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X.2 Practice Practices

Reasoning & proof

Conjecture, justify, generalise and critique reasoning.

Reasoning is the habit of asking why a mathematical fact is true and being able to say so out loud. Proof is what you get when your reasons are airtight — a chain of steps anyone can follow that leaves no room for doubt.

What it means

Most of maths is not about getting an answer; it is about being sure the answer is right and being able to convince someone else. This concept names the four moves that do that work.

  • Conjecture — making a careful guess about what is true, usually after noticing a pattern. “I think every number you get by adding two even numbers is also even.” A conjecture is a claim you have not yet checked.
  • Justify — giving reasons for a claim. Not “it just is,” but “here is why.” A justification can be a picture, an example that captures the general case, or a step-by-step argument.
  • Generalise — moving from a few cases to a claim about all cases. You checked 2 + 4, 6 + 8, 10 + 2; generalising means saying it holds for every pair of even numbers, and saying why the pattern can’t break.
  • Critique — judging whether an argument actually works. Does it cover every case? Is there a gap? A single counterexample — one case where the claim fails — is enough to knock a conjecture down.

The big idea for a child: checking three examples shows a pattern might be true; explaining why it must be true is a different, stronger thing. Maths prizes the second. Reasoning is a mathematical practice, so it runs through every strand — counting, shape, chance — rather than living in one topic.

Worked examples

1. Conjecture and justify. Add two odd numbers: 3 + 5 = 8, 7 + 1 = 8, 9 + 3 = 12. Conjecture: odd + odd is always even. Justify with a picture — an odd number is a row of pairs with one dot sticking out. Two odd numbers give two sticking-out dots, and those two lonely dots pair up. No leftovers, so the total is even. This argument covers every odd pair at once, not just the three we tried.

2. Critique with a counterexample. Claim: “every number that ends in a 5 or 0 is in the 10 times table.” Test it: 15 ends in 5 but 15 ÷ 10 is not whole. One counterexample, and the claim is dead. The fix: numbers ending in 0 are in the 10s; ending in 5 only puts them in the 5s.

3. Generalise carefully. The sums 1, 1+3=4, 1+3+5=9, 1+3+5+7=16 give 1, 4, 9, 16 — the square numbers. Conjecture: adding the first n odd numbers gives n × n. You can see why: each new odd number is exactly the L-shaped border that grows one square into the next.

The generative-art connection

Reasoning is what turns a rule into a picture you can trust. In Parable of the Polygons you set one small rule for how shapes move, then watch a large pattern emerge — and the fun is explaining why that rule forces that outcome. That is conjecture-and-justify made visual.

The same loop lives in our own tools. In dot-multiplier, one dot becomes a symmetric starburst by a single repeated rule. A child can conjecture “if I set 6 arms I’ll count 6 dots on each ring,” then run it and check — a live experiment where the art is the argument. Change the rule, predict the new picture, verify. When your prediction and the picture agree every time, you have reasoned your way to a small proof.

Common misconceptions

  • “I found three examples, so it’s proved.” Examples can suggest a pattern but rarely prove it. A general reason — a picture or argument that covers every case — is what settles it.
  • “A counterexample is just bad luck.” One genuine counterexample is decisive: it means the claim, as stated, is false. That is a result, not a failure.
  • “Justifying means restating the answer.” Saying “it’s even because it’s even” is not a reason. A justification explains what makes it true.

What a learner should be able to solve

The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.

  1. You notice something and make a careful guess: `10 + 20 = 30`, `40 + 30 = 70`, `50 + 50 = 100`. You say, "I think two multiples of ten always add up to a multiple of ten." A friend asks, "How do you KNOW it works for every pair, not just these three?" Give a reason that covers ALL cases at once, not just more examples.

    Answer

    A general justification, e.g.: "A multiple of ten is some whole tens with zero ones. Add two of them and you still have only whole tens and zero ones, so the total lands on a tens mark — it is a multiple of ten." Success = the child gives a general reason (not just extra examples) and can say why the pattern cannot break.

    Art hook Number line as a strip of ten-blocks: click to drop coloured 10-bars end to end; the running total always lands on a labelled tens-tick and the ones-column stays empty — the picture shows why the sum can never miss a tens mark.

  2. Someone claims: "If you double a number, the answer is always bigger than the number." Decide if this is always true. If it is not, find ONE example that breaks it (a counterexample) and explain why one example is enough to settle it.

    Answer

    False. Simplest counterexample: `double 0 = 0`, which is not bigger — it stays the same. (If your class works with negative numbers, `double -3 = -6` is even smaller.) Success = child produces a genuine counterexample (0 is the cleanest) and states that one true counterexample makes the whole claim false.

    Art hook A doubling machine: a dot at position n on a number line; press a button to send a second dot to 2n. For most inputs the new dot jumps right; drag the input to 0 and it stays put — the counterexample is the frame where the arrow has length zero.

  3. Look at this pattern of squares made with matchsticks: 1 square needs `4` sticks, 2 squares in a row need `7`, 3 need `10`, 4 need `13`. Conjecture the rule, predict how many sticks `10` squares in a row need, and explain WHY your rule works (what is each new square adding?).

    Answer

    Rule: start with 4, then add 3 for each extra square, i.e. sticks `= 3n + 1`. For `10` squares: `3×10 + 1 = 31`. Why: the first square costs 4 sticks, and every square after it shares one side with the square before it, so it only adds 3 new sticks. Success = correct prediction (`31`) AND a reason tied to the shared side.

    Art hook Canvas grid of squares in a row; a slider adds squares one at a time. Each added square lights its 3 NEW sticks in a fresh colour and greys the shared stick — the +3 rule becomes visible as the colour count grows.

  4. A friend says: "I proved that adding two numbers next to each other, like `4 + 5` or `7 + 8`, always gives an odd answer, because I tried `4+5=9`, `7+8=15`, and `2+3=5` — all odd!" Is their reasoning airtight? Say what is strong and what is missing, and give a reason that would make it certain.

    Answer

    Critique: the three examples show it MIGHT be true but do not prove it — there are infinitely many pairs left untested. A general reason: two numbers next to each other are always one even and one odd, and even + odd = odd, so the sum must always be odd. Success = child recognises examples aren't a proof AND supplies (or accepts) a why-it-must-hold reason.

    Art hook Two neighbouring numbers shown as dot-bars: one always pairs up cleanly (even) and one always has a single leftover dot (odd). Stack them and the lone leftover has no partner, so the whole set stays odd — the even+odd = odd rule made visible.

  5. Two children get different answers to the same question and both feel sure. What are two fair things you could do to find out who is right — WITHOUT just insisting louder? Describe your steps.

    Answer

    Any two sound strategies, e.g.: (1) redo it a different way or use the inverse operation to check; (2) test a smaller, easier case where you already know the answer; (3) draw a picture of it; (4) walk through each other's steps together to find the exact place they differ. Success = child names concrete checking moves, treating a mistake as information to trace rather than blame.

    Art hook A split-screen canvas: both children's methods run as two animations side by side toward the same target dot; the frame where the two paths first diverge flashes, pointing to the exact step to re-examine.

Training exercises

Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.

  1. True or false, and say why in one sentence: `3 + 4 = 4 + 3`. Then say whether swapping the order changes the answer for ANY two numbers you add.

    Answer

    True — both equal `7`. Swapping the order of two numbers you add never changes the total (addition can be done in any order). This is a first taste of generalising from one case to all cases.

    Art hook Two coloured dot-bars (3 red, 4 blue) on a line; a button swaps their order and the combined length stays the same — the order flips but the total dot-count does not.

  2. Here is a claim: "All the numbers I say when I count in twos from 2 are even: `2, 4, 6, 8, 10`." Add the next three numbers, and say whether they keep the pattern going.

    Answer

    Next three: `12, 14, 16` — all even, so the pattern continues. (This is continuing a pattern; a full proof of 'always even' would come later.)

    Art hook Skip-counting ring: dots at 2, 4, 6, ... placed around a circle; each even landing glows and each odd gap stays dark, drawing an even-only necklace.

  3. Which one does not belong, and why? `4`, `6`, `9`, `10`. There is more than one good answer — pick one and give a clear reason.

    Answer

    Any correctly justified answer, e.g.: `9` is the only odd one (the others are even); `10` is the only two-digit number / the only multiple of 5; `4` is the only one below 5. (Note: `9` is NOT 'the only square' — `4` is a square too, so that reason does not separate it.) Success = a reason that truly separates the chosen number from the other three.

    Art hook Four dot-figures drawn on canvas (a 2x2 block, a 2x3 block, a 3x3 block, a 2x5 block); tap one to highlight it and the tool shows the property that sets it apart, such as odd count or number of digits.

  4. Spot the mistake. A friend says: "`5` is even, because `4` is even and `5` comes right after `4`." What is wrong with this reasoning?

    Answer

    The reasoning is faulty: numbers alternate odd, even, odd, even, so the number right after an even number is always ODD. `5` is odd — it cannot be split into two equal whole groups (`2 + 2` leaves 1 over). Success = child identifies the flawed step (the 'comes right after' reason), not just the wrong label.

    Art hook A row of 5 dots trying to pair up two-by-two; the leftover single dot flashes to show 5 can't be split evenly, so it's odd.

  5. Investigate: is `7 + 7` odd or even? Is `9 + 9`? Is `3 + 3`? Make a conjecture about odd + the SAME odd number, and give a reason it must be even.

    Answer

    `14`, `18`, `6` — all even. Conjecture: an odd number added to itself is always even. Reason: adding a number to itself is doubling it, and any whole number doubled splits into two equal groups, so it is even. Success = pattern spotted AND a doubling/pairing reason (not just three more examples).

    Art hook Mirror-double canvas: place n dots on the left and they reflect to the right; every dot gets a twin, so the whole set pairs perfectly — evenness you can see.

  6. A shape rule: to grow a triangle staircase you add a new row of dots each time. Row sizes are `1, 2, 3`, giving totals `1, 3, 6` dots. Predict the total after adding a 4th row, and explain how you got it from the previous total.

    Answer

    Add a row of 4 dots: `6 + 4 = 10` dots. Rule: each new total is the last total plus the size of the next row. (Generalising a growing pattern by naming its step.)

    Art hook Triangular dot-stack (the triangular numbers); a slider adds a bottom row, the new row lights up, and the running total updates — the +row-size step is visible each time.

  7. Test this claim carefully: "Every number in the 3 times table is odd." Try a few cases. Is the claim true? Back up your decision.

    Answer

    False. `3` and `9` are odd, but `6`, `12`, `18` are even, so the 3 times table mixes odd and even numbers. The counterexample `6` is enough to settle it. Success = at least one even multiple of 3 found and named as the decisive counterexample.

    Art hook Multiples of 3 placed around a colour wheel, coloured by odd/even; the two colours alternate around the ring, instantly disproving 'all odd'.

  8. You and a partner both work out `27 + 15`. One says `42`, the other says `32`. Describe a quick way to check who is right using an estimate first, then pick the correct answer.

    Answer

    Estimate: `27 ≈ 30` and `15 ≈ 15`, so the answer should be near `45` — much closer to `42` than to `32`. Exact: `27 + 15 = 42`. So `42` is correct and `32` is unreasonable. Success = uses an estimate to judge reasonableness first, then confirms with the exact sum.

    Art hook A number line with a shaded 'reasonable zone' around 45; both proposed answers drop in as dots — one lands inside the zone, one far outside, flagging the wrong one.

  9. Conjecture challenge: add any two numbers that both end in `5`, like `15 + 25` or `35 + 45`. What does the answer always end in? Try three pairs, make a conjecture, then explain why it happens.

    Answer

    They always end in `0` (`15+25=40`, `35+45=80`, `55+25=80`). Reason: the ones digits are `5 + 5 = 10`, so the ones column finishes at `0` and a ten carries over — no matter what tens you started with, the ones always land on 0. Success = correct conjecture AND a reason about `5 + 5` making a full ten.

    Art hook Two spinners of numbers ending in 5; each spin drops dots whose ones-columns (5 red + 5 red) snap together into a full ten-bar, leaving the ones column empty and adding one bar to the tens.

  10. Open challenge: make up your OWN 'always, sometimes, or never' claim about numbers or shapes (for example, 'a square always has 4 equal sides'). Then decide which of the three it is and give one reason (for 'always' or 'never') or one counterexample (for 'sometimes' or 'never') to back your decision.

    Answer

    Answers vary. A strong response states a clear, testable claim, correctly labels it always / sometimes / never, and supports it — a general reason for 'always' or 'never', or a breaking example for 'sometimes' or 'never'. E.g. the claim 'every rectangle is a square' is FALSE, and a `2` by `3` rectangle is a counterexample that shows it. Success = a testable claim with a justification that matches its label.

    Art hook A claim-tester canvas: type or pick a claim, then drag example shapes or dot-sets into an 'always / sometimes / never' bin; the tool tallies supporting versus breaking cases as coloured tokens.

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