Revision Notes. Name Class

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1 M8 Revision Notes Name Class 1

2 fractions WHEN OR TURN TO TOP HEAVY multiply do both top and bottom = = = 4 10 = divide = 3 4 flip over second number then multiply 3 4 = = =

3 Add/subtract deal with whole numbers first = = = use the grid method(mathswatch) or the crossover method = = 1 12 you could still turn to top heavy first but numbers large to work with 3

4 Cubic/reciprocal graphs y = mx + c y = x 2 straight line TYPES OF GRAPHS quadratic cubic y = x 3 y = 1 x reciprocal 4

5 example of exam question x y (a) Complete the table 1 3 =1 2 3 =8 3 3 = = = = 27 This table shows some values of x and y for the equation y = x ( 27+5= 22) (27+5=32) [1] (b) Draw the graph of y = x from x = 3 to x = 3 [2] (c) Use your graph to solve the equation x = 10 see working on the graph 1.6 (c) x =... [2] 5

(b) y 40 to solve x 3 + 5 = 10 draw a line across at 10 then read off the x

6 (b) y 40 to solve x = 10 draw a line across at 10 then read off the x value x show dashed lines to help read off the graph

7 probability MATHS 0.7 pass ENGLISH tree diagrams pass fail remember to multiply probabilities = = fail 0.6 pass = 0.18 must 0.4 fail add up = 0.12 to 1 pass Maths pass English = = 0.42 pass one subject only = = = gives

8 STARTER must add up to 1 same rule applies for fractions soup salad MAIN beef fish beef fish remember to multiply probabilities 2 1 = = = = soup for starter and fish for main = 2 2 =

9 Solving equations involving divides eg if you see a it means do the opposite which will be a x + 6 = x 4 4 = spare part goes first 20 x = 9

10 Harder solving equations eg 6x + 8 = x x + 8 = 2x remember to both terms by 2 2x 4x = 4x = x = 2x

11 Be careful of negatives eg 12 3x = x = x = x = 8 11

12 Solving Inequalities < > less than more than < > less more than or than or equal to equal to to solve inequalities, solve like equations then keep the < or > in eg solve 4n + 1 < n < n < 3 12

13 percentage increase/decrease 15% means 15/100 = 0.15(decimal) so 15% increase = % decrease = 0.85 amount eg Increase 400 by 15% eg Decrease 320 by 7% take 15% from 100% first decimal = =

14 finding the original A wage has been increased by 20%. The new wage is 420. Find the original wage? use 1.2 for 20% increase when finding original just amount decimal = 350 finding the % increase/decrease eg In a sale a TV has been reduced from 400 to 380 Find the percentage decrease. new old = 0.95 =95% so 100% 95% which means a 5% decrease 14

15 Shading/inequalities on a grid y y = 2 y = x o x = 4 there are three lines choose < or > above the line y = 2 y > 2 x < 4 y < x below the line x = 4 use a test point to help below the line y = x + 1 x 15

16 Things to remember vertical lines of the form x = horizontal lines of the form y = diagonal lines of the form use < or > if a broken line use < or > if a solid line y = mx + c see y = mx + c topic 16

Cumulative frequency weights (kg) 0 and less than 10 10 and less than 20 20 and less than 30 30 and less than 40 cumulative frequency is a running total frequency 15 30 10 5 Median = 14 half of 60 is

17 Cumulative frequency weights (kg) 0 and less than and less than and less than and less than 40 cumulative frequency is a running total frequency Median = 14 half of 60 is 30 cumulative frequency Interquartile range = = plot on the bigger number in the group weights(kg) How many people weighed more than 35kg? cumulative frequency upper quartile lower quartile upper quartile median lower quartile read the weight axis first this time = 3 17

18 Box plots CLASS A CLASS B lowest lower median upper highest value quartile quartile value test results class A class B Make two comments bigger median better bigger interquartile range more varied class A better overall class A more varied(inconsistent) 18

19 Repeated % increase/decrease amount decimal eg1 In a sale reduced by 12% every week. How much would a 250 coat cost after 4 weeks? remember 12% decrease is 0.88 as a decimal power button determines number of repeats = = use scientific calculator power button x round to 2 decimal places (nearest pence) 19

20 eg2 Compound interest is 4% If I started with after how many years will it be more than ? trial different years to help compound interest means 4% increase = = = years 20

21 Reflect possible lines of symmetry x axis y axis vertical line x = horizontal line y = diagonal lines y = x y = x Transformations y = x C x = 4 4 C to D is a reflection in the line x = 4 D O y A B y = x x A to B is a reflection in the line y = x 21

22 so Translate translate means move y and ( ) left +right down + up ( 2) 4 using vectors ( ) 1 5 means move 2 right, 4 down means move 1 left, 5 up 7 squares left 9 squares down B O translate shape A 7 to get B A ( ) 9 use a dot to help you count x 22

23 Rotate centre (c) NOT ALWAYS (0,0) y A to B has been rotated 90 o clockwise centre (1,2) size 90 o 180 o quarter half direction clockwise anti clockwise use tracing paper D centre ( 5, 6) C centre (1,2) o A B C to D has been rotated 90 o anti clockwise centre ( 5, 6) x 23

24 Reflect/ rotate/ translate C y A y = x centre ( 1,2) B o x reflect A in the line y = x to give B rotate A 90 o anti clockwise ( 1,2) to give C translate A 2 ( 6) to give D D 24

25 Enlargement given a CENTRE C scale factor 2 y axis on a grid use lines to help C o x axis coordinates of centre (0,1) scale factor = 3 25

26 Decimal scale factors scale factor 1.5 extend by half original means 1 and a half original add half original so originally 2 then = 3 4 then = 6 C

27 Reduction scale factors scale factor 0.5 means half of original 0. scale factors reduce the shape new shape will be smaller and must go between centre and original so originally 8 then half is 4 6 then half is 3 C

28 Negative scale factors SF 2 for a negative scale factor invert the projection lines c so SF 2 enlarges shape by 2 but also rotates 180 o 28

29 Simultaneous equations eg1 a) number equations b) c) d) e) rewrite equations eliminate y solve to find x substitute back into equation 2x + 3y = 12 5x y = 13 2x + 3y = 12 15x 3y = 39 17x = x = 3 f) solve to find y y = 6 + 3y = 12 3y = add equations (signs are different) 29

30 number 3x + 2y = 13 eg2 a) equations 2x + 5y = 5 b) rewrite equations 15x +10y = 65 4x +10y = subtract this time (signs same) c) d) e) f) eliminate y solve to find x substitute back into equation 1 solve to find y 11x = x = 5 + 3y = 12 2y = 2 y = 1 remember signs different add same subtract answers could be negatives or decimals 30

31 Standard form used to write large or small numbers need to know to about 10,100 the number with a. in a 10 n has to be between 1 and 10 number of places moved large numbers positive power = = = = small numbers negative power = = = =

32 Standard form calculations without a calculator ( ) + ( ) turn each to ordinary numbers use the with a calculator 10 x button = do a written sum to help ( ) + ( ) ( ) ( x x 4) x = = = always put back into standard form 32

33 straight line graphs y = mx + c y = mx + c gradient eg y = 3x + 4 y 3 4 o x gradient is a measure of steepness y intercept gradient = y intercept is where line crosses the y axis 3 coordinates of y intercept = (0,4) make sure you put(0, ) for the coordinates 33

34 how to find the equation of a straight line y use y = mx + c find m gradient find c y intercept gradient = x 4 the equation is 4 3 y = x 1 line cuts at 3 so y intercept is 3 34

35 eg Which of the two lines are parallel? A y = 2x 3 gradient = 2 B y = 3 2x gradient = 2 C 2y = 5x 6 rearrange to get in the form y = y = 2.5x 3 D 3y 6x = 12 3 lines are parallel if the equations have the same gradient gradient = 2.5 or 5 2 y 2x = 4 +2x +2x y = 2x + 4 gradient = 2 2 is not parallel to 2 A and D are parallel 35

36 Similar shapes(triangles) similar means enlarged/reduced o 23 q 8cm Find angle a 3.6cm cm 12cm angles remain the same so 23 o and 62 o are in the triangle so = 85 o so o = 95 o a o 62 p scale factor = new old Find length p Find length q scale factor = 12 = 12 8 = = 5.4cm = 7cm 36

37 Be careful when triangles NOT facing the same way F 10cm C A 11cm 5cm B? 16cm 8cm D Explain why these triangles are similar check scale factors Find length EF = 17.6cm E AC = 16 = 1.6 DF 10 AB = 8 = 1.6 DE 5 scale factors same so must be similar 37

38 Comparing data When making comparisons 1. Comment on the average (who did better) 2. Comment on the range(varied/inconsistent) Class A mean = 67% range = 40% Class B mean = 48% range = 10% Make two comparisons mean 67% so bigger than 48% Class A s results were better overall Class A s results were more varied(inconsistent) range 40% so bigger than 10% 38

39 Box plots test results A B class A bigger median so test results better class A bigger interquartile range so results more varied(inconsistent) 1. Use an average (mode,median,mean) to see who did better 2. Use a range (range,interquartile range) to see who was more spread out/ varied 3. If less spread out then results consistent If more spread out then results inconsistent 4. Don't put class A median higher say class A better overall 5. Have to find averages/ranges from a) lists b) stem and leaf diagrams c) box plots d) cumulative frequency curves e) frequency tables revise these from M5,M6,M7,M8 39

40 Trigonometry labelling hyp(h) opp(o) S O H A C H O T A adj(h) Using trigonometry to find a missing side eg hyp 17cm 34 o? S O H opp 1 Label sides 2 Choose triangle 3 Calculation S H = sin(34) 17 = 9.5cm (rounded to 1dp) 40

41 Harder side example O A O S H C H T A eg adj 10cm 52 o A C H? hyp 1 Label sides 2 Choose triangle 3 Calculation A = A C = 10 C = cos(52) 16.2cm (rounded to 1dp) 41

42 Using trigonometry to find an angle O A O S H C H T A eg opp 6cm 10cm adj Find angle y A O T A = y 1 Label sides 2 Choose triangle 3 Calculation 4 Use SHIFT O = O A = 6 10 = 0.6 SHIFT tan(0.6) = 31 O (rounded to nearest whole number) 42

43 Trigonometry problems Flag pole A o 50 Isosceles Find angle x B D 1 x 2 opp 8m x 12.4cm 8.3cm hyp 12m 6.2cm(half of 12.4cm) 8.3cm O S H a) Calculate the angle that BC makes with the horizontal opp 8m b) Find the length AD opp O H C 50 o hyp 12m adj?? 8m O S H = 8 12 O T A O H then SHIFT = 42 o O T = 8 tan(50 o ) = 6.7m always draw a sketch of each right angled triangle with all sides/angles and? on = then SHIFT so x = 2 = 48 o 48 = 96 o 43

44 Bearings N A bearing is the angle measured clockwise from the North line. It has to be three figures. highlight the actual bearing first then decide which angle to find using trigonometry A 25km then decide what else to do eg (add 90 o ) to find total bearing 31km Find the bearing of B from A A C H B A adj? 31km 25km hyp A = A H = H = SHIFT so = 126 o = 36 o B 44

45 Dimensions 3ab 2 r 2 h = area (length length) = volume (length length length) Length, area or volume? length = length area = length length volume = length length length 2L + 2W = length (length + length) numbers and not affected? +? must be the same or None 3(ab + bc) =3ab + 3 bc multiply out bracket = area (area + area) 2r 2 + 3h = none (area + length) 45

46 Quadratics Multiply out (x 8) (x + 3) Method 1 GRID Method 2 FOIL x +3 (x 8) (x + 3) x 8 x 2 8x 3x 24 x 2 8x + 3x 24 x 2 5x 24 (x 8) (x + 3) x 2 + 3x 8x 24 x 2 5x 24 be careful with negatives 46

47 Multiply out (x 2) (x 5) Method 1 GRID Method 2 FOIL x 5 (x 2) (x 5) x 2 x 2 5x 2x +10 x 2 2x 5x + 10 x 2 7x + 10 (x 2) (x 5) x 2 2x 5x + 10 x 2 7x + 10 remember 2 5 is

48 Factorising quadratics factorise means put back into double brackets set up double brackets and then find the two missing numbers difference or total x 2+ 4x 21 answer = (x + ) (x ) be careful of negatives

49 Other examples on factorising quadratics 5 3 x 2 + 2x 15 = (x + 5) (x 3) 4 3 x 2 x 12 = (x 4) (x + 3) check negative signs carefully 6 5 x 2 11x 30 = (x 6) (x 5) 49

50 Difference of two squares eg square number on the end 5 5 x 2 25 difference is 0 answer = (x + ) 5 (x ) 5 so factorising x 2 49 gives (x + ) 7 (x ) 7 so factorising x 2 81 gives (x + ) 9 (x ) 9 50

51 Solving quadratic equations eg 2 Solve x + 5x 14 = 0 factorise first (x + 7) (x 2) = 0 so = x or x 2 = 0 7 so x = or x = 2 do the opposites of the signs in the brackets 51

52 Forming/solving quadratics x + 5 x 2 length is x + 5 metres width is x 2 metres area is 44 square metres(m 2 ) Show that x 2 + 3x 54 = 0 area = length width (x + 5) (x 2) x 2 + 3x 10 form an equation x 2 + 3x 10 = 44 Hence solve x 2 + 3x 54 = so x 2 + 3x 54 = 0 factorising gives (x + 9) (x 6) = 0 so x = 9 or 6 (opposites) hence find the length and width of the rectangle if x = 6 then x + 5 = = 11 length = m 11 if x = 6 then x 2 = 6 2 = 4 width = m 4 take the positive solution check 11 4 = 44m 2 x = 6 52

53 Quarter Year 1st 2nd 3rd 4th Year 1 Year 2 Year 3 Plot the last two moving averages on the grid Time series/moving averages moving average = add up 4 numbers 4 start here for one from the last moving average last moving average is the last four plot point halfway between quarters = = What is the general trend? fluctuates but a slight increase What is the seasonal trend? peaks every 3 rd quarter, dips every 1 st quarter 53

54 Rearranging formulas two step y = 5 + 3x y 5 = 3x 3 x = y 5 3 (x) do the opposites like solving equations show working out v = m 2r (m) +2r h v + 2r = m h +2r h h m = h(v + 2r) use brackets to show h 54

55 rearranging powers p = 4x p = x 2 4 (x) do the opposites of a 2 and a 3 which are and 3 v = w v = w 3 (w) 3 3 x = p 4 w = 5v 3 55

56 rearranging where subject appears twice t = 3a + ab (a) 2y + v = 6y + 3 (y) factorise single bracket t = a (3 + b) now by (3 + b) to get a on its own (3+b) a = (3+b) t (3 + b) get all y s to one side 2y 2y v = 4y v 3 = 4y 4 4 y = v 3 4 y s can be collected so subject appears now once 56

Mathematics GCSE Higher Tier Taster Pages

Mathematics GCSE Higher Tier Taster Pages Question 14 (June 2011 4306/1H) a) on the cumulative frequency diagram you can work out the lower quartile, median and upper quartile. These have been got by using the dashed red lines. Draw them across