EdexcelGCSE (9-1) Mathsfor post-16

Fiona Mappwith Su Nicholson

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Contents Introduction 5 How to use this book 13

1 Numbers 141.1 Positive and negative numbers 151.2 Square, cube and triangular numbers 181.3 Factors, multiples and primes 201.4 Indices 231.5 Standard index form 24

2 Fractions and decimals 282.1 Fractions 292.2 Calculating with fractions 312.3 Fraction problems 332.4 Decimals and fractions 34

3 Calculations 383.1 Multiplying and dividing numbers

by power of 10 393.2 Written methods 413.3 Order of operations 443.4 Using a calculator 46

4 Accuracy 504.1 Rounding 514.2 Estimating 54

5 Formulae and expressions 58

5.1 Algebraic terms and conventions 595.2 Simplifying expressions 605.3 Laws of indices 625.4 Substituting into formulae 635.5 Writing formulae 65

6 Algebraic manipulation 696.1 Expanding single brackets 706.2 Expanding two pairs of brackets 716.3 Factorisation 726.4 Rearranging formulae 73

7 Equations and inequalities 767.1 Linear equations of the form

ax + b = c 777.2 Linear equations of the form

ax + b = cx + d 79

7.3 Linear equations with brackets 807.4 Equation problems 817.5 Inequalities 82

8 Sequences 868.1 Sequences 878.2 Finding the nth term of an

arithmetic sequence 88

9 Straight-line graphs 929.1 Coordinates 939.2 Drawing straight-line graphs 949.3 Calculating the gradient of a

straight-line graph 96

10 Curved graphs 10210.1 Quadratic graphs 10310.2 Using graphs to solve quadratic

equations 10510.3 Graph shapes 107

11 Interpreting graphs 11311.1 Conversion graphs 11511.2 Real-life graphs 11611.3 Distance–time and velocity–time

graphs 119

12 Percentages 13012.1 Percentage of a quantity 13112.2 Increasing and decreasing

quantities by a percentage 13212.3 One quantity as a percentage

of another 13312.4 Percentage change 13412.5 Repeated percentage change 13612.6 Interest and tax 13712.7 Reverse percentage problems 139

13 Ratio and proportion 14313.1 Simplifying ratios 14413.2 Sharing a quantity in a given ratio 14513.3 Problem solving 14713.4 Increasing and decreasing in a

given ratio 149

14 Measurement 15314.1 Units of measurement 15414.2 Maps and diagrams 15714.3 Compound measures 159

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15 Similarity and congruence 165

15.1 Similar shapes 16615.2 Congruent shapes 169

16 2D and 3D shapes 17516.1 Properties of 2D shapes 17716.2 Symmetry 18016.3 Properties of 3D shapes 18116.4 Plans and elevations 184

17 Angles 18917.1 Classifying angles 19117.2 Angle facts 19217.3 Angles in parallel lines 19517.4 Angles in polygons 19817.5 Bearings 20117.6 Scale drawings 203

18 Pythagoras’ theorem 21118.1 Calculating an unknown length 21218.2 Calculating the length of a line

segment on a coordinate grid 21418.3 Solving problems 216

19 Trigonometry 22319.1 Trigonometric ratios 22419.2 Calculating a length 22619.3 Calculating an angle 228

20 Perimeter and area 23520.1 Plane shapes 23620.2 Compound shapes and problem

solving 24020.3 Circles 24220.4 Arc length and sector area 244

21 Surface area and volume 251

21.1 Simple 3D shapes 25221.2 More complex solids 25421.3 Converting units of area and

volume 258

22 Vectors and transformations 264

22.1 Translations 26622.2 Reflections 268

22.3 Rotations 27022.4 Enlargements 273

23 Probability 28223.1 The probability scale 28423.2 Calculating probabilities

associated with single events 28523.3 Probability that an event will

not happen 28723.4 Experimental probability 28823.5 Possible outcomes of two or

more events 29023.6 Probabilities of multiple events

and tree diagrams 292

24 Sets and Venn diagrams 299

24.1 Set notation 30024.2 Venn diagrams 30124.3 Using Venn diagrams to solve

probability questions 303

25 Statistics 31025.1 Collecting data 31325.2 Organising data 31525.3 Displaying data 31625.4 Pie charts 321

26 Averages 32926.1 Comparing data: averages and

spread 33026.2 Finding averages from a

frequency table 33226.3 Averages of grouped data 33426.4 Stem and leaf diagrams 335

27 Time series and scatter graphs 342

27.1 Scatter graphs and correlation 34327.2 Time-series graphs 347

Online stretch lessons 353Answers 355Formulae sheet 397Glossary 398Index 402

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6 Algebraic manipulation

Objectives

Before you start this chapter, mark how confi dent you feel about each of the statements below:

I can expand a single pair of brackets.

I can expand two pairs of brackets.

I can square a linear expression, e.g. (x + 1)2.

I can factorise algebraic expressions by taking out common factors.

I can factorise quadratic expressions of the form x2 + bx + c.

I can factorise a quadratic expression of the form x2 – a2 using the difference of two squares.

I can change the subject of a formula.

Check-in questions

• Complete these questions to assess how much you remember about each topic. Then mark your work using the answers at the back of the book.

• If you score well on all sections, you can go straight to the Revision checklist and Exam-style questions at the end of the chapter. If you don’t score well, go to the chapter section indicated and work through the examples and practice questions there.

1 Expand and simplify these expressions.

a t(3t – 4) b 4(2x – 1) – 2(x – 4) Go to 6.1

2 Expand and simplify these expressions.

a (x + 3)(x – 2) b 4x(x – 3) c (x – 3)2 Go to 6.2

3 Factorise these expressions.

a 12xy – 6x2 b 3a2b + 6ab2 c x2 + 4x + 4

d x2 – 4x – 5 e x2 – 100 Go to 6.3

4 Factorise these expressions. Expand and simplify first if necessary.

a y2 + y b 5p2q – 10pq2

c (a + b)2 + 4(a + b) d x2 – 5x + 6 Go to 6.2, 6.3

5 Make u the subject of the formula v2 = u2 + 2as. Go to 6.4

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6.1 Expanding single bracketsMultiplying out brackets helps to simplify algebraic expressions.

The term outside the bracket is multiplied by each of the terms inside the bracket.

5(x + 6) = 5x + 30

1

2

Exam tips When you multiply out brackets, check that the signs are correct for each term. If the signs are the same, it is +. If the signs are different, it is −.

Practice questions 1 Expand and simplify.

a 3(x + 5) b 7(y – 2) c 13(z + 5)

2 Expand and simplify.

a –2(x + 4) b 2(2x + 7) c –3(5x + 3y) d 5x(x + 4)

3 Expand and simplify.

a 2(x + 4) + 3(x + 1) b 3(y – 2) + 3(y + 3)

c 4(m + 5) – 2(m + 6) d 3(2m + 3) – 2(4m – 12)

4 Deepak expands 3(4f – 2). His result is 7f – 2.

Identify two errors that he has made.

5 Show that 3(y + 2) – 2(y – 3) ≡ 2(y + 6) – y.

Hint

Expand both sides and show that they give the same result.

This sign means ’is equivalent to’.

Expand and simplify these expressions.

a –2(2x + 4) b 5(2x – 3) c 8(x + 3) + 2(x – 1) d 3(2x – 5) – 2(x – 3)

a –2(2x + 4) = –4x – 8

b 5(2x – 3) = 10x – 15

c 8(x + 3) + 2(x – 1) = 8x + 24 + 2x – 2

= 10x + 22

d 3(2x – 5) – 2(x – 3) = 6x – 15 – 2x + 6

= 4x – 9

Collect like terms.

Example

1Q

A

Collect like terms.

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6.2 Expanding two pairs of bracketsEvery term in the second bracket must be multiplied by every term in the first bracket.

Often, but not always, the two middle terms are like terms and can be collected together.

(x + 4)(x + 2) = x2 + 2x + 4x + 8

= x2 + 6x + 8

12

3 4

Expand and simplify these expressions.

a (x + 4)(2x – 5) b (2x + 1)2 c (3x – 1)(x – 2)

d (x – 4)(3x + 1) e (2x + 3y)(x – 2y)

a (x + 4)(2x – 5) = 2x2 – 5x + 8x – 20

= 2x2 + 3x – 20

b (2x + 1)2 = (2x + 1)(2x + 1)

= 4x2 + 2x + 2x + 1

= 4x2 + 4x + 1

c (3x – 1)(x – 2) = 3x2 – 6x – x + 2

= 3x2 – 7x + 2

d (x – 4)(3x + 1) = 3x2 + x – 12x – 4

= 3x2 – 11x – 4

e (2x + 3y)(x – 2y) = 2x2 – 4xy + 3xy – 6y2

= 2x2 – xy – 6y2

Remember that x2 means x multiplied by itself.

Example

2Q

A

Exam tips Remember to check that the signs are correct for each term.

Practice questions 1 Expand and simplify.

a (y + 2)(y + 3) b (x + 4)(x + 2) c (m + 3)(m + 5)

2 Expand and simplify.

a (x + 4)(x – 1) b (y + 6)(y – 2) c (m – 2)(m + 8) d (x + 5)(x – 5)

3 Expand and simplify.

a (3x + 2)(x + 4) b (5y – 2)(5y + 4) c (7 – 2y)(3 – 8y)

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4 Match the expression on the left with the correct expansion on the right.

(x – 2)(x + 2) 2x2 – x – 1

(x + 3)(x – 1) x2 – 4

(x + 4)(x – 2) x2 + 2x – 3

(2x + 1)(x – 1) x2 + 2x – 8

5 Tash expands and simplifies (x + 2)(x – 3). Her result is x2 – 1.

Identify any errors she has made.

6.3 FactorisationFactorisation is the opposite of expansion. It involves looking for common factors and putting them outside brackets.

One pair of bracketsTo factorise 4x + 6:

• Identify the HCF of the terms. 2

• Divide the expression by the HCF. 2x + 3

• Write the HCF outside the brackets and the rest of the expression inside the brackets. 2(2x + 3)

When it is expanded or multiplied, it will equal the original expression.

Factorise fully.

a 8x – 16 b 3x + 18 c 5x2 + x d 4x2 + 8x

a 8(x – 2) b 3(x + 6) c x(5x + 1) d 4x(x + 2)

Example

3Q

A

Two pairs of bracketsTwo pairs of brackets are obtained when a quadratic expression of the type ax2 + bx + c is factorised.

Factorise these expressions.

a x2 + 4x + 3 b x2 – 7x + 12 c x2 + 3x – 10

a Find two numbers with a product of 3 (the constant) and a sum of 4 (the coefficient of x): 3 and 1 (3 × 1 = 3 and 3 + 1 = 4)

(x + 1)(x + 3)

b Find two numbers with a product of 12 (the constant) and a sum of –7 (the coefficient of x): –4 and –3 (–4 × –3 = 12 and –4 + –3= –7)

(x – 3)(x – 4)

c Find two numbers with a product of –10 (the constant) and a sum of 3 (the coefficient of x): –2 and 5 (–2 × 5 = –10 and –2 + 5 = 3)

(x + 5)(x – 2)

Example

4Q

A

Note that although –5 × 2 = –10, –5 + 2 = –3

Exam tips Remember to check that you have put all the common factors outside the bracket, otherwise the expression is not fully factorised.

Check there is no common factor inside the bracket for your final answer.

Exam tips

Questions 1 to 3 ask you to ‘Expand and simplify’. After expanding remember to collect like terms.

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Factorise these expressions.

a x2 – 64 b 81x2 – 25y2

a x2 – 64 = (x + 8)(x – 8)

b 81x2 – 25y2 = (9x + 5y)(9x – 5y)

Example

5Q

A This is known as the ‘difference of two squares’. In general, x2 – a2 = (x + a)(x – a).

Practice questions 1 Factorise these expressions.

a 4x + 10 b 15x + 35 c 14x – 120 d 6x – 30

2 Factorise fully these expressions.

a 5y + 10x + 25z b 16x – 48x2 + 56xz c 14x – 42y + 63z d –9m + 12n – 21p

3 Factorise these quadratic expressions.

a x2 + 5x + 6 b x2 + 10x + 16 c x2 – 10x + 21 d x2 + 6x – 16

4 Factorise these expressions.

a x2 – 9 b y2 – 49 c m2 – 1 d 9t2 – 121y2

5 In an exam, Ravjeet is asked to fully factorise 25x2 – 40x + 15xy.

As her answer, she writes 5(5x2 – 8x + 3xy).

She does not score full marks. Explain why.

6.4 Rearranging formulaeThe subject of a formula is the letter that appears on its own on one side of the formula.

Any letter in a formula can become the subject if you rearrange the formula. It is important to do the same things to both sides of the formula as you rearrange it.

Make c the subject of the formula: b = c – a

b = c – a

b + a = c

So c = b + a

Example

6Q

A Add a to both sides.

Make a the subject of the formula: b = (a – 3)2

b = (a – 3)2

± b = a – 3

± b + 3 = a

a = ± b + 3

Deal with the power fi rst. Square root both sides of the formula. There will be a positive and negative solution, shown as ± (see Chapter 1).

Add 3 to both sides of the formula.

Example

7Q

A

6.4 Rearranging formulae 73

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Make x the subject of the formula: 5(y + x) = 8x + 3

5(y + x) = 8x + 3

5y + 5x = 8x + 3

5y − 3 = 8x – 5x

5y − 3 = 3x

x = 5 3−3

y

Example

8Q

A

Divide both sides by 3 to make x the subject.

Collect the x terms on one side.

Exam tips Show all steps in your working. This will help you gain marks for your method even if your fi nal answer is incorrect.

Practice questions 1 Rearrange each formula to make a the subject.

a x = a + 6 b y = a – 6 c z = 6 – a

2 Rearrange each formula to make b the subject.

a x = 4b + 6 b y = 5b –3 c z = 6 – 2b

3 Rearrange each formula. The subject is given in brackets.

a V = IR (R) b A = bh2

(b) c P = mgh (g)

4 Make y the subject of the formula. 5y + m = 4(y – 3)

When the subject occurs on both sides of the equals sign, they need to be collected on one side. It is easier to collect the terms involving the new subject on the side where there are more of them.

REVISION CHECKLIST ● To expand single brackets, multiply the term outside the brackets by everything inside

the brackets.

● To expand two pairs of brackets, multiply each term in the second bracket by each term in the fi rst bracket.

● Factorising is the opposite of expanding brackets. Look for the highest common factor (HCF).

● You can rearrange a formula to make any letter the subject. When rearranging a formula, you must do the same thing to both sides.

Exam-style questions 1 Which of these are true?

a 5(a + 3) = 5a + 3 b 3(b + 2) = 3b + 6

c 5c(c – 2) = 5c2 – 2c d 2d(d + 4) = 2d2 + 8d

2 Factorise this expression. 3pr + 12qr – 21r2

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3 Ted factorises 5x – 40. His answer is 5(x – 40).

Show, by expanding his answer, that he is incorrect.

4 Expand and simplify this expression. 4(x + 3) + 2(x + 1)

5 Expand and simplify. (x + 8)(2x – 3)

6 Expand and simplify. (2y + 6)(3y – 2)

7 Write an expression for the perimeter of the rectangle. Fully expand and simplify your answer.

8 Write an expression for the area of this square. Fully expand and simplify your answer.

9 The area of this rectangle is 8m + 16. Write an expression for the length of the rectangle if the width is 8.

10 Rearrange this formula to make m the subject. y = mx + c

11 Rearrange this formula to make p the subject. W = prq

12 Expand and simplify this quadratic expression. (2x + 3)(5x – 4)

13 Show that 3(x + 4) + 2(x – 3) ≡ 5(x + 1) + 1.

14 Factorise this expression. x2 – 144

15 Factorise. 9p2 – 25q2

16 Rearrange this formula to make F the subject. 5 329

( )F C− =

17 Show that (2x – 4y)(2x + 4y) = 4x2 – 16y2.

18 Rearrange this formula to make r the subject. S = 4πr2

19 Rearrange this formula to make r the subject. V = 43

πr2

20 Factorise this expression. x2 – x – 20

Now go back to the list of objectives at the start of this chapter.

How confident do you now feel about each of them?

5(y + 2)

3y

(m – 2)

8 8m + 16

75Exam-style questions

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EdexcelGCSE (9-1) Mathsfor post-16

Achieve success in a year at Edexcel GCSE (9-1) Maths with resources that build exam con dence and help you master maths.

• Covers all the key topics that you need for a Grade 5 pass, with supportive teaching, worked examples and plenty of focused practice to bring you up to the right level.

• Realistic exam-style questions for each topic give you con dence you’re ready for assessment.

• Exam tips and techniques from an Edexcel examiner help you pick up smart marks and get the most out of every question.

• Take control of your learning with self-diagnostic charts and questions that make it easy to identify skills and knowledge gaps, track progress and target areas to work on.

• A exible course that can be used for a fast-track resit in November or full-year course with exams in June.

• Bonus online material for students who want to go further!

Written by Fiona Mapp, Head of Mathematics in a large 11-18 comprehensive school. She has over twenty years’ experience of being a head of department in addition to being a mathematics consultant for a local education authority. Fiona has been an Edexcel Assistant Principal Examiner and is the author of over fty mathematics books.

Introduction and exam tips by Su Nicholson, an experienced maths teacher in the post-16 sector, and Ofqual External Expert. A Senior Associate with Edexcel, Su is an examiner for Edexcel GCSE Maths and an accomplished mathematics author and trainer.

ISBN 978-0-00-822722-7

9 780008 227227

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