Edexcel post 16 maths sample chapter

Edexcel

GCSE (9-1) Maths for post-16

Fiona Mapp

with Su Nicholson

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Contents Introduction How to use this book

1

Numbers

1.1 1.2 1.3 1.4 1.5

Positive and negative numbers Square, cube and triangular numbers Factors, multiples and primes Indices Standard index form

2

Fractions and decimals

2.1 2.2 2.3 2.4

Fractions Calculating with fractions Fraction problems Decimals and fractions

3

Calculations

3.1 3.2 3.3 3.4

Multiplying and dividing numbers by power of 10 Written methods Order of operations Using a calculator

4

Accuracy

4.1 4.2

Rounding Estimating

5

Formulae and expressions

5 13

14 15 18 20 23 24

28 29 31 33 34

38 39 41 44 46

50 51 54

58

5.1 5.2 5.3 5.4 5.5

Algebraic terms and conventions Simplifying expressions Laws of indices Substituting into formulae Writing formulae

6

Algebraic manipulation

69

6.1 6.2 6.3 6.4

Expanding single brackets Expanding two pairs of brackets Factorisation Rearranging formulae

70 71 72 73

7

Equations and inequalities 76

7.1

Linear equations of the form ax + b = c Linear equations of the form ax + b = cx + d

7.2

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59 60 62 63 65

77 79

7.3 7.4 7.5

Linear equations with brackets Equation problems Inequalities

8

Sequences

8.1 8.2

Sequences Finding the nth term of an arithmetic sequence

80 81 82

86 87 88

9

Straight-line graphs

92

9.1 9.2 9.3

Coordinates Drawing straight-line graphs Calculating the gradient of a straight-line graph

93 94

10

Curved graphs

10.1 10.2 10.3

Quadratic graphs Using graphs to solve quadratic equations Graph shapes

11

Interpreting graphs

11.1 11.2 11.3

Conversion graphs Real-life graphs Distance–time and velocity–time graphs

12

Percentages

12.1 12.2

12.4 12.5 12.6 12.7

Percentage of a quantity Increasing and decreasing quantities by a percentage One quantity as a percentage of another Percentage change Repeated percentage change Interest and tax Reverse percentage problems

13

Ratio and proportion

13.1 13.2 13.3 13.4

Simplifying ratios Sharing a quantity in a given ratio Problem solving Increasing and decreasing in a given ratio

12.3

14

Measurement

14.1 14.2 14.3

Units of measurement Maps and diagrams Compound measures

96

102 103 105 107

113 115 116 119

130 131 132 133 134 136 137 139

143 144 145 147 149

153 154 157 159

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15

Similarity and congruence

165

15.1 15.2

Similar shapes Congruent shapes

166 169

16

2D and 3D shapes

16.1 16.2 16.3 16.4

Properties of 2D shapes Symmetry Properties of 3D shapes Plans and elevations

17

Angles

17.1 17.2 17.3 17.4 17.5 17.6

Classifying angles Angle facts Angles in parallel lines Angles in polygons Bearings Scale drawings

18

Pythagoras’ theorem

18.1 18.2 18.3

Calculating an unknown length Calculating the length of a line segment on a coordinate grid Solving problems

19

Trigonometry

19.1 19.2 19.3

Trigonometric ratios Calculating a length Calculating an angle

20

Perimeter and area

20.1 20.2

Plane shapes Compound shapes and problem solving Circles Arc length and sector area

20.3 20.4

21 21.1 21.2 21.3

22 22.1 22.2

Surface area and volume Simple 3D shapes More complex solids Converting units of area and volume

Vectors and transformations Translations Reflections

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175

22.3 22.4

Rotations Enlargements

23

Probability

23.1 23.2

The probability scale Calculating probabilities associated with single events Probability that an event will not happen Experimental probability Possible outcomes of two or more events Probabilities of multiple events and tree diagrams

177 180 181 184

23.3

189

23.6

191 192 195 198 201 203

211 212 214 216

223 224 226 228

235

23.4 23.5

24 24.1 24.2 24.3

251 252 254 258

Set notation Venn diagrams Using Venn diagrams to solve probability questions

25

Statistics

25.1 25.2 25.3 25.4

Collecting data Organising data Displaying data Pie charts

26

Averages

26.1

Comparing data: averages and spread Finding averages from a frequency table Averages of grouped data Stem and leaf diagrams

26.2

236 240 242 244

Sets and Venn diagrams

26.3 26.4

27 27.1 27.2

Time series and scatter graphs Scatter graphs and correlation Time-series graphs

Online stretch lessons Answers Formulae sheet Glossary Index

270 273

282 284 285 287 288 290 292

299 300 301 303

310 313 315 316 321

329 330 332 334 335

342 343 347 353 355 397 398 402

264 266 268

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6

Algebraic manipulation

Objectives Before you start this chapter, mark how confident 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) 2

4

5

Go to 6.1

Expand and simplify these expressions.

a (x + 3)(x – 2) 3

b 4(2x – 1) – 2(x – 4)

b 4x(x – 3)

c (x – 3)2

Go to 6.2

Factorise these expressions.

a 12xy – 6x2

b 3a2b + 6ab2

d x2 – 4x – 5

e x2 – 100

c x2 + 4x + 4 Go to 6.3

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

Make u the subject of the formula v2 = u2 + 2as.

Go to 6.2, 6.3 Go to 6.4

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

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

1

5(x + 6) = 5x + 30 2

Example Q Expand and simplify these expressions.

1

a –2(2x + 4)

b 5(2x – 3)

c 8(x + 3) + 2(x – 1)

d 3(2x – 5) – 2(x – 3)

A 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

Collect like terms.

d 3(2x – 5) – 2(x – 3) = 6x – 15 – 2x + 6 Collect like terms.

= 4x – 9

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)

2

4

c 13(z + 5)

Expand and simplify. a –2(x + 4)

3

b 7(y – 2)

b 2(2x + 7)

c –3(5x + 3y)

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)

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

Identify two errors that he has made.

5

d 5x(x + 4)

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’.

70

6 Algebraic manipulation

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6.2 Expanding two pairs of brackets Every 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.

2 1

2 + 2x + 4x + 8 (x + 4)(x + 2) = x 3 4

= x2 + 6x + 8 Example Q Expand and simplify these expressions.

2

a (x + 4)(2x – 5)

b

(2x + 1)2

d (x – 4)(3x + 1)

e

(2x + 3y)(x – 2y)

c (3x – 1)(x – 2)

A 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

Remember that x2 means x multiplied by itself.

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

Exam tips

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

Practice questions 1

Expand and simplify.

a (y + 2)(y + 3) 2

c (m + 3)(m + 5)

b (y + 6)(y – 2)

c (m – 2)(m + 8)

b (5y – 2)(5y + 4)

c (7 – 2y)(3 – 8y)

Expand and simplify.

a (x + 4)(x – 1) 3

b (x + 4)(x + 2)

d (x + 5)(x – 5)

Expand and simplify.

a (3x + 2)(x + 4)

6.2 Expanding two pairs of brackets

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4

5

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 +

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

2x – 8

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

Identify any errors she has made.

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

One pair of brackets To 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.

Example Q Factorise fully.

3

Exam tips

a 8x – 16

b 3x + 18

c 5x 2 + x

d 4x2 + 8x

A a 8(x – 2)

b 3(x + 6)

c x(5x + 1)

d 4x(x + 2)

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.

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

Example Q Factorise these expressions.

4

a x2 + 4x + 3

b x2 – 7x + 12

c x2 + 3x – 10

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

3 and 1 (3 × 1 = 3 and 3 + 1 = 4)

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

–4 and –3 (–4 × –3 = 12 and –4 + –3= –7)

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

72

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

6 Algebraic manipulation

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Example Q Factorise these expressions.

5

a x2 – 64

b 81x2 – 25y2

A a x2 – 64 = (x + 8)(x – 8) b

81x2

–

25y2 =

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

(9x + 5y)(9x – 5y)

Practice questions 1

Factorise these expressions.

a 4x + 10 2

b 15x + 35

c 14x – 42y + 63z

d –9m + 12n – 21p

b x2 + 10x + 16

c x2 – 10x + 21

d x2 + 6x – 16

c m2 – 1

d 9t2 – 121y2

Factorise these expressions.

a x2 – 9 5

b 16x – 48x2 + 56xz

Factorise these quadratic expressions.

a x2 + 5x + 6 4

d 6x – 30

Factorise fully these expressions.

a 5y + 10x + 25z 3

c 14x – 120

b y2 – 49

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 formulae The 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.

Example Q Make c the subject of the formula:

6

A

b=c–a

b=c–a Add a to both sides.

b+a=c So c = b + a

Example Q Make a the subject of the formula:

7

A

b = (a – 3)2 ± b =a–3 ± b +3=a a=± b +3

b = (a – 3)2

Deal with the power first. 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.

6.4 Rearranging formulae

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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.

Example Q Make x the subject of the formula:

8

5(y + x) = 8x + 3

A 5(y + x) = 8x + 3 5y + 5x = 8x + 3 5y − 3 = 8x – 5x

Collect the x terms on one side.

5y − 3 = 3x x = 5y − 3 3

Exam tips

Divide both sides by 3 to make x the subject.

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

Practice questions 1

Rearrange each formula to make a the subject.

a x=a+6 2

4

c z=6–a

Rearrange each formula to make b the subject.

a x = 4b + 6 3

b y=a–6

b y = 5b –3

c z = 6 – 2b

Rearrange each formula. The subject is given in brackets. a V = IR (R) b A = bh (b) c P = mgh 2 Make y the subject of the formula.

(g)

5y + m = 4(y – 3)

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 first 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

2

74

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

Factorise this expression.

3pr + 12qr – 21r2

6 Algebraic manipulation

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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.

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

9

4(x + 3) + 2(x + 1)

5(y + 2)

3y

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

The area of this rectangle is 8m + 16.

Write an expression for the length of the rectangle if the width is 8.

(m – 2)

8

8m + 16

10

Rearrange this formula to make m the subject.

y = mx + c

11

Rearrange this formula to make p the subject.

W=

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.

15

Factorise.

16

Rearrange this formula to make F the subject.

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=

20

Factorise this expression.

pr q

x2 – 144

9p2 – 25q2 5 ( F − 32) =C 9

4 2 πr 3

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?

Exam-style questions

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Edexcel GCSE (9-1) Maths for post-16 Achieve success in a year at Edexcel GCSE (9-1) Maths with resources that build exam conwdence 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 conwdence 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 ƃexible 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 wfty 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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