Class B

ook

5Anita Straker, Tony Fisher, Rosalyn Hyde, Sue Jennings and Jonathan Longstaff e

N5.1 Powers and roots 11 Integer powers of numbers 12 Estimating square roots 33 Prime factor decomposition 6How well are you doing? 9

A5.1 Sequences and graphs 111 Generating sequences 112 Making generalisations 153 Using computers 184 Sketching linear graphs 205 Rearranging linear equations 246 Graphs using real-life contexts 27How well are you doing? 30

G5.1 Measures and mensuration 331 Perimeter and area 332 Finding � 373 Area of a circle 404 Solving circle problems and using � 425 Volume of prisms 456 Surface area of prisms 48How well are you doing? 51

Functional skills 1 54

N5.2 Proportional reasoning 561 Adding and subtracting fractions 562 Multiplying fractions 593 Dividing fractions 604 Percentage change 625 Ratio 656 Direct proportion 69How well are you doing? 73

S5.1 Enquiry 1 751 Stem-and-leaf diagrams 752 Starting a statistical investigation 1 793 Completing a statistical investigation 1 814 Data collection sheets 835 Starting a statistical investigation 2 876 Completing a statistical investigation 2 90How well are you doing? 93

A5.2 Equations and formulae 951 Multiplying out brackets 952 Factorising expressions 973 Substituting into formulae 994 Changing the subject of a formula 1015 Solving linear equations 1046 Trial and improvement 107How well are you doing? 110

Functional skills 2 112

G5.2 2D and 3D shapes 1141 Exploring angles and lines 1142 Solving problems 1173 Solving longer problems 1194 Drawing 3D objects 1225 Drawing plans and elevations 1246 More plans and elevations 1267 Solving problems using surface area and

volume 1298 Surface area and volume of prisms 132How well are you doing? 134

N5.3 Calculations and calculators 1371 Powers of 10 1372 Rounding and approximation 1393 Mental calculations with decimals 1434 Written calculations with decimals 1455 Using a calculator 1476 Problems involving measures 150How well are you doing? 153

S5.2 Probability 1 1551 Simple probability 1552 Equally likely outcomes with two events 1593 Mutually exclusive events 1614 Practical probability experiments 1635 Simulating probability experiments 166How well are you doing? 168

A5.3 Functions and graphs 1701 Generating linear graphs using ICT 1702 Sketching graphs 172

Contents Tier5

Tier 5 Class book Contents | iii

3 Drawing accurate graphs 1754 Direct proportion 1775 Refl ecting graphs in y = x 1806 Simple quadratic graphs using ICT 183How well are you doing? 185

G5.3 Transformations 1871 Planes of symmetry 1872 Combined transformations 1913 Islamic patterns 1944 Enlargements 1955 Enlargements in real-life applications 1996 Length, area and volume 203How well are you doing? 208

A5.4 Using algebra 2111 Using graphs to solve problems 2112 Using algebra in geometry problems 2143 Using algebra in investigations 216How well are you doing? 218

Functional skills 3 220

S5.3 Enquiry 2 2221 Calculating statistics 2222 Line graphs for time series 2243 Scatter graphs 2274 Collecting and organising data 2305 Analysing and representing data 2326 Interpreting data 2347 Reporting and evaluating 235How well are you doing? 239

G5.4 Angles and constructions 2421 Angles in polygons 2432 Regular polygons 2463 Regular polygons and the circle 2474 Angle problems and polygons 2495 Polygons and parallel lines 2536 Constructions 2577 Constructing triangles 2608 Loci 2649 More loci 267How well are you doing? 270

A5.5 Equations, formulae and graphs 2721 Factorising 2722 Working with algebraic fractions 2763 Working with formulae 2794 Forming equations 2815 Visualising graphs 2846 Interpreting graphs 2877 Matching graphs to real-life situations 2908 Using graphs to solve problems 292How well are you doing? 296

Functional skills 4 298

S5.4 Probability 2 3001 Theoretical and experimental probability 3002 Mutually exclusive events 3053 Using experimental probability 3094 Choice or chance? 312How well are you doing? 314

N5.4 Solving problems 3161 History of our number system and zero 3162 Number puzzles based on 3 by 3 grids 3173 Exploring fractions 3194 Problems involving properties of numbers 3215 Using algebra and counter-examples 323How well are you doing? 327

R5.1 Revision unit 1 3291 Using a calculator 3292 Using percentages to compare proportions 3323 Sequences, equations and graphs 3364 Angles and polygons 3415 Charts and diagrams 345

R5.2 Revision unit 2 3511 Ratio and proportion 3512 Solving number problems 3553 Expressions, equations and formulae 3604 Circles and enlargements 3645 Probability 369

Answers 376

Index 380

iv | Tier 5 Class book Contents

Powers and roots

This unit will help you to:

calculate whole-number powers of numbers;

estimate square roots;

write a number as the product of its prime factors;

fi nd the highest common factor (the HCF) of two numbers;

fi nd the lowest common multiple (the LCM) of two numbers.

This lesson will help you to work out integer powers of numbers and use the power keys of a calculator.

1 Integer powers of numbers

N5.1

The short way to write 2 � 2 � 2 � 2 � 2 is as 25, or ‘2 to the power 5’.

The small number 5 is called the index. An index can be negative as well as positive. For example:

9�2 � 1 _____ 9 � 9 � 1 ___ 81

The calculator key to fi nd powers of numbers looks like this: xy .

Example 1 Find the value of 64.

Key in 6 xy

4 � . The display shows the answer: 1296

To multiply two numbers in index form, add the indices, so am � an � am � n.

Example 2 Simplify 34 � 32.

34 � 32 � 34 � 2 � 36

To divide two numbers in index form, subtract the indices, so am � an � am � n.

Example 3 Simplify 53 � 52.

53 � 52 � 53 � 2 � 51 � 5

Exercise 1

N5.1 Powers and roots | 1

1 Write each expression in index form.

a 2 � 2 � 2 b 4 � 4 � 4 � 4 � 4

c 3 � 3 � 3 � 3 � 3 � 3 � 3 � 3 d (�1) � (�1) � (�1) � (�1)

e 1 _____ 5 � 5 f 1 __ 6

2 Work out each value without using your calculator. Show your working.

a 26 b (�3)5 c 44 d (�2)7

e 120 f (�1)17 g 4�2 h 5�3

3 Use your calculator to work out each value.Where appropriate, give your answer correct to two decimal places.

a 74 b 56 c 113 d 39

e (�2)10 f 1.56 g 31.83 h 1.785 � 10.34

4 Simplify these.

a 25 � 23 b 34 � 3 c 102 � 102 d a5 � a3

e 56 � 52 f 125 � 12 g 84 � 84 h b5 � b2

5 A palindromic number reads the same forwards and backwards.Copy and complete this cross-number puzzle.Across Down1 A square number 1 23 � 53 A square palindromic number 2 A multiple of 134 A cube number

6 Some numbers can be written as the sum of three square numbers. For example:

35 � 52 � 32 � 12

Write each of these numbers as the sum of three square numbers.

a 19 b 41 c 50

d 65 e 75 f 94

7 Rachel is 3 years older than her sister Hannah.The sum of the squares of their ages in years is 317.How old are Rachel and Hannah?

1

3

4

2

2 | N5.1 Powers and roots

Extension problem8 Find the two smallest whole numbers where the diff erence of their squares is a cube, and

the diff erence of their cubes is a square.

Points to remember The number 2 raised to the power 4 is 24 or 2 � 2 � 2 � 2.

4 is called the index or power, and 24 is written in index form.

To multiply numbers in index form, add the indices, so am � an � am � n.

To divide numbers in index form, subtract the indices, so am � an � am � n.

A negative number raised to an even power is positive. A negative number raised to an odd power is negative.

This lesson will help you to estimate square roots and to use the root keys of a calculator.

2 Estimating square roots

√ __

n is the square root of n. For example, √ __

81 � �9.

You can fi nd positive square roots on a calculator.

Example 1To fi nd √

__ 81 , press 8 1 √

_ . The display shows the answer: 9

On some calculators you press the square-root key fi rst: √ _ 8 1

Some calculators have a cube-root key 3 √ _ .

Example 2To fi nd 3 √

__ 64 , press 8 1 3 √

_ . The display shows the answer: 4

Exercise 2

N5.1 Powers and roots | 3

You can estimate the positive square root of a number that is not a perfect square.

Example 3Estimate the value of √

___ 70 .

√ ___

70 must lie between √ ___

64 and √ ___

81 , so 8 � √ ___

70 � 9. Since 70 is closer to 64 than to 81, we expect √

___ 70 to be closer to 8 than to 9.

An estimate is 8.4.

You can estimate the value of a square root more accurately using trial and improvement.

Example 4Solve a2 � 135.

Value of a Value of a2

11 121 too small

12 144 too big a is between 11 and 12.

11.5 132.25 too small a is between 11.5 and 12.

11.6 134.56 too small a is between 11.6 and 12.

11.7 136.89 too big a is between 11.6 and 11.7.

11.65 135.7225 too big a is between 11.6 and 11.65.

So a must lie on the number line between 11.6 and 11.65.

Numbers between 11.6 and 11.65 round down to 11.6 to 1 d.p, so a � 11.6 to 1 d.p.

8 9

8164 70

11.6 11.65 11.7

1 Write two solutions to each of these equations.

a x2 � 9 b x2 � 49 c x2 � 144 d x2 � 1

2 Write the value of each of these expressions. Use a calculator to help you.Where appropriate, give your answer correct to two decimal places.

a √ __

2 b 3 √ __

10 c 3 √ _____

(�64) d √ ____

0.04

e 3 √ ______

(�125) f √ ___

1.5 g √ __

3 h 3 √ _____

(�1)

4 | N5.1 Powers and roots

3 Estimate the integer that is closest to the positive value of each of these.

a √ __

6 b √ ___

45 c √ ____

115 d √ ___

70

4 Use your calculator to fi nd the positive value of each of the square roots in question 3.Give your answers correct to one decimal place.

5 A square patio is to be pavedwith square paving slabs. Only whole slabs will be used.

The paving slabs come in packs of 50. 15 packs of slabs are needed to makesure that there are enough slabs.

How many slabs are used?

6 Solve these equations by using trial and improvement. Make a table to help you. Give your answers to one decimal place.

a a2 � 95 b a2 � 152 c a2 � 415

7 The area of this square rug is 15 m2. Use trial and improvement to fi nd the length of

a side correct to two decimal places.

Extension problem8 What is the smallest square number that begins with three 8s?

Points to remember √

__ n is the square root of n, for example √

___ 81 � �9.

3 √ __

n is the cube root of n, for example 3 √ ___

125 � 5, 3 √ ____

�27 � �3.

N5.1 Powers and roots | 5

This lesson will help you to:

write a number as the product of its prime factors;

find the highest common factor (the HCF) of two numbers;

find the lowest common multiple (the LCM) of two numbers.

3 Prime factor decomposition

Did you know that…?The Greek mathematician Euclid proved in about 300 BC what is called the Fundamental Theorem of Arithmetic.

This shows that every integer can be written as a product of prime factors in only one way.

You can use a division or ladder method to fi nd the prime factors of a number.

Example 1 The prime factors of 75 are 5 � 5 � 3 � 52 � 3.

The prime factors of 24 are 3 � 2 � 2 � 2 � 3 � 23.

You can also use a tree method to fi nd the prime factors of a number.

Example 2The prime factors of 48 are 2 � 2 � 2 � 2 � 3 � 24 � 3.

You can use prime factors to fi nd the highest common factor (HCF) and the lowest common multiple (LCM) of two numbers.

Example 3The prime factors of 72 are 2 � 2 � 2 � 3 � 3.The prime factors of 60 are 5 � 3 � 2 � 2.These are shown on the Venn diagram.The overlapping or common prime factors give the HCF:

2 � 2 � 3 � 22 � 3 � 12

All the prime factors give the LCM:2 � 2 � 2 � 3 � 3 � 5 � 23 � 32 � 5 � 360

Exercise 3

48124 4

22

2

23

72 60

522

3

3

2

3 755 255 5

1

3 242 82 42 2

1

6 | N5.1 Powers and roots

1 The number 18 can be written as the product of prime factors.

18 � 2 � 3 � 3 � 2 � 32

Write each of these numbers as the product of prime factors.

a 12 b 15 c 21 d 24 e 27 f 54

2 a List all the factors of 50.

b Write 50 as the product of prime factors.

3 a List all the factors of 45.

b Write 45 as the product of prime factors.

4 Find a number bigger than 50 that has the same number of factors as 50.

5 Using the Venn diagrams below, work out the HCF and LCM of:

a 72 and 30 b 50 and 80 c 48 and 84

6 The prime factors of 120 are 2, 2, 2, 3 and 5. The prime factors of 75 are 3, 5 and 5.Show these numbers on a Venn diagram.Use the diagram to work out the HCF and LCM of 120 and 75.

7 The prime factors of 40 are 2, 2, 2 and 5.The prime factors of 90 are 2, 3, 3 and 5.Show these numbers on a Venn diagram.Use the diagram to work out the HCF and LCM of 40 and 90.

8 a Which prime numbers are factors of both 42 and 54?

b What is the biggest number that is a factor of both 42 and 54?

c What is the smallest number that is a multiple of both 42 and 54?

9 Find the HCF and LCM of:

a 28 and 40 b 200 and 175 c 36 and 64

10 A four-digit number is a multiple of 21 and a multiple of 35.What is the smallest number that it could be?

72 30

52

3

3

2

2

50 80

2

22

2

55

48 84

722

3

2

2

N5.1 Powers and roots | 7

Extension problems11 a What is the smallest number with exactly 3 factors?

b What is the smallest number with exactly 5 factors?

c What is the smallest number with exactly 7 factors? Exactly 9 factors? 11 factors? 13 factors?

12 Seven friends are having lunch at the same café.

The fi rst one eats there every day,the second every other day,the third every third day,the fourth every fourth day,the fi fth every fi fth day,the sixth every sixth day,and the seventh once a week on the same day.

The next time they all meet at the café they are planning to have a lunch party.In how many days from now will the lunch party be?

Points to remember Writing a number as the product of its prime factors is its prime factor

decomposition.

For example, 24 � 2 � 2 � 2 � 3 or 23 � 3.

The highest common factor (HCF) of a pair of numbers is the largest number that is a factor of each number.

For example, 8 � 2 � 2 � 2 and 12 � 2 � 2 � 3. The highest common factor is 2 � 2.

The lowest common multiple (LCM) of a pair of numbers is the smallest number that is a multiple of each number.

For example, 8 � 2 � 2 � 2 and 12 � 2 � 2 � 3. The lowest common multiple of 8 and 12 is 2 � 2 � 2 � 3 � 24.

8 | N5.1 Powers and roots

Can you: work out whole-number powers of numbers?

estimate square roots?

write a number as the product of its prime factors?

fi nd the highest common factor (the HCF) of two numbers?

fi nd the lowest common multiple (the LCM) of two numbers?

Powers and roots (no calculator)

1 2006 level 6

a Put these values in order of size with the smallest fi rst.

52 32 33 24

b Look at this information.

55 is 3125

What is 57?

2 2001 level 6

a Look at these numbers.

Which is the largest?

Which is equal to 92?

b Which two of the numbers below are not square numbers?

How well are you doing?

16 25 34 43 52 61

24 25 26 27 28

N5.1 Powers and roots | 9

3 Work out the value of each expression.

a 37 � 35 b 24 � 23 c 34 � 25

_______ 32 � 24

4 Look at these equations.

a 24 � 3 � 2a

What is the value of a?

b 28 � 7 � 2b

What is the value of b?

5 a Find the highest common factor of 84 and 60.

b Find the lowest common multiple of 16 and 36.

Powers and roots (calculator allowed)

6 Suzy thinks of a number.She uses her calculator to square the number and then adds 5.Her answer is 29.01.What is Suzy’s number?

7 The three numbers missing from the boxes are diff erent prime numbers bigger than 3.

c � c � c � 1045

What are the missing prime numbers?

10 | N5.1 Powers and roots