Direte Numers Curriulum Rey www.mthletis.om
Numers ome in ll sizes n forms. They n e positive or negtive, whole numers, frtions or eimls n rtionl or irrtionl. Before you strt, investigte these terms n write rief esription out them in the oxes. Positive n negtive numers: Whole numers, frtions n eimls: Rtionl n irrtionl numers: Give this go! Q Clulte the vlue of: (-) - + [ # (+ ) ] + { } Work through the ook for gret wy to solve this Mthletis Pssport P Lerning H SERIES TOPIC
How oes it work? Types of numers Nturl numers re ll the positive ounting numers ",,,...,. Integers re ll the positive (+) n negtive (-) whole numers (inluing zero). Cirle ll the numers elow tht elong to the fmily known s integers. (psst!: some of them nee to e simplifie y lulting their vlue first) - 5 0. 0-50. - 7-0 5 5.987509... -6-9 6-0. o - 8-8 0 0 986. Ple eh of these numers into their orret group elow to lssify them. Irrtionl numers n t e written s frtion Rtionl numers n e written s frtion Integers: the positive n negtive whole rtionl numers inluing 0. Put the speil - types of rtionl numers into their orret group Nturl numers: the ounting rtionl numers (not inluing 0). H SERIES TOPIC Mthletis Pssport P Lerning
How oes it work? Mesuring instruments n integers There re mny pplitions for irete numers. Thermometers re gret exmple euse the temperture n go elow zero. The temperture t miy ws 0 egrees Celsius. During the fternoon, the temperture rose egrees efore flling 9 egrees in the lte fternoon. (i) She in the thermometer showing the lte fternoon temperture. Strting t 0 C, temperture hnge + C then-9c rose =+ n flling =- Temperture t miy 0C Temperture y lte fternoon 5C 5 o o o o o 0 o 9 o 8 o 7 o 6 o 5 o o o o 0 o 9 o 8 o 7 o 6 o 5 o o o o o 0 o +C -9C Centigre o C (ii) Wht irete numer gives the overll hnge in temperture etween the two reings? Overll hnge =+ C- 9C =-5C Digitl sles nee to use irete numers to lulte hnges in mss. Use this informtion to lulte the mss of olives purhse y Mri t her lol Deli: Mri ske for 500 g of olives. After first weighing 575 g of olives, the Deli owner took out 98 g n then put k 6 g. The irete numer sentene for this is: + 575g- 98g+ 6g Inrese = + n erese = - ` Mri purhse + 575g- 98g+ 6g = 5g of olives If the first term is positive, we usully on't write in the plus sign. Mthletis Pssport P Lerning H SERIES TOPIC
How oes it work? Mesuring instruments n integers She in eh thermometer elow to urtely isply these tempertures: (i) C (ii) - 5C (iii) 6 C (iv) 0 C (v) 8 C ove zero (vi) C elow zero 0 o 0 o 0 o 0 o 0 o 0 o 5 o 5 o 5 o 5 o 5 o 5 o 0 o 0 o 0 o 0 o 0 o 0 o 5 o 5 o 5 o 5 o 5 o 5 o 0 o 0 o 0 o 0 o 0 o 0 o -5 o -5 o -5 o -5 o -5 o -5 o -0 o -0 o -0 o -0 o -0 o -0 o Centigre o C Centigre o C Centigre o C Centigre o C Centigre o C Centigre o C This thermometer shows reing of C t the strt. Write the new reings tht woul e isplye fter eh of these hnges in temperture: C 0 o 5 o 0 o 5 o 0 o 5 o 0 o -5 o -0 o Centigre o C (i) Inreses y C (ii) Beomes 8 C ooler (iii) Dereses y C (iv) Wrms up y (v) Rose y C (vi) Droppe 8C 9C (vii) Derese y 7 C then inrese y C (viii) Wrms y C n then further 9C (ix) Cools 5 C efore wrming y 9C (x) Chnges to 0 C efore ropping C C C C C C C C C C C H SERIES TOPIC Mthletis Pssport P Lerning
MEASURING INSTRUMENTS AND INTEGERS * How oes it work? Mesuring instruments n integers The mss of glss ontiner with some wter in it is shown elow:.../.../0... 00 grms Write own the new reing on the sles fter ny one of the following hnges ourre. (i) 0 g of wter poure into the glss grms (ii) g of wter ws spille out of the glss grms (iii) 88 g of flvour syrup ws stirre in n then 50 g ws runk grms (iv) A similr glss with the sme mount of wter ws ple on the sles next to this one. grms (v) A up fille with 6 g of wter ws emptie into the glss n uring the y, 7 g evporte wy euse the room ws hot. grms Try this triky one! A glss ontiner holing ol wter elow ws ple on the igitl sles fter eing move from C room into C room. The glss h ollete grms of wter (through proess lle onenstion) on the outsie for every egree rise in room temperture. Wht ws the originl mss of the wter fille glss if grms of wter ws spilt uring the move? 60 o 55 o 0 o 0 o C 0 o 0 o 0 o Centigre o C 058 grms Mthletis Pssport P Lerning H SERIES TOPIC 5
How oes it work? Mgnitue n iretion of integers Integers re positive (+) n negtive (-) whole numers, inluing zeros. They tell us the size (mgnitue) n iretion of vlues ssoite with hnge. - + It is importnt to first eie wht will e the positive iretion. If up ( ) is positive, then the opposite iretion own ( ) is negtive If right ( ) is positive, then the opposite iretion left ( ) is negtive If hotter is positive (+), then the opposite oler is negtive (-) GROW, FORWARD, ABOVE, FURTHER, MORE, FASTER, EXTRA, ASCEND, HIGHER, OVER, INCREASE, WARMER, NORTH, EAST, REVERSE, BELOW, CLOSER, LESS, SLOWER, REDUCE, DESCEND, LOWER, UNDER, DECREASE, COOLER, SOUTH, WEST, SHRINK, POSITIVE NEGATIVE These re some other wors usully ssoite with positive n negtive irete numers Only negtive numers must hve the sign written in front. Write own the irete numer for eh of these sttements: (i) 5 egrees elow the zero: =-5 The mgnitue of the temperture is 5 egrees (ii) 0 egrees ove the zero: =+ 0 The mgnitue of the temperture is 0 egrees = 0 The positive sign (+) n e hien (iii) $ extr money: =+ $ = $ The mgnitue of the money is $ The irete numer of n mount left fter some hnges represents the overll hnge. One y Mtt strte with no money n ws given $0. He spent $ of it ownloing musi. Wht is his overll hnge of money on this y? The hnge in money for Mtt uring this y ws: $ 0 + $0 - $ = $6 left ` the overll hnge in money for Mtt on this y is +$6 The finl irete vlue = overll hnge 6 H SERIES TOPIC Mthletis Pssport P Lerning
MAGNITUDE AND DIRECTION OF INTEGERS * How oes it work? Mgnitue n iretion of integers For eh of these sttements, write own: (i) the irete numer tht mthes it n, (ii) the mgnitue of the irete numer..../.../0... The tree in Jmie s yr grew metres in one yer. (i) Direte numer: (ii) Mgnitue of the tree s growth: The sme tree ws then trimme whih reue the height of the tree y metres. (iii) Direte numer: (iv) Mgnitue of the reution in height: Aki wlke 50 m West from her strting point. (i) Direte numer: (ii) Mgnitue of the istne Aki wlke: Aki then turne roun n wlke nother 600 m Est (iii) Direte numer: (iv) Mgnitue of this istne wlke y Aki: Sen s nk ount lne erne interest n inrese y $. (i) Direte numer: (ii) Mgnitue of Sen's nk lne inrese: Sen s nk ount lne then hnge ue to eing hrge $ in nk fees. (iii) Direte numer: (iv) Mgnitue of the hnge in Sen's nk lne: Think refully for eh of these sttements n write own the irete numer tht mthes the overll hnge. Pip ws given $ n then spent $5 on ress the sme y. Direte numer for the overll hnge in money tht y: The temperture initilly inrese y 6 egrees n then inrese y further egrees. +$9 Direte numer for the overll hnge in temperture: Nigel went own 6 rungs on ler n then up rungs. Direte numer for the overll hnge in ler rungs: Cmeron's oolness rose y points when he plye guitr, n further 0 points when he i mths. Direte numer for the overll hnge in Cmeron's oolness fter oing oth tivities: e Shiromee hike km North, km Est n then 9 km South. North is the positive iretion. Direte numer for the overll hnge in Shiromee's North-South movement: f Aele's hir grew from 0 m to 0 m long t the k. After hir ut it ws only 7 m long. Direte numer for the overll hnge in Aele's hir length from the initil 0 m: Mthletis Pssport P Lerning H SERIES TOPIC 7
How oes it work? Asening n esening orer When ompring irete numers, it often helps to rrnge them into numeril orer. Asening Desening Asening orer = lowest to highest Desening orer = highest to lowest Don t e trike y the mgnitue of numer. Whih of these two vlues is higher? C -0C - 0C represents very ol temperture (elow/less thn 0 C) C is ol temperture, however it is (ove/greter thn 0 C) ` C is higher vlue thn -0C C -0C 0 - - - - -5-6 -7-8 -9-0 - - -0 is igger numer in mgnitue thn, however the negtive sign mkes it lower in vlue Centigre o C Let s look t n sening n esening orer exmple. For the numers -,0,.5, -6,5.,, - 0,5. : (i) Arrnge them into sening orer -0, -6, -,0,,5.,5.,.5 Lowest negtive Highest positive (ii) Arrnge them into esening orer.5,5.,5.,,0, -, -6, -0 Highest positive Lowest negtive Orer is opposite to sening 8 H SERIES TOPIC Mthletis Pssport P Lerning
ASCENDING AND DESCENDING ORDER * How oes it work? Asening n esening orer Cirle the wor tht represents the orer of the vlues in these sttements:.../.../0... e f Shortest to Tllest Asening Desening Longest to shortest Asening Desening Closest to frthest Asening Desening Wrmest to Coolest Asening Desening Heviest to lightest Asening Desening Thinnest to wiest Asening Desening Arrnge the following groups of numers into sening orer (lowest to highest).,0,7,,,,,5,0,9,,,,, 0, 8,,, 6 - - - - -,,,,.,.,.6,0, - -,,,, e -,,, -.5,, 5,,,,, Arrnge the following groups of numers into esening orer (highest to lowest).,6,5,8,,, 0,,,8,9,,,, 9,, 6, 8, - - - - -,,,, e.6,.9,.,.0,. - - 0, -, -, -.,., -0,,,,,,,,, Mthletis Pssport P Lerning H SERIES TOPIC 9
COMBO TIME * COMBO TIME * COMBO TIME * How oes it work? Como Time! Elevtor Riing Asening n esening orer A eprtment store elevtor opertor strte work on the r floor of the 5 storey uiling. For the first 5 minutes of work, the opertor trvelle to the following floors in the orer written: th floor 5 0 7 8 9 5 6 G > < > < 8 th floor Groun floor n floor Groun floor 5 th floor th floor 0 th floor 5 th floor If up is positive n own is negtive, write numers tht represent the movement of the elevtor opertor uring the first 5 minutes of work.,,,,,,,, Arrnge the irete movements into esening orer.,,,,,,,, During the first 5 minutes, i the opertor mostly sen or esen in the elevtor? Explin you nswer..../.../0... 0 H SERIES TOPIC Mthletis Pssport P Lerning
How oes it work? The numer line Direte numers n e plotte on numer line to instntly see their orer of vlue. Vlues erese (esening) Vlues inrese (sening) -5 - - - - -0-9 -8-7 -6-5 - - - - 0 5 6 7 8 9 0 5 Lower numers re further left n higher numers re further right. Disply the numers -, -, 7 on numer line. -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 Use the plotte points to ompre the vlues of: (i) -, -: - is further left thn -, `- - (ii) -, 7: - is further left thn 7, `- 7 Rememer: mens 'less thn' n mens 'greter thn' This exmple requests numers to e plotte using given rule. Disply ll the even positive integers etween -5 n on numer line. Even n positive -6-5 - - - - 0 5 6 7 8 9 0 Write these numers in esening orer. Re the numers from right to left ( ) ` The numers in esening orer re:,0,8,6,, Hlf vlues re plotte y pling the ot hlf-wy etween the integers on either sie. Disply the numers, -,, - 55., 9, 0 on numer line. -9-8 -7-6 -5 - - - - 0 5 6 7 8 9-5.5 is hlf wy etween -5 n -6 is hlf wy etween n 5 Write these numers in sening orer. Re the numers from left to right ( ) ` The numers in sening orer re: -5.5,-,0,,, 9 Mthletis Pssport P Lerning H SERIES TOPIC
THE NUMBER LINE * THE NUMBER LINE * How oes it work? The numer line Insert the orret symol (less thn) or (greter thn) for eh of these..../.../0... 6 - -5 - - e 8-8 f 7 - -7 g - 0 h 9 9.9 i -5. -5.8 j - 8.. k -.5 -.5 l List the numers isplye elow in sening orer -5- - - - -0-9 -8-7 -6-5 - - - - 0 5 6 7 8 9 0 5,,,, -9-8 -7-6 -5 - - - - 0 5 6 7 8 9,,,, List the numers isplye elow in esening orer -8-7 -6-5 - - - - -0-9 -8-7 -6-5 - - - - 0 5 6 7 8 9 0,,,, -9-8 -7-6 -5 - - - - 0 5 6 7 8 9,,,, H SERIES TOPIC Mthletis Pssport P Lerning
How oes it work? The numer line Disply the following sets of numers on numer line. 5, 8,,, 6. - - = 0.5-9 -8-7 -6-5 - - - - 0 5 6 7 8 9 8.5,.5,, 7.5, 5. - - -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 -,, 6, -, 5. -9-8 -7-6 -5 - - - - 0 5 6 7 8 9-7, 7.5,, 0,,. -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 5 Disply the following sets of numers on numer line. The first six o integers ove -8. -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 All the negtive even integers less thn 7 ut greter thn negtive 8. -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 All the multiples of higher thn 7 ut lower thn 8. 0 5 6 7 8 9 0 5 6 7 8 All the integers tht re or 7 whole numers wy from -. -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 Mthletis Pssport P Lerning H SERIES TOPIC
How oes it work? Aition n sutrtion using numer line Strting from zero eh time, lultions involving irete numers n e me using numer line. Negtive iretion Positive iretion Use numer line to lulte these: Rememer Alwys strt from 0 n move using eh irete numer one t time. (i) 8 + + -5 - - - - -0-9 -8-7 -6-5 - - - - 0 5 6 7 8 9 0 5 (ii) -6-7 -7 Strting t +8, to the right finishes t right (+) 8 ` + = Strt Finish -8-7-6-5-----0-9 -8-7 -6-5 - - - - 0 5 6 7 8 9 0 Finish Strt Strting t -6, then 7 to the left finishes t left (-) ` -6-7 =- (iii) + (-) (iv) (-) + Negtive numers re often written insie prentheses - Sme s (iii) ut in ifferent orer + - - - 0 - - - 0 Finish Strt Strt Finish Strting t + then to the left finishes t right (+) Strting t -, to the right finishes t right (+) ` + (- ) = ` (- ) + = (v) 0-6+ -6 + -8-7-6-5-----0-9 -8-7 -6-5 - - - - 0 5 6 7 8 9 0 Finish Strt Strting t +0, then 6 to the left followe y to the right finishes t left (-) ` 0-6 + =- H SERIES TOPIC Mthletis Pssport P Lerning
How oes it work? Aition n sutrtion using numer line Show the lultion for eh of these on numer line n write own the nswer. 5 + = -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 (- ) + 8 = -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 (-)- = -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 7 5 - = -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 e 6 - + = 0 5 6 7 8 9 0 5 6 7 8 Write own the lultion to get the given nswer shown on eh of these numer lines: =- -5 - - - - -0-9 -8-7 -6-5 - - - - 0 =- -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 = -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 Mthletis Pssport P Lerning H SERIES TOPIC 5
How oes it work? Aition n sutrtion using numer line Show the lultion for eh of these on numer line n stte the nswer..5 + = + -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 (-5. )- 5. = Strt Finish - - -0-9 -8-7 -6-5 - - - - 0 5 6 - + 05. = -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 8-55. = -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 Try these trikier ones! Write own the lultion shown on eh of these numer lines: = ADDITION AND SUBTRACTION USING A NUMBER LINE *.../.../0... -9-8 -7-6 -5 - - - - 0 =-5. -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 = -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 = -9-8 -7-6 -5 - - - - 0 5 6 7 8 9 6 H SERIES TOPIC Mthletis Pssport P Lerning
Where oes it work? The mysterious Dr Thermos! Dr Thermos ontrols the temperture of his lortory using lrge hot or ol spheres. + C -C Hot Sphere Col Sphere Count the numer of eh sphere type to fin the overll temperture of his lortory. Clulte the temperture of the lortory for these sphere omintions: (i) (ii) 7 hot spheres n ol spheres `+ 7C hot spheres n ol spheres `+ C n ` lortory is n ` lortory is +C -C -C -C +7 n - omine to mke + + n - omine to mke - To hnge the temperture, Dr Thermos s or removes spheres. Show two wys Dr Thermos n rise the temperture y + C for the omintion of spheres elow. 9 hot spheres n 6 ol spheres `+ 9C n -6C ` lortory is urrently +C +9 n -6 omine to mke + (i) (ii) Dr Thermos oul two hot spheres 9 hot spheres + hot spheres n 6 ol spheres `+ C 9 hot spheres n 6 ol spheres - ol spheres `+ 9C n ` lortory is now ` hnge of Dr Thermos oul remove two ol spheres n +C ` lortory is now ` hnge of +C -6C +5C -C +5C + n -6 omine to mke +5 +9 n - omine to mke +5 Mthletis Pssport P Lerning H SERIES TOPIC 7
Where oes it work? The mysterious Dr Thermos! + C -C Hot Sphere Col Sphere Write own the temperture of Dr Thermos lortory for these sphere omintions: THE MYSTERIOUS DR.THERMOS! THE MYSTERIOUS DR.THERMOS!.../.../0... Lortory temperture: C Lortory temperture: C Lortory temperture: C Lortory temperture: C Write the numer n type of hot or ol spheres tht woul nee to e e to the lortory to use these hnges in temperture. +C -9C hot spheres -6C +0C Write the numer n type of hot or ol spheres tht woul nee to e remove from the lortory to use these hnges in temperture. -7C +C -5C +8C 8 H SERIES TOPIC Mthletis Pssport P Lerning
Where oes it work? The mysterious Dr Thermos! For eh of these omintions of hot n ol spheres, explin two ifferent wys Dr Thermos n hnge the temperture of the lortory to equl the mount given in the squre rkets. [ + C] (i) First wy: (ii) Seon wy: [-C] (i) First wy: (ii) Seon wy: Try these trikier ones! 5 If Dr Thermos oul only use the spre spheres shown elow eh given omintion, explin two ifferent wys he oul hnge the temperture of the lortory to equl the mount in the squre rkets. You must use ll of the spre spheres for t lest one of the methos [ +6C] (i) First wy: Spre spheres: (ii) Seon wy: Using ll the spre spheres [-C] (i) First wy: Spre spheres: (ii) Seon wy: Using ll the spre spheres Mthletis Pssport P Lerning H SERIES TOPIC 9
Where oes it work? Puzzle time! The mysterious Dr Thermos! 6 Dr Thermos hs puzzle room t the entrne to his lortory to keep his experiments seret. The oor will open when the isply in the puzzle room res - C. The isply urrently res + C. Eh hot ( + C) or ol (- C) step n e use only one. Only siewys ( ) or forwr ( ) steps n e tken. Steps in the white setion re swithe on (e to the puzzle room). Steps in the grey setion re swithe off (remove from the puzzle room). (i) Drw pthwy tht will llow you to enter the lortory of Dr Thermos. (ii) See if you n fin nother pth! Finish nywhere long here Strt nywhere long here 0 H SERIES TOPIC Mthletis Pssport P Lerning
Where oes it work? Aing n sutrting irete numers The exerises in Dr Thermos lortory were exmples of ing n sutrting irete numers. Here is wht hppene: Aition sutrtion When When When When =+ C =- C ws e, the temperture went up ws e, the temperture went own ws remove (Sutrte), the temperture went own ws remove (Sutrte), the temperture went up The rules for ing n sutrting represent the sme thing. Use the pttern ove to lulte these: = + n = - Aition Sutrtion 5 + = 5 + (+) = 5 + = 6 5 + = 5 + (-) = 5 - = 5 - = 5 - (+) = 5 - = 5 - = 5 - (-) = 5 + = 6 + + = + (Sme signs together mens plus) + - = - (Opposite signs together mens minus) - + = - (Opposite signs together mens minus) - - = + (Sme signs together mens plus) These rules work for ll kins of irete itions n sutrtions Clulte these itions n sutrtions: (i) + (- ) = - = + - = - (ii) (- ) + =- + = Sme s (i) ut in ifferent orer (iii) 6 -(- 5. ) = 6+ 5. = 85. - - = + (iv) (-) -(- ) =- + - - = + =-7 (v) + (- ) = - =- + - = - (vi) (-.) + (- 5.) =-.-5. + - = - =-7. (vii) 95. -(- 85. ) + 6 = 95. + 85. + 6 = (viii) +- ^ h -- 6 6 ` = - + j = 0 Mthletis Pssport P Lerning H SERIES TOPIC
ADDING AND SUBTRACTING DIRECTED NUMBERS * Where oes it work? Aing n sutrting irete numers Clulte these itions n sutrtions of integers without lultor. - 5+ - + 7 - -.../.../0... 8 - - e 6 - f 6 - g 6 ( ) + - h 0 ( 7) + - i 0 + (-8) j ( 5) - + - k 0 ( 9) - + - l -(-5) m 0 -(-) n 8 ( 6) - - - o -5 -(-6) 8 - + 9 6 8 - + - 8 7 - + - - 7 ( ) - + - + e -( -6)- 7 f 5+ (-)-(-) H SERIES TOPIC Mthletis Pssport P Lerning
Where oes it work? Aing n sutrting irete numers Simplify these itions n sutrtions: Use the numer line on the sie if it helps. g e 55. - 8-6 -9 05. + (-55. ) -- 7+ 5. + (- 5. ) + 5. -85. - 6 `- +8 j e h - + 5. 95. - 6 - -(- ) f + 05. -6-7 + ^-9h - - ` j - + ^-h - - - ` j f i --75. 75. + (-8) 0- ( + 8) -(-6) - - -0-9 -8-7 -6-5 - - - - 0 5 6 7 8 9 0 Mthletis Pssport P Lerning H SERIES TOPIC
Where oes it work? Aing n sutrting irete numers For trikier questions, the lultor n e use. Use the ( ) utton to hnge etween positive n negtive when entering numers into the lultor. Sometimes the utton my look like -9. -( - 9. ) + (- 9. 6) =- 9. +. 9-96. ( ) 9. +. 9 9. 6 = =-6 Simplify using the rules Enter into lultor Use your lultor (only if you nee to) for these: 5 5 89-9 - + + (-5) - 5 + (-9) e -9 -(-8) f - 6 + (-87)-(-) g - 6. + 0. h -05. -67. i - 0. + (-56. ) j 66 -. 6+ (-. ) k 06. + 5. -97. l - 0. + 76. m (- 7. ) + (-5. 8) n -0. -(-509. ) o 98. -(-. 8) -. p 9 - +5 q -0-66 `- j r 00+-5-6 ` j H SERIES TOPIC Mthletis Pssport P Lerning
Where oes it work? Multiplying n iviing irete numers The rules for multiplying n iviing re similr to ing n sutrting: Multiplition Positive # Positive = Positive Positive # Negtive = Negtive Negtive # Positive = Negtive Negtive # Negtive = Positive + # + =+ + # - = - - # +=- - # - =+ (Sme signs men nswer is positive) (Opposite signs men nswer is negtive) (Opposite signs men nswer is negtive) (Sme signs men nswer is positive) Division The ext sme rules pply euse ivision is the opposite opertion to multiplition! These rules lwys pply to ll multiplition n ivision lultions. Clulte these multiplitions n ivisions: (i) - # =- - # +=- (ii) - ' =-7 - ' +=- (iii) # (-.5) =- 8 + # - = - (iv) ' (- 8. ) =-5 + ' - = - (v) -8 # (- ) = - # - =+ -6.8 ' (- 8) =.6 (vi) - ' - =+ (vii) # - 6 =- - # +=- (viii) - ' `- = 0 5 j - ' - =+ Be reful squring (or uing) irete numers s prentheses ffet the question. (ix) (- ) = (-) # (- ) = 9 (x) - =- ( # ) =-9 If more thn two terms re multiplie or ivie, simplify y lulting in orer from left to right. Simplify these mixe questions: (i) 6 ( ) 9 # ' - =-8 ' 9 =- Clulte 6 # (- ) first - ' +=- + # - = - (ii) (-)' 0.5 # (-) =- # (-) = Clulte (- ) ' 05. first - # - =+ - ' +=- (iii) (-) = (-) #(-) #(-) = # (-) =- Clulte (-) # (- ) first Clulte # (- ) next - ' - =+ + ' - = - (iv) 8 # 5 '(-)'(-) 5 = '(-)'(-) =- ' (-) = Clulte 8 # 5 first 5 Clulte ' (- ) next - ' - =+ + # + =+ + ' - = - Mthletis Pssport P Lerning H SERIES TOPIC 5
Where oes it work? Multiplying n iviing irete numers Simplify these without using lultor: ( 7) # - -6 # 8-6 ' 8 ' (-) e -6 # (-) f - # (-) g - ' (-) h ( ) ' - i ( ) 6 # ' - Simplify these without using lultor: - ' ' (-8) -8 ' (-6) # -5 # #(-) # -5 e (-) f ( ) - g - h ' (-) i ( - ) #- j 6 ' (-) k '- l 9 '- # () 6 H SERIES TOPIC Mthletis Pssport P Lerning
MULTIPLYING AND DIVIDING DIRECTED NUMBERS * Where oes it work? Multiplying n iviing irete numers Simplify these using lultor: 7 ( ) # - - # - # 9.../.../0... -95 ' (-9) e 5 ( 9) # - f -6' 8 g -5 # 6 h (-9)' (-) i -5 # 5 ' (-) Use your lultor to simplify these: 5 ( 65) # ' - - -500'(-5)' 5-55' (-) # (-8) # (-6) ' 5 e.5 5 ( 0.5) # ' - f - ' ( - ) # 5 # ( - ) 5 g.6 7.9 # # ' - h -0. 7' (-6. )' ' 97. i 60 '(-6) # ' 0.6 5 j (-7) k - 0 l (-8) - -0 m (-9) n --6 ( ) o --5 (. ) 5 Mthletis Pssport P Lerning H SERIES TOPIC 7
Wht else n you o? Comining the si opertions Questions tht mix multiplying/iviing with ing/sutrting nee to e one in ertin orer. Multiplition or ivision opertions re rnke the highest n therefore must e omplete first. Clulte these omine opertions questions: (i) - + # 5 - + # 5 =- + 5 Do this first = A - n 5 (ii) 5 ( 5) ' - - 5 ' (-5)- =-7- Do this first =-8 Sutrt from -7 If more thn one multiply/ivie sign, work left to right. Do the sme for ition/sutrtion. Clulte these omine opertions questions: (i) - 8 ' 7 # + 6-8 ' 7 # + 6 =- # + 6 Do this first =- + 6 =-6 Multiply - y A - n 6 (ii) 7 --(-0) # 7 --(- 0) # = 7 --(-0) Do this first = -(-0) = Sutrt from 7 Sutrt -0 from ( = + 0 ) (iii) ' + (-5) # ' + (- 5) # = 8 + (-5) # Do this first = 8+ (-0) =- Multiply -5 y A 8 n -0 (iv) 6 # (-) -0' 0 # (-) 6 #(-)-0' 0 #(- ) =-8-0' 0 # (-) Do this first =-8-7 # (-) =- 8 + =- Divie -0 y 0 Multiply -7 y - A -8 n 8 H SERIES TOPIC Mthletis Pssport P Lerning
COMBINING THE BASIC OPERATIONS * Wht else n you o? Comining the si opertions Simplify these omine opertions without lultor: - + 6' 8 6 # (-8)-.../.../0... 50 ' (- 5) + 0-8-5 # e 5+ 7 # (-6) f - #- + 8 g -9 ' (-) h -0 - # 5 Simplify these omine opertions without lultor: ( ) 8 # - + - 6 9 # ' - + - 9+ 8' (- ) + ' - # (-) Use lultor to simplify these omine opertions: 8 ' (- ) + # 7 0 ' (-5) # -8 5 6 ( ) # ' - - - 0 #(-5) -5 ' #(-6) Mthletis Pssport P Lerning H SERIES TOPIC 9
Wht else n you o? Orer of opertions For more omplex lultions involving prentheses n exponents, we use similr onvention: P E D M A S Prentheses Exponents Division Multiplition Aition Sutrtion Do these first Then o these Do these lst Simplify these irete numer lultions using the orret orer of opertions: (i) (- 8) # 7 (- 8) # 7 = # 7 = 8 Sutrt 8 from insie the prentheses Multiply n 7 (ii) - ' ( + 5) + - '( + 5) + =- ' 7+ =- + = 0 A n 5 insie the prentheses Divie - y 7 A - n (iii) ' (-) ( 8-6) (-) '(8-6) = (-) ' = 6 ' = 8 Sutrt 6 from 8 insie the prentheses Squre - Divie 6 y (iv) 5 # (- 5+ ) - 0 5 #(- 5+ ) - 0 = 5 #(-) -0 = 5 # (-8)- 0 =-0-0 =-0 A - n 5 insie the prentheses Cue - Multiply 5 y -8 Sutrt 0 from -0 (v) 9 -(- # 5) + 9 -(- # 5) + = 9 -( - 0) + = 9 -(- 7) + = 9 -(- 7) + 9 = 6 + 9 = 5 Multiply - y 5 insie the prentheses Sutrt 0 from insie the prentheses Squre Sutrt -7 from 9 A 6 n 9 0 H SERIES TOPIC Mthletis Pssport P Lerning
Wht else n you o? Orer of opertions Simplify these irete numer lultions using the orret orer of opertions without lultor: (6-0) # ' ( 6+ (-8)) 5 (7 9) # ' - (--(-9)) # e -5 # ( -9) - 5 f (6 ) ( ) # ' - - g 8 + ( 6 ' + 0) h ( 8 ' -( -)) - 7 Simplify these lultions ontining exponents using the orret orer of opertions without lultor: ' (-) (-) (-7-5) # (-) ( 8-) -6 ( - ) ' 8 Mthletis Pssport P Lerning H SERIES TOPIC
ORDER OF OPERATIONS * ORDER OF OPERATIONS * Wht else n you o? Orer of opertions Try simplify these trikier ones using the orret orer of opertions: 6 ' ( + # 6) + (-).../.../0... -00 # ( - '(-)) ' 0 ( # - 0) ' 9+ 8 9 (( 8) 8 6) # ' - - + - e 00 '(-) #(-)'(6 - #(-)) H SERIES TOPIC Mthletis Pssport P Lerning
Wht else n you o? Orer of opertions with grouping symols For questions ontining multiple pirs of grouping symols, omplete eh pir seprtely. Grouping symol nmes: ( ) = prentheses [ ] = squre rkets { } = res Simplify these omine opertions questions ontining multiple rkets: (i) (0 ) ( 7 ) # ' - - ( 0 # ) '(-7- ) = 0' (-0) =- Simplify the prentheses Divie 0 y -0 The sme question n e written s frtion. 0 # (0 # ) ' ( 7 ) -7- = - - numertor enomintor = (numertor)' (enomintor) (ii) - # ' (-) - # ' (-) = (- # ) '('(-)) = ' (-) =- Re-write with prentheses Simplify the prentheses Divie y - When questions ontin rkets within other rkets, work from the insie to the outsie. Simplify these omine opertions questions ontining multiple rkets: (i) 6 ( 6) ' 5@ # - # 6( -6) ' 5@ = # 6-5' 5@ = #- =- Simplify the prentheses first Simplify the squre rkets Multiply y - (ii) 0 -" 8-66 ' ( - 7) @, 0 -" 8-66 '( -7)@, = 0 -" 8 + 66 '( -7) @, Simplify the prentheses = 0 -" 8 + 66 ' (-)@, = 0 -" 8 + -, = 0-6 = Simplify the squre rkets Simplify the res Sutrt 6 from 0 Mthletis Pssport P Lerning H SERIES TOPIC
ORDER OF OPERATIONS WITH GROUPING SYMBOLS * Wht else n you o? Orer of opertions with grouping symols Simplify these questions: (6 9) ( 7) + + - ( -9) -( - 8).../.../0... (7 - ) # (+ ) ( 8 -)' (--(-7)) e 6-9 5- = ( 6-9)' ( 5-) = 7 ' = 9 f - # 0 8 ' g ( 50 + 5 ' 9) '(-' ) h 8 # 5- ' (-6) Simplify these questions with multiple grouping symols: - ' 6 ( 7+ ) ' 9@ 60 + (- ' ) @# 8 6-6 # ( -(-7))@' 5 6 ( 9) # @ # - H SERIES TOPIC Mthletis Pssport P Lerning
Wht else n you o? Simplify these: Orer of opertions with grouping symols { } [ + ( 6 # 8) ]' -5 (-) - + [ # (+ ) ] + { } Rememer me? Give these three triky ones go to ern n wesome stmp. (-8) # (-) 5 -(-) * AWESOME *.../.../0... * AWESOME * { - # - 7 + (-6) ' [( # (-5)) -(- ) ]} { } - # [(--(- )) + (-) ]' #(-) Mthletis Pssport P Lerning H SERIES TOPIC 5
Chet Sheet Here is summry of the importnt things to rememer for irete numers Types of numers Positive n negtive numers re numers tht hve oth size n iretion. Eg: + or - Rtionl numers n e written s frtion. They inlue whole numers n terminting or reurring eimls. Eg: 8,.,.5 o,,. Irrtionl numers nnot e written s frtion. These inlue eimls tht keep going without following pttern, n roots of numers tht re not perfet squres/ues et. Eg:,.98698..., r. Integers re whole numers only. They re rtionl n hve no eiml or frtion prts. Eg:, -, 00. Mgnitue n iretion of numers Direte numers re positive (+) or negtive (-) whole numers. They show oth the size (mgnitue) n iretion of vlues ssoite with hnge. It is importnt to first eie wht will e the positive iretion. Asening n esening orer Asening orer = lowest to highest Desening orer = highest to lowest Aing n sutrting irete numers + + = + (Sme signs together mens plus) Aition + - = - (Opposite signs together mens minus) - + = - (Opposite signs together mens minus) Sutrtion - - = + (Sme signs together mens plus) Multiplying n iviing iret numers Multiplition Positive # Positive = Positive Positive # Negtive = Negtive Negtive # Positive = Negtive Negtive # Negtive = Positive + # + =+ + # - = - - # +=- - # - =+ (Sme signs men nswer is positive) (Opposite signs men nswer is negtive) (Opposite signs men nswer is negtive) (Sme signs men nswer is positive) Division The ext sme rules pply euse ivision is the opposite opertion to multiplition! Orer of opertions The wor PEDMAS helps to rememer the orer in whih we nee to perform lultions. P E D M A S Prentheses Exponents Division Multiplition Aition Sutrtion Do these first Then o these Do these lst 6 H SERIES TOPIC Mthletis Pssport P Lerning
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