Problem-solving
Model, decompose, choose operations, persevere and check reasonableness.
Problem-solving is how you turn a question you have never seen before into an answer you can trust. It is less a topic than the way of working that carries you through every other topic.
What it means
Most maths on this site teaches a tool — adding, fractions, symmetry. Problem-solving is the habit of choosing and using those tools well. A good problem-solver runs a loop, roughly in this order.
Model. Turn the words and pictures into maths. Ask: what am I actually being asked, and what do I know? Draw it, write down the numbers, name the unknown. “Three baskets, four apples each” becomes 3 × 4.
Decompose. Break a hard problem into smaller, easier pieces. If you cannot do the whole thing, do a part of it. “How many legs on 7 spiders and 5 birds” splits cleanly into the spiders, then the birds, then a sum.
Choose operations. Decide which maths fits — add to combine, subtract to take away or compare, multiply for equal groups, divide to share or measure out. The hardest part of a word problem is usually picking the operation, not doing the arithmetic.
Persevere. When the first idea stalls, try another: guess and check, work backwards, try a smaller case, look for a pattern. Being stuck is normal and is not the same as being wrong.
Check reasonableness. Ask whether the answer could be true. Estimate first so you have something to compare against. If 3 children share 12 sweets and you get 40 each, something broke — 40 is far more than the 12 you started with.
You do not always run these steps once, in order. You loop: model, try, check, and go back when the check fails.
Worked examples
A market stall. “Pears cost 30p each. I have £2. How many can I buy?”
- Model: I have 200p; each pear is 30p.
- Choose operation: sharing 200p into groups of 30 means divide,
200 ÷ 30. - Compute:
30 × 6 = 180,30 × 7 = 210. Six fit, seven do not. - Check: 6 pears cost 180p, leaving 20p — I cannot afford a 7th. Reasonable. Answer: 6.
Estimate first. “What is 48 × 5?” Before computing, 50 × 5 = 250, so expect just under 250. Now compute exactly: 48 × 5 = 240. It sits just below 250 — the estimate confirms it.
Decompose a shape count. How many small squares tile a 4-by-3 rectangle? Rather than counting one by one, see 3 rows of 4: 3 × 4 = 12.
The generative-art connection
Making generative art is the problem-solving loop, made visible. To draw a starburst you must model it (how many arms? at what angles?), decompose it (one arm, then repeat it around a circle), choose operations (360 ÷ n degrees between arms), and check reasonableness (if the arms overlap or leave a gap, the count or the angle is wrong — and you can see it instantly).
The dot-multiplier tool is exactly this: you set a number, and one dot multiplies into that many, spread evenly around a ring. Ask for 6 and get a clean hexagon of dots; ask for 5 and the symmetry changes. The picture is immediate feedback on your reasoning — a wrong count looks wrong. That tight make-look-fix cycle is what perseverance and checking feel like when the maths is something you can watch.
Common misconceptions
- “Getting stuck means I’m bad at maths.” Getting stuck is the job. Every solver tries things that fail; the skill is trying another approach, not never being stuck.
- “The number is the answer.” An answer with no check is a guess. Always ask whether it could be true — an estimate is the cheapest check there is.
- “Find the keyword, do that operation.” Rules like “‘altogether’ means add” fail often — “3 rows of 4 altogether” is multiplication. Model the situation, then choose; do not pattern-match on a single word.
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.
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Read this problem, then before you work it out, say roughly what size the answer should be: "A class of `28` children go to the zoo in minibuses. Each minibus holds `6` children. How many minibuses do they need?" First give a rough guess, then work it out exactly, then check your answer makes sense.
Answer
Rough guess: about `30 ÷ 6 = 5`, so around 5. Exact: this is grouping into buses, so divide `28 ÷ 6`. `6 × 4 = 24` (not enough), `6 × 5 = 30`. So 4 buses carry 24 children and 4 children are left over, needing a 5th bus. Answer: `5` buses. Check: 5 is near the guess, and 4 buses would leave 4 children stranded, so 5 is right. Success = models it as division AND rounds UP because of the leftover.
Art hook Pack N dots into rows of 6 on a canvas; each full row of 6 lights up as one 'bus', and the leftover dots start a new partly-full bus in a new colour, so the child sees why a remainder of 4 means a 5th bus.
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Here is a problem someone got wrong: "There are `4` tables. Each table has `5` chairs. Someone said there are `4 + 5 = 9` chairs in the room." Is that right? If not, explain what went wrong and give the correct answer.
Answer
Wrong. '4 tables of 5 chairs' is equal groups, so it should be multiplied, not added: `4 × 5 = 20` chairs. Adding `4 + 5` treats the tables and the chairs as two piles to combine, which is not what is happening. Answer: `20` chairs. Success = spots that the operation is wrong (× not +) and can say why.
Art hook Draw 4 tables as 4 dots in a row; clicking a table sprouts 5 chair-dots around it. A counter contrasts 4+5=9 against 4×5=20, making 'groups of' visibly different from 'combine two amounts'.
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Solve this by working backwards: "I thought of a number. I doubled it, then added `3`, and got `17`. What was my number?" Show your steps and then check your answer works.
Answer
Work backwards, undoing each step in reverse order. The last step was +3, so undo it with −3: `17 − 3 = 14`. Before that it was doubled, so undo with halving: `14 ÷ 2 = 7`. So the number was `7`. Check forwards: `7` doubled is `14`, plus `3` is `17`. Correct. Success = reverses the operations in the right order and checks by running it forwards.
Art hook A number-line animation: a dot starts at the unknown, jumps to double its position, then hops +3, landing on 17. Running it backwards plays the arrows in reverse to reveal the starting dot at 7.
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Read this carefully and decide whether every number is needed: "A shop has `12` red balloons and `8` blue balloons. `5` balloons pop. How many balloons are left altogether?" Say which numbers you use, estimate first, then solve it.
Answer
Estimate: about 20 balloons, lose about 5, so roughly 15. Here you DO need all three numbers. Combine the balloons `12 + 8 = 20`, then take away the popped ones `20 − 5 = 15` left. Answer: `15`. Check: 15 is close to the estimate and fewer than the 20 we started with, so it is reasonable. The trap is thinking 'altogether = just add' and forgetting the popping. Success = combines then subtracts, and does not blindly add on the keyword 'altogether'.
Art hook Scatter 12 red + 8 blue dots; a 'pop' button removes 5 with a burst animation; a live tally shows the total then the remaining count, so both operations (combine, then take away) are visible.
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Try a smaller case first: "How many small squares are on a chessboard that is `8` squares across and `8` squares down?" It is hard to count 64 one by one, so first work out a board that is `2` across and `2` down, then `3` across and `3` down, spot the rule, and use it.
Answer
Small cases: 2 by 2 has `2 × 2 = 4` squares; 3 by 3 has `3 × 3 = 9`. The rule is 'across × down' (equal rows). So 8 by 8 is `8 × 8 = 64` squares. Check: 64 is much bigger than 9, which fits a much bigger board. Success = uses small cases to find 'across × down' rather than counting one by one, then applies it.
Art hook Grid tool: a slider n draws an n-by-n grid of squares and prints n × n = total. Colour each square by (row+col) parity to reveal the checkerboard pattern as n grows.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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Match the story to the maths. "There are `6` nests with `2` eggs in each nest." Which of these fits: `6 + 2`, `6 - 2`, or `6 × 2`? Then work out the answer.
Answer
`6 × 2` fits — it's 6 equal groups of 2. Answer: `12` eggs. Adding or subtracting doesn't match 'groups of'.
Art hook Place 6 dots (nests) on a canvas; clicking each pops out 2 egg-dots. A counter climbs 2, 4, 6, ... to show 6 groups of 2 = 12.
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Before doing any exact maths, choose the best rough answer for `19 + 22`. Is it closer to `10`, `40`, or `100`? Explain your choice.
Answer
Closer to `40`. Round each: `19` is about `20` and `22` is about `20`, so about `20 + 20 = 40`. (Exact is `41`.) 10 is too small and 100 is far too big.
Art hook A number line where two draggable dots at 19 and 22 snap to their nearest ten (both to 20); the point at their sum lights up right beside the 40 mark.
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Draw or describe a picture for this problem, then solve it: "`3` children share `12` stickers equally. How many does each child get?"
Answer
Picture: 3 boxes, deal stickers one at a time into each until all 12 are gone — each box ends with 4. This is sharing, so divide: `12 ÷ 3 = 4`. Each child gets `4` stickers.
Art hook Deal 12 dots one by one into 3 columns (an animated 'dealing cards' loop); each column fills to 4, showing equal sharing as division.
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Which operation do you need — add, subtract, multiply, or divide? "Mia has `15` marbles. Sam has `9` marbles. How many MORE does Mia have than Sam?" Choose the operation, then answer.
Answer
Subtract — 'how many more' is a comparison, so `15 - 9 = 6`. Mia has `6` more marbles.
Art hook Two vertical stacks of dots (15 and 9) side by side; the extra 6 dots above the shorter stack glow to show the difference visually.
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Spot the mistake. A child solved "`24` sweets shared equally between `4` friends" and wrote `24 × 4 = 96` each. What went wrong, and what is the correct answer?
Answer
Sharing means divide, not multiply — and 96 each is impossible when there are only 24 sweets to begin with. Correct: `24 ÷ 4 = 6` sweets each. The reasonableness check (96 > 24) alone shows it's wrong.
Art hook Show 24 dots; an '×4' button wrongly floods the screen to 96 (clearly too many), while a 'share' button neatly splits them into 4 groups of 6 — the contrast makes the error obvious.
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Estimate first, then compute, then compare. "A box holds `39` crayons. How many crayons in `4` boxes?" Give your rough estimate, then the exact answer, then say whether they agree.
Answer
Estimate: `39` is about `40`, and `40 × 4 = 160`, so expect a bit under 160. Exact: `39 × 4 = 156`. They agree — 156 is just under 160, so it's reasonable.
Art hook Four rectangles each fill with 39 dots; a running total ticks up to 156, with a faint line at 160 showing the estimate to compare against.
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Break this into smaller pieces: "How many legs altogether on `5` dogs and `3` birds?" Solve the dogs and the birds separately, then combine.
Answer
Decompose. Dogs: `5 × 4 = 20` legs. Birds: `3 × 2 = 6` legs. Combine: `20 + 6 = 26` legs. Splitting the problem makes each part easy.
Art hook Two mini-scenes: 5 four-legged dot-creatures and 3 two-legged dot-creatures; each scene shows its subtotal (20, 6), then they merge into one total 26.
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Work backwards. "After I gave away `6` cards, I had `10` left. How many did I start with?" Solve it, then check by playing it forwards.
Answer
Undo the 'gave away 6' by adding it back: `10 + 6 = 16`. I started with `16`. Check forwards: `16 - 6 = 10`. Correct.
Art hook A number line: a dot at the unknown hops left by 6 to land on 10; pressing 'undo' hops it back right by 6 to reveal 16.
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Persevere with guess-and-check. "Two numbers add to `10` and one is `4` more than the other. What are they?" Try a guess, check it, and adjust until it works.
Answer
Guess 6 and 4: they add to 10, but 6 is only 2 more than 4 — difference too small, adjust. Try 7 and 3: they add to 10, and 7 is 4 more than 3. That works. Answer: `7` and `3`.
Art hook A seesaw with two draggable dot-piles; it stays level only when the piles total 10, and a label shows their difference, so the child tunes toward difference = 4.
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Read carefully — one number is not needed. "A film starts at `3` o'clock and lasts `2` hours. There are `40` seats in the cinema. What time does the film end?" Which number can you ignore? Answer the question.
Answer
Ignore the `40` seats — it has nothing to do with the ending time. `3 + 2 = 5`, so the film ends at `5` o'clock.
Art hook A clock face where an arc sweeps 2 hours from the 3 position to the 5 position; the '40 seats' fact sits greyed-out to show it's unused.
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Is this answer reasonable? "A pencil costs `30p`. Tom buys `3` pencils and the shopkeeper asks for `£9`." Without finding the exact price first, decide if `£9` could be right, then find the true cost.
Answer
Not reasonable — 3 pencils at about 30p should cost around 90p, nowhere near £9. £9 is roughly ten times too big. True cost: `3 × 30 = 90p`. A quick estimate catches the error before any exact working.
Art hook Three coin-stacks of 30p animate a running total to 90p, with a '£9' marker sitting way off to the side to show how far off the claim is.
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Solve this two-step problem, choosing an operation at each step: "A baker has `20` buns. She sells `8`, then bakes `15` more. How many buns does she have now?" Show both steps and check your answer is sensible.
Answer
Step 1 (sold, take away): `20 - 8 = 12`. Step 2 (baked more, combine): `12 + 15 = 27`. Answer: `27` buns. Check: she ended with more than she started (20), which makes sense because she baked more than she sold. Reasonable.
Art hook A tray of 20 dots: a 'sell' action removes 8 (fade out), then a 'bake' action adds 15 (fade in), with the count updating live at each step to 12 then 27.