Year 12 Retrieval: Proof by Exhaustion and Counterexample

Active learning works here because students often confuse inverse operations with reciprocals and misapply range rules to composite functions. Hands-on activities make abstract notation concrete, turning 'f^-1(x)' from a symbol into a physical reversal of steps. Role play and debates help students rehearse correct reasoning before formalizing it on paper.

• Year 13
• Mathematics
• 3 activities
• 20 min–35 min

Learning Objectives

1. Evaluate the efficiency of proof by exhaustion versus counterexample for a given conjecture, referencing the size of the case space.
2. Analyze the completeness of a proof by exhaustion, identifying any overlooked cases.
3. Construct a minimal counterexample to refute a universal statement, explaining its sufficiency.
4. Differentiate between a proof by exhaustion and a proof by counterexample, articulating the conditions under which each is most appropriate.

Role Play

⏱ 25 min·Small Groups

Role Play: The Function Machine Factory

Students act as different 'components' in a composite function machine. One student applies function f, the next applies function g, and they observe how the order of operations changes the final output, illustrating why fg(x) is not the same as gf(x).

Prepare & details

Evaluate whether proof by exhaustion or a counterexample is the more efficient strategy for a given conjecture, justifying your choice with reference to the size of the case space.

Facilitation Tip: During the Function Machine Factory, circulate while students act out inverses using props like socks and shoes to catch notation mix-ups in real time.

Setup: Open space or rearranged desks for scenario staging

Materials: Character cards with backstory and goals, Scenario briefing sheet

• Apply
• Analyze
• Evaluate
• Social Awareness
• Self-Awareness

Formal Debate

⏱ 20 min·Whole Class

Formal Debate: To Invert or Not to Invert?

The class is given several functions (like y=x^2). One side must argue why the function cannot have an inverse, while the other side proposes specific domain restrictions that would make an inverse possible. They must use the 'horizontal line test' as evidence.

Prepare & details

Analyse the completeness of a proof by exhaustion, determining whether every case has been rigorously covered and identifying any that have been overlooked.

Facilitation Tip: In the debate, assign students clear roles (pro-inversion, anti-inversion) and require them to cite domain/range evidence from their flowchart diagrams.

Setup: Two teams facing each other, audience seating for the rest

Materials: Debate proposition card, Research brief for each side, Judging rubric for audience, Timer

• Analyze
• Evaluate
• Create
• Self-Management
• Decision-Making

Inquiry Circle

⏱ 35 min·Small Groups

Inquiry Circle: Modulus in the Real World

Groups are given scenarios involving tolerances in manufacturing or distances between moving objects. They must construct modulus functions to model these situations and use graphing tools to find where the 'v-shape' vertex occurs and what it represents in context.

Prepare & details

Construct a minimal counterexample to refute a plausible conjecture, explaining precisely why a single instance is sufficient to disprove a universal statement.

Facilitation Tip: For the modulus investigation, provide graph paper and colored pencils so students can visually trace how f(|x|) alters symmetry and intercepts.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Teaching This Topic

Teach this by starting with physical models before moving to abstract symbols. Research shows that students grasp composite functions better when they trace values through multiple machines in sequence. Avoid rushing to formal definitions; let students discover the need for domain restrictions through failed inversion attempts. Use peer teaching during debates to surface misconceptions early.

What to Expect

Successful learning looks like students distinguishing inverse functions from reciprocals in the factory activity, justifying inversion decisions in the debate, and accurately mapping modulus transformations in the real-world task. They should articulate why domain and range constraints matter when composing functions.

Watch Out for These Misconceptions

Common Misconception

During the Function Machine Factory, watch for students treating f^-1(x) as 1/f(x).

What to Teach Instead

Ask them to physically reverse a two-step process (e.g., adding 3 then multiplying by 2) and notice how this differs from dividing by 2 then subtracting 3.

Common Misconception

During the Structured Debate: To Invert or Not to Invert?, watch for students assuming the range of fg(x) is the same as f(x).

What to Teach Instead

Have them trace specific values through both machines in a flowchart, marking where the range narrows or expands after each stage.

Assessment Ideas

Quick Check

After the Function Machine Factory, present students with the conjecture: 'All prime numbers are odd.' Ask them to identify whether proof by exhaustion or a counterexample is the more efficient strategy and to provide their reasoning. Then, ask them to provide the counterexample if applicable.

Discussion Prompt

During the Structured Debate: To Invert or Not to Invert?, pose the conjecture: 'For any integer n, n^2 + n + 41 is prime.' Ask students to discuss in pairs: 'How would you attempt to prove this? What are the potential challenges with proof by exhaustion here? Can you find a counterexample, and if so, how did you find it?' Collect their notes on domain and range constraints used in their reasoning.

Exit Ticket

After the Collaborative Investigation: Modulus in the Real World, give each student a different conjecture, e.g., 'The sum of two even numbers is always even,' or 'The product of two negative numbers is always positive.' Ask them to state whether they would use proof by exhaustion or a counterexample, write down their chosen proof strategy, and execute it to confirm or deny the conjecture.

Extensions & Scaffolding

• Challenge: Ask students to design a modulus function that has exactly three distinct roots and justify their choice using the graph.
• Scaffolding: Provide partially completed mapping diagrams for composite functions, leaving blanks for students to fill in domain and range constraints.
• Deeper exploration: Have students research how composite functions model real-world processes, such as temperature conversions or currency exchange, and present their findings with diagrams.

Key Vocabulary

Proof by Exhaustion	A method of proof that involves demonstrating a statement is true by checking every possible case.
Counterexample	A specific instance that shows a general statement or conjecture to be false.
Conjecture	A mathematical statement that is believed to be true but has not yet been formally proven.
Case Space	The complete set of all possible scenarios or values relevant to a mathematical statement or conjecture.
Universal Statement	A statement that claims something is true for all members of a particular set or for all possible instances.