Mathematics · Year 13 · Year 12 Retrieval: Proof by Deduction · Autumn Term

Year 12 Retrieval: Proof by Exhaustion and Counterexample

Exploring proof by exhaustion for finite cases and disproving statements using counterexamples.

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

About This Topic

Functions and Modeling in Year 13 extends the basic concept of mappings into the realm of composite, inverse, and modulus functions. Students must grapple with the formal definitions of domain and range, understanding how these constraints dictate whether a function can be inverted. This is a fundamental part of the Algebraic Thinking strand of the National Curriculum, preparing students for the rigors of calculus and real-world data analysis.

Modeling allows students to apply these abstract concepts to physical scenarios, such as population growth or cooling rates. By examining the limitations of these models, students develop a critical eye for how mathematics represents reality. Students grasp this concept faster through structured discussion and peer explanation of how domain restrictions affect the 'one-to-one' nature of a function.

Key Questions

1. 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.
2. Analyse the completeness of a proof by exhaustion, determining whether every case has been rigorously covered and identifying any that have been overlooked.
3. Construct a minimal counterexample to refute a plausible conjecture, explaining precisely why a single instance is sufficient to disprove a universal statement.

Learning Objectives

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

Before You Start

Introduction to Proofs

Why: Students need a foundational understanding of what constitutes a mathematical proof and the concept of a conjecture.

Properties of Numbers (Integers, Primes, Evens, Odds)

Why: Familiarity with number properties is essential for constructing and evaluating cases in proofs and counterexamples.

Algebraic Manipulation

Why: The ability to manipulate algebraic expressions is necessary for simplifying cases or demonstrating relationships within a proof.

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.

Watch Out for These Misconceptions

Common Misconception

Confusing the inverse function f^-1(x) with the reciprocal 1/f(x).

What to Teach Instead

This is a notation error. Using 'Function Machines' where students physically reverse the steps of a process (e.g., putting on socks then shoes vs. taking off shoes then socks) helps distinguish the inverse operation from a simple reciprocal.

Common Misconception

Thinking the range of a composite function is simply the range of the last function applied.

What to Teach Instead

The range of fg(x) depends on the range of g(x) being used as the domain for f(x). Mapping diagrams and peer-led 'flowchart' activities help students track the values through both stages.

Active Learning Ideas

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).

25 min·Small Groups

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.

20 min·Whole Class

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.

35 min·Small Groups

Real-World Connections

• Software engineers developing algorithms for financial modeling must rigorously test for edge cases. A single counterexample, like an incorrect calculation for a specific transaction amount, can lead to significant financial losses.
• Cryptographers designing secure communication protocols rely on proving the infeasibility of certain attacks. Proof by exhaustion might be used for small, discrete key spaces, while counterexamples help identify weaknesses in proposed encryption methods.

Assessment Ideas

Quick Check

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

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?'

Exit Ticket

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.

Frequently Asked Questions

How do I find the range of a function quickly?

The most reliable way is to sketch the function within its given domain. Look for stationary points and end-points. Working in pairs to sketch and then compare ranges helps students catch errors in vertical shifts or asymptotes.

What is the difference between a mapping and a function?

A mapping is any rule that relates inputs to outputs. A function is a specific type of mapping where every input has exactly one output. In class, use a 'many-to-one' vs 'one-to-one' sorting activity to make this distinction clear.

Why do we use modulus functions?

Modulus functions are essential for dealing with absolute differences and magnitudes where direction doesn't matter. They are used extensively in error analysis, engineering tolerances, and calculating distances in coordinate geometry.

How can active learning help students understand functions?

Active learning turns abstract mappings into concrete processes. By using 'Human Graphs' where students stand at coordinates or 'Logic Flowcharts' to build composite functions, students internalize the sequence of operations. This physical involvement helps them remember the importance of order and the impact of domain restrictions.