advertisement

Edexcel GCSE MATHEMATICS Foundation Keith Pledger The Publishers would like to thank the following for permission to reproduce copyright material. Photo credits Acknowledgements Every effort has been made to trace all copyright holders, but if any have been inadvertently overlooked, the Publishers will be pleased to make the necessary arrangements at the first opportunity. Although every effort has been made to ensure that website addresses are correct at time of going to press, Hodder Education cannot be held responsible for the content of any website mentioned in this book. It is sometimes possible to find a relocated web page by typing in the address of the home page for a website in the URL window of your browser. Hachette UK’s policy is to use papers that are natural, renewable and recyclable products and made from wood grown in sustainable forests. The logging and manufacturing processes are expected to conform to the environmental regulations of the country of origin. Orders Bookpoint Ltd, 130 Park Drive, Milton Park, Abingdon, Oxon OX14 4SE. Telephone: (44) 01235 827720. Fax: (44) 01235 400454. Email education@bookpoint.co.uk Lines are open from 9 a.m. to 5 p.m., Monday to Saturday, with a 24-hour message answering service. You can also order through our website: www.hoddereducation.co.uk ISBN: 978 1 4718 8246 3 © Keith Pledger 2016 First published in 2016 by Hodder Education, An Hachette UK Company Carmelite House 50 Victoria Embankment London EC4Y 0DZ www.hoddereducation.co.uk Impression number 10 9 8 7 6 5 4 3 2 1 Year 2019 2018 2017 2016 2015 All rights reserved. Apart from any use permitted under UK copyright law, no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or held within any information storage and retrieval system, without permission in writing from the publisher or under licence from the Copyright Licensing Agency Limited. Further details of such licences (for reprographic reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Cover photo Typeset in Printed in A catalogue record for this title is available from the British Library. Get the most from this book Everyone has to decide his or her own revision strategy, but it is essential to review your work, learn it and test your understanding. These Revision Notes will help you to do that in a planned way, topic by topic. Use this book as the cornerstone of your revision and don’t hesitate to write in it — personalise your notes and check your progress by ticking off each section as you revise. You can also keep track of your revision by ticking off each topic heading in the book. You may find it helpful to add your own notes as you work through each topic. Revision planner page Tick to track your progress Use the revision planner on pages 4 and 5 to plan your revision, topic by topic. Tick each box when you have: l revised and understood a topic l tested yourself l checked your answers Example page Features to help you succeed Rules The key rules you need to follow when answers questions on this topic. Worked examples Several worked examples are given for each topic. Key terms Exam-style questions Practice exam questions are provided for each topic. Use them to consolidate your revision and practise your exam skills. Answers Check how you’ve done using the answers at the back of the book. Key terms are highlighted in each section. Exam tips Expert tips are given throughout the book to help you polish your exam technique in order to maximise your chances in the exam. iii My revision planner Number Number: pre-revision check xx BIDMAS xx Multiplying decimals xx Dividing decimals xx Using the number system effectively xx Understanding standard form xx Calculating with standard form xx Rounding to decimal places, significance and approximating xx Limits of accuracy xx Multiplying and dividing fractions xx Adding and subtracting fractions and working with mixed numbers xx Converting fractions and decimals to and from percentages xx Applying percentage increases and decreases to amounts xx Finding the percentage change from one amount to another xx Reverse percentages xx Repeated percentage increase/decrease Mixed exam-style questions xx Sharing in a given ratio xx Working with proportional quantities xx The constant of proportionality xx Working with inversely proportional quantities xx Index notation and rules of indices xx Prime factorisation Mixed exam-style questions Number: summary of rules iv Edexcel GCSE Maths Revision Guide Foundation Algebra Algebra: pre-revision check xx Working with formulae xx Setting up and solving simple equations xx Using brackets xx Solving equations with the unknown on both sides xx Solving equations with brackets xx Simplifying harder expressions and expanding two brackets xx Using complex formulae and changing the subject of a formula xx Identities xx Linear sequences xx Special sequences xx Quadratic sequences xx Geometric progressions Mixed exam-style questions xx Real-life graphs xx Plotting graphs of linear functions xx The equation of a straight line xx Plotting quadratic and cubic graphs xx Finding equations of straight lines xx Quadratic functions xx Polynomial and reciprocal functions xx Linear inequalities xx Solving simultaneous equations by elimination and substitution xx Using graphs to solve simultaneous equations xx Factorising quadratics of the form x2 + bx + c xx Solve equations by factorising Mixed exam-style questions Algebra: summary of rules v Geometry Geometry: pre-revision check xx Bearings and scale drawings xx Compound units xx Working with compound units xx Types of quadrilateral xx Angles and parallel lines xx Angles in a polygon xx Congruent triangles and proof xx Proof using similar and congruent triangles xx Circumference xx Pythagoras theorem xx Arcs and sectors Mixed exam-style questions xx Constructions with a pair of compasses xx Loci xx Enlargement xx Similarity xx Trigonometry xx Trigonometry for special angles xx Finding centres of rotation xx Understanding nets and 2D representation of 3D shapes xx Volume and surface area of cuboids and prisms xx Enlargement in two and three dimensions xx Constructing plans and elevations xx Surface area and 3D shapes xx Vectors Mixed exam-style questions Geometry: summary of rules vi Edexcel GCSE Maths Revision Guide Foundation Statistics and probability Statistics and probability: pre-revision check xx Using frequency tables xx Using grouped frequency tables xx Vertical line charts xx Pie charts xx Displaying grouped data xx Scatter diagrams and using lines of best fit Mixed exam-style questions xx Single event probability xx Combined events xx Estimating probability xx The multiplication rule xx The addition rule Statistics and probability: exam-style questions Statistics and probability: summary of rules Exam preparation xx xx xx xx xx xx The language used in mathematics examinations Exam technique Key topics for revision One week to go Answers Index vii Algebra: pre-revision check 1 This formula gives the value of p in terms of q and r: p = 2q – 3r. Find the value of p when q = 10 and r = 4. F S1 U4 S = ut + 1 at 2 2 a Find the value of S when u = 5, t = 4 and a = 10. b Make a the subject of the formula. F/H S1 U10 2 Solve the equations a a+4=6 b b =5 3 c 5c + 4 = 6 d 15 – 3e = 24 F S1 U5 2 8 Prove that the sum of the three consecutive numbers (n – 1), n and (n + 1) is a multiple of 3. F/H S1 U11 3 a Expand these brackets i 5(2a + 3) ii h(3h – 6) iii 3x(4x – 2y) b Factorise fully i 6y + 12 ii 6p2 – 9p iii 5e2 + 10ef iv 8x2y – 12xy2 F S1 U6 9 Here are the first 5 terms of a linear sequence. 4 10 16 22 28… a Find the nth term of the sequence. b Work out the 50th term in the sequence. c Explain if 900 is a member of the sequence. F S2 U3 F/H S1 U8 6 a Simplify i a4 × a6 x8 ii 5 x 12e6f 7 iii 8e 9f 5 b Expand and simplify i (t + 2)(t + 5) ii (v – 7)(v + 5) iii (y – 6)(y – 5) F/H S1 U9 10 a Write down the first 5 terms of the quadratic sequence with nth term 2n2 – 3. b Find the nth term of the quadratic sequence that has the first 5 terms 3, 8, 15, 24, 35. F S2 U4 11 The nth term of quadratic sequence is n2 + 5. The nth term of a different quadratic sequence is 80 – 2n2. Find the number that is in both sequences. F/H S2 U5 12 Here is a geometric sequence. 5 15 45 135 … a Find the common ratio. b Find the 10th term of the sequence. F/H S2 U6 13 Here is the graph that shows the depth of water in a harbour. A ship needs to enter the harbour between 0800 and 2000. It needs a 4 metre depth of water in the harbour. Between what times can the ship enter the harbour? F S3 U1 10 Depth of water (m) 4 Solve these equations a 5x – 6 = 2x + 3 b 7 – 2p = 6p + 13 y y c 2 – 3 = 5 + 5 F/H S1 U7 2 4 5 Solve. a 5(3g – 2) = 35 b 4(5h + 7) = 3(2h + 8) c 2(5k + 8) – 6 = 4(2k – 1) 7 This formula is used to find the distance, S, travelled by an object. 5 0 00:00 04:00 08:00 12:00 16:00 20:00 24:00 Time Algebra 1 Algebra: pre-revision check 14 a On a coordinate grid drawn with values of x from –3 to +3 and values of y from –6 to +8, draw the graph of y = 2x + 1. b Find the value of x when y = 6 F S3 U2 15 Find the equation of a straight line graph that passes through the point (0, –2) and has a gradient of 3. F S3 U3 22 Here is the graph of the line y + 2x = 3. y 6 16 a On a coordinate grid drawn with values of x from –2 to +4 and values of y from –6 to +8, draw the graph of y = x2 – 3x – 2. b Find the values of x when x2 – 3x – 2 = 0. F S3 U4 17 Find the equation of a straight line graph that passes through the point (–2, 3) and is parallel to the line x + 2y = 8. F/H S3 U5 20 a Write down the inequality shown on this number line. x 1 2 3 4 2 −3 −2 −1 0 1 2 3 x −2 Find graphically the solution to the simultaneous equations: y + 2x = 3 y – 2x = 1 F/H S4 U5 19 a On a coordinate grid drawn with values of x from –3 to +3 and values of y from –10 to +30, draw the graph of y = x3 + x2 – 3x. b Find the values of x when x3 + x2 – 3x = 0. F/H S3 U7 −5 −4 −3 −2 −1 0 4 −4 18 a Sketch the graph of the quadratic function y = x2 – 4x + 3 for values of x from 0 to 5. b Write down the roots of the equation x2 – 4x + 3 = 0. c Write down the line of symmetry of the graph. F/H S3 U6 5 b Solve these inequalities i 2x + 5 < 9 ii 24 + 2t > 30 – 3t iii 5(y – 3) 3y – 6 F/H S4 U2 2 21 Solve this pair of simultaneous equations. 5x + 2y = 8 2x – y = 5 F/H S4 U 3 & 4 Edexcel GCSE Maths Revision Guide Foundation 23 a Expand and simplify i (x + 4)(x – 5) ii (y + 8)(y – 8) iii (6 – a)(a + 6) b Factorise i x2 + 7x + 12 ii e2 – 3e – 10 iii b2 – 25 F/H S5 U1 24 Solve the equations a x2 – 5x + 6 = 0 b x2 – 2x = 15 c p2 – 49 = 0 F/H S5 U2 DIFFICULTY LEVEL mID Rules 1 2 3 You can replace words or letters in a formula with numbers. Use BIDMAS to find the value of the missing word or letter. Use inverses to write the formula or equation so that the missing letter is on its own on one side of the formula or equation. Worked examples Look out for a 2w means 2 × w Here is a formula to ﬁnd the perimeter of a rectangle: P = 2l + 2w. Find the value of P when l = 6 and w = 4. P = 2l + 2w b 1 P=2×6+2×4 2 P = 12 + 8 = 20 Algebra: pre-revision check Working with formulae So if w = 5 then 2w is 2 × 5 = 10 and not 25 Cars 2U Key terms Tom hires a car from Cars 2U. Formula i How much does it cost to hire a car for 7 days? ii Ben has £100. For how many days can he hire a car? Substitute Variable Equation £20 plus £30 a day You must explain your answer. i Cost = 20 + 7 × 30 Cost = 20 + 210 2 Cost = £230 1 ii 100 = 20 + N × 30 1 100 – 20 = N × 30 3 80 = N × 30 so N = 80 ÷ 30 = 2.666… 2 Ben can hire the car for 2 days. This costs £80. 3 days cost £110 which is too much. Exam tips Always show your working when answering algebra questions. Do not use trial and error methods as you may well lose marks. Exam-style questions 1 Bobbie uses this number machine to work out the number of cartons of orange juice she needs for a party. Number of people ÷5 +2 Number of cartons a How many cartons will Bobbie need for 40 people? b How many people are in a party that uses 20 cartons? 2 This formula gives the time taken T minutes to cook a chicken of weight w kg. T = 40w + 20 a How long does it take to cook a chicken of weight 2.5 kg? b It takes 3 hours 20 minutes to cook a different chicken. How heavy was the chicken? [2] [2] [2] [2] Algebra 3 Algebra: pre-revision check Setting up and solving simple equations DIFFICULTY LEVEL mid Rules 1 2 3 4 Always use the inverse operations to solve an equation. + and – are the inverse of one another. × and ÷ are the inverse of one another. To set up an equation a variable must be defined. Worked examples a 2 3 b 2 2p + 5 = 17 (–5 is the inverse operation of + 5) 2p + 5 – 5 = 17 – 5 (subtract 5 from each side of the equation) Key terms 2p = 12 (÷ 2 is the inverse of × 2) Equation p = 6 (divide each side of the equation by 2) Inverse operation nn is two years younger than Ben. Clara is twice as old as A Ben. The total of their ages is 58. Work out their ages. Solve Variable Let Ben’s age be x, Firstly, set up the equation: Ann’s age will be x – 2, x + (x – 2) + 2x = 58 Clara’s age will be 2x Now collect like terms: Exam tips 4x – 2 = 62 Always use algebraic methods and show your working to gain full marks. 4x = 60 (Add 2 to each side) (Divide each side by 4) x = 15 Always check your answer to make sure it is correct. Ann will be 13, Ben 15 and Clara 30. Exam-style questions 1 Solve the equations b a a – 3 = 7 [1]; c b = 3 [1]; 5 5d + 4 = 29 [2]; d e 6 – 2e = 3 [2] 2 2 Here is a rectangle. 2x + 5 The length is 2x + 5. The width is x – 3. The perimeter is 46 cm. Work out the area of the rectangle in cm2. [4] 4 Edexcel GCSE Maths Revision Guide Foundation 3c + 9 = 7 [2]; x–3 DIFFICULTY LEVEL mid Rules 1 2 When you expand a bracket you multiply what is inside the bracket by the number or variable outside the bracket. When you factorise an algebraic expression you take out the common factor from each term of the expression and put it outside the bracket. Worked examples a Expand i 4(3x + 5); 1 Look out for ii t(3t – 4) x × x = x2 using index laws 1 Key terms t × (3t − 4) t × 3t − t × 4 3t2 − 4t 4 × (3x + 5) 4 × 3x + 4 × 5 12x + 20 Brackets Variable Expression b Factorise i 4p + 6 2 × 2 × p + 2 × 3 2 is in 4 and in 6 2 2(2p + 3) Algebra: pre-revision check Using brackets Expand ii 6p2q – 9pq2 3 × p × q × 2 × p – 3 × p × q × 3 × q 3 and p and q are in both terms 2 3pq(2p – 3q) Factorise Exam-style questions 1 Beth is 3 years older than Amy. Cath is twice as old as Beth. The total of their ages is 41. How old are the three girls? [3] 2 PQRS is a rectangle. The length of the rectangle is (2x – 5) cm (2x – 5) cm. P Q The width of the rectangle 6 cm is 6 cm. The area of the rectangle R S is 72 cm2. Work out the perimeter of the rectangle. [4] Exam tips Always check your answer by multiplying out the brackets. Always define your variable e.g. let Amy’s age be x. Then set up your equation. Algebra 5 Mixed exam-style questions (Strands 1, 2 and 3) 1 The diagram show the position of two villages. N Redford Redford is on a bearing of 050° from Brownhills. Karen walks from Brownhills to Redford. She walks at an average speed of 6 km/h. Brownhills She takes 1 h 30 mins to cover the distance. a Work out the distance between Brownhills and Redford. [2] b Using a scale of 1 cm to 4 km, make an accurate scale drawing showing the position of the two villages. [3] 2 The diagram shows a square attached to four similar regular polygons. Calculate the number of sides on the polygons. [3] 3 ABC is an isosceles triangle. D and E are points on BC. AB = AC, BD = EC Prove that triangle ADE is isosceles. [3] A B D E C 4 The wheel on a bicycle has a diameter of 70 cm. John cycles 15 km on the bicycle. a How many revolutions will the wheel make during the journey? b The journey takes John 1 h 20 mins. Calculate John’s average speed. 5 In the diagram PQ is parallel to ST. QX = XS Prove that triangles PQX and STX are congruent. [4] [2] P S X 6 The diagram shows a plan of a garden design. Q T A circular pond with a diameter of 1.5 m is dug in the lawn. The centre of the pond is in the centre of the lawn. a Make an accurate scale drawing of the plan using a scale of 2 cm: 1 m.[4] b Calculate the area of the lawn. [4] 2m 4m patio pond lawn 8m 6 Edexcel GCSE Maths Revision Guide Foundation [3] [4] A B 3 cm C 8 A paper cone is made from a folding a piece of paper in the shape of sector of a circle. The angle at the centre of the sector is 100°. The radius of the sector is 6 cm. a Calculate the length of the arc of the sector. [2] b Calculate the diameter of the base of the finished cone [2] 6 cm 100° 9 The diagram shows a trapezium PQRS. PQ is parallel to SR. PS = QR. Show that triangles PXS and QXR are similar. P Mixed exam-style questions 7 The diagram shows a right-angled triangle drawn inside a quarter circle. The chord AC is 3 cm. a Calculate the radius of the circle. b Calculate the area of the segment ABC. [3] Q X S R 10 ABCD is a rhombus. AC = 10.8 cm, BD = 15.6 cm. Calculate the length of the sides of the rhombus. A [4] B 10.8 cm 15.6 cm D C Geometry 7 Exam technique ● Be prepared and know what to expect. ● Read each question carefully. ● Don’t just learn key points. ● Show stages in your working. ● Work through past papers. Start from the back and work towards the easier questions. Your teacher will be able to help you. ● Check your answer has the units. ● Work steadily through the paper. ● Practice is the key, it won’t just happen. ● Skip questions you cannot do and then go back to them if time allows. ● Read the question thoroughly. ● Use marks as a guide for time 1 mark = 1 min. ● Present clear answers at the bottom of the space provided. ● Go back to questions you did not do. ● Read the information below the diagram – this is accurate. ● Use mnemonics to help remember formula you will need, for example: SOH sin = opposite/hypotenuse CAH cos = adjacent/hypotenuse TOA tan = opposite/adjacent or ‘silly old hens cackle and hale, till old age’. for the order of operations, BIDMAS: Brackets, Indices, Division, Multiply, Add, Subtract formula triangles for the relationship between three parameters e.g. speed, distance and time ● Cross out answers if you change them, only give one answer. ● Underline the key facts in the question. ● Estimate the answer. ● Is the answer right/realistic? ● Have the right equipment. Calculator Pens Pencils Ruler, compass, protractor Eraser Tracing paper Spares ● Never give two different answers to a question. ● Never just give just an answer if there is more than 1 mark. ● ● Never measure diagrams; most diagrams are not drawn accurately. D S D S Never just give the rounded answer; always show the full answer in the working space. Edexcel GCSE Maths Revision Guide Foundation T D S 8 T T Distance = Speed × Time Time = Distance Speed Speed = Distance Speed Exam preparation The following formulae will be provided for students within the relevant examination questions. Perimeter, area, surface area and volume formulae Where r is the radius of the sphere or cone, l is the slant height of a cone and h is the perpendicular height of a cone: Curved surface area of a cone = π rl Volume of a cone = l 1 2 πr h 3 h r Surface area of a sphere = 4π r 2 Volume of a sphere = 4 3 πr 3 r All other formulae and rules must be learnt. Exam technique 9 Key topics for revision Here are some topics that students frequently make errors in during their exam. Number Rules 1 2 3 A factor is a number that divides into another number, e.g. 2 is a factor of 6. A multiple is a member of the multiplication table of that number, e.g. 6 is a multiple of 2. A prime number is one that can only be divided by 1 and itself, e.g. 2, 3, 5, 7, 11 … Highest Common Factor (HCF) Question Working Find the HCF of 24 and 36. Use factor trees to find all the factors of 24 and 36: Answer 24 = 2 × 2 × 2 × 3 36 = 2 × 2 × 3 × 3 The common factors are: 2 × 2 × 3 = 12 Find the LCM Lowest of 9 and 12. Common Multiple (LCM) Adding 12 List the multiples of 9 and 12: 9, 18, 27, 36, 45 12, 24, 36, 48 36 Question Working Answer 2 1 + 3 4 Write equivalent fractions: 2 4 6 8 1 2 3 = = = and = = 3 6 9 12 4 8 12 12 is the LCM of 3 and 4 so write the fractions in 12ths: 8 3 8 + 3 11 + = = 12 12 12 12 Subtracting 2 1 − 3 4 You use the same method as adding but just take away so we get: 8 3 8−3 5 − = = 12 12 12 12 Multiplying 2 3 × 5 8 5 12 Multiply the tops together and then the bottoms of the fractions: 2×3 6 = then cancel by 2 5 × 8 40 10 11 12 Edexcel GCSE Maths Revision Guide Foundation 3 20 Changing a fraction to a decimal Write the first fraction down and turn the second fraction upside down and multiply: 5 2 ÷ 12 3 5 3 15 × = then cancel be 3 12 2 24 5 8 Question Working Answer 3 as a 8 decimal. Divide the top number by the bottom number so divide 3 by 8: Write 0.3 7 5 3 Question Finding a fraction of an amount 3 of 5 £4.80. Exam preparation Dividing 6 4 8 3. 0 0 0 0.375 Working Answer 3 × £4.80 5 There is a simple rule for this calculation, which is ‘Divide by the bottom and Times by the top’ This can be written as Find You can sing it to the ‘Wheels on the bus’ song to help you remember it. £4.80 ÷ 5 = 0.96 then 0.96 × 3 = £2.88 Finding a percentage of an amount Estimating Work out 60% of £4.80. £2.88 60 × £4.80. 100 You can use the same rule so you divide by 100 and times by 60: This can be written as £4.80 ÷ 100 × 60 = £2.88 £2.88 Question Working Answer Estimate Write each number to one significant figure so that: 76.15 × 0.49 19.04 76.15 becomes 80 0.49 becomes 0.5 19.04 becomes 20 Remember that the size of the estimate needs to be similar to the original number. Question Using a calculator Work out 2 76.15 + 5.62 19.04 So 80 × 0.5 = 40 and 40 ÷ 20 = 2 2 Working Answer You either need to enter the whole calculation into your calculator using the fraction button or work out the top first then divide the answer by the bottom. 76.15 + 5.622 = 107.7344 107.73 ÷ 19.04 = 5.658319327 5.658319327 Key topics for revision 11 Exam preparation Algebra Rules 2y × y = 2y2; 2y + y = 3y; 2y − y = y; y ×y =y y ÷y =y m n m+n m n 2y ÷ y = 2 (y ) = y m−n m n Question mn Working Answer = 2ab + 3ab − 1ab 4ab = 3y2 − 1y2 2y2 Question Working Answer Simplify =x x9 Collecting like Simplify terms a 2ab + 3ab − ab b 3y2 − y2 Index laws a x4 × x5 b = a7 a3 c (y2)3 Multiplying out the brackets Factorising expressiona a7 a3 = a7 ÷ a3 = a7− 3 = y2×3 d 7f g × 2f g 4 3 4+5 a4 y6 14f 7g4 = 7 × 2 × f 4 + 3 × g3+1 3 Question Working Answer Expand = 3 × 4p + 3 × 5 12p + 15 a 3(4p + 5) = 7p − 4 × p − 4 × −q = 7p − 4p + 4q 3p + 4q b 7p − 4(p − q) =y×y+y×−4+3×y+3×−4 y2 − y − 12 c (y + 3)(y − 4) = y2 − 4y + 3y − 12 Question Working Answer Factorise completely =5×a×b+2×5×b×c 5b(a + 2c) a 5ab + 10bc =3×4×e×e×f−3×3×e×f×f 3ef(4e − 3f) b 12e f – 9ef = x + (3 + 4)x + 3 × 4 (x + 3)(x+ 4) Question Working Answer Solve 3t = 4 + 2 so 3t = 6 t=2 a 3t − 2 = 4 4 + 3= 5f − 3f so 7 = 2f or 2f = 7 f = 3.5 b 3f + 4 = 5f − 3 5x + 10 = 3 so 5x = 3 − 10 or 5x = −7 x = −1.4 c 5(x + 2) = 3 (y + 2)(y − 5) = 0 so y + 2 = 0 or y − 5 = 0 y = −2 or y = 5 2 2 2 c x2 + 7x + 12 Solving equations d y − 3y − 10 = 0 2 12 Edexcel GCSE Maths Revision Guide Foundation Exam preparation Geometry and measure Rules The perimeter of a shape is the distance around its edge. You add all the side lengths together. The area of a shape is the amount of flat surface it has. You multiply two lengths. The volume of a shape is the amount of space it has. You multiply three lengths. Alternate angles are in the shape of a letter Z. Corresponding angles are in the shape of a letter F. Allied angles or co-interior angles are in the shape of a letter C. Question Perimeter of a Find the perimeter of this shape. shape Working Answer For a rectangle you need to add the lengths of the four sides. 16 cm 3 + 5 + 3 + 5 = 16 3 cm 5 cm Area of a shape Question Working Find the area of this shape. For this right-angled triangle you need to use 9 cm2 the formula: Area = ½ base × vertical height 3 cm So the area = ½ × 6 × 3 = 9 6 cm Volume of a solid Answer You have multiplied two lengths Question Working Answer Find the volume of this shape with radius 5 cm and height 12 cm in terms of π For this cylinder you need to use the formula: 300π cm3 Volume = π × r2 × h So volume is π × 5 × 5 × 12 = 300π You have multiplied three lengths Angles between parallel lines Question Working and answer Find the missing angles in this diagram. Give reasons for your answer. a = 50° (Alternate angles are equal) c = 50° (Corresponding angles are equal) 50° a b b = 130° (Allied angles add to 180° (supplementary)) c Key topics for revision 13 Exam preparation Question Working and answer ABC is an isosceles triangle. BCD is Finding a straight line. missing angles and Find, giving reasons, angle ACD. giving reasons A 50° Angle ABC = (180 – 50) ÷ 2 = 65° (The three angles of a triangle add to 180°) Angle ACB = Angle ABC = 65° (Base angles of an isosceles triangle are equal) Angle ACD = 180 – 65 = 115° (Sum of the angles on a straight line = 180°) D C B Statistics and probability Mean from a grouped frequency table Pie chart 14 Question Working Answer Work out an estimate of the mean age from this frequency table Multiply the mid value of the age groups by the frequency 24 5 × 4 = 20 Age f 0 a < 10 4 10 a < 20 6 35 × 5 = 175 20 a < 30 12 45 × 3 = 135 30 a < 40 5 Divide the total of age × frequency by the total frequency: 720 ÷ 30 40 a < 50 3 15 × 6 = 90 25 × 12 = 300 Note: Don’t forget to divide by the total frequency and not the number of groups (5) Question Working Answer Draw a pie chart from this information As pie charts are based on a circle then we need to divide the number of degrees in a whole turn (360°) by the total frequency which is 20. So 360° ÷ 20 = 18° Red = 7 × 18° = 126° Favourite colour f Red 7 Blue 4 Green 2 Yellow 3 Black 4 Blue = 4 × 18° = 72° Green = 2 × 18° = 36° Yellow = 3 × 18° = 54° Black = 4 × 18° = 72° The angle for each colour is then Then draw the circular pie calculated by multiplying its chart frequency by 18° Edexcel GCSE Maths Revision Guide Foundation One week to go You need to know these formulae and essential techniques. Number Topic Formula When to use it Negative numbers ++=+ ––=+ Two signs next to each other +–=– –+=– Multiplying integers +×+=+ –×–=+ Dividing integers +×–=– –×+=– +÷+=+ –÷–=+ +÷–=– –÷+=– Order of operations BIDMAS Percentages 20% of 50 = Simple interest SI for 5 years at 3% on £150 If you have to carry out a calculation. You use the order Brackets, Indices, Division, Multiplication, Addition and Subtraction 20 × 50 100 3 × 150 × 5 100 Compound interest Standard form Approximating CI for 2 years at 3% on £150 Year 1 3 × 150 = £4.50 100 Year 2 3 × (150 + 4.50 ) 100 To find the percentage of an amount e.g. 20% of 50. To find the simple interest you find the interest for one year and multiply by the number of years. For compound interest you find the percentage interest for one year, add it to the initial amount and find the interest on the total and so on. You can also do this using geometric progressions and write £150 × (1.03)2 = 2.5 × 10 3 = 2500 A number in standard form is 2.5 × 10 −3 = 0.0025 (a number between 1 and 10) × (a power of 10) Decimal places You round to a number of decimal places by looking at the next decimal place and rounding up or down. Significant figures The first non-zero digit is always the first significant figure and you count the number of significant figures then look at the next figure and round up or down. You should always keep the idea of the size of the number. One week to go 15 Exam preparation Algebra Topic Formula When to use it Rules of indices y ×y =y m n m+n ym ÷ yn = ym−n (ym)n = ymn Straight line graph y = mx + c When you multiply you add the indices or powers. When you divide you subtract the indices or powers. When you raise a power to a power you multiply the indices or powers. m is the gradient and (0, c) the intercept on the y-axis Geometry and measure Topic Formula When to use it Parallel sides Parallel lines are shown with arrows Equal sides Equal lines are shown with short lines Perimeter Add lengths of all sides To find the perimeter of any 2D shape Areas of 2D shapes Area = l × w Area of a rectangle is length × width w Area = ½b × h Area = b × h Area = ½ (a + b) × h l Area of a triangle is ½base × vertical height h b Area of a parallelogram is base × vertical height h b Area of a trapezium is a b h ½ the sum of the parallel sides × the vertical height Circumference and area of a circle C = π × D or C = π × 2r A = π × r2 Circumference or the perimeter of a circle is: pi × diameter or pi × double the radius Area of a circle is pi × radius squared 16 Edexcel GCSE Maths Revision Guide Foundation r D V=l×w×h Volume of a cuboid is: V = πr h Length × width × height 2 Exam preparation Volumes of 3D shapes Volume of a cylinder is: Area of circular end × height Pythagoras’ theorem h = a2 + b2 The hypotenuse of a right-angled triangle can be found by finding the square root of the sum of the squares of the two shorter sides. A shorter side of a right-angled triangle can be found by finding the square root of the difference between the hypotenuse squared and the other shorter side squared. Trigonometry sin = o a o ;cos = ;tan = h h a You can find a missing side or a missing angle by selecting and using one of these formulae. h a o You use the trigonometry ratio that has two given pieces of information and the one you have to find. Statistics and probability Topic Formula When to use it Probability P(A and B) = P(A) × P(B) You use this when you have two independent events P(A or B) = P(A) + P(B) P(A or B) = P(A) + P(B) − P(A) × P(B) You use this when you have mutually exclusive events You use this when you do not have mutually exclusive events One week to go 17