高中数学双语教案.docx
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高中数学双语教案.docx
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高中数学双语教案
§1Arithmetic算术
Thenumbers0,1,2,3……arecalledwholenumbersorintegers.So23isanintegerbut1.5isnotaninteger.
Anumbergreaterthanzeroiscalledapositivenumber.
Anumbersmallerthanzeroiscalledanegativenumber.
0isanintegerbutitisneitherpositivenornegative.
Anintegerthatisdivisibleby2isaneveninteger.Anintegerthatisnotdivisibleby2isanoddinteger.
Foranytwonumbersonthenumberline,thenumberontheleftislessthanthenumberontheright,forexample,2<3and-4<-3.
Theabsolutevalueofanumberistheequivalentpositivevalue.
|+2|=+2,|-3|=+3
Ifxisanintegerdivisiblebyanotherintegery,thenxissaidtobeeamultipleofy,andyissaidtobeafactorordivisorofx.
Thefactorsof12are:
1,2,3,4,6,12.
Themultipleof5are:
5,10,15,20……
Ifpisawholenumberwhichhasnootherfactorsbesidesitselfand1,pisaprimenumber.2,3,5,7,11,…areprimenumbers.But1isnotconsideredaprimenumber.
Theleastcommonmultipleoftwonumbersisthesmallestnumberwhichisacommonmultipleofbothnumbers.
Afractionisanumberwhichrepresentsaratioordivisionoftwonumbers.Afractioniswrittenintheform
.Thenumberonthetop,a,iscalledthenumerator;thenumberonthebottom,b,iscalledthedenominator.
Afractioncannothave0asadenominator,sincedivisionby0isnotdefined.
Tocomparethevaluesoftwofractions.Is
greaterorlessthan
?
Firstmultiply4by9andget36;thenmultiply5by7andget35,asshownhere:
364535
--
79
Since36isgreaterthan35,
isgreaterthan
.(Itisnotnecessarytomultiply7by9asthecommondenominator.)
Adecimalfractionisafractionwhosedenominatorissomepowerof10(suchas10,100,1000ete.)
Thedigitsofanumberarenamedasfollows:
2,078.146
千百十个十百千
位位位位分分分
数数数数位位位
数数数
thousands’digit千位数
hundreds’digit百位数
tens’digit十位数
units’digit个位数
tenths’digit十分位数
hundredths’digit百分位数
thousandths’digit千分位数
Roundingoffnumberscanhelpyougetquick,approximateanswers.Examples:
Roundoff538,423.98tothenearesthundred.Theansweris538,400.
Roundoff24.76tothenearesttenth.Theansweris24.8.
Thepartsofadivisionproblemareindicatedintheexamplebelow.
Thequotientis5
Theaverageorarithmeticmeanof12,3,6is:
(12+3+6)÷3=7.
Ifagroupofnumbersisarrangedinorder,themiddlenumberiscalledthemedian.
Tofindthemedianof12,3,6,arrangetheminorder:
3,6,12.Themedianis6.
Ifthenumberofitemsinthegroupiseven,themedianistheaverageofthetwomiddlenumbers.Forexample,tofindthemedianof6,3,30,14,arrangetheminorder:
3,6,14,30.Themedianis(6+14)÷2=10.
Ingeneral,themedianandtheaverageofacollectionofnumbersaredifferent.
Powers,Exponents,andRoots:
Ifbisanynumberandnisawholenumbergreaterthan0,b
meantheproductofnfactorseachofwhichisequaltob.Thus,b
=b*b*b*…*b,wheretherearencopiesofb.
Theexpressionb
iscalledthenthpowerofb,niscalledtheexponent;biscalledthebase.
Anoddintegerpowerofanegativenumberisnegative,andanevenintegerpowerofanegativenumberispositive.Forexample,(-2)
=-8,but(-2)
=16.
Youshouldknowthefollowinglaws:
b
=1(b≠0)
b
=b
b
*b
=b
=b
(b≠0)
(b≠0)
(b≠0)
Ifyouraiseanumberdtothenthpowerandtheresultisb,thendiscalledthenthrootofb.
Noticethefollowingrelations:
Exercise
1.AllthefollowingnumbersarepositiveEXCEPT
(A)5.3(B)0.07(C)0(D)2(E)5
2.Thenumberofthefactorsof24is
(A)9(B)8(C)7(D)6(E)5
3.Thedistinctprimefactorsof24are
(A)1,2,3(B)2,3(C)2,3,6(D)8,3
4.Howmanyofthepositiveintegerslessthan25areintegermultiplesof4?
(A)7(B)6(C)5(D)4(E)3
5.Theleastcommondenominatorof
and
is
(A)
(B)
(C)
(D)48(E)96
6.Thetenths’digitof3,610.48is
(A)3(B)6(C)1(D)4(E)8
7.Theunits’digitof5
is
(A)20(B)10(C)5(D)4(E)0
8.Round508.64tothenearesttenth.Theresultis
(A)510(B)508(C)508.6(D)508.7(E)509
9.360dividedby37,theremainderis
(A)27(B)333(C)0(D)9(E)
10.Theaverageorarithmeticmeanof2,4,6,10is
(A)6(B)5(C)5.5(D)4(E)3
11.Themedianof2,4,6,10is
(A)6(B)5(C)5.5(D)4(E)3
12.10
÷10
=
(A)100(B)10(C)0.1(D)0.01(E)-2
13.If87million=8.7*10
thenn=
(A)5(B)6(C)7(D)8(E)9
14.
(A)
(B)5
(C)3
(D)15
(E)3
15.Whichofthefollowinghasthegreatestvalue?
(A)
(B)0.3(C)0.01п(D)
(E)
16.7lessthan5x.Thismeans
(A)7-5x(B)5x-7(C)7<5x(D)7>5x(E)7-5+x
17.8islessthan2y.Thismeans
(A)8-2y(B)2y-8(C)8<2y(D)2y<8(E)8-2+y
§2Algebra代数
Thelettersinanalgebraicexpressionarecalledvariablesorunknowns.Whenavariableismultipliedbyanumber,thenumberiscalledthecoefficientofthevariable.Thus,intheexpression3x
-5y,thecoefficientofx
is3,andthecoefficientofyis–5.
Theonlyalgebraictermswhichcanbeaddedendsubtractedareliketerms.Forexample,2x
+3x-5x
=-3x
+3x.
Anequationisastatementthatsaystwoalgebraicexpressionsareequal.
Alinearequationoranequationofthefirstdegreeisoneinwhichvariablesareraisedonlytothefirstpower.Forexample,2x+3=9.
Aquadraticequationisanequationwhichcontainssquaresoftheunknownaswellastermsofthefirstdegree.Forexample,3x
+2x-4=0,x
-9=0。
Youcansolveaquadraticequationbyfactoring.
Aratioisacomparisonoftwonumbersbydivision.Theratioofatobiswrittenasa:
bor
.
Aproportionisastatementthattworatiosareequal.Forexample,2:
3=10:
15,or
.
Intheproportiona:
b=c:
d,thetermsontheoutside(aandd)arecalledtheextremes,andthetermsontheinside(bandc)arecalledthemeans.
Twovariables,aandb,aredirectlyproportional,iftheysatisfyarelationshipoftheforma=kb,wherekisanumber.
Twovariables,aandb,areindirectlyproportional,iftheysatisfyarelationshipoftheformab=k,wherekisanumber.
Sometimestheword“directly”isomitted.So“aandbareproportional”means:
aandbaredirectlyproportional.
Aninequalityisastatementthatoneexpressionisnotequaltoanother.
Tofindthesolutionsetforaninequality,youmustremembertheseprinciples.
1.Youcanaddthesamequantitytoorsubtractthesamequantityfrombothsidesofaninequalitywithoutalteringitssolutionset.
If5>3,then5+2>3+2.If5>3,then5-4>3-4.
2.Youcanmultiplyordividebothsidesofaninequalitybythesamenegativequantitywithoutalteringitssolutionset,youmustchangethedirectionoftheinequality.
Iif5>3,then5(-2)<3(-2).If5>3,then
.
Exercise
1.Weknowthat
≠-3.If
theny=
(A)+3only(B)–3only(C)+9only(D)–9only(E)+3or-3
2.IfP=EI,andE=IR,findPintermsofEandR.
3.If5x+2<2x+5,thenallofthefollowingmaybeavalueofxEXCEPT
(A)0(B)1(C)–1(D)–2
4.Abagofchickenfeedwillfeed18chickensfor54days.Forhowmanydayswillitfeed12chickens?
(A)36(B)38(C)54(D)72(E)81
5.7x-5y=13;2x-7y=m26;9x-12y=?
(A)13(B)26C)39(D)42(E)52
6.Howmanysolutionsdoestheequationa-3=
have?
(A)0(B)1(C)2(D)3(E)4
What’stheChineseforthefollowingword?
1.integer__________2.numerator__________3.roundoff__________4.decimal__________5.fraction__________6.positive__________7.multiple__________8.power__________9.remainder__________10.median__________11.denominator__________12.oddinteger__________13.exponent__________14.arithmeticmean__________15.quotient__________16.primenumber__________17.absolutevalue__________18.divisor__________19.negative__________20.root__________
Howtodescribe“
”___________________________“
”_______________
§Geometry
Anacuteanglehasameasuregreaterthan0°butlessthan90°.
Arightanglehasameasureof90°.
Anobtuseanglehasameasuregreaterthan90°butlessthan180°
Astraightangle(whichlookslikeastraightline)hasameasureof180°.
Twoanglesthattogetherhaveameasureof90°arecalledcomplementaryangles.
Twoanglesthattogetherhaveameasureof180°(thatis,twoanglesthattogetherformastraightline)arecalledsupplementaryangles.
Iftwolinesintersect,thefouranglesformedareeitherequalorsupplementary.Theanglesoppositeoneanotherarecalledverticalanglesandareequal.Theanglesnexttooneanotheraresupplementary.
Iftwoparallellinesareintersectedbyathirdline,calledatransversal,theeightanglesformedareeitherequalorsupplementary.
Inthefigure,linePQisparalleltolineRS.
Thenangles1and2arecalledcorrespondinganglesandareequal.
Angles2and3arecalledalternateinterioranglesandareequal.
Atrianglethathastwoequalsidesiscalledanisoscelestriangle.
Ifallthreesidesofatrianglearethesamelength,thetriangleiscalledanequilateraltriangle.
Thesumofthemeasuresofthethreeinterioranglesofatriangleis180°.
Ifthevertexofanangleisatthecenterofacircle,thentheangleiscalledacentralangleanditsmeasureisequaltothemeasureofthearcitintercepts.
Ifthevertexofanangleliesonacircle,thentheangleiscalledaninscribedangleinthecircleanditsmeasureisequaltohalfthemeasureofthearcitintercepts.
Ifthevertexofanangleliesonacircle,thentheangleiscalledaninscribedangleinthecircleanditsmeasureisequaltohalfthemeasureofthearcitintercepts.
IncircleO,angle
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