Generalising with repeated reasoning

Activities to try

1. 1

   Polygon angle-sum explorer: build, tabulate, abstract, verify

   Why it works: The angle sum is (n−2)×180° BECAUSE the polygon decomposes into n−2 triangles — a rule explained by structure, and confirmed by testing a new polygon rather than trusted on faith.

   1. Start with a triangle (angle sum 180°). Add one side at a time to make a quadrilateral, then a pentagon (convex shapes); for each, split the polygon into triangles by drawing diagonals from one vertex. Keep the hexagon back as a fresh test case.
   2. Record in a running table: number of sides, number of triangles, and angle sum. Notice the sum climbs by a steady +180° each time while the triangle count stays at (sides − 2).
   3. Explain WHY: in a convex polygon, from one vertex you can reach the n−3 non-adjacent vertices by diagonals, so an n-sided polygon splits into exactly n−2 triangles, and its angle sum is (n−2) lots of 180° — reason the rule from the structure, don't just fit it to the numbers.
   4. Write the general rule in words ('two fewer than the number of sides, times 180°') then in symbols: (n−2)×180°.
   5. Verify on a case you did not build the rule from: test a hexagon — (6−2)×180° = 720° — and check it matches the measured/added angles. Then stretch it: a decagon (10 sides, 1440°) and a 100-gon — does it still make sense?
2. 2

   nth-term rule tester for a linear sequence

   Why it works: A general nth-term rule lets you jump to any term directly, and the offset (start value) matters — direct proportion from the first term is a classic error that verification against a known term catches.

   1. Take a linear sequence such as 5, 9, 13, 17, … and tabulate position (1, 2, 3, 4) against term so the constant difference (+4) and the start value are visible.
   2. Move from the term-to-term rule ('add 4') to a position-to-term rule: difference × n + offset. For 5, 9, 13, 17 this is 4n + 1.
   3. Predict a far term you cannot easily count to — the 20th term (81) and the 100th term (401) — using the rule, not step-by-step continuation.
   4. Verify: first check your rule reproduces every listed term (e.g. the 4th term should give 4×4+1 = 17), then test a fresh case not used to build it — extend the sequence by one step to get the 5th term (21) and check that 4×5+1 = 21. If a conjectured rule breaks any term, it is wrong — fix it.
   5. Explain why direct proportion fails here: the 20th term is NOT 20 × the first term, because the sequence does not pass through zero at position zero (the rule 4n+1 gives 1 at n=0, not 0).
3. 3

   Generalise a structural fact and find the case that disproves a false rule

   Why it works: Generalising means stating a rule, justifying why it holds from structure, and being able to produce the single case that would disprove a false version — all three are part of mathematical reasoning.

   1. Generalise an operation property with a reason, e.g. 'any whole number m divided by a non-zero whole number n can be written as the fraction m/n, so dividing by n gives a denominator of n (before any simplifying)' — state it, then justify why it holds, not just that it works on examples.
   2. Explore the cancelling regularity in products like (x−1)(x+1), (x−1)(x²+x+1), … and describe the general pattern that emerges; an area model or algebra tiles may help visualise the simpler binomial product (x−1)(x+1), but are harder to apply to the cubic case, so treat them as an optional aid rather than a required method.
   3. Practise verification-by-counterexample: given a false claim such as 'a polygon's angle sum is n×180°', name the specific polygon that shows it is wrong and state what it gives versus the true value (e.g. pentagon: rule gives 900°, truth is 540°).
   4. Answer the discriminating prompt: 'Why n−2 and not n — what does the −2 correspond to in the shape?'