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© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

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Ready-Ed

Acknowledgements i. Clip art images have been obtained from Microsoft Design Gallery Live and are used under the terms of the End User License Agreement for Microsoft Word 2000. Please refer to www.microsoft.com/permission. ii. Corel Corporation collection, 1600 Carling Ave., Ottawa, Ontario, Canada K1Z 8R7.

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Publications

Title: Working On Number And Algebra © 2011 Ready-Ed Publications Printed in Australia Author: Mirella Trimboli Illustrator: Rod Jefferson

Copyright Notice

The purchasing educational institution and its staff have the right to make copies of the whole or part of this book, beyond their rights under the Australian Copyright Act 1968 (the Act), provided that: 1.

The number of copies does not exceed the number reasonably required by the educational institution to satisfy its teaching purposes;

2.

Copies are made only by reprographic means (photocopying), not by electronic/digital means, and not stored or transmitted;

3.

Copies are not sold or lent;

4.

Every copy made clearly shows the footnote, ‘Ready-Ed Publications’.

educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under Act. For details of the CAL licence for educational institutions contact: Copyright Agency Limited Level 19, 157 Liverpool Street Sydney NSW 2000 Telephone: (02) 9394 7600 Facsimile: (02) 9394 7601 E-mail: info@copyright.com.au

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

The Act allows a maximum of one chapter or 10% of the pages of this book, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that

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Except as otherwise permitted by this blackline master licence or under the Act (for example, any fair dealing for the purposes of study, research, criticism or review) no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher at the address below.

o c . che e r o t r s super Published by: Ready-Ed Publications PO Box 276 Greenwood WA 6024 www.readyed.com.au info@readyed.com.au

ISBN: 978 1 86397 829 3 2

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Any copying of this book by an educational institution or its staff outside of this blackline master licence may fall within the educational statutory licence under the Act.

Reproduction and Communication by others


Contents Teacher Notes National Curriculum Links

Integers

4 5

30 31 32 33 34 35

6 7 8 9 10 11 12 13 14 15

Using Index Numbers Place Value Revisited Expressions With Indices Prime Factor Trees Easy Calculations Using Prime Factors A Different Approach to the Lowest Common Multiple A Different Approach to the Highest Common Factor Square Numbers Square Root

17 18 19 20 21

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Teachers' Notes Imagining Negative Numbers Part 1 Imagining Negative Numbers Part 2 Where Am I? Integer Addition Integer Subtraction Walking Up and Down the Number Line Which is Larger? Multiplication of Signed Numbers Multiply and Divide

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Scientific Formulas Create Your Own Formula Catching a Taxi Part 1 Catching a Taxi Part 2 Electrician and the Plumber Part 1 Electrician and the Plumber Part 2

Fractions, Decimals and Percentages Teachers' Notes Equivalent Fractions Adding and Subtracting Fractions Multiplying and Dividing Fractions Ratios are Fractions Fractions and Percentages Decimals and Percentages Fractions and Decimals What is my Test Score as a Percentage? What’s the Discount? Best Buy

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22

Linear Equations

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© ReadyEdPubl i cat i ons Indices, Squares and Square Roots f o r evi ew16pur posesonl y• Teachers'• Notes r

Teachers' Notes Equations Versus Expressions One-Step Equations Backtracking Two Step Equations Checking Solutions Solving Real Life Problems 1 Solving Real Life Problems 2

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Calculations and Algebraic Generalisations Teachers' Notes How We Calculate Get the Order Right Calculations With Formulas

26 27 28 29

Answers

36 37 38 39 40 41 42 43 44 45 46

47 48 49 50 51 52 53 54

55 - 58

3


Teachers’ Notes This resource focuses on the Number and Algebra Strand of the Australian Curriculum for students in aged between 11 and 13 years old. Each section provides students with the opportunity to explore a key area of their numerical and algebraic understanding, often with the opportunity to explore their real life contexts or extend their exploration further.

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The section entitled Integers exposes students to working with directed numbers and examines their uses in calculations and their real life applications. Students are encouraged to work on this section using mental skills and may check their solutions with a calculator.

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The section entitled Indices, Squares and Square Roots teaches students the use of index numbers to simplify calculations and expressions. Students may also practice finding the lowest common multiple and the highest common factor using prime factors and index notation. The section Calculations and Algebraic Generalisations focuses on calculating using the correct order of operation (BIMDAS) and the real life use of formulas. Students learn the use of formulas, the ability to substitute into formulas and how to derive their own formula from given information. Fractions, Decimals and Percentages is the next section it encourages students to move fluidly between each of these three representations of numbers. Students will learn a variety of skills to deal with each type of representation mentally and to perform calculations in real life situations.

© ReadyEdPubl i cat i ons Linear Equations isr ther final section. It exposes students to the difference between • f o e v i e w p u r p o s e s o nl y• expressions and equations. Students will learn to solve linear equations using a variety of strategies and also to apply these strategies to real life problems.

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Each section is also prefaced by a Teachers' Notes page, explaining the idea and purpose behind each activity. Included here are methods to extend the activities or modify the activities based on individual student ability.

The majority of activities are scaffolded into two sections: Task A introduces the general skills to be mastered, usually enabling students competence in a given skill or an understanding of the basic number sequence. Task B explores the skill further with a more in-depth investigation or consideration and often extends the concept further.

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Most activities contain a challenge at the bottom of the page. These challenges range from individual challenges, through to research and small group challenges. Each of these are designed to complement the activity page, yet extend the material. They are designed to engage student interest and appreciation for mathematics as well as expose students to the idea that mathematics can be a creative and investigative pursuit. Challenges can be included in the lesson of the day, or used as a stand-alone lesson when time permits. Many can be set as homework or assignment tasks over a longer period of time. Research tasks do tend to include the use of internet resources and it is advisable that computer resources are organized in advance. It is hoped that Working On Number And Algebra will be used to help guide teachers in their teaching strategies and methods of presentation. While some activities are designed to be extra practice for students, many others can be used to present and teach students new concepts. 4


National Curriculum Links Number and Place Value • Investigate index notation and represent whole numbers as products of powers of prime numbers (ACMNA149) • Investigate and use square roots of perfect square numbers (ACMNA150) • Apply the associative, commutative and distributive laws to aid mental and written computation (ACMNA151) • Compare, order, add and subtract integers (ACMNA280)

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Real Numbers • Compare fractions using equivalence. Locate and represent fractions and mixed numerals on a number line (ACMNA152) • Solve problems involving addition and subtraction of fractions, including those with unrelated denominators (ACMNA153) • Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154) • Express one quantity as a fraction of another, with and without the use of digital technologies (ACMNA155) • Round decimals to a specified number of decimal places (ACMNA156) • Connect fractions, decimals and percentages and carry out simple conversions (ACMNA157) • Find percentages of quantities and express one quantity as a percentage of another, with and without digital technologies (ACMNA158) • Recognise and solve problems involving simple ratios (ACMNA173)

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Money and Financial Maths • Investigate and calculate ‘best buys’, with and without digital technologies (ACMNA174)

Patterns and Algebra • Introduce the concept of variables as a way of representing numbers using letters (ACMNA175) • Create algebraic expressions and evaluate them by substituting a given value for each variable (ACMNA176) • Extend and apply the laws and properties of arithmetic to algebraic terms and expressions (ACMNA177)

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Linear and Non-Linear Relationships • Given coordinates, plot points on the Cartesian plane, and find coordinates for a given point (ACMNA178) • Solve simple linear equations (ACMNA179)

5


Teachers’ Notes

Integers   Imagining Negative Numbers

This activity exposes students to the real life uses of negative numbers in two contexts. Students can further extend their understanding by a study of international temperatures, as given in Task C.

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In this activity students have the opportunity to familiarise themselves with the concept of a number line. Students should be able to fluidly explain the position of certain numbers relative to other numbers on the number line. Students can consolidate their understanding with Task C, and can make a game of it.

This is a page of mental maths questions for students to practise their grasp of the addition and subtraction of directed numbers. Students are to be encouraged to attempt this task without using a number line, to help them develop their mental visualisation.

  Which is Larger?

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  Where Am I?

  Walking Up and Down the Number Line

This activity allows students to practise their understanding of ordering and analyzing directed numbers. Students are also encouraged to use the correct inequality symbols. Task C provides students with another internet resource to practise their skills.

© ReadyEdPubl i cat i ons This activity teaches students a method for   Multiplication of Signed •f o rr e i ew u r posesonl y• adding directed numbers. Forv example, -3 p Numbers

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+ 7 means to start at -3 on the number line, turn towards the positive direction and move 7 spaces up the number line to 4. As another example, 4 – (-3) means to start at 4, turn towards the negative direction and then turn again (the second negative reverses the direction) and move 3 spaces up the number line to 7. Task C is an extension activity that students may like to explore using their calculators.

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This activity is an investigative task for students to explore the rules behind multiplying and dividing directed numbers. Task B allows students to summarise their findings and develop a set of rules. Task C is an important task for students to explore to ensure they understand that the same rules apply for division as for multiplication.

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  Integer Addition

o c . che   Integer Subtraction e r o t r s super

Similar to the previous activity, students use the strategy of moving up and down the number line to subtract directed numbers. For example, -5 – 7 means starting at -5, turning towards the negative direction and moving 7 spaces down the number line to -12. Task C is an opportunity for students to test their skills with one of the numerous free maths games available on the internet.

6

  Multiply and Divide

This is another mental maths task for students to test their skills with multiplication and division. This task can be used as an assessment task to test the progress achieved by students to date.


Imagining Negative Numbers

*

  Task a

The hilly town of Siena in Tuscany has a special sort of multi-level shopping centre. Look at the store directory sign right and study it carefully before answering the following questions.

a. What number could you use to represent the level that the Butcher and Bakery are on?

Part 1

Siena Shopping Village Directory

r o e t s Bo r e p o Store u k S Appliances

__________________________________________________

b. What do the negative level numbers represent?

__________________________________________________

c. If you park in Car Park A and travel on the lift to the Medical Centre, how many floors will you pass?

__________________________________________________

d. You leave the Post Office and travel 4 levels down on the lift. Do you arrive at the Laundromat?

__________________________________________________

f. Maria parks in Car Park A, travels up 4 floors, then up 3 more floors, down one floor, up 3 floors and then down 9 floors. Write down all the places that she visited.

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4

Post Office/Newsagent

3

Greengrocer

2

Supermarket

1

© ReadyEdPubl i caButcher/Bakery t i ons •f orr evi ew pur poses onl y• Delicatessen __________________________________________________

e. If you leave the Laundromat and travel up the lift 5 floors, where do you end up?

__________________________________________________ __________________________________________________

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Medical Centre

G

-1

Laundromat

-2

Car Park A

-3

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Level

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__________________________________________________

g. Gianni starts on level G, travels to level -2, then to level 3, followed by level 1, then back to level G. Describe Gianni’s movements on the lift.

__________________________________________________

__________________________________________________

__________________________________________________

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Part 2

      Imagining Negative Numbers

*  Task b

Take a look at Robert’s Savings Account statement for the last few days.

Date

Item

Amount

Total

ATM Withdrawal

-$200

$5800

3/5/11

Salary

$1855

$7355

4/5/11

Electricity

-$124

$7231

5/5/11

Cash Deposit

$450

$7681

6/5/11

Mortgage

-$1250

$6431

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a. What do the negative amounts represent?

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3/5/11

_____________________________________________________________________________

© ReadyEdPubl i cat i ons _____________________________________________________________________________ •f orr evi ew pur posesonl y•

b. How much money did Robert have in his account before the ATM Withdrawal?

c. How are the amounts in the Total column calculated? _____________________________________________________________________________

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d. What is the total amount that Robert spent during these few days?

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_____________________________________________________________________________

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e. How much more or less money does Robert have in his account at the end of the above bank statement?

_____________________________________________________________________________

*  Task c: Personal Challenge

See if you can find out the average maximum and minimum temperatures in January for the following cities in the world: New York, Paris, Sydney, Beijing, Moscow, Johannesburg, Tokyo, Vancouver. Order the cities from lowest minimum to highest minimum.

8


Where Am I?

*  Task a

Use the number line to help you answer each of the following questions.

a. 5 is eight places above

f. -8 is five places above

b. 2 is seven places below

g. 15 is twenty places above

d. 0 is ten places above

i. -12 is four places below

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h. 3 is sixteen place below

e. -7 is thirteen places below

*  Task b

j. -1 is eleven places above

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c. -3 is five places below

You may like to use the number line to answer each of these questions.

a. 12 more than 3 is b. 10 less than 5 is

e. 2 more than -5 is

which is 3 less than

f. 7 less than 2 is

which is 4 more than

g. 9 more than 1 is

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which is 8 less than

h. 50 less than 10 is

which is 6 more than

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i. 86 more than 17 is

j. 37 more than -14 is k. 12 less than -62 is l. 150 less than -210 is

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© ReadyEdPubl i cat i ons d. 12 more than -18 is •f orr evi ew pur posesonl y• c. 6 less than -4 is

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which is 10 less than

which is 15 less than

which is 21 more than

which is 325 more than

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

ask c: class Challenge *  TEach member of the class is to write down one clue, similar to those above, which represents an integer value. Each member of the class will then state their clue to the class and the others in the class will write down the number they were thinking of. Once all class members have given their clue, go through the answers with your teacher.

9


Integer Addition When we add two numbers together, we can think of moving from one number to the other number by moving further up the number line. For example, -2 + 7 means starting at -2 on the number line, moving 7 places up the number line, to the number 5. So -2 + 7 = 5.

*  Task a

Calculate each addition sums.

a. 7 + 12 =

r o e t s Bo r e p ok u S g. 24 + -8 =

b. -20 + 6 =

h. -21 + 32 =

i. -125 + 42 =

d. 25 + -30 =

j. 32 + -18 =

(Hint: rewrite this as -30 + 25). k. 5 + -12 =

e. 18 + -6 =

l. -11 + 16 + -2 + 8 =

f. -8 + 3 =

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c. -100 + 90 =

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

© ReadyEdPubl i cat i ons f or r e w ap ur pos e s nl y• When we see• a calculation that asksv usi toe subtract negative number, what we areo really

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*  Task b

a. 7 – (-4) =

b. (-2) – (-8) =

c. 25 – (-42) = d. -18 – (-11) =

Rewrite sum Answer

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7+4

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Rewrite sum Answer

e. (-20) – (-53) =

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g. -54 – (-36) =

h. (-78) – (-99) =

class Challenge *  TUseaskyourc: calculator to investigate what happens when we calculate the powers of negative numbers. Try each of these and write two sentences about what you have found. (-2)3, (-2)5, (-2)8, (-1)2, (-1)11, (-5)4, (-4)2, (-5)3, (-4)7 10

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doing is adding those two numbers together. A subtraction sign means we change direction, and two subtraction signs means we change direction twice.


Integer Subtraction Subtracting one number from another means starting with the first number and then moving down the number line. For example, 5 – 12 means starting with 5 and then moving down the number line 12 places to the number -7. So 5 – 12 = -7

*  Task a

Calculate each subtraction sum.

a. 6 – 2 =

r o e t s Bo r e p ok u S g. -42 – 16 =

b. -8 – 10 =

h. 82 – 31 =

i. 16 – 12 – 7 – 3 =

d. -23 – 13 =

j. -10 – 3 – 6- 29 =

e. 50 – 24 =

k. 3 – 15 – 24 – 37 =

f. -19 – 7 =

l. 42 – 18 – 21 – 65 =

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c. 5 – 22 =

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

When we add a negative number to another number, really what we are doing is subtracting that number. The addition sign means that you will move up the number line, but the negative right after it means to change direction. So -2 + (-8) means -2 – 8 = -10

*  Task b

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a. 8 + (-6) =

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b. -12 + (-9) = c. 25 + -36 = d. -17 + -21 =

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Rewrite Sum

Answer

2

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Calculate each of the following:

Rewrite Sum

e. 103 + (-92) =

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

Answer

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g. -32 + -65 + -20 =

h. 25 + -82 + -204 =

Challenge *  TAtaskhome,c: Personal or in your spare time, play the following online game and see how many you can get correct in 15 minutes. 4http://tinyurl.com/2f6df8x

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Walking Up and Down the Number Line Use the skills that you have learned to try to work out these 40 calculations mentally.

1. 16 – (-7)

21. 123 – 87

2. 24 – 18

22. -245 – 82

r o e t s B – (-75) r e -3 + 5 24. -412o p ok u -8 – (6) 25. 750 – 245 S

3. 34 – 54 4. 6.

-12 – (-8)

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5.

23. 305 + (-31)

26. – 336 + (-170)

7. 15 + -7

27. – 456 + 210

8. -4 – 29

28. 389 – (-621)

9. 33 + (-8)

29. 1450 + (- 830)

© ReadyEdP30. ub–l i cat i ons 2250 + 832 •f orr evi ew pur posesonl y•

10. -16 + (-3)

31. -545 – 781

12. 32 – (-15)

32. -480 + (-1312)

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11. -25 – 11

33. 764 + (-105)

17. -62 + 31

37. -816 – 620

18. 43 – (-15)

38. -38 + 24 – (-41)

19. 86 + (-73)

39. -72 + (-60) – 54

20. -31 – 46

40. 103 – 42 + 23 – (-16)

13. 42 + 17

34. 580 + 982 . t o 15. -21 –e (-14) 35. -2456 – 803 c . c e 16. 76 – (-15) h 36. 975 +r er o t s (-626) super 14. -15 – 24

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Which is Larger?

*  Task a

Insert the correct symbol, < or >, to make each statement true.

a. 23

14

d. 6

-14

b. -10

5

e. 7

9

c. -8

-11

f. -21

-15

Write the numbers in ascending order.

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*  Task b

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a. -23, 31, -18, 1, 0, -5, 7, -40, 8, -9, 12, 20, -21

_ _____________________________________________________________________________

b. 102, -98, 45, 67, -32, 81, -79, -135, 5, -7, -116, 21

_ _____________________________________________________________________________

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• d. 243, -108, -356, 428, 910, -672, 154, -487, 590, -462, 392 c. 12, -18, 63, -42, 10, -8, -62, 54, -75, 84, -25, 43

_ _____________________________________________________________________________

_ _____________________________________________________________________________

a. 24 – 12

b. 8 – 5

-11 – (-8)

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e. -15 + 19

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Perform each calculation below and then insert the correct symbol, <, >, = to make each statement true.

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*  Task c

30 + (-26)

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-12 – (-9)

f. -5 – (-38)

45 – 21

c. 6 – 18

-20 + 9

g. 17 – 39

-24 – (-18)

d. 32 – 47

-25 + (-3)

h. 12 – (-4)

48 + (-32)

Personal challenge *  TAsaska fund:method of revision try out these online integer games http://www.onlinemathlearning.com/integergames.html. Try playing side by side with a friend or record each of your personal best scores. To test whether you really understand all areas of this topic, try the MathCar Racing. 13


Multiplication of Signed Numbers When we multiply two numbers together, what we are really doing is adding a number together many times. When we say 5×3 we are really saying “add 3 five times” or 3+3+3+3+3 = 15. This works in the same way for multiplying positive and negative numbers too.

*  Task a

Complete the following:

Subtract 4 three times.

-4 -4 -4

Subtract (-5) three times.

–(-5) –(-5) –(-5)

c. 8 × (-4) =

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e. -5 × 7 = f. -4 × 6 =

g. -3 × (-5) = h. -3 × (-9) = i. -4 × (-7) =

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-2 + -2 + -2 + -2

b. 2 × (-6) =

d. -3 × 4 =

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Add (-2) four times.

a. 4 × (-2) =

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

We can summarise what we have discovered in Task A and we can write down a set of rules that will help us multiply positive and negative numbers.

*  Task b

In each space below, use the word “positive” or “negative” to complete the statement.

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a. A positive number multiplied by another positive number gives a ___________ answer.

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b. A positive number multiplied by a _____________ number gives a negative answer. c. A negative number multiplied by a positive number gives a _________ answer.

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d. A negative number multiplied by a __________ number gives a positive answer.

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e. If we multiply three negative numbers together we will get a ___________ answer. f. If we multiply six negative numbers together we will get a __________ answer.

g. If there are an even number of negative numbers being multiplied together we will get a _____________ answer. h. If there are an odd number of negative numbers being multiplied together we will get a _____________ answer.

c: Personal Challenge *  DoTask the rules that we have found in Part B work for division? Use your calculator to investigate at least ten different division calculations, involving a mixture of positive and negative numbers. Compare your findings with your partner. 14


Multiply and Divide You now know the rules for multiplying positive and negative numbers and from your investigation you know that dividing positive and negative numbers follow exactly the same rules. Try out your skills on the mental calculations below.

1. 3×4

r o e t s Bo r 100÷5 23. (-2) e p ok u 24. -7×9 -200÷(-4) S

2. -2×10 3.

22. -3×-3×-3 4

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4.

21. 500÷20

25. -360÷(-30)

6. 20×(-2)

26. 550÷(-11)

7. 24÷(-3)

27. -12×(-2)

8. -52÷4

28. -5×4

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5. -7×(-3)

© ReadyEdPub l i c at i ons 29. -32×3 10. 30. 400÷(-50) •-3×11 f orr evi ew pur po se sonl y• 9. (-9)2

12. -60÷(-12)

32. -3×4×(-5)

13. 5×(-15)

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33. -42÷(-7)÷(-2)

14. -6×(-120)

34. (-3)2×(-2)3

17. 480÷(-80)

37. 5×(-10)×(-6)

18. -9×(-12)

38. -500÷25×(-2)

19. -6×(-8)

39. 4×(-12)÷(-6)

20. -32÷(-8)

40. -8×5÷(-10)

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31. -36÷4×(-2)

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11. 248÷(-3)

o c . 35. 10×(-3)×(-4) 15. 8×(-9) c e her r o t s super 36. -200÷(-10)÷5 16. -7×6

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Teachers’ Notes

Indices, Squares and Square Roots   Using Index Numbers

r o e t s Bo r e p ok u S

  Place Value Revisited

Students are encouraged to explore how place value can be expressed using index notation. This task will help students progress to using and understanding scientific notation. Task C allows students to explore how the powers of 10 are used in real life to explore the scale of our universe. This might be a nice activity to explore as a class.

  A Different Approach to the Lowest Common Multiple

Many students determine the LCM for two or more numbers by the more inefficient method of listing numbers. Here students are shown how Prime Factor Trees can make the task easier and more efficient.

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This activity explores how index numbers can be used to simplify numerical expressions. Task A asks students to simplify expressions while Task B asks students to expand simplified expressions. Task C can be used to allow more capable students further exploration of the index laws.

prime factors. By expressing two numbers in a multiplication calculation by their prime factorisation, the calculation can be made easier and more efficient.

  A Different Approach to the Highest Common Factor

Many students determine the HCF for two or more numbers by the more inefficient method of listing numbers. Here students are shown how Prime Factor Trees can make the task easier and more efficient.

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

  Expressions with Indices

  Square Numbers

This task allows students to revisit their use and understanding of prime factor trees. This activity extends this understanding further by asking students to simplify their answers using index notation.

In this activity students explore the inverse of the squaring operation, namely the square root. Students are encouraged to estimate values and also employ technology to calculate values to an appropriate number of decimal places.

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This activity serves as a refresher exercise for looking at square numbers. Task A is a simple recall task while Task B allows students to explore the x2 function on their calculator. Task C is slightly more challenging, allowing students to explore a pattern involved in squaring a particular type of number. Discussion of their findings with the class is encouraged.

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This activity extends using index notation in a more generalised context, using variables to represent numbers. Task B is more advanced and will demonstrate whether students have a strong grasp of these concepts. Task C allows students to use their creativity while also demonstrating their success in consolidating these core concepts.

o c . ce e   Square Root r  Prime Factor Trees h o r st super

  Easy Calculations Using Prime Factors

This activity exposes students to the power of expressing numbers in terms of their 16


Using Index Numbers Instead of writing out long calculations, we can sometimes use index numbers or powers to write a shorter expression.

*

  Task a

For each of the following expressions, write a shorter, simplified expression. Questions a and c has been partially completed for you.

r o e t s Bo r e p ok u S =

b. 5×5×6×5×5×7×7×6×5

=

c.

d.

24 x

e.

7x7x7x2x2x3x7x2 3x3x2x7x7x3

=

f. 3x2x3x3x4x2 x 4x2x3x2x4 =

3x3x2x3x2x4x2 2x2x3x4

=

10x10x4x10x4x4x6x6 10x4x4x6x10

32 x

g. 6x4x2x6x6x4x4x2 x 43x64x25 =

ew i ev Pr

Teac he r

a. 2×2×3×2×3×3×2

= 5 5 x3 2 x4 7

h. 3x3x4x4x4x3x5x5x5x5 =

© ReadyEdPubl i cat i ons b Write each of the following expressions (which are in index form) in expanded form. In other f o rwrite r e vi w p ur po s ewere so nl y• *  Task• words, them ase they would appear before they are simplified.

e

w ww

c

43 x 52 x 67

. te

36 x 103 x 152

b

d

(-3)4 x 122 x 73

m . u

a

(23 x 62)2

o c . che e r o t r s super

10 2 x 154

(72 x 45 x 33)4

f

(42 x 113)2 (3 4 x 72) 3

Challenge *  TInaskthe c:workResearch that you have done in Task A and Task B, you have discovered a few of what we call the Index Laws. In small groups, research as many Index Laws as you can find. Create a poster showing all these Index Laws and make sure that you include some examples to show how each one works. 17


Place Value Revisited

*  Task a

The powers of 10 are important for everyday calculations and our understanding of size and perspective in our universe. We can express any number and its place value using the powers of 10. Fill in the missing spaces in the table below. A few have been done for you. 1010

100 1 000 000 000

108

0.1

r o e t s Bo r e p ok u S 0.0001

10-5

102

0.00000001

101

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

  Task b

Express each of the following numbers using place value and the powers of 10. The first one has been done for you. is the same as

b. 10 781

is the same as

c. 3 457 682

is the same as

d. 18.25

is the same as

w ww

a. 2375

. te

2 x 103 + 3 x 102 + 7 x 101 + 5 x 100

m . u

*

ew i ev Pr

Teac he r

105

o c . che e r o t r s super

e. 0.00768

is the same as

f. 2 034.74

is the same as

g. 5.006032

is the same as

*  Task c: Internet Challenge

• To gain a real perspective of how large and small powers of 10 are in our everyday life, check out this site and be patient while the java applet loads. It will take you on a tour of our universe in relation to the powers of 10. http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/ • While you are using the internet, see if you can find out what powers of 10 each of the following words represent: Milli, Tera, Zepto, Nano, Zetta, Googol.

18


Expressions With Indices We can work with algebraic expressions that involve indices in the same way that we work with numerical expressions involving indices. We can also work with expressions that involve both. For example, we know that 32 × 45 × 43 × 36 simplifies to 38 × 48. In the same way we can say that m4 × n4 × m2 × n7 simplifies to m5 × n11.

*  Task a

Simplify each expression below.

r o e t s Bo r e p ok u S

a. 7 6 × 4 2 × 4 8 × 7 2

d. d 3 × p 2 × t × d × p 6 × t 4

__________________________________

e. 5 2 × z 3 × 4 2 × z 7 × 4 3 × 5 9

__________________________________

f. 2 × f 3 × g 4 × g 5 × 3 × f 8

__________________________________

*  Task b

a

__________________________________

Try simplifying each expression below which involve multiplication, division and brackets. Be sure to show all your working.

eu © ReadyEdP bl i cat i ons 7 × 10 (3 vs ) ×e (vs × 3o )n orr evi ew pur p×o l y• 10 ו 7 f 4

6

2

4

3

b

4 5

2

5 3

f 62 × g4 × g7 × 63 62 × g5

(5 × 3 ) 7

4 2

w ww

c

__________________________________

c. 3 5 × q 4 × 3 2 × q 5

ew i ev Pr

Teac he r

b. a 8 × b 2 × a 3 × b 5

__________________________________

m ×n n2 × m5 3

7

d

. te

g

m . u

o c . che e r o t r s super (8 4 × h 2) 5 83 × h4

h

(p 7 × b 4) 5

(b 3 × q 2 × y 3) 4 (q 5 × b 3 × y) 2

ask c: Partner Challenge *  TCreate a ten question quiz with questions similar to those above. Be sure to create a variety of questions, some more difficult than others. On a separate sheet of paper create the marking key for your quiz, including the steps to work out the answers and the number of marks you would give for each question. Ask another member of the class to try your quiz, or perhaps your teacher might like to randomly hand out each quiz to students in the class. 19


Prime Factor Trees Do you remember constructing a prime factor tree? Let’s look at an example below. Example:

48 4

Choose any two numbers that multiply to give you 4.

x

2x2

12

Choose any two numbers that multiply to give you 48.

3x4

Choose any two numbers that multiply to give you 12.

r o e t s Bo r e p ok u S2 x 2 = 2 x 3 48 = 2 x 2 x 3 x 2x2

b.

d.

e.

Here we have expressed 48 as a product of its prime factors in the SIMPLEST form.

ew i ev Pr

Teac he r

4

The branches stop once we have a prime number.

Using the above example as a guide, express each of the following numbers as a product of their prime factors in the simplest form. a.

36

50

c.

72

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

w ww

. te

g.

20

98

250

f.

320

m . u

120

o c . che e r o t r s su r e p 1220 135 h.

i.


Easy Calculations Using Prime Factors To make the following calculations below easier, break up each calculation into the following steps: 1. Write each number as a product of its prime factors (you might like to use a prime factor tree to do this). 2. Simplify the overall product of the prime factors. The first one has been done for you. a.

32 x 10

32 4

x

10 x

2

10 = 5 x 2

32 = 25 32 x 10 = 25 x 5 x 2

d.

24 x 33

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

w ww

e.

81 x 27

m . u

c.

= 26 x 5

ew i ev Pr

2x2

54 x 16

r o e t s Bo r e p ok u S 5

2x4

Teac he r

2x2

8

b.

. te o 100 x 50 90 . xc 48 che e r o r st super f.

21


A Different Approach to the Lowest Common Multiple When we are trying to find the Lowest Common Multiple (LCM) of two or more numbers we can use the prime factors of those numbers to help us find this multiple quickly. Let’s say we want to find the LCM of the numbers 8 and 10. First we can express each number as a product of its prime factors.

8

Example:

2

10 4

2

5

8:

2 x 2 x 2

10:

2 x 5

r o e t s Bo r e p ok u S 2 2

8= 2x2x2

10 = 2 x 5

LCM = 2 x 2 x 2

x 5 = 40

Teac he r

ew i ev Pr

For the LCM we need to line up these prime factors in columns, as shown in the diagram. We then take the numbers from each column and multiply these together. As you can see we don’t count the 2s in the first column twice, just once. So the LCM for 8 and 10 is 40.

Use this method to find the LCM for each of the following. You may want to use prime factor trees to help you. a.

6 and 10

b.

8 and 12

c.

9 and 10

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

4 and 15

w ww

. te

g.

22

10 and 12 and 8

e.

6 and 4

f.

5 and 9

m . u

d.

o c . che e r o t r s 6 and 12 and 4 super 3 and 4 and 9 h.

i.


A Different Approach to the Highest Common Factor When we are trying to find the Highest Common Factor (HCF) of two or more numbers we can use the prime factors of those numbers to help us find this factor quickly. Let’s say we want to find the HCF of 12 and 8. We begin by finding the prime factors of each number and then we examine them to find the HCF.

12

Example:

3

8 4

2

4

12:

3 x 2 x 2

8:

2 x 2 x 2

r o e t s Bo r e p ok u S 2 2

HCF = 2 x 2 = 4

ew i ev Pr

Teac he r

Once we have our prime factors we then circle what they have in common, one number at a time. As we can see, 12 and 8 have a two and then another two in common. We then multiply together the numbers that are in common. So we get a HCF of 4.

Use this method to find the HCF for each of the following. You may want to use prime factor trees to help you. a.

24 and 18

b.

48 and 36

c.

50 and 120

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• 52 and 45

w ww

. te

g.

24 and 46

e.

40 and 84

f.

32 and 46

m . u

d.

o c . che e r o t r s su 54p ande 72r 36 and 42 h.

i.

23


Square Numbers

*  Task a

A square number is a number calculated by multiplying any number by itself. For example, 100×100 = 10 000. So 10 000 is a square number. Use your calculator to write down the first 20 square numbers in the space below.

r o e t s Bo r e p ok u S

*

  Task B

Use the x2 button on your calculator to square the numbers below.

a. 2.3

c. 8.42

e. 6.82

ew i ev Pr

Teac he r

1000

© Read yEdPubl i cf. at i ons b. 10.5 d. 3.17 22.3 •f orr evi e w pur poses_____________ onl y• _____________ _____________ _____________

_____________

*

a. 1 2 = 1

_____________

Use your calculator to square the numbers below. The first two have been done for you.

w ww

  Task c

b. 11 2 = 121

. te

c. 111 2 = _______

m . u

d. 1111 2 = _______

o c . che e r o t r s super

Describe the pattern and how it works.

_____________________________________________________________________________

*  Task d

Use your pattern to calculate the following, without a calculator:

a. 111111 2 = _____________________

b. 11111112 = _____________________

Will this pattern work forever? Explain your answer. _____________________________________________________________________________

24


Square Root When we want to calculate the square root of a number, we are asking ourselves what other number, when multiplied by itself, will give this number. For example if we want to find the square root of 49, or √49 , we ask ourselves, what number, when multiplied by itself, gives 49? The answer is 7×7 = 49.

*  Task a

r o e t s Bo r e p ok u √ √ S

Without using a calculator, calculate each of the following:

_____________

Teac he r

√ 16

b. √ 100

_____________

c.

25

_____________

d. √ 1

_____________

e.

169

ew i ev Pr

a.

_____________

f.

√400

_____________

Without your the approximate value ofo each ofs the following: Rusing ea dcalculator, yEestimate dP u bl i ca t i n *  Task B © √ 110p √n orr evi e ur posese. o y• c. w 80 l a. √ 10 •f b. √ 40

_____________

d. √ 147

w ww

_____________

. t   Task c e *

_____________

_____________

f.

√28

_____________

m . u

_____________

o c . che e r o t r s s r u e p √ √

Use your calculator to calculate the square root of the numbers below. Round each answer to two decimal places.

a.

√ 11

_____________

b. √ 30

_____________

c.

52

_____________

d. √ 95

_____________

e.

180

_____________

f.

√205

_____________

25


Teachers’ Notes

Calculations and Algebraic Generalisations

r o e t s Bo r e p ok u S

  How We Calculate

Teac he r

  Get the Order Right

This activity builds on the previous activity and allows students to further practice using the correct order of operations. Task B will help students understand the benefit of showing all their working out. Students are to examine these “solutions” for errors and to then correct the errors that they see. Students will have achieved excellent consolidation with their success in this task.

In this activity students are asked to read information and transform it into a formula. Many of these formulas appear in real life and contexts which students can understand. Translating real life mathematical processes into mathematical symbols is an important higher order skill.

  Catching a Taxi

ew i ev Pr

This activity exposes students to the correct use of BIMDAS in conducting calculations. Students are very strongly encouraged to show all their working out to ensure that they can demonstrate the method clearly.

  Create Your Own Formula

The first activity allows students to explore the visual representation of a commonly used function and exposes students to the fact that functions are represented by an equation, a table of values and a graph. This activity makes a strong introduction to future function work.

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

w ww

. te

Similar to the “Catching a Taxi” task, this activity provides students with another readily accessible example of the various representation of functions.

m . u

  Calculations with Formulas

This task shows students the real life use of formulas. Task A teaches students how to substitute into formulas and calculate correctly using BIMDAS. Task B exposes students to a variety of different formulas. Task C is a research task where students can explore various financial formulas.

   Electrician and the Plumber

o c . ce e  Scientific Formulas h r o t r s super

This activity builds on the skills learned in the previous task. Students will use and substitute into a variety of real scientific formulas which will complement the work that they undertake in the science curriculum. Task C extends these concepts by asking students to informally solve equations. This concept is visited in detail in the final section of the book.

26


How We Calculate When we have a few calculations to perform, all in the same question, how do we know which ones to do first? We follow the mathematical rules of BIMDAS. For example, if we want to calculate -10÷5×3+(7-4) 2 , we follow the rules of BIMDAS as shown below.

r o e t s Bo r e p ok u S

= -10÷5×3+(3) 2 = -10÷5×3+9 = -2×3+9 = -6+9 =3

Inside the brackets first.

Use the power, calculate 32.

Working left to right, we divide first. Multiply next.

Calculate last.

Remember: When there is a string of addition and subtraction or a string of multiplication and division, we simply calculate from left to right.

ew i ev Pr

Teac he r

Brackets Indices (powers) Multiplication Division Addition Subtraction

Calculate each sum below using the laws of BIMDAS. Set out your working as shown above. a. 2- (5-2)3

b. -7 + 2 × (-4) -5

c. 23 – 10 ÷ 5

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

w ww

. te

e. 100 ÷ 20 × -3 - 52

f. 10 – 3 + 4 - 2 × (-12)

m . u

d. 14×(-2)÷7-8×3

o c . che e r o t r s super

g. (4×5-13)2 + 3 × (-2)

h. 12 - 24÷(-3)×2

i. 5×(-6)÷15 – 7- 3 + 12

27


Get the Order Right

*  Task a

Now that you’ve learnt the rules of BIMDAS, let’s try some more difficult examples. Calculate each sum below following the rules of BIMDAS, and be sure to show all your working out.

Remember: When there is a string of addition and subtraction or a string of multiplication and division, we simply calculate from left to right.

r o e t s Bo r e p ok u S

a. (-2 + 5) × -3 × -2

d. -24 ÷ (-6) + 3 × (-4)

b. 12 – ((-2)2 × -3 + 5)

c. -3 + -7 × 2 - 8

e. -25 – (24 – 2 × 3)

f. -30 -5 × (-3) + -2

ew i ev Pr

Teac he r

Brackets Indices (powers) Multiplication Division Addition Subtraction

w ww

  Task b * In each calculation below, a student has made at least one error.

m . u

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

Examine each problem carefully, circle the error and then recalculate the sum, this time correctly!

. te

a. -3 + 2 × (-5) + 7

= -1 × (-5) + 7

= 5 + 7

= 12

o c . che e r o t r s super b. 2 4 ÷ (-2) × 3 – 100

= -24 ÷ -6 – 100

= -4 – 100

= -104

c. (-4×2 -12) 2

d. (-20 – 15 +7) × (-2) – 25

= (-8 – 12) 2

= (-20 – 22) × (-2) -25

= (-8) 2 - 12 2

=-42 × -2 – 25

= 64 – 144

= -84 – 25

= -75

= -59

28


Calculations With Formulas In everyday life we use formulas and equations to calculate quantities. Even when we use these formulas, the rules of BIMDAS must still be used. The formula used to calculate the surface area of a cylinder is A = 2∏r 2 + 2∏rh where r is the radius and h is the height.

*

  Task a

Calculate the surface area of the cylinders with the specified dimensions showing all working. You may use your calculator to find the final answer.

a. r = 2 cm

r o e t s Bo r e p ok u S h = 10 cm

b. r = 5 mm

= 25.13 + 125.66

= 150.79 cm2

Teac he r

A = (2 x ∏ x 22) + (2 x ∏ x 2 x 10)

c. r = 25 cm

h = 8 cm

d. r = 50 mm

ew i ev Pr

h = 20 mm

h = 12 cm

© ReadyEdPubl i cat i ons o rr e vi eeach wformula pu r p ose sworking. onl y• Calculate the result for below, showing all your   Task• bf *

W = 5 cm

A = 2(l x w + w x h + h x l)

w ww

b. a = 3

H = 4 cm

c=

. te

b = 4

√ a 2 + b2

m . u

a. L = 2 cm

o c . che e r o t r s super

c. a = 100 b = 4 a – 2b P= bc

c = 3

Challenge *  SeeTaskif youc: canResearch find the formulas for the financial concepts below. Write down the formula you find and show with examples, how you can use this formula. Profit Simple Interest Compound Interest

29


Scientific Formulas Let’s calculate some real life quantities using formulas that we see everyday in science. If we want to measure the weight of an object in Newtons (a unit used in physics) we use this formula: Weight = mass (kg) x acceleration due to gravity (9.8 m/s2) = m x g For example, if your mass is 52 kg, then your weight, in Newtons (N) would be W=52×9.8=509.6 N

*  Task a

Calculate the weight in Newtons for each of these objects:

r o e t s Bo r e p ok u S

a. A cat with mass 4.5 kg.

c. An adult with mass 65 kg.

______________________________

______________________________

Teac he r

d. A vehicle with mass 1.5 tonnes.

______________________________

ew i ev Pr

b. A fish with mass 2.8 kg.

______________________________

If we want to measure how fast something is travelling, (its speed), then we use this formula:

Speed =

For example, if you walk 5 km in 2 hours then your speed in km/h would be:

S=

Distance D = Time T

5 = 2.5 km/h 2

© ReadyEdPubl i cat i ons * a. A car that travels inv 2 hours. b. r A runner who runs 100 in 3 • minutes. • f o150 rkm r e i ew pu p os es o nml y

Calculate the speed of these objects:

__________________________

a.

b.

__________________________

Use these scientific formulas to calculate the unknown quantity:

w ww

*  Task c

Density =

Mass (grams) m = Volume (cm3) v

. te

m = 500 g v = 40 cm3

m . u

  Task b

o c . che e r o t r s super

Electric Power = Voltage x Current

V x I V = 12

I = 10

Personal Challenge *  UseTaskthed:formulas on this worksheet to answer the questions. i. If a car is travelling at 80 km/h and has travelled for 90 minutes, how far has it travelled? ii. If the weight of a person is 833 N, what is their mass in kg? iii. If you cycle at a speed of 415 m/min and you cycle 2 km, how long does it take? 30


Create Your Own Formula In each situation below you must create a formula and use it to perform a calculation. a. Area of a rectangle with a length of 20 cm and a width of 5 cm.

e. The cost of travelling 45 km in a taxi that charges $1.10/km with an initial fee of $2.20.

r o e t s Bo r e p ok u S

Formula:_ ________________________

Formula:_ ________________________

b.

f.

The cost of buying 1.8 kg of bananas when they cost $3.50/kg. Formula:_ ________________________

The number of litres of petrol you bought if you spend $57 and the price of petrol is $1.35/L. Formula:_ ________________________

© ReadyEdPu bl i cat i ons Calculation:_______________________ •f orr evi ew pur posesonl y• g.

Converting your age to minutes if you are exactly 12 years old.

w ww

Formula:_ ________________________

. te

The height of a rectangular garden bed if the length and width are both 1.2 m and the volume of soil in the bed is 1.152m3.

m . u

Calculation:_______________________

c.

Calculation:_______________________

ew i ev Pr

Teac he r

Calculation:_______________________

Formula:_ ________________________

o c . che e r o t r s super

Calculation:_______________________

d. The time taken to wrap 10 presents if you can wrap presents at a speed of 4 presents/hour. Formula:_ ________________________ Calculation:_______________________

Calculation:_______________________

h. The number of hours a plumber has worked at your house if he charged you a total of $430 and he charged $95/hour with a call-out fee of $50. Formula:_ ________________________ Calculation:_______________________ 31


      Catching a Taxi

Part 1

  Task a * When you catch a taxi you have to pay a flagfall amount and then an amount for each kilometre you travel. The

flagfall is similar to a booking fee. When travelling in a taxi on a weekday the flagfall amount is $2.20 and then you pay $0.30 per kilometre you travel. The rule for this can be expressed as: Cost (C) = 0.30 x D + 2.20

r o e t s Bo r e p ok u S

1. Explain how you would use this rule to calculate how much your taxi ride costs.

* Use the rule to fill in the table below.

Distance (D)

10 km

20 km

30 km

40 km

50 km

60 km

Cost (C)

ew i ev Pr

Teac he r

____________________________________________________________________________

70 km

80 km

2. If you plot these values what do you think the shape of the graph would look like?

© ReadyEdPubl i cat i ons *Plot your table ofr values ons the graph below. •f orr evi ew pu po e so nl y• PLot the points

____________________________________________________________________________

w ww

m . u

35 30

Cost (C)

25 20

. te

15

o c . che e r o t r s super

10 5

10

20

30

40

50 Distance (D)

32

60

70

80

90


      Catching a Taxi

*  Task B

Part 2

On a Saturday night the cost of catching a taxi can be modeled by this rule: Cost (C) = 0.40 x D + 3.80

r o e t s Bo r e p ok u S * Create a table of values, similar to the one in Part A, to show how much it costs to travel different distances. 10 km

Cost (C)

20 km

30 km

40 km

50 km

1. How much would it cost to travel 55 km on a Saturday night?

60 km

70 km

80 km

ew i ev Pr

Teac he r

Distance (D)

____________________________________________________________________________

© ReadyEdPubl i cat i ons ____________________________________________________________________________ •f orr evi ew pur posesonl y• 2. How much would it cost to travel 55 km on a weekday?

m . u

3. If you only had $22, how far could you travel on a weekday and how far could you travel on a Saturday night?

w ww

____________________________________________________________________________

*  Task c . t

e

o c . che e Cost (C) = 0.25 x D + 4120 r o r st super

1. Create your own table of values for this taxi service whose costs on a weekday are given by : ____________________________________________________________________________ 2. Which taxi service is cheaper? Explain. ____________________________________________________________________________ ____________________________________________________________________________

33


Part 1

      Electrician and the Plumber

*  Task a

You have hired an electrician to come to your house and install some extra power points and network cables. She says to you, “I charge a call out fee of $70 and then I charge $55 for each half hour of work that I do.”

1. Describe what you think is meant by a “call out fee”. ____________________________________________________________________________

r o e t s Bo r e p ok u S

2. Circle which of these two rules you think would calculate the total cost of the work completed by the electrician. Explain why you chose your answer. Explanation:_ _____________________

Cost (C) = 70 x Number of Half Hours + 55

Rule 2

_______________________________

Cost (C) = 55 x Number of Half Hours + 70

ew i ev Pr

Teac he r

Rule 1

_______________________________

* Use the rule you chose to complete the table of values below.

Number of Half Hours (N)

0.5

1

1.5

2

2.5

3

3.5

4

235 © ReadyEdPubl i cat i ons *Plot your table ofr values ons the graph below. •f orr evi ew pu po e so nl y• PLot the points Cost (C)

w ww

m . u

400 350

. te

300

Cost (C)

250 200 150

o c . che e r o t r s super

100 50

0.5

1

1.5

2

2.5

3

Number of Half Hours (N) 34

3.5

4

4.5


Part 2

      Electrician and the Plumber

*  Task B

You have also called a plumber to come to your house to do the plumbing for a new bathroom. He says to you, “I charge a call out fee of $80 and I charge $50 for each half hour of work I do.

r o e t s Bo r e p ok u S

Teac he r

Rule:

Number of Half Hours (N)

*Use your rule to fill in this table of values. 0.5

1

1.5

2

2.5

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1. Write a rule, similar to the one you chose for the electrician, to describe the cost of hiring the plumber.

3

3.5

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Cost (C)

PLot the points

4

*Plot your values on the graph in Task A (page 34).

m . u

w ww

2. Over the long term, who is cheaper, the plumber or the electrician? Explain how you chose your answer.

____________________________________________________________________________

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o c . che e r o t r s super

____________________________________________________________________________ 3. For how many hours of work do the electrician and plumber charge exactly the same amount? ____________________________________________________________________________ 4. Another plumber charges for his work according to the rule: C = 60 x N. Which plumber is cheaper? Explain your answer. ____________________________________________________________________________ ____________________________________________________________________________ 35


Teachers’ Notes

Fractions, Decimals and Percentages   Equivalent Fractions

Simplifying fractions is a core skill and this activity serves as extra practice for students. Task B explores equivalent fractions to ensure students understand that there are infinitely many representations of the one fraction.

r o e t s Bo r e p ok u S

This activity serves as another task for students to practise their skills with their fractions. All questions contain fractions with different denominators and many involve mixed numbers. The questions develop in difficulty and students are encouraged to show their working out.

  Multiplying and Dividing Fractions

  Decimals and Percentages

This is the second in a series of tasks asking students to move fluidly between fractions, decimals and percentages. Task C allows students to explore the use of mathematics in health and the BMI. It also allows students an opportunity to familiarise themselves with the nutritional information contained on food packaging.

ew i ev Pr

  Adding and Subtracting Fractions

Teac he r

when wanting to compare two or more quantities.

  Fractions and Decimals

This is the third in a series of tasks asking students to move fluidly between fractions, decimals and percentages. Task C is a challenging investigation, exploring the concept of recurring decimals and their representation as fractions. More able students can work on this task while other students practice their short division skills.

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

w ww

 Ratios are Fractions

This activity teaches students that ratios are another representation of fractions and are often favoured over fractions in real life situations. Task B allows students to see that ratios can be simplified in the same way that we simplify fractions. Task C is a research activity that enables students to explore the concept of odds and ratios.

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This question is often asked by students when they receive their test score back. This activity begins with showing students how they can record their test score as a percentage. Task B exposes students to a variety of other quantities that can be expressed as a percentage.

o c . che e r o t r s super

  Fractions and Percentages

This activity is the first in a series of activities that teach students to move fluidly between fractions, decimals and percentages. In each activity students are asked to convert one to the other. Task C informs students of the fact that often percentages are more useful than fractions

36

  What is my Test Score as a Percentage?

m . u

Task A teaches students the simple ideas behind multiplying two fractions. Task B then teaches students the method behind dividing two fractions. The key to Task C, as with Tasks A and B, is to simplify the fractions and expressions before performing the calculations.

  What’s the Discount?

This activity shows students mental strategies that they can employ to calculate the discount of certain sale items.

  Best Buy

This activity allows students to explore a common task that we all engage in when doing the weekly shopping. A common use of rates and ratios; exploring which item is the best buy is a real life application for students to consider.


Equivalent Fractions

*  Task a

Simplify each fraction to make it into a smaller, equivalent fraction.

a.

b. 45 ÷ 5 60 ÷ 5

=

9 ÷ 3 12 ÷ 3

d.

=

3 4

c. 8 24

16 36

r o e t s Bo r e p ok u S e.

55 75

g.

110 250

h.

36 48

ew i ev Pr

Teac he r

21 28

f.

i. 54 90

32 46

For every fraction youy can think of, there are infinitely many equivalent fractions that you © R e a d E d P u b l i c a t i o n s can make. Fill in the spaces to make the equivalent fractions. * •f orr evi e w pur poses onl y• b. c. a.

  Task b

=

16

40

=

20

50

80

=

100

w ww

d.

50

=

=

3

10

=

6 9

=

7

g.

=

f.

6

=

=

3

=

=

60

=

2 12

15 40

70

=

4

25 = 75 100

=

35

=

=

o c . che e r o t r 5 s supe 10r 22 12 15

24

h.

1

2

60

e.

. te

4 20

=

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4 5

=

21 30

=

200

i.

11

=

33

=

55

=

15

=

=

40

=

16 60

ask c: Partner Challenge *  TCreate a task of ten questions, similar to those in Task B, for your partner to answer. Make sure you work out the answers before giving the worksheet to your partner!

37


Adding and Subtracting Fractions Perform each calculation and be sure to show all your working. Leave your answers as mixed numbers in the most simplified form.

b.

1 = 3

+

e.

f.

h.

i.

j.

38

+

7 1 − = 10 4

m.

n.

7 1 − = 8 7

3 5

4 = 7

2 89 + 3 45

=

2 13 =

3 45 2 15

4 = 5

o. 1 l 4a © R e a d y E d P u i c t i ons + 1b − = 3 5 •f orr evi ew pur posesonl y•

1 12

1 13 =

w ww

g.

5 = 6

+

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d.

3 8

4 7

r o e t s Bo r e p ok u S 1 +3 l.

4 1 − = 5 2

Teac he r

c.

1 4

k.

+

3 = 30

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15 1 − = 20 5

3 2 12

4 20 30

+

1 13 =

1 16 =

p.

q.

2 12 −

7 = 8

3 15 + 2 34

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a.

=

o c . che e 4 − r o r st super r.

s.

t.

1 12

5 = 6

5 14 − 2 23

=

10 25 − 4 79

=


Multiplying and Dividing Fractions Multiplying two fractions together is easier than adding two fractions together! All you need to do is multiply the numerators together and multiply the denominators together. Then just simplify your answer. 2 ×3 6 2 2 3 = = For example, if we want to multiply and we can work out the answer like this: 3 × 5 15 5 3 5

*  Task a

Calculate each of the following:

1 4

2 = 5

×

5 8

3 11

×

4 = 6

5 10

×

i.

2 12 × 3 57

=

4 = 7

f.

h.

g.

1 1 45 × 1 23

3 4

e.

2 = 9

×

2 = 6

ew i ev Pr

Teac he r

d.

6 8

r o e t s Bo × r × e p ok u S c.

b.

a.

5 = 6

4 16 × 2 34

=

=

© ReadyEdPubl i cat i ons 1 = 4 ×2 = 8 = 13 ×l •f orr evi ew pur poseso45n 2y 5 ו 1 5 5

Dividing two fractions is easy! We simply flip the second and multiply. 4 1 For example, if we want to divide by we can work out the answer like this: 5 2

  Task b * a. 2 = 3

3 4

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d.

5 11

÷

1 = 3

÷

1 = 6

5 9

1 13 ÷ 14

2 = 7

o c . che e r o t r s super f.

e.

7 8

÷

1 = 4

10 12

2 58 ÷ 1 12

=

÷

2 = 5

i.

h.

g.

÷

m . u

÷

c.

b.

w ww

4 5

Calculate each of the following:

10 35 ÷ 2 15

=

=

Personal Challenge *  TUseaskyourc: skills learned on this page to calculate, without a calculator, this sum: 2 5

×

5 3

×

4 10

÷

1 10

×

3 4

4 ÷ × 10

4 6

×

2 6 39


Ratios are Fractions You use ratios often in everyday life but you probably didn’t realise that they are just another way of representing fractions. For example, in a lolly bag the ratio of snake lollies to banana lollies is 3:1. This means that for every banana lolly there are three snake lollies. The fraction of snake lollies in the bag is ¾. We know this because we can see that three parts are snakes and one part is banana lollies, so altogether there are four parts. Let’s look at each common use of ratios.

*  Task a

r o e t s Bo r e p ok u S

Teac he r

When Jo makes cordial she mixes water and cordial in the ratio 5:1. In a glass of cordial mixture: ii. what fraction is water?

iii. what fraction is cordial?

________________________

________________________

________________________

*  Task b

ew i ev Pr

i. how many parts are water?

For the annual sausage sizzle fundraiser the number of adults to children who attend is usually 2:3. i. If there are 50 adults at the fundraiser, how many children will there be?

ii. If there are 90 children at the fundraiser, how many adults will there be?

© ReadyEdPubl i cat i ons ____________________________________ •f orr evi ew p____________________________________ ur posesonl y•

w ww

  Task c * a. a. 4:8

b. 36:24 c. 60:12:48

Simplify each of the following ratios:

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m . u

We can simplify ratios in the same way that we can simplify fractions. For example if we have the ratio 10:12:2, we can divide each number by the highest common factor of 2 and we get 5:6:1. We simplify them so that they are easier to use.

o c . che e r o t r s super d. 110:240

g. 42:21:49

e. 36:9:18

h. 16:48:96

f. 750:1000:1500

i. 132:22:110

ask c: Research Challenge *  TResearch the use of ratios when we talk about the “odds” of winning something. Write a short

paragraph describing what we mean when we say that a horse has a 45:1 chance or odds of winning a race. Also describe what we would mean by “good odds” and “bad odds”.

40


Fractions and Percentages

*

  Task a

Convert each of the following percentages to fractions. Be sure to simplify your answers. The first one has been done for you. c.

b.

a. 85% =

85 ÷ 5 100 ÷ 5

=

17 20

20%

r o e t s Bo r e p ok u S f.

e.

d.

100%

64%

i.

h.

g.

130%

96%

86%

54%

ew i ev Pr

Teac he r

42%

Convert fraction toa percentage. Theu first one has been done forn you. Reach ea dy EdP b l i c at i o s *  Task b © c. o b. w p a. •f orr evi e ur poses nl y•

w ww

g. 3 50

. te

3 10

f.

e.

d.

3 4

1 2

4 5

7 25

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20 2 × 100 = 2 × 20 = 40% 5

o c . che e r o t r s super i.

h.

1 3

2 3

c: Personal Challenge *  ToTaskconvert fractions whose denominator does not divide evenly into 100 we can use short division. See if you can use short division to convert each of the following fractions to percentages: 3 2 7 3 1 , , , , 8 15 8 14 24 41


Decimals and Percentages

*  Task a

Convert each decimal to a percentages. The first one has been done for you.

a.

b. 0.03 × 100 = 0.03 = 3%

d.

c. 0.78

0.45

r o e t s Bo r e p ok u S e.

f.

0.01

2.04

g.

h.

i.

0.067

0.002

0.105

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Teac he r

1.3

d.

e.

w ww

2.1%

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0.4%

52%

107%

g.

f.

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each percentage to a decimal. The first one has been done for you. *  Task b Convert © Reab.dyEdPubl i c at i ons a. c. •f r evi ew pur pose4% sonl y• 24% = 24 ÷ 100 =o 24r = 0.24 95%

o c . che e r o t r s super h.

i.

0.15%

33.3%

Task c: Small Group Challenge *  When we measure our Body Mass Index or BMI we use percentages. In a small group, each of you calculate your BMI using the formula BMI =weight (kgs) ÷height2 (cms). Also in small groups choose three of your favourite foods that you can buy at the supermarket. Record the energy, sugar, protein and fat content that is found in each of these three products as a percentage.

42


Fractions and Decimals

*

  Task a

Convert each decimal to a fraction. Be sure to simplify your answers. The first one has been done for you. c.

b.

a. 0.42 =

42 21 = 100 50

r o e t s Bo r e p ok u S f.

e.

d.

Teac he r

h.

  Task b

0.015

Convert eac fraction to a decimal using short division. The first one has been done for you.

© Readb.yEdPubl i ca i ons c. t 0.375 1 4 • evi ew p u r p o s e s nl y• 8 f 3o . 0r 0 0r 4 5o 3 6 4

e.

d.

w ww

3 10

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f. 1 7

1 9

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3 8

i. 0.09

0.82

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g.

a.

0.004

1.1

0.25

*

0.6

0.05

o c . che e r o t r s super   Task c: Investigative Challenge * g.

1 3

i.

h.

5 8

5 6

Converting recurring decimals to fractions is difficult if you don’t First Fraction Fraction = just know the answer. We can use this formula to help us work it out: 1 – Fraction Ratio For example if we look at 0.111111111 We can see that this is 0.1 +0.01 + 0.001 + … Using the formula we would have: Use this formula to convert these recurring decimals to fractions: 0.666666666 0.4444444444 0.16161616161616

1 1 10 = 1 10 1– 10

×

10 1 = 9 9

43


What is my Test Score as a Percentage?

How many times have you annoyed your teacher by asking them what your test score is as a percentage? Converting your test score to a percentage is exactly like converting a fraction to a percentage. We take our mark, divide it by the total number of marks in the test and then multiply by 100. For example, if I scored 45 marks out of 50 this is what I would do to change it to a percentage. 2 45 × 100 = 45 × 2 = 90% 50 1

*

  Task a

r o e t s Bo r e p ok u S

Change each of the test scores to a percentage. Check your answers with a calculator.

b. 8 out of 10

c. 18 out of 20

e. 8 out of 15

d. 37 out of 40

f. 49 out of 60

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Teac he r

a. a. 15 out of 25

w ww

b. There is 220 grams of sugar left in a 1 kilogram bag. What percentage of sugar is left?

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m . u

each of these quantities to a percentage. *  Task b Convert © e yE dPubl i cat i ons a. a. Three people in aR class ofa 25d have red hair. What percentage of the class have red hair? •f orr evi ew pur posesonl y•

Sugar

o c . che e r o t r s super

c. In a school of 1500 students, the 250 Year 11 students are going on camp. What percentage of students will remain at school?

d. During a TV program that goes for an hour, there are 12 minutes of commercials. What percentage of the program is television commercials?

Task c: Personal Challenge *  Begin to record your test results for every subject this year as a percentage score. Also record the average mark for each test. Over time you will be able to see if your marks are improving.

44


      What’s the Discount?

One of the most familiar areas where we use percentages every day is when we go shopping at the sales. You’ll see lots of signs telling you the percentage discounts available. For example, if you want to buy a pair of jeans that are on sale at 20% off the normal retail price of $180, we can work out the sale price like this: 20 × 180 = $36 Sale Price = 180 - 36 = $144 100

  Task a or * eB t s r Calculate the discount available on the sale items below.

Teac he r

a. 10% off $3000

oo k

b. 35% off $250

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e p u S

Calculate the sale price for the items below. c. 15% off $850

d. 8% off $400

a. 5% of 200 grams

b. 10% of 54 km

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d. 12% of 3L

c. 25% of 800 m

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e. 84% of 5000 cm2

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adyEdPubl i cat i ons   Task b © Re * Using the same method as the one to calculate a discount, calculate the percentage of each of the following orr evi ew pur posesonl y• amounts: •f

o c . che e r o t r s super f. 27% of 10 hours

ask c: Partner Challenge *  TCreate a mental maths quiz for your partner. The quiz should be fifteen questions

long and should have a mix of questions using fractions, decimals and percentages. Make sure you work out the answer, without a calculator, before giving your quiz to your partner. If it’s too hard for you then it will be too hard for them!

quiz 45


Best Buy When you go shopping for groceries it can be confusing to determine which brand is the “better buy”. For example if you see 1.5L of softdrink for $2.20 and 2L of softdrink for $3.10, it is not immediately obvious which is the better buy. First we see what each costs for 1L. 3.10 ÷ 2 = $1.55/L 2.20 ÷ 1.5 = $1.47/L So the better buy is the 1.5 L softdrink.

*  Task a

r o e t s Bo r e p ok u S

Determine which of the following is the “better buy”.

c. 30 m of plastic wrap for $3.90 or 45 m of plastic wrap for $4.50.

b. 2 cans of baked beans for $1.90 or 3 cans of baked beans for $2.95.

ew i ev Pr

Teac he r

a. 1L of icecream for $5.10 or 600ml of icecream for $2.80.

d. 40 g of tuna for $1.05 or 250g of tuna for $5.80.

© ReadyEdPubl i cat i ons o rr ev i e w u s sitems on y• the supermarket they often claimp that if r youp buyo two ore three of al particular *  Task b•Atf product then you will save money.

w ww

m . u

a. Do you think you are always really saving money if you buy two or three items instead of just one? _________________________________________________________________________

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b. A loaf of bread costs $4.70 but if you buy two loaves it will cost you $8.00.

c. A cake-mix costs $2.40 but if you buy two cake mixes it costs $4.

i) How much money do you save?

i) How much money do you save?

ii) What percentage do you save?

ii) What percentage do you save?

o c . che e r o t r s super

*  Task c: Class Challenge

Each student in the class is to present two products, like those in Task A. Each student can write their two products on the board and every other member of the class is to determine which product is the "best buy".

46


Teachers’ Notes

Linear Equations   Equations Versus Expressions

r o e t s Bo r e p ok u S

  One Step Equations

This activity develops the strategy of using inverse operations to solve simple one step equations. Task C asks students to consider the relationship between BIMDAS and solving equations, namely that solving equations involves using BIMDAS in reverse.

An important concept when solving equations is for students to realise that they can of course check their solutions without needing a set of solutions. Building upon their substitution skills learned earlier in this book, students are asked to check whether they have solved an equation correctly. Task B asks students to examine the working out for errors and to resolve these errors.

ew i ev Pr

In Task A students are asked to explore their understanding of the difference between an expression and an equation. Task B allows students to develop their abilities in transforming worded information into a mathematical equation. Task C explores a common use of simple linear equations; foreign currency exchange.

Teac he r

  Checking Solutions

  Solving Real Life Problems 1 and 2

These two activities require students to build upon the skills developed in this section and to apply them to real world problems. Students can be encouraged to share their findings and working out with their peers to both ensure shared understanding and to enhance mathematical communication.

  Backtracking

w ww

This activity allows students to explore a different strategy to solve linear equations. The key to this strategy is to build up the equation using the correct BIMDAS operations and then to unravel this set up to solve the equation.

. te

m . u

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

o c .   Two Step Equations che e r o t r s super

This task builds upon the previous two, and extends solving equations into equations involving two steps. The strategy demonstrated in this task is the traditional use of inverse operations and the emphasis needs to be on how students set out their working.

47


Equations Versus Expressions It is important to understand the difference between an equation and an expression. The easiest way to think about this is: an equation involves an equal sign, a sense of balancing both sides, while an expression doesn’t have an equal sign.

*  Task a

For each of the following circle those that are expressions and underline those that are equations:

a. 2x - 3 b. 5y - 1 = 6 m+5 2

g. 3p - 2 = p + 5

e. a + b - c

h.

5 = 20 g

f. 3p2 - 2 = p + 5

ew i ev Pr

Teac he r

c.

r o e t s Bo r e p ok u S d. w2 = 4

We use equations to solve real life problems and it’s important to be able to turn statements into equations.

*

  Task b

Write an equation to represent each statement. The first one has been done for you.

a. a. I think of a number, add 5

to it and then divide the answer by 3. The result is 6.

a. a. d. I add 6 to a number and g. If I subtract 2 from a a. multiply the result by 3. The a. number, multiply the result final answer is 50.

by 4 and divide this answer by -3 I end up with -4.

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

y+5 =6 3

w ww

. te

c. Twice a number plus 4 is 24.

e. If I subtract 12 from a number and divide the result by 4 I end up with 12.

h. Four less than twice a number gives a result of 8.

f. A number plus 7 is the same as twice the number subtract 13.

i. 3 more than a number, multiplied by -2 is the same as 18 less than this number.

m . u

b. A number minus 10 gives a result of 45.

o c . che e r o t r s super

Task c: Small Group Challenge *  Important yet easy equations are those used to convert one currency to the next. In small groups of

three or four students, examine each rate below and create two equations for each, one to turn it into Australian dollars (AUD) and one to change it out of Australian dollars. 1 AUD = 0.71 EUR (European Dollar) 1 AUD =0.99 US (American Dollar) 1 ZAR (South African Rand) = 0.15 AUD

48

1GBP (British Pound) = 1.57 AUD 1 AUD = 44.89 INR (Indian Rupee) 1 JPY (Japanese Yen) = 0.012 AUD


One Step Equations Solve each equation involving either addition or subtraction, by undoing each sum. The first one has been done for you.

*

  Task a

a. a. y – 5 = 13

a. b. m + 12 = 20 a.

y = 13 + 5 y = 18

r o e t s Bo r e p ok u S

a. g. 10 – q = 32

a. e. p + 3 – 6 = 10 a.

a. f. 5 + y – 3 = -10 a.

a. h. 20 = d + 37 a.

a. i. 55 = 30 – k a.

ew i ev Pr

Teac he r

a. d. g – 8 = 210

a. c. 7 + p = 35 a.

Solve each equation involving multiplication and division, by undoing each sum. The first *  Task b © Rbeen ea d y EdPubl i cat i ons one has done for you. a. p •f a. w p a. o r r e v i e p o s e s l y• a. = -4 o c. -5gn = 50 b. 3 × m = u 18 r a. 3 p = -4 × 3

p = -12

w ww

a. f d. = 12 -2

a. 18 =9 g. a

. te

a.

a. e. -7y = -42 a.

m . u

a. x f. =5 -9 a.

o c . che e r o t r s super a. -24 =6 h. a. k

a. 1 h = 16 a.i. 2

ask c: Class Challenge *  TWhen we perform a normal, everyday calculation, we follow the rules of BIMDAS. What rules do you

think we follow to solve an equation and how do these relate, if at all, to the BIMDAS rules? Share your thinking with the class, be sure to provide examples to explain your reasoning. 49


Backtracking A useful method for solving equations, involving more than one step, is to backtrack, or undo the equation step by step. d–3 + 1 = 8. For example, let’s solve the equation: 2 d

Then we mirror the same diagram, this time doing the opposite of each step, or undoing each step.

17

d–3

d–3 2

d–3 +1 2

r o e t s Bo r e p ok u S –3

+3

14

÷2

×2

7

+1

-1

As you can see, the answer is d = 17.

Solve each of the following using the backtracking method: a. y + 2 = 10

b.

1  m = 6 3

8

ew i ev Pr

Teac he r

First we build up the equation step by step, as shown in the diagram.

w ww

e.

x + 7 = 25 -2

. te

g. 2(v – 1) = 20

i. 8 – 2q = 0

50

f. 6 + 10w = -24

m . u

© ReadyEdPubl i cat i ons t +o 4 s c. -3p + 5• = -1f orr evi ew pu r p sonl y• =2 e d. 3

o c . che e r o t r s super h. 5 – k = -20


Two Step Equations There are many ways to approach solving equations and this method is the most commonly used and the easiest and neatest way to set out all of your working. Let’s solve the equation: 3y – 4 = 11 We can see that we have two steps in this equation. We have a “multiply by 3” and also a “subtract 4”. According to the rules of BIMDAS we know that the “multiply by 3” happened first and the “subtract 4” happened last.

r o e t s Bo r e p ok u S

Teac he r

When we solve equations we use BIMDAS in reverse. So here is how we would solve this equation: 3y = 11 + 4 According to BIMDAS we move the 4 first, and because we are undoing the equation we add instead of subtract. 3y = 15 Here we work out the calculation. y = 15÷ 3 We now move the 3, and because we are undoing the equation we divide instead of multiply. y = 5 Our final calculation gives us the answer.

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Remember, when we set out our working we have a movement step followed by a calculation step, and we keep doing this until we have our final answer.

Solve each of the following equations using the method shown above: 4m + 3 = 23 4m = 4m = m= m=

a. b. 5x – 7 = 23 a.

a. y a.c. 5 + 1 = 9

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

a. w d. –6=2 4

w ww

. te

a. g. 6 – a = 8

a. j. -2(z + 4) = 20

a. h–2 =4 e. a. 3

a. k + 1 a.f. -8 = 2

m . u

a. a.

o c . che e r o t r s super a. h. 10 – 4f = 14 a.

a. a.i.

a. 9+b =4 k. a. -3

a. -2 a.l. 3 p + 7 = -3

1 g – 5 = -2 4

51


Checking Solutions When we solve an equation, how do we know if we have the correct answer? There’s a very easy way to check and see whether we got the answer right. x If I solve the equation – 10 = -5 , I get an answer of x = 20. Am I right? 4 20 To see if I got the correct answer, all I need to do is substitute my answer back – 10 4 into the equation and see if I can balance both sides. We do it like this: = 5 – 10 So as we can see, the left hand side equals -5 and the right hand side equals -5. = -5 Since we have the same answer on both sides, my solution is correct!

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Check each answer to see if it is correct. Show all your working out.

a. g + 4 = 9

a. n + 4 d. =8 3

g=5

a. m b. =6 a. -2

n = 28

a. z – 7= 5 e. a. -3

m = 12

z = -36

a. a.c. 3y – 10 = 26

y = 12

a. a.f. 2 (f – 5) = 12

f = -7

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  Task a * a.

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a. a.

In each calculation below, a student has made at least one error when solving the equation. Examine each problem carefully and circle as many errors as you can find. Solve each equation correctly, showing all your working.

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3g + 8 = 11 3g = 11 – 8 3g = 3 g=3×3 g=9

a. y + 9 = 12 d. -4 y = 12 – 9 -4 y =4 -4 y = 4 × -4 y = -16 52

b.

x – 12 =7 4 x = 7 + 12 4 x = 19 4 x = 19 × 4 x = 75

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*

  Task b

c.

7 – 2p = 9 2p = 9 – 7 2p = 2 p = 2 ÷ -2 p = -1

f.

n – 15 = 2 25 n – 15 = 2 × 25

n – 15 = -50

n = 15 + 50

n = 65

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e. -3(x –7) = 21

x – 7 = -3 ÷ 21 -1 x–7= 7 -1 x= +7 7 1 x=7 7


Solving Real Life Problems 1 Each real life problem involves a rate of some kind, and mirrors a situation that you might come across in everyday life.

  Problem 1 * To catch a taxi it costs $1.20 for each kilometre travelled plus an initial fee (or a flagfall fee) of $2.25. i. How much does it cost to travel 55km?

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_ __________________________________________________________________________ ii. If the total cost of a taxi ride is $26.25, how far have you travelled?

_ __________________________________________________________________________

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iii. If the total cost of a taxi ride is $103.05, how far have you travelled?

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_ __________________________________________________________________________

  Problem 2 * An electrician charges $65 for each hour of work (or part thereof) and an initial or callout fee of $55. i. How much does the electrician charge for a job that takes 3 hours?

_ __________________________________________________________________________

© ReadyEdPubl i cat i ons iii. If thef job $185, how long did p the u job r take? • ocosts rr ev i e w posesonl y• ii. How much does the electrician charge for a job that takes 4.2 hours?

_ __________________________________________________________________________

_ __________________________________________________________________________ iv. If the job costs $445, how long did the job take?

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*  Problem 3

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_ __________________________________________________________________________

When using an overseas calling card, you pay a connection fee and then a certain amount per minute for your call. To call Paris from Australia for a particular brand of calling card the total cost of the call is given by the equation: Cost($) = 0.03M + 0.25

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i. What does the 0.25 represent?

_ __________________________________________________________________________ ii. How much does it cost per minute to call Paris?

_ __________________________________________________________________________ iii. How much does a call lasting 3 minutes cost? _ __________________________________________________________________________ iv. If a call cost $0.73, how long did the call last? _ __________________________________________________________________________ 53


Solving Real Life Problems 2   Problem 1 * In many countries in the Northern Hemisphere, temperature is measured in Fahrenheit, rather than in 9 degrees Celsius. An easy linear equation is used to convert between one and the other: F =  C + 32. 5 i. If it is 40˚C, what is the temperature in Fahrenheit?

_ ___________________________________________________________________

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ii. If it is 72F, what is the temperature in degrees Celsius?

_ ___________________________________________________________________ iii. If it is 20F, what is the temperature in degrees Celsius?

  Problem 2 * A street performer tells you that he can read your mind. This is what he says to you:

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_ __________________________________________________________________________

“Think of any number you like, just don’t tell me what it is. Add 10 to your number. Have you done that? Now multiply your answer by 4. Now divide this answer by 2. Okay, are you ready for the last step? Subtract twice the number you first thought of. Now keep that number in your mind while I concentrate on reading your mind.” There is a pause while the performer reads your mind, and he then says, “The number you have is 20!”

© ReadyEdPubl i cat i ons _ __________________________________________________________________________ •f orr evi ew pur posesonl y• ii. Choose your own number and see if the trick still works. i. Let’s say you chose the number 8 to start with, is the street performer right? Show how you followed the steps.

_ __________________________________________________________________________

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iii. Write the steps of the trick as an equation. Can you now explain how the trick works?

_ __________________________________________________________________________

  Problem 3 * . There is a special formula t called Heron’s Formula, which is used to find the area

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of any triangle. It is an ancient formula and easy to use. Part of the formula requires you to calculate the value of s: The variables a, b and c represent the three sides of the triangle.

a+b+c s=   2

i. If the sides of the triangle are 3, 4, and 5, what is the value of s?

_ __________________________________________________________________________ ii. If s = 10.52, a = 8 and b = 7.5, what is the length of the third side? _ __________________________________________________________________________ iii. If s = 12.93, c = 12 and a = 6.36, what is the length of side b? _ __________________________________________________________________________ 54


Answers Integers Imagining Negative Numbers Part 1 p7 a. 0 b. Below ground level c. 6 d. No e. Post Office f. -4 + 4 + 3 – 1 + 3 – 9 Butcher/Bakery, Post Office/Newsagent/ Greengrocer, Appliances, Car Park A g. Down 2, up 5, down 2, down 1

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f. g. h. i. j.

-13 -5 19 -8 -12

g. h. i. j. k. l.

10, 18 -40, -46 103, 113 23, 38 -74, -95 -360, -685

Walking Up and Down the Number Line p12 31. -1326 21. 36 11. -36 1. 23 32. -1792 12. 47 22. -327 2. 6 33. 659 23. 274 13. 59 3. -20 34. 1562 24. -337 14. -39 4. 2 35. -3259 25. 505 15. -7 5. -14 36. 349 26. -506 16. 91 6. -20 37. -1436 27. -246 17. -31 7. 8 38. 27 28. 1010 18. 58 8. -33 29. 620 39. -186 19. 13 9. 25 30. -1418 40. 100 20. -77 10. -19

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Imagining Negative Numbers Part 2 p8 a. Amount he withdrew b. $6000 c. Add each amount in the Amount column d. 324 + 1250 = $1574 e. $431 more

Where Am I? p9 Task A a. -3 b. 9 c. 2 d. -10 e. 6 Task b a. 15 b. -5 c. -10 d. -6 e. -3, 0 f. -5, -9

Integer Subtraction p11 Task A a. 4 g. -58 b. -18 h. 51 c. -17 i. -6 d. -36 j. -48 e. 26 k. -73 f. -26 l. -62 Task b a. 2 e. 11 b. -21 f. -112 c. -11 g. -117 d. -38 h. -261

© ReadyEdPubl i cat i ons •f orr evi ew puWhich r po ses nl y• is Larger? p13o c. > d. >

e. < f. <

Task b a. -40, -23, -21, -18, -9, -5, 0, 1, 7, 8, 12, 20, 31 b. -135, -116, -98, -79, -32, -7, 5, 21, 45, 67, 81, 102 c. -75, -62, -42, -25, -18, -8, 10, 12, 43, 54, 63, 84 d. -672, -487, -462, -356, -108, 154, 243, 392, 428, 590, 910 Task c a. > d. > g. < b. > e. = h. = c. < f. >

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Integer Addition p10 Task A a. 19 g. 16 b. -14 h. 11 c. -10 i. -83 d. -5 j. 14 e. 12 k. -7 f. -5 l. 11 Task b a. 11 e. 33 b. 6 f. 105 c. 67 g. -18 d. -7 h. 21

Task A a. > b. <

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Multiplication of Signed Numbers p14 a. -8 b. -6 + -6 = -12 c. -4 + -4 + -4 + -4 + -4 + -4 + -4 + -4 = -32 d. -12 e. -35 f. -24 g. 15 h. 27 i. 28

55


Task b a. positive b. negative c. negative d. negative

e. f. g. h.

negative positive positive negative 31. 18 32. 60 33. -3 34. 72 35. 120 36. 4 37. 300 38. 40 39. 8 40. 4

Expressions With Indices p19 Task a a. 78 x 410 b. a11 x b7 c. 37 x q9 d. d4 x p8 x t5 e. 511 x z10 x 45 f. 6 x f11 x g9 Task b a. 7 x 104 b. 514 x 38 n5 c. 2 m d. p35 x b20 e. 320v20 x v6 x 315 = 335 x v26

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Indices, Squares and Square Roots Using Index Numbers p17 Task a a. 33 b. 52 x 62 x 7 c. 2 d. 10 x 4 x 6 e. 72 x 22 ÷ 32 f. 34 x 24 x 43 g. 67 x 46 x 27 4 h. 5 x 4 3 Task b a. 4x4x4x5x5x6x6x6x6x6x6x6 b. -3x-3x-3x-3x12x12x7x7x7 3x3x3x3x3x3x10x10x10x15x15 c. 10x10x15x15x15x15 d. 2x2x2x2x2x2x6x6x6x6 e. 7x7x7x7x7x7x7x7x4x4x4x4x4x4x4x4x4x4x4 4x4x4x4x4x4x4x4x4x3x3x3x3x3x3x3x3x3x3 x3x3

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Multiply and Divide p15 21. 25 11. -16 1. 12 22. -27 12. 5 2. -20 13. -75 23. 16 3. 20 24. -63 4. 50 14. 720 15. -72 25. 12 5. 21 26. -50 6. -40 16. -42 27. 24 17. -6 7. -8 28. -20 8. -13 18. 108 29. -96 19. 48 9. 81 20. 4 30. -8 10. -33

Task b b. 1x104+7x102+8x10+1x100 c. 3x106+4x105+5x104+7x103+6x102+8x10+2x 100 d. 1x10+8x100+2x10-1+5x10-2 e. 7x10-3+6x10-4+8x10-5 f. 2x103+3x10+4x100+7x10-1+4x10-2 g. 5x100+6x10-3+3x10-5+2x10-6

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4x4x4x4x11x11x11x11x11x11 f. 3x3x3x3x3x3x3x3x3x3x3x3x7x7x7x7x7

Place Value Revisited p18 1010 109 108 107 106 105 104 103 102 101

56

10 000 000 000 1 000 000 000 100 000 000 10 000 000 1 000 000 100 000 10 000 1 000 100 10

100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9

Prime Factor Trees p20 a. 36 = 22 x 32 b. 50 = 2 x 52 c. 72 = 23 x 32 d. 120 = 23 x 3 x 5 e. 250 = 2 x 53 f. 320 = 26 x 5 g. 98 = 2 x 72 h. 135 = 33 x 5 i. 1220 = 22 x 5 x 61

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© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• f. 63g6 820 x h10 g. 3 4 = 817 x h6 8 xh b12 x q8 x y12 b6 x y10 h. q10 x b6 x y2 = q2

1

0.1

0.01 0.001

0.0001 0.00001 0.000001 0.0000001 0.00000001 0.000000001

Easy Calculations Using Prime Factors p21 b. 54 x 16 2 x 33 x 24 = 25 x 33 c. 81 x 27 34 x 33 = 37 d. 24 x 33 23 x 3 x 3 x 11 = 23 x 32 x 11 e. 100 x 50 22 x 52 x 2 x 52 = 23 x 54 f. 90 x 48 2 x 32 x 5 x 24 x 3 = 25 x 33 x 5


A Different Approach to the Lowest Common Multiple p22 a. 6 + 10 = 30 b. 8 + 12 = 24 c. 9 + 10 = 90 d. 4 + 15 = 60 e. 6 + 4 = 12 f. 5 + 9 = 45 g. 10 + 12 + 8 = 120 h. 3 + 4 + 9 = 36 i. 6 + 12 + 4 = 12

d. 9.75 e. 13.42 f. 14.32

Calculations and Algebraic Generalisations How We Calculate p27 a. -25 f. 35 b. -20 g. 43 c. 6 h. 28 d. -28 i. 0 e. -40

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Square Numbers p24 Task a 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225, 256,289,324,361,400 Task b a. 5.29 d. 10.0489 b. 110.25 e. 46.5124 c. 70.8964 f. 497.29

Get the Order Right p28 Task a a. -18 d. -8 b. 19 e. -43 c. -25 f. -17

Task b a. Error: Answer: b. Error: Answer: c. Error: Answer: d. Error: Answer:

-3 + 2 -6 -2 x 3 -24 ÷ -6 -64 (-8)2 – 122 64 – 144 400 -15 + 7 -42 x -2 -84 – 25 31

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A Different Approach to the Highest Common Factor p23 a. 24, 18 =6 b. 48, 36 = 12 c. 50, 120 = 10 d. 52, 45 =1 e. 40, 84 =4 f. 32, 46 =2 g. 24, 46 =2 h. 54, 72 = 18 i. 36, 42 =6

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Task c a. 3.32 b. 5.48 c. 7.21

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Task c c. 12321 d. 1234321 "n" ones, goes from 1 up to "n" down to 1 Task d a. 12345654321 b. 1234567654321 No, after 9 ones, pattern will not work

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Square Root p25 Task a a. 4 d. 1 b. 10 e. 13 c. 5 f. 20 Task b a. 3.2 b. 6.3 c. 10.5

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© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Calculations With Formulas p29 Task a b. 785.40 cm2 c. 5183.63 cm2 d. 53407.08 mm2 Task b a. 76 cm2 b. 5 c. 7.67 Scientific Formulas p30 Task a a. 44.1 N b. 27.44 N c. 637 N d. 14700 N Task b a. 75 km/h b. 33.3 m/mm Task c a. 12.5 b. 120

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d. 12.1 e. 8.9 f. 5.3

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20 11.8

30 15.8

40 19.8

50 23.8

h.

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Hours = 4

Catching a Taxi Part 1 p32 Task a 1. Substitute the distance into the D position and calculate 5.2, 8.2, 11.2, 14.2, 17.2, 20.2, 23.2, 26.2 2. Straight line

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1. 25.80 2. 18.70 3. Weekday: 66km Saturday: 45.5 km Task c 1. Check student answers. 2. Over long distances, this second taxi service. Electrician and the Plumber Part 1 p34 Task a 1. What you pay for the tradesperson to come to your house 2. C = 55×N+70 97.5 , 125, 152.5, 180, 207.5, 262.5, 290

Electrician and the Plumber Part 2 p35 Task b 1. C=50×N+80 105, 130, 155, 180, 205, 230, 255, 280 2. Plumber 3. 2 hours 4. In the long term, the first plumber

o c . che e r o t r s super Fractions, Decimals and Percentages Equivalent Fractions p37 Task a b. 1/3 c. 4/9 d. 3/4 e. 11/15 f. 11/25 g. 3/4 h. 3/5 i. 16/23

35 30

25 20 15 10 5

20

80 35.8

© ReadyEdPubl i cat i ons •Charge f or r evi ew pur posesonl y• – Call Out Fee

Hourly Rate 430 – 50 Hours = 95

10

70 31.8

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Hours =

60 27.8

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H=

58

10 7.8

volume LxW H = 1.152 1.2 x 112 H = 0.8 m

g.

Catching a Taxi Part 2 p33 Task b

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Create Your Own Formula p31 a. A = L x W A = 20 x 5 A = 100cm2 b. C = 3.50 x number of kgs C = 3.50 x 1.8 C = $6.30 c. A = years x 365 x 24 x 60 A = 12 x 365 x 24 x 60 A = 6 307 200 minutes (ignore leap years) number of presents d. T = 4 10 T= 4 T = 2.5 hours e. C = $1.10 x number of kms + 2.20 C = 1.10 x 45 + 2.20 C = $51.70 cost of petrol f. N = 57 1.35 N = 1.35 N = 42.2 L

30

40

50

60

70

80

90


Task b 2 b. = 3 2 c. = 8 10 d. = 50 6 e. = 16 70 f. = 100

10 15 1 4 7 35 9 24 35 50

40 = 60 75 = 300 3 = 15 3 = 8 140 = 200

1 10 5 = = 6 60 30

h.

2 6 22 = = 11 33 121

i.

4 12 40 = = 15 45 150

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Adding and Subtracting Fractions p38 7 7 13 5 f. 2 k. 1 a. p. 1 12 15 35 8 3 7 41 19 q. 5 b. g. l. 10 30 56 20 5 11 6 1 c. 1 m. 5 r. 3 h. 24 20 35 4 9 7 31 7 i. 3 n. 6 s. 2 d. 20 12 45 12 5 1 8 28 e. 3 j. 3 t. 5 o. 6 2 15 45

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g.

Ratios are Fractions p40 Task a i. 5 ii. 5/6 iii. 1/6 Task b i. 75 ii. 60 Task c a. 1:2 b. 3:2 c. 5:1:4 d. 11:24 e. 4:1:2 f. 3:4:6 g. 6:3:7 h. 1:3:6 i. 6:1:5

Fractions and Percentages p41 Task a 1 16 43 b. e. h. 5 25 50 24 27 c. 1 f. i. 25 50 21 3 d. g. 1 50 10 Task b f. 28% b. 50% g. 6% c. 30% h. 33.3% d. 75% i. 66.6% e. 80%

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Decimals and Percentages p42 Task a f. 1% b. 45% g. 10.5% c. 78% h. 0.2% d. 130% i. 6.7% e. 204%

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Multiplying and Dividing Fractions p39 Task a 1 5 a. f. 10 12 5 19 g. 1 b. 24 27 3 2 h. 9 c. 7 7 1 11 i. 11 d. 6 24 2 e. 11 Task b 1 1 f. 2 a. 1 5 12 1 1 b. 4 g. 5 2 3 17 3 c. 1 h. 1 18 4 4 9 d. 1 i. 4 11 11 1 e. 3 2

o c . che e r o t r s super Task b b. 0.95 c. 0.04 d. 1.07 e. 0.52

f. g. h. i.

0.004 0.021 0.0015 0.333

Fractions and Decimals p43 Task a 1 1 e. 1 b. h. 20 10 3 1 c. f. i. 5 250 1 41 d. g. 4 50

9 100 3 200

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Task b b. 0.25 c. 0.8 d. 0.3 e. 0.142857

f. g. h. i.

Two Step Equations p51 a.m=5 b.x=6 c.y=40 d.w=32 e.h=14 f.k=-17 g.a=-2 h.f=-1 i.g=12 j.z=-14 k.b=-21 l.p=15

0.2 0.3 0.625 0.83

Checking Solutions p52 Task a a.correct b.incorrect c.correct d.incorrect e.correct f.incorrect Task b a. Error: Line 4 Answer: g = 1 b. Error: Line 2, Line 6 Answer: x = 40 c. Error: Line 2, Line 5 Answer: p = 1 d. Error: Line 3 Answer: x = 0 e. Error: Line 2, Line 5 Answer: x = 0 f. Error: Line 2, Line 4 Answer: n = 65

What is my Test Score as a Percentage? p44 Task a a. 60% c. 90% e. 53.3% b. 80% d. 92.5% f. 81.6% Task b a.12% b.22% c.83.3% d.20%

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What’s the Discount? p45 Task a a.$300 b.$87.50 c.$722.50 d.$368 Task b a.10g b.5.4 km c.200 m d.0.36L e.4200cm2 f.2.7 hours

Best Buy p46 Task a a.600ml b.$1.90 for 2 c.45m d.250g Task b a.Class Discussion b. (i)$1.40 (ii)14.9% c. (i)$0.80 (ii)16.7%

Solving Real Life Problems 1 p53 Problem 1 i. $68.25 ii. 20 kms iii. 84 kms Problem 2 i. $250 ii. $380 iii. 2 hours iv. 6 hours Problem 3 i. connection fee ii. 3 cents iii. $0.34 iv. 16 minutes

2

6

Equations: 5y – 1 = 6, w = 4, 3p – 2 = p + 5, 5/g = 20 Task b b. x – 10 = 45 c. 2x + 4 = 24 d. 3(x + 6) = 50 x – 12 e. 4 = 12 f. x + 7 = 2x – 13 4(x – 2) g. -3 = -4 h. 2x – 4 = 8 i. -2(x + 3) = x – 18 2

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One Step Equations p49 Task a b.m = 8 c.p = 28 d.g = 218 e.p = 13 f.y = -12 g.q = -22 h.d = -17 i.k=-25 Task b b.m = 6 c.g = -10 d.f = 24 e.y = 6 f.x = -45 g.a = 2 h.k = -4 i.h = 32

Backtracking p50 a.y=8 b.m=18 c.p=2 d.t=2 e.x=36 f.w=-3 g.v=11 h.k=25 i.q=4 m.=3 60

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© ReadyEdPubl i cat i ons Equations Versus Expressions p48 Task a •f orr evi ew pur posesonl y• Expressions: 2x – 3, m + 5/2, a + b + c, 3p + 5t Linear Equations

Solving Real Life Problems 2 p54 Problem 1 i. 104F ii. 22.2oC iii. -6.7oC Problem 2 i. yes ii. Check student work 4(x + 10) – 2x 2 2(x + 10) – 2x 2x + 20 – 2x 20 Problem 3 i. 6 ii. 5.54 iii. 7.5

Number Strategies Series: Working on Number and Algebra  

The Number Strategies Series helps students develop their skills and abilities in topics on Number and Algebra. The books contain a blend of...