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1. GCSE Mathematics Revision C/D borderline

2. C/D borderlineChoose your session! • Session 11 • Session 12 • Session 13 • Session 14 • Session 15 • Session 16 • Session 17 • Session 18 • Session 19 • Session 20 • Session 1 • Session 2 • Session 3 • Session 4 • Session 5 • Session 6 • Session 7 • Session 8 • Session 9 • Session 10

3. Back to sessions Session 1 • Expand and simplify 5(3x + 4) + 6(2x – 2) • A circle has radius 3cm. Write the area and circumference of the circle in terms of π. • a) Find the nth term of the sequence 4, 7, 10, 13, 16, ... b) Hence, or otherwise, find the 50th term in this sequence. 4. Simplify 5a2b3 x 4a4b2 Ans. 1 Ans. 2 Ans. 3 Ans. 4

4. Expand and simplify 5(3x + 4) + 6(2x – 2) Expand the first bracket 5(3x + 4) = 15x + 20 Expand the second bracket 6(2x – 2) = 12x – 12 Replace the brackets with their expansions 15x + 20 + 12x – 12 Simplify by collecting like terms 27x + 8 Back

5. A circle has radius 3cm. Write the area and circumference of the circle in terms of π. Recall the formula for the area of a circle A = πr2 Replace the r with the value for the radius A = π x 32 Simplify A = 9 π Circumference formula C = πd Replace the d with the value for the diameter C = π x 6 Simplify C = 6 π Back

6. 3. a) Find the nth term of the sequence 4, 7, 10, 13, 16, ... b) Hence, or otherwise, find the 50th term in this sequence. • Find the difference between the terms. • In this case 3 • The rule is linked to the three times table and starts with 3n • Find out what needs to be added/subtracted to 3n to make the rule work • 3n + 1 • The rule is 3n +1 • For the 50th term the value of n is 50 • Substitute • 3 x 50 + 1 = 151 Back

7. Simplify 5a2b3 x 4a4b2 Consider numbers first: 5 x 4 = 20 Consider the a’s: a2 x a4 = a6 Consider the b’s: b3 x b2 = b5 Put it all back together 20a6b5 Back

8. Back to sessions Session 2 • Find the volume of the shape below 2. Solve the equation 4x + 5 = 15 • Write 60 as a product of prime numbers • Increase £120 by 30% 3cm 5cm 6cm Ans. 1 Ans. 2 Ans. 3 Ans. 4

9. 1. Find the volume of the shape below 3cm 5cm 6cm Recall the formula for the volume of a prism Volume of a prism = area of cross section x length Find the area of the cross section (triangle) 6 x 3 = 9cm2 2 Multiply this by the length 9 x 5 = 45cm3 Back

10. 2. Solve the equation 4x + 5 = 15 We need to get the x’s on their own Subtract 5 from each side 4x = 10 Now divide by the number before x x = 10 4 x = 2.5 Back

11. 3. Write 60 as a product of prime numbers Draw a factor tree to help 60 30 2 15 2 5 3 So the answer is 2 x 2 x 3 x 5 Which simplifies to 22 x 3 x 5 Your tree may look different Back

12. 4. Increase £120 by 30% To increase by 30% we have to start by finding 30% of the amount and then add it on. 10% of £120 = £12 30% of £120 = £12 x 3 = £36 Now add it on: £120 + £36 = £156 Back

13. Back to sessions Session 3 Ans. 1 Ans. 2 Ans. 3 Ans. 4 Put these fractions into order, starting with the smallest Write the number 2 400 000 in standard form Calculate the size of an interior angle of a regular hexagon. Factorise 8x + 12

14. Put these fractions into order, starting with the smallest Start by changing the fractions to equivalent ones with the same denominator. Is there a number that is a multiple of 8, 20, 4, 5 and 40? 40 The new fractions you get are Putting them in order gives Back

15. Standard form must start with a number between one and ten. 2.4 Then it must be multiplied by ten to the power of something. 2.4 x 106 Write the number 2 400 000 in standard form Back

16. Calculate the size of an interior angle of a regular hexagon. First calculate the exterior angle of an hexagon. A hexagon has six sides. Exterior angles add to 360o All exterior angles on a regular shape are equal. 360o ÷ 6 = 60o Exterior and interior angles lie on a straight line so must add to 180o Interior angle = 180o – 60o = 120o Back

17. Factorise 8x + 12 Start by finding a common factor. Is there a number that goes into 8 and 12? 4 Take this value outside the bracket, then work out what needs to go inside the bracket to make it work when expanded. 4(2x + 3) Back

18. Back to sessions Session 4 • Write three values that satisfy the inequality x – 2 < 4. • One angle in an isosceles triangle is 40o. Find the possible sizes of the other angles. • David worked out that 21.082 x 9.86 = 438.15 (2dp) Explain why David must be wrong. 4. Find the average speed of a car which travels 75 miles in one and a half hours Ans. 1 Ans. 2 Ans. 3 Ans. 4

19. Write three values that satisfy the inequality x – 2 < 4. We need three numbers that when we subtract two, give an answer less than 4 Try numbers until you get one that works. Possible answers include: 5, 4, 3, 2, 1, 0, ... Back

20. Angles in a triangle add to 180o 40o yo 40o 40o xo xo y = 100o x = 70o 2. One angle in an isosceles triangle is 40o. Find the possible sizes of the other angles. What do we know about isosceles triangles? 2 sides the same and two angles the same Sketch the two different types of triangle Back

21. 3. David worked out that 21.082 x 9.86 = 438.15 (2dp) Explain why David must be wrong. Use estimation to find an approximate answer Round each number to one significant figure 202 x 10 400 x 10 = 4000 The answer is approximately 4000 so cannot possibly be 438.15 Back

22. Formula method: Distance = speed x time 75 = speed x 1.5 75 = speed 1.5 Speed = 50mph Scaling method: 75 miles in 1 and a half hours 25 miles in half an hour 50 miles in one hour 50mph Easier to use when calculators are allowed Find the average speed of a car which travels 75 miles in one and a half hours Back

23. Back to sessions Session 5 7m 2m • Find the value of x in the diagram below • Work out 7.562 + 9.52 14.36 – 3.61 a) write the full calculator display b) give your answer to two decimal places • Write the equation of a line parallel to y = 5x + 2 • Rearrange the equation in question 3 to make x the subject x Ans. 1 Ans. 2 Ans. 3 Ans. 4

24. 7m 2m x Recognise the skill required: Pythagoras’ Theorem 1. Find the value of x in the diagram below Recall Pythagoras’ Theorem a2 + b2 = c2 (where c is the hypotenuse) Substitute the values x2 + 22 = 72 Simplify x2 + 4 = 49 Solve x2 = 45 x = √(45) = 6.71m (2 d.p.) Back

25. = 6.202195349 2. Work out 7.562 + 9.52 14.36 – 3.61 a) write the full calculator display b) give your answer to two decimal places Either enter this into your calculator as it appears in the question or work out the numerator and denominator first: a) 66.6736 10.75 b) 6.20 Back

26. Write the equation of a line parallel to y = 5x + 2 What would be true about any line parallel to the one given? It would have the same gradient What is the gradient of the line given? 5 (the number attached to x when the equation is in the form y = mx + c) Possible answers: y = 5x + 1 y = 5x + 10 y = 5x – 2 etc. Back

27. 4. Rearrange the equation in question 3 to make x the subject Equation is: y = 5x + 2 We need to get the x on its own on one side of the equation Subtract 2 from both sides y – 2 = 5x Divide both sides by 5 y – 2 = x 5 Back

28. Back to sessions Session 6 • Factorise x2 + 5x • Solve the equation 2x + 3 = 14 • Write a fraction equivalent to 0.3 • The length of a rectangle is 3 more than its width. a) Show that the perimeter of the rectangle can be written as 4x + 6 b) The perimeter is 20cm. Find the dimensions of the rectangle Ans. 1 Ans. 2 Ans. 3 Ans. 4

29. 1. Factorise x2 + 5x Look for the common factor in the terms x Take this value outside the bracket. x( ) Find what x must be multiplied by to make the original terms x (x + 5) Back

30. 2. Solve the equation 2x + 3 = 14 To get the 2x on its own we need to get rid of the +3. To do this we subtract three from each side. 2x = 11 Now we need to divide both sides by two. x = 5.5 Back

31. Write a fraction equivalent to 0.3 The column after the decimal point is called the tenths column. We have three of these, so we have three tenths. 3 10 Back

32. x x + 3 Width = 3.5cm Length = 6.5cm 4.The length of a rectangle is 3 more than its width. a) Show that the perimeter of the rectangle can be written as 4x + 6 b) The perimeter is 20cm. Find the dimensions of the rectangle Sketch the rectangle a) The perimeter is the total distance around the outside of the shape: x + x + 3 + x + x + 3 Which simplifies to 4x + 6 b) Solve the equation 4x + 6 = 20 Subtract 6 from each side 4x = 14 Divide each side by 4 x = 3.5 Back

33. Back to sessions Session 7 • Write down the first five multiples of 6 • A ball bounces to 80% of its previous height. I drop the ball from 2m, calculate the height after its first bounce • Calculate the area of a circle with radius 4.5cm. Give your answer to 3 significant figures. • £1 = \$1.65 Use this to find the cost in pounds of sunglasses bought in America for \$96.97 Ans. 1 Ans. 2 Ans. 3 Ans. 4

34. Write down the first five multiples of 6 The first five multiples of six are the first five numbers in the six times table. 6, 12, 18, 24, 30 Back

35. 10% of 2m = 20cm 80% of 2m = 160cm New height = 160cm or 1.6m 80% as a decimal is 0.8 Height after first bounce 2 x 0.8 = 1.6m 2. A ball bounces to 80% of its previous height. I drop the ball from 2m, calculate the height after its first bounce Back

36. Calculate the area of a circle with radius 4.5cm. Give your answer to 3 significant figures. Recall the formula for the area of a circle: A = πr2 Substitute the value of r and calculate: A = π x 4.52 = 63.61725124 Round to 3 significant figures A = 63.6cm2 Back

37. 4. £1 = \$1.65 Use this to find the cost in pounds of sunglasses bought in America for \$96.97 To convert dollars into pounds we divide by 1.65 So \$96.97 ÷ 1.65 = £58.77 (rounded to d.p. for money) Back

38. Back to sessions Session 8 • Expand 3(4x + 2) • Solve the equation 7x – 5 = 3x + 27 • You are given the formula v = u + at Find the value of v when u = 3, a = 4 and t = 2 • Make t the subject of the formula in question 3 Ans. 1 Ans. 2 Ans. 3 Ans. 4

39. Write the terms out: 12x + 6 12x 6 Expand 3(4x + 2) Use a suitable method for expansion e.g. grid method 4x +2 3 Back

40. 2. Solve the equation 7x – 5 = 3x + 27 Use a suitable method e.g. Do the same thing to both sides to keep the equation balanced add 5 to each side 7x = 3x + 32 subtract 3x from each side 4x = 32 divide both sides by 4 x = 8 Back

41. 3. You are given the formula v = u + at Find the value of v when u = 3, a = 4 and t = 2 Substitute the values in for the correct letter. Remember that at means a x t v = 3 + (4 x 2) v = 3 + 8 v = 11 Back

42. 4. Make t the subject of the formula in question 3 The formula was v = u + at subtract u from each side of the equation v – u = at divide each side by a v – u= t a Back

43. Back to sessions Session 9 x 20o • Find the size of the angle marked x • Find the length of the side marked y • A circular pond has diameter 5m, find the area of the pond’s surface 4. The shapes below have the same area. Find the value of h y m 7.2m 7.6m 2.5cm h cm Ans. 1 Ans. 2 Ans. 3 Ans. 4 8cm 8cm

44. x 20o 1. Find the size of the angle marked x Angles in a triangle add up to 180 degrees We already have 90o + 20o = 110o So, x = 180o – 110o = 70o Back

45. y m 7.2m 7.6m Find the length of the side marked y Recall Pythagoras’ Theorem a2 + b2 = c2 (where c is the hypotenuse) Substitute the values 7.62 + 7.22 = y2 Simplify 109.6= y2 Solve y = √(109.6) = 10.47m (2 d.p.) Back

46. 3. A circular pond has diameter 5m, find the area of the pond’s surface Recall the formula for the area of a circle A = πr2 Replace the r with the value for the radius A = π x 2.52 A = 19.63m2 (2 d.p.) Back

47. Find the area of the parallelogram A = base x height A = 8 x 2.5 = 20cm2 Recall the formula for the area of a triangle: A = base x height 2 20 = 8 x h 2 Solve the equation: 40 = 8h h = 5cm 2.5cm h cm 8cm 8cm 4. The shapes below have the same area. Find the value of h Back

48. Back to sessions 40o xo Session 10 Ans. 1 Ans. 2 Ans. 3 Ans. 4 From the numbers 3 9 24 40 Write down (i) A prime number (ii) A square number (iii) A multiple of 6 2. A straight line has a gradient of 2 and passes through the point (0,3). What is the equation of the line? 3. 60 = 22 x 3 x 5. Write 600 as a product of primes. 4. Find the size of the angle marked x

49. From the numbers 3 9 24 40 Write down (i) A prime number (ii) A square number (iii) A multiple of 6 (i) A prime number has exactly two factors, one and itself 3 (ii) A square number is the what you get when you multiply any whole number by itself 9 (because 3 x 3 = 9) (iii) We need a number in the six times table 24 Back

50. 2.A straight line has a gradient of 2 and passes through the point (0,3). What is the equation of the line? Straight line graphs are in the form y = mx + c where m is the gradient and c is the y-intercept Gradient = 2 y-intercept = +3 Equation: y = 2x + 3 Back