Constructing mathematical arguments

Activities to try

1. 1

   The confident conjecture champion (adversarial dialogue)

   Why it works: A universal claim is destroyed by one honest counter-example, while confirming cases cannot prove it unless every possible case is checked (impossible for claims about all numbers); constructing arguments and critiquing flawed ones are the same reasoning practice, and a claim is not the same as its converse.

   1. Face an in-role character who asserts a universal claim with total confidence: 'Multiplying ALWAYS makes a number bigger — I've tried 3×4, 5×6, 10×10, works every time. So it's proved, right?'
   2. Decide whether piling up more agreeing examples settles it, or whether you must HUNT for a case that breaks the claim.
   3. Produce and state a decisive counter-example, e.g. '6 times a half is three — that's smaller, so it's not always,' and explain why that single case is enough.
   4. When the champion patches the claim ('okay, multiplying always makes bigger OR the same') test the patch again: × 1/2 still shrinks it (so it breaks the patch), while × 1 only reaches the 'same' boundary the patch now allows.
   5. When the champion defends a flawed fraction division ('just flip the FIRST fraction') locate the faulty step, say why it fails, and give the correct method (invert the divisor — the second fraction — then multiply).
2. 2

   Conjecture-testing and proof-building workspace

   Why it works: Chaining established facts yields a proved value that holds generally, not just for the drawn diagram; a well-chosen single case breaks an 'always' claim; and locating a faulty step is more productive than rejecting an answer for looking odd.

   1. Sort 'always / sometimes / never' claims and use a counter-example finder: drag values and watch a claim collapse when, say, × 1/2 makes the product shrink. Use claims whose truth value is determinate — e.g. 'multiplying makes bigger' (an 'always' claim broken by a factor below 1 such as × 1/2). If including 'the more sides a shape has, the bigger its angles', state inside the task which angle is meant (e.g. each interior angle): for regular polygons each interior angle does grow with the number of sides, but it fails for irregular shapes, so it sorts as 'sometimes' — do not present it as a straightforwardly false 'always' claim.
   2. Build an angle-fact proof chain: snap facts ('angles on a line sum to 180°', 'angles in a triangle sum to 180°', 'vertically opposite angles are equal') into an if/therefore track to reach a proved angle such as 60°, reserving measuring only to check — and only when the chosen givens genuinely force that value.
   3. Critique a peer's fraction division: tap the mis-inverted step and select the correct method, then state in your own words why the flawed step fails.
3. 3

   Annotated sound-vs-flawed proofs and conjecture cards

   Why it works: Seeing a flawed step laid out beside a correct one, and cards that reveal breakers or sketches, can seed critique vocabulary and prime the claim/converse distinction before learners construct their own arguments.

   1. Compare an annotated correct fraction division with a flawed one side by side, and discuss where and why the flawed one goes wrong (it inverts the first fraction rather than the divisor).
   2. Work through a set of true/false conjecture cards; tap a card to reveal either a counter-example that breaks it or a short proof sketch that supports it.
   3. Use these as a springboard for talk: say aloud whether each revealed case is a proof or just an example, and whether the converse of a true claim also holds.