

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

86

D

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

L

Re ga le du ca tio n

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

M

N

LMN =

LNM

es.

These figures are all isosceles triangles.

A

Triangle ABC is both an isosceles and an equilateral triangle.

F

C

B

Q

P

P

R

T

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

U

V

87

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

Re ga le du ca tio n

We use se different ngs on each eac markings sidee to show the they are not equ equal.

T

These figures are scalene triangles.

N

Q

12 in

14 m

17 m

M

P

18 in

20 m

R

8 in

O

A

D

This scalene ttriangle is also a right-angled triangle!

5 cm

3 cm

E

88

4 cm

F

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

Re ga le du ca tio n

K 79º

What is the sum m of the angles?

J

57º

44º

L

These figures are acute triangles.

Q

81º

11 cm

This acute triangle is also a scalene triangle!

12 cm

47º

52º

P

R

15 cm

38º

This acut acute trianglee is also an isosceles triangle! riangle

71º

71º

60º

60º

60º

This acute triangle is also an equilateral triangle!

89

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

Re ga le du ca tio n

X

Which angle iss greater tha than 90º??

96º

33º

51º

Z

Y

These figures are obtuse triangles.

This obtuse triangle is also a scalene triangle!

30º

131º

19º

A

98º

41º

41º

B

90

C

Triangle ABC is also an isosceles triangle!

Let’s Practice Use a ruler to measure the sides of the triangles.

Check to classify the triangle. You may check more than an 1 box. ox.

Re ga le du ca tio n

1.

(a)

Equilateral al

66º

Isosceles eles es

48º

66º

Scalene calene ne

(b)

Equilateral Equilate

45º

Isosceles Isos Isosc

90º

Scalene S

45º

(c)

Equilateral

60º

Isosceles

60º

60º 60

Scalene

(d)

Equilateral

68º

Isosceles

59º

53º

Scalene

91

2.

Use a protractor to measure the internal angles of the triangles.

Check to classify the triangle. You may check more than 1 box.

Re ga le du ca tio n

(a)

B

Equilatera Equilateral

Right-angled Right-a Right-an Acute Acu Acut

C

A

Obtuse O

(b)

Equilateral

Right-angled

T

Acute

Obtuse

(c)

Equilateral

Right-angled

31º

127º M

22º

Acute

Obtuse

(d)

P

Equilateral

Right-angled

Q

R

92

Acute

Obtuse

Use a ruler and pencil to draw and label the following figures on the dot paper. (a) (b) (c) (d) (e)

Right-angled triangle Scalene triangle Acute triangle Isosceles triangle Obtuse triangle

Re ga le du ca tio n

3.

93

Hands On

Re ga le du ca tio n

Work in pairs to classify each triangle.

41º

75º

109º

C

A

73º

B

32º

30 30º

62º

75º

D

E

75º

F

59º

59º

30º

64º

91º

G

49º

H

67º

59º

30º

I

94

Re ga le du ca tio n 45º

L

K

J

90º

45º

Triangle

Type(s) pe(s

A B

C

D E F

G H I

J

K L

95

At Home Classify each triangle.

Re ga le du ca tio n

1.

(a)

Right-angled Right-an Scalene Scale

Isosceles Is

(b)

Right-angled Scalene

Isosceles

(c)

Right-angled Scalene

Isosceles

(d)

Right-angled Scalene

Isosceles

96

2.

Write one classification per triangle. (b)

Re ga le du ca tio n

(a)

(c)

(d)

(e)

(f)

97

Anchor Task

98

n

Angles of Triangles

Let’s Learn

Re ga le du ca tio n

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

Line ST is a straight line.

Add SVU and UVT.

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

U

S

T

V

180o

SVU +

UVT = 126o + 54o = 180o

126o

54o

You have learned that the angles always combine to form a gles es of a triangle tr tri straight line.

The sum of the internal inter angles of a triangle is 180º

Find the unknown own angle of each ea ttriangle. (a)

A

C

23º

31º

B

180o

23o

ABC C = 180 180º – 23º – 31º = 126º

31o

ABC ?

99

Q

Re ga le du ca tio n

(b)

34º

R

43º

P

180º

43o

34o

PQR ?

PQR = 180º – 43º – 34º = 103º

(c)

W

66º

c

Y

180 180º

90o

66o

c

?

c = 180º – 90º – 66º = 24º

100

X

S

(d)

This is an ngle. equilateral triangle. es a All of the angles are the same!

Re ga le du ca tio n

a

b

c

U

T

a+ a=

b+ b=

c = 180º c = 180º ÷ 3 = 60º

Each angle is 60º.

(e)

This is an isosceles triangle.

Two of the angles are the same!

X

32º 2º

Y

Z

180º – 32º = 148º 14 ZXY +

ZXY =

XZY = 148º

XZY = 148º 1 ÷2 = 74º 74

1 01

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

Re ga le du ca tio n

1.

(a)

E

75º

37º

D

EDF =

F

(b)

x

90º

31º

x=

(c)

B

A

30º

73º

C

ABC =

(d)

t

t=

1 02

(e)

Re ga le du ca tio n

70º

a

a=

(f)

m

68º

m=

(g)

42º

p=

p

(h)

T

U

V

TVU =

1 03

2.

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

b.

Re ga le du ca tio n

(a)

a

a= b=

b

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

A

w

y

B

x

x=

C

(c)

w=

y=

p is

5 the size of o 6

q.

114º

p

q

p=

q=

104

Hands On

Re ga le du ca tio n

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

Work with your partner to draw these Pythagorean triples. riples.

(5, 12, 13), (6, 8, 10) and (8, 15, 17) Can you find more?

5 cm

3 cm

4 cm

1 05

At Home

Re ga le du ca tio n

Find the unknown angle. Show your working. (a)

t

41º

t=

(b)

v

v=

(c)

B

62º 2º

C

39º

ABC =

A

(d)

z

31º

106

z=

Solve It!

Re ga le du ca tio n

Find the unknown angles.

(a) LMO is a triangle. LN is a straight line.

L

a

67º

49º

a=

N

M

O

(b) ABC is an equilateral triangle.

AD is a straight line A

39º

D

B

(c)

u=

u

60º

C

PQR is an isosceles osceles triangle. trian PS is a straight ght line.

P

50º 5

40º

Q

S

PSQ =

R

1 07

Angles of Quadrilaterals

Re ga le du ca tio n

Anchor Task

108

Re ga le du ca tio n

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

The sum of the angles of a triangle is 180º.

The sum of 2 triangles must be 360º!

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

A

B

C

D

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

1 09

Re ga le du ca tio n

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. e para quadrilaterals

trapezoids

• at least 1 pair of parallel sides

parallelograms

• 2 pairs of parallel sides des

• no o parallel pa sides

e equal ual • opposite sides are

qua • opposite angless are equal

rrhombi h

rectangles

• all angles are e equal equ

• all sides are equal

squares sq

• all sides and angles are equal

110

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

Re ga le du ca tio n

A B

72º

360 6 o 60

99 º

77º

72o

77o

99o

ABC

C

D

?

ABC = 360º – 72º – 77º– 99º

= 112º

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

L

y

x

M

360o

z

38o

38º

O

x

y

z

N

Opposite angles are e equal. al So, y = 38º x+

z = 360ºº – 38º – 38º = 284º 4º

x=

z = 284º 4º ÷ 2 = 142º 42

x = 142º,

y = 38º, 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

Re ga le du ca tio n

A

33º

D

C

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

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

I

H

43º

K

In a parallelogram, are equal. am, opposite pposite angles an IJK = KHI = 43º HIJ + HIJ =

112

HKJ KJ = 360º 60º – (2 x 43º) 43 = 274º HKJ KJ = 274º ÷ 2 = 137º

J

Let’s Practice

Re ga le du ca tio n

Find the unknown angle. Show your working. (a)

L

61º

M

110º

O

(b)

N

R

LMN M =

S

73º

110º

115º

U

T

RST =

(c)

F

E

91º

H

G

EFG =

FGH =

GHE =

1 13

(d)

B

A

Re ga le du ca tio n

92º

C

D

BCD D=

(e)

G

H

78º

I

GHI =

J

(f)

M

N

60º

P

O

PMN =

MNO = NOP =

114

(g) B

A

Re ga le du ca tio n

155º

C

D

BCD =

(h)

CDA =

DAB =

R

Q

51º

S

T

QRS =

STQ =

TQR =

1 15

Hands On

Re ga le du ca tio n

Work in pairs to classify each quadrilateral.

B

A

C

D

E

F

116

Re ga le du ca tio n G

H

J

I

Quadrilateral

Type(s) Type

A B

C

D E F

G H I

J

1 17

At Home

Re ga le du ca tio n

Find the unknown angle. Show your working. (a)

W

X

72º

92º

104º

Y

WZ = WZY

Z

(b)

A

B

72º

112º

77º

C

D

(c)

ADC =

D

68º 8º

E

G

F

DEF =

118

A

(d)

B

x

Re ga le du ca tio n

120º

y

z

C

D

x= y= z=

(e)

M

L

55º

N

O

LMN =

MNO = NOL =

1 19

Solve It!

Re ga le du ca tio n

(a) OPQR is a parallelogram.

SP is a straight line. Find OPS. O

118º

P

20º

R

S

Q

(b) GHIJ is a parallelogram.

HJ is a straight line. Find m.

G

56º

H

m

J

44º

I

120

Looking Back Find the unknown angle. Show your working.

eg al ed uc ati on

1.

(a)

E

68º

37º

D

DEF =

F

(b)

59º

x

x=

(c)

B

77º

A

30º

C

BCA =

(d)

q

q=

121

2.

Classify each figure. Choose one classification per figure. (b)

Re ga le du ca tio n

(a)

122

(c)

(d)

(e)

(f)

3.

Find the unknown angle. Show your working.

A

B

Re ga le du ca tio n

(a)

73º

62º

115º

D

C

ADC =

(b)

44º

y

y=

(c)

X

Y

120º 0º

W

Z

XYZ =

YZW =

ZWX =

123

Anchor Task

on

8

Measurement

1,200 g = 1.2 kg 2 km 600 m = 2.6 km m 5 weeks =

1 24

Converting Measurement Units

Re ga le du ca tio n

Let’s Learn

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

Length

Metric

Customary

1 centimeter (cm) = 10 millimeters (mm)

1 foot (ft) = 12 inches (in) (i

1 decimeter (dm) = 10 centimeters

1 yard (yd) = 3 feet

1 meter (m) = 100 centimeters

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

1 kilometer (km) = 1,000 meters

Mass

Metric

Customary

1 pound (lb) = 16 ounces (oz)

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

1 ton (T) = 2,000 pounds

Volume Volum

Metric

Customary

1 cup (c) = 8 fluid ounces (fl oz) 1 pint (pt) = 2 cups

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

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

Re ga le du ca tio n

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 mad in pints?

How much money will they raise if they sell all of their lle lemonade?

4 gallons = 4 x 8 pt = 32 pt

So, Sophie and Riley made 32 pints of lemonade. lemona 32 x $3 = $96

Sophie and Riley will raise aise ise $96 $ if they hey sell se all of their lemonade.

Jordan's newborn n kitten ten weighed 275 grams. It now ow weighs 12 times tim as much as it did as a newborn. newbo rams does How many kilograms Jordan's kitten weigh now?

275 g x 12 = 3,300 g

300 g ÷ 1,000 = 3.3 k 3,300 kg

Jordan's weighs 3.3 kilograms now. dan's kitten k we

1 26

Mrs. Jenkins has 10

1 pounds of 2

Re ga le du ca tio

cooked rice. She needs 4 ounces of rice to make a serving of Thai green curry. If she makes 40 servings of Thai green curry, how many ounces of rice will Mrs. Jenkins have left?

1

10 2 lb x 16 = 168 oz

Mrs. Jenkins has 168 ounces of rice.

40 x 4 oz = 160 oz

168 oz – 160 oz = 8 oz

ce left.

Mrs Jenkins will have 8 ounces of rice

Wyatt has 1.5 liters of water in n his drink nk bottle. bottle After a run, he drinks 525 milliliters water. ers of water wate How much water is left in his is drink rink bottle? bottle 1.5 l = 1.5 x 1,000 ml = 1,500 ml

Before the run, there mll o of water in here was 1,500 m Wyatt's drink bottle. e.

1,500 ml – 525 m mll = 975 m mll

There iss 975 5 ml ml of water left le in Wyatt's drink bottle.

127

Let’s Practice Convert the customary units of length. Show your working. g.

Re ga le du ca tio n

1.

1

(a) 2 2 ft =

(c)

5 mi =

1

(e) 2 4 mi =

2.

(b) 66 in =

yd

(d) 366 6 ft =

yd

(f)

2 yd =

ft

yd

in

Convert the metric units off length. Show your working.

S Sh

(a) 2.6 km =

m

(b) 12,345 m =

(c)

m

(d) 390 cm =

mm

(f)

25 dm =

(e) 6.32 m =

1 28

in

296 mm =

km

dm

cm

3.

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

(b) 256 oz =

lb

Re ga le du ca tio n

(a) 17 lb =

(c)

4.

4,000 lb =

T

(d) 128 ozz =

lb

(e) 9,000 lb =

T

(f)

lb

352 oz =

Convert the metric units of mass. ss. Show your y working. (a) 17 kg =

(c)

g

(b) 3,250 g =

8.54 kg =

g

(d) 13,400 g =

(e) e)) 10.11 .11 k kg g=

g

(f)

1,270 g =

kg

kg

kg

129

5.

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

(b) 32 fl oz =

c

Re ga le du ca tio n

(a) 15 pt =

(c)

44 pt =

(e) 1.5 qt =

6.

gal

(d) 12 gal al =

qt

fl oz

(f)

pt

Convert the metric unitss of volume. Show Sho your working. l

(a) 1,200 ml =

l

(b) 620 ml =

2.25 l =

mll m

(d) 9,900 ml =

(d) 0.22 l =

ml

(e) 3.11 l =

(c)

130

66 cu =

l

ml

At Home Convert the customary units of measurement. Show yourr working. orking.

Re ga le du ca tio n

1.

(a) 4.5 lb =

(c)

oz

T

11,000 lb =

(b) 96 oz =

lb

(d) 6.2 2T =

lb

(e) 192 in =

ft

(f)

(g) 51 yd =

ft

(h) 2.5 mi =

(i)

13 pt =

fl oz

(j)

24 qt =

(k)

10.5 gal =

(l)

12 fl oz =

pt

72 7 ft =

in

yd

gal

pt

1 31

2.

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

(b) 0.02 kg =

g

Re ga le du ca tio n

(a) 12.1 kg =

(c)

1,250 g =

kg

l

(e) 22,200 ml =

(g) 0.13 l =

132

(i)

91 dm =

(k)

12.8 8 km =

(d) 15,300 00 0g=

ml

mll m

m

(f)

1.22 1 l=

(h) 950 ml =

(j)

92.5 cm =

(l)

0.41 m =

kg k

ml

l

mm

cm

Word Problems

Re ga le du ca tio n

Let’s Practice

1.

Ethan fills a watering can with 3 gallons of water. After watering his garden, there are 2 quarts of water remaining in the watering can.

How many quarts of water did Ethan use?

2.

A bottle contains 32 fluid flu ounces es of ketchup.

How many pintss of ketchup etchup are a in 4 bottles?

1 33

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.

Re ga le du ca tio

3.

4.

e athletics hletics track trac tra at school. ool.

Sophie ran 9 laps of the 1

She ran a total distance tan of 4 2 kilo tance kilometers.

ters is 1 lap o of the track?

How many meters

1 34

Jordan and Wyatt make 12 liters of lemonade to sell at the local market. e.

Each cup 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?

Re ga le du ca tio

5.

6.

es. es trawberrie A farmer has 24 pounds of strawberrie strawberries.

He packs the strawberries punnets ries into punnet punne s weighing 12 ounces each. ch. How many man punnets of strawberries err can n the farmer pack?

1 35

A bag of potatoes weighs 64 ounces.

Mr. Whyte buys 4 bags of potatoes.

How many pounds of potatoes did Mr. Whyte buy?

Re ga le du ca tio n

7.

8.

1 36

Wyatt is giving away one balloon on to each friend who attendss his birthday party. On each balloon n he ties 2.5 feet fe of string. How many yards ds of string will Wyatt need if 12 attend his 2 friends fri birthday party?

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?

Re ga le du ca tio

9.

10. Railway workers install 58 meters rs of train track per day. It takes them em 47 days to install in tall a track between Beach Station on and Sunnybank S Sun bank e between the th t Station. Find the distance stations in kilometres e and meters. es

1 37

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

Re ga le du ca tio n

11.

12.

1 38

Halle buys ribbon to tie bowss on some gifts. She uses 8 lengthss of ribbon that th are each 36 cm in length and nd has 22 cm of ribbon left. Find the e total to length ngth gth of ribbon she bought in meters. eters. ers.

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?

Re ga le du ca tio

13.

14.

uit Sophie and Halle made 4.5 gallons of fru fruit punch to sell at the town wn fair. air. They plan to sell the fruit punch for $2.5 per pint. How H much money will Sophie and Sop nd Halle make e if they sell all of the punch?? he e fruit p

1 39

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

Re ga le du ca

1.

140

A bathtub contains 120 liters of water.

Ethan pulls the plug and the water drainss 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?

Re ga le du ca t

2.

1 41

At Home Keira is building a square picture frame me with a side length of 15 inches. She cuts the sides of the frame from a piece of wood thatt is 6 feet long. What is the length of the wood leftover?

Express your answer in inches.

Re ga le du ca ti n

1.

2.

142

Danny the bricklayer loads pounds oadss 9,000 po pou unds of bricks onto his truck. He 2,000 e uses 2,00 2,0 0 pounds of bricks to wall. o build bu a retaining t i taini What is the mass left on ss off the bricks b br n his truck? Expresss your ur answer in i tons. t

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

Re ga le du ca tio

3.

4.

Blake is on summer break for 6 weeks. On O the first day of summer break, ak, he goes on a diving trip with his father er for or 12 days. He H then stays at his grandmother's er's house fo for 2 weeks.

How long before Blake ake goess back to t school? hool?

Express your answer and wer in weeks w an days. s.

1 43

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.

Re ga le du ca tio n

5.

6.

1 44

There is a dance performance mance at the local loca loc department store. There e are re 3 shows of o the same duration with h a 15-minute 5-minute break b between each show. The first rst show starts at midday. The last finished stt show sh shed at 4:00 p.m. What is the e length leng of 1 show in hours and minutes? utes? s?

7.

It takes Riley 3 minutes to walk 250 meters. ers.

How far does Riley walk in 1

1 hours?

2

Re ga le du ca tio

Express your answer in kilometers.

8.

A painter needs 4 gallons of paint aint to complete painting a living room. om. m. He finds 3 old tins of paint. The first tin in contains ontains 2 quarts of paint. The second nd tin contains 5 pints of paint. The third d tin n contains 1 gallon of paint. Does the he painter ainter hav have enough paint to finish painting inting ting the living lil room? If no, how much muc more e paint pain will he need? If yes, how will be left w much paint pa p over? Express you u answer in pints. pi

1 45

Solve It!

Re ga le du ca tio n

A computer store sells data cables in 2 price plans.

Plan 1:

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

Coilss of of cable cabl ble p please lea ase N atcha Natcha

Plan 2: 40¢ per meter for any length.

4 km of data cable. cabl 5

1.

Find the cheaper plan for buying

2.

How much on the cheaper plan? uch money is saved sa

146

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

Re ga le du ca tio n

3.

4.

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

1 47

Looking Back Convert the units of measurement. Show your working.

Re ga le du ca tio n

1.

(a) 12.5 km =

(c)

1.2 gal =

(e) 8.23 m =

2.

m

pt

mm

(b) 12,345 m =

(d) 6.5 T =

(f)

56 fl f oz =

lb

qt

A baker bakes 20 baguettes ettes es that ar are each 1 foot and 1 inch in length.

He cuts each baguette uette iinto slices ces that are 3

baguette slices can n the bak baker make?

148

km

1 inch in length. How many 4

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 omple d to run to t laps of the running track. How many more laps does he need cover a total distance of 6 kilometers?

Re ga le du ca tio n

3.

4.

The school football coach of sports drink for his ach prepared 3 gallons g team of 14 players. After player drank 3 cups of sports er the he game, each e drink. How many pints int of sports ports drink dri are remaining?

1 49

Volume

Volume and Unit Cubes Anchor Task

150

on

9

Let’s Learn

Re ga le du ca tio n

Compare the tennis ball and the soccer ball. Which takes up more e space?

The soccer ball all is bigger and ta takes up more space.

The soccer ball takes up more space than ball. The amount han the tennis ba b of space an object takes up is its volume. volume of the soccer me. The he volum ball is greater than the volume of the he tennis ball.

Compare the volume of the boxes. xes.

Box ox A

Box B

The volume of Box A is greater the volume of Box B. grea than t The volume than the volume of Box A. e of Box B is smaller sma We compared the volumes of the boxes. v How H can we measure the volumes?

1 51

Re ga le du ca tio n

The solids below are rectangular prisms.

Which prism has the greater volume?

Prism A

Prism B

To find out which prism has the greater volume,, we can divide tthem into unit cubes. Each edge in a unit cube is 1 unit in length. has a th. A unit cube cu c volume of 1 cubic unit.

Alll sides of thee cube have tthe me len same length.

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

Prism B can be divided into 6 unit cubes. It has a volume of 6 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 row of 4 unit cubes.

Re ga le du ca tio n

4 unit cubes 4 unit cubes 4 unit cubes

The prism is made up of 12 unit cubes. Its volume e is 12 2 cubic units. units unit Let's find the volume of this Solid A.

ube 1 unit cube

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

2 unit cubes cub

6 unit cubes

Solid A

Solid A is made up of 9 unit cubes. bes. Its volume vo volu is 9 cubic units.

Let's find the volume off this Solid So B.

1 unit cube c

Can you divide the solid into layers another way?

3 unit cubes

5 unit cubes

Solid B

9 unit cubes

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

153

Solid E

ed uc ati on

Let's find the volume of these solids.

Solid E is made up of 15 unit cubes.

Its volume is 15 cubic units.

Solid G

Re ga

Solid G is made up of 9 unit un cubes. bes. es.

Its volume is 9 cubic units. nits.

Solid I

Solid d I is m made up of 14 unit cubes.

Its volume cubic units. me is 14 c

154

olid F Solid

2 unit cubes.

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

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 Compare the volumes of the objects.

Check the object with the greater volume.

Re ga le du ca tio n

1.

(a)

(b)

(c)

(d)

155

Compare the volumes of the rectangular prisms. Fill in the blanks.

Circle the prism with the greater volume.

Re ga le du ca tio n

2.

(a)

Prism A

Prism B

Prism A has a volume of

cubic units. s.

Prism B has a volume of

cubic bic units. its

(b)

Prism C

Prism D

Prism C has a volume volu o off

cubic units.

Prism D hass a volume ume of

cubic units.

(c)

Prism E

156

Prism F

Prism E has ha a volume of

cubic units.

Prism F has a volume of Pr

cubic units.

Re ga le du ca tio n

(d)

Prism G

Prism rism H

Prism G has a volume of

cubic bic units. nits

Prism H has a volume of

cubic c units.

(e)

Prism mI

Prism I has a volume volume of o

Prism of sm J has a volume volu o

Prism J

cubic units.

cubic units.

157

Hands On With a classmate, use unit cubes to build each rectangular ar prism. rism.

Record the volume of each prism in the table.

Re ga le du ca tio n

1.

Prism P

Prism Q

Prism R

Prism T

Prism rism S

Prism

Volume (cubic ubic uni units)

P

Q R S

T

U

2.

1 58

Prism U

Take a cuboid apart apa and build another cuboid of the same volume. a Did the volume of volu o the cuboid change when you re-built it?

Explain your answer. you an

Let’s Practice Compare the volumes of the solids. Fill in the blanks.

Circle the solid with the smaller volume. (a)

Solid A

ed uc ati on

1.

Solid B

Solid A has a volume of

cubic ubic c units. unit

Solid B has a volume of

cubic ubic units.

(b)

Solid C

Solid D

olume o Solid C has a volume of

cubic units.

Solid D has as a volume of of

cubic units.

Re g

(c)

Solid So E

Solid F

has a volume of So E h Solid

cubic units.

d F has a volume of Solid

cubic units.

159

Re ga le du ca tio n

(d)

Solid G

Solid H

Solid G has a volume of

cubic units. s.

Solid H has a volume of

cubic bic units. nits.

(e)

Solid I

Solid I has a volume off lu

Solid J has a volume ume of of

Solid J

cubic units.

cubic units.

(f)

S Solid K

160

Solid L

has a volume of So K h Solid

cubic units.

d L has a volume of Solid

cubic units.

Match the solids with the same volume.

Re ga le du ca tio n

2.

1 61

Hands On With a classmate, use unit cubes to build each solid.

Record the volume of each solid in the table below.

Re ga le du ca tio n

1.

Solid U

Solid V

Solid S W

Solid X

lid Y Solid

Solid Z

Solid

Volume (cubic ubic units)

U

V

W X Y Z

2.

Arrange the solids in order from the smallest volume to the s greatest eatest volume. vol volume

smallest volume

1 62

greatest volume

At Home Circle the object with the smaller volume.

Re ga le du ca tio n

1.

(a)

(b)

2.

Arrange the balls alls in n order from fr fro the greatest volume to the smallest volume. me.

bowling wling ball

greatest volume olu

beach ball

basketball

smallest volume

1 63

Find the volume of each prism. Circle the prism with the greater volume.

Circle both solids if they have the same volume.

Re ga le du ca tio n

3.

(a)

cubic units

cubic u units

cubic bic units

cubic units

c cubic units

cubic units

(b)

(c)

1 64

Find the volume of each solid. Circle the solid with the smaller volume.

Circle both solids if they have the same volume.

Re ga le du ca tio n

4.

(a)

cubic units

cubic unit units

cubic c units un

cubic units

cubic units

cubic units

(b)

(c)

1 65

Solve It! Compare the solids.

Re ga le du ca tio n

1.

Solid W

Solid dX

(a) What is the volume of Solid W?

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

2.

Compare the solids.

Solid olid Y

Solid Z

(a) What is the vo volume of Solid Y?

(b) How m man many y unit cubes need to be taken away from Solid Y to make So Z? Solid

(c)

1 66

What is the volume of Solid Z?

Halle is using unit cubes to build stairs.

Re ga le du ca tio n

3.

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

(b) Halle wants to build the stairs irs to a height o of 6 steps. Complete the table to show how many will need to add any ny unit cubes Halle H for each step.

Steps

Volume me (cubic units) u

1

3

2

3 4 5 6

(c) c)

what will be the total volume of Halle's stairs?

Once nce competed, c competed omp

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

1 67

Anchor Task

168

n

Volume of Rectangular Prisms

Let’s Learn

Re ga le du ca tio n

Riley makes a rectangular prism from 1-centimeter cubes.

1 cm

1 cm

Each cube has a side length of 1 cm.

The volume of each cube is 1 cubic ubic c cm.

3 We write: 1 cm We say: 1 cubic centimeter eter t

1 cm

2 units = 2 cm

m 3 units = 3 cm

4 units = 4 cm

2 cm

4 x 3 x 2 = 24.

The rectangular prism has a volume of 24 cm3.

3 cm

4 cm

Volume of a rectangular tangular pr 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!

1 69

Re ga le du ca tio n

Let's identify the length, width and height of these rectangular prisms.

Make sure to include the correct units.

3 cm

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

8 cm

12 cm

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

6 in

6 in

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

6 in

8 ft

8 ft

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

20 ft

4m

9m

12 2m

170

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 0x4 = 240 ft 3

Re ga le du ca tio n

4 ft

5 ft

12 ft

(b)

6 cm

olum = l x w x h Volume =8x7x6 = 56 x 6 = 336 cm3

7 cm

5 6 x 6 3 3 6

8 cm

(c)

8m

Volume = l x w x h =8x8x8 = 64 x 8 = 512 m3

8m

8m

x

6 4 8 5 1 2

(d)

7 in

9 in

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

23 in 2

1 71

Let’s Practice Find the length, width and height of the rectangular prisms. ms.

Re ga le du ca tio n

1.

(a)

1 cm

1 cm

Length gth = Width dth

=

Height eight

=

(b)

Length gth =

(c)

Length = Width idth

=

Height He Heig

=

(d)

Length =

172

Width

=

Height

=

Width Wid Widt

=

Height H

=

(e) 11 cm

Re ga le du ca tio n

Length th = Width h

=

Height

=

9 cm

14 cm

(f)

9 in

Length = Le

n 5 in

Width

=

Height

=

13 in

(g)

12 m

Length = Width

=

Height

=

10 m

28 8m

(h)

15 ft

Length = Width

=

Height

=

12 ft

32 ft

1 73

2.

Find the volume of the rectangular prisms. Show your working.

Re ga le du ca tio n

(a)

14 cm

12 cm

16 cm

Volume =

(b)

25 5 cm

c 12 cm

Volume =

7 cm

(c)

10 in

12 in

19 in

Volume =

174

Re ga le du ca tio n

(d)

16 m

16 m

Volume m =

16 m

(e)

9 cm

14 cm

11 cm

Volume =

(f)

9 ft

7 ft

Volume =

3 ft

1 75

At Home Complete the table. Show your working.

Re ga le du ca tio n

1.

1 cm

Prism B

1 cm

6 cm

11 cm

21 cm m

Prism C

Prism A

6 in

Prism D

10 in

17 in

22 ft

1 in

20 cm

1 in

10 ft

Prism E

7 cm

1 ft

4 cm

Prism F

176

Re ga le du ca tio n Prism

Length ength

Width

Height

Volume

A B

C

D E F

1 77

2.

Find the volume of the rectangular prisms. Show your working.

Re ga le du ca tio n

(a) 3m

8m

1m

Volume =

(b)

16 in

16 6 in

16 in

Volume =

(c)

10 ft

8 ft

9 ft

Volume =

178

(d)

Re ga le du ca tio n

2m

10 m

6m

Volume m =

(e)

m 7 cm

8 cm

20 cm

Volume =

(f)

10 m

4m

Volume =

4m

1 79

Solve It!

Re ga le du ca tio n

Wyatt cut some wood to make a rectangular-shaped block. (a) Read the clues to find the dimensions of the block.

• 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. timeters. ers

Length

Width h

Height

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

180

Volume and Capacity

Re ga le du ca tio n

Let’s Learn

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

10 cm

10 cm

10 cm

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

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

The container co

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

Capacity: acity: 1,000 c 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

1 81

Re ga le du ca tio n

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

20 cm

How much more water ter do we need ne fi this too add to fill containe container?

10 cm

10 cm

10 cm

10 cm

10 cm

The volume of liquid in the container is 1,000 0 cm3.

The capacity of the container is 2,000 0 cm3.

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

10 cm

12 cm

4 cm

15 cm

Capacity apacity = 15 x 12 x 10 = 1,800 cm c 3 = 1,000 1,00 cm3 + 800 cm3 = 1 l 800 ml

182

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

Let’s Practice

Re ga le du ca tio n

Find the capacity and volume of each rectangular tank in liters rs and nd milliliters. millilite (a)

5 cm

5 cm

2 cm

13 cm

Capacity =

Volume Volum =

(b)

9 cm

20 cm

6 cm

30 c cm

Capacity =

Volume =

1 83

(c)

Re ga le du ca tio n

9 cm c

13 3 cm c

6 cm

36 cm

Capacity =

Volume olume =

(d)

8 cm

15 cm

4 cm

45 cm

Capacity =

184

Volume =

At Home

Re ga le du ca tio n

Find the capacity and volume of each rectangular tank. (a)

12 cm

14 cm

7 cm

40 cm

Capacity =

Volume =

(b)

9 cm

4 cm

10 cm

45 cm

Capacity acity = ac

Volume =

1 85

Word Problems

Re ga le du ca tio n

Let’s Learn

The base of a rectangular prism measures 20 cm by 14 cm. m.

3 If the volume of the prism is 5,600 cm , find its height.

?

Volume = 5,6003

14 cm

20 cm

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

Volume = l x w x h 5,600 = 20 x 14 x height height = 5,600 600 ÷ (20 x 14) = 5,600 00 ÷ 280 0 ÷ 28 = 560 = 20 cm

The rectangular prism is 20 cm. he height of the rec

186

Both numbers are multiples of 10. We can simplify!

5,600 ÷ 280

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1 A rectangular tank measuring 30 cm by 30 cm by 12 cm is filled w with 3 water. Find the volume of the water. Give your answer in liters.

12 cm

30 cm

1 3

30 cm

First, find the height of the water.

1 of 12 = 12 ÷ 3 3 =4 The height of the water is 4 cm.

Now, let's find the volume.

Volume = l x w x h

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

= 30 x 30 x 4 = 3,600 cm 3

3,600 cm3 = 3.6 l

1 87

Re ga le du ca tio n

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 ug iss used to 5 move water from Container A to an identical Container B. Container A

Container B

28 cm

28 c cm

40 cm

40 cm

40 0 cm

40 cm m

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

Container A

x5

Container B C

6 cm

The water level B is now 6 cm from the brim. evel of the Container Containe Con Find the capacity y of the jug. jug

First, let's et's find nd the volume of water that is originally in Container A.

Volume olum = 40 x 40 x 28 olume = 44,800 44 80 cm3

Container er A originally orig contained 44,800 cm3 of water.

188

Next, find the height of the container.

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

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

Find the volume of water in Container B.

Height of water in Container B is 35 cm – 6 cm m = 29 cm

Volume = 40 x 40 x 29 = 46,400 cm3

Find the volume of water added. ded.

46,400 – 44,800 = 1,600 cm3

The jug was filled 5 times. mes. Divi Divide D by y 5 to find the capacity of the jug.

1,600 ÷ 5 = 320

e jug is 320 cm3.

The capacity of the

Can you find the answer another way?

1 89

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

2.

The base of a rectangular 20 cm by 25 cm. If the ular prism measures mea m 3 volume of the cuboid oid is 7,500 500 00 cm , find its height. Use the space provided to draw and show your working. w a diagram diag dia

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1.

190

Ethan has 2 identical coolers. Each one measures 50 cm long, 40 cm wide and 60 cm high. What is the total volume of the e2 coolers in liters? Use the space provided to draw a diagram ram and show your working.

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3.

4.

Peter has 2 identical bookshelves stacked one on top of the other. ookshelves shelves sta st The combined capacity is 72,000 cm3. If the area ac off the bookshelves boo of the base is 1,200 00 0 cm2, how tall all is each bookshelf? Use the space provided to draw aw a diagram diagra and show your working.

1 91

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 om the ter did tank, the water level dropped to 9 cm. How many liters of water gram m and an Halle pour out? Use the space provided to draw a diagram show your working.

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5.

192

A rectangular container is 16 cm wide and 30 cm in length. Its height is equal to its width. The container is completely filled with water.

Half of the water in the container is poured out. How much h water ater is left in the container? Use the space provided to draw a diagram agra and show your working.

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6.

1 93

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

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7.

1 94

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

2.

Riley built a house for her hamster boxes. One box measures mster ster using 2 box bo 16 cm by 10 cm by 12 cm and 8 cm by 12 cm by d the e other measures mea 6 cm. What is the total volume hamster's house? Use the space ume of the ham provided to draw a diagram gram m and show your working.

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1.

1 95

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.

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3.

196

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

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4.

(a) Find the volume of the solid. ea which is painted paint pai (b) If the solid is painted white, find the total area white.

Use the space provided to draw a diagram m and d show your you working.

1 97

The figure shows a cuboid consisting of 12 cubes.

The area of the shaded face is 100 cm2.

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5.

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

1 98

Looking Back Find the volume of each prism. Circle the prism with the greater ater volume. volum Circle both prisms if they have the same volume.

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1.

(a)

cubic units

cubic units

cubic bic units

cubic units

cubic units

cubic units

(b)

(c)

1 99

Find the volume of each solid. Circle the solid with the smaller volume.

Circle both solids if they have the same volume. (a)

(b)

ed uc ati on

2.

cubic units

cubic unit units

cubic c units un

cubic units

Re g

(c)

cubic units

20 0

cubic units

3.

Find the volume of the rectangular prisms. Show your working.

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(a)

12 cm

10 cm

10 cm

Volume =

(b)

6 ft

7 ft

8 ft

Volume =

(c)

5 in

10 in

20 in

Volume =

2 01

Re ga le du ca tio n

(d)

12 m

12 m

Volume m =

12 m

(e)

9 cm

10 cm

13 cm

Volume =

(f)

1 m 12

8m

Volume =

5m

20 2

Find the capacity and volume of each rectangular tank in liters and milliliters. (a)

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4.

8 cm

4 cm

m 5 cm

10 cm

Capacity =

Volume Volum =

(b)

6 cm

m 2 cm

4 cm

20 cm

Capacity =

Volume =

2 03

(c)

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8 cm

6 cm

12 cm

15 cm

Capacity =

Volume =

(d)

15 cm

10 cm

15 cm

25 c cm

Ca Capacity =

20 4

Volume =

The volume of a rectangular container is 512 cm3. Its width is 2 times its height. Its length is 2 times its width. Find its height. Use the he space spa provided to draw a diagram and show your working.

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5.

6.

ainers. rs.

X and Y are 2 rectangular containers.

2 The base area of X is 40 cm and nd the base bas of Y has dimensions as shown. wn. X contains 3 1,000 cm of water and Y iss empty. pt

8 cm

X

20 cm Y

(a) What is the height water level in X? g of the he wate (b) Blake pours some water from X into Y until the height of the water in ome w both containers How much water did Blake pour into Y? ners is the same. sa

2 05

Line Plots Anchor Task

ca tio n

10 Data and Graphs Hours Spent Reading Per Night (min) n)) 30

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15

206

45

60

75

90

Let’s Learn

n

A line plot, sometimes called a dot plot, is a way to show how frequently data ot occurs along a number line. An 'X' or a dot is placed above a number each time it occurs in the data set.

ed uc a

Riley emptied her pencil case on her desk k and measured the length of each object 1 to the nearest inch. She recorded the 4 data in the table below.

Length of Pencil Case Objects (in (in) 3

1 2

3

2

3 4

6

4

4

1

1 4

4

2

1 2

3

4

3 4

2

Let's represent the data in a line interpret the data. e plot and in Length of Objects cts in Pencil P Case

Re g

3

1 2

3

3 4

4

4

1 4

4

1 2

4

3 4

inches

2 07

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

3 inches in n length. len 4

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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

The shortest objects are 3

What is the difference between the shortest and longest ongest object? obje 4

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

The difference between the shortest estt and longest object is 1

Find the total length of the objects are 4 ects that ar a

4

1 inches in length.

2

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

The total length off the e4

1 1 inch ob objects is 13 inches. objec 2 2

What are some other ways we can interpret the data in the line plot?

208

1 inches.

4

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

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1.

Hours of Exercise

2

2

1 2

3

3

1 2

4

4

1 2

5

Hours urs

(a) How many students exercised cised sed for 3

1 hours? ho hou 2

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

How many students nts exercised xercised for 3 hours or more?

(d) What was the total ttota o exercise cise titime of the students that spent 3

1 hours exercising? rcising?

2

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

2 09

2.

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

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Fruit Sold

22

22

1 2

23

3 23

1 2

24

24

1 2

Pounds

(a) What was the least amount of fruitt sold on any a day?

(b) What was the most amount ountt of fruit sold on any day? (c)

On how many days did Mr. Whyte sell se s more than 24 pounds of fruit?

(d) On how many ny y day days did Mr.r W Whyte sell more than 22 pounds of fruit?

(e) Whatt is the he total to t tal al mass ma of o fruit Mr. Whyte sold over the 15-day period?

Mr M Mr. W Whyte sold

210

pounds of fruit over the 15-day period.

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

Fishing Hook Sizes

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3.

1 2

3 4

1

1

1 4

1

1 2

1

3 4

Inches

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

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

What is the total length gth of the

1 -inch -inc nc hooks?

2

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

(e) e) What hat is the difference difff 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. ine plo

3 4

1

3

1

4

1

5.

1 4

4

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1 2

tio n

Sports Drink Consumed (pints)

1

1 2

3

1

3 4

3

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

Le

1

212

3 4

2

2

5

2

1 4

4

2

1 2

2

2

3 4

3

3

5

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

uc ati on

6.

1 cup 2

1 cup 2

3 cup 4

3 cup 4

1 cup p 4

1 cu cup 4

1 cup 2

3 cup 4

1 cup 2

3 cup up 4

1 cup c 4

1 cup 4

Re ga

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

1 cup of water?

2 1 (c) How ow many beakers beake contain more than cup of water?

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

2 (b) How ow man many any beakers eake co contain

cups

213

1 cup cu of water?

4

Re ga le du ca tio n

(e) What is the total volume of water in the beakers with

cups

(f)

What is the total volume of water in the beakers eakers kers with

3 cup of water? cu 4

cups

(g) What is the total volume of water in the beakers with more than beak 1 cup of water?

4

cups

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

If redistributed equally, each beaker would contain red redistrib of wate water.

21 4

cups

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

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

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7.

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

(b) How many strings are

1 inch lo long? lon 2

3 inch?

4 1 (d) How many ny strings are longe lo longer than inch?

4 (c)

How many strings rings are a shorter t than

(e) What at is the total len length o of the strings?

The e total tota length of the strings is

.

215

Hands On Place your hand on a sheet of paper. Stretch your fingers apart art as far fa 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 an to o the nearest neares near one eighth of an inch.

Re ga le du c

ion

1.

2.

Share and collect data on the hand and spans of your yo y classmates.

Record the data in the table below. elow.

Hand Span pan Lengths Leng

Name

21 6

Hand Span n (in.)

Name

Hand Span (in.)

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

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3.

(a) What is the most common n hand span width? wid

(b) What is the least common mon hand span spa width? (c)

What is the combined ned length of th the two shortest hand spans?

in.

at is the co combin mbi (d) What combined length of the two longest hand spans?

in.

217

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

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1.

Sugar in Coffees

1 4

1 2

3 4

1

1

1 4

1

1 2

1

3 4

Teaspoons spoons

(a) What is the most amount nt off sugar in a coffee? c

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

tsp.

(c)

What was the amount of sugar in coffees with he combined combin ombined a 1 or more teaspoons of su sugar?

tsp.

21 8

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

Fruit Sold

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2.

8

8

1 2

9

9

1 2

10

10

1 2

Kilograms

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

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

(c)

What is the combined ombin mb mass ass of the flour when 8 kilograms of flour

was used??

(d) What is the he total to al mass ma of the flour used?

kg

219

3.

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

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

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

(b) How many insects are re (c)

1 inch long long??

2

How many insects are e shorter than th

1 inch?

2

(d) What is the total otal length leng len of al all insects longer than

1 inch?

2

in.

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

1 inch?

2

in.

22 0

Bob's Burger Shack sells burgers with patties of different weights.

The weights of the patties sold during lunch time are shown n below. below 7 1 1 3 5 1 3 1 3 3 lb, lb, lb, lb, lb, lb, lb, lb, lb, lb 8 2 8 4 8 2 8 8 4 4 1 1 3 1 5 1 7 3 1 7 lb, lb, lb, lb, lb, lb, lb, lb, lb, lb 8 2 8 4 8 2 8 8 8 8

Re ga le du ca tio n

4.

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

1 pound or heavier?

2 3 (c) How many patties are re lighter than th pound?

4 1 (d) What is the total al weight weigh weig of the -pound patties?

2 (b) How many patties are

lb

er Shack used u (e) If Bob's Burger the same amount of meat to make pound patt patties, h only half-pound how many patties could they make?

221

Hands On Collect 16 objects from around your home that are between een 1 and 2 inches in length. Write the names of the objects in the table able below. belo Use a ruler to measure the length of each object to the he nearest earest quarter-inch. Write the length of each object in the table. e.

Re ga le du ca tio n

1.

2.

Objects at Home

Object

222

Length (in.)

Objectt

Length (in.)

Le

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

Re ga le du ca tio n

3.

(a) What is the most common n object length?

(b) What is the least common mon object length? leng len (c)

What is the combined ned length of th the 4 shortest objects?

in.

at is the co combin mbi (d) What combined length of the 4 longest objects?

in.

223

Graphing Equations

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Anchor Task

12 11

10 9 8 7

6

5 4

3 2 1

0

224

1

2

3

4

5

6

7

8

9

10

Let’s Learn

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A coordinate grid can be used to locate points on a plane. A coordinate rdinate grid is made up of a horizontal number line, called the x-axiss and d a vertical vertica line called the y-axis. sing an ordered ordere Each location on a coordinate grid can be described using pair of numbers. The first number in an ordered pair is the x-coordinate x-coordin and is located on the x-axis. The second number in pair is the n an n ordered pai p y-coordinate and is located on the y-axis.

Two points are marked on the coordinate grid pair for A d below. ow. The ordered ord is (2, 8). The ordered pair for B is (7, 4). y

10 9

A (2, 8)

8 7

6

5

B (7, 4)

4

3 2 1

0

1

2

3

4

5

6

7

8

9

10

x

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.

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10 9 8 7

6

5 4

3 2 1

0

22 6

2

1

3

4

5

6

7

8

(1, 8)

(8, 9)

(4, 6)

(2, 2)

(7, 4)

(7, 1)

9

10

You can use a coordinate plane to graph an equation.

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

Cost of Cupcakes

Number of Cupcakes

1

2

3

4

5

Cost ($)

2

4

6

8

10

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

Drawing a line through the plots formss a straight aight line lin graph. The straight line graph represents an equation that hat shows how tthe x-axis and y-axis are connected. The equation is y = 2x 2x. x.

11

10 9 8

Cost ($) C

7

6

5 4

3 2 1

0

1

2

3 4 5 6 7 8 Number of Cupcakes

9

10

227

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

Re ga le du ca tio n

Cost of Wood

Length of Wood (m)

3

6

9

12 2

15

Cost ($)

12

24

36

48

60

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

132

120

108 96

Cost ($)

84 72

60 48

36 24

12

0

3

6

9 12 15 18 21 Length of Wood d (m)

Blake's 21 meters of wood to ake's father ne needs 2 Using the straight line build a ne new fence. U graph, see that the total cost of 21 h, we can se meters of wood is $84. w

22 8

24 27 30

How much does 27 m of wood cost?

Let’s Practice Use the ordered pairs to draw and color circles at the correct rrectt places on the coordinate grid.

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1.

10 9 8 7

6

5 4

3 2 1

0

1

2

3

4

5

6

7

8

(1, 5) 5

(2, 9)

(6, 6)

(8, 8) 8

(9, 5)

(8, 1)

9

10

229

The coordinate grid shows the location of each child's house.

Name the ordered pair for each child's house.

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2.

10 9 8 7

6

5 4

3 2 1

0

2 30

1

2

3

4

(a)

(b)

(c) c)

(d)

(e)

(f)

(g)

(h)

5

6

7

8

9

10

A supermarket sells watermelons for $6 each. The cost of watermelons is plotted on the coordinate plane using the equation y = 6x.

Watermelons Purchased

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3.

66

60 54 48

Cost ($)

42

36

30 24 18

12 6

0

1

2

3

4 5 6 7 Watermelons

8

9

10

(a) How much does 2 watermelons waterm wat cost?

(b) How w much uch does 5 watermelons wat cost? (c)

How w much does 9 watermelons cost?

(d) Riley spent spent $36 on watermelons. How many did she buy? (e) Blake h has $ $50. How many watermelons can he buy? much money will he have left?

Ho m How

2 31

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

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4.

(a) Complete the table.

Perimeter of a Square (cm)

Side Length (s)

1

Perimeter (P)

4

2

3

16

20

(b) Use the equation P = 4s to create a straight aight ht line graph.

Perimeter of a Square quare

40 36 32

Perimeter (cm)

28

24

20 16 12 8

4

0

(c) c)

1

2

3 4 5 6 7 Side Length (cm)

8

9

10

square A squa squar eh has a side length of 2 cm. What is its perimeter?

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

23 2

A square has a perimeter 16 cm. What is its side length?

A painter needs 27 liters of paint to paint a house.

Paint is sold in cans of 3 liters for $9 each.

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5.

(a) Complete the table.

Cost of Paint

Paint (l)

3

Cost ($)

9

6

12

27

15

45

118

21

63

(b) Use the equation y = 3x to create a straight aight ht line graph.

Cost of Paintt

90 81

72

Cost ($)

63 54 45

36 27 18 9

0

(c) c)

3

6

9

12 15 18 Paint (l)

21

24 27 30

How ow much will 6 liters of paint cost?

(d) How m much will 9 liters of paint cost?

(e) e) Th The painter pain has $20. How much more money does he need to buy the paint he needs?

2 33

At Home Use the ordered pairs to plot the points on the coordinate e grid. d.

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1.

(a) T (3, 3)

(b) A (1, 9)

(c)

Z (3, 9)

(d) W (2, 7)

(e) C (6, 9)

(f)

3, 6) 6 H (3,

(g) S (7, 7)

(h) B (5, 8)

(i)

O (1, 5)

10 9 8 7

6

5 4

3 2 1

0

2 34

1

2

3

4

5

6

7

8

9

10

The coordinate grid shows the location of the cars.

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2.

10 9 8 7

6

5 4

3 2 1

0

1

2

3

4

5

(a)

(b)

(c) c)

(d)

(e)

(f)

(g)

(h)

6

7

8

9

10

2 35

A hardware store sells metal chain for $6 per meter.

The minimum length is 4 meters.

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3.

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

60 54 48

Cost ($)

42

36

30 24 18

12 6

0

1

2

7 3 4 5 6 Length of Leng o Chain (m)

8

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

What is the co cost st of 5 meters of metal chain?

(d) d)) What hat is the cost cost of 8 meters of metal chain? (e) What le leng length th of chain can you buy with $48? (f) f)

23 6

What le W length of chain can you buy with $54?

9

10

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

Re ga le du ca tio n

4.

ndwiches ches using tthe (a) Create a straight line graph to show the cost of sandwiches wer the question questio equation y = 2x. Use the data in the graph to answer questions.

Cost of Submarine Sandwich

28

26 24

Cost ($)

22

20 18

16 14 12

10

0

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

14

(b) What at is the cost cost of tthe sshortest submarine sandwich?

(c)

What is the co cost st of a 13-inch submarine sandwich?

(d) What is the longest submarine sandwich she d) Halle alle has $ $23. W can buy buy?

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

2 37

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

Re ga le du ca tio n

1.

Fun Run Distances

3 4

1

1

1 4

1

1 2

1

3 4

2

2

1 4

Miles les

(a) How many students ran 2 miles or more? more

(b) How many students ran an further rther than 1 mile? (c)

ined d distance ra What is the combined ran by the students who ran 1 mile or less?

mi

he co c mbined d mbine (d) Whatt is the combined distance ran by the students who ran 3 further 1 miless or further?

4

mi

238

Use the ordered pairs to plot the points 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)

6, 5) O (6,

Re ga le du ca tio n

2.

10 9 8 7

6

5 4

3 2 1

0

1

2

3

4

5

6

7

8

9

10

2 39

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.

Re ga le du ca tio n

3.

(a) Complete the table.

Cost of Flour

Flour (lb)

2

Cost ($)

4

4

110

8

12

20

12

14

28

(b) Use the equation y = 2x to create a straight aight ht line graph.

Cost of Flourr

36 32

Cost ($)

28

24

20 16 8

4

0

(c)

2

4

6

8 10 12 Flour (lb)

14

16

18

20

How much of flour cost? uch will 8 pounds p pou

(d)) How ow much will 14 pounds of flour cost?

(e) (e Mrs. T Taylor aylo has ha $50. How much money will she have left when she buys the flour she needs?

24 0

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

Re ga le du ca tio n

4.

(a) Complete the table.

Perimeter of an Equilateral Triangle (in.))

Side Length (s)

3

Perimeter (P)

9

5

12

18 1

21

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

Perimeter of a Equilateral ral Triangle angl

30 27

24

Perimeter (cm)

21 18

15 12 9

6

3

0

(c) c)

1

2

3 4 5 6 7 Side Length (cm)

8

9

10

triangle has a side length of 2 cm.

An equilateral equilate tr

perimeter?

What is its p

(d) d) An equi equilateral triangle has a perimeter of 21 cm.

What is the length of each side?

2 41

Problem Solving

Re ga le du ca tio n

11

Act It Out

Example 1 Wyatt has 2 empty water bottles.

Wyatt wants to give 5 liters of water to his friend, Halle. alle. e.

How can he measure out 5 liters of water using the he bottles? ottles?

7L

4L

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

4L

7L

7L

Step 2 Fill the 4-liter bottle tle again and pour po it into the 7-liter bottle.

Wyatt now has 1 liter left in the 4-liter bottle.

4L

4L

7L

4L

7L

4L

1L

242

4L

7L

Re ga le du ca tio n

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

7L

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

4L

4L

7L

7L

1L

Wyatt now has 5 literss of water to t give giv to Halle.

5L

7L

24 3

Re ga le du ca tio n

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

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

A

D

B

A

B

C

D

E

E

F

F

AB ABC ABE A ABF

ACD ACF

ADE ADF AEF

BCD BCE BDF

CDE CDF CEF DEF

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

244

C

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 he boat boa cross the river?

Re ga le du ca tio n

1.

24 5

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 me spot?

Re ga le du ca tio n

2.

A

B

Boss

C

D

246

In a mathematics lesson, the teacher wants to arrange 8 students to form 4 straight lines. The teacher wants 3 students in each ch line by intersecting some of the lines. The students form lines as shown wn below. belo

Re ga le du

ati on

3.

The teacher now wants to arrange rrange nge 10 students studen to form 5 straight lines, with 4 students in each line.. Draw w a diagram diagra to show how the students should be arranged.

24 7

Draw a Model

Re ga le du ca tio n

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

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.

ed, add dd the number num nu To find the total number of matches played, of matches played in each round.

8 + 4 + 2 + 1 = 15

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

Draw a diagram when you are gi given complex infor information.

Diagrams are a great way to communicate and present ideas.

248

Round 1 Round 2

Re ga le du ca tio n

Team 1 Team 2

Round 3

Team 3 Team 4

Round 4 R Ro

Team 5 Team 6 Team 7

Team 8

Team 9

Team 10 Team 11

Team 12

Team 13 Team am 14

Team 115 Team 16

24 9

The cost of a present was shared among Ethan, Halle and Sophie.

Ethan paid one fifth the cost of the present. Sophie paid $15 5 more than one third of the remaining amount. Halle paid $30 more than Sophie Sophie.

What was the price of the present?

Re ga le du ca tio n

1.

Ethan

Sophie

25 0

Halle

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 est of the students each received 1 yellow star. Express as a fraction n the number numbe warded. ed. of blue stars awarded over the total number of stars awarded.

Re ga le du ca tio n

2.

251

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 iss equal equa to three fifths the number of lions. If there are 15 more zebras as than an lions, lions how many deer are there?

Re ga le du ca tio n

3.

252

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 secon after that, the speed at which the parachute falls is halved until it is 89.75 m teady y rate of from the ground. After which, the parachute falls at a steady ch the ground? groun grou 1.25 m/s. How long would it take for the parachute to reach

Re ga le du ca tio n

4.

25 3

Guess-and-Check

Re ga le du ca tio n

Example acher.r.

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 ed to combine their th short of the amount needed to buy the gift. They decided money and buy the gift together. After buying the gift, they had $3 left le over.

How much is the gift?

Cost of the gift

Wyatt's amount

Blake's amount

$40

40 - 23 = 17

$41

Total

Left over ove

Check

40 - 18 = 22

17 + 22 = 39

39 - 40 4 = -1

no

41 - 23 = 18

41 - 18 = 23

18 + 23 3 = 41

41 4 - 41 = 0

no

$42

42 - 23 = 19

42 - 18 = 24

19 + 24 = 43

43 - 42 = 1

no

$43

43 - 23 = 20

43 - 18 = 25

20 + 25 = 45 45 - 43 = 2

no

$44

44 - 23 = 21

44 - 18 = 26

2 = 47 211 + 26

yes

Therefore, the gift costs $44.

4.

Keep guessing and checking until you arrive at the correct answer

254

47 - 44 = 3

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 moon-shaped lights are 5 meters long. During a parade, 174 meters of o lights were used for decoration. How many packs of star-shaped ar-shaped aped and moon-shaped lights were used?

Re ga le du ca tio n

1.

25 5

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

Re ga le du ca tio n

2.

25 6

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 e's yard?

Re ga le du ca tio n

3.

25 7

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

Re ga le du ca tio n

4.

25 8

Make a List

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

Bb Bk Br

Kb Kk Kr

blue sharpener ener (b) (b b b) black sharpener arpenerr (k) ( red sharpener arpener ner (r)

ga le d

List key blue pencil (B) black pencil (K) red pencil (R)

ca tio n

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 ed sharpener.

In how many different ways can she pair a pencil with a sharpener? rpener?

Rb Rk Rr

Therefore, there are 9 possible combinations. comb c nations.

25 9

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

Re ga le du ca tio n

1.

260

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 w many man different ways can she choose 3 beads?

Re ga le du ca tio n

2.

26 1

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 th than 1 cm and the side length of the biggest square is less than n 10 cm, what wha is the perimeter of the figure formed by the 3 squares?

Re ga le du ca tio n

3.

262

A rectangle has an area that is equal to its perimeter. Its length and breadth are whole numbers. The length is greater than the breadth bread and both are less than 10 cm. Find the length and breadth of the he rectangle. rectangl ectangl

Re ga le du ca tio n

4.

26 3

Look for Patterns

uc ati on

Example A group of students are at a farewell gift exchange ceremony. ony. Each ach student studen has one gift. Each student then exchanges a gift with every ry other ther student. studen (a) If 4 students attend the ceremony, how many times es are gifts exchanged? exch exc

Re ga l

Let's look at the number of exchanges between en the students and try to identify a pattern.

3+2+1=6

There between the 4 students. ere are 6 exchanges exch

264

Re ga le du ca tio n

Starting with 1 less than the number of students, the number of exchanges decreases by 1 each time the students exchange gifts

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

12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 78 Gifts are exchanged 78 times.

(c)

ow many students studen are If gifts are exchanged 105 times, how at the ceremony?

Let's try 14 students.

13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 91 This is too low.

105 – 91 = 14 105 = 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 ents at a the t ceremony.

So, there are 15 students

26 5

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

Re ga le du ca tio n

1.

1 row

266

2 rows

3 rows

2.

Look at the pattern below.

Re ga le du ca tio n

MATHEMATICSMATHEMATICSMATHEMATICS... (a) What is the 38th letter?

(b) What is the 125th letter? etter? r?

267

Dominic wrote all the numbers between 1 and 85 on the whiteboard.

How many digits did he write in total?

Re ga le du ca tio n

3.

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

26 8

4.

What are the next two numbers in the pattern? ,

Re ga le du ca tio n

2, 6, 12, 20, 30, 42, 56, 72, 90,

26 9

Work Backwards

?

Interchange

ed uc ati on

Example A bus left an interchange with some passengers on board.

At the first bus stop, half of the passengers got off and 2 people At ple got on. A the second stop, a quarter of the passengers got off and d 3 people got on. o When the bus left the second stop, there were 12 people ple le on board.

How many people were on the bus when it left the interchange? nterchange? terchange?

– 1 2 +2

– 1 4 +3 12 passengers passeng passen

First Stop

Secon eco Stop Second

Working backwards, the last event that was 3 people got on the hat occurred wa w bus. Let's subtract 3 from the final number of pas passengers on the bus. pass 12 - 3 = 9.

We know that 9 passengerss were ere on the th bus b before the 3 people boarded.

a

?

Interchange change e

– 1 4

– 1 2 +2

9 passengers

First Stop Firs

Second Stop

Re

Moving back, ck, a quarter of the t people got off the bus. So, three quarters remain on the number of passengers on the bus by n the e bus. Let's multiply mult mu four thirds. rds.

9x

4 36 = 3 3 = 12 1

There were on the bas before it arrived at the second stop. e 12 passengers p

270

Interchange

First Stop

Re ga le du ca tio n

?

– 1 2 +2 12 passengers

Second d Stop op

Working back, 2 people got on at the first stop. Let's subtract ubtract btract 2 people. people 12 - 2 = 10

– 1 2

?

10 passengers ers

Interchange

First Stop

Second Stop Se

Finally, the first event that occurred ed was half the passengers got off, leaving half the passengers on board. Let's multiply th the number of people by 2.

10 x 2 = 20.

So, there were 20 people le on o the bus at the interchange.

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

27 1

Sophie has some savings in her bank account. During the day y she spends $74.50 on shoes, $80 on food and deposits $535 into o her account. At the end of the day she sees that her balance is $851.80.

851.80.

How much money did she have in the bank at the beginning nning g of the day?

Re ga le du ca tio n

1.

272

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 er of her h remaining masks to Halle and then took 6 masks back. Finally, nally, y, Riley 0 masks sks left. gave half of the remaining masks to Blake. Riley had 30 How many masks did she have at first?

Re ga le du ca tio n

2.

27 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, Monday he shared his stickers equally amongst himself and his 4 friends. nds. How many stickers did Wyatt have at the start of Monday?

Re ga le du ca tio n

3.

274

I am thinking of a 7-digit number.

The millions digit is the difference between the ten thousands ds digit dig g and the ones digit.

Re ga le du ca tio n

4.

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

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

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

The ones digit is 3.

What is the number that I am thinking hinking of? o

27 5

Simplify the Problem

Re ga le du ca tio n

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

Let's find the perimeter of 1 unit nit of the he pattern 4,872 ÷ 28 = 174

So, one unit pattern has ass a p perimeter eter o of 174 cm.

The smaller triangle all have the same side length as e and nd the 2 squares squa sq denoted by the markings. kings.

Perimeter of smaller aller triangle and 1 square = 7 units

The larger that is half of the perimeter of the er triangle ngle has a perimeter pe peri smaller triangle square. ngle and 1 squa 1

Perimeter triangle = 2 of 7 units rimeter larger tr triang 1 = 3 2 units

276

1

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

Re ga le du ca tio n

= 14 2 units

1

14 2 units = 174 cm 1

1 unit = 174 ÷ 14 2 = 12 cm

1

Perimeter of larger triangle = 3 units 2 1

= 3 x 12 2 = 42 cm

Divide by 3 to find the length of 1 side.

42 ÷ 3 = 14

The larger triangle has a side length h of 14 cm. cm

That wass a lot easier orking with 28 2 than working it patterns! unit

27 7

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 squ q is twice that of square B. The perimeter of the figure is 75 cm.. Find the th area of square C.

Re ga le du ca tio n

1.

C

B

A

278

3

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

Re ga le du ca tio n

2.

27 9

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 en arranged arr the cubes to form the given figure. After arranging the cubes, paints ubes,, he pain e. Each face fac all of the exposed faces including the underside of the figure. lete the whole takes 1 minute to paint. How long did he take to complete process in hours and minutes?

Re ga le d

ati on

3.

280

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?

Re ga le du ca tio n

4.

28 1

Solve Part of the Problem

Re ga le du ca tio n

Example Ethan saves a sum of money on Monday. Every subsequentt day, y, he saves one third more than he did the day before. By Thursday, he had ad saved $128. $12 $ How much did he save on Monday?

Monday

Tuesday

This diagram am will help us to solve rent parts the different of the problem.

Wednesday

Thursday

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

4 parts of Wednesday of Thursday y = 3 parts pa p Th 3 x 32 = 96 96 ÷ 4 = 24 1 part of Wednesday sday = 24 4 parts of Tuesday Wednesday esday ay = 3 parts of W 3 x 24 = 72 72 ÷ 4 = 18 8 1 part of Tuesday esday = 18 1 Monday nday = 3 parts of o Tuesday Tu 3 x 18 = 54

So, Ethan $54 than saved sa $ on Monday

282

The diagram shows part of a 3-tunnel system. 2 parts of the tunnel system run North to South while another part runs East to West. T Two tunnels meet at a traffic light junction. The width of each tunnel nel is 1.9 m.

Find the area of road covered by the tunnel system shown own in n the th diagram.

Re ga le du ca tio n

1.

North

21.8 m

North h

Junction

East

West

16.1 m

South

South

28 3

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

Re ga le du ca tio n

2.

y

284

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

Re ga le du ca tio n

3.

15 cm

26 cm

28 5

The diagram shows 1 large circle surrounded by 6 identical smaller circles. Straight lines are extended from the center of the larger circle rger c onto the sides of the smaller circles. Straight lines are also o extended ended from the center of each of the 6 smaller circles onto the e side e of the larger circle. They form a right angle with the lines that at extended tended from fro the large circle. Find the sum of the angles.

Note that all 6 angles are equal.

Re ga le du ca tio n

4.

286

Before-After Concept

Re ga le du ca tio n

Example The ratio of the number of sweets Riley had to the number of sweets weets Wyatt Wyat had was 5 : 3. When Riley gave 40 sweets to Wyatt, the ratio atio became 7 : 17.

1 How many sweets did Riley have at first?

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

Riley

Before

Wyatt

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

Riley

After

Wyatt

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

To make the 2 ratios have amount of units, we can multiply the ave an equal e before ratio by 3.

Riley

Before

Wyatt yatt

Before the he exchange: xchange: 15 : 9 – which ch is a total of 24 2 units.

Riley

After A

Wyatt

287

Re ga le du ca tio n

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

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

Riley

Before

Wyatt

8 units nits

Riley

After

Wyatt

nits.

So, 40 sweets is equal to 8 units. ets in n 1 unit, we w divide 40 by 8.

To find the number of sweets 40 ÷ 8 = 5

1 unit = 5 sweets

Riley had 15 units its att first. first weets. ts.

5 x 15 = 75 sweets.

y had d 75 sweets at ffirst.

So, Riley

288

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?

Re ga le du ca tio n

1.

28 9

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 anges and how many pears does she have?

Re ga le du ca tio n

2.

290

There are some coins in 2 boxes – 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 e twice as many coins as Box B. How many coins are there in Box A at first? rst?

Re ga le du ca tio n

3.

29 1

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 ets to mber of Sophie’s was 4:3. After Dominic used 55 of his tickets, the number ny more tickets Ethan had was four fifths that of Dominic’s. How many tickets does Dominic have than Sophie now?

Re ga le du ca tio n

4.

292

Make Suppositions

Re ga le du ca tio n

Example Blake sold a total of 40 large and small mangoes in a particular ular 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?

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

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

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

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

So, when we change and replace a large mango with a hange ge our supposition supposit supp small mango, decrease by $2. We need to find the number o, the e outcome will w d of small mangoes our supposition outcome by $32. angoes es that will decrease dec d 32 ÷ 2 = 16

Exchanging with small mangoes, we will reach $168. changing 16 large mangoes ma 40 0 - 16 = 24.

Therefore, sold 24 large mangoes. e, Blake e B

29 3

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 e ther her farm?

Re ga le du ca tio n

1.

294

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 mals a total of 120 legs in the zoo. How many zebras are there in the zoo?

Re ga le du ca tio n

2.

29 5

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

Re ga le du ca tio n

3.

296

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 and tricycles, how much would d the nd tricycles? cyc shopkeeper have earned if he sold all of the bicycles and

Re ga le du ca tio n

4.

29 7

Re ga le du ca tio n

Re ga le du ca tio n

Re ga le du ca tio n © Bluee Ring Media dia Pty Ltd ACN 161 16 590 496 2013 - 2021.

This his publication lication would not have been possible without the tireless effort of our production team.

Special thanks to: Daniel Cole, Matthew Matthe Cole, Col Wang Hui Guan, Kevin Mahoney, Winston Goh, Jesse Singer, Joseph eph Anderson, Anderson Halle Taylor-Pritchard, Sophie Taylor-Pritchard, Tejal Thakur, Nakapat,Varasinun Mathanattapat, Kanungnit Pookwanmuang, Saijit Lueangsrisuk Natchanuch Nak Nakapat,V

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