Contents

ii

Exam board specification map iv

Introduction vi

Topic checker x

Topic checker answers xv

Number

The decimal number system 2Fraction calculations 4Fractions, decimals and percentages 6Powers and roots 8Surds 10Standard index form 12Ratio and proportion 14Percentage calculations 16

Algebra

Algebraic expressions 18 Formulae and substitution 20

Rearranging formulae 22Using brackets in algebra 24Factorising quadratic expressions 26Algebraic fractions 28Solving equations 30Equations of proportionality 32Trial and improvement 34Quadratic equations 36Functions 38Inequalities and regions 40Number patterns and sequences 42Sequences and formulae 44Lines and equations 46Curved graphs 48Transforming graphs 50Linear simultaneous equations 52Mixed simultaneous equations 54

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Contents

iii

Shape, space and measures

Rounding and accuracy 56Dimensions 58Speed and motion graphs 60Properties of shapes 62Circle geometry 64Pythagoras’ Rule and basic trigonometry 66The sine rule 68The cosine rule 70Trigonometric graphs: angles of any size 72Problems in three dimensions 74Calculating areas 76Circle calculations 78Volume calculations 80Congruence and similarity 82Constructions 84Loci 86Vectors 88Transformations 90

Handling data

Pie charts 92Histograms 94Finding averages 96Cumulative frequency 98Comparing sets of data 100Probability 102Scatter diagrams and correlation 104

Exam questions and model answers 106

Complete the facts *

Answers to complete the facts *

Answers to practice questions 118

Glossary 126

Web links *

Last-minute learner 129

* Only available in the CD-ROM version of the book.

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Each place in a decimal number represents a power of ten: , , 1, 10, 100, etc.

B Decimal calculations

key factTo multiply a number by 10, 100 or 1000, simply move

its digits 1, 2 or 3 places to the left:

45.67 10 456.7 45.67 100 4567 45.67 1000 45 670.

key fact>> To divide by 10, 100 or 1000, move digits 1, 2 or 3 places to the right:

45.67 10 4.567 45.67 100 0.456 7 45.67 1000 0.045 67.

Multiplying by 0.1 is the same as dividing by 10.

You can use the result of a whole number multiplication to fi nd the answer to many decimal multiplications:

12 2 24, so 12 0.2 2.4 12 0.02 0.24 and 1.2 2 2.4 1.2 0.2 0.24 1.2 0.02 0.024, etc.

If one of the numbers is made ten times smaller, the answer will be ten times smaller.

A similar rule works with division, but if the number you are dividing by gets smaller, the answer gets bigger:

12 2 6, so 12 0.2 60 12 0.02 600 and 1.2 2 0.6 1.2 0.2 6 1.2 0.02 60, etc.

2

The decimal number system

2

A Place value

The digit 2 means something different in each of these numbers:

C Negative numbers: addition and subtraction

key factAdding a negative number gives the same result as subtracting a

positive number.

Examples: 4 (3) 4 3 1 brackets are used to make it clearer 3 (7) 3 7 4 you could use a number line to help

Number Position of ‘2’ Value of ‘2’

125 tens 20

2 130 559 millions 2 000 000

3.28 tenths

0.0026 thousandths

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>> practice questions1 Do a whole number calculation first, then use the result to answer the question.

(a) 2.5 3 (b) 0.3 1.2 (c) 6.4 8 (d) 1.44 0.03

2 Work these out without a calculator.

(a) 3 10 (b) 4 (2) (c) 13 (6) (d) (5) 9

(e) (5) (9) (f) (2) (7) (g) 4 (8) (h) (3) (7)

(i) (30) 6 (j) (24) (3) (k) (10)2 (l) (3)3

3 Use your calculator to find these.

(a) 138 272 (b) 67 (125) (c) (3320) 2671 (d) (6.93) (4.63)

(e) (255) 30 (f) 27 (0.54) (g) (8) (200) (h) (0.05)2

The decimal number system

33

D Negative numbers: multiplication and division

The rules for multiplication and division are very simple:

key fact>> negative positive negative: negative negative positive negative positive negative: negative negative positive

Examples: 3 (5) (3 5) 15 (4) (10) 4 10 40 (16) 2 (16 2) 8 (39) (3) 39 3 13

You can remember this with the ‘word’ SPON:

‘Same (signs) Positive, Opposite (signs) Negative’. Otherwise, use this table:

key factThe square of a negative number is positive.

Example: (7)2 (7) (7) 49

key fact>> The cube of a negative number is negative. Example: (2)3 (2) (2) (2) 4 (2) 8

/

Another way to do the last example is to notice that when you swap the numbers in a subtraction, you change the sign of the answer: 7 3 4 and so 3 7 4.

This can be useful when larger numbers are involved: 25 (42) 25 42. 42 25 17, so 25 42 17.

key factSubtracting a negative number gives the same result as

adding a positive number. Examples: 2 (4) = 2 4 = 6.

(12) (36) = (12) 36 = 24 use a ‘rough’ number line to help

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The numerator is the top number in a fraction and the denominator is the bottom number.

Create equivalent fractions by multiplying or dividing both numbers by the same thing.

Change mixed numbers to improper fractions before calculating.

Fraction calculations

4

A Equivalent fractions and mixed numbers

key fact Fractions that contain different

numbers, but represent the same amount, are equivalent. Fractions that contain the smallest possible whole numbers are in lowest terms.

key factFractions in which the numerator is bigger than the

denominator are called improper or ‘top-heavy’. Improper fractions can be written as mixed numbers: 3

7 36 3

1 2 31 2 3

1 .

Changing mixed numbers to improper fractions is similar: 3 43 = 4

12 + 43 = 4

15 .

You can write fractions in order of size by writing them as equivalent fractions with a common denominator.

Example: Order these: 54 , 4

3 , 127 , 10

7 , 32 .

Look at the denominators: 5, 4, 12, 10 and 3 will all divide into 60.

Rewritten, the fractions are 6048 , 60

45 , 6035 , 60

42 , 6040 .

So the order is 127 , 3

2 , 107 , 4

3 , 54 .

B Adding and subtracting fractions

key fact To add or subtract fractions, write them using a

common denominator. If you can fi nd the lowest common denominator (LCD), this keeps the numbers

small and you are less likely to make a mistake.

Sometimes, the LCD will be the denominator of one of the fractions in the question.

Example: 32 9

4 you can use 9 as the LCD.

32 9

4 96 9

4 910 9

11 .

Example: 2013 12

5

The LCD is 60: 2013 12

5 6039 60

25 6014 30

7 .

3045

69

23

= =

÷3 ÷5

÷3 ÷5

lowest terms

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Fraction calculations

>> practice questions

5

C Multiplying and dividing fractions

key fact To multiply fractions, just multiply the numerators and

multiply the denominators. Change mixed numbers to improper form first. Example: 4

3 611 4

3 67 24

21 87 .

You can keep the numbers small by cross-cancelling before you multiply.

Example: 85 25

12 : 5 cancels with 25 and 12 cancels with 8.

key factTo divide fractions, just invert the second fraction (replace it

by its reciprocal) and multiply. Example: 2 2

1 1 41 2

5 45 2

5 54 1

2 2.

D Fractions of an amount

key fact To find a fraction of an

amount, divide by the denominator and multiply by the numerator.

Example: What is 52 of £36?

£36 5 £7.20 5 to fi nd 51

£7.20 2 £14.40 2 to fi nd 52

key factTo express an amount

as a fraction of another, write the amounts as a fraction and cancel to lowest terms.

Example: What fraction of 7.5 km is 4.8 km?

..

7 54 8 75

48 top and bottom by 10: now it uses integers

7548 25

16 . cancel

310

1225

58

=

1

52

3

x

(a) (b) (c) (d) 1

(e) 2 3 (f) 5 (g) (h)

(i) (j) 3 (k) 2 1 (l) 6

710

210

58

716

34

45

12

56

38

12

29

13

25

712

57

56

23

12

57

12

12

34

815

1115

35

45

12

58

12

14

712

, , , ,1 Write in order of size, smallest to biggest:

2 Calculate the following:

3 Find these amounts:

(a) of 35 km (b) of 240 ml (c) of £16.50 (d) 2 times 450 grams

4 What fraction of the second amount is the first?

(a) 25 cm, 75 cm (b) £3.60, £5.40 (c) 60 cl, 1 litre (d) 630 kg, 1.5 tonnes

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