AQA A Level Mathematics: Overarching Themes — Revision Notes

Condensed for the final weeks. For the full explanation, use the Overarching Themes study guide.

Mathematical argument and proof

Method	Approach
Deduction	Argue directly from known results
Exhaustion	Check every possible case
Counter-example	One case disproves a general claim
Contradiction	Assume the negation, derive an impossibility

A single counter-example disproves a statement, but no number of examples proves one. Saying that explicitly is often worth a mark in itself.

Proof by contradiction must have visible structure: assume the opposite, derive a contradiction, conclude the original. The classic results are the irrationality of √2 and the infinitude of primes.

Worked example. Prove by contradiction that √2 is irrational.

Assume root 2 is rational: root 2 = a/b in lowest terms, b not 0.
Square:      2 = a^2 / b^2   ->   a^2 = 2b^2
So a^2 is even, therefore a is even. Write a = 2k.
Then (2k)^2 = 2b^2  ->  4k^2 = 2b^2  ->  b^2 = 2k^2
So b^2 is even, therefore b is even.

But a and b are both even, contradicting "lowest terms".
Therefore root 2 is irrational.

Notation carries marks:

=>   implies
<=   is implied by
<=>  if and only if (implies both ways)

Using ⟹ where ⟺ is required, or vice versa, is a common and avoidable loss. x = 2 ⟹ x² = 4 is true, but the reverse is not, so ⟺ would be wrong.

A condition can be necessary without being sufficient. Having four equal sides is necessary for a square (every square has them) but not sufficient (a rhombus also has them without being a square) — a necessary and sufficient condition needs four equal sides and four right angles.

Worked example — proof by exhaustion. Prove that no square number ends in 2, 3, 7 or 8.

Every integer ends in a digit 0-9.
Squaring each final digit gives final digits: 0,1,4,9,6,5,6,9,4,1
None of these is 2, 3, 7 or 8, and a square's final digit depends only
on the final digit of the original number, so the statement holds for
every integer.

Worked example — proof by deduction. Prove that the product of two consecutive even numbers is a multiple of 8.

Let the numbers be 2n and 2n + 2.
Product = 4n(n + 1)
Of any two consecutive integers n and n+1, one must be even, so
n(n+1) is a multiple of 2. Hence 4n(n+1) is a multiple of 8.

Modelling

The modelling cycle: real problem → assumptions → mathematical model → solve → interpret → validate → refine.

Assumptions must be stated, and their effect assessed. Common ones and what they do:

• Ignore air resistance — overestimates range and speed.
• Treat as a particle — ignores size, rotation and air resistance.
• Assume the surface is smooth — ignores friction, so overestimates motion.
• Assume constant acceleration — allows suvat, but is often unrealistic.
• Assume the string is light and inextensible — tension is equal throughout and length is fixed.

Questions asking “criticise the model” want the assumption named, and the direction of the resulting error stated — not just “it is unrealistic”.

Limitations of a model are not failures. A model is judged by whether it is useful for its purpose, not by whether its assumptions are literally true.

Problem solving

Where the route is not given, the reliable approach is:

1. Identify what is given and what is required.
2. Sketch a diagram — almost always worth doing.
3. Choose a strategy: form an equation, use a known result, work backwards, or consider special cases.
4. Carry out the work, showing every step.
5. Check the answer for reasonableness — right order of magnitude, sensible sign, correct units.

Method marks are awarded for a correct approach even when the final answer is wrong, so showing working is not optional. An unsupported answer earns nothing if it is incorrect.

Working accurately

• Do not round partway through — carry full accuracy and round only at the end.
• Match significant figures to the data given; quoting more implies precision you do not have.
• State units in the answer.
• Where a question says “show that”, the answer is given — so the marks are entirely for the method, and you must reach the stated result convincingly.
• Where it says “hence”, you must use the previous part; “hence or otherwise” allows a fresh start.

Calculator use

Know what the calculator can and cannot do. It can solve equations, integrate numerically and handle statistics — but an unsupported answer from a calculator earns no method marks where working is required. Use it to check, not to replace, the working.

Exam traps

• Using ⟹ where ⟺ is needed.
• Offering examples as proof.
• Naming an assumption without stating its effect.
• Rounding partway through a calculation.
• Omitting units.
• Giving a bare answer with no working in a “show that” question.

Self-test

1. Why does a counter-example disprove but examples not prove?
2. State the structure of a proof by contradiction.
3. Distinguish ⟹ from ⟺ with an example.
4. What must a criticism of a model include beyond naming the assumption?
5. Why does working matter even when the answer is wrong?

Answers: 1. A general statement claims something for all cases, so one failing case refutes it; but confirming any finite number of cases leaves the remaining ones untested. 2. Assume the opposite of the statement, derive a logical contradiction, and conclude that the original statement must be true. 3. ⟹ means implies in one direction only: x = 2 ⟹ x² = 4, but x² = 4 does not imply x = 2, so ⟺ would be false there. 4. The direction of the resulting error — for example, ignoring air resistance overestimates the range. 5. Method marks are awarded for a correct approach independently of the final answer, so working can earn most of the marks even after an arithmetic slip.