9789174241365

Östberga komvux Matte för vuxna med särskilda behov Författare: Ebrahim Isaacs & Joel Uhr Illustrationer: Olle Ljungström Bonnier Fakta

%RN LQGE

>>> 6:;),9.(264=<? :, )6550,9 -(2;( ^^^ IVUUPLYMHR[H ZL ,IYHOPT 0ZHHJZ VJO 1VLS <OY 0SS\Z[YH[PVULY! 6SSL 3Q\UNZ[Y T 6TZSHN! 1VOHU 4LSIP 0USHNH! 3PII` =HUKLY7SVLN ;Y`JR[ OVZ! 3P]VUPH 3L[[SHUK 0:)5!

%RN LQGE

Hej och välkommen kZ[ILYNH 2VT]\_ OHY Wr RVY[ [PK L[HISLYH[ ZPN ZVT LU H] :]LYPNLZ TLZ[ U`[pURHUKL VJO MYHTNrUNZYPRH ZRVSVY =rY ]LYRSPNOL[ZM YHURYHKL ¹KL[ OpY M YZ[rY QHN KPYLR[ ]HK QHN OHY M Y U`[[H H]¹ WLKHNVNPR OHY YLKHU NL[[ [PV[HSZ LSL]LY ]LYR[`NLU H[[ RYHTH KL[ IpZ[H \Y [PSS]HYVU 0 kZ[ILYNH 2VT]\_ ¶ 4H[[L M Y ]\_UH TLK ZpYZRPSKH ILOV] WSVJRHY ]P Y\ZZPULU \Y TH[[LRHRHU VJO WYLZLU[LYHY Wr L[[ H[[YHR[P][ Zp[[ O\Y K\ VTZp[[LY NY\UKSpNNHUKL TH[LTH[PRR\UZRHW P YLKH WLUNHY ,SSLY HUUH[ ZVT NLY SP]L[ N\SKRHU[ 5\ pY KL[ ZUHY[ K\ ZVT RHU N YH KL IpZ[H KLHSHYUH -P_H KL MLZ[SPNHZ[L MLZ[LYUH /H KL ZR UHZ[L ZR UPUNHYUH +L[ LUKH ZVT Z[rY TLSSHU KPN VJO HSS[ KL[[H pY KLUUH IVR

Styrelsen, Östberga komvux

%RN LQGE

INNEHÅLL Taluppfattning 9 Mätning, rum och geometri 37 Mönster och samband 51 Statistik och sannolikhet 71

Facit 84

%RN LQGE

TALUPPFATTNING 'HW NDQ MX YDUD EUD DWW KD NROO Sn WDOHQ $WW YHWD YLOND ROLND WDO GHW ÀQQV KXU GH IXQNDU RFK YDG GH lU YlUGD , GHW KlU NDSLWOHW GUDU YL LJHQRP GH YLNWLJDVWH ² GHW GX EHK|YHU NXQQD I|U DWW NODUD GLJ EUD GlU XWH

Naturliga tal $OOD SRVLWLYD KHOWDO LQNOXVLYH lU QDWXUOLJD +HOWDO EOLU SRVLWLYDUH MX VW|UUH GH lU , DOOD IDOO RP GHW JlOOHU VWnODU .RQVWLJW QRJ lU lYHQ PlQJGHQ EXNIHWW DQWDO SDUNHULQJVE|WHU RFK VnQW SRVLWLYD WDO

Negativa tal bU GX VN\OGLJ QnJRQ VWnODU Vn lU GHW HWW QHJDWLYW WDO ² L DOOD EHPlUNHOVHU bU GX VN\OGLJ 6LOYLR IUnQ +|NLV VWnODU Vn lU GHW VMXNW QHJDWLYW

Bråk 'HW lU YLNWLJW DWW YHWD DWW EUnN LQWH EHW\GHU MLGGHU Sn PDWWHVSUnN HQ WlOMDUH lU LQWH HQ VRP WlOMHU PHG NQLYHQ RFK HQ QlPQDUH lU LQWH HQ WMDOODUH %UD DWW NXQQD OLWH RP EUnN RP GX VND VSOLWWD QnW PHG QnQ NROOD HWW UHFHSW RFK VnQW

Decimaler %UD DWW NXQQD RP GHW LQWH UlFNHU WLOO KHOD JUHMRU

Procent 6MXNW YLNWLJW DWW NXQQD 'X VND NXQQD GHW VRP HWW ULQQDQGH YDWWHQ Vn DWW GX LQWH VHU XW VRP HWW InQ QlU GX VND J|UD EUD GHDODU +XU PnQJD SURFHQW InU GX"

Räknesätt 9LVVD NDOODU GHW I|U DGGLWLRQ VXEWUDNWLRQ PXOWLSOLNDWLRQ RFK GLYLVLRQ PHQ SOXV PLQXV JnQJHU RFK GHODW lU GHW PDQ VlJHU L GHW YHUNOLJD OLYHW L YHUNOLJKHWHQ

9

%RN LQGE

1.

)XOO RFK NnW NDQ GX WlQND GLJ DWW OLJJD PHG DY GHW PRWVDWWD N|QHW , Q\NWHUW WLOOVWnQG lU PRWVYDUDQGH VLIIUD +XU PnQJD NDQ GX VXSD VQ\JJD L HWW UXP PHG SHUVRQHU DY GHW PRWVDWWD N|QHW"

10

%RN LQGE

2.

7RPSD ( VLWWHU VRP VW\UHOVHRUGI|UDQGH L I\UWLRWYn DNWLHERODJ )|U YDUMH nUVUHGRYLVQLQJ KDQ PHG KHGHU RFK VDPYHWH VLJQHUDU InU KDQ WYn VMXWWLVDU +XU PnQJD OLWHU VWDUNW InU YnUW VW\UHOVHSURIIV L HUVlWWQLQJ SHU nU"

11

%RN LQGE

3.

(NYDWRULDOJXLQHD lU HWW IDWWLJW ODQG *HQRPVQLWWVLQYnQDUHQ InU L VLJ NFDO SHU GDJ /DQGHWV EHIRONQLQJVPlQJG lU LQYnQDUH , 6YHULJH lWHU YL VHPORU YDUMH nU (Q VHPOD LQQHKnOOHU NFDO +XU PnQJD GDJDU VNXOOH YL NXQQD QlUD (TXDWRULDOJXLQHDV KHOD EHIRONQLQJ PHG GHVVD VHPORU"

12

%RN LQGE

4.

(Q PlQQLVND VRP YlJHU NJ EHK|YHU NRQVXPHUD FO UHQ DONRKRO Sn HQ NYlOO I|U DWW EOL DONRKROI|UJLIWDG /DUV OLJJHU Sn 0DULDSRO I|U DWW PDJSXPSDV HIWHU DWW KD GUXFNLW HQ SURFHQWLJ D KE Sn GHQ SHUVRQOLJD UHNRUGWLGHQ PLQXWHU /DUV YlJHU NJ bU KDQ VSULWI|UJLIWDG HOOHU EDUD ULNWLJW SDFNDG"

13

%RN LQGE

5.

&RQQ\ RFK 6RQQ\ KDU YDULW Sn VW|OGWXUQp L V|GUD 6YHULJH RFK EHÃ&#x20AC;QQHU VLJ QX KRV VLQ KlODUH /LLVD L +DOORQEHUJHQ /LLVD VlJHU DWW GH InU NU I|U VP\FNHQD NU I|U HOHNWURQLNHQ NU I|U LQVWUXPHQWHQ RFK VSlQQ I|U GHQ IXOD WDYODQ a) +XU P\FNHW InU &RQQ\ RFK 6RQQ\ RP GH JRGWDU /LLVDV EXG" b) +RV SDQWEDQNHQ KDGH &RQQ\ RFK 6RQQ\ InWW NU I|U VDNHUQD +XU P\FNHW PHU VNXOOH GH KD InWW KRV SDQWEDQNHQ lQ KRV /LLVD" c) 2P &RQQ\ RFK 6RQQ\ VNXOOH VlOMD VDNHUQD Sn %ORFNHW VNXOOH GH DQWDJOLJHQ In LQ NU 9DG VNXOOH GH KD WMlQDW Sn GHW MlPI|UW PHG /LLVD UHVSHNWLYH 3DQWEDQNHQ"

14

%RN LQGE

6.

$QWRQ lU L cUH RFK nNHU VNLGRU Sn VSRUWORYHW +DQ lU OLWH VPnIXOO QlU KDQ I|U WLGLJW OlPQDU VLWWOLIWHQ )DOOHW lU Sn PHWHU 3n VMXNKXVHW L gVWHUVXQG NRQVWDWHUDU PDQ DWW $QWRQ KDU KHOD EHQ L NURSSHQ ,QQDQ RO\FNDQ KDGH $QWRQ KHOD EHQ +XU PnQJD SURFHQW DY EHQHQ KDU JnWW DY L $QWRQV NURSS"

15

%RN LQGE

7.

3HUQLOOD KHWVlWHU QlU KRQ PnU GnOLJW (IWHU HQ MREELJ GDJ N|SHU 3HUQLOOD HQ SnVH GLOOFKLSV I|U NU HWW SDNHW %DOOHULQDNH[ I|U NU J VPnJRGLV I|U NU WUH V\OWPXQNDU j NU HQ EXUN MRUGQ|WVVP|U I|U NU RFK HQ ÁDVND KDOORQVRGD I|U NU DWW VN|OMD QHG JRGVDNHUQD PHG (IWHU DWW KD lWLW XSS DOOW GHW JRGD WDU GHW LQWH OnQJ WLG LQQDQ 3HUQLOOD VHU WLOO DWW NUlNDV XSS DOOWVDPPDQV a) +XU P\FNHW NUlNV 3HUQLOOD" $QJH VYDUHW L NURQRU b) DY GHWWD lU VNDWW L IRUP DY PRPV +XU P\FNHW KDU VWDWHQ WMlQDW Sn 3HUQLOODV VMXNGRP" c) +DQGODUHQ OlJJHU Sn Sn JURVVLVWSULVHW I|U DWW In IUDP XWSULVHW Sn HQ YDUD L VLQ EXWLN +XU P\FNHW KDU KDQGODUHQ WMlQDW Sn 3HUQLOODV LQN|S" $QJH VYDUHW H[NOXVLYH PRPV

16

%RN LQGE

8.

'RULV lU HQ OXULJ IDQ +RQ KDU NRPPLW Sn DWW KRQ InU VWXGHQWUDEDWW Sn 6/ RFK (ULNVGDOVEDGHW RP KRQ DQPlOHU VLJ WLOO QnJRQ VNLWNXUV Sn 8QLYHUVLWHWHW RFK InU NnUOHJ (WW WHUPLQVNRUW Sn 6/ NRVWDU NU XWDQ UDEDWW RFK NU PHG 7UlQLQJVNRUWHW NRVWDU NU WHUPLQ XWDQ UDEDWW RFK NU PHG .nUDYJLIWHQ lU Sn NU +XU P\FNHW WMlQDU 'RULV Sn VLQ NOXULJKHW"

17

%RN LQGE

9.

2OOH (EUDKLP RFK -RHO VND UnQD EDQNHQ Sn *XOOPDUVSODQ -RHO RFK (EUDKLP XWI|U VMlOYD UnQHW PHGDQ 2OOH VLWWHU L ELOHQ RFK YDNWDU (IWHUVRP 2OOH lU HQ PLOM|PHGYHWHQ PDQ VWlQJHU KDQ DY PRWRUQ XQGHU GH WUH PLQXWHU -RHO RFK (EUDKLP XWUlWWDU VLQD EDQNDIIlUHU 8WVOlSSHQ IUnQ HQ 5HQDXOW lU J &2 SHU PLQXW YLG WRPJnQJVN|UQLQJ +XU P\FNHW WMlQDU PLOM|Q Sn 2OOHV PHGYHWHQKHW"

18

%RN LQGE

10. )|U VMX nUV SV\NRDQDO\V InU GX EHWDOD NU L UXQGD VOlQJDU 0HQ RP GX nNHU WLOO .DPHUXQ RFK WDU ,ERJDLQ PHG KMlOS DY HQ VKDPDQ XSSQnU GX VDPPD UHVXOWDW Sn WUH GDJDU RFK GHW NRVWDU HQGDVW NU PHG UHVD RFK DOOW +XU P\FNHW VSDUDU GX Sn GHW" $QJH VYDUHW L SURFHQW

19

%RN LQGE

11.

6LOYLR lJHU GHQ ORNDOD WY VWDWLRQHQ L +|NDUlQJHQ , SURJUDPPHW +|NLV L YnUW KMlUWD QlPQV NLRVNHQ 7XUEXWLNHQ L QHJDWLYD RUGDODJ IHPWRQ JnQJHU XQGHU YHFND /LPSDQ FLJDUHWWHU VRP 6LOYLR YLOOH KD VRP UHNYLVLWD PHQ LQWH Ã&#x20AC;FN DY lJDUHQ WLOO EXWLNHQ NRVWDGH NU +XU P\FNHW KDGH GHW NRVWDW 7XUEXWLNHQ DWW XQGYLND GH QHJDWLYD RUGDODJHQ" 6YDUD L NURQRU SHU LQVODJ

20

%RN LQGE

12.

.HFNH RFK )LPSHQ GUHY HWW VWlGERODJ VRP VDQHUDGH NORWWHU L WXQQHOEDQDQ 9DUMH XWU\FNQLQJ JDY LQWlNWHU Sn NU SOXV PRPV (Q GHO DY LQWlNWHUQD JLFN WLOO NXOWXUVSRQV DY 6WUHHWDUW NRQVWQlUHUQD 6RQLF RFK 6S\ 6RP VDQQD NRQVWQlUHU YlJUDGH 6RQLF RFK 6S\ KnOOD VLJ LQRP GH VQlYD UDPDU VRP HWDEOLVVHPDQJHW VDWW XSS I|U YDG VRP DQVnJV YDUD JRGWDJEDUD SODWVHU I|U NXOWXUXW|YDQGH $Y HQ KlQGHOVH UnNDGH GHW IDOOD VLJ Vn DWW WROY DY GHUDV YHUN YDUMH PnQDG XSSI|UGHV L GHW GLVWULNW GlU .HFNH RFK )LPSHQ KDGH DYWDO PHG WXQQHOEDQDQ RP VDQHULQJ +XU P\FNHW RPVDWWH ERUWWDJDQGHW DY GXRQV NRQVWYHUN L IHEUXDUL" $QJH VYDUHW LQNOXVLYH PRPV

21

%RN LQGE

13.

'DYLG VlOMHU IHMNDGH UHSRUWDJH +DQ InU L VQLWW NU SHU LQI|UG VLGD RFK NDQ J|PG EDNRP PHGGHODQGHVN\GGHW DOGULJ nND IDVW 'RMMH UnQDU PLQLOLYVDU L V|GHURUW +DQ InU L VQLWW LKRS NU SHU EXWLN +XU PnQJD PLQLOLYVDU PnVWH 'RMMH UnQD I|U DWW In LKRS OLND P\FNHW VRP 'DYLG Ã&#x20AC;FN I|U VLWW VHQDVWH WUHVLGLJD UHSRUWDJH L I\UIlUJ"

22

%RN LQGE

14.

3MRWLU J|U GXEEHOEXVLQHVV RFK LPSRUWHUDU EnGH WUlVSULW RFK EOLQGNlSSDU IUnQ HQ NRQWDNW L /HWWODQG 3n VSULWHQ WMlQDU KDQ NU O RFK Sn NlSSDUQD NU VW 6WDWLVWLVNW VHWW EOLU HQ SHUVRQ EOLQG SHU OLWHU NRQVXPHUDG WUlVSULW 3MRWLU VlOMHU NlSSDU WLOO DOOD VRP EOLU EOLQGD RFK WMlQDU NU Sn GHVVD +XU P\FNHW KDU 3MRWLU KXQQLW WMlQD WRWDOW Gn VDPEDQGHW XSSGDJDV RFK KDQ VN\QGVDPW KRSSDU Sn IlUMDQ WLOO 5LJD"

23

%RN LQGE

15.

$KPHG DO +DVQDH ILUDU DWW KDQ JnWW XW .7+ PHG ILQILQD EHW\J PnQDGHU RFK DQV|NQLQJVEUHY VHQDUH KDU $KPHG InWW Jn Sn LQWHUYMXHU 1lU KDQ EHWDODW DYLQ IUnQ 359 RFK PHG VLWW Q\D QDPQ $QGHUV +DQVVRQ VNLFNDU LYlJ DQV|NQLQJVEUHY Vn OHGHU GHWWD WLOO LQWHUYMXHU +XU VWRU lU EODWWHNRHIÃ&#x20AC;FLHQWHQ"

24

%RN LQGE

16.

7KH PDQ ZLWK WKH *ROGHQ 'LFN KHWHU HQ OHJHQGDULVN QDWXUILOP 0HGHOOlQJGHQ Sn HQ VYHQVN SHQLV lU FD FP 0HGHOOLYVLQNRPVWHQ KRV VYHQVND PlQ lU NU RFK KRV VYHQVND NYLQQRU NU +XU P\FNHW lU HQ VYHQVN SHQLV YlUG" $QJH VYDUHW L NURQRU SHU FP

25

%RN LQGE

17.

1lU &DUO )UHGULN YnOGWRJ 7HVVDQ Sn HQ IHVW L 7lE\ YDU KRQ EnGH |YHUI|UIULVNDG RFK OlWWNOlGG 6MlOYNODUW Ã&#x20AC;FN & ) NRUWDUH VWUDII Sn JUXQG DY GHVVD HQOLJW nNODJDUHQ I|UPLOGUDQGH RPVWlQGLJKHWHU RFK EHK|YGH EDUD VLWWD PnQDGHU +XU OlQJH VNXOOH & ) In VLWWD RP 7HVVDQ YDULW Q\NWHU RFK KDIW HQ NQlOnQJ NMRO"

26

%RN LQGE

18.

1lU VNDOERODJVKlUYDQ L *|WHERUJ UXOODV XSS RFK KROGLQJERODJHW +XPPHUQ RFK .UDEEDQ JnU L JUDYHQ YLVDU GHW VLJ DWW I|UJUHQLQJDUQD JnU lQGD WLOO 7lE\ +lUYDQV WUH KXYXGPlQ nULQJHQ nULQJHQ RFK nULQJHQ KLQQHU LQQDQ GHQ NRPSLFHUDGH UlWWHJnQJHQ lU |YHU EOL nULQJHQ nULQJHQ RFK nULQJHQ 0HG KXU P\FNHW KDU GHUDV VDPPDQODJGD nOGHU |NDW" $QJH VYDUHW L SURFHQW

27

%RN LQGE

19.

$Y .QLYV|GHUV LQYnQDUH DY PDQOLJW N|Q KHMDU KLVQDQGH Sn %DMHQ L IRWEROO RFK /HNVDQG L KRFNH\ +XU PnQJD lU DOOWVn LQÁ\WWDGH IUnQ ODQGVE\JGHQ"

28

%RN LQGE

20. %|UMH I|UV|NHU NUlQJD HWW JUDP DIIH WLOO 5RIIH $Y HQ KlQGHOVH KDU 5RIIH HQ YnJ L YlVNDQ RFK GHW YLVDU VLJ EDUD YDUD J L GHW SU\GOLJD OLOOD NXYHUWHW +XU P\FNHW I|UV|NHU %|UMH EOnVD 5RIIH Sn" $QJH VYDUHW L a) JUDP b) SURFHQW

29

%RN LQGE

21.

0DQJH 0DV -RUG (ULN RFK 0DWWH 0XVVOD KDU HQ ULNWLJ NDUDWHI\OOD RFK J|U Sn QDWWHQ HQ NLFN DQG JUDE NXSS PRW ,QWHUQVSRUW L +|JGDOHQ 'H NRPPHU |YHU YDURU WLOO HWW YlUGH DY NU RFK GH InU DOOD GDJVE|WHU i PLQLPLEHORSSHW RP NU VDPW VNDGHVWnQG VRP WlFNHU JODVPlVWDUHQV UlNQLQJ Sn NU SOXV PRPV HIWHU DWW GH WRUVNDW +XU P\FNHW KDU YDU RFK HQ I|UORUDW Sn NXSSHQ"

30

%RN LQGE

22. 2VV\ %HUWLO RFK %RVVH VLWWHU L VDQGOnGDQ RFK EUnNDU RP YHPV SDSSD VRP lU VWDUNDVW 'HW WDU IHP PLQXWHU RFK VMX WnUDU DWW UHGD XW -XGHQGRPHQ XSSVWRG I|U nU VHGDQ NULVWHQGRPHQ I|U nU VHGDQ RFK LVODP I|U nU VHGDQ ² DOOD WUH KDU VDPPD SDSSD 'H KDU DOOD EUnNDW L nU RP YHP SDSSD JLOODU PHVW +XU P\FNHW OlQJUH WLG KDU GHVVD WUH UHOLJLRQHU EUnNDW" $QJH VYDUHW L SURFHQW

31

%RN LQGE

23. .HFNH RFK )LPSHQ Ã&#x20AC;UDU YLQVWHQ Sn 9DOOD (IWHU DWW XQGHU NYlOOHQ KD VDWVDW WRWDOW NU KLWWDU GH WLOO VOXW HQ YLQQDUH )XWXUH +DPEXUJ L ORSS VRP JHU NU WLOOEDND *HQRP DWW VlOMD YLQVWERQJHQ WLOO 6LOYLR VRP DOOWLG OHWDU HIWHU VSnUEDUD LQWlNWHU VRP ULGn I|U VLQ PHVWDGHOV URVWIULD YHUNVDPKHW Vn |NDU GH VLQD LQWlNWHU PHG \WWHUOLJDUH +XU P\FNHW KDU GH WRUVNDW XQGHU NYlOOHQ"

32

%RN LQGE

24. 1lU 0DNDUQD 7UROOH XSSQnWW PD[JUlQVHQ I|U EnGH 587 RFK 527 \UNDU GH Sn NURQRU L 5g7 5HMlOW gYHUNODVV 7LOOlJJ 'HW PRWVYDUDU HQ WUHGMHGHO DY HQ WHUPLQVDYJLIW Sn YRQ GHQ )LQNHOV LQWHUQDWVNROD I|U EOLYDQGH VDPKlOOVWRSSDU GlU SDUHWV VRQ 5nODQG JnU VLQ VLVWD WHUPLQ 9DG NRVWDU VH[ WHUPLQHU Sn YRQ GHQ )LQNHOVND"

33

%RN LQGE

25. .HFNH RFK )LPSHQ KDU HQ 9LQ9LQ VLWXDWLRQ L %M|UQV 7UlGJnUG .HFNH KlOOHU FO SURFHQWLJ 0DULQHOOD L )LPSHQV ÁDVND LQQHKnOODQGHV FO SURFHQWLJ 'LDPDQW +XU PnQJD FHQWLOLWHU UHQ DONRKRO ÀQQV QX L GHQ JHPHQVDPPD ÁDVNDQ"

34

%RN LQGE