Teacher page: Deductive proof

The one thing to do with the animation

Ask the class when they would be willing to call it proved.

Run the squares one at a time and stop around n = 20 to take a vote. Most will accept it by then. Keep going to 39, take the vote again, then press once more.

1681 = 41 x 41. Forty confirmations counted for nothing and one case ended the claim permanently. That asymmetry is the entire reason proof exists, and no amount of explaining lands it the way the red square does.

The layout that loses every mark

Writing the statement down and rearranging both sides to 0 = 0 assumes the thing being proved. It is the commonest way to score nothing on a question the student can actually do.

Start from one side. Arrive at the other. Never touch both. Say it as a rule, enforce it in every example, and mark it strictly from the first lesson, because the habit is easy to form and hard to break.

The answers

1. n² + n + 41 at n = 40	1600 + 40 + 41 = 1681, which is 41².
2. Counterexamples needed	1. One is enough and it is final.
3. Rearranging to 0 = 0	B, it assumes what it is proving.

Where the marks go

1 markStarting from one side only.

1 markEach line following from the one above by a stated step.

1 markA closing statement that the two sides are equal.

What each wrong answer tells you

They give	What it means
1641 (Q1)	Dropped the n term.
0 (Q2)	Thinks no counterexample exists, which is what the first forty values suggested.
More than 1 (Q2)	Has not grasped that one failure ends an "always" claim completely.
A (Q3)	Reached a true statement from an assumed one and did not notice the direction.

Other things they will say

"But it worked every time I tried." So did Euler's formula, forty times. That is the answer, and it is better than any abstract one.

"How many cases would be enough?" None, ever, for a claim about all n. That is what makes proof a different activity from testing.

"Can I use a calculator to check?" To check, yes. To prove, no. Worth keeping those two words separate out loud all year.

A possible order

	What is happening
1	Run the squares. Vote at 20, vote at 39, then reveal.
2	The asymmetry: confirmation against counterexample.
3	LHS to RHS layout on two algebraic identities.
4	A numerical proof, and marking each other's layout.
5	Disproof by counterexample as a technique in its own right.

Two things not to say

Do not accept a proof that starts by writing the statement. Mark it at zero once and it stops happening.

Do not say "we can see that" in a worked proof. If it can be seen it can be written, and students copy the phrase to cover gaps.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

Expand and simplifyShow that (2n + 1)² − (2n − 1)² is always a multiple of 8.
It expands to 8n, which is a multiple of 8 for every integer n.
Check itVerify that at n = 3.
49 − 25 = 24, which is 8 × 3.

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

Parity argumentProve that n² − n is always even.
n² − n = n(n − 1), the product of two consecutive integers, so one of them is even and the product is even. At n = 7 it gives 42.
Odd and evenProve that the sum of any two consecutive integers is odd.
n + (n + 1) = 2n + 1, which is odd by definition for integer n.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

A counter-example is enoughA student claims n² + n + 41 is prime for every positive integer n. Disprove it.
At n = 40 it gives 1681 = 41². One counter-example settles a universal claim, and no amount of checking small cases can establish one.
ContradictionOutline a proof by contradiction that √2 is irrational, and name the step that does the work.
Assume √2 = p/q in lowest terms. Then p² = 2q², so p is even, so p = 2m and q² = 2m², so q is even too. Both even contradicts lowest terms. The work is done by the assumption of lowest terms, which is what the contradiction breaks.

Practicalities

Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.