Parents Guide to helping your child with Higher Maths

Transcription

1 Prets Guide to helpig your child with Higher Mths The essece of mthemtics is ot to mke simple thigs complicted, but to mke complicted thigs simple. S. Gudder Arithmetic is beig ble to cout up to twety without tkig off your shoes M. Mouse

2 Itroductio Help Your Child With Higher Mths We ve desiged this booklet so tht you c use it with your child throughout the sessio, s he/she moves through the Higher course, i order to help them remember key fcts d methods. There re seprte sectios coverig the three uits of the Higher, s well s oe o Credit/Itermedite revisio. Obviously there s o poit tryig to revise Uit 3 work whe your child hs t yet covered it i clss, so it s worth givig rough timetble for the course: Uit is typiclly completed by mid-october Uit is typiclly completed by Jury Uit 3 is typiclly completed by April (The sectio o Credit/Itermedite revisio is fir gme t y time of the course your child should lredy kow it ll!) The booklet is ot ehustive summry of the cotet of the Higher seprte booklet is vilble for tht, should you wish to red it! or is it met to replce the set of much more detiled Higher otes which your child should be buildig up over the sessio. But the booklet does summrise the key fcts d methods which your child will eed to be fmilir with, if they re to hve chce of pssig the Higher. Regulr revisio of these fcts d methods will py rel divideds: it is much better to cotiully revise throughout the yer, th to ttempt to crm it ll i t the lst miute. How to use the booklet Your child could use this booklet o their ow, but we thik it would be much better if they hd someoe else to test them o the cotet. Bsiclly, ll you hve to do is red out the questio o the left-hd colum of the pge, d ll they hve to do is give the correct swer (more or less) which is show o the right. If you prefer, you could simply show your child the swer d sk them wht they thik the questio is. So by the ed of Uit (mid-october), for emple, you could test your child o the etire Uit cotet. However we thik it s much better to revise more regulrly th tht, so we d suggest tht you fid out from your child which sectio they re o (or hve lredy completed) d test them o tht, t resobly regulr itervls. If you c do this the you will be mkig rel cotributio to your child s kowledge of the course. I sese, you will be helpig them to remember their lies for the performce tht will be the Higher Mthemtics emitio. If you lck cofidece i mths yourself Do t worry! We hve tried to epli the termiology d ottio s we go log, but if you re i doubt, the either simply show the questio to your child isted or sk them to epli to you how to sy it. You my fid tht some of your child s swers differ slightly from wht is here: this my ot ecessrily me tht they re wrog, s differet techers will turlly tech thigs i slightly differet wys. If i doubt, check with your child. If still i doubt, plese feel free to get i touch with us t the school. Thks i dvce for your help, d good luck!

3 Revisio from C redit/itermedite Wht is the Sie Rule? b c si A si B si C Wht is the Cosie Rule? b c bc cos A Wht is the formul for the re of trigle? How do you solve qudrtic equtio? How do you fctorise qudrtic? Are bsi C Mke oe side zero, the fctorise the other (or use the qudrtic formul) Look for:. Commo fctor. Differece of squres 3. Double brckets Wht is the qudrtic formul? b b 4c How do you fid the solutio to simple trig equtio, with solutios from 0 to 360 degrees? Fid the cute gle the use ASTC digrm to fid the solutios (usully two) How do you write s power of? ( to the power hlf) How do you write m s power of? (the th root of to the power m) m Wht is 0? How do you write power? with positive

4 Revisio from C redit/itermedite Wht is the Sie Rule? b c si A si B si C Wht is the Cosie Rule? b c bc cos A Wht is the formul for the re of trigle? How do you solve qudrtic equtio? How do you fctorise qudrtic? Are bsi C Mke oe side zero, the fctorise the other (or use the qudrtic formul) Look for:. Commo fctor. Differece of squres 3. Double brckets Wht is the qudrtic formul? b b 4c How do you fid the solutio to simple trig equtio, with solutios from 0 to 360 degrees? Fid the cute gle the use ASTC digrm to fid the solutios (usully two) How do you write s power of? ( to the power hlf) How do you write m s power of? (the th root of to the power m) m Wht is 0? How do you write power? with positive

5 Qudrtic Fuctios How do you sketch qudrtic curve (prbol)? Completig the squre: Why do we complete the squre? Wht is the process for completig the squre? Wht form must the epressio be i before you c complete the squre? The discrimit: Wht is the coditio for equl roots? two distict rel roots? rel roots? o-rel roots? (or o rel roots). Fid the shpe hppy or sd?. Fid the roots (if they eist) ie. where the curve cuts the -is (solve y 0) 3. Fid where the curve cuts the y-is ( 0) 4. Use symmetry to fid the turig poit (or use differetitio) To llow us to mke quick sketch of the prbol, which llows us to fid the turig poit Idetify the -coefficiet Hlve it Squre it Add it o/tke it wy OR: epd brckest d equte coefficiets Must be... d ot etc, so tke out commo fctor if you hve to b 4c 0 b 4c 0 b 4c 0 b 4c 0 How do you show tht lie is tget to curve? Wht does it me to sy tht qudrtic is irreducible? Substitute the lie ito the curve d solve the equtio to show tht there re equl roots (or show tht b 4c 0) It cot be fctorised

6 Polyomils How do you show tht f ()? How do you fctorise cubic? is fctor of How do you sketch the grph of polyomil? Use sythetic divisio (with ) to show tht the remider is zero, or show tht f ( ) 0 First fid lier fctor, usig sythetic divisio, the fctorise the qudrtic from the bottom row of the tble.. Fid where the curve crosses the -is ( y 0 ) d the y-is ( 0) dy. Differetite d solve 0 to fid the d sttiory poits 3. Use ture tble to determie ture 4. Sketch the grph

7 Itegrtio How do you itegrte? How do you prepre for itegrtio? Whe itegrtig idefiite itegrl (oe with o limits), wht must we lwys remember? Why do we itegrte? Wht do we hve to remember whe the eclosed re is below the -is? Wht do we hve to remember whe the re is prtly bove d prtly below the -is? How do we fid the re betwee two curves or lie d curve? How do we fid where the curves meet? Wht do we get if we itegrte ccelertio? Wht do we get if we itegrte speed? Icrese the power by oe, the divide by the ew power Chge y roots ito powers must ot be o the deomitor (bottom) of y frctio Ay pirs of brckets should be epded +C To fid the re uder curve, or to recover f () from f () The swer will be egtive, so we epli this fct d chge the swer to positive We hve to work out the res seprtely (oe bove -is, oe below) the dd ( curve bove curve below) d Use Speed Distce y y d solve Compoud Agle Formule cos(a + B) =? cos(a B) =? si(a + B) =? si(a B) =? Whe sked to fid the ect vlue of si, cos or t, wht should you look for? If right-gled trigle is ot ivolved, wht should you do? cosacosb siasib cosacosb + siasib siacosb + cosasib siacosb cosasib Right-gled trigles Try to mke epressio up which ivolves right-gled trigles d ect vlues you kow (eg 30, 45, 60 degrees)

8 If you re give si, cos or t d told tht the gle is cute ( 0 90, 0 ), how c you fid the other rtios s ect vlues? sia =? cosa =? (three possible swers) How c you epd cos4a, si4a etc? How c you epd cos3a etc? Whe solvig trig equtio, wht twostep process shoud you follow? How do you recogise d solve stright-forwrd solve? How do you recogise d solve double-gle solve? Wht should you lwys check t the ed of trig questio? Drw right-gled trigle, use Pythgors to fid the missig side, the use SOHCAHTOA si Acos A cos A si cos A si A A Write s (A+A) the epd usig the formule Write s (A+A) the epd usig the formule Is it stright-forwrd solve? If ot, the double gle solve si, cos or t ppers oce oly Solve to fid cute gle, the use ASTC Look for double gle d sigle gle (eg A d A) Replce the double-gle formul with pproprite sigle epressio, the mke oe side zero d fctorise i order to solve Should the swer be give i degrees or rdis?

9 The Circle Wht kid of circle hs equtio y r? ( r ) ( y b)? y g fy c 0? How do you fid the equtio of circle? Do you eed to epd the brckets d tidy up your swer? How c you show tht equtio does N O T represet circle? How do you fid where lie meets circle? How do you show tht lie is tget to circle? How do you show tht lie does ot meet circle t ll How do you fid the equtio of tget to circle? Wht is commo tget? How do I show tht two circles touch eterlly? Wht is met by cogruet circles? Wht is met by cocetric circles? cetre (0, 0), rdius r cetre (, b), rdius r cetre ( g, f ), rdius g f c Fid the cetre d rdius, the use ( ) ( y b) r No!! Try to fid the rdius you should be left with the squre root of egtive umber, which is impossible, or zero Rerrge the lie ito the form y or (whichever is esier) the substitute this ito the circle d solve As bove you should fid equl roots, ie oly oe poit of cotct (ltertively, show tht b 4c 0) As bove this time show tht there re o rel roots, ie b 4c 0 Fid the grdiet of the rdius Use m m to fid the grdiet of the tget The use y b m( ) A lie which is tget to two circles Show tht the distce betwee the two cetres is equl to the sum of the two rdii Circles tht re the sme size Circles with the sme cetre

10 Vectors Wht is the differece betwee vector d sclr? Wht is met by givig vector i compoet form? How would you write this i i, j, k form? A vector hs mgitude (size) d directio, wheres sclr oly hs mgitude Writig the swer s colum vector with brckets, eg b c i b j ck How do you fid the mgitude (legth) of vector u = b? c u b c How do you fid vector AB? How do you show tht two vectors re prllel? If poit P divides A B i the rtio m:, how do you fid the coordites of P? Wht re the two forms of the sclr (or dot) product? How do you fid the gle betwee two vectors? How do you show tht two vectors re perpediculr? Useful rules:? ( b c)? AB b Show tht oe vector is multiple of the other Use the Sectio Formul: p ( mb) m d the write out the coordites of P (or use rtios to crete equtio d solve) b b cos (for this versio, remember tht the vectors must NOT be ose-to-til ) b b b 3b3 Use the dot product d solve to fid b - or use formul cos b Show tht b 0 b c

11 Further Clculus Wht do you get if you differetite: si? cos? Wht do you get if you itegrte: si? cos? Wht is the chi rule for differetitio? (How do you differetite f ( g( ))?) Wht do you get if you differetite: si( b)? cos( b)? Wht do you get if you itegrte: si( b)? cos( b)? cos si cos c si c f ( g( )) g ( ) (differetite roud the brckets, the multiply by the derivtive of wht is iside the brckets) cos( b) si( b) cos( b) c si( b) c Wht you get if you itegrte ( b)? ( b) ( ) c

12 The Wve Fuctio How do you epress cos bsi i the form k cos( ) or k si( )? Epd the brckets Equte coefficiets Solve to fid k (squre d dd to get Solve to fid (divide to get t ) k ) How do you kow which qudrt is i? Look t the sigs for k cos d k si - if both re positive the is cute, otherwise you eed to do ASTC digrm Give the choice, which versio of the wve fuctio should you use? How do you fid the mimum or miimum vlues of wve fuctio If it strts with cos, use k cos( ) If it strts with si, use k si( ) Use the versio which keeps both coefficiets positive, if possible Thik of the grph: whe is cos (or si) t mimum or miimum, the djust s ecessry How do you solve cos bsi c? Put the left-hd side ito wve fuctio form, the solve i the usul wy Wht if the questio hs or 3 etc? You still solve the problems i the usul wy with k d foud s before but t the ed you will eed to divide y swers to fid

13 Epoetil d Logrithmic Fuctios Wht poits does the grph of y lwys pss through? (y equls to the power ) Wht poits does the grph of y log lwys pss through? (y equls the log of, bse ) ( 0,) d (, ) (,0) d (,) How do you rewrite form? y log i power y How do you solve equtio where is the power? (eg 4 0) How do you solve log equtio? Log rules: log log y? log log y? log? log? log? How do you get log to bse e (the turl log) o your clcultor? How do you get log to bse 0 o your clcultor? If the grph of log y gist log is stright lie, how do you fid y i terms of? If the grph of log y gist is stright lie, how do you fid y i terms of? Tke logs of both sides the use log rules to work out Epress ech side s sigle log the ccel the logs Or, get logs to oe side d umbers to the other, the rewrite usig power form log y log y log 0 l butto log butto y k The vlues of k d c be foud from the grph y b The vlues of d b c be foud from the grph