activemaths_10_ace_sb_30697_look_inside

9mm

Australian Curriculum

Australian Curriculum edition

The teacher book features: £ introductory material on the topics in all books, showing alignment with the Australian Curriculum £ simple suggestions on ways the program can be used in your school £ a quick teacher reference guide that includes a list of: − skillsheets and a breakdown of concepts covered − investigations, with descriptions and problem-solving behaviours highlighted − technology tasks, with descriptions and details of the technologies used in each task £ all student tasks from the student book, with answers included for easy marking £ a simple record sheet that can be photocopied to record individual student progress £ certificates that can be photocopied and modified to your school needs.

The program has a simple and flexible design, enabling it to successfully complement any other Mathematics resource being used. It is written by teachers for teachers, and has been successfully trialled and used throughout Australia.

Series titles

www.matildaeducation.com.au

Active Maths 7 Australian Curriculum edition ISBN 978 1 4202 3063 5

Active Maths 8 Australian Curriculum edition ISBN 978 1 4202 3065 9

Active Maths 9 Australian Curriculum edition ISBN 978 1 4202 3067 3

Active Maths 10 Australian Curriculum edition ISBN 978 1 4202 3069 7

Active Maths 7 Teacher book Active Maths 8 Teacher book Active Maths 9 Teacher book Active Maths 10 Teacher book Australian Curriculum edition Australian Curriculum edition Australian Curriculum edition Australian Curriculum edition ISBN 978 1 4202 3064 2 ISBN 978 1 4202 3066 6 ISBN 978 1 4202 3068 0 ISBN 978 1 4202 3070 3

AM_10_ACE_30697_cvr_prs.indd 1

10

Australian Curriculum edition

Each student book covers the core Mathematics topics taught at that level, and incorporates skill sheets, investigations and technology tasks.

Active Maths 10

ACTIVE MATHS

program featuring a student book and teacher book at Years 7 to 10.

The series has been revised to address all the content descriptions and achievement standards contained in the Mathematics Australian Curriculum.

Active Maths 10 Australian Curriculum edition Homework Program

Active Maths Homework Program is a topic-based homework

m a r g o Pr k r o w e m o H Monique Miotto Tracey MacBeth-Dunn

11/18/20 1:34 PM

Australian Curriculum

Australian Curriculum edition

ACTIVE MATHS

10

201000_AM10HP_SB_00.indd 1 201000 ActiveMaths10 ACE ttl pg-CS55.indd 1

m a r g o r P k r o w e m Ho Series editor and lead author

Monique Miotto Authors

Tracey MacBeth-Dunn Natalie Caruso Contributing authors

Dina Antoniou Lee Brandt Loretta Carter Bozenna Graham Andrew Mark 8/08/12 8/08/12 11:14 11:03 AM AM

This edition published in 2021 by

Internet addresses At the time of printing, the internet addresses appearing in this book were correct. Owing to the dynamic nature of the internet, however, we cannot guarantee that all these addresses will remain correct.

Matilda Education Australia, an imprint of Meanwhile Education Pty Ltd Level 1/274 Brunswick St Fitzroy, Victoria Australia 3065 T: 1300 277 235 E: customersupport@matildaed.com.au www.matildaeducation.com.au

While every care has been taken to trace and acknowledge copyright, the publishers tender their apologies for any accidental infringement where copyright has proved untraceable.

They would be pleased to come to a suitable arrangement with the rightful owner in each case.

Copyright © Macmillan Education 2012 The moral rights of the author have been asserted.

First edition published in 2012.

All rights reserved.

Except under the conditions described in the Copyright Act 1968 of Australia (the Act) and subsequent amendments, no part of this publication may be reproduced, 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 the copyright owner.

Educational institutions copying any part of this book for educational purposes under the Act must be covered by a Copyright Agency Limited (CAL) licence for educational institutions and must have given a remuneration notice to CAL. Licence restrictions must be adhered to. For details of the CAL licence contact: Copyright Agency Limited, Level 15, 233 Castlereagh Street, Sydney, NSW 2000.

Telephone: (02) 9394 7600. Facsimile: (02) 9394 7601.

Email: info@copyright.com.au Publication data Author: Miotto, Monique and others Title: Active Maths 10: Homework Program (Australian Curriculum edition) Workbook ISBN: 9781420230697 Publisher: Project editor: Editor: Illustrator:

Colin McNeil and Peter Saffin Claire Lavin Joanne Gower Paul Lennon (cartoons), Andy Craig and Nives Porcellato (technical diagrams) Cover designer: Dim Frangoulis Text designer: Sunset Digital Production control: Loran McDougall Permissions clearance: Elizabeth Sim Typeset in Times Ten Roman 10/13 pt by Sunset Digital and Nikki M Group

The authors and publisher are grateful to the following for permission to reproduce copyright material: © Australian Curriculum, Assessment and Reporting Authority 2012. For all Australian Curriculum material except elaborations: This is an extract from the Australian Curriculum.

Elaborations: This may be a modified extract from the Australian Curriculum and may include the work of the author(s). ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that: The content descriptions are solely for a particular year and subject; All the content descriptions for that year and subject have been used; and The author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject. You can find the unaltered and most up to date version of this material at http://www.australiancurriculum.edu. au/. This material is reproduced with the permission of ACARA, iv; Graph, ‘Annual mean temperature anomalies for Australia’ from ‘Australian Annual Mean Temperature Anomalies’, Bureau of Meterology, Australian Government <http://www.bom.gov.au/ climate/change/amtemp.shtml>, 46; Casio ClassPad screenshots reproduced with permission from Shriro Australia Pty Ltd, 9, 10, 12, 61, 62, 81, 82, 83, 107, 108, 109, 110, 125, 126, 135, 136, 153, 154, 155, 156; Extract from ‘The science behind southeast Australia’s wet, cool summer’, Climate Commission, 48; Extract from ‘Climate change still a reality despite soggy summer, warns report’ by David Wroe, Sydney Morning Herald, 15 March 2012, 45; TI-Nspire screenshots reproduced with permission from Texas Instruments Australia Pty Ltd, 13, 14, 16, 63, 64, 85, 86, 87, 111, 112, 113, 114, 127, 128, 137, 138, 157, 158, 159, 160; Graph, ‘Global average temperature from 1950 to 2011, shown as the variation (anomaly) in degrees Celsius from the 1961–1990 average. La Nina years shown in purple, others in red’, World Meteorological Organization, 47.

The author and publisher would like to acknowledge the following: Microsoft Excel screenshots © 2012 Microsoft Corporation.

All rights reserved, 30, 31, 32, 51, 70, 71, 161, 162.

While every care has been taken to trace and acknowledge copyright, the publisher tenders their apologies for any accidental infringement where copyright has proved untraceable.

They would be pleased to come to a suitable arrangement with the rightful owner in each case.

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Record sheet Name: ....................................................... Teacher: ....................................................... Class: ...........

TEACHER FEEDBACK

Chance

17

3

Congruence and similarity

33

4

Data

39

5

Expanding and factorising

53

6

Financial mathematics

65

7

Linear equations

73

8

Linear graphs

89

9

More equations and graphs

95

10

Pythagoras and trigonometry

115

11

Quadratic equations

129

12

Surface area and volume

141

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

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Technology task

2

Investigation

1

Skill sheet 3

Algebra

Skill sheet 2

Skill sheet 1

1

Permission is granted for this page to be photocopied by the purchasing institution.

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Skill sheet

Macmillan Active Maths 10

Algebra 1 Name: .............................................................................................. Due date: .........../.........../...........

For 1–4, match the following expressions and terms to the statements in the questions. x + y xy 1

[Algebra: terms]

8

[Substitution]

x x – y y

9

[Substitution]

The product of x and y.

m+y

5x – 2y

mx y

10 2

[Algebra: terms]

3

[Algebra: terms]

4

[Algebra: terms]

[Substitution]

The difference between x and y.

The sum of x and y.

11

[Substitution]

PrT , find I when P = 4000, r = 7 100 [Substitution: formula] and T = 3.

12

6

[Algebra: terms]

If I =

For 5–6, write an algebraic expression for:

[Algebra: terms]

The ratio of x to y.

5

If x = −5 and y = 4, what is the value of 3(x + 2y)?

17 less than m

13

[Substitution: formula]

If W = 220 + 15h, find W if h = 7

1 .

2

8 more than triple x

If the radius of a circle can be found c using the formula r = , where C is [Substitution: 2π formula] the circumference of the circle, calculate the radius of a circle with a circumference of 22.4 cm. Answer correct to 1 decimal place.

14 For 7–10, find the value of the expressions if x = 3, y = −2 and m = 5.

7

[Substitution]

4y

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

NUMBER AND ALGEBRA Algebra

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For 15–30, simplify fully: 15

[Simplify like terms]

16

[Simplify like terms]

24

[Simplify expression]

6a + 5b - a

25

[Simplify expression]

3x + 4a + 2x – 6a

[Simplify term]

–6x ÷ 3xy

- 10 x2 y

26

4x – 2y + 3x + 4y

17

40am ÷ 4a

- 2 xy2

[Simplify term]

18

[Simplify like terms]

19

[Simplify like terms]

6a2 – 7m + 3a2 – 2m

[Simplify term]

7m2 – 3a + 4m2 + 2a

[Simplify like terms]

21

[Simplify expression]

[Simplify term]

5xy – 3y – 7xy + y

–2x × 3y

29

[Expand: distributive law]

23

6 pr 12 pm

2(d – 11) + 9

4 k 2 l × 3l k × 2l2

30

[Simplify expression]

-

3a2 × (–5ab)

22

[Simplify term]

28

20

−4 x × 3 x 6x

27

[Simplify expression]

5ap × 3a

Student comment

Guardian comment/signature

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

Teacher feedback

NUMBER AND ALGEBRA Algebra

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Skill sheet

Macmillan Active Maths 10

Algebra 2 Name: .............................................................................................. Due date: .........../.........../...........

For 1–8, simplify fully: 1

[Simplify expression]

2

[Simplify expression]

5x × 3x

4

[Simplify expression]

5

[Simplify expression]

[Add algebraic fractions]

–3x × 5xy × (–4x)

3y y 5 7

10

[Add algebraic fractions]

8 a2 b 4a

p 3p + 3 4

11

3x + 4 × 2x

[Add algebraic fractions]

5 × 2y – 3 × 4y

5x 2x 6 3

12

[Subtract algebraic fractions]

6 m × 4 am 2 am

6

[Simplify expression]

x x + 2 3

9

3

[Simplify expression]

For 9–20, simplify the algebraic fractions:

7 3 2x x

13

[Subtract algebraic fractions]

− 3 xy × 2 x

7

12 xy2

[Simplify expression]

x 4 × 8 3x

14

8 [Simplify complex expression]

[Multiply algebraic fractions]

5 xy - 3 xy - 6 x2 y2

4m 3 × 5 2 am

15

[Multiply algebraic fractions]

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

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23

a a ÷ 8 4

16

[Divide algebraic fractions]

[Indices: mixed]

(3y2)-3

24

[Indices: mixed]

8 m 5m ÷ 3 6

17

[Divide algebraic fractions]

25

[Indices: mixed]

5x 2m x ÷ × 6 15 2

18 [Mixed algebraic fractions]

[Complex algebraic fractions]

20

[Complex algebraic fractions]

10m2 n × 2mn 15m0 n4

26

19

(2n5)2 ÷ (5n3)0

[Indices: mixed]

(3x4)-2

( a + 5)( a − 3) a × 5a ( a + 5) 27

[Indices: mixed]

 4  −2   5

9 ÷ 18 ( 5 - p) ( 5 - p) 28

[Indices: mixed]

(4a3)3 ÷ (4a6)2

For 21–30, simplify fully, giving your answers without negative indices.

29

4x10 6x2

21

[Indices: mixed]

( a2 m− 4 )3 × ( a− 3 m2 )− 2 ( a3 m4 )− 1

[Indices: mixed]

1

30

2m-6 × 3m5

22

[Indices: mixed]

[Indices: mixed]

Student comment

( p4 )− 2 q3 × p0 (q6 ) 2

Guardian comment/signature

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

Teacher feedback

NUMBER AND ALGEBRA Algebra

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Investigation

Macmillan Active Maths 10

Radioactive decay Name: .............................................................................................. Due date: .........../.........../...........

The aliens who live on Planet Zorg mine three different types of radioactive elements, which they then use to power their spacecraft. These aliens conduct extensive investigations of their galaxy and need to power the spaceships with the correct type of fuel for their missions. For long journeys, they need to use an element for fuel that decays slowly, whereas for shorter voyages they can use a fuel that decays more rapidly.

1

[Interpret a graph]

The alien scientists know that one kilogram of radioactive substance decays exponentially according to a rule of the form y = 2-kx, where y represents the amount of substance present, in kilograms, after x years and k is referred to as the decay constant.

Consider the graph of y = 2-x for x ≥ 0. y 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

5

6

7

8

9

10

x

a

What is the value of k in the equation y = 2-x?

b

Use the graph to determine what y is equal to when x = 0.

c

There is one kilogram of radioactive substance to begin with. Use the graph to determine the weight of substance that is left after one year.

d

How much of the substance remains after two years?

e

Use the rule to determine how much substance is left after 10 years.

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

NUMBER AND ALGEBRA Algebra

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The x-axis is referred to as an asymptote of this graph. This means that the graph approaches but never touches this line. What does this mean about the amount of substance that will remain in the long run?

f

2

The three radioactive elements that the aliens on Planet Zorg use to power their spaceships are Groanium, Dashium and Astrodium. One kilogram of each element decays according to the rule y = 2-kx, with each element having a different decay factor, k.

Complete the following table.

a

[Use substitution]

Rule y = 2-kx

Groanium Dashium Astrodium

y = 2-0.2x y = 2-0.1x y = 2-0.05x

k value (decay constant)

Which element has the:

b

3

Element

i largest k value?

ii smallest k value?

The graphs of y = 2-0.2x, y = 2-0.1x and y = 2-0.05x are shown together on the axes below. y 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

2

4

6

8

10

12

14

16

18

20

x

a

Use the rules to show why each graph has a y-intercept at (0, 1).

b

Complete the table below.

Rule

y = 2-0.2x

y = 2-0.1x

y = 2-0.05x

x y (x, y)

10

10

10

(10, )

(10, )

(10, )

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

NUMBER AND ALGEBRA Algebra

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Plot each of the points (x, y) from the table above on the graph above and, hence, determine the rule and element corresponding to each curve.

c

Top curve rule:

Middle curve rule: Bottom curve rule: 4

[Read a graph]

The half-life of a radioactive substance is the time taken for the amount of substance to decrease by half. So if you start with one kilogram of a radioactive element, then the half-life of the element is the time it takes for the weight, y, to decrease to 0.5 kg. y 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

2

4

6

8

10

12

14

16

18

20

x years

a

On the graph locate 0.5 on the y-axis and draw a horizontal line across until it touches the bottom curve. Draw a vertical line from this point to the x-axis. The value on the x-axis is the half-life of Groanium. What is the half-life of Groanium?

b

What is the half-life of Dashium?

c

What is the half-life of Astronium?

d

How does the value of k, the decay constant, in the rule y = 2-kx, affect the half-life of the element?

5

[Interpret a graph]

Which radioactive element would be the best choice to use in spacecraft used for long journeys?

Give a reason for your answer.

6

There are many other radioactive elements on Planet Zorg and the aliens are keen to put these to good use in their space explorations. Their scientists have studied three of these substances, which also decay exponentially, and their research has shown that each substance decays according to a different rule.

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

NUMBER AND ALGEBRA Algebra

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The name of each substance and the rule according to which it decays is shown in the table below. In each case y represents the amount of substance present in kilograms after x years.

a

Substance

Decay rule

Planetonium Pluranium Plusonium

y = 2–0.03x y = 2–0.15x y = 2–0.3x

Complete each of the following tables of values quoting answers correct to 2 decimal places. i Planetonium 0

x y

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

ii Pluranium 0

x y

iii Plusonium 0

x y

b

On the axes below, plot three graphs of amount present, y, in kilograms after x years.

Label each graph with the name of the substance. y 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

c

1

2

3

4

5

6

7

8

9

10

x

In order for a radioactive substance to be useful for space travel, it must have a half-life of at least four years. Which of Planetonium, Plusonium and Pluranium would be suitable for such use?

Student comment

Guardian comment/signature

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

Teacher feedback

NUMBER AND ALGEBRA Algebra

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Technology task—Casio ClassPad

Macmillan Active Maths 10

Fraction machine Name: .............................................................................................. Due date: .........../.........../...........

In this task, you will investigate a machine that feeds on fractions. The fraction machine is 1− f , where f is the original fraction and f1 = 1+ f f1 is a new fraction.

For example, if you feed in

3 , the machine 5

3 2 5 = 5 = 1. calculates 3 8 4 1+ 5 5 1−

1

Using the main application on your calculator, find the output when f =

2

a

[The symbol | Type f1 = (1 – f) / (1 + f) | f = 1 / 4 and press E. means ‘with’ and is used for substitution.

To locate |, tap k and then OPTN.]

1 .

4

3 is fed into the machine. The output is fed back into the machine, then the new 5 result is fed back into the machine, and so on for 1000 processes altogether. Describe what happens and predict the final result.

Suppose

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

NUMBER AND ALGEBRA Algebra

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[There is no need to type the formula again. Highlight the original entry and press G to copy.

Tap the next entry line and tap H to paste. Then edit the entry and press E.]

3

b

Repeat a, starting with the fraction

2 .

3

If the original fraction is

a , predict the result after one million processes. b

To check your answer, use f1 = (1 – f) / (1 + f) | f = a / b. Substitute the output for f. To simplify the result, tap Actions, then Transformation, then Simplify. Copy the previous output, paste after simplify( and close the brackets. Press E.

4

Summarise your findings about the fraction machine.

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

NUMBER AND ALGEBRA Algebra

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5

Now investigate what happens when you feed a negative fraction into the machine.

[Use the

Simplify command to test whether an output in terms of pronumerals can be simplified.]

6

3 When f = − , the machine returns 4. Find some other proper fractions (in simplest terms) that 5 produce a positive whole number. Describe, in general terms, the fractions that return a whole number.

7

Show that the output for -

a a+b is equal to . Use this to explain your results from 6. b b–a

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

NUMBER AND ALGEBRA Algebra

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Another fraction machine produces the fraction f2:

Try this!

f + f2 =

1 f

1 − f2

.

Find outputs for

1 1 1 1 1 1 3 3 and - . What do you notice? and - , and - , and - , 2 2 4 4 5 5 7 7

a a and f = – . Use the results to explain why the b b outcome always has the same sign as the original fraction.

Now find and simplify the outputs for f =

Student comment

Guardian comment/signature

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

Teacher feedback

NUMBER AND ALGEBRA Algebra

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Technology task—TI-Nspire

Macmillan Active Maths 10

Fraction machine Name: .............................................................................................. Due date: .........../.........../...........

In this task, you will investigate a machine that feeds on fractions. The fraction machine is 1− f f1 = , where f is the original fraciton and 1+ f f1 is a new fraction: For example, if you feed in

3 , the machine 5

3 2 1 5 calculates = 5 = .

3 8 4 1+ 5 5 1−

1

Open a new Calculator page and make sure the calculator is in Exact mode. Press b11

[Press I for to select Define from the Actions menu. Type f 1 = (1 – f) ÷ (1 + f) and press ·. To find the the symbol 1 output when f = , enter f 1 | f = 1 ÷ 4. that means 4 ‘with’ and is used for substitution.]

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

NUMBER AND ALGEBRA Algebra

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2

a

3 is fed into the machine. The output is fed back into the machine, then the new 5 result is fed back into the machine, and so on for 1000 processes altogether. Describe what happens and predict the final result.

Suppose

b

Repeat a, starting with the fraction

2 .

3

3

If the original fraction is

a , predict the result after one million processes. b

To check your answer, enter f1 | f = a ÷ b. Then, to substitute the output for f, press £ until the output is highlighted and press ·.

© Macmillan Education Australia 2012 ISBN: 978 1 4202 3069 7

NUMBER AND ALGEBRA Algebra

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