1. Reasoning & problem solving for learning and teaching
A-level mathematics

2. Including: Proof In-depth learning & Derivations

3. The assessment will test students’ ability to construct rigorous mathematical arguments and proofs through use of precise statements, logical deduction and inference…

4. …and by the manipulation of mathematical expressions, including the construction
of extended arguments for handling substantial problems presented in unstructured form.

6. What are the gradients of the tangents where cosx and tanx intersect?

7. ?

8. What are the implications for developing students’ mathematical reasoning and problem skills?

9. ?

10. Greater learner involvement

11. Changing the nature of teacher talk

12. Posing accessible tasks

13. Providing extension tasks

14. Using a range of strategies

15. Student presentations
Some strategies:
Think, pair, share
Group work
Flipped classroom
Student presentations
Mantle of the expert

16. Using a range of resources

17. - function/graph plotters - graphical calculators - you tube clips
Hi-tech resources
- spreadsheets
- function/graph plotters
- graphical calculators
- you tube clips

18. - paper and card for the folding thereof
Lo-tech resources:
- card sorts
- examiners’ reports
- paper and card for the folding thereof
- pegs and pegboards

19. the sequence of folds on
Prove whether
the sequence of folds on
A-size paper produces
a regular pentagon
… or not

21. Transformations of straight line graphs Using different origins

22. “There is so much content we don’t have time for students to explore and find out for themselves.”

23. Would students benefit by ‘covering’ 100% of the content superficially or 70% in greater depth?

24. What prior knowledge do students need to begin to develop an understanding of integration?

25. ?

26. Functions
Differentiation
Limits

27. At issue is how in-depth learning, at an earlier age, enables A-level students the opportunity to explore more

28. In-depth learning is a key driver of the new NC, supported by the NCETM and regional Maths Hubs

29. In-depth learning, in contrast to ‘acceleration’, is being strongly encouraged in Primary schools and at KS3

30. At KS4 the emphasis, via GCSE papers containing more problem solving questions, will be a further driver for us to help learners develop mathematical thinking skills

31. Depth and progression from EYFS to A-level

32. Choose two whole numbers which sum to 10

34. 8 + 2
and
2 + 8 =

35. How many different solutions are there?

36. How can we be sure we have found them all?

38. Is that a proof?

39. Suppose we turn these partitions into co-ordinate pairs, e. g
Suppose we turn these partitions into co-ordinate pairs, e.g. (8, 2) and (2, 8)

40. …and graph them

42. When are we ‘allowed’ to join the points together to form a graph?

44. Find the products for each pair of partitions

45. 2 x8 = 16
and
8 x2 = 16

46. Suppose we create co-ordinate pairs from partitions and their products, e.g. (8, 16) and (2, 16)

47. Does this routine work for
non-integer values, e.g. if the two values are 2.5 and 7.5?

48. What does the graph look like when pairs of values are plotted as co-ordinate pairs?
E.g. (2, 16) and (8, 16)

50. What is the function?

51. If the original two values a and b sum to S
what is the general function?

52. GeoGebra

54. Deriving the product rule

55. f(x) = (2x+4)(3x+5)

56. f(x) = 6x2 +22x + 20

57. f’(x) = 12x +22

58. The unit circle

59. Sin 30
Cos 50

60. E θ A D B 1 C

61. Find line segments for: sin⩉ cos⩉ tan⩉ cosec⩉ sec⩉ cot⩉

62. What trig identities can you derive using the diagram?

63. D F C A 1 E β α B

64. Sin (A+B) ?

65. Sin (A-B) ?

66. Draw a picture which shows why: tan 75∘= 2 + √3

67. Prove all primes >3 can be written in the form 6n-1 or 6n-5

68. Locating prime numbers 3<p<50

69. 6n-5
6n-4
6n-3
6n-2
6n-1 6n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

70. 4m - 1 4m - 3 6n - 1 6n - 5 Prime Intersections 11, 23, 47 13, 37
5, 17, 29, 41
6n - 5
7, 19, 31, 43
13, 37

71. MEI Core 3 Jan 2008, Question 5
[4 marks]
(i) Verify the following statement: ‘ 2p -1 is a prime number for all prime numbers p less than 11.’
(ii) Calculate 23 x 89, and hence disprove this statement: ‘ 2p -1 is a prime number for all prime numbers p.’

72. MEI Core 3 Jan 2013, Question 7 [4 marks]
(i) Disprove the following statement: 3n + 2 is prime for all integers n ≥ 0
(ii) Prove that no number of the form 3n (where is a positive integer) has 5 as its final digit.

73. Change one aspect of 𝑦 = 𝑥2 + 6𝑥 + 8 so the graph touches the 𝑥-axis
Change one aspect of 3𝑦 + 𝑥 = 4 so it is perpendicular to 𝑦 = 2𝑥 – 1
Change one ordinate of 𝐴 (2, 5) or
𝐵 (−1, 4) so 𝐶 (5, 2) lies on the line through 𝐴B

74. You are given three expressions a, b and c as follows:
a: 𝑥2 - 3𝑥, b: 4𝑥 – 6 c: 4
Find values for 𝑥 so they can be arranged in each of the orders:
a>b>c, a>c>b, b>c>a, b>a>c, c>a>b and c>b>c