Make the Grade. A Programme for Success. Target Grade A

Transcription

1 Make the Grade A Programme for Success Target Grade A

2 MAKE THE GRADE NUMBER 1. a) Find the Highest Common Factor of 36 and 84 b) Find the Least Common Multiple of 28 and a) Write the following numbers in standard form: i) ii) 4 million iii) iv) (4) 3. a) Write the following numbers in standard form: i) ii) iii) 5 10 iv) (4)

3 4. Calculate the following, giving your answer in standard form: 17 5 a) (4 10 ) (8 10 ) 12 5 b) (9 10 ) (3 10 ) -5-7 c) (5 10 )+(7 10 ) 5. Estimate: 5 8 ( ) ( ) 8 ( ) (5)

4 6. A Memory Stick holds 128Mb of information. A hard drive of a computer has a capacity of 40Gb. Estimate the number of memory sticks that are needed to hold the same information as a full hard drive of the computer. 6 9 (Hint: 1Mb = 1 10 and 1Gb = 1 10 ) (5) 7. Evaluate: i) -2 3 ii) -3 2 iii) 0 2 iv) 0 b (1) (1) 3 y y v) 4 y -2

5 8. A shop has a sale of 20% off everything. A shirt costs 32 in the sale. What was the original price of the shirt? 9. Phil wants to increase a number by 35% and then decrease the answer by 28%. He says a quick way to do this is to multiply by 1 35 and then multiply by Do you agree with Phil s method? Give reasons for you answer. 10. Olga sells her TV for a profit of 25%. She sold the TV for 60. How much did Olga pay for the TV?

6 11. Calculate: , leaving your answer as mixed number. (4) Hannah measures the length of her curtains to be 1 m in length. How 4 2 many lengths of curtain can she make from 11 m of material? 3 (5) 13. Express the following as decimals: i) 8 5 ii) 9 5 (4)

7 Did you Make the Grade? Skill Qu I can find the HCF of two numbers 1a I can find the LCM of two numbers 1b I can write numbers in standard form 2 I can write numbers in standard form as ordinary numbers 3 I can perform calculations involving standard form 4 Do you know how to using a calculator? I can estimate calculations involving standard form 5,6 I can evaluate negative indices 7 i, ii, v I can evaluate the zero index 7 iii, iv I can solve reverse % problems 8,10 I understand % multipliers 9 I can perform calculations involving mixed numbers 11,12 I can express fractions as decimals 13i I can express fractions as recurring decimals 13 ii Yippee!! - I got all the questions correct. I made mistakes and need to practise this topic more. Top 3 topics I need to revise are If your score is: Well done. You are definitely working at grade B with your number skills Promising work make sure you ask for help and revise the topics that you had difficulty with. You can still get that B!!! Serious revision and help is needed if you are going to get that B. Sort it out now. Don t wait any longer

8 MAKE THE GRADE NUMBER 1. Evaluate: i) ii) 1 ( ) iii) ( ) 3-3 iv) v) vi) vii) viii) (0.01) ix) (2 ) x) ( 25 ) 1-2 xi) x) (22) Robin measures the dimensions of his bedroom. The length is 3 4m and the width is 220cm. Both dimensions are measured to the nearest 10cm. i) Calculate the upper and lower bounds of the perimeter of his bedroom. (4)

9 3. Heather thinks of two numbers, x and y. x = 4 to the nearest whole number y = 2 3 to one decimal place. i) Find the lower bound of the sum of x and y. ii) Find the upper bound of the difference of x and y. 4. Simplify: (6) i) 12 ii) 50 iii) 72 iv) v) vi) Expand and simplify the following: i) 4 ( ) (18) ii) 2 ( 2 + 8) iii) 3 ( ) (7)

10 iv) (4 + 3)(4-3) v) (2 + 5)(4 + 3) vi) (3 + 6)(2-3) (13) 6. Rationalise and simplify the following: i) 1 3 ii) 3 5 iii) 2 6 (9)

11 7. Draw two triangles with one of the sides equal to 5 in each case. (4) 8. Convert the following recurring decimals to fractions in their simplest form: i) ii) 0 3 (1) iii) (1) iv) (5)

12 Did you Make the Grade? Skill Qu I can evaluate numbers involving negative powers 1 i, ii, iii I can evaluate numbers involving unit fractional powers 1 iv - x I can evaluate numbers involving more complex fractional powers 1 xi, xii I can calculate upper and lower bounds 2,3 I can simplify surds 4 I can expand expressions involving surds 5 I can rationalise simple surds 6 I can represent surds diagrammatically 7 I can convert recurring decimals to fractions 8 Yippee!! - I got all the questions correct. I made mistakes and need to practise this topic more. Top 3 topics I need to revise are If your score is: Well done. You are definitely working at grade A with your number skills. Bring on the A*! Promising work make sure you ask for help and revise the topics that you had difficulty with. You can still get that A!!! Serious revision and help is needed if you are going to get that A. Sort it out now. Don t wait any longer

13 MAKE THE GRADE ALGEBRA 1 Simplify the following: a) 24k 6 c) (2y 3 ) 2 6k² b) 8h 4 5h 2 d) 4h 5 3 a 2 b 4 7ab 3 (12) 2. Expand and simplify: a) (y +2)(y + 6) d) (x 1)² b) (x -3)(x + 4) e) (2m + n)(3m - 2n) c) (p 5)(p 2) (4)

14 3. Solve the following equations: 4 + 2p a) = 2 6 2y b) + 6 = 2 3 c) 5 + 4m m = 2 4. Solve the following pairs of simultaneous equations: (4) a) 3x + y = 26 b) 4p 2q = 20 x + y = 10 p 2q = 2 x =. y = p =. q =.

15 c) 5x + y = 24 d) 3x 3y = 24 2x + 2y = 8 x 2y = 9 x =. y =. x =. y =. (4) (4) 5. The graph of y=5x - 4 is shown below: a) On the same axes, draw the graph of x + y = 5 b) Using the graphs, find the solution of the simultaneous equations: y=5x 4 x + y = 5

16 6. A y = 2x + 6 B y = 3 2x C 2y = 4x + 6 D y = 2 + 4x Using the equations of the graphs in the four boxes given above: a) Write down two graphs that are parallel to each other b) Write down two graphs that cross the y-axis at the same point. c) Write down any graphs that go through the point (2,10) d) Write down any graphs that cross the x-axis at x = -3. (6) 7. Find the equation of the line parallel to the line y = 4 + 3x and passing through the point (-4,-1) (4) 8. Find the equation of the line that passes through the points (1,-1) and (3,3). (5)

17 9. Match the equations with the correct graph using the list below Note: You will have to find a missing equation and complete the blank graph. y = 3 x y = x 3-3 y = x 2-3 x + y = - 3 Equation: Equation: Equation: Equation: Equation: (6)

18 10. Solve 4x 5 7 2x Show your answer on the number line below: 11. a) On the axes below, mark the region R that satisfies the inequalities: x < 2 y 1 y x + 3 x + y < 4 (6) b) Write down the integer co-ordinate pairs that lie within R

19 12. a) Complete the table of values for y = x 3 + 3x 2 3. X y 1 1 b) Draw the graph of y = x 3 + 3x 2 3 c) Using your graph, or otherwise, solve: i) x 3 + 3x 2 3 = 0 ii) x 3 + 3x 2 3 = Make t the subject of the equation in the following: (4) t a) v = + s 2 b) R = t b

20 14. Factorise completely: i) 3pq + 2p² ii) 4st 14t³ 15. Factorise completely: (4) i) x² + 5x + 6 iv) p² - 9p + 20 ii) x² + 2x 8 v) t² - 49 iii) y² - y 12 (15) 16. a) Factorise x² - y² b) Hence, or otherwise, evaluate: 53² - 47² 17. Simplify: 2 3(x + 1) a) x + 1 b) x 2 + 5x + 4 x + 1 (1)

21 18. Solve x² + 13x + 12 = 0 (4) 19. x -2 x + 5 a) Write down an expression for the area, A, of the rectangle in terms of x. b) If A = 18cm², find the longest side of the rectangle. (5)

22 Skill Qu I can simplify involving indices expressions 1 I can expand two linear brackets 2 I can expand and simplify two linear brackets 2 I can solve linear equations involving fractions 3 I can solve simple simultaneous equations algebraically 4a,b I can solve more complex simultaneous equations algebraically 4c,d I can draw the graph of a linear equation 5a, 9,11a I can solve simultaneous equations graphically 5b I can interpret the equation of a straight line, y=mx+c 6 I can find the equation of a straight line y=mx+c 7,8 I can identify linear, quadratic, cubic and reciprocal graphs 9 I can solve inequalities with unknowns on both sides 10 I can solve inequalities graphically 11 I can complete tables of values and draw graphs of cubics 12a,b I can solve cubic equations graphically 12c I can change the subject of harder fomulae 13 I can factorise simple expressions 14 I can factorise quadratics 15i - iv I can identify, factorise and use the difference of two squares 15v, 16a,b I can use factorizing to simplify algebraic fraction 17 I can solve quadratic equations 18 I can form and solve quadratic equations 19 Did you Make the Grade? Yippee!! - I got all the questions correct. I made mistakes and need to practise this topic more. Top 3 topics I need to revise are If you r sco re is: Well done. You are working at grade B with your algebra skills Promising work make sure you ask for help and revise the topics that you had difficulty with. You can still get that B!!! Serious revision and help is needed if you are going to get that B. Sort it out now. Don t wait any longer

23 MAKE THE GRADE QUESTIONS 1-9 ALGEBRA 1. Find the equation of the line passing through (6,5) and perpendicular to the line y = 3x + 4. (4) 2. Rearrange the following expressions in descending order if: a a a a i) a = 16 1 ii) a = 81 (5) (5)

24 3. Make t the subject of the formula in each case: i) 4 P = t ii) 2 + 3t W = t (1) iii) s - t F = t 4. The volume of a splogoid is directly proportional to the cube of the diameter When the volume is 324cm³, the diameter is 6cm. i) Find a formula for the volume of a splogoid, V, in terms of the diameter, d. ii) Find the diameter of a splogoid with volume 12cm³

25 5. w is inversely proportional to the positive square root of t. When w = 5, k = 100 i) Find a formula linking w and k ii) Using your formula find: a) w when t = 4 1 b) k when w = Prove that the sum of three consecutive numbers is always a multiple of 3. (4) (4)

26 7. The graph of y = x² + 2x 8 is shown below: Use the graph to solve the following equations: i) x² + 2x 8 = 0 ii) x² + 2x 8 = -4 iii) x² + 2x 10 = 0 iv) x² + 2x 8 = x v) x² + x 3 = 0 (17)

27 8. Factorise the following: i) x² - 36 = ii) 2a² - 50 = iii) 4s² - 9t² = iv) 2x² + 7x + 3 = v) 3y² + y - 4 = vi) 5x² - 11x +2 = (21) 9. i) Simplify x x + 2 (4)

28 4 6 ii) Hence, or otherwise, solve + = 2 x x + 2 (5) 10. Solve the following, giving your answers to 1 decimal place. i) x² + 5x + 1 = 0 CALCULATORS ARE NOW ALLOWED ii) y² -10y +4 = 0 iii) 2x² +13x 5 = 2 (15)

29 11. Meryl is solving the quadratic equation 2x² - 10x 8 = 0 using the quadratic formula. The first part of her solution is shown below: x = -10 ± = -10 ± Find four mistakes with Meryl s solution. 12. Solve the following by completing the square, giving your answers to 1 decimal place. ii) x² + 6x + 2 = 0 (4) iii) y² + y 1 = 0 (10)

30 13. 3x + 2 A B 4 2x - 4 2x + 4 a) Show that the area of the shape is given by the expression 6x² +12x 16. b) If the area of the shape is 20cm², find the length of the longest side. (5)

31 14. Match the quadratic function to the correct graph, giving reasons for your answer. y= x² + 8x + 16 y = x² - 6x + 16 y = x² + 10x + 16 y = x² - 9x + 16 Equation: Reason: Equation: Reason: Equation: Reason: Equation: Reason: (12)

32 Did you Make the Grade? Skill Qu I can find the equation of a perpendicular line 1 I can evaluate algebraic expressions involving negative and fractional powers 2 I can rearrange more complex formulae including when variables are given twice 3 I can solve direct proportion problems 4 I can solve inverse proportion problems 5 I can prove simple statements 6 I can solve quadratic equations graphically 7 I can factorise quadratics using the difference of two square 8 i,ii,iii I can factorise harder quadratics (a>1) 8 iv, v, vi I can simplify and solve simple equations involving algebraic fractions 9 I can solve quadratic equations using the quadratic formula 10,11,13 I can solve simple quadratic equations using completing the square 12,(13) I can sketch the graphs of quadratic functions of the form y = x²+ bx + c 14 Yippee!! - I got all the questions correct. I made mistakes and need to practise this topic more. Top 3 topics I need to revise are If your score is: Well done. You are definitely working at grade A with your algebra skills. Bring on the A*! Promising work make sure you ask for help and revise the topics that you had difficulty with. You can still get that A!!! Serious revision and help is needed if you are going to get that A. Sort it out now. Don t wait any longer

33 MAKE THE GRADE QUESTIONS 1-6 SHAPE, SPACE & MEASURES 1. a) Reflect triangle ABC in the x axis. Label the new triangle A B C (1) - 4 b) Translate triangle A B C ( ). Label the image A B C (1) 0 c) Describe the single transformation that will map triangle ABC onto triangle A B C

34 2. Trapezia A and B are similar. NOT TO SCALE yº 30º 10cm 15cm x x 3cm Calculate the values of: a) x b) y (4) 3. Calculate the value of x in the diagram below: NOT TO SCALE 5cm x 2cm 5cm 6cm

35 4. Find the value of a in the diagram below: NOT TO SCALE 12cm 4cm 3cm a (4) 5. x, y and z represent lengths in the following formulae. By considering dimensions, which of the formulae below could represent length (L), area (A) or volume (V)? x+y+z xyz 3xy x(y+z) 4π x² y²+2z x + y z (6)

36 6. Find the missing angles in the diagrams below, giving reasons for your answers: a) NOT TO SCALE a 50º O a =. because b) 50º NOT TO SCALE O b c b =. because c =. because

37 c) NOT TO SCALE d e 20º g 15º f d =. because e =. because f =. because g =. because

38 7. Find the missing side indicated in each the following triangles, giving your answer to a sensible degree of accuracy: NOT TO SCALE a) 12cm a 30º CALCULATORS ARE ALLOWED FOR QUESTIONS 7-10 b) b 40º 10cm c) c 30º 40º 12m d) 8cm 50º d

39 e) 50º e 5m f) 4cm f 70º 8. Find the missing angle indicated in each the following triangles, giving your answer to a sensible degree of accuracy: NOT TO SCALE a) 12cm 8cm a b) 9cm 5cm b

40 c) 80cm c 30º 1.2m d) 8cm d 4cm 9. A step ladder can only be safely climbed if the angle of elevation to the ground is less than 75º. How far away from a wall must the ladder be to safely reach 3m above the ground? 10. A ship sails from Harbour due East for 4km. It then sails due South for a further 6km. Calculate the bearing of the ship from the harbour. (4) (4)

41 Skill Qu I can transform shapes using reflections 1a I can transform shapes using translations 1b I can describe transformations 1c I can find missing sides in simple similar shapes 2a I can find missing sides in more complex similar shapes 3,4 I can find missing angles in similar shapes 2b I can use dimensions to identify types of formulae 5 I can solve circle problems using angle properties 6 I can state the circle properties correctly 6 I can find missing sides using trigonometry in simple cases 7a - d I can find missing sides using trigonometry in harder cases 7e - f I can find missing angles using trigonometry in simple cases 8a,b I can find missing angles using trigonometry in harder cases 8c - d I can solve problems using trigonometry 9,10 Did you Make the Grade? Yippee!! - I got all the questions correct. I made mistakes and need to practise this topic more. Top 3 topics I need to revise are If your score is: Well done. You are working at grade B with your shape and space skills Promising work make sure you ask for help and revise the topics that you had difficulty with. You can still get that B!!! Serious revision and help is needed. Sort it out now. Don t wait any longer

42 MAKE THE GRADE Questions a) Calculate the area of the following sectors, leaving your answer in terms of π : SHAPE, SPACE & MEASURES SHAPE, SPACE AND MEASURES (i) 2cm (ii) o θ = 72 5cm (iii) 45º 6cm (6) b) Find the length of the arc in each case, leaving your answer in terms of π : (6)

43 2.. a) Enlarge triangle ABC by a factor of -1 5, centre (0,0). Label the image DEF. b) How many times bigger is the area of triangle DEF than the triangle of ABC? 3. 10cm 15cm A B Cones A and B are similar. The volume of cone A is 400cm³. Find the volume of cone B.

44 4. In the following diagrams, are the two triangles congruent? Justify your answers in each case. NOT TO a) B D SCALE 35º 60º F 35º 85º A C E b) B D 35º 60º E 60º 4cm 35º A 4cm C F c) B D 4cm 6cm 60º 6cm F 60º A 4cm C E (9)

45 5. C NOT TO SCALE D B A E In the diagram above, BÂD = 60º and BÂE = 45º. Find, giving reasons for your answer: a) A D B Reason: b) DC B Reason: c) CD A Reason: (6)

46 6. a) On the axes below, sketch the graphs of: i) y = sinx ii) y= tanx b) i) On the axes below, sketch the graph of y= cosx ii) Using your graph and cos60 = 0 5, solve t he following equations in the range 0 <x<360 : cosx = 0 5 cosx = -0 5 x = x = (10)

47 7. Find the value of x in the following triangles: i) 70º x 30º 4cm NOT TO SCALE NOTE Calculators are now allowed ii) 8cm 40º x 110º iii) 13cm x 120º 5cm (10)

48 8. a) Find the value of x in the following triangles: i) x 12cm 4cm NOT TO SCALE 14cm ii) x 12cm 50º 8cm b) Find the possible values of y in the following triangles: 8cm 12cm y 40º (11)

49 9. Calculate the area of the following triangles: a) An equilateral triangle of side 5cm b) 40º 8cm c) 6cm 7cm 30º 8cm (6)

50 10. Find the area of the shaded region. 55º 8cm (4) 11. Will Pam s snooker cue of length 1.5m fit inside her cupboard? 1 2m 70cm 60cm (5)

51 12. Calculate the distance between (1, -2, 5) and (-1, -3, 3) 13. Chocolate balls are made by pouring molten chocolate into spherical moulds of diameter 4cm. How many can be made from 5 litres of molten chocolate? 14. a) How much ice cream can be fill the following cone? 12cm 8cm

52 b) Find the curved surface area of the cone. 15. Which is greater the total surface area of a hemisphere, radius 5cm or the total surface area of a cone, diameter 6cm, sloping edge 5cm? (7) 16. Find the radius of a sphere with a volume of 500cm³. (4)

53 Did you Make the Grade? Skill Qu I can calculate the area of sectors 1a, (10) I can calculate the length of circular arcs 1b I can enlarge shapes using negative scale factors 2a I understand the effect of enlargements on area and volumes 2b, 3 I understand and use the conditions for congruent triangles 4 I can solve circle problems using angle properties, in particular the 5 Alternate Segment Theorem. I can sketch the graphs of sine, cosine and tangent 6a, bi I can solve simple trigonometrical equations I can solve problems using the sine rule 6bii 7i,7ii, 8ii, 8iii I can solve problems using the cosine rule 7iii, 8i I can find the area of any triangle 9, (10) I can solve Pythagoras problems in 2D and 3D 11, 12,14b I understand and use the formula for the volume of a sphere 13,16 I understand and use the formula for the surface area of a sphere 15 I understand and use the formula for the volume of a cone 14a I understand and use the formula for the surface area of a cone 14b,15 Yippee!! - I got all the questions correct. I made mistakes and need to practise this topic more. Top 3 topics I need to revise are If your score is: Well done. You are definitely working at grade A with your shape, space and measures skills. Bring on the A*! Promising work make sure you ask for help and revise the topics that you had difficulty with. You can still get that A!!! Serious revision and help is needed if you are going to get that A. Sort it out now. Don t wait any longer

54 MAKE THE GRADE HANDLING DATA 1. Using the Box and Whisker Plot, complete the table: Minimum value Lower Quartile Median Interquartile Range Upper Quartile Maximum value Range (5) 2. Bob and Mary are driving instructors. The number of minor errors made by their students in the driving test are shown in the box and whisker plots below: Bob Mary a) Who is the best driving instructor? Give reasons for your answer.

55 b) What percentage of Mary s students had ten or more minor errors? 3. Match the frequency diagram with the correct Box and Whisker plot. Complete the missing Box and whisker plot. (1)

56 (7) 4. The table below shows the scores of students on the Non-calculator GCSE paper. Score(x) Frequency Cumulative Frequency 0<x <x <x <x <x <x <x <x <x <x a) Complete the cumulative frequency column. b) Draw a cumulative frequency curve for the results (4)

57 c) Using your graph, find the median score for the Non-calculator paper. d) If the minimum score needed for a grade C is 52, using your graph, how many students achieved a grade C or higher? e) 20% of the students achieved a grade B. Using your graph, what is the minimum score needed to achieve a grade B? f) The Box and whisker plot shows the scores of the students on the Calculator paper. Using your cumulative frequency curve, draw the box and whisker plot for the Non-calculator paper scores. g) Using the box and whisker plots, comment how students have preformed on the non-calculator and calculator papers. (4)

58 5. Lance is drawing a cumulative frequency curve. Length Frequency 1<x 2 0 2<x 3 5 3<x <x <x <x <x <x 9 5 9<x <x <x Is he correct? Give at least two reasons for your answer. 6. Using the cumulative frequency curve below, complete the table: 65 Length (L) Frequency

59 (4) 7. The number of detentions Phil receives each week for poor behaviour are shown below: Week Detentions a) Calculate 4 point moving averages for this data. b) Plot your 4 point moving averages on the time series below: (5) c) Using your graph, comment on the trend of Phil s behaviour (1) d) Using the trend, estimate the number of detentions Phil will receive in Week 9. (1)

60 8. Bob has a tube of Millions (small coloured sweets). He is counting the sweets to see if there are millions if sweets in his packet. He records the colour of the first ten sweets: Sweet Colour Blue Yellow Blue Yellow Red Blue Blue Blue Red Blue Bob makes the following statements: - There are only blue, yellow and red sweets in my packet - There are more blue sweets than any other coloured sweet in my packet - The next sweet out will highly likely be blue or red Do you agree with Bob s statements? Explain your answers. 9. Helen has two regular dice. She rolls each one 42 times and records the results. Number Die 1 frequency Die 2 frequency (6) a) Helen thinks one of her dice is biased. Do you agree? Give reasons for your answer. 10. The probability of a spinner landing on a two is 0 2. If the spinner is spun 150 times, how many times would you estimate not to get a two?

61 11. Meryl has been revising hard using past papers and Make the Grade assessments. She has worked out the probability of passing the calculator and non-calculator papers (getting at least a grade C). Complete the following probability tree diagram: Pass 7 5 Pass Pass Pass.. Fail..... Fail 1 4 Pass. Fail... (5) CALCULATOR PAPER NON- CALCULATOR PAPER What s your probability of passing both papers?

62 Did you Make the Grade? Skill Qu I can draw box and whisker plots 1 I can compare two sets of data using box and whisker plots 2,4g I can match frequency diagrams with the correct box and whisker plot 3 I can calculate the cumulative frequency 4a,6 I can draw a cumulative frequency curve 4b,5 I can use a cumulative frequency curve to find the median 4c I can use a cumulative frequency curve to find the upper and lower quartiles 4f I can use a cumulative frequency curve to find estimates 4d-e I can calculate moving averages 7a I can plot moving averages 7b I can use trends to make predictions 7c-d I can interpret relative frequencies 8,9 I can use theoretical probabilities to make estimates 10 I can complete probability tree diagrams correctly 11 Yippee!! - I got all the questions correct. I made mistakes and need to practise this topic more. Top 3 topics I need to revise are If your score is: Well done. You are working at grade B with your handling data skills Promising work make sure you ask for help and revise the topics that you had difficulty with. You can still get that B!!! Serious revision and help is needed. Sort it out now. Don t wait any longer