Let's Do Mathematics 6 – Worktext B by Regal Education

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Workt ext

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for learners 11 - 12 years old

Aligned to the US Common Core State Standards

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Workt ext

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for learners 11 - 12 years old

Let’s Do Mathematics

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

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

Integers

Anchor Task

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

o ChicFinae g

Cheesecake Recipe

–8ºC h: 4º

2º Low: –1

Serves 10 people Prep Time Cook Time Cooling Time

Hig

Let’s Learn

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30 minutes 1 hour 15 minutes

Ingredients Cheesecake

–12

y Monda y Tuesda

sday Wedne ay

–2

–14

–7

Thursd

Crust

2 1 lbs cream cheese 2 3 cup sour cream 4 1 tsp salt 2 3 1 cups sugar 4

–2

14 Graham Crackers 1

1 cup pecans 2

4 tbsp butter 1 cup sugar 4 1 tsp cinnamon 2

3 1 tbsp flour 2

5 eggs 2 egg yolks

Friday

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

Fractions

Anchor Task

Key features of the series include:

Let’s Learn

When findi ng the area of triangles, height. we first

need to find

Lets find the

the base and

base and heig

hts of som

(a)

e more trian

gles.

The height must be perpendic ular to the base.

In this triangle, the height is not a side length.

height

C We can choo se any side of the trian let's take the gle to be the base to be base. For the BC. triangle ABC, The height of the trian gle is given chosen base by the perpendi . This is a right cular heig height. -angled trian ht to our gle, so AB is the perpendi cular BC is the base and AB is the height.

(b)

base

If we choose the base to be MO, the is perpendi cular to the height is given base. by the line NP

which

height height S base B base

For a base SU, the perp endicular heig triangle at line TV. ht is

located outs ide

the

Let’s Practice

(d)

Let’s Practice

height of the Identify the base and

height =

(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.

base =

triangles.

base = height = F

(e) Q

(b)

base = base =

height =

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(c)

(f)

base =

base = height =

height =

At Home

Complete the followin g. Show your workin in its simplest form. g and write your answer (a)

4 x 7 5

At Home 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)

5 x 8 12

Multiply the mixed numbers. Show your working and write answer in its simples your t form.

(a) 3 2 x 5 5

(b) 6 x 2 3

(d) 10 x 5 6

(b) 2 x 3 5 8

(c)

(e) 6 x 3 7

(f)

4x22 3

(d) 7 x 3 3 5

7 x 9 5

(e) 12 x 4 1 8

8x53 12

(f)

1 12

Solve It!

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

with the of a rectangle Hands On half the area a triangle is the area of Show that cm. and height. height of 12 same base a and 16 cm a width of below has The rectangle of 192 cm2 . height of 12 cm and a It has an area a base of 16 site page has on the oppo The triangle d lines so the cm. along the dotte neatly fill up to . Then cut w page belo the grid triangle from nge the pieces in the Cut out the 3 pieces. Arra triangle is in rectangle. half of the

Hands On

Solve It! 1.

The figures are made up of semicircles (half circles) and straight lines. Can you find the area of each figure? Take π = 3.14 and round off your answer to 1 decimal place.

(a)

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

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1 13

(b) 2 cm

2 cm 5 cm

The figure below is made from a square of side length 6 cm. The circular hole in the middle has a diamete r that is 2 the side length of the square. Find the 3 area of the figure. Take π = 3.14 and round off your answer to 1 decima l place.

Looking Back 1.

Looking Back

Express the percentage as a decimal

(a) 12%

What percentage of each square is

colored?

(a)

(b)

(c)

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

(c)

(e) 70%

(d)

Color 14% of the square.

42%

and fraction in its simplest form. (b) 28%

(d) 86%

(f)

50%

percentage. Express the fraction as a decimal and 4 17 (b) 20 (a) 100

Color 45% of the square.

(c)

15 60

(d)

66 88

235 234

iii

Contents Geometry Properties of Circles Area of Circles Area of Triangles Area of Composite Figures

8 Time Expressing 12-hour and 24-hour Time Duration of Time Word Problems

9 Speed Speed and Distance Average Speed Word Problems

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10 Pie Charts Reading and Interpreting Pie Charts Word Problems

2 2 16 24 46

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66 68 69 74 85 94 94 109 120 138 138 160

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174 174 180 186 192 197 201 206 210 216 221

226 226 234 241

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12 End-of-year Exam Section A Section B Section C

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

Geometry

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Properties of Circles

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Anchor Task

Green circle Blue circle Red circle

Diameter

Circumference

Let’s Learn Let's look at the different parts of a circle.

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O is a point at the center of the circle. AB is a straight line that passes through the center point O. B

AB is a diameter of the circle.

OC is a straight line from the center of the circle to its perimeter.

OC is a radius of the circle.

DE is a straight line that does not pass through the center of the circle.

DE is a chord of the circle.

The diameter of a circle is 2 x the radius.

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OA and OB are also radii of the circle. Radii is plural of radius.

Recall that the path around a shape such as a rectangle or triangle is called the perimeter. The perimeter of a circle has a special name – circumference.

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The circumference of a circle is its perimeter.

Dominic placed a piece of string so that it fits around a circle. He then measured the length of the string to find the circumference of the circle.

Dominic used string to find the circumference of different sized circles. He recorded his findings in a table.

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Circle 1 Circle 2 Circle 3

Diameter (cm) 5 10 20

Circumference (cm) 15.7 31.4 62.8

Doubling the diameter also doubles the circumference!

Dominic uses a calculator to divide the circumference by the diameter for each circle. He notices that the quotient is the same for every circle.

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Diameter Circumference Circumference ÷ (cm) (cm) Diameter Circle 1 5 15.7 3.14 Circle 2 10 31.4 3.14 Circle 3 20 62.8 3.14

decimal 3.14 or the fraction

22 . 7

The circumference of any circle divided by its diameter is always the same value. This value is represented by the symbol π, which is a letter of the Greek alphabet. We say this symbol as 'pie'. We can approximate π as the

Circumference = π x d = πd

πd means π x d.

The diameter of a circle is 2 times its radius. So we can also write:

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Circumference = 2 x π x r = 2πr

2xπxr means 2πr.

Find the circumference of each circle. Take π = 3.14 and round off your answer to 1 decimal place. The diameter of the circle is 10 cm.

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Circumference = πd = 3.14 x 10 cm = 31.4 cm 10 cm

Make sure you write the correct unit of length.

The radius of the circle is 4 in.

4 in

Circumference = 2πr = 2 x 3.14 x 4 in = 8 x 3.14 in = 25.1 in

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The diameter of the circle is 5 m.

Circumference = πd = 3.14 x 5 m = 15.7 m

Find the circumference of each circle. Take π =

22 . 7

The diameter of the circle is 7 cm.

22 x 7 cm 7 = 22 cm =

7 cm

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Circumference = πd

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14 7 =2

4 cm 5

4 22 = x2 5 7 22 14 = x 5 7 22 x 14 = 7x5 22 x 2 = 5 44 4 = =8 5 5 The circumference is 8

4 cm. 5

Can you express the circumference as a decimal?

Let’s Practice 1. The center of each circle is point O. Labeled lines are straight lines. Identify the radius of each circle.

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(a) (b) L

radius =

P L

M Q

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radius =

R N

2. O is a point on the center of each circle. Labeled lines are straight lines. Identify the diameter of each circle. (a) (b)

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W U

Q S

diameter =

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diameter =

X U

Y O W

L V

diameter =

3. The grid below is made up of 1 cm by 1 cm squares. Find the diameter and circumference of each circle. Take π = 3.14 and round off your answer to 1 decimal place. Circle B

Circle C

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Circle A

Circle E

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Circle D

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4. A mini pizza has a diameter of 8 cm. Find the circumference. Take π = 3.14 and round off your answer to 1 decimal place.

5. The radius of a round hot tub is 2 meters. Find the circumference. Take π = 3.14 and round off your answer to 1 decimal place.

22 . 7

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Take π =

6. The diameter of a sports arena is 70 meters. Find the circumference.

7. A round place mat has a radius of 14 cm. Find the circumference. Take π =

22 . 7

Solve It!

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The figures are made up of semicircles (half circles) and straight lines. Can you find the perimeter of each figure? Take π = 3.14 and round off your answer to 1 decimal place. (a)

4 cm

(c)

4 cm

(b)

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4 cm

10 cm

At Home 1. The center of each circle is point O. Labeled lines are straight lines. Identify the radius of each circle. H

K I L

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(a) (b)

radius =

R EG radius =

Q R

T O

radius =

2. O is a point on the center of each circle. Labeled lines are straight lines. Identify the diameter of each circle. (a) (b) Q

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C U

diameter =

Q (c) (d)

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diameter =

N L

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3. A discus has a diameter of 22 cm. Find the circumference. Take π = 3.14 and round off your answer to 1 decimal place.

4. A round table has a radius of 50 cm. Find the circumference in meters. Take π = 3.14 and round off your answer to 1 decimal place.

22 . 7

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Take π =

5. A round swimming pool has a diameter of 14 ft. Find the circumference.

6. A round place mat has a radius of 14 cm. Find the circumference. Take π =

22 . 7

Area of Circles

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Anchor Task

Let’s Learn

πr πr

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Let's find the area of a circle. Dominic divides a circle into 4 equal parts. He cuts the parts and arranges them as shown below.

He continues to divide circles into more equal parts and arranges them as shown.

πr

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πr

More divisions results in the arrangement looking more like a rectangle!

As we continue to divide the circle, the arrangement of the parts forms a rectangle of height r and width πr. The area of the rectangle is π x r x r. Area of circle = π x radius x radius = π x r x r

Let’s Practice 1. Find the area of each circle. Take π = 3.14 and round off your answer to 1 decimal place.

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(a) 2 cm O

(b)

(c)

5 ft

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(d)

14 in O

2. Find the area of each circle. Take π =

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(a)

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7 cm O

(b)

28 in

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(d)

56 cm

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3. A round swimming pool has a radius of 15 m. Find its area. Take π = 3.14 and round off your answer to 1 decimal place.

4. A circular sports field has a diameter of 100 m. Find its area. Take π = 3.14 and round off your answer to 1 decimal place.

22 . 7

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5. A coin has a diameter of 14 mm. Find the area. Take π =

6. A round place mat has a diameter of 35 cm. Find the area. Take π =

22 . 7

Solve It!

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1. The figures are made up of semicircles (half circles) and straight lines. Can you find the area of each figure? Take π = 3.14 and round off your answer to 1 decimal place. (a)

(b) 2 cm

5 cm

2 cm

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2. The figure below is made from a square of side length 6 cm. 2 The circular hole in the middle has a diameter that is the side length 3 of the square. Find the area of the figure. Take π = 3.14 and round off your answer to 1 decimal place.

At Home 1. Find the area of each circle. Take π = 3.14 and round off your answer to 1 decimal place.

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(a) 12 cm O

(b) O

18 m

(a)

2. Find the area of each circle. Take π =

28 in

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(b)

70 cm O

22 . 7

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3. A wall clock has a diameter of 32 cm. Find the area. Take π = 3.14 and round off your answer to 1 decimal place.

4. A pizza has a radius of 8 in. Find the area. Take π = 3.14 and round off your answer to 1 decimal place.

22 . 7

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5. A coat button has a diameter of 21 mm. Find the area. Take π =

6. An archery target has a radius 49 cm. Find the area. Take π =

22 . 7

Area of Triangles

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Anchor Task

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Let’s Learn

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When finding the area of triangles, we first need to find the base and height. A

The height must be perpendicular to the base.

e can choose any side of the triangle to be the base. For the triangle ABC, W let's take the base to be BC. he height of the triangle is given by the perpendicular height to our T chosen base. This is a right-angled triangle, so AB is the perpendicular height.

BC is the base and AB is the height.

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height

base

Let's find the base and height of some more triangles. (a)

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In this triangle, the height is not a side length. height

base

If we choose the base to be MO, the height is given by the line NP which is perpendicular to the base. T

(b)

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height

base

For a base SU, the perpendicular height is located outside the triangle at line TV.

Find the area of triangle ABC. 1 cm

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1 cm

he triangle is inside a rectangular grid where each grid square is T 1 cm by 1 cm. o find the area of the triangle, we need to choose a base and find T the height.

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et's choose BC as the base. L The height of the triangle is perpendicular to the base. This is a right-angled triangle, so the height is AB.

This triangle has a base of 10 cm and a height of 5 cm.

height 5 cm

base 10 cm

The area of a triangle is given by:

For our triangle: Area of triangle =

1 x base x height 2

1 x 10 cm x 5 cm 2

1 = x 50 cm2 2 = 25 cm2

The area of the triangle is half the area of rectangle ABCD.

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Area of a triangle =

ur triangle is enclosed in a rectangle ABCD. To find the area of a rectangle O we multiply the length by the width.

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Area of rectangle = length x width = 10 cm x 5 cm = 50 cm2 Visually, we can see that the triangle divides the rectangle into halves. So, the area of the triangle is half of the area of the rectangle. A

D 1 area of rectangle 2

1 area of rectangle 2 B

6 cm

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1. Each square in the grid is 1 cm by 1 cm. Find the area of the triangle.

1 x base x height 2

1 x 7 cm x 6 cm 2

Area of triangle =

7 cm

1 = x 42 cm2 2

= 21 cm2

2. Find the area of the triangles.

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(a)

Area of triangle =

1 x base x height 2

1 x7mx8m 2

1 = x 56 m2 2 = 28 m2

(b)

1 x base x height 2 1 x 8 in x 10 in 2

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Area of triangle = =

10 in

1 = x 80 in2 2 = 40 in2 8 in

30 cm

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(c)

15 cm

Area of triangle =

1 x base x height 2

1 x 15 cm x 30 cm 2

1 = x 450 cm2 2 = 225 cm2

Let’s Practice 1. Identify the base and height of the triangles.

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(a)

base =

height =

A Q

(b)

base =

height =

R M

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(c)

base = height =

(d)

base =

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height =

(e)

base = height =

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(f)

base = height =

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2. Each square in the grids is 1 cm by 1 cm. Find the area of the triangles. (a) 1 Area of triangle = x base x height 2 1 = x x 2 1 = x 2

(b)

1 x base x height 2 1 = x x 2 1 = x 2

Area of triangle =

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(c)

1 x base x height 2 1 = x x 2 1 = x 2 Area of triangle =

1 x base x height 2 1 = x x 2 1 = x 2

(d)

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Area of triangle =

(e)

1 x base x height 2 1 = x x 2 1 = x 2

Area of triangle =

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(f)

1 x base x height 2 1 = x x 2 1 = x 2 Area of triangle =

3. Find the area of each triangle. Show your working in the space provided.

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(a)

12 cm

5 cm

cm2

Area =

(b)

10 m

Area =

in2

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(c)

8 in

(d)

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6 cm

4 cm

Area =

(e)

12 ft

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7 ft

(f)

Area =

cm2

Area =

ft2

2 cm

14 cm

cm2

Hands On Show that the area of a triangle is half the area of a rectangle with the same base and height.

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The rectangle below has a width of 16 cm and a height of 12 cm. It has an area of 192 cm2.

The triangle on the opposite page has a base of 16 cm and a height of 12 cm.

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Cut out the triangle from the page. Then cut along the dotted lines so the triangle is in 3 pieces. Arrange the pieces in the grid below to neatly fill up half of the rectangle.

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This page is blank to allow for a cut-out on the previous page.

At Home 1. Identify the base and height of the triangles. A

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(a)

base =

height =

(b)

height =

R O

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(c)

base =

base = height =

(d)

base =

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height =

base =

height =

(e)

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(f)

base = height =

2. Each square in the grids is 1 cm by 1 cm. Find the area of the triangles. Show your working in the space provided.

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(a)

Area =

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(c)

(b)

Area =

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(d)

Area =

(e)

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(f)

Area =

Solve It!

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Ethan is designing a mask for a dress-up party. The mask is made from a piece of square cardboard of side length 20 cm He cut a 12 cm by 3 cm rectangular hole for the mouth. For the eyes, he cut 2 right-angled triangles of the same size. They each had a base and height of 8 cm.

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Find the area of Ethan's mask.

Ethan's mask has an area of

cm2.

Area of Composite Figures

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Anchor Task

Let’s Learn The figure below is inside a 1 cm by 1 cm grid. Let's find its area.

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We can divide the figure into a triangle and a square.

Find the area of the composite shapes and add to find the total area of the figure.

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4 cm

3 cm

1 x 3 cm x 4 cm 2 1 = x 12 cm2 2 Area B =

C 3 cm

Area C = 3 cm x 3 cm = 9 cm2

= 6 cm2

Area A = Area B + Area C = 6 cm2 + 9 cm2 = 15 cm2

6 cm

3 cm

4 cm

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Find the area of the figure.

6 cm

We can break the figure into 2 triangles and a rectangle.

P 3 cm

4 cm

Area of figure = Area P + Area Q + Area R Area P =

1 x 3 cm x 6 cm 2

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= 9 cm2

Area Q = 4 cm x 6 cm = 24 cm2 Area R =

Is there another way to break up the figure to find the area?

1 x 4 cm x 6 cm 2

= 12 cm2

Area of figure = 9 cm2 + 24 cm2 + 12 cm2 = 45 cm2

R 4 cm

Find the area of figure ABCD.

2m D

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Area of figure ABCD = Area of triangle ABD – Area of triangle CBD 1 x7mx5m 2 1 = x 35 m2 2

Area of triangle ABD =

= 17.5 m2

1 x7mx2m 2 1 = x 14 m2 2

Area of triangle CBD =

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= 7 m2

Area of figure ABCD = 17.5 m2 – 7 m2 = 10.5 m2

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Find the area of the yellow figure.

2 in 8 in

12 in

4 in

Area of yellow figure = Area of triangle – Area of rectangular hole

1 x 12 in x 8 in 2 1 = x 96 in2 2 Area of triangle =

= 48 in2

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Area of rectangle = 2 in x 4 in = 8 in2 Area of yellow figure = 48 in2 – 8 in2 = 40 in2

Let’s Practice

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1. The pink figure is drawn with straight lines inside a rectangle. Find the area of the pink figure.

6 cm

2 cm 4 cm

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11 cm

The area of the pink figure =

cm2.

2. Find the area of the figure.

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20 cm

6 cm

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5 cm

Area =

cm2

The side of a factory wall needs to be painted. The dimensions of the wall are shown in the grid below. Each square on the grid represents 1 square meter. Find the total area of the wall.

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The area of the wall =

4. Find the area of the colored part of figure LMNOP. M 2 cm N

2 cm

5 cm

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7 cm

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12 cm

Area of colored figure =

Solve It!

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1. ABCDE is composed of the triangles ABE, BCD and BDE. Triangle BDE is twice the area of triangle BCD. Find the area of the figure ABCDE. C

5 cm

4 cm

3 cm

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Area ABCDE =

2. Figure PQRST is composed of triangle PQR and square PRST. The area of the colored part of PQRST is 38 cm2. Find the side length of the square PRST.

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Q 2 cm P

2 cm 2 cm

2 cm

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2 cm

The side length of the square PRST =

At Home

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1. The figure below is a triangle with a 1 cm by 1 cm square hole. Find the area of the figure.

25 cm

7 cm

6 cm

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18 cm

Area =

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2. Find the area of the blue figure.

5 ft

4 ft

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2 ft

Area =

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3. Each square in the grid has a side length of 1 inch. Find the area of the figure.

Area =

4. The green figure below is drawn inside a rectangle. Find the area of the figure. 9m

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Area =

Looking Back

(a) (b) Q U

R O

diameter =

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1. O is a point on the center of each circle. Labeled lines are straight lines. Identify the diameter of each circle.

2. O is a point on the center of each circle. Labeled lines are straight lines. Identify the radius of each circle. (a) (b)

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I J

radius =

3. Find the area of each circle. Take π = 3.14 and round off your answer to 1 decimal place. (a)

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20 cm O

(b) O

9 in

4. Find the area of each circle. Take π = (a)

7 ft

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(b)

63 mm O

22 . 7

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5. A pie has a diameter of 11 cm. Find the area. Take π = 3.14 and round off your answer to 1 decimal place.

6. A round sticker has a radius of 16 mm. Find the area. Take π = 3.14 and round off your answer to 1 decimal place.

22 . 7

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7. A plate has a diameter of 28 cm. Find area. Take π =

8. A dart board has a radius 14 in. Find the area. Take π =

22 . 7

9. Identify the base and height of the triangles. (a) W

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base = height =

(b)

base =

height =

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(c)

base = height =

10. Each square in the grids is 1 in by 1 in. Find the area of the triangles. Show your working in the space provided.

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(a)

Area =

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(c)

(b)

Area =

11. Find the area of the figure. 4 cm

Area =

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12. Find the area of the figure.

Area =

14 cm

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2 cm

1m 8m

13. Find the area of the figure.

3 cm

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2 cm

3 cm

2 cm

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4 cm

Area =

Time

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Anchor Task

globalair.ae

08 25 Dubai (DXB)

09 10

13 15

Departures

Global Airlines

Dubai (DXB)

21 55 Dubai (DXB)

Frankfurt (FRA)

14 00 San Francisco (SFO)

23 40 Cairo (CAI)

Arrivals

15 20

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Amsterdam (AMS)

Dubai (DXB)

21 15

05 15+1

23 00

19 45+1

Melbourne (MEL)

New York (JFK)

23 59 Dubai (DXB)

Dubai (DXB)

Expressing 12-hour and 24-hour Time Let’s Learn

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The times on the departure board are shown in 24-hour time. When we use 24-hour time, we do not need to use a.m. or p.m.

When using 24-hour time, midnight is expressed as 00 00. 6:00 a.m.

00 00

06 00

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12:00 a.m.

In 24-hour time, you normally don’t say o’clock.

12:00 noon 12 00

6:00 p.m. 18 00

12:00 a.m. 00 00

For 18 00, we say eighteen hundred hours.

To convert times between noon and midnight into 24-hour time, you add 12 to the hours. To convert times between noon and midnight from 24-hour time to 12-hour time, you subtract 12 from the hours and add p.m.

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Ethan goes fishing at 3:20 p.m. Express the time Ethan goes fishing in 24-hour time.

3 + 12 = 15

24-hour time: 15 20 fifteen twenty

21 – 12 = 9

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Ethan goes to bed at 21 45. Express the time Ethan goes to bed in 12-hour time.

12-hour time: 3:20 p.m. three twenty p.m.

24-hour time: 21 45 twenty-one forty-five 12-hour time: 9:45 p.m. nine forty-five p.m.

Let’s Practice 1. Write the times in 24-hour time.

AT IO N

(a) (b)

Morning:

Afternoon:

Evening:

Morning:

Night:

Morning: Night:

R EG

(e) (f)

Morning:

Night:

Afternoon:

2. Write the times in 12-hour time using a.m. or p.m.

AT IO N

(a) (b)

(e) (f)

3. Complete the table.

Flight Departures – San Francisco International Airport (SFO)

R EG

City

Denver

Departure

(12-hour)

(24-hour time)

11:45 a.m.

Chicago

New York

San Diego

00 05 10:20 a.m. 20 10

At Home Complete the table.

Activity

12-hour Time

Headmaster’s opening speech.

9:15 a.m.

High jump

10:50 a.m.

Shot put

12 30

1:35 p.m.

4 x 100 m relay

24-hour Time

09 45

Discus

Javelin

AT IO N

School Athletics Day Program

2:25 p.m.

18 45

Awards ceremony

20 05

R EG

100 m sprint

Duration of Time Let’s Learn

AT IO N

Halle and Sophie went for a picnic in the park at 11 30. They left at 13 55. How long were Halle and Sophie in the park?

25 min

13 30

13 55

11 30

2 h + 25 min = 2 h 25 min Halle and Sophie were in the park for 2 h 25 min.

R EG

Dominic and Jordan rode their bikes to the lake. They arrived at 14 40 and hiked around the lake until 16 25. How long was their hike?

20 min

14 40

15 00

1 h + 20 min + 25 min = 1 h 45 min Dominic and Jordan hiked for 1 h 45 min.

25 min 16 00

16 25

AT IO N

Blake and his family are taking the train to the beach. The train departs at 11 30. They arrive at the beach 3 hours 35 minutes later. What time did they arrive at the beach?

35 min

14 30

15 05

11 30

3 hours after 11 30 is 14 30. 35 min after 14 30 is 15 05. Blake and his family arrived at the beach at 15 05.

R EG

Riley played ice hockey for 2 hours 45 minutes. She finished playing at 13 30. What time did Riley start playing ice hockey?

15 min

10 45

30 min

11 00

2h 11 30

13 30

2 hours before 13 30 is 11 30. 30 minutes before 11 30 is 11 00. 15 minutes before 11 00 is 10 45. Riley started playing ice hockey at 10 45.

Complete the word problems. Use a timeline to show your working.

AT IO N

1. Keira started doing her homework at 16 25. She stopped for dinner at 18 35. How long did Keira spend doing her homework?

2. A train left Boston at 09 15. It arrived in New York City at 13 22. How long was the train ride?

R EG

3. Blake visited his uncle at 11 38. He left his uncle’s house at 13 55. How long did Blake spend visiting his uncle?

AT IO N

4. Ethan and Dominic started playing chess at 13 26. They played for 2 hours 34 minutes. What time did they finish playing chess?

5. Sophie arrived at school at 08 25. Her English class started 2 hours 43 minutes after she arrived. What time did Sophie’s English class start?

R EG

6. Wyatt washed the car with his father for 1 hour 47 minutes. They started washing the car at 11 30. What time did they finish washing the car?

AT IO N

7. Riley chatted with her aunt on the phone for 1 hour 16 minutes. She finished the conversation at 13 05. At what time did she start chatting with her aunt?

8. A tennis match lasted 3 hours 28 minutes. The match finished at 21 25. What time did the tennis match start?

R EG

9. On Sunday, Sophie did housework for 2 hours 45 minutes. She finished the housework at 15 15. What time did Sophie start doing housework?

Solve It!

R EG

AT IO N

1. On Saturday, Halle had singing lessons at 10 15. Her lesson went for 1 hour 25 minutes. After her lesson, she walked home. The walk took 38 minutes. When she arrived home, she had lunch. It took her 45 minutes to finish lunch. She then read a book for 2 hours 10 minutes. At what time did Halle finish reading her book?

2. Wyatt went on a holiday to Resort Island with his family. The map on the opposite page shows the nearby islands and the travel time by boat. Use the map to answer the questions. Show your working.

AT IO N

(a) Wyatt took the boat from Resort Island to Jungle Island at 10 50. He went sightseeing for 1 hour 15 minutes. He then took the boat to Desert Island for lunch. What time did he arrive at Desert Island?

(b) Wyatt took the boat from Resort Island to Icy Island and took photographs for 45 minutes. He then took the boat to Volcano Island and went hiking for 1 hour 15 minutes. Finally, Wyatt took the boat to Jungle Island and arrived at 18 05 in time for dinner with his family. What time did Wyatt leave Resort Island?

R EG

(c) Wyatt was on Volcano Island and wanted to return to Resort Island. To do this, he had to stop at some islands to change boats. Each change of boat took 5 minutes. It took Wyatt 2 hours 7 minutes to return to Resort Island. Which islands did Wyatt stop at on his return trip?

Rocky Island

Resort Island

AT IO N

29 min

34 min

16 min

25 min

Jungle Island

Icy Island

R EG

19 min

25 min

28 min 42 min

Volcano Island 48 min

Desert Island

At Home Complete the word problems. Use a timeline to show your working.

AT IO N

1. Ms. Kang started playing golf at 08 35. She finished playing golf at 11 55. How long did Ms. Kang spend playing golf?

R EG

2. Jessica works at the ice-creamery on Saturday. She started her shift at 11 30 and worked for 5 hours 50 minutes. What time did Jessica finish her shift?

3. A school play goes for 2 hours 45 minutes. The play finishes at 21 15. What time did the school play start?

AT IO N

4. Mr. Pritchard ran a marathon in 3 hours 18 minutes. He started running at 13 52. At what time did Mr. Pritchard finish the marathon?

5. Fabio opened his barber shop at 10 45. He closed the shop at 20 30. How long was Fabio’s barber shop open?

R EG

6. Mr. Whyte raced his car for 2 hours 55 minutes. He finished the race at 16 45. What time did the car race start?

Hands On

AT IO N

Work in pairs. Use the map below to create a story problem for your partner. Pick a start time and list the places your hiked. Have your partner work out the time you arrived at the final destination. When they answer correctly, switch roles.

42 min

60 min

28 min

30 min

R EG

25 min

1 h 15 min

29 min

Word Problems Let’s Learn

3 h 18 min

2 h 37 min

train ride

ferry ride ?

AT IO N

Sophie is going on vacation with her family to an island resort. To get there, they spend 3 hours 18 minutes on a train. They then take the ferry to the island. The ferry trip is 2 hours 37 minutes. How long did it take to get to the resort?

3 h 18 min + 2 h 37 min = 5 h + 18 min + 37 min = 5 h 55 min

It took 5 hours 55 minutes to get to the resort.

The flight from Las Vegas to Phoenix takes 1 hour 28 minutes. It takes 4 hours 5 minutes to drive. How much faster is it to fly from Las Vegas to Phoenix than to drive? 1 h 28 min

R EG

fly

drive

Regroup 1 hour into 60 minutes. Then subtract.

4 h 5 min

4 h 5 min – 1 h 28 min = 3 h + 65 min – 1 h – 28 min = 2 h 37 min It is 2 hours 37 minutes faster to fly from Las Vegas to Phoenix than to drive.

Wyatt reads for 47 minutes every day. How long does Wyatt spend reading in 1 week?

AT IO N

47 min

? 4

x 3

7 7

329 min = 5 h 29 min

In 1 week, Wyatt spends 5 hours 29 minutes reading.

Mr. Gil walks to work everyday. In 5 days he walks for a total of 3 hours 10 minutes. If he walks for the same duration each day, how long does it take him to walk to work?

R EG

First, convert the hours into minutes.

190 min

190 min ÷ 5 = 38 min Each day, Mr. Gil spends 38 minutes walking to work.

Let’s Practice Complete the word problems. Show your working.

AT IO N

1. Michelle ran for 1 hour 28 minutes. She then cycled for 2 hours 43 minutes. How long did Michelle spend exercising in all?

2. It took Jordan 3 hours 17 minutes to reach the top of a mountain. Blake reached the mountain top 1 hour 45 minutes faster than Jordan. How long did it take Blake to reach the mountain top?

R EG

3. Sophie spent 1 hour 48 minutes cleaning her bedroom. Chelsea spent 2 hours 12 minutes cleaning her bedroom. How much longer did Chelsea spend cleaning her bedroom than Sophie?

AT IO N

4. Ethan takes 5 minutes 42 seconds to walk around his neighborhood block. How long will it take Ethan to walk around the block 7 times?

5. It takes Mrs. Siew 17 minutes to fold a tray of dumplings. How long will it take Mrs. Siew to fold 12 trays of dumplings?

R EG

6. On a hike, Halle and Sophie stopped for a break 7 times. Each break was the same number of minutes. The total break time was 3 hours 23 minutes. How long was each break?

At Home Complete the word problems. Show your working.

AT IO N

1. Blake took 2 hours 25 minutes to revise for his history test. He took 3 hours 12 minutes to revise for his mathematics test. How much longer did he take to revise for his mathematics test than his history test?

2. Keira made a series of 8 video clips for a school presentation. Each clip ran for the same length of time. The whole series ran for 1 hour 12 minutes. How long was each video clip?

R EG

3. Halle swims one lap of a swimming pool in 3 minutes 26 seconds. How long does it take for her to swim 12 laps of the swimming pool?

Solve It! Complete the word problems. Show your working.

AT IO N

1. Jordan took 42 minutes to walk from his house to the library. He read in the library for 2 hours 38 minutes and left at 16 50. What time did Jordan leave his house?

R EG

2. A baker takes 12 minutes to prepare a cake mix, 25 minutes to bake the cake and 15 minutes to add the icing. She needs to bake 12 cakes by 13 15. What is the latest time the baker should start making the cakes?

Looking Back 1. Write the times in 24-hour time.

AT IO N

(a) (b)

Morning:

Afternoon:

Morning:

Afternoon:

R EG

Night:

Morning:

2. Write the times in 12-hour time using a.m. or p.m. (a) (b)

AT IO N

(e) (f)

3. Complete the table.

City

Departures – Midtown Bus Terminal

The Bronx

Departure

(12-hour)

(24-hour time)

6:45 a.m. 17 25

R EG

Newark

Yonkers

Staten Island

3:40 p.m. 23 20

AT IO N

4. After school, Halle plays chess with her friend for 1 hour 34 minutes. She then walks 46 minutes back to her home. She arrives home at 18 00. What time did Halle start playing chess with her friend?

5. A teacher takes 18 minutes to mark an exam. How long will it take her to mark 13 such exams?

R EG

6. It took Riley 2 hours 51 minutes to jog around the lake 9 times. How long does Riley take to jog 1 time around the lake?

Speed

Speed and Distance

AT IO N

Anchor Task

100 meters

Sophie 14.1 s Dominic 15.5 s Halle Wyatt

R EG

Ethan 15.4 s Blake 14.2 s Michelle 15.1 s Riley 16.0 s

Let’s Learn

AT IO N

A truck and a car are traveling in the same direction. They are traveling at different speeds. Speed is how fast something is moving. It is the distance something moves per unit of time.

The speed of the truck is 60 kilometers per hour. Traveling at this speed, the truck will cover a distance of 60 kilometers in 1 hour. We write this as 60 km/h.

The speed of the car is 100 kilometers per hour. Traveling at this speed, the car will cover a distance of 100 kilometers in 1 hour. We write this as 100 km/h.

We can calculate speed by dividing distance by time. speed = distance ÷ time

R EG

The greater the distance per unit time, the faster the speed. The speed of a brisk walk is about 6 km/h.

The speed of a galloping horse is about 45 km/h.

speed = distance ÷ time = 50 ÷ 5 = 10 The speed of the caterpillar is 10 cm/min.

AT IO N

A caterpillar crawls 50 centimeters in 5 minutes. Find the speed of the caterpillar in cm/min.

A plane travels a distance of 3,200 kilometers in 4 hours. Find the speed of the plane in km/h.

The speed of the plane is 800 km/h.

3,200 ÷ 4 = 800

99 ÷ 9 = 11

An owl flies 99 meters in 9 seconds. Find the speed of the owl in m/s.

The speed of the owl is 11 m/s.

R EG

Wyatt jogged 750 meters in 3 minutes. Find Wyatt's speed in m/min. 750 ÷ 3 = 250

Wyatt jogged at a speed of 250 m/min. When we know an object's speed, we can calculate the distance it will cover for a given amount of time. distance = speed x time

Riley skates at a speed of 4 m/s for 20 seconds. How far does Riley skate?

AT IO N

distance = speed x time = 4 x 20 = 80 Riley skates 80 m. 1

A car took 2 4 h to travel from Springfield to Bakersfield at a speed of 60 km/h. How far is Springfield from Bakersfield? 1

60 x 2 4 = 60 x 4 = 135

Springfield is 135 km from Bakersfield.

32 x 2 2 = 32 x 2 = 80

A cyclist rides at a speed of 32 km/h. 1 How far does he ride in 2 2 hours?

R EG

The cyclist rides 80 km in 2 2 h. A fighter jet flies at a speed of 400 m/s. How far can the fighter jet fly in 1 minute? 1 m = 60 s 400 x 60 = 2,400 2,400 m = 2.4 km The fighter jet can fly 2.4 km in 1 minute.

When we know the speed an object is moving and the distance it needs to cover, we can calculate the time it will take to cover that distance.

A plane travels at a speed of 900 km/h. How long does it take for the plane to fly 2,700 km from Singapore to Hong Kong? time = distance ÷ speed = 2700 ÷ 900 = 3

AT IO N

time = distance ÷ speed

The plane takes 3 h to fly from Singapore to Hong Kong.

2.4 m = 240 cm 240 ÷ 8 = 30

A snail crawls at a speed of 8 cm/min. How long does it take for the snail to cross a path 2.4 m in length?

It takes the snail 30 minutes to cross the path.

R EG

The distance between Harbortown and Slicks Creek is 385 km. The top speed of Mr. Whyte's motorcycle is 110 km/h. What is the shortest time he can ride from Harbortown to Slicks Creek?

385 ÷ 110 = 3.5

The shortest time Mr. Whyte can ride from Harbortown to Slicks Creek is 3.5 h.

Let’s Practice

AT IO N

1. The distance from a shoe factory to a warehouse is 240 km. It takes a truck 3 h to travel between the factory and the warehouse. Find the speed of the truck.

2. A tortoise covered a distance of 600 m in 50 min. Find the speed of the tortoise.

R EG

3. Sophie lives 8.5 km from her school. To get from her home to school takes 10 min. At what speed does Sophie travel to school?

AT IO N

4. Station A is 270 km from Station B. A train takes 3 h to travel from Station A to Station B. Find the speed of the train.

5. Traveling at top speed, it takes 3 hours for a fighter jet to cover a distance of 6,500 km. What is the top speed of the fighter jet?

R EG

6. A bird flew 156 m in 12 s. What was the speed of the bird?

100

AT IO N

7. A cyclist rides for 4 hours at a speed of 27 km/h. How far did the cyclist ride?

8. Find the distance covered by a cheetah that runs at a speed of 20 m/s for 2 min.

R EG

9. Keira rolled a ball the width of her classroom at a speed of 75 cm/s. It took the ball 10 s to roll from one side to the other. What is the width of Keira's classroom?

1 01

AT IO N

10. 2 laps of the running track at Blake's school is 400 m in length. If Blake runs at a speed of 4 m/s, how long will it take him to complete 7 laps of the running track?

11. A car is traveling on a motorway at the speed of 85 km/h. How long will it take the car to cover a distance of 255 km?

R EG

12. Sophie walks to school at a speed of 55 m/min. She leaves home at 7:45 a.m. and arrives at school at 8:05 a.m. How far is Sophie's house from her school?

1 02

Solve It! The table below shows the speeds of different commercial plane models and the distances they can cover in a given time.

What is the speed of a B707 plane? How long does it take a B777 plane to fly 4,320 km? How far does a B787 fly in 7 hours? What is the speed of the B950 plane? Distance (km)

Time (min)

B707

6,300

540

B777

4,320

11,400

Speed

720 km/h 825 km/h

720

B950

420

B787

Model

R EG

1. 2. 3. 4.

AT IO N

Answer the following questions to complete the table.

103

1. In small groups, go into your schoolyard and use a trundle wheel to measure and mark a distance of 100 m. Use a stopwatch to time how long it takes to complete each activity shown in the table. Calculate your speed to complete the table.

walk

100

job

100

run

100

hop

100

Time (s)

Speed

Distance (m)

Activity

AT IO N

Hands On

R EG

2. Use a trundle wheel to measure the length of the school basketball court. Use the stopwatch to time how long it takes for a soccer ball to move from one side of the basketball court to the other when it is pushed in the different ways shown in the table. Activity

Roll slowly

Roll quickly Throw Kick

104

Distance (m)

Time (s)

Speed of Ball

At Home

AT IO N

1. To escape a predator, a gazelle ran 420 m in 70 s. What was the speed of the gazelle?

2. A truck takes 9 h to drive 720 km from Melbourne to Sydney. Find the speed of the truck.

R EG

3. The swimming pool at Halle's school is 25 m in length. Halle takes 7 2 minutes to swim 9 lengths of the pool. At what speed does Halle swim?

105

AT IO N

4. Ethan took 8 min to walk from his home to the local park. He walked at a speed of 42 m/min. How far is Ethan's house from the local park?

5. An albatross flies at the speed of 11 m/s for 4 2 hours. What distance does the albatross cover?

R EG

6. Wyatt constructed a dirt track for his remote-controlled car. One lap of the track is 75 m. Traveling at a speed of 35 km/h, how long will it take for the car to complete 7 laps?

106

AT IO N

7. A plane flies at a speed of 665 km/h. The plane leaves Los Angeles at 6:10 a.m. and flies 3,990 km to New York. What time does the plane arrive in New York?

8. A rabbit hops at a speed of 6 m/s. How long will it take to cross a field 450 m wide?

R EG

9. A peregrine falcon spots prey that is 175 m away. It dives towards the prey at 70 m/s. How long will it take for the peregrine falcon to reach its prey?

107

Solve It!

R EG

AT IO N

The students in Grade 6 are going on a school camp in two buses, Bus A and Bus B. The distance from the school to the camp is 150 km. Bus A left for the camp at 8:00 a.m. Bus B left for the camp at 8:30 a.m. Both buses arrived at the camp at 10:30 a.m. Find the speed of the buses.

108

Average Speed Let’s Learn

AT IO N

When an object is moving, its speed usually changes. A moving car may slow down to turn a corner, or speed up as it moves from a road onto a motorway.

We can calculate the average speed an object moves by dividing the total distance traveled by the total time taken. average speed = total distance traveled ÷ total time taken

Ascent

Distance = 3,750 m Time = 5 min

Distance = 3,000 m Time = 10 min

During a bicycle race, riders must ride up one side of a mountain and down the other side. The distance up the mountain is 3,000 m. The distance down the mountain is 3,750 m. It takes a cyclist 10 min to cycle up the mountain and 5 min to cycle down the mountain. What was the average speed of the cyclist?

Descent

? m/min

To find the average speed, we first need to find the total distance traveled and the total time taken.

R EG

Total distance = 3,000 + 3,750 = 6,750 m Total time = 10 + 5 = 15 min average speed = total distance traveled ÷ total time taken = 6,750 ÷ 15 = 450 The average speed of the cyclist was 450 m/min.

109

Chelsea usually walks from her home to her school. On Monday, she left home at 8:10 a.m. She walked the first 800 m to the market in 8 min. Realizing she may be late, she ran the remaining 700 m and arrived at school at 8: 22 a.m. Find Chelsea's average speed from her home to her school.

Home

Distance = 700 m Time = 4 min

AT IO N

Distance = 800 m Time = 8 min

School

Market ? m/min

Total distance = 800 + 700 = 1,500 m

Total time = 8 + 4 = 12 min

average speed = total distance traveled ÷ total time taken = 1,500 ÷ 12 = 125 Chelsea's average speed from her home to her school was 125 m/min.

A bus traveled from Town A to Town D. The bus left Town A at 8:45 a.m. and arrived in Town D at 12:45 p.m. What was the average speed of the bus from Town A to Town D? Town A

R EG

Town B

64 km

Town C

90 km

Total distance = 64 km + 90 km + 110 km = 264 km Total time taken = 4 h Average speed = 264 ÷ 4 = 66 The average speed of the bus was 66 km /h.

110

Town D 110 km

Let’s Practice Draw a diagram and show your working for each question.

R EG

AT IO N

1. A salesman took 2 h to drive 108 km from Town A to Town B. He took 3 h to drive 216 km from Town B to Town C. What was the average speed of the salesman for the whole trip?

111

R EG

AT IO N

2. The school running track has a length of 480 m. Wyatt ran 2 laps of the school running track. He ran the first lap in 4 minutes. He ran the second lap in 6 minutes. Find Wyatt's average speed for the 2 laps.

112

3. A train left Station Y at 6:15 a.m. and traveled at an average speed of 1

80 km/h for 1 2 hours. The train then increased its average speed to

90 km/h and arrived at Station Z at 9:45 a.m. Find the distance from

R EG

AT IO N

Station Y to Station Z.

113

R EG

AT IO N

4. Riley walked 480 meters at a speed of 120 m/min to get from her school to the park. She then took 6 minutes to walk from the park to her home. The distance from the park to Riley's home is 520 m. Find Riley's average speed from her school to her home.

114

Hands On

AT IO N

Use some cones to mark distances in your school as shown.

The distance between Cone A and Cone B is 100 meters. The distance between Cone B to Cone C is 20 meters. The distance from Cone C to Cone D is 120 meters.

R EG

Use a stopwatch to time how long it takes you to complete the activities shown in the table. Calculate your speed to complete each activity, then calculate your average speed from Cone A to Cone D. Activity

Distance (m)

Walk from Cone A to Cone B

100

Crawl from Cone B to Cone C

Run from Cone C to Cone D

120

Time (s)

Speed

Average Speed

115

At Home Draw a diagram and show your working for each question.

R EG

AT IO N

1. Ethan walked 440 meters at an average speed of 80 m/min. 1 Jordan took 1 2 minutes more to cover the same distance. What was Jordan's average speed?

116

2. Chelsea took 8 minutes to kayak from the campsite to the lake at a speed of 48 m/min. She took 6 minutes to kayak back from the lake to the campsite.

R EG

AT IO N

(a) Find the distance from the campsite to the lake. (b) What was Chelsea's average speed for the whole journey?

117

3. A plane is flying at a height of 3,200 feet. A skydiver leaps from the plane and falls at an average speed of 320 ft/s for 8 seconds. He then opens his parachute and lands 32 seconds later.

R EG

AT IO N

(a) What was the skydiver's average speed from when he opened the parachute to when he landed? (b) What was the skydiver's average speed from the plane to the ground?

118

4. Halle and Sophie competed in a 420 m race. Sophie ran at an average speed of 8 m/s. Halle ran at an average speed of 7 m/s.

R EG

AT IO N

(a) How long did it take for Sophie to finish the race? (b) How far had Halle ran when Sophie crossed the finish line?

119

Word Problems Let’s Practice

AT IO N

Draw a diagram and show your working for each word problem.

R EG

1. Mrs. Wong took 3 hours to drive from Town A to Town B at an average speed of 70 km/h. On her way back, she drove at an average speed of 60 km/h. How long did Mrs. Wong take to drive back from Town B to Town A?

120

2. Wyatt started cycling at 8:30 a.m. By 10:30 a.m. he had covered a distance of 15 km.

R EG

AT IO N

(a) Find his average speed in km/h. (b) If he cycled a further distance of 7.5 km at the same average speed, what would the time be? (c) The next day, Wyatt started cycling at the same time. This time, his average speed was double that of the day before. What distance will he cover if he finishes cycling at 10:00 a.m.?

121

R EG

AT IO N

3. Halle traveled from her home to her cousin's house. She covered 2 of 1 the trip in the 1st hour and 3 of the trip in the 2nd hour. It took 1 hour to travel the remaining 15 km. Find her average speed for the whole trip.

1 22

4. Mr. Robinson drove from his home to his beach house. He traveled the first 96 km at an average speed of 72 km/h. He traveled the remaining 36 km at an average speed of 54 km/h.

R EG

AT IO N

(a) How far is Mr. Robinson's house to his beach house? (b) Find his average speed for the whole journey.

123

R EG

AT IO N

5. Halle took 3 hours to cover 3 of a journey to her grandmother's house. She covered the remaining 90 km in 2 hours. Find her average speed for the whole journey.

124

R EG

AT IO N

6. A motorist took 2.5 hours to travel from Town X to Town Y. His average 2 speed for the whole journey was 60 km/h. For the first 3 of the journey, he traveled at an average speed of 50 km/h. Find his average speed for the remaining journey.

125

Solve It! 3

AT IO N

A motorist traveled from Town A to Town B. After traveling 4 of the journey at an average speed of 50 km/h, he continued to travel another 90 km to reach Town B.

R EG

1. Find the distance between the two towns. 2. If his average speed for the whole journey was 60 km/h, find his average speed for the last part of the journey.

126

At Home Draw a diagram and show your working for each word problem.

R EG

AT IO N

1. Keira and Blake took 2 hours to hike from the lake to the waterfall at an average speed of 70 m/min. On their way back, they hiked at an average speed of 60 m/min. How long did it take Keira and Blake to hike from the waterfall back to the lake?

127

2. Wyatt took 5 minutes to walk from his house to the sports store at an average speed of 45 m/min. He then took another 10 minutes to walk to the restaurant at an average speed of 30 m/min.

R EG

AT IO N

(a) How far did he walk altogether? (b) Wyatt rides the same distance on his bicycle at an average speed 3 times faster than the speed he walked. How long will it take Wyatt to cover the same distance?

1 28

3. A motorcyclist took 3 hours to travel from Town X to Town Y at an average speed of 70 km/h. A truck took 4 hours for the same journey.

R EG

AT IO N

(a) Find the average speed of the truck. (b) At the same average speed, what distance can the truck travel in 6 hours?

129

R EG

AT IO N

4. Wyatt cycled from Town A to Town B. He covered 6 of the trip in the 1 first hour and 4 of the trip in the second hour. He took 2 hours to cycle the remaining 35 km. Find his average speed for the whole trip.

130

R EG

AT IO N

5. Bus A and Bus B left Portsea College at midday for an interschool sports competition. Bus A traveled at an average speed of 60 km/h. When Bus A arrived at the stadium at 12 30, Bus B was 2 km from the stadium. Find the average speed of Bus B.

131

R EG

AT IO N

6. Blake and Ethan started cycling from the same starting point in opposite directions. Ethan cycled at an average speed of 30 km/h. Blake cycled at an average speed that was 10 km/h slower than that of Ethan. They stopped cycling after 18 minutes. Find the distance between them now.

1 32

Solve It!

AT IO N

1. John and Leon started driving from Town A to Town B at the same 3 time. After 2 hours, John reached Town B while Leon had completed 4 of the journey. (a) If John’s average speed for the whole journey was 70 km/h, find 3 Leon’s average speed for the first 4 of the journey. 1

R EG

(b) If Leon’s average speed for the last 4 was increased by 7.5 km/h, 1 how long did he take for the last 4 of the journey?

133

Looking Back

AT IO N

1. The distance from a toy factory to a toy shop is 260 km. It takes a truck 4 h to travel between the factory and the shop. Find the speed of the truck.

2. Riley took 9 min to walk from her house to her grandmother's house. She walked at a speed of 63 m/min. How far is Riley's house from her grandmother's house?

R EG

3. A plane is traveling at a speed of 930 km/h. How long will it take for the plane to cover a distance of 2,325 km?

1 34

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4. Jordan took 3 hours to walk 12 km to his uncle's house. He took 2 hours to walk back to his house. What was Jordan's average speed for the return journey?

5. The distance around a lake is 400 meters. Halle took 13 minutes to cycle 2 around the lake 6 3 times.

R EG

(a) Find the total distance Halle cycled. (b) What was Halle's average speed?

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6. Ethan walked on a beach in 1 direction for 3 hours at an average speed of 75 m/min. He then turned around and walked back at an average speed of 60 m/min.

R EG

AT IO N

(a) How long did the return trip take? (b) What was Ethan's average speed for the whole journey?

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R EG

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7. Halle and Riley started jogging from the park in opposite directions. Halle jogged at an average speed of 80 m/min. After 10 minutes, they were 1,800 m apart. Find Riley's jogging speed.

137

10 Pie Charts

Reading and Interpreting Pie Charts

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Anchor Task

running 3 swimming 8 tennis 5 badminton 2 football 6

R EG

What is your favorite sport, Jordan?

Favorite Sport

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I love tennis!

Let’s Learn

We can represent the data in a pie chart.

The size of each portion in a pie chart is relative to its value.

Aquarium 60%

Zoo 40%

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Mrs. Jones asked the students in her science class where they would like to go on a field trip. 60% of the students chose the aquarium. The remaining 40% of the students chose the zoo.

The combined parts in the pie chart represents 100%, or 1 whole. In 1 day, a greengrocer sold 60 pieces of fruit. He sold 21 apples, 12 pears and 9 oranges. The rest of the fruit sold were mangoes. Apples

Pears

Oranges

Mangoes

Number Sold

Fruit

R EG

Let's use a pie chart to represent and interpret the data The largest portion in the pie chart is apples.

Mangoes ? Oranges 9

Apples 21

Pears 12

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Which type of fruit did the greengrocer sell the most? We can see that the largest portion of the pie chart is apples. So, the greengrocer sold apples the most.

AT IO N

Which type of fruit did the greengrocer sell the least? We can see that the smallest portion of the pie chart is oranges. So, the greengrocer sold oranges the least. How many mangoes did the greengrocer sell?

We know that the greengrocer sold 60 pieces of fruit in all. 60 – 21 – 12 – 9 = 18 The greengrocer sold 18 mangoes.

Fishing

Mr. Whyte spent $700 on different activities whilst on a holiday. The pie chart shows the amount of money spent on each activity.

1 1 1 + + =1 4 4 2

Diving

Dining

R EG

By interpreting the data in the pie chart, we can see that Mr. Whyte spent half of his money on diving. 1 = $350 2 Mr. Whyte spent $350 on diving. $700 x

We can also see the other half of his money was spent equally on fishing and dining. $350 ÷ 2 = $175 Mr. Whyte spent $175 on fishing and $175 on dining.

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Jordan asked 40 students in Grade 6 how they get to school. He created a pie chart to represent the data he collected.

Half of the students walk or ride. The other half come to school in a car or bus.

Ride 4

AT IO N

Walk

Car 6

How many students walk to school? 40 – 4 – 6 – 14 =16 16 students walk to school.

Bus 14

What fraction of the students ride to school? 1 4 = 40 10 1 of the students ride to school. 10

What fraction of the students take a bus to school?

R EG

14 7 = 40 20 7 of the students take a bus to school. 20

What percent of students ride or come to school in a car? 1 10 = = 25% 40 4

25% of students ride or come to school in a car.

1 41

A group of 60 students in Grade 6 were asked which sport they'd prefer to play in an interschool competition. The pie chart shows how many students chose each sport. What fraction of the students prefer to play basketball?

AT IO N

Golf Basketball Soccer 1 5

Tennis 2 5

What was the most preferred sport of the Grade 6 students? Tennis was the most preferred sport. What fraction of the students preferred to play golf? 1 1 20 8 4 5 3 2 – – = – – – = 5 5 4 20 20 20 20 20

1–

3 of the students preferred to play golf. 20

R EG

How many students preferred to play basketball? 1 1 of 60 = x 60 4 4 60 = = 15 4 15 students preferred to play basketball. What percent of students preferred to play tennis? 2 4 = = 40% 5 10

40% of students preferred to play tennis.

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Blake and his friends put their money together to buy a gift for Ethan. The amount each friend contributed is shown in the pie chart below.

Wyatt (20%)

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Sophie

What percent of the money did Sophie and Halle contribute?

Blake (36%)

Halle

What percent of the money did Halle contribute? 50% – 36% = 14% Halle contributed 14% of the money.

What percent did Blake, Wyatt and Halle contribute altogether? 36% + 20% + 14% = 70% Blake, Wyatt and Halle contributed 70% of the money altogether.

What percent of the money did Sophie contribute? 100% – 70% = 30% Sophie contributed 30% of the money.

R EG

Sophie contributed $90. How much was the gift for Ethan? 30% $90 1% $3 100% $300 The gift for Ethan cost $300. What is the ratio of the amount of money Blake contributed to the amount of money Sophie contributed? 36 : 30 = 6 : 5 The ratio of the amount of money Blake contributed to the amount of money Sophie contributed is 6 : 5.

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Let’s Practice

Sheep 9

Ducks 3

AT IO N

1. The pie chart below shows the number of animals at a petting zoo.

Rabbits 12

Goats 6

(a) How many ducks and goats are at the petting zoo?

(b) How many more rabbits than ducks are at the petting zoo?

(c) How many fewer ducks than goats are at the petting zoo? (d) How many animals are at the petting zoo altogether? (e) What fraction of the animals at the petting zoo are goats?

(f) What fraction of the animals at the petting zoo are either sheep or rabbits?

R EG

(g) What fraction of the animals at the petting zoo are not rabbits?

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2. Riley saved $900 in 4 months. The pie chart below shows how much she saved each month.

Jan $105

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Feb

Mar

Apr

(a) In which month did Riley save the most money?

(b) How much did Riley save in February?

(d) How much did Riley save in March? (e) What fraction of the money was saved in March and April?

(f) What fraction of the money was saved in January? (g) What fraction of the money was saved in February and March

R EG

altogether?

145

3. On school camp, Chelsea took 450 photographs. She transferred them to her computer and put them into folders based on different themes. The pie chart shows the number of photographs in each folder.

AT IO N

Beach Lake 125

Forest

Friends 160

(a) How many forest photographs are there?

(b) How many beach photograph are there? (c) What fraction of the photographs are in the 'Lake' folder? (d) What fraction of the photographs are in the 'Friends' folder?

(e) What percent of the photographs are in the 'Beach' folder? (f) What percent of the photographs are in the 'Forest' folder?

R EG

(g) What is the ratio of the number of beach photographs to photographs of Chelsea's friends?

146

4. 60 students were graded on their English essays. The pie chart shows the number of students that received each grade.

AT IO N

D A 5 12

C 1 5 B

(a) What was the most common grade students received?

(b) What fraction of students received a D?

(d) How many students received a grade lower than a B?

(e) What percent of students received a C? (f) What is the ratio of the number of students who received a B to the total number of students?

R EG

(g) What is the ratio of the number of students who received a C to the number of students who received an A?

147

5. The pie chart shows the number of the different type of vehicles parked in Sunshine College in an afternoon. Express the number of each type of vehicle as a fraction of the total number of vehicles in the car park.

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Buses (13) Cars (45)

Motorcycles (20)

(b) Vans:

(a) Cars:

Vans (22)

(d) Buses:

6. The pie chart shows the amount of money Keira and her friends spent at the aquarium gift shop. Express the amount of money each child spent as a percentage of the total amount they spent in all.

R EG

Wyatt ($84)

Keira

(a) Keira:

1 48

Chelsea ($56)

Halle ($100)

(b) Wyatt: (d) Halle:

7. The pie chart shows the percentage of items sold in a clothes shop. Hats

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Dresses Skirts Shirts (38%)

(a) What percent of clothes sold were skirts?

(b) What percent of clothes sold were dresses? (c) What percent of clothes sold were hats?

(d) What fraction of clothes sold were skirts or dresses? (e) What is the ratio of the number of hats sold to the total number of

items sold?

R EG

(f) 60 hats were sold. Find the number of dresses, skirts and shirts sold.

Dresses sold:

Skirts sold:

Shirts sold:

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Hands On 1. The table below shows the number of different snacks sold during a lunch break. Nuts

Chips

Chocolate

Fruit

Number Sold

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Snack

R EG

Use the Matholia chart tool to create a pie chart to represent the data in the table. Print and paste your pie chart below. Label the pie chart and express the values as percentages.

150

2. The table below shows the number of students learning different languages at an international school. English

Arabic

Chinese

Spanish

Number of Students

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Language

R EG

Use the Matholia chart tool to create a pie chart to represent the data in the table. Print and paste your pie chart below. Label the pie chart and express the values as fractions.

1 51

At Home

Daisies 27

AT IO N

1. The pie chart below shows the number of bunches of flowers a florist sold in 1 week.

Tulips 81

Lilies 45

Roses 108

(a) How many bunches of roses and tulips were sold?

(b) How many more roses than daisies were sold? (c) How many fewer lilies than roses were sold? (d) How many bunches of flowers were sold in all?

(e) What fraction of the bunches of flowers sold were daisies? (f) What fraction of the flowers sold were either lilies or roses?

R EG

(g) What fraction of the flowers sold were not roses?

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2. Mrs. Jenkins spent $1,000 on gifts for her children. The pie chart below shows how much she spent on each item. Bat $75 Guitar

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Dollhouse $175

Surfboard

(a) How much more was the dollhouse than the bat?

(b) What percent of the money spent was the guitar?

(c) What fraction of the money spent was the surfboard? (d) What was the total cost of the guitar and the surfboard? (e) What fraction of the money was spent on the dollhouse?

(f) What is the ratio of the cost of the surfboard to the cost of

R EG

the guitar?

1 53

3. Riley kept a tally of the number of different colored butterflies in her garden. She spotted 50 butterflies in all. She created the pie chart below to represent her data.

Blue 8

AT IO N

Yellow 6 Green 19

Orange

(a) How many orange butterflies did Riley spot?

(b) What fraction of the butterflies Riley spotted were green?

(c) What fraction of the butterflies Riley spotted were yellow? (d) What percentage of the butterflies Riley spotted were blue? (e) What percentage of the butterflies spotted were not green?

(f) What was the ratio of blue butterflies spotted to yellow butterflies

R EG

spotted?

154

4. The pie chart shows the amount of different dishes sold at a Thai restaurant in one day. There were 180 dishes sold in all.

AT IO N

Curry 1 6 Soup 2 9

Rice

Salad

(a) What fraction of the dishes sold were salads?

(b) What fraction of the dishes sold were soups or curries?

(c) What fraction of the dishes sold were curries or rice? (d) How many rice dishes were sold? (e) How many curries and soups were sold?

(f) What was the ratio of curries to salads sold?

R EG

(g) What was the ratio of rice dishes to soups sold?

155

Rolls (98)

Pies

Donuts (38)

Cakes (32)

AT IO N

5. The pie chart shows the number of different items sold in a bakery in 1 day. Express the number of each item as a fraction of the total number of items sold.

(b) Pies:

(d) Rolls:

(a) Cakes:

Fish (30kg)

6. The pie chart shows the mass of different meats sold by a butcher over the weekend. Express the mass of each type of meat sold as a percentage of the total mass of meat sold.

R EG

Chicken

Beef

Lamb

(a) Fish:

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(b) Beef: (d) Chicken:

7. The pie chart shows Mr. Woods' expenses for 1 month. Food Transport (15%)

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Utilities

Rent (40%)

(a) What percent of Mr. Woods' expenses are on food?

(b) What percent of Mr. Woods' expenses are on utilities?

(d) What is the ratio of Mr. Woods' expenses on rent and transport to his total expenses for 1 month?

R EG

(e) Mr. Woods spends $2,200 on rent. Calculate his other expenses.

Utilities:

Food:

Transport:

1 57

Solve It!

Thailand United States

Spain (15%)

China

AT IO N

1. On Saturday, 800 people visited the City Museum. The pie chart shows the different countries of the visitors. The number of visitors from Canada was the same as the number of visitors from Spain.

Australia

Canada

R EG

(a) Express the number of visitors from each country as a fraction.

(b) How many visitors were from China or Thailand?

158

2. The pie chart shows the colors of candies in a jar.

Yellow

Blue

Green 1 5

Orange

AT IO N

Purple 1 6

R EG

The total number of candies in the jar is 1,200. How many candies of each color are there?

159

Word Problems Let’s Practice

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1. Students in Grade 6 were asked to name their favorite type of movies. Their responses were recorded in a pie chart.

15%

50%

Animation Horror Comedy Drama

30%

R EG

(a) What fraction of the students liked horror movies? (b) A total of 120 students gave their responses. How many more students liked animation movies than dramas? (c) How many fewer students like horror movies than comedies?

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2. The pie chart shows how pupils of Southport School travel to school.

Walk

AT IO N

Cycle (10%)

Public Transport (35%)

School bus (50%)

R EG

(a) What percentage of the pupils walk to school? (b) What fraction of the pupils travel to school by public transport? (c) Given that there are 60 pupils who walk to school, how many pupils attend Southport School?

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3. The pie chart shows how Ethan spends his weekly pocket money of $20. Stationery (5%)

AT IO N

Savings (15%)

Transport (20%)

(a) How much money does Ethan spend on food and drink? (b) How much money does he spend on stationery? (c) What fraction of the money does he save? (d) How much more money does Ethan spend on food and drink than transport?

R EG

Food and Drink (60%)

1 62

AT IO N

4. The pie chart shows the number of different colored marbles that Riley has. She has 50 blue marbles.

Blue

Orange

Red

Gray 1 5

R EG

(a) How many marbles does Riley have altogether? (b) What is the ratio of the number of orange marbles to the total number of marbles she has? (c) What fraction of the marbles are either gray or red?

163

AT IO N

5. The pie chart shows the different menu items that customers ordered at a restaurant one evening. Each customer ordered one menu item. There were 6 customers who ordered pasta.

Fish and Chips (15%)

Grilled Chicken (35%)

Pasta (10%)

Seafood Platter (15%)

Salad (25%)

R EG

(a) How many customers ordered the seafood platter? (b) What is the ratio of the number of customers who ordered the salad to the total number of customers that evening? (c) What fraction of the menu is fish and chips?

164

AT IO N

6. The pie chart shows the types of fruits consumed by the students at High Ridge school in a day. The number of apples consumed was 4 times the number of oranges consumed.

Oranges

Pears

Apples (60%)

Strawberries (10%)

R EG

(a) What fraction of the fruits consumed were oranges? (b) What percentage of the fruits consumed were pears? (c) 48 apples were consumed. How many strawberries were consumed?

165

At Home

AT IO N

1. The pie chart shows the types of animals 40 pupils kept as pets.

Rabbits (10%)

Birds (30%)

Fish (40%)

(a) (b) (c) (d)

What fraction of the pupils kept fish? What fraction of the pupils kept birds? What percentage of the pupils keep cats? How many pupils kept rabbits?

R EG

Cats

166

Food (45%)

AT IO N

2. The pie chart shows how Sophie spent her money in a week. The ratio of the amount of money she spent on transport to the amount she spent on clothes was 1 : 2. She spent $60 more on food than on shoes.

Shoes (25%)

Transport (10%)

Clothes

R EG

(a) How much did Sophie spend on transport? (b) How much did she spend on shoes? (c) What fraction of the money was spent on clothes?

1 67

Strawberry

Banana

AT IO N

3. The pie chart shows the types of flavored ice cream sold at a shop last month. The shop sold 1,200 ice creams in all. Half of the ice creams sold were chocolate. The shop sold 3% more strawberry flavored ice creams than banana flavored ice creams.

Chocolate

Vanilla (17%)

R EG

(a) How many vanilla flavored ice creams did the shop sell? (b) What percentage of the ice cream sold last week were strawberry flavored? (c) What fraction of the flavored ice cream is banana?

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4. A survey was conducted in a food mall to find out the favorite food of a group of tourists in Singapore. The pie chart below shows the results of the survey.

AT IO N

Satay

Chilli Crab (40%)

Claypot Rice

Chicken Rice (30%)

R EG

(a) What percentage of the tourists chose satay as their favorite local food? (b) The number of tourists who chose chicken rice was 30 more than the claypot rice. What was the total number of tourists who took part in this survey?

169

Looking Back

Hiking (20%)

AT IO N

1. At Florida Gardens College, there are 720 students. Each student is required to choose one club to join. The pie chart shows the percentage of students in each club.

Photography (35%)

Yoga

Reading (30%)

(a) What percentage of the students are in the yoga club?

(b) How many students are in the hiking club? (c) How many students are in the photography club?

(d) How many more students are in the photography club than the

R EG

reading club?

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2. In 1 week, a fast food restaurant sold 450 items in total. The pie chart shows the number of each item sold.

Drinks

Nuggets 1 5

AT IO N

Burgers 4 15 Fries 1 6

(a) What fraction of the items sold were drinks?

(b) What percentage of the items sold were nuggets?

R EG

(d) How many more burgers than fries were sold in 1 week?

171

3. Wyatt has a jar containing 500 marbles. He sorted them by color into 5 containers. The pie chart shows the number of marbles in each jar. Blue (45) Yellow (55) Green (135)

Orange

(a) How many marbles are orange?

AT IO N

Purple (110)

(b) What percentage of the marbles are yellow or blue?

(c) What fraction of the marbles are purple? (d) What fraction of the marbles are not green? (e) What is the ratio of green marbles to purple marbles to yellow

R EG

marbles?

172

How many boys wear glasses? What fraction of the girls wear glasses? How many pupils are there in the class? What percentage of the boys do not wear glasses?

(a) (b) (c) (d)

R EG

Boys with glasses

Girls without glasses (10%)

Boys without glasses

Girls with glasses (30%)

AT IO N

4. The pie chart shows the number of pupils with and without glasses in a class. There are 24 boys in the class.

173

Problem Solving

AT IO N

Act It Out

Example 14 cards, labeled from 1 to 14, are grouped in pairs. The sums of the numbers in each pair are 4, 6, 13, 14, 20, 21 and 27. What are the 7 number pairs?

10 11 12 13 14

Let's draw a table to keep track of the different sums. Sum 6 13

(1, 3)

R EG

Possible Pairs

(2, 4)

(5, 8) (6, 7) (5, 9) (6, 8) (13, 1)

(8, 12) (9, 11) (14, 6)

(9, 12) (10, 11) (13, 8)

(13, 14)

17 4

There is only 1 way to make a sum of 4, 6 and 27. We can remove them from the table. Possible Pairs

(5, 8) (6, 7)

(5, 9) (6, 8)

(8, 12) (9, 11)

(9, 12) (10, 11)

AT IO N

Sum

Every card must be used once only. The 10 card is only used with 11 to make a sum of 21. So, we can remove the 11 card from other possible pairs. Possible Pairs (5, 8) (6, 7)

(5, 9) (6, 8)

(8, 12)

Sum

(10, 11)

Similarly, the 9 card is only used with 5 to make the sum of 14. So, we can remove the 5 card from other possible pairs.

R EG

We have found our 7 number pairs.

1 3 5 9

2 4

6 7 13 14

8 12

10 11 175

R EG

AT IO N

1. Boats A, B, C, D and E went trawling for fish. Boat A and Boat D caught the least number of fish. Boat B caught more fish than Boat D but fewer fish than Boat C. Boat E caught fewer fish than Boat B. Which boat caught the most number of fish?

176

R EG

AT IO N

2. Ethan has 24 poles that are each 1 meter in length. How many different rectangles can he make using all of the poles?

177

R EG

AT IO N

3. In a classroom, there is a row of 5 chairs. The chair in the middle is empty. The boys and girls want to swap sides. To do this, the boys can only move to the next chair to their left. Similarly, the girls can only move to the next chair to their right. What is the least number of moves needed for both the boys and girls to swap sides?

178

R EG

AT IO N

4. A BBQ shop has 7 full gas bottles, 7 half-full has bottles and 7 empty gas bottles. Each gas bottle is equal in size. 3 people are looking to buy some gas bottles for their BBQ. How can the shop distribute the gas bottles equally between the 3 people such that each gets the same amount of gas and the same number of bottles?

179

Draw a Model Example

AT IO N

2 Ethan had some books to give to his friends. He gave 13 less than of the 5 3 books to Wyatt. He then gave 8 more than of the remaining books to 8 Halle and had 32 books left over. How many books did Ethan have at first? Work backwards from when Ethan had 32 books. Let's draw a bar model. books given to Halle

book remaining

3 of his books to Halle and was left with 32 books. 8

Ethan gave 8 more than

40 ÷ 5 = 8

5 parts of the bar model is equal to 32 + 8 = 40 books. We can find the value of 1 part by dividing by 5.

To find the total number of books Ethan had before he gave some to Halle, we need to find the value of 8 parts. So, we multiply 1 part by 8. 8 x 8 = 64

R EG

Ethan had 64 books before he gave some to Halle.

180

Let's draw another bar model to find how many books Ethan had at first. 13

Ethan gave 13 less than

AT IO N

books given to Wyatt

2 of the books to Wyatt. 5

8 green parts = 64 books. 64 - 13 = 51

3 blue parts = 8 green parts – 13.

From our bar model, we can see that 8 green parts make up 3 blue parts plus 13.

3 blue parts = 51 books 1 blue part = 51 ÷ 3 = 17

5 blue parts = 17 x 5 = 85

R EG

So, Ethan had 85 books at first.

18 1

R EG

AT IO N

1. A goldsmith wants to sell a bracelet and make a profit of 50%. If the bracelet was sold at a discount of 10%, he would make a profit of $25. If he sells it at a discount of 40%, he would make a loss of $35. How much must the goldsmith sell the bracelet to gain a 50% profit?

182

R EG

AT IO N

2. Dominic, Jordan and Riley had $216 between them. Dominic gave Jordan some money, and Jordan's amount tripled. Jordan then gave some money to Riley and Riley's sum of money doubled. The 3 friends now have an equal amount of money. How much more money did Dominic have than Riley at first?

18 3

R EG

AT IO N

3. There are 120 students in Grade 6 at Burleigh Junior School. 70 students can speak German and 80 can speak French. 30 of them cannot speak German or French. How many students can speak both German and French?

184

R EG

AT IO N

4. Mr. Enkel took 3 hours to drive from Singapore to Malacca at an average speed of 80 km/h. From Malacca, he took another 2 hours to travel to Kuala Lumpur. His average speed for the whole journey was 75 km/h. Find his average speed for the journey from Malacca to Kuala Lumpur.

18 5

Guess-and-Check

AT IO N

Example Four classes A, B, C and D participated in a dance contest. Each class received a whole number score from 1 to 10 by the judges. The average score of classes B, C and D was 6. The average score for classes A, B and D was 5. Class D scored more than Class B. Class C received the highest score of 9. Find the score of each of the classes.

The average score of classes B, C and D is 6. Multiply the average score by the number of classes to find their combined score.

R EG

6 x 3 = 18.

Class C has a score of 9 so we can subtract. 18 - 9 = 9.

So, the combined score of classes B and D is 9.

186

The average score of classes A, B and D is 5. Multiply the average score by the number of classes to find their combined score.

AT IO N

5 x 3 = 15. The sum of classes B and D is 9. So, we can subtract to find Class A's score. 15 - 9 = 6. Class A has a score of 6.

To find the remaining scores of classes B and D, let's use guess-and-check. The score of Class B and class D can't be greater than 10. The sum of the scores of Class B, C and D is 18. The sum of the scores of Class A, B and D is 15.

Total of A, B and D

1 + 9 + 9 = 19

6 + 1 + 9 = 16

2 + 9 + 9 = 20

6 + 2 +9 = 17

2 + 9 + 8 = 19

6 + 2 + 8 = 16

2 + 9 + 7 = 18

6 + 2 + 7 = 15

Total of B, C and D

Guess of Class D's score

Guess of Class B's Score

R EG

The scores for each class are: Class A = 6 Class B = 2 Class C = 9 Class D = 7

187

R EG

AT IO N

1. There are 629 students participating in the American Mathematics Competition at Palm Beach Junior School. Each class has the same number of students. There must be at least 20 students in a class. How many classes are there?

188

R EG

AT IO N

2. Halle's math test had a total of 50 questions. For every question answered correctly, 2 marks were awarded. 5 marks were deducted for each incorrect answer. If a question is not answered, no marks are awarded or deducted. The number of questions not attempted by Halle was equal to the number of incorrectly answered questions. How many questions did she get correct if she scored a total of 82 marks?

18 9

3. Riley puts 2 coins in her money box each day. The value of each coin is either 10 cents or 20 cents. Her mother also puts a 50-cent coin in the box every 7 days. The total value of the coins after 84 days was $34.20.

R EG

AT IO N

(a) How many coins were there altogether? (b) How many of the coins were 20 cents coins?

19 0

R EG

AT IO N

4. A bicycle shop has a total of 37 bicycles and tricycles. Each bicycle sells for $189 while each tricycle sells for $99. There are a total of 90 wheels on the bicycles and tricycles. How much will the shopkeeper earn if all the bicycles and tricycles are sold?

19 1

Make a List

AT IO N

Example The diagram shows 2 types of squares. The 4 smaller squares are identical. The side length in centimeters of all the squares is a whole number. The total area of 1 large square and 1 small square is 100 cm2. What is the perimeter of the figure?

We need to find the 2 areas that add to 100. Area of large square

Total area

check

4 x 4 = 16

5 x 5 = 25

25 + 16 = 41

5 x 5 = 25

6 x 6 = 36

25 + 36 = 61

R EG

Area of small square

6 x 6 = 36

7 x 7 = 49

36 + 49 = 85

6 x 6 = 36

8 x 8 = 64

36 + 64 = 100

yes

So, the smaller squares have a side length of 6 cm and the larger square has a side length of 8 cm. To find the perimeter, we add.

P = 6 + 6 + 6 + 1 + 1 + 6 + 6 + 6 + 1 + 1 + 6 + 6 + 6 + 1 + 1 + 6 + 6 + 6 + 1 + 1 = 80 The figure has a perimeter of 80 cm.

192

R EG

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1. 3 pipes, A, B and C are connected to an empty container. Pipe A can fill 1 1 of the container in 1 hour. Pipe B can fill of the container in 1 hour. 10 8 1 However Pipe C drains of the container in 1 hour. At first, pipe A and 16 pipe B were turned on. After 1 hour, pipe C was turned on for 2 hours before being turned off. How long did it take to fill up the container?

19 3

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2. Dominic has the same number of candies and chocolates. If he gave each of the 18 students in Class A an equal number of candies, he would have 7 candies left. If he gave each of the 12 students in Class B an equal number of chocolates, he would have 1 chocolate left. What is the total number of candies and chocolates Dominic has?

19 4

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3. Mr. Olsen put aside $1,533 to spend. Every day he spent twice the amount spent the day before. If he spent $3 on the first day, which was a Monday, on what day would he have spent all of his money?

19 5

Look for Patterns

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Example There are 139 marbles of 3 colors arranged in the following pattern. How many more green than blue marbles are there in all?

There are 4 marbles in each repeating part of the pattern. To find the total number of sets, we divide.

139 ÷ 4 = 34 R 3

The remaining 3 marbles are red, blue and green.

Let's find the number of blue and green marbles in 34 sets. In one set, there is one blue and two green marbles. Total green = 34 x 2 = 68

Total blue = 34 x 1 = 34

So, there are 68 green marbles and 34 blue marbles in a set of 34.

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There are 1 of each color remaining. So we add 1 to both green and blue totals. There are 69 green marbles and 35 blue marbles. 69 – 35 = 34

So, there are 34 more green marbles than blue marbles.

19 6

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1. Mrs. Wong installed new tiles in her rectangular living room measuring 6 m by 3 m. She used triangular tiles as shown in the diagram below. The contractor charged $3 to cement each tile or part of a tile. The cost of cutting a tile was $5. The labor cost for tiling the living room was $300. How much did Mrs. Wong pay in total to install the new tiles?

50 cm

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60 cm

19 7

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2. A dictionary has 1,348 pages. How many pages of the book have the digit(s) 1 in the page number?

198

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3. A ball is thrown off a building from a height of 30 m. Each time the 3 ball bounces, it reaches a new maximum height of its previous 5 maximum height. What is the total vertical distance the ball would have traveled when it touches the ground for the 5th time?

19 9

4. Find the 62nd digit in the repeating digit pattern below.

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3 7 8 0 3 7 8 0 3 7 8 0 3 ...

2 00

Work Backwards Example

City A

1 3

1 4

1 h 12 min

City B

total distance traveled total time taken

Recall that average speed =

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1 Ethan took a train from City A to City B. He covered of the distance in the 3 1 first hour. In the next hour, he covered of the total distance of the journey. 4 He then took 1 h 12 min to travel the remaining 80 km. Calculate his average speed in km/h for the whole journey.

covered in the final stage. 1 5 1 – = 3 4 12

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5 = 80 km 12 1 = 80 ÷ 5 = 16 km 12 12 = 16 x 12 = 192 km 12

Last stage = 1 –

Let's work backwards and find the fraction of the journey that was

The total distance between City A and City B is 192 km.

The total time taken = 1 h + 1 h + 1 h 12 min = 3 h 12 min = 3.2 h average speed =

192 = 60 3.2

The average speed for the whole journey was 60 km/h. 201

1 1. Wyatt and Jordan had some pens. Wyatt gave of his pens to Jordan. 2 1 Jordan then gave of his pens back to Wyatt. In the end, Wyatt had 3 3x pens and Jordan had 2x pens.

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(a) How many pens did Wyatt have at first? Give your answer in terms of x. (b) What was the total number of pens they had when x = 10?

202

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5 2. Some water is held in 2 containers – A and B. of the water in A is 6 7 poured into B. of the new amount of water in B is then poured back 9 into A. In the end, A contained 160 liters of water and B contained 40 liters of water. What was the volume of water in A initially?

203

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3. Sophie, Halle and Riley had 180 hairpins. When Sophie gave some of her hairpins to Halle, the number of hairpins Halle had was doubled. Halle then gave some of her hairpins to Riley and the number of hairpins Riley had was doubled. The 3 girls had an equal number of hairpins at the end. How many hairpins did each girl have at first?

20 4

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4. Mrs. Farrugia gave 2 more than half of her pies to her daughter, Kathy. Mrs. Farrugia then gave 4 more than half of the remaining pies to her grandson, Luke. 3 more than half of the leftover pies were given to her granddaughter, Ella. In the end, Mrs. Farrugia had 2 pies for herself. How many pies did she have at first?

205

Simplify the Problem

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Example Mr. Tan has some potted plants. He arranged the plants along the perimeter of 2 squares as shown in Diagram A. The distance between each potted plant was 2 m. After arranging the plants, he placed a string of lights over the plants as shown in Diagram B. The total length of the string of lights was 112 m. How many potted plants did Mr. Tan have?

Diagram B

Diagram A

For each potted plant, 4 meters of lights are needed.

The potted plants at the junctions of the square are connected to by the lights to 3 potted plants as shown. All other potted plants are connected by the lights to 2 potted plants.

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The 2 plants at the junction each use an additional 2 m of string light. Let's subtract 4 m from the total length of the lights. 112 m – 4 m = 108 m Divide by the length of string light required for 1 potted plant to find the total number of potted plants. 108 ÷ 4 = 27 Mr. Tan has 27 potted plants.

206

3 cm

1m 4 cm

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1. What is the maximum number of triangles that can be cut from a rectangular sheet of wood as shown below?

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1.3 m

207

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2. The figure in Diagram A is made from a single length of wire bent to form a square of area 81 cm2. The wire is then straightened and bent again to form the shape in Diagram B. Each of the connectors connecting the three identical circles measure 4 cm. 22 Find the radius of each of the circles. Take π as . Leave your answer 7 as an improper fraction in cm.

Diagram B

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Diagram A

208

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3. The price for a bento box meal at a restaurant was $21. During the Black Friday sale, the price for a bento box meal was reduced. The number of bento box meals sold was tripled and the amount of money collected was doubled. How much was the bento box meal after the price reduction?

209

Solve Part of the Problem

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Example The figure below shows 3 overlapping circles. The overlaps of 2 circles form the areas B and C. The ratio of area A to area B is 1 : 2. Area C is 5% less than area B. If the area of C is 20 cm2, what is the area of the green circle?

C A

area of C = 20 cm2 area of B = 105% of area C = 105% x 20 cm2 = 21 cm2

area A : area B = 1 : 2 2 units = 21 cm2 1 unit = 21 ÷ 2 = 10.5 cm2 area A = 10.5 cm2

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area of green circle = area A + area B = 10.5 + 21 = 31.5 cm 2

210

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1. The diagram shows 4 squares of side length 25 cm. The curved sections were formed from a semicircle and a quarter-circle. Find the area of the shaded portion. Take π as 3.14.

21 1

2. The figure QRSTU shows a square B enclosed in a rectangle. Based on the diagram, the area A is 12 cm2 more than the sum of areas B and C. The area A is equal to that of C. The length RS is twice that of ST. Find the length of square B. R D

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2 12

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3. It takes 5 men, each working 4 hours per day to complete building a hut in 3 days. On the 1st day, all the men were present and worked as planned. However, on the 2nd day, only 3 men turned up to work and each of them only worked for 3 hours. On the 3rd day, only 4 men turned up to work. How many hours does each worker have to work on the 3rd day to complete building the hut?

213

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4. Tap A takes 3 minutes to fill a tank. Tap B takes 9 minutes to fill the same tank. If both taps are turned on at the same time, how long will it take to fill up the tank?

214

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5. Mr. Craven wanted to buy a watch. The watch dealer told Mr. Craven that all of the watches available cost less than $1,000. The cheapest watch available was $100. The prices of all the watches sold had an even number digit in the ones place, and the digit in the tens place was less than 8. Given that all of the watches were priced in whole dollars, how many watches did the watch dealer have if every watch was priced differently?

215

Before-After Concept

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Example Riley has 20% more stickers than Dominic. If Riley gives 40 stickers to Dominic, he will have 20% more stickers than Riley. How many stickers do they have in all? Before

Riley (120%)

Dominic (100% – 1 unit)

After

Riley (100% – 1 unit)

20% of 1 unit = 40 1 unit = 5 x 40 = 200

Dominic (120%)

Total number of stickers = 220% x 200 = 440

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They had 440 stickers in all.

2 16

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1 1. Jordan had 2 boxes of cards. He transferred of the cards from 3 1 Box 1 to Box 2. Then he transferred of the cards from Box 2 back 3 1 to Box 1. Finally, he transferred of the cards from Box 1 back to Box 3 2 again. There are now 68 cards in Box 1 and 90 cards in Box 2. How many cards were in each box at the beginning?

217

1 2. Sophie gave of her money to her sister, Chelsea. Chelsea then gave 2 1 1 of her money back to Sophie. Sophie then gave of her money to 4 3 Chelsea again. Finally Sophie had $975 and Chelsea had $1,325.

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How much money did Sophie have at first?

2 18

5 her brother’s age. Elaine is 28 years old now. 7 In how many years time will her brother be 27 years older than 1 Elaine’s age? 2

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3. 3 years ago, Elaine was

219

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4. Wyatt had some green beans and red beans in the ratio 8 : 5. He had 126 more green beans than red beans. After he sold an equal number of each type of bean, the ratio of the number of green beans to red beans he had became 3 : 1. How many beans did he sell in total?

22 0

Make Suppositions

First, let's find the initial cost price. 286 x $0.85 = $243.10

12 fruits were rotten and could not be sold. 286 – 12 = 274

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Example A fruit seller bought a total of 286 apples and oranges from a farmer. He bought each fruit at $0.85. Upon bringing the fruits back to his shop, he threw away 12 rotten fruits. He then sold the remaining oranges for $1.10 and each and the remaining apples for $1.30 each. If he made a profit of $87.10, how many apples did he sell?

Find the total amount of money he received from sales. Income = expenses + profits = $243.10 + $87.10 = $330.20 He received $330.20 from sales.

Assuming all the fruits sold were apples: $1.30 x 274 = $356.20 $356.20 - $330.20 = $26

This is $26 more than the money received from sales.

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$1.30 - $1.10 = $0.20 The price difference in apples and oranges is $0.20 $26 ÷ $0.20 = 130 He sold 130 oranges 274 – 130 = 144

He sold 144 apples. 221

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1. Ethan has a total of 35 $5 and $2 notes. If he has a total of $115, how many $5 notes does Ethan have?

222

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2. In a mathematics competition, each pupil has to answer 10 questions. 5 points were given for each correct answer and 2 points were deducted from each wrong answer. Halle answered all the questions and scored a total of 36 points. How many questions did she answered correctly?

223

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3. Andy paid $4.40 for 2 pens and 1 marker pen. The marker pen cost 20 cents more than a pen. How much is the cost of 1 pen?

22 4

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4. Blake bought 50 pens and rulers altogether. Each pen costs $1.20 and each ruler costs $2.40. The difference between the total cost of the pens and the total cost of the rulers is $12. How many pens and rulers did Blake buy?

225

End-of-year Exam

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Section A – Multiple Choice Questions Questions 1 – 20 carry 1 mark each. There are 4 options given in each question – (a), (b), (c) and (d). Shade the letter that best matches the answer. 1. The center of the circle is point O. Labeled lines are straight lines. Which lines represent the radius and diameter respectively? B

D O

OB and CE AO and OB OB and AD AO and CE

(a) (b) (c) (d)

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2. Evaluate the algebraic expression when b = 4. 36 – 9b (a) 0 (b) 27 (c) 32 (d) 23

Sub-total

22 6

(a) 11 42 (b) 10:42 p.m. (c) 11:42 a.m. (d) 11:42 p.m.

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3. Express the time shown on the clock in 12-hour time.

U ED

(a) 1.2 km (b) 120 km (c) 20 m (d) 120 m

4. A car travels at a speed of 20 m/s for 1 minute. How far does it travel?

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5. What percentage of the square grid is colored?

(a) 20% 1 (b) 4 (c) 25% (d) 75%

Sub-total

227

6. 120 students were asked about their favorite subjects. Their responses were recorded in the pie chart below. How many students chose science as their favorite subject?

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English Science Art 1 5

Math 2 5

(a) 40 (b) 20 (c) 120 (d) 30

7. Find the area of the triangle.

1 cm

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1 cm

(a) 32 m2 (b) 32 cm2 (c) 16 cm2 (d) 16 in2

Sub-total

228

8. Find

2 1 x . Give your answer in its simplest form. 4 7

1 14 3 (b) 11 2 (c) 28 3 (d) 28

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(a)

9. Halle and Sophie rode their bikes to the lake. They arrived at 13 20 and hiked around the lake until 16 10. How long was their hike?

(a) 1 h 50 min (b) 2 h 50 min (c) 3 h 10 min (d) 3 h 20 min

10. What is 20% of 360?

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(a) 80 (b) 720 (c) 18 (d) 72

Sub-total

229

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11. A man takes 3 h to run 39 km. Find his average speed. (a) 13 km/h (b) 42 km/h (c) 13 m/s A (d) 42 m/s

Apples 8

Melons 8

Pears 4

Mangoes 12

12. The pie chart below shows the different fruits stocked by a fruit seller. What fraction of the fruits are mangoes? Give your answer in its simplest form.

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1 (a) 4 3 (b) 8 12 (c) 36 1 (d) 8

Sub-total

230

13. Arrange the numbers from the smallest to the greatest. –3, |–8|, 0, 7, –4

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(a) |–8|, –3, 0, 7, –4 (b) |–8|,–4, –3, 0, 7 (c) –4, –3, 0, 7, |–8| (d) –4, –3, |–8|, 0, 7

22 . 7

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(a) 21 cm (b) 462 cm (c) 66 cm (d) 33 cm

21 cm

14. Find the circumference of the circle. Take π =

15. A school play goes for 2 hours 35 minutes. The play finished at 21 05. What time did the school play start? (a) 18 30 (b) 17 30 (c) 18 35 (d) 18 05

Sub-total

231

16. Simplify the algebraic expression. 10m – n – m – 2m + 5n

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(a) 7m + 4n (b) 10m + 4n (c) 7m – 4n (d) 8m + 4n

14 : 84 14 cm : 84 cm 7 : 42 1:6

(a) (b) (c) (d)

17. Express 14 cm to 84 cm as a ratio in its simplest form.

18. Express 24 ÷ 16 as a mixed number in its simplest form.

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1 (a) 1 8 1 (b) 1 2 1 (c) 2 2 1 (d) 16 2

Sub-total

232

19. Find the area of the circle. Take π = 3.14 and round off your answer to 1 decimal place.

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3 cm O

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(a) a = 5 (b) a = 6 (c) a = 7 (d) a = 8

6a – 20 = 22

20. Solve the equation.

(a) 28.3 cm2 (b) 9 cm2 (c) 113 cm2 (d) 18.8 cm

End of Section A

Sub-total

233

Section B – Short Answer Questions 21 – 40 carry 2 marks each. Show your working and write your answer in the space provided.

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21. A truck takes 13 h to drive 1,118 km from Miami to Nashville. Find the average speed of the truck.

Answer:

22. The temperature changed from –9ºC to 2ºC. How much did the temperature rise?

Answer:

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23. A round table has a radius 49 cm. Find the area. Take π =

22 . 7

Answer: Sub-total

23 4

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24. Sophie takes 8 minutes 42 seconds to walk around her school. How long will it take Sophie to walk around her school 8 times?

Answer:

Rabbits

Lizards

Answer:

Cats (10%)

Fish (25%)

Birds (35%)

25. The pie chart below shows the pets kept by class 6C. Half of the class kept birds or rabbits. What fraction of the class kept lizards?

26. Solve the equation.

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100 – 12x = 52

Answer: Sub-total

235

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27. Express 78% as a fraction in its simplest form.

Answer:

28. Find the area of the triangle.

60 cm

40 cm

Answer:

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29. A rabbit hops at a speed of 7 m/s. How long will it take to cross a field 462 m wide? Give your answer in minutes and seconds.

Answer: Sub-total

236

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30. Mrs. Olsen takes 12 minutes to wrap a gift. How long will it take her to wrap 23 such gifts? Give your answer in hours and minutes.

Answer:

Walk

Car (30%)

31. The pie chart below shows how students travel to school. 45 students ride to school. How many students walk to school?

Bus (10%)

Ride

Answer:

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32. The ratio of Ethan's savings to Blake's savings to Jordan's savings is 1 : 4 : 2. They have $56 in total. How much money does Jordan have?

Answer: Sub-total

237

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33. A round swimming pool has a radius of 17 m. Find its area. Take π = 3.14 and round off your answer to 1 decimal place.

Answer:

34. Solve the equation.

Answer:

42 – 4r = 2r

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35. Jordan took 18 minutes to walk from his house to the library. He studied in the library for 2 hours 27 minutes and left at 13 50. What time did Jordan leave his house?

Answer: Sub-total

23 8

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36. What is 65% of 80?

Answer:

37. Divide. 2 9

6 ÷

38. A triangle has a height that is

Answer:

3 that of its base. Find the area of the 8

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triangle given its base is 24 cm.

Answer: Sub-total

239

39. Simplify the algebraic expression.

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10a + 3a – d – 2d – 3d + 5a

Answer:

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40. The ratio of the mass of a watermelon to the mass of a mango is 12 : 5. The mass of the mango is 700 g. Find the mass of the watermelon.

Answer: End of Section B

Sub-total

240

Section C – Word Problems Questions 41 – 50 carry 4 marks each. Show your working and write your answer in the space provided.

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41. The blue figure below is drawn in a rectangle. Find the area of the blue figure. 9 cm

3 cm

5 cm

1 cm

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5 cm

Answer: Sub-total

2 41

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42. A pizza chef takes 12 minutes to prepare the pizza base, 5 minutes to add the toppings and 18 minutes to bake in the oven. He needs to make 8 pizzas by 18 15. What is the latest time that the pizza chef should start making the pizzas?

Answer: Sub-total

2 42

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43. Wyatt took 2 hours to cover 4 of a journey to his grandmother's house. He covered the remaining 31 km in 1 hour 6 minutes. Find his average speed for the whole journey.

Answer: Sub-total

243

44. The pie chart shows the colors of flowers spotted in a nursery. There were 720 flowers spotted in total. How many flowers of each color were spotted?

Blue

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Orange

1 5

Pink Green Yellow

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1 5

Answer :

Sub-total

2 44

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45. The figure below is composed of 2 semi-circles and a square. Find the area of the figure. Take π = 3.14 and round off your answer to 1 decimal place.

2 cm

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2 cm

Answer: Sub-total

245

46. Mr. Hogan earns $m per month. Mr. Booker earns 3 times as much as Mr. Hogan.

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(a) Express Mr. Hogan's and Mr. Booker's total income over a period of 12 months. (b) How much does each person earn in 1 year when m = $2,680?

(b):

Answer (a):

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47. During a sale, a notebook computer is discounted by 15% and a computer bag is discounted by 10%. The sale price of the notebook computer is $918 and the sale price of the computer bag is $36. Find the regular price of both items.

Answer:

Sub-total

2 46

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48. Find the shaded area of the figure.

12 cm

2 cm

4 cm

R EG

16 cm

Answer: Sub-total

247

49. A triangle has a base of b cm and a height of 2 b.

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(a) Express the area of the triangle in terms of b. (b) Find the area of the triangle when b = 32.

Answer (a): (b): Sub-total

2 48

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50. Halle and Sophie took 2 hours 20 minutes to hike from their campsite at the top of a hill down to the river at an average speed of 60 m/min. On their way back up, they hiked at an average speed of 40 m/min. How long did it take Halle and Sophie to hike from the cave back to the campsite? Give your answer in hours and minutes.

Answer: End of Exam

Sub-total

249

End-of-year Exam Results Correct

Score

/ 20

/ 40

/ 10

/ 40

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Total

25 0

Comments

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Section

/ 50

/ 100

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AT IO N C U ED AL

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