Mathematics · Year 12

Active learning ideas

Proof by Contradiction and Disproof

Proof by contradiction and disproof rely on precise reasoning that students often find abstract. Active learning lets them test assumptions by arguing them out loud, hunting for flaws, and physically tracing logical steps. These activities move the mental work of negation and counterexamples into visible, collaborative forms.

• National Curriculum Attainment Targets
• A-Level: Mathematics - Proof

• 20–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Socratic Seminar
• 30 min · Pairs

Pairs: Assumption Debate

Assign statements like '√3 rational.' One student assumes true, other false; they build argument chains until contradiction emerges. Switch roles after 10 minutes, then pairs share strongest proofs with class.

Design a proof by contradiction for the irrationality of a number.

Facilitation Tip

During Assumption Debate, circulate and ask each pair: ‘Show me the first step’s assumption, then the next move. Does the contradiction connect back to that start?’

What to look for

Present students with the statement: 'All prime numbers are odd.' Ask them to write one sentence explaining why this statement is false and provide the specific counterexample. Then, ask them to explain what a proof by contradiction would look like for the statement 'All prime numbers are odd'.

• Analyze
• Evaluate
• Create
• Social Awareness
• Relationship Skills

Activity 02

• Socratic Seminar
• 40 min · Small Groups

Small Groups: Counterexample Hunt

Provide 5 statements to disprove, such as 'n² + 1 always even.' Groups find and verify 2-3 counterexamples each, record with explanations, then rotate to critique others' work on posters.

Critique common errors in attempting to disprove a statement.

Facilitation Tip

In Counterexample Hunt, hand each group a different statement set and say: ‘Find the smallest counterexample first—then explain why it breaks the universal claim.’

What to look for

Pose the following: 'Imagine a student is trying to prove that the sum of two even numbers is always odd. They start by assuming the sum is odd and then show that this leads to the conclusion that an odd number must equal an even number. What is the flaw in their approach, and how would you correct it using proof by contradiction?'

• Analyze
• Evaluate
• Create
• Social Awareness
• Relationship Skills

Activity 03

• Socratic Seminar
• 25 min · Whole Class

Whole Class: Proof Relay

Project √2 irrational proof skeleton. Students line up; each adds one logical step verbally or on board, classmates vote yes/no with reasons before next turn, correcting as a group.

Analyze how the assumption of the opposite leads to a logical inconsistency in proof by contradiction.

Facilitation Tip

For Proof Relay, post the chain steps on separate sheets around the room so students physically move from one to the next, forcing them to check each predecessor’s logic before adding their own.

What to look for

In pairs, have students write a statement about numbers (e.g., 'The square of any rational number is rational'). One student writes a proof by contradiction, and the other writes a disproof by counterexample. They then swap and critique each other's work, checking for logical flow and accuracy of the contradiction or counterexample.

• Analyze
• Evaluate
• Create
• Social Awareness
• Relationship Skills

Activity 04

• Socratic Seminar
• 20 min · Individual

Individual: Custom Disproof

Students pick a false claim from handout, like 'all triangles equilateral,' craft counterexample with diagram and proof sketch, then pair-share for peer review.

Design a proof by contradiction for the irrationality of a number.

Facilitation Tip

During Custom Disproof, remind students to write the original statement, its negation, and the one clean counterexample that seals the deal.

What to look for

Present students with the statement: 'All prime numbers are odd.' Ask them to write one sentence explaining why this statement is false and provide the specific counterexample. Then, ask them to explain what a proof by contradiction would look like for the statement 'All prime numbers are odd'.

• Analyze
• Evaluate
• Create
• Social Awareness
• Relationship Skills

A few notes on teaching this unit

Teachers should normalize the discomfort of assuming the opposite and following it to a dead end. Emphasize that the contradiction must be a logical impossibility tied directly to the assumption. Use color-coding: one color for the assumption, another for the chain of deductions, and a third for the contradiction. This visual separation helps students see how the assumption unravels.

Students will articulate why a false assumption breaks down and recognize when a single counterexample is enough. They will compare proof by contradiction with disproof, choosing the right tool for each claim. Participation will show clear chains of logic and precise identification of contradictions or counters.

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Watch Out for These Misconceptions

• During Assumption Debate, watch for students declaring any inconsistency as a contradiction, even if it doesn’t follow from the assumption.

  Circulate during Assumption Debate and ask each pair to underline the exact assumption they started with, then circle each deduction that follows from it. If they point to an unrelated inconsistency, redirect them to the chain: ‘Trace back—does this inconsistency link back to your starting assumption?’

• During Counterexample Hunt, some students think any odd number serves as a counter to an even-number claim.

  During Counterexample Hunt, hand each group a list of candidates and say: ‘Your counter must fit the statement’s form; if the claim is “all even numbers greater than 2 are sums of two primes,” test 26, not 3.’

• During Proof Relay, students believe contradiction can replace any direct proof automatically.

  At the end of Proof Relay, hold a two-minute debrief: ‘Which statements felt easier to disprove by counterexample? Which needed contradiction? Why did the contradiction chain feel necessary there?’

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Methods used in this brief

• Socratic Seminar