Teacher page: Proof by induction

The one thing to do with the animation

Run all three settings before writing any algebra.

With both halves, everything falls. With the step only, nothing falls, which surprises students who think the base case is a formality. Break the step at k = 6 and the first six are genuinely proved while everything beyond stands untouched.

Three pictures, three different failures. A student who has seen the chain stop does not write "assume true for all n" afterwards, which is the single most common fatal phrase in an induction proof.

Assume for n = k, not for all n

"Assume true for all n" assumes the result and scores nothing. It must be assume true for n = k, a single unspecified value.

And the assumption must be used. Examiners look for the moment it is substituted in. A step that reaches the right answer without ever using the k case has not done induction, however correct the algebra.

The answers

1. n(n+1)/2 at n = 10	55.
2. (9³ - 1)/8	728/8 = 91.
3. Step but no base case	B, nothing is proved.

Where the marks go

1 markA base case evaluated on both sides.

1 markThe assumption used explicitly in the step.

1 markA closing statement naming both halves.

What each wrong answer tells you

They give	What it means
110 (Q1)	Forgot to halve.
728 (Q2)	Stopped before dividing by 8.
A (Q3)	Treats the step as the whole proof. Run the no-base-case setting again.
C (Q3)	Thinks n = 2 is safe without n = 1. It is not: each case needs the one before it.
"Assume true for all n"	Not a question answer but the fatal phrase to watch for in written work.

Other things they will say

"Isn't the base case obvious?" Often, and it is still a mark, and without it the proof proves nothing. The dominoes make that concrete.

"Where do I use the assumption?" Wherever the k case appears inside the k+1 expression. Write the k+1 statement down as a target first; the substitution point then becomes obvious.

"Can the base case be n = 0 or n = 3?" Yes, whatever the claim starts at. The proof is then for all n from there on, and saying so is part of the conclusion.

A possible order

	What is happening
1	All three domino settings. Collect what each failure means.
2	The four-part structure, written as a template.
3	The sum formula in full, with the target written first.
4	A divisibility proof, which feels different and is not.
5	Marking each other's conclusion sentence.

Two things not to say

Do not let "assume true for all n" past, ever. It is the difference between a proof and a circle.

Do not skip the conclusion because the algebra is finished. It is a mark and it is the one most often dropped.

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.

Check the base caseFor the claim that the sum of the first n cubes is n²(n + 1)²/4, verify n = 1.
The sum is 1 and the formula gives 1 × 4 / 4 = 1.
And a few moreVerify it for n = 2, 3 and 4.
9, 36 and 100, and the formula gives 9, 36 and 100.

2In context

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

Write the inductive stepState what must be assumed and what must then be shown.
Assume the formula holds for n = k. Then show the sum to k + 1, which is the assumed sum plus (k + 1)³, equals (k + 1)²(k + 2)²/4.
DivisibilityProve by induction that 4ⁿ − 1 is divisible by 3, and check at n = 5.
4^k+1 − 1 = 4(4^k − 1) + 3, both terms divisible by 3. At n = 5: 1023 = 3 × 341.

3Challenge

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

Why the base case is not optionalExplain what goes wrong in an induction with a valid step but no base case.
The step only passes truth along a chain. With nothing true to start from, the chain is empty and any false statement can have a correct inductive step. The base case is what connects the argument to reality.
Name the missing wordsA student's proof ends "so it is true for k + 1, therefore true for all n". State what is missing from the conclusion.
They never said the assumption was used, nor stated the conclusion properly: since it holds for n = 1, and holding for k implies holding for k + 1, it holds for all positive integers n. The form of the conclusion carries a mark.

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.