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1、Greg Byrd,Lynn Byrd and Chris PearceCoursebookCambridge CheckpointMathematics8cambridge university press Cambridge,New York,Melbourne,Madrid,Cape Town,Singapore,So Paulo,Delhi,Mexico CityCambridge University Press The Edinburgh Building,Cambridge CB2 8RU,UKwww.cambridge.org Information on this title
2、:www.cambridge.org/9781107697874 Cambridge University Press 2013This publication is in copyright.Subject to statutory exception and to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written permission of Cambridge University Press.Fi
3、rst published 2013Reprinted 2013Printed in the United Kingdom by Latimer TrendA catalogue record for this publication is available from the British LibraryISBN 978-1-107-69787-4 PaperbackCover image Cosmo Condina concepts/AlamyCambridge University Press has no responsibility for the persistence or a
4、ccuracy of URLs for external or third-party internet websites referred to in this publication,and does not guarantee that any content on such websites is,or will remain,accurate or appropriate.3IntroductionWelcome to Cambridge Checkpoint Mathematics stage 8The Cambridge Checkpoint Mathematics course
5、 covers the Cambridge Secondary 1 mathematics framework and is divided into three stages:7,8 and 9.This book covers all you need to know for stage 8.There are two more books in the series to cover stages 7 and 9.Together they will give you a firm foundation in mathematics.At the end of the year,your
6、 teacher may ask you to take a Progression test to find out how well you have done.This book will help you to learn how to apply your mathematical knowledge and to do well in the test.The curriculum is presented in six content areas:?Number?Geometry?Algebra?This book has 18 units,each related to one
7、 of the first five content areas.Problem solving is included in all units.There are no clear dividing lines between the five areas of mathematics;skills learned in one unit are often used in other units.Each unit starts with an introduction,with key words listed in a blue box.This will prepare you f
8、or what you will learn in the unit.At the end of each unit is a summary box,to remind you what youve learned.Each unit is divided into several topics.Each topic has an introduction explaining the topic content,?there is an exercise.Each unit ends with a review exercise.The questions in the exercises
9、 encourage you to apply your mathematical knowledge and develop your understanding of the subject.As well as learning mathematical skills you need to learn when and how to use them.One of the most important mathematical skills you must learn is how to solve problems.When you see this symbol,it means
10、 that the question will help you to develop your problem-solving skills.During your course,you will learn a lot of facts,information and techniques.You will start to think like a mathematician.You will discuss ideas and methods with other students as well as your teacher.These discussions are an imp
11、ortant part of developing your mathematical skills and understanding.Look out for these students,who will be asking questions,making suggestions and taking part in the activities throughout the units.HassanDakaraiShenXavierJakeAndersRaziSashaMahaMiaAliciaHarshaZalikaOditiTaneshaAhmad4ContentsIntrodu
12、ction 3Acknowledgements 6Unit 1 Integers,powers and roots 71.1 Arithmetic with integers 81.2 Multiples,factors and primes 111.3 More about prime numbers 131.4 Powers and roots 15End-of-unit review 17Unit 2 Sequences,expressions and formulae 182.1 Generating sequences 192.2 Finding rules for sequence
13、s 212.3 Using the nth term 232.4 Using functions and mappings 242.5 Constructing linear expressions 262.6 Deriving and using formulae 27End-of-unit review 30Unit 3 Place value,ordering and rounding 313.1 Multiplying and dividing by 0.1 and 0.01 323.2 Ordering decimals 343.3 Rounding 363.4 Adding and
14、 subtracting decimals 373.5 Dividing decimals 383.6 Multiplying by decimals 393.7 Dividing by decimals 403.8 Estimating and approximating 41End-of-unit review 43Unit 4 Length,mass and capacity 444.1 Choosing suitable units 454.2 Kilometres and miles 47End-of-unit review 49Unit 5 Angles 505.1 Paralle
15、l lines 515.2 Explaining angle properties 545.3 Solving angle problems 57End-of-unit review 60Unit 6 Planning and collecting data 616.1 Collecting data 626.2 Types of data 656.3 Using frequency tables 66End-of-unit review 69Unit 7 Fractions 707.1 Finding equivalent fractions,decimals and percentages
16、 717.2 Converting fractions to decimals 737.3 Ordering fractions 747.4 Adding and subtracting fractions 757.5 Finding fractions of a quantity 777.6 Multiplying an integer by a fraction 787.7 Dividing an integer by a fraction 797.8 Multiplying and dividing fractions 80End-of-unit review 82Unit 8 Shap
17、es and geometric reasoning 838.1 Recognising congruent shapes 848.2 Identifying symmetry of 2D shapes 868.3 Classifying quadrilaterals 888.4 Drawing nets of solids 908.5 Making scale drawings 92End-of-unit review 94Unit 9 Simplifying expressions and solving equations 959.1 Collecting like terms 969.
18、2 Expanding brackets 989.3 Constructing and solving equations 99End-of-unit review 101Unit 10 Processing and presenting data 10210.1 Calculating statistics from discrete data 10310.2 Calculating statistics from grouped or continuous data 10510.3 Using statistics to compare two distributions 107End-o
19、f-unit review 1095Unit 11 Percentages 11011.1 Calculating percentages 11111.2 Percentage increases and decreases 11311.3 Finding percentages 11511.4 Using percentages 117End-of-unit review 119Unit 12 Constructions 12012.1 Drawing circles and arcs 12112.2 Drawing a perpendicular bisector 12212.3 Draw
20、ing an angle bisector 12412.4 Constructing triangles 126End-of-unit review 128Unit 13 Graphs 12913.1 Drawing graphs of equations 13013.2 Equations of the form y=mx+c 13213.3 The midpoint of a line segment 13413.4 Graphs in real-life contexts 136End-of-unit review 139Unit 14 Ratio and proportion 1401
21、4.1 Simplifying ratios 14114.2 Sharing in a ratio 14314.3 Solving problems 145End-of-unit review 147Unit 15 Probability 14815.1 The probability that an outcome does not happen 14915.2 Equally likely outcomes 15015.3 Listing all possible outcomes 15215.4 Experimental and theoretical probabilities 154
22、End-of-unit review 157Unit 16 Position and movement 15816.1 Transforming shapes 15916.2 Enlarging shapes 161End-of-unit review 164Unit 17 Area,perimeter and volume 16517.1 The area of a triangle 16617.2 The areas of a parallelogram and trapezium 16717.3 The area and circumference of a circle 16917.4
23、 The areas of compound shapes 17117.5 The volumes and surface areas of cuboids 17317.6 Using nets of solids to work out surface areas 175End-of-unit review 177Unit 18 Interpreting and discussing results 17818.1 Interpreting and drawing frequency diagrams 17918.2 Interpreting and drawing pie charts 1
24、8218.3 Interpreting and drawing line graphs 18418.4 Interpreting and drawing stem-and-leaf diagrams 18618.5 Drawing conclusions 188End-of-unit review 191End-of-year review 192Glossary and index 196 Contents6AcknowledgementsThe publisher would like to thank ngel Cubero of the International School San
25、to Toms de Aquino,Madrid,for reviewing the language level.Cover image Cosmo Condina concepts/Alamyp.7b pressureUA/iStock;p.18tl Jon Arnold Images Ltd/Alamy;p.18mr Maksim Toome/Shutterstock;p.18br forestpath/Shutterstock;p.26b ilyast/iStock;p.31b Antonio Mo/Iconica/Getty Images;p.37b Christopher Stee
26、r/iStock;p.38b DAJ/Getty Images;p.44mr Chris Ryan/OJO Images/Getty Images;p.44b NASA;p.46tr Lynn Byrd;46mr Aspen Photo/Shutterstock;p.63m dundanim/Shutterstock;p.83t Diego Cervo/Shutterstock;p.83mr Francesco Dazzi/Shutterstock;p.83br Peter Kirillov/Shutterstock;p.93mr mbbirdy/iStock;p.95tr pidjoe/iS
27、tock;p.95mr Liz Van Steenburgh/Shutterstock;p.95br Aleksandar Petrovic/iStock;p.114ml a40757/Shutterstock;p.114bl Pakhnyushcha/Shutterstock;p.129tr Portrait Essentials/Alamy;p.140mr RosetteJordaan/iStock;p.140br Mark Bowden/iStock;p.143b Ferenc Szelepcsenyi/Shutterstock;p.146tr kryczka/iStock;p.147b
28、r design56/Shutterstock;p.158br Geoff Brightling/Peter Minister/Dorling Kindersleyl=left,r=right,t=top,b=bottom,m=middle1 Integers,powers and roots7The first primes are 2 3 5 7 11 13 17 19 23 29.Prime numbers have just two factors:1 and the number itself.Every whole number that is not prime can be w
29、ritten as a product of prime numbers in exactly one way(apart from the order of theprimes).8=2 2 2 65=5 13 132=2 2 3 11 2527=7 19 19It is easy to multiply two prime numbers.For example,13 113=1469.It is much harder to do the inverse operation.For example,2021 is the product of two prime numbers.Can
30、you find them?This fact is the basis of a system that is used to encode messages sent across the internet.The RSA cryptosystem was invented by Ronald Rivest,Adi Shamir and Leonard Adleman in 1977.It uses two large prime numbers with about 150 digits each.These are kept secret.Their product,N,with ab
31、out 300 digits,is made public so that anyone canuse it.If you send a credit card number to a website,your computer performs a calculation with N and your credit card number to encode it.The computer receiving the coded number will do another calculation to decode it.Anyone else,who does not know the
32、 factors,will not be able to do this.Prime numbers more than 200 are 211 223 227 229 233 239 241 251 257 263 269 271.1 Integers,powers and rootsMake sure you learn and understand these key words:integerinversemultiplecommon multiplelowest common multiple(LCM)factorcommon factorhighest common factor(
33、HCF)prime numberprimefactor treepowerindex(indices)squarecubesquare rootcube rootKey words1 Integers,powers and roots81.1 Arithmetic with integers1.1 Arithmetic with integersIntegers are whole numbers.They may be positive or negative.Zero is also an integer.You can show integers on a number line.012
34、34554321Look at the additions in the box to the right.The number added to 2 decreases,or goes down,by 1 each time.The answer also decreases,or goes down,by 1 each time.Now see what happens if you subtract.Look at the first column.The number subtracted from 5 goes down by 1 each time.The answer goes
35、up by 1 each time.Now look at the two columns together.You can change a subtraction into an addition by adding the inverse number.The inverse of 3 is 3.The inverse of 3 is 3.For example,5 3=5+3=8.Look at these multiplications.The pattern continues like this.You can see that negative integer positive
36、 integer=negative answer.Now look at this pattern.The pattern continues like this.You can see that negative integer negative integer=positive answer.2+3=5 2+2=4 2+1=3 2+0=2 2+1=12+2=02+3=12+4=25+3=25+2=3 5+1=4 5+0=5 5+1=6 5+2=7 5+3=8 5 3=2 5 2=3 5 1=4 5 0=5 5 1=65 2=75 3=8 1 5=52 5=103 5=154 5=203 5
37、=152 5=10 1 5=50 5=0 3 1=33 2=63 3=93 4=123 5=153 4=123 3=93 2=6 3 1=33 0=0Worked example 1.1aWork these out.a 3+7 b 5 8 c 3 9a 3+7=4 Subtract 7 from 3.3 7=4b 5 8=13 The inverse of 8 is 8.5 8=5+8=13c 3 9=6 The inverse of 9 is 9.3 9=3+9=61 Integers,powers and roots91.1 Arithmetic with integersHere is
38、 a simple rule,which also works for division.Warning:This rule works for multiplication and division.It does not work for addition or subtraction.Exercise 1.11 Work out these additions.a 3+6 b 3+8 c 10+4 d 10+7 e 12+42 Work out these additions.a 30+20 b 100+80 c 20+5 d 30+70 e 45+403 Work out these
39、subtractions.a 4 6 b 4 6 c 6 4 d 6 6 e 2 104 Write down additions that have the same answers as these subtractions.Then work out the answer to each one.a 4 6 b 4 6 c 8 2 d 4 6 e 12 105 Work out these subtractions.a 7 2 b 5 3 c 12 4 d 6 6 e 2 106 Here are some addition pyramids.Each number is the sum
40、 of the two in the row below it.Copy the pyramids.Fill in the missing numbers.a 2513 b 235 c 462 d 323 e 7627 Here is a subtraction table.Two answers have already been filled in:4 4=8 and 4 2=6.Copy the table and complete it.second number42024first number4820246When you multiply two integers:if they
41、 have same signs?positive answer if they have different signs?negative answerIn part a,3+5=2Worked example 1.1bWork these out.a 12 3 b 8 5 c 20 4 d 24 6a 12 3=36 12 3=36 The signs are different so the answer is negative.b 8 5=40 8 5=40 The signs are the same so the answer is positive.c 20 4=5 20 4=5
42、 The signs are different so the answer is negative.d 24 6=4 24 6=4 The signs are the same so the answer is positive.1 Integers,powers and roots101.1 Arithmetic with integers8 Work out these multiplications.a 5 4 b 8 6 c 4 5 d 6 10 e 2 209 Work out these divisions.a 20 10 b 30 6 c 12 4 d 50 5 e 16 41
43、0 Write down two correct division expressions.a 4 10 b 20 5 c 20 5 d 40 8 e 12 411 Here are some multiplications.In each case,use the same numbers to write down two correct division expressions.a 5 3=15 b 8 4=32 c 6 7=4212 Here is a multiplication table.Three answers have already been filled in.3210
44、1233621201323a Copy the table and complete it.b Colour all the 0 answers in one colour,for example,green.c Colour all the positive answers in a second colour,for example,blue.d Colour all the negative answers in a third colour,for example,red.13 These are multiplication pyramids.Each number is the p
45、roduct of the two in the row below it.Copy each pyramid.Fill in the missing numbers.a 6322 b 451 c 48123 d 6416214 a What integers will replace the symbols to make this multiplication correct?=12b How many different pairs of numbers can you find that give this answer?15 Work these out.a 5 3 b 5+3 c
46、4 5 d 60 10 e 2+18 f 10 416 Write down the missing numbers.a 4 =20 b 2=6 c 5=2 d 3=12 e 2+=2 f 4=3The product is the result of multiplying two numbers In part a,2 3=61 Integers,powers and roots111.2 Multiples,factors and primes1.2 Multiples,factors and primesThe multiples of 6 are 6,12,18,24,30,36,T
47、he multiples of 9 are 9,18,27,36,45,54,The common multiples of 6 and 9 are 18,36,54,72,The lowest common multiple(LCM)of 6 and 9 is 18.The factors of a number divide into it without a remainder.The factors of 18 are 1,2,3,6,9 and 18.The factors of 27 are 1,3,9 and 27.The common factors of 18 and 27
48、are 1,3 and 9.The highest common factor(HCF)of 18 and 27 is 9.Some numbers have just two factors.Examples are 7(1 and 7 are factors),13(1 and 13 are factors)and 43.Numbers with just two factors are called prime numbers or just primes.The first ten primes are 2,3,5,7,11,13,17,19,23 and 29.6 1=6 6 2=1
49、2 6 3=18 9 1=9 9 2=18 9 3=27 18 36 54 are in both lists of multiples.3 6=18 so 3 and 6 are factors of 18Worked example 1.2aa Find the factors of 45.b Find the prime factors of 48.a The factors of 45 are 1,3,5,9,15 45=1 45 so 1 and 45 are factors.(1 is always a factor.)and 45.Check 2,3,4,in turn to s
50、ee if it is a factor.2 is not a factor.(45 is an odd number.)45=3 15 3 and 15 are factors.4 is not a factor.45=5 9 5 and 9 are factors.6,7 and 8 are not factors.The next number to try is 9 but we already have 9 in the list of factors.You can stop when you reach a number that is already in the list.b