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1、8GregByrd,Lynn Byrd and Chris PearceCambridge CheckpointMathematicsCoursebookcambrid ge university pressCambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SoPaulo,Delhi,MexicoCityCambridge University PressThe Edinburgh Building,Cambridge CB2 8RU,UKwww.cambridge.orgInformation on this title:www.ca
2、mbridge.org/9781107697874 Cambridge University Press 2013Thispublication is in copyright.Subject tostatutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.First publish
3、ed 2013Reprinted2013Printed 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/AlamyCambridgeUniversityPresshasnoresponsibilityforthepersistence oraccuracyofURLs forexte
4、rnalorthird-partyinternetwebsites referredtointhis publication,and does not guarantee that any content on such websites is,or will remain,accurate or appropriate.3WelcometoCambridgeCheckpointMathematicsstage8The Cambridge Checkpoint Mathematics course covers the Cambridge Secondary 1 mathematicsfram
5、ework and is divided into three stages:7,8 and 9.This book covers all you need to know forstage 8.There are two more books in the series to cover stages 7 and 9.Together they will give you a firmfoundationin mathematics.At the end of the year,your teacher may ask you to take aProgression testto find
6、 out how well you havedone.This book will help you to learn how to apply your mathematical knowledge and to do well in thetest.The curriculum is presented in six content areas:r Numberr.FBTVSFr Geometryr Algebrar)BOEMJOHEBUBr 1SPCMFNTPMWJOH.This book has 18 units,each related to one of the first fiv
7、e content areas.Problem solving is included inall units.There are no clear dividing lines between the five areas of mathematics;skills learned in oneunitareoftenused inotherunits.Eachunitstartswithanintroduction,withkeywordslistedinabluebox.Thiswillprepareyouforwhatyou will learn in the unit.At the
8、end of each unit is asummarybox,to remind you what youve learned.Each unit is divided into several topics.Each topic has an introduction explaining the topic content,VTVBMMZXJUIXPSLFE FYBNQMFT.)FMQGVMIJOUT BSF HJWFO JOCMVFSPVOEFE CPYFT.UUIFFOE PGFBDIUPQJDthere is an exercise.Each unit ends with a re
9、view exercise.The questions in the exercises encourage youto 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 mostimportantmathematicalskillsyoumustlearnishowtosolveproblems.W
10、hen you see this symbol,it means that the question will help you to develop your problem-solvingskills.During your course,you will learn a lot of facts,information and techniques.You will start to think likea mathematician.You will discuss ideas and methods with other students as well as your teache
11、r.Thesediscussions are an important part of developing your mathematical skills and understanding.Look out for these students,who will be asking questions,making suggestions and taking part in theactivities throughout the units.IntroductionXavierMiaDakaraiOditiAndersSashaHassanHarshaJakeAliciaShenTa
12、neshaRaziMahaAhmadZalika4ContentsIntroduction3Acknowledgements6Unit1Integers,powersandroots71.1Arithmetic with integers81.2Multiples,factors and primes111.3More about prime numbers131.4Powers and roots15End-of-unit review17Unit2Sequences,expressionsandformulae182.1Generatingsequences192.2Finding rul
13、es for sequences212.3Using the nth term232.4Using functions and mappings242.5Constructing linear expressions262.6Deriving and using formulae27End-of-unit review30Unit3 Placevalue,orderingandrounding313.1Multiplying and dividing by 0.1 and 0.01323.2Orderingdecimals343.3Rounding363.4Adding and subtrac
14、ting decimals373.5Dividing decimals383.6Multiplying by decimals393.7Dividing by decimals403.8Estimating and approximating41End-of-unit review43Unit4Length,massandcapacity444.1Choosing suitable units454.2Kilometres and miles47End-of-unit review49Unit 5 Angles505.1Parallellines515.2Explaining angle pr
15、operties545.3Solving angle problems57End-of-unit review60Unit6 Planning andcollecting data616.1Collectingdata626.2Types of data656.3Using frequency tables66End-of-unit review69Unit 7 Fractions707.1Finding equivalent fractions,decimalsandpercentages717.2Converting fractions to decimals737.3Orderingfr
16、actions747.4Adding and subtracting fractions757.5Finding fractions of a quantity777.6Multiplying an integer by a fraction787.7Dividing an integer by a fraction797.8Multiplying and dividing fractions80End-of-unit review82Unit8Shapesandgeometricreasoning838.1Recognising congruent shapes848.2Identifyin
17、g symmetry of 2D shapes868.3Classifyingquadrilaterals888.4Drawing nets of solids908.5Making scale drawings92End-of-unit review94Unit 9 Simplifying expressions andsolvingequations959.1Collecting like terms969.2Expandingbrackets989.3Constructing and solving equations99End-of-unit review101Unit10 Proce
18、ssing andpresenting data10210.1 Calculating statistics from discrete data10310.2 Calculating statistics from grouped orcontinuousdata10510.3 Using statistics to compare two distributions107End-of-unit review1095ContentsUnit 11 Percentages110Unit16Positionandmovement15811.1 Calculating percentages111
19、16.1 Transforming shapes15911.2 Percentage increases and decreases11316.2 Enlarging shapes16111.3 Finding percentages115End-of-unit review16411.4 Using percentages117End-of-unit review119Unit17Area,perimeterandvolume16517.1 The area of a triangle166Unit 12 Constructions12017.2 The areas of a paralle
20、logram12.1 Drawing circles and arcs121andtrapezium16712.2 Drawing a perpendicular bisector12217.3 The area and circumference12.3 Drawing an angle bisector124of a circle16912.4 Constructing triangles12617.4 The areas of compound shapes171End-of-unit review12817.5 The volumes and surfaceareas of cuboi
21、ds173Unit 13 Graphs12917.6 Using nets of solids to work out13.1 Drawing graphs of equations130surfaceareas17513.2 Equations of the form y=mx+c132End-of-unit review17713.3 The midpoint of a line segment13413.4 Graphs in real-life contexts136Unit 18 Interpreting and discussing results178End-of-unit re
22、view13918.1 Interpreting and drawing frequencydiagrams179Unit14 Ratioandproportion14018.2 Interpreting and drawing pie charts18218.3 Interpreting and drawing line graphs18418.4 Interpreting and drawing stem-and-leafdiagrams18618.5Drawingconclusions188End-of-unit review191nothappen14915.2Equally like
23、ly outcomes15015.3 Listing all possible outcomes15215.4 Experimental and theoretical probabilities154End-of-unit review157End-of-year review192Glossary and index19614.1 Simplifying ratios14114.2 Sharing in a ratio14314.3 Solving problems145End-of-unit review147Unit 15 Probability14815.1 The probabil
24、ity that an outcome does6AcknowledgementsThe publisher would like to thank ngel Cubero of the International School Santo Toms de Aquino,Madrid,forreviewing thelanguagelevel.Cover image Cosmo Condina concepts/Alamyp.7b pressureUA/iStock;p.18tl Jon Arnold Images Ltd/Alamy;p.18mr Maksim Toome/Shutterst
25、ock;p.18br forestpath/Shutterstock;p.26b ilyast/iStock;p.31b Antonio Mo/Iconica/Getty Images;p.37b Christopher Steer/iStock;p.38b DAJ/Getty Images;p.44mr Chris Ryan/OJO Images/Getty Images;p.44b NASA;p.46tr LynnByrd;46mrAspenPhoto/Shutterstock;p.63m dundanim/Shutterstock;p.83t Diego Cervo/Shuttersto
26、ck;p.83mr Francesco Dazzi/Shutterstock;p.83br Peter Kirillov/Shutterstock;p.93mr mbbirdy/iStock;p.95tr pidjoe/iStock;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 Rosett
27、eJordaan/iStock;p.140br Mark Bowden/iStock;p.143b Ferenc Szelepcsenyi/Shutterstock;p.146tr kryczka/iStock;p.147br design56/Shutterstock;p.158br Geoff Brightling/Peter Minister/Dorling Kindersleyl=left,r=right,t=top,b=bottom,m=middle1Integers,powers and roots7The firstprimesare 2 3 5 7 11 13 17 19 23
28、 29.Prime numbers have just two factors:1 and the number itself.Every whole number that is not prime can be written as a productofprimenumbersinexactlyoneway(apart fromtheorderoftheprimes).8=2 2 265=5 13132=2 2 3 112527=7 19 19It is easy to multiply two prime numbers.For example,13 113=1469.Itismuch
29、hardertodotheinverseoperation.Forexample,2021 is the product of two prime numbers.Can you find them?This fact is the basis of a system that is used to encode messagessentacross theinternet.The RSA cryptosystem was invented by Ronald Rivest,Adi Shamir and Leonard Adleman in 1977.It uses twolarge prim
30、e numbers with about 150 digits each.Theseare kept secret.Their product,N,with about 300 digits,is madepublicso thatanyonecanuseit.Ifyousenda creditcardnumbertoawebsite,yourcomputerperformsacalculation with N and your credit cardnumbertoencodeit.Thecomputerreceivingthecodednumberwilldoanother calcul
31、ation to decode it.Anyoneelse,who does not know the factors,willnot be ableto dothis.Primenumbersmorethan200are 211 223 227 229 233 239 241 251 257 263 269 271.1Integers,powers and rootsKeywordsMake sure you learn andunderstandthesekeywords:integerinversemultiplecommonmultiplelowest common multiple(
32、LCM)factorcommonfactorhighest common factor(HCF)prime numberprimefactor treepowerindex(indices)squarecubesquare rootcube root101Integers,powers and roots1.1 Arithmetic with integers34=1233=932=631=330=035=1525=1015=505=0Work these out.a 3+7b 5 8c 3 9a3+7=4b5 8=13c3 9=6Subtract 7 from 3.The inverse o
33、f 8 is 8.The inverse of 9 is 9.3 7=45 8=5+8=133 9=3+9=6Worked example 1.1a1.1ArithmeticwithintegersIntegersare whole numbers.They may be positive or negative.Zero is also an integer.Youcanshowintegersonanumberline.54321012345Look at the additions in the box to the right.The number added to 2 decreas
34、es,or goesdown,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 answergoes up by 1 each time.Now look at the two columns together.You can change a subtraction i
35、nto an addition by adding theinversenumber.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 integer=negative answer.Now look at this pattern.The pattern continues like this.You c
36、an see that negative integer negative integer=positive answer.31=332=633=934=1235=1515=525=1035=1545=205+3=25+2=35+1=45+0=55+1=65+2=75+3=85 3=25 2=35 1=45 0=55 1=65 2=75 3=82+3=52+2=42+1=32+0=22+1=12+2=02+3=12+4=21.1 Arithmetic with integers1Integers,powers and roots9When you multiply two integers:i
37、f they have same signs positive answerif they have different signs negative answerWork these out.a 12 3b 8 5c 20 4d 24 6Worked example 1.1bHere is a simple rule,which also works for division.a12 3=3612 3=36The signs are different so the answer is negative.b8 5=408 5=40The signs are the same so the a
38、nswer is positive.c20 4=520 4=5The signs are different so the answer is negative.d24 6=424 6=4The signs are the same so the answer is positive.Warning:This rule works for multiplication and division.It does not work for addition or subtraction.Exercise1.14 Write down additions that have the same ans
39、wers as these subtractions.Then work out the answer toeach one.a 4 6b 4 6c 8 2d 4 6e12 105 Work outthesea 7 2subtractions.b 5 3c 12 4d 6 6e2 106Here are some addition pyramids.Each number is the sum of the two inthe row below it.Copy the pyramids.Fill in the missing numbers.abcde7Here is a subtracti
40、on table.Two answers have already been filled in:4 4=8 and 4 2=6.Copy thetable andcompleteit.secondnumber42024firstnumber4820246267In parta,3+5=223512352461 Workouttheseadditions.a 3+6b 3+8c 10+4d 10+7e 12+42 Workouttheseadditions.a 30+20b 100+80c 20+5d 30+70e 45+403 Workoutthesesubtractions.a 4 6b
41、4 6c 6 4d 6 6e 2 103231.1 Arithmetic with integers101Integers,powers and roots8Work out these multiplications.a 5 4b 8 6c 4 5d 6 10e 2 209Work out these divisions.a 20 10b 30 6c 12 4d 50 5e 16 410Write down two correct division expressions.a 4 10b 20 5c 20 5d 40 8e 12 411Here are some multiplication
42、s.In each case,use the same numbers to write downtwo correct division expressions.a 5 3=15b 8 4=32c 6 7=4212Here is a multiplication table.Three answers have already been filled in.32101233621201323aCopy the table and complete it.bColour all the 0 answers in one colour,for example,green.cColour all
43、the positive answers in a second colour,for example,blue.dColour all the negative answers in a third colour,for example,red.13These are multiplication pyramids.Each number is the product of thetwo in the rowbelow it.Copy each pyramid.Fill in the missing numbers.abcd14a What integers will replace the
44、 symbols to make this multiplication correct?0=12b How many different pairs of numbers can you find that give this answer?15Work these out.a 5 3b 5+3c 4 5d 60 10e2+18f10 416Write down the missing numbers.a 4=20b 2=6c 5=2d 3=12e 2+=2f 4=3The product isthe resultofmultiplying two numbersIn part a,2 3=
45、6623245148123641621.2 Multiples,factors and primes1Integers,powers and roots11aFind the factors of 45.bFind the prime factors of 48.aThe factors of 45 are 1,3,5,9,15and45.45=1 45 so 1 and 45 are factors.(1 is always a factor.)Check 2,3,4,in turn to see if it is a factor.2 is not a factor.(45 is an o
46、dd number.)45=3 153 and 15 are factors.4 is not a factor.45=5 95 and 9 are factors.6,7 and 8 are notfactors.The next number to tryis 9 but we already have 9inthe listoffactors.You canstop whenyoureach anumberthat isalready inthe list.bThe prime factors of 48 are 2 and 3.You only need to check prime
47、numbers.48=2 24 2 is a prime factor.24 is not.48=3 16 3 is a prime factor.16 is not.5 and 7 are not factors.Because 7 7 is bigger than 48,you can stop there.Worked example 1.2aFind the LCM and HCF of 12 and 15.The LCM is 60.The HCF is 3.The multiples of 12 are 12,24,36,48,60,.The multiples of 15 are
48、 15,30,45,60,75,60 is the first number that is in both lists.The factors of 12 are 1,2,3,4,6 and 12.The factors of 15 are 1,3,5 and 15.3 is the largest number that is in both lists.Workedexample1.2b1.2Multiples,factors and primesThemultiplesof6 are6,12,18,24,30,36,The multiples of 9 are 9,18,27,36,4
49、5,54,Thecommonmultiplesof6 and9 are18,36,54,72,Thelowestcommon multiple(LCM)of 6 and 9 is 18.Thefactorsof a number divide into it without a remainder.Thefactorsof18are1,2,3,6,9 and18.The factors of 27 are 1,3,9 and 27.Thecommonfactorsof 18 and 27 are1,3 and 9.Thehighestcommonfactor(HCF)of18and27 is9
50、.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 calledprimenumbersor justprimes.The first ten primesare2,3,5,7,11,13,17,19,23 and29.18 3654 are in both lists of multiples.3 6=18 so 3 and 6 are factors of 186 1=6