student journal it’s simply a smarter approach dart …€¦ · time 12-hour time 50, 51, 54, 55...

STUDENT JOURNAL

Product Code: SSJ 226 16

STUD

ENT JO

UR

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L6

ORIGO Stepping Stones is an award-winning mathematics program developed by curriculum specialists for Australian primary schools.

This revolutionary online program integrates print and digital technology to deliver comprehensive coverage of the F-6 Australian Curriculum – and even more!

THIS BOOK BELONGS TO

ORIGO Stepping Stones was developed by mathematics specialists for Australian primary schools to:

• make maths more focused and coherent

• foster students’ thinking and reasoning skills

• deliver multiple ways to differentiate classroom instruction

• provide a valuable source of professional learning for the teacher

• offer methods to assess deep understanding and skills

• provide online and print resources that engage all students

IT’S SIMPLY A SMARTER APPROACH

Digitthe wombat

Millithe possum

Splitthe kangaroo

Dartthe echidna

Platothe platypus

Tallythe turtle Cubit

the koala

ORIGO Stepping Stones Student Journal Year 6

Copyright 2014 ORIGO Education

Senior Authors: James Burnett, Calvin Irons

Contributing Authors: Debi DePaul, Peter Stowasser, Allan Turton

Program Editors: James Burnett, Beth Lewis, Donna Richards, Stacey Lawson

For more information, visit www.origoeducation.com.

All rights reserved. Unless specifically stated, no part of this publication may be reproduced, copied into,

stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying,

recording, or otherwise, without the prior written permission of ORIGO Education.

ISBN: 978 1 922246 22 6

10 9 8 7 6 5 4 3 2 1 IN

DEX Measurement (cont’d)

Time12-hour time 50, 51, 54, 5524-hour time 52, 53Elapsed time 54, 55, 196, 197Language 196Timelines 196-199Timetables 50-55Word problems 232, 233

Temperature 106, 107Volume

Dimensions for given volume 222, 223Formal unit Cubic centimetre 220, 221Related to capacity 246, 247Rule for calculating 220, 221

MoneyDiscounts 264-267Fractions of 260, 261Language 46, 47Payment plans 110, 111Percentage of 262, 263Transactions 42, 43, 46, 47, 94, 95Unit costs 182-185

MultiplicationArea model 56, 58, 60, 62, 63Decimal fractions 56-65, 140-143, 188-191Equivalent fractions 154-157Factors 66, 67, 158, 159, 191Formal algorithm 28-31, 142, 143Four-digit numbers 30, 31Mental strategies 26-28, 30, 56-65, 136-143Multiples 67Patterns 57, 61, 188-191Three-digit numbers 28, 29Two-digit numbers 26-31, 136-139Word problems 26, 27, 29

NumberComposite 66, 67Integers 104, 105Language 66-73Oblong 70, 71Prime 66, 67Square 68, 69Triangular 72, 73

Number lineRecording mental strategies 36, 37, 56, 57, 60, 61,

80, 81, 86-89, 94, 95, 204-206, 208, 209, 248, 250, 252, 253

Relative positionCommon fractions 8, 9Decimal fractions 8, 9, 14-17Whole numbers 104, 105

Decimal fractionsExpanded notation 10-12Symbolic 12, 13Word 12, 13

Order of operationsBrackets 282, 283Language 274, 276, 278, 282Parentheses 278-283Rule 274, 278With one operation 272, 273With multiple grouping symbols 282, 283With multiple operations 274-281Word problems 276, 277, 281

OrderingCommon fractions 162, 163Decimal fractions 49Integers 105

PatternsAddition 68, 69Division 188-191Geometric 128, 129Multiplication 57, 61, 69, 188-191Number 130-135Oblong numbers 70, 71, 132Square numbers 68, 69, 132Triangular numbers 72, 73, 133

PercentageArea model 224, 225As an amount per hundred 226, 227As hundredths of one whole 224, 225, 228-231Greater than 100% 234, 235Of a quantity 262-267Related to common fractions 228-231, 234, 235Related to decimal fractions 228-231, 234, 235Symbol 224, 225Word problems 232, 233

Place value Partitioning

Decimal fractions (standard form) 10-12

ProbabilityRecording (as a fraction) 290, 291, 293, 295Prediction 290-293

Relative positionDecimal fractions 8, 9Integers 104, 105

RoundingDecimal fractions 14-17

ShapeThree-dimensional objects

Drawing 222, 270, 271Nets 268-271

TransformationLanguage 236, 238Refl ections 236-239Rotations 238, 239Tessellations 240, 241Translations 238, 239

Two-dimensional shapesDiagonal lines 192, 193Drawing 194, 195Language 192, 194, 195Parts of a circle 194, 195Symmetry Line 236

SubtractionCommon fractions 248-255Decimal fractions 80-95Formal algorithm 84-93Five-digit numbers 34, 35Four-digit numbers 32-35Integers 106-109Mental strategies 32-35, 80, 81, 86-89, 94, 95Mixed numerals 252-255Three-digit numbers 32, 33Word problems 33

James Burnett

Beth Lewis

Donna Richards

Stacey Lawson

PROGRAM Editors

James Burnett

Calvin Irons

SENIOR AUTHORS

Debi DePaul

Peter Stowasser

Allan Turton

contributing authors

STUDENT JOURNAL

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MODULE 1

1.1 Reviewing the Relationship Between Decimal and Common Fractions

8

1.2 Regrouping Tenths, Hundredths and Thousandths

10

1.3 Reading and Writing Decimals 12

1.4 Rounding Thousandths 14

1.5 Rounding Decimal Fractions 16

1.6 Adding Tenths 18

1.7 Adding Hundredths 20

1.8 Adding Tenths and Hundredths 22

1.9 Solving Word Problems Involving Decimal Fractions

24

1.10 Revising the Partial-Products Strategy for Multiplication

26

1.11 Revising the Formal Algorithm for Multiplication

28

1.12 Extending the Formal Algorithm to Multiply Two-Digit Numbers by Four-Digit Numbers

30

MODULE 2

2.1 Using Mental Strategies to Add and Subtract Whole Numbers

32

2.2 Adding and Subtracting Four- and Five-Digit Whole Numbers

34

2.3 Adding Decimal Fractions 36

2.4 Adding Decimal Fractions (with Regrouping) 38

2.5 Using the Formal Algorithm to Add Two Decimal Fractions

40

2.6 Using a Compensation Strategy to Add Dollars and Cents

42

2.7 Using a Written Method to Add More Than Two Decimal Fractions

44

2.8 Estimating Total Cost 46

2.9 Solving Addition Problems Involving Decimal Fractions

48

2.10 Interpreting Information in Timetables 50

2.11 Using a Timetable Involving 24-Hour Time 52

2.12 Calculating Elapsed Time 54

MODULE 3

3.1 Multiplying Decimal Fractions (Tenths) 56

3.2 Using a Partial-Products Strategy to Multiply Decimal Fractions (Tenths)

58

3.3 Multiplying Decimal Fractions (Hundredths) 60

3.4 Using a Partial-Products Strategy to Multiply Decimal Fractions (Hundredths)

62

3.5 Multiplying Whole Numbers and Decimal Fractions (Hundredths)

64

3.6 Identifying Prime and Composite Numbers 66

3.7 Investigating Square Numbers 68

3.8 Investigating Oblong Numbers 70

3.9 Investigating Triangular Numbers 72

3.10 Creating and Interpreting Two-Way Tables 74

3.11 Interpreting Side-by-Side Column Graphs 76

3.12 Comparing Different Displays of the Same Data

78

MODULE 4

4.1 Subtracting Decimal Fractions (Tenths or Hundredths)

80

4.2 Subtracting Decimal Fractions (Tenths and Hundredths)

82

4.3 Using Written Methods to Subtract Decimal Fractions

84

4.4 Subtracting Decimal Fractions Involving Tenths (Decomposing Ones)

86

4.5 Subtracting Decimal Fractions Involving Hundredths (Decomposing Tenths)

88

4.6 Subtracting Decimal Fractions (Decomposing Multiple Places)

90

4.7 Consolidating Strategies to Subtract Decimal Fractions

92

4.8 Calculating Change 94

4.9 Exploring the Relationship Between Kilograms and Grams

96

4.10 Converting Between Grams and Kilograms 98

4.11 Working with Thousandths of a Kilogram 100

4.12 Solving Multi-Step Word Problems Involving Conversions of Mass

102

MODULE 5

5.1 Introducing Integers 104

5.2 Working with Integers 106

5.3 Solving Problems Involving Integers 108

5.4 Revising Partitioning to Divide Mentally 110

5.5 Extending Partitioning to Divide Mentally 112

5.6 Using the Formal Division Algorithm (Dividing Before Regrouping)

114

5.7 Using the Formal Division Algorithm (Regrouping Before Dividing)

116

5.8 Solving Division Problems Involving Remainders

118

5.9 Solving Division Problems 120

5.10 Exploring the Relationship Between Metres and Millimetres

122

5.11 Exploring the Relationship Between Metres, Centimetres and Millimetres

124

5.12 Solving Word Problems Involving Length 126

MODULE 6

6.1 Identifying and Describing Geometric Patterns

128

6.2 Identifying and Describing Number Patterns 130

6.3 Identifying Relationships Between Two Numerical Patterns

132

6.4 Creating Number Patterns Involving Decimal Fractions and Common Fractions

134

6.5 Using Mental Strategies to Multiply Two-Digit Numbers

136

6.6 Comparing Mental Strategies for Multiplication

138

6.7 Using a Double-and-Halve Strategy to Multiply Dollars and Cents

140

6.8 Using Multiplication Strategies with Decimal Fractions

142

6.9 Estimating and Measuring Angles 144

6.10 Examining Angles Around a Point 146

6.11 Exploring Angles in Right Angles and on Straight Lines

148

6.12 Investigating Vertically Opposite Angles 150

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MODULE 77.1 Identifying Equivalent Fractions 152

7.2 Using Multiplication to Find Equivalent Fractions

154

7.3 Reinforcing Mental Strategies for Finding Equivalent Fractions

156

7.4 Writing Common Fractions in Their Simplest Form

158

7.5 Comparing Common Fractions (Related Denominators)

160

7.6 Ordering Common Fractions (Related Denominators)

162

7.7 Identifying Misleading Data 164

7.8 Investigating Misleading Data 166

7.9 Exploring the Connection Between Perimeter and Area

168

7.10 Relating Perimeter and Area 170

7.11 Investigating Area of Right Triangles 172

7.12 Developing a Rule to Calculate Area of Triangles 174

MODULE 88.1 Using Partial Quotients with Decimal Fractions 176

8.2 Extending the Partial-Quotients Strategy with Decimal Fractions

178

8.3 Solving Division Problems Involving Dollars and Cents

180

8.4 Calculating Unit Costs (Whole Dollars) 182

8.5 Calculating Unit Costs (Dollars and Cents) 184

8.6 Working with Speed (Kilometres per Hour) 186

8.7 Multiplying and Dividing by 10, 100 and 1000 188

8.8 Comparing Multiplication and Division Involving Decimal Fractions

190

8.9 Analysing Diagonals of 2D Shapes 192

8.10 Identifying Parts of a Circle 194

8.11 Interpreting a Timeline and Working with Elapsed Time

196

8.12 Drawing and Interpreting a Timeline 198

MODULE 99.1 Converting Mixed Numerals to

Improper Fractions200

9.2 Converting Improper Fractions to Mixed Numerals

202

9.3 Revising Addition of Common Fractions (Same Denominators)

204

9.4 Adding Common Fractions (Related Denominators)

206

9.5 Adding Mixed Numerals (with Bridging) 208

9.6 Adding Mixed Numerals (Related Denominators)

210

9.7 Converting Between Centimetres and Metres 212

9.8 Converting Between Millimetres and Centimetres

214

9.9 Converting Between Millimetres and Metres 216

9.10 Converting Between Metres and Kilometres 218

9.11 Developing a Formula to Calculate Volume of Prisms

220

9.12 Finding the Dimensions of Prisms with a Given Volume

222

MODULE 1010.1 Introducing and Interpreting Percentage

as Hundredths of One224

10.2 Interpreting Percentage as an Amount Per Hundred

226

10.3 Relating Percentage to Common Fractions and Decimal Fractions (Area Model)

228

10.4 Relating Percentage to Common Fractions and Decimal Fractions (Linear Model)

230

10.5 Using Common Fractions, Decimal Fractions and Percentages in Everyday Contexts

232

10.6 Relating Fractions and Percentages Greater Than 100%

234

10.7 Working with Refl ections 236

10.8 Working with Multiple Transformations 238

10.9 Exploring Tessellations 240

10.10 Working with Units of Capacity 242

10.11 Adding Mixed Units of Capacity 244

10.12 Connecting Volume and Capacity 246

MODULE 1111.1 Subtracting Common Fractions

(Same Denominators)248

11.2 Subtracting Common Fractions (Unrelated Denominators)

250

11.3 Subtracting Mixed Numerals (Same Denominators)

252

11.4 Subtracting Mixed Numerals (Related Denominators)

254

11.5 Finding a Unit Fraction of a Quantity 256

11.6 Finding Simple Fractions of Quantities 258

11.7 Working with Fractions of Money 260

11.8 Working with Percentages of Money 262

11.9 Using Percentage to Find Discounts 264

11.10 Applying Discounts to Calculate Selling Price 266

11.11 Examining Nets 268

11.12 Creating Nets 270

MODULE 1212.1 Investigating Order with One Operation 272

12.2 Investigating Order with Multiple Operations 274

12.3 Consolidating Order with Multiple Operations (without Parentheses)

276

12.4 Introducing Parentheses 278

12.5 Consolidating Order with Multiple Operations (with Parentheses)

280

12.6 Investigating Order with Multiple Grouping Symbols

282

12.7 Introducing Coordinates 284

12.8 Exploring the Coordinate Plane 286

12.9 Exploring Shapes on All Four Quadrants 288

12.10 Using Fractions to Describe Probabilities 290

12.11 Conducting Chance Experiments 292

12.12 Working with Probability 294

Index 296

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ORIGO Stepping Stones • Year 66.1

Step Up

Identifying and Describing Geometric PatternsStep In

What do you notice?What patterns do you see?

Complete this table to match the pattern.

Picture number 1 2 3 4

Number of squares 1 2

Number of toothpicks 4 7

How could you work out the number of toothpicks in the 7th picture? What pattern rule could you use?

Draw more dots to continue the toothpick pattern on the graph.

This growing pattern was made with toothpicks.

a. Complete the table. If necessary, draw more pictures on scrap paper.

Picture number 1 2 3 4 5

Number of squares 1

Number of toothpicks 4

c. Draw dots on the graph to show the pattern.0 1 2 3 4 5 6 87 9 10

0

24

22

20

18

16

14

12

10

8

6

4

2

Number of squares

Num

ber

of t

ooth

pick

s

26

28

30

1 2 3 4

1 2 3

1. Look at this pattern made with toothpicks.

b. Describe the pattern that you see.

0 1 2 3 4 5 6 870

151413121110987654321

Number of squares

Num

ber

of t

ooth

pick

s

Picture number

Number of squares

Number of

Describe the pattern that you see.

Complete the table. If necessary, draw more pictures on scrap paper.

1 2 3

1

made with toothpicks.

How could you work out the number of toothpicks in the 7th picture? What pattern rule could you use?


Complete the table. If necessary, draw more pictures on scrap paper.

Look at this pattern made with toothpicks.


0 2 301

Number of squares

129

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ORIGO Stepping Stones • Year 6 6.1

Step Ahead

2. Look at this pattern made with toothpicks.

Picture number 1 2 3 4 5

Number of triangles 1

Number of toothpicks 3

a. Complete the table. If necessary, draw more pictures on scrap paper.

Continue the pattern. Then draw red dots to represent the pattern on the graph in Question 2 above.

0 2 4 6 8 10 12 140

24

22

20

18

16

14

12

10

8

6

4

2

Number of triangles

Num

ber

of t

ooth

pick

s

30

26

28

16 18

3 toothpicks 9 toothpicks toothpicks

1 2 3 4

c. Draw blue dots on the graph to showthe pattern.

d. Use the pattern to work out the number of toothpicks for Picture 6 and Picture 7. Then draw dots on the graph to continue the pattern.

b. Describe the pattern that you see.

Picture 1 Picture 2 Picture 3

Picture 4

toothpicks

Step Ahead

Picture 1

Continue the pattern. Then draw red dots to represent the pattern on the graph in Question 2 above.

of toothpicks for Picture 6 and Picture 7. Then draw dots on the graph to continue the pattern.

6

4

Num

ber

of t

ooth

pick

s

Draw blue dots on the graph to show

Use the pattern to work out the number of toothpicks for Picture 6 and Picture 7. Then draw dots on the graph to continue

18

16

14

12

Num

ber

of t

ooth

pick

s

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Step Up

Identifying and Describing Number PatternsStep In

1. Look at this pattern. Write the missing numbers in the table below.

How do the number patterns below describe the pictures of the squares?

What will be the next three terms in each student’s pattern?

How did you decide?

What number patterns can you fi nd in these pictures?

Length of side 1 2 3 4 5 10

Dots along one side

Dots around the triangle

4, 8, 12, ...

Anna

1, 2, 3, ...

Haroon

2, 3, 4, ...

Daniel

1, 4, 9, ...

Mio

Step Up Look at this pattern. Write the missing numbers in the table below. Look at this pattern. Write the missing numbers in the table below.

What will be the next three terms in each student’s pattern?

2, 3, 4, ...

Daniel

How do the number patterns below describe the pictures of the squares?

MioMio

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Step Ahead

Quadrilateral frames Number of quadrilaterals Number

of dotsNumber of lines

2. Complete this table to describe this pattern of quadrilaterals.

Annaa. Haroonb.

Danielc. Miod.

A B

3. Write the number of quadrilaterals, dots and lines for the next six frames.

4. Describe the pattern in these columns.

A

B

Work out the 20th term for the patterns each student found in the picture of squares at the top of page 130.Step Ahead

Anna

Work out the 20th term for the patterns each student found in the picture of squares at the top of page 130.

132

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Step Up

Identifying Relationships Between Two Numerical PatternsStep In

Look at this growing pattern. What types of numbers are represented?

What numbers should be written in the second row of this table to describe the pattern?

How did you work out the numbers to write in the table?

What do you notice about the number you wrote for each picture?

Picture number 1 2 3 4 5 6 7

Total number of counters

1 2 3 4

1. Look at the pictures for these oblong numbers.

a. Complete the table below to show the total number of counters in each picture of this pattern.


Total number of counters 2

1 2 3 4

b. Write how you worked out the numbers to keep the pattern going.

Complete the table below to show the total number of counters in each picture of this pattern.

Picture number

Look at the pictures for these oblong numbers.






4

What numbers should be written in the second row of this table to describe the pattern?

133

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Step Ahead

2. Look at the pictures for these triangular numbers.

This pattern of ‘houses with roofs’ was made by joining the shape in the pattern above and the shape in the pattern at the top of page 132. The fi rst row of the table matches the number of rows of counters in the square part of the ‘house’.

b. Complete the table below to show the total number of counters in the pictures of this pattern.



a. Draw the next picture in the pattern.

1 2 3 4




1 2 3

3. Look at the number of counters for each term in Questions 1a and 2. Write how the patterns are related.

table matches the number of rows of counters in the square part of the ‘house’.

Draw the next picture in the pattern.

This pattern of ‘houses with roofs’ was made by joining the shape in the pattern This pattern of ‘houses with roofs’ was made by joining the shape in the pattern above and the shape in the pattern at the top of page 132. The fi rst row of the table matches the number of rows of counters in the square part of the ‘house’.

Draw the next picture in the pattern.

Look at the number of counters for each term in Questions 1a and 2.

6

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How much does this jar weigh?

How would the total mass change if there were 2, 3 or 4 jars on the scale?

Imagine several jars are placed on the scale and the total mass is now 2.8 kg.

How many jars were placed on the scales? How do you know?

Complete this table.

How could you show the same pattern with common fractions?

Creating Number Patterns Involving Decimal Fractions and Common FractionsStep In

Step Up 1. a. The picture shows the mass of one jar. Complete the table.

b. Write how you worked out the numbers to keep the pattern going.

0.4 kg

0.3 kg

Jars 1 2 3 5 7 10

Mass (kg) 0.4 0.8

Number of jars 1 2 3 5 7 10

Total mass (kg) 0.3

The mass will increase by 0.4 kg each term. That«s 0.4 kg, 0.8 kg, 1.2 kg…

Step Up

Number of jars

Total mass (kg)


1. a. The picture shows the mass of one jar. Complete the table.

1


The picture shows the mass of one jar. Complete the table.

3

0.8

How many jars were placed on the scales? How do you know?

0.4 kg

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Step AheadZookeepers are monitoring the weight gain of four tiger cubs. They are concerned about the weight gain of one cub. Look for patterns in the table to identify the tiger cub they are concerned about. Explain your thinking.

2. a. Complete the table.


Total mass (kg) 0.25

Weight (kg)

Tiger cub Week 1 Week 2 Week 3 Week 4 Week 5

Cia 3.6 4.1 4.6 5.1 5.5

Samba 3.5 3.7 3.8 4 4.1

Chilli 3.2 3.7 4.1 4.6 5

Simba 4.1 4.5 5 5.4 5.9

b. Write the rule you used to work out the masses.


Total mass (kg) 610

3. This table shows some terms in a diff erent pattern.

a. Write the rule you would use to continue the pattern. Then complete the table.

b. Use your rule to work out the total mass in these numbers of jars.

20 jars kg 50 jars kg 80 jars kg

Tiger cub

Cia

Samba

Chilli

Zookeepers are monitoring the weight gain of four tiger cubs. They are concerned about the weight gain of one cub. Look for patterns in the table to identify the tiger cub they are concerned about. Explain your thinking.

Zookeepers are monitoring the weight gain of four tiger cubs. They are Zookeepers are monitoring the weight gain of four tiger cubs. They are concerned about the weight gain of one cub. Look for patterns in the table to identify the tiger cub they are concerned about. Explain your thinking.

Use your rule to work out the total mass in these numbers of jars.

kg 50 jars

Write the rule you would use to continue the pattern. Then complete the table.

Use your rule to work out the total mass in these numbers of jars.

kg 80 jars

Write the rule you would use to continue the pattern. Then complete the table.

12

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Step Up

Using Mental Strategies to Multiply Two-Digit NumbersStep In

Use the data in this table to complete Question 1 below and Questions 2, 3 and 4 on page 137.

1. Work out the cost of making these calls from Australia.

a.Jordan for 3 minutes

$

b.India for 6 minutes

$

c.Fiji for 4 minutes

$

d.Iran for 8 minutes

$

It depends on the country you call, the time that you make the call and how long you talk.

Grace used place value to work it out.

4 × 40 + 4 × 5

Luke used a doubling-and-halving strategy.

4 × 45 is the same as 2 × 90

Layla used the double double strategy.

Double 45 then double 90

Jianna used factors like this.

4 × 45 is the same as 4 × 5 × 9

What does it cost to make an international phone call?

How could you work out the cost of a four-minute call to Ireland?

How did you convert your answer to dollars and cents?

How much would it cost for a four-minute call to each of the other countries?What strategies could you use to work it out?

How could you work out the cost of a 15-minute call to France?

Home Phone International Call Costs per Minute from Australia

Greece Iran Denmark Egypt Fiji Jordan Kenya India

45c 55c 40c 50c 70c 75c 45c 70c

Canada 25c

France 32c

Germany 28c

Ireland 45c

Italy 30c

Step Up Use the data in this table to complete Question 1 below and Questions 2, 3 and 4 on page 137.


Iran

Use the data in this table to complete Question 1 below and Questions 2, 3 and 4 on page 137.

What strategies could you use to work it out?


Luke used a doubling-and-halving strategy.

How did you convert your answer to dollars and cents?

How much would it cost for a four-minute call to each of the other countries?What strategies could you use to work it out?




Jianna used factors like this.

4 × 45 is the same as 4 × 5 × 9




Italy 30c

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Step AheadSometimes phone calls cost more at certain times of the day. These times are called peak times. Work out how much you would save if you made a 15-minute off -peak call to these countries.

a. Greece

d. Egypt

b. Iran

e. Fiji

c. Denmark

f. India


To Greece Iran Denmark Egypt Fiji India

Off -Peak 39c 48c 39c 49c 69c 67c

Peak 45c 55c 40c 50c 70c 70c

3. Work out the cost of a 15-minute call from Australia to these countries.

Denmarkb.

$

Iranc.

$

Kenyaa.

$

Greeceb.

$

Egyptc.

$

Indiaa.

$

a.Kenya for 4 minutes

$

b.Egypt for 7 minutes

$

c.Greece for 9 minutes

$

d.Denmark for 6 minutes

$

4. Work out the cost of a 12-minute call from Australia to these countries.

2. Work out these call costs.

Refer to the data table at the bottom of page 136 to answer these questions.

Step Ahead


To

Sometimes phone calls cost more at certain times of the day. These times are called peak times. Work out how much you would 15-minute

$

Greece

Work out the cost of a 12-minute call from Australia to these countries. Work out the cost of a 12-minute call from Australia to these countries.

$

Work out the cost of a 12-minute call from Australia to these countries.

Iran

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Step Up

a.

Comparing Mental Strategies for MultiplicationStep In

Read about these strategies used to mentally calculate 36 × 50.

I multiplied 36 by 10. Then I multiplied my answer by 5 because 50 is 5 x 10.

I multiplied 36 by 100. Then I halved my answer because 50 is one-half of 100.

Mariam used place value and multiplied the tens and then the ones.

10 × 25 + 2 × 25

Soma used a doubling-and-halving strategy.

12 × 25 is the same as 6 × 50

Liam used factors.

12 × 25 is the same as 3 × 4 × 25

Think about some of the diff erent situations in which you use multiplication.

Multiplication is used to work out the cost of accommodation, car hire or telephone calls.

12 m

25 m

Imagine you have to buy carpet for this fl oor area. Look at how these students work out the area that has to be covered.

Use a strategy you like to calculate the area of a rectangle measuring 15 cm × 24 cm.

Is there another way you could work it out? Which method do you like best? Why?

Step Up Read about these strategies used to mentally calculate 36 × 50.Read about these strategies used to mentally calculate 36 × 50.



12 × 25 is the same as 6 × 50



12 × 25 is the same as 3 × 4 × 25

25 m25 m

12 m

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Step Ahead Write the missing numbers in each machine.

2. Use a method you like to mentally calculate these.

a. 72 × 50 = b. 34 × 50 =

c. 64 × 50 =

d. 42 × 50 = e. 28 × 50 =

f. 31 × 50 =

4. Use the same method to mentally calculate these.

a. 24 × 25 = b. 16 × 25 =

c. 28 × 25 =

d. 44 × 25 = e. 12 × 25 =

f. 36 × 25 =

5. Complete number sentences you can solve using your method.

a. × 50 = b. 50 ×

=

3. Write how you would mentally work out 32 × 25.

1. Try to use more than one method from page 138 to solve these.

a. 16 × 50 = b. 24 × 50 = c. 25 × 50 =

d. 27 × 50 = e. 15 × 50 = f. 14 × 50 =

IN

12

OUT

6300

a.

× 15022

IN

12

OUT

7500

b.

× 50022

Step Ahead

IN

Write the missing numbers in each machine.

Complete number sentences you can solve using your method.

× 50 =

Use the same method to mentally calculate these.

16 × 25 =

e. 12 × 25 =

Complete number sentences you can solve using your method.

b.

Use the same method to mentally calculate these.

64 × 50 =

31 × 50 =

140

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Step Up 1. Double one number and halve the other to make a problem that is easier to solve. If necessary, repeat this step then write the product.

Using a Double-and-Halve Strategy to Multiply Dollars and CentsStep In

c. 6 × $1.25 = $

×

×

b.6 × $2.50 = $

×

×

a.6 × $1.15 = $

×

×

How could you work out the cost of buying two issues of this comic book?

$1.25

How could you work out the cost of buying 12 issues?

Eva used a double-and-halve strategy. Write how you think she multiplied.

12 × $1.50 = $

is the same as

× =

Use Eva’s strategy tocalculate this product.

I double doubled 1.25 to work out the cost of 4 issues. That's 5 and 3 x 5 is 15.

Three issues cost 3.75. The cost of 12 issues is double double 3.75. That's 15.

1. Doubleto solve. If necessary, repeat this step then write the product.

12 × $1.50 =

Eva used a double-and-halve strategy. Write how you think she multiplied.Eva used a double-and-halve strategy. Write how you think she multiplied.

3.75. That's

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Step Ahead

4. Write some number sentences involving money that you can solve by doubling and halving.

a. × =

c. × =

b. × =

d. × =

3. Use the same strategy to work out these.

b. 8 × $1.25 = $

e. 16 × $0.75c = $

a. 4 × $1.45 = $

d. 16 × $0.50 = $

c. 8 × $2.50 = $

f. 24 × $0.25 = $

c.

24 × $0.75 = $

×

×

b.

$1.25 × 16 = $

×

×

a.

6 × $0.75 = $

×

×

f.

8 × $1.75 = $

×

×

e.

32 × $1.50 = $

×

×

d.

$0.25 × 16 = $

×

×

Moran has $20 to spend on comic books. If he buys some of each type, work out a possible combination of issues he could buy with the least amount of change. You can use a calculator to help.

issues

$1.75

issues

$1.45

issues

$2.25

2. Double and halve to solve these.

a.

c. ×

Write some number sentences involving money that you can solve by doubling and halving.

=

Use the same strategy to work out these.

8 × $1.25 = $

Write some number sentences involving money that you can solve by doubling and halving.

e. 16 × $0.75c =

×

$1.75 = $

×

142

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Step Up

Using Multiplication Strategies with Decimal FractionsStep In

This large rectangle represents an area that has to be paved.

1. Split the decimal fraction to complete these number sentences.

a. 24 × 1.25 is the same as 24 × + 24 × =

b. 8 × 3.25 is the same as 8 × + 8 × =

c. 12 × 1.5 is the same as 12 × + 12 × =

d. 14 × 2.5 is the same as 14 × + 14 × =

e. 16 × 2.75 is the same as 16 × + 16 × =

I know 0.5 is one-half, o.25 is one-quarter and 0.75 is three-quarters.

1.5 m

14 m

How could you work out the number of square metres that will be covered?

How are these steps similar to other algorithms you have used?

I split 1.5 into two parts. 14 x 1.5 is the same as 14 x 1 + 14 x 0.5.

Aiden used the formalalgorithm. He followed these steps.

Are the steps easy to follow?

Step 2

1 4× 1.5 7 .0 1 4.0

2

Multiply the ones.

Step 3

1 4× 1.5 7 .0 1 4.0 2 1 .0

2

Add the partial products.

Step 1

1 4× 1.5 7.0

2

Multiply the tenths.

Step Up

is the same as

8 × 3.25

Split the decimal fraction to complete these number sentences.

How are these steps similar to other algorithms you have used?How are these steps similar to other algorithms you have used?How are these steps similar to other algorithms you have used?

7.0

Step 2

1 4× 1.5 7 .0

Multiply the ones.

is the same as 14 x 1 + 14 x 0.5.

Multiply the ones. Add the partial

I split 1.5 into two parts. 14 x 1.5

Step 3

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Step Ahead

Multiply the numbers in the circles by the number in the middle. Write your answers around the outside.

2. Use the thinking from Question 1 to calculate each product.

a. 28 × 1.25 = b. 12 × 2.25 = c. 20 × 1.75 =

d. 22 × 1.5 = e. 18 × 3.5 = f. 24 × 2.75 =

3. Use the formal multiplication algorithm to calculate these.

32 × 1.8a. 12 × 2.9b. 14 × 1.7c.

25 × 2.3d. 15 × 2.6e. 18 × 1.9f.

12

24

15

18

20

40

1.25

Step Ahead

15 × 2.6

144

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Step Up 1. Write acute, obtuse or refl ex to describe each angle below.

How could you measure this angle?

Estimating and Measuring AnglesStep In

Addison and Jamal know that a straight angleis 180° because it is half of a full turn. Knowing that part, they turned the protractor upside down to measure the rest of the angle.

Jamal then used addition to calculate the total angle. Addison used subtraction. They both worked out the correct answer. What steps do you think they followed?

I would start by estimating. I know it is greater than a three-quarter turn, so it must be between 270 and 360 degrees.

Measuring angles with a 360û protractor is easy. I'm not sure about how to use a 180û protractor though.

90

18

0

80

170

70

160

60

150

50

140 40

130

30

120

20

110

10

100

0

a. b. c.

d. e. f.

Write acute

b.

obtuse or refl ex to describe each angle below.

Jamal then used addition to calculate the total angle.

What steps do you think they followed?

170

160

150

140

Measuring angles with a 360û protractor is easy. I'm not sure about how to use a 180û protractor though.

145

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turns turnsturns

900° 1350° 480°

Step Ahead Karin turned on the spot two times. She turned a total of 720° (2 × 360 = 720). Complete these. You can use a calculator to help.

2. Estimate and write the size of each angle in Question 1. Use the picture below to help.Do not use a protractor.

4. Use a protractor and ruler to draw each shape.

3. Use a protractor to measure the angles in Question 1. Write your answers below.

° ° ° ° ° °a. b. c. d. e. f.

270°

45°135°90°

° ° ° ° ° °a. b. c. d. e. f.

a triangle with two angles that are 40°

a. a quadrilateral with two angles that are 65°

b.

is the same as

1 12 turnsa.

°

is the same as

2 14 turnsb.

°

is the same as

3 23 turnsc.

°

is the same as

d.

is the same as

e.

is the same as

f.

Step Ahead

is the same as

112 turns

Karin turned on the spot two times. She turned a total of 720° (2 × 360 = 720). Complete these. You can use a calculator to help.

Use a protractor and ruler to draw each shape.

a quadrilateral with two angles b.

° e.

a quadrilateral with two angles that are 65°

Use a protractor to measure the angles in Question 1. Write your answers below.

° f.

146

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Step Up 1. a. The design above used shapes like the one below. Complete the chart to show other ways of combining the angles around a point to total 360º.

What do you know about the shapes in this design?

What do you know about the angles of the shapes?

What do you notice about the angles at the centre of the design?

Examining Angles Around a PointStep In

2. Choose one of your combinations from the chart in Question 1. Draw congruent kites to show how they fi t around the point below. Use a ruler and protractor to help you.

Combination of angles that total 360 degrees

60 + 60 + 60 + 60 + 60 + 60

b. Measure and write the length of each side of the kite below.

mm

mm mm

mm

90°

120°

90°

60°

Choose one of your combinations from the chart in Question 1. Draw congruent kites to show how they fi t around the point below. Use a ruler and protractor to help you.Choose one of your combinations from the chart in Question 1. Draw congruent kites to show how they fi t around the point below. Use a ruler and protractor to help you.

mm

120°

90°

The design above used shapes like the one below. Complete the chart to show other ways of combining the angles around a point to total 360º.

Measure and write the length of each side

147

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Step Ahead

a. Calculate two combinations of kite angles from Question 1 that could be arranged to form a straight angle.

3. Use what you know about the angles around a point to work out the unknown angles in each picture below. You can use a calculator but do not use a protractor.

b. Draw a combination of kites to show your thinking.

Angle d =

°

115°

24°

60°d

Angle e = °

65°

82°

75°

e

23°

Angle f =

°

65°

39°96°

f76°19°

Angle a =

°

255°

a

Angle b =

°

337°

b

Angle c =

°

110°

c

140°

Step Ahead

Calculate two combinations of kite angles

Angle°

Angle

75°

e

23°

65°

82°

96°

c =

148

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Step Up 1. Use what you know about right angles to work out the unknown angles in each picture below. You can use a calculator but do not use a protractor.

What do you know about the shapes in this picture?

What do you know about the angles of the shapes?

What do you notice about the angles that meet at the bottom of the picture?

Exploring Angles in Right Angles and on Straight LinesStep In

How could you work out the size of each angle at the bottom of the design?

I can see that all the angles that meet at the bottom make a straight line. The total must be 180û.

If six of those angles makes 180û then three of them must have a total that is half of that. That means together they make a right angle.

Angle sum = 90°

Angle c =

°

22° c34°

Angle sum = 90°

Angle b = °

52°

b

Angle sum = 90°

Angle a = °

25°

a

Use what you know about right angles to work out the unknown angles in each picture below. You can use a calculator but do Use what you know about right angles to work out the unknown angles in each picture below. You can use a calculator but do

How could you work out the size of each angle at the bottom of the design?

that is half of that. That means together they make a right angle.

If six of those angles makes 180û then three of them must have a total that is half of that. That means together they make a right angle.

149

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Step Ahead Solve each problem. Show your thinking.

2. Use what you know about angles on a straight line to work out the unknown angles in each picture below. You can use a calculator but do not use a protractor.

Angle f = °

Angle a =

°

45°

a

Angle e =

°

13°48°67°

e24°

Angle d =

°

40°

18°

65°

d

Angle b =

°

80°

b75°

Angle c =

°

50°

c

70°

27°

Angles f and g are equal.

83°

f

27°

g

Angles f and g form a straight line. Angle g is 134°. What is the difference in size between Angle f and Angle g?

a. Angles q, r and s form a straight line. Angle r is a right angle. Angle s is

13 of Angle r. What size is Angle q?

b.

° °

Step Ahead

f and f and fAngle g is 134°. What is the difference in size between Angle

Solve each problem. Show your thinking.

Angle

48°24°

83°

f

27°

Angle c =

150

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Step Up1. Each picture below is made by intersecting straight lines. Use what you

know about vertically opposite angles to work out the unknown angles. Do not use a protractor.

Investigating Vertically Opposite AnglesStep In

Adjacent angles share an angle arm and a vertex.

Angle a and Angle b are adjacent angles.

Which angle arm do they have in common?

Angle a and Angle c are not adjacent even thoughthey share a vertex.

When two lines intersect four angle arms are made.

Angle d and Angle f are vertically opposite angles.

Which other angles are vertically opposite?

Which angles are adjacent?

Vertically opposite angles do not have to be vertical. The word vertical comes from vertex and vertically opposite angles are on opposite sides of a vertex.

e

g

f

d

a b

c

Angle a =

°

Angle b =

°

65°

b

Angle c = °

Angle d =

°

124°

d

Angle e = °

Angle f =

°

133°

e

ac

f

e

115°

56°47°

b

Each picture below is made by intersecting straight lines. Use what you know about vertically opposite angles to work out the unknown angles. Do not use a protractor.

Each picture below is made by intersecting straight lines. Use what you know about vertically opposite angles to work out the unknown angles.

Vertically opposite angles do not have to be vertical. and vertically

opposite angles are on opposite sides of a vertex.

d

f

c

151

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2. Use what you know about angles around a point and on a straight line to work out the unknown angles in each picture below. You can use a calculator but do not use a protractor.

Step AheadFour angles are around a point. Angles c and d are adjacent angles.Angles a and b are adjacent angles. Angle a is 71° and Angle d is 119°.Are there any vertically opposite angles? Explain how you know.

Angle a =

°

Angle b =

°

Angle c =

°

Angle d =

°

Angle e =

°

Angle f =

°

Angle g =

°

Angle h =

°

Angle i =

°

Angle j =

°

c

b

104°

a e

d

152°f

hg

70°j

i

76°

Angle k =

°

Angle m =

°

Angle l =

°

Angle n =

°m

l

53°

k

n

96°

Step AheadFour angles are around a point. Angles Angles and b Are there any vertically opposite angles? Explain how you know.

53°

Angle k =

Angle m =

°

Angle

Angle j =

Angle g =

=

STUDENT JOURNAL

Product Code: SSJ 226 16

STUD

ENT JO

UR

NA

L

6

ORIGO Stepping Stones is an award-winning mathematics program developed by curriculum specialists for Australian primary schools.

This revolutionary online program integrates print and digital technology to deliver comprehensive coverage of the F-6 Australian Curriculum – and even more!

THIS BOOK BELONGS TO

ORIGO Stepping Stones was developed by mathematics specialists for Australian primary schools to:

• make maths more focused and coherent

• foster students’ thinking and reasoning skills

• deliver multiple ways to differentiate classroom instruction

• provide a valuable source of professional learning for the teacher

• offer methods to assess deep understanding and skills

• provide online and print resources that engage all students

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