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

Including: Proof In-depth learning & Derivations

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

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

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

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

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Greater learner involvement

Changing the nature of teacher talk

Posing accessible tasks

Providing extension tasks

Using a range of strategies

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

Using a range of resources

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

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

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

Transformations of straight line graphs Using different origins

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

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

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

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Functions Differentiation Limits

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

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

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

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

Depth and progression from EYFS to A-level

Choose two whole numbers which sum to 10

8 + 2 and 2 + 8 =

How many different solutions are there?

How can we be sure we have found them all?

Is that a proof?

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)

…and graph them

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

Find the products for each pair of partitions

2 x8 = 16 and 8 x2 = 16

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

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

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

What is the function?

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

GeoGebra

Deriving the product rule

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

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

f’(x) = 12x +22

The unit circle

Sin 30 Cos 50

E θ A D B 1 C

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

What trig identities can you derive using the diagram?

D F C A 1 E β α B

Sin (A+B) ?

Sin (A-B) ?

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

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

Locating prime numbers 3<p<50

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

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

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.’

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.

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

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