The times tables (2, 5, 10 → 3, 4, 8)
Build & recall the 2, 5, 10 tables (then 3, 4, 8); odd/even.
A “times table” is the list of answers you get when you count in equal jumps of one number: the 5 table is 5, 10, 15, 20, and so on. Knowing these by heart is what turns slow counting into fast multiplication.
What it means
Multiplication is repeated addition of equal groups. 4 × 5 means “four groups of five”, which is 5 + 5 + 5 + 5 = 20. A times table collects every answer for one of the numbers in that pair, so the 5 times table is what you land on when you keep adding 5: 5, 10, 15, 20, 25, 30…. This is called skip-counting — counting in jumps of the same size instead of by ones.
The order of the two numbers never changes the answer: 4 × 5 and 5 × 4 both equal 20. This is the commutative property, and it roughly halves how much you must memorise.
Some tables have friendly patterns that make them the natural place to start:
- 2s — every answer is even:
2, 4, 6, 8, 10…. An even number is one you can split into two equal whole groups; an odd number always has one left over. - 10s — just write the counting number and add a zero:
10, 20, 30, 40…. - 5s — every answer ends in 5 or 0, and 5s are exactly half of the matching 10s.
Once 2, 5 and 10 feel automatic, the 3, 4 and 8 tables follow, and each leans on an easier one: the 4s are the 2s doubled (4 × 6 is 2 × 6 doubled = 24), and the 8s are the 4s doubled again.
Worked examples
Skip-count the 5 table on your fingers. Say a multiple of 5 for each finger: 5, 10, 15, 20, 25 — five fingers, so 5 × 5 = 25.
Use commutativity to shrink the work. If you already know 2 × 7 = 14, then 7 × 2 = 14 for free — same fact, flipped.
Double your way up. Build 8 × 4 from an easy table:
2 × 4 = 8 (the 2s)
4 × 4 = 16 (double the 2s)
8 × 4 = 32 (double the 4s)Odd or even? 6 × 3 = 18 ends in 8, so it is even. 5 × 7 = 35 ends in 5, so it is odd. A product is even whenever at least one factor is even.
The generative-art connection
An array of dots is literally a multiplication: 4 × 5 is four rows of five dots, and counting the rectangle gives 20. Building that rectangle and watching it grow is the clearest picture of what a times fact means — the dot-multiplier tool turns one dot into exactly this kind of structured, countable array.
Tables become genuinely beautiful on a circle. Mark 10 points around a ring, number them, and for each point n draw a line to point 2 × n (wrapping past the last point). The 2 table produces a cardioid — a crisp heart shape woven from straight lines. Change the multiplier and the woven curve changes with it, so the times table you chose is something you can see. The Coding Train’s Times Tables Cardioid challenge shows how to make it.
Common misconceptions
- “
3 × 4is different from4 × 3.” They give the same product. Three rows of four and four rows of three are the same rectangle, just turned. - “Multiplying always makes a bigger number.” True for the whole numbers here, but not a rule to lean on —
× 1leaves a number unchanged and× 0gives 0. - “Every 5-times answer ends in 5.” They alternate:
5, 10, 15, 20…end in 5 or 0. Ending in 5 means odd; ending in 0 means even.
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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Answer these times-table facts quickly, straight from memory: `6 × 5`, `10 × 9`, `2 × 8`, and `7 × 10`.
Answer
`6 × 5 = 30`, `10 × 9 = 90`, `2 × 8 = 16`, `7 × 10 = 70`. These are core 2/5/10 facts a child should recall without counting up.
Art hook A fixed 4-cell strip; each cell draws its own dot grid (cell 1: 6 rows of 5 dots, cell 2: a 10x9 grid, cell 3: 2 rows of 8, cell 4: 7 rows of 10). When the child's recalled answer equals the dots in that cell, the grid fills with colour.
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Fill in the missing numbers in the `3` times table: `3, 6, 9, __, __, 18, __, 24`.
Answer
`12, 15, 21`. Skip-counting in 3s: 3, 6, 9, 12, 15, 18, 21, 24.
Art hook A number line from 0 to 24 with a dot that hops in equal jumps of 3; each landing spot lights up. Missing spots start grey and glow when the child names them.
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Solve these `4` and `8` table facts: `4 × 7`, `8 × 6`, and `8 × 4`. For the last one, show how doubling helps.
Answer
`4 × 7 = 28`, `8 × 6 = 48`, `8 × 4 = 32`. Doubling: `4 × 4 = 16`, then double to get `8 × 4 = 32` (8s are 4s doubled).
Art hook Two stacked dot-arrays: the top shows 4 rows of 4, the bottom mirrors it to show the doubled 8-rows-of-4 version, so a child literally sees one array double into the other while a counter jumps 16 -> 32.
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Sort these numbers into odd and even: `12, 15, 20, 27, 8, 45`.
Answer
Even: `12, 20, 8` — they end in 0/2/4/6/8 and split into two equal groups. Odd: `15, 27, 45` — they end in 1/3/5/7/9 and always leave one over when you try to pair them up.
Art hook Each number appears as a row of dots the child tries to pair up two-by-two: even numbers pair perfectly (calm colour), odd numbers leave one lonely dot glowing red.
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`5 × 8 = 40`. Use that ONE fact to write down `8 × 5` and to help you find `10 × 8`.
Answer
`8 × 5 = 40` (commutativity — the same fact flipped). `10 × 8 = 80`, because 10s are double the 5s: `40` doubled is `80`.
Art hook A dot rectangle you can rotate 90 degrees: 5x8 turns into 8x5 with the same 40 dots, showing the flip changes nothing about the count.
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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Count in 10s out loud from 0. Write the first six numbers you say.
Answer
`10, 20, 30, 40, 50, 60`. Each jump adds 10; the counting number gets a zero on the end.
Art hook A vertical stack of ten-frames: each time the child says a multiple of 10, a fresh full frame of 10 dots snaps into place beside the number.
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Count in 2s and circle whether each answer is odd or even: `2, 4, 6, 8, 10`. What do you notice?
Answer
All are even. The 2 times table only ever gives even numbers, because you are counting in equal pairs.
Art hook Place N dots and try to link them in pairs. Multiples of 2 always pair up with none left over — the animation shows every dot finding a partner.
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`5 × 3` means five, added three times (three groups of five). Draw or imagine the groups and give the total.
Answer
`5 × 3 = 15`. Three groups of 5: `5 + 5 + 5 = 15`.
Art hook Three circles, each holding 5 dots; a running counter ticks 5, 10, 15 as each group appears.
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Which of these belong to the `5` times table? `15, 22, 30, 34, 45`.
Answer
`15, 30, 45` belong (multiples of 5 end in 5 or 0). `22` and `34` do not.
Art hook Points spaced every 5 units around a ring glow gold; the child taps numbers and correct multiples snap onto a glowing point, wrong ones bounce off.
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You know `2 × 6 = 12`. Use it to find `4 × 6` by doubling.
Answer
`4 × 6 = 24`. The 4s are the 2s doubled: `12` doubled is `24`.
Art hook A 2x6 dot-array that duplicates and stacks beneath itself, morphing into a 4x6 array while a counter jumps 12 -> 24.
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Skip-count in 4s to fill the gaps: `4, 8, __, 16, __, 24, __`.
Answer
`12, 20, 28`. Counting in 4s: 4, 8, 12, 16, 20, 24, 28.
Art hook A dot hops along a number line in equal jumps of 4, leaving a coloured trail; gap positions pulse until filled.
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Spot the mistake. A friend says: 'The 5 table is 5, 10, 15, 20, 24, 30.' What did they get wrong?
Answer
`24` is wrong — after 20 comes `25`, not 24. Every multiple of 5 ends in 5 or 0, and 24 ends in 4.
Art hook The ring-of-fives from before: the wrong number 24 tries to land but there is no glowing point for it, so it visibly falls off the circle.
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Write two multiplication facts that both equal `20` using the 2, 4, 5 or 10 tables.
Answer
For example `4 × 5 = 20` and `2 × 10 = 20` (also `10 × 2`, `5 × 4`). Any correct pair is fine.
Art hook A target number 20 in the middle; the child builds different dot-rectangles (4x5, 2x10) that all rearrange into the same 20 dots, proving they match.
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Build `8 × 3` step by step from an easier table by doubling twice.
Answer
`2 × 3 = 6`, double to `4 × 3 = 12`, double again to `8 × 3 = 24`. So `8 × 3 = 24`.
Art hook A dot-array that doubles twice in sequence: 6 dots -> 12 dots -> 24 dots, each doubling shown as the array mirroring itself.
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Is `9 × 5` odd or even? How can you tell without working it out fully?
Answer
`9 × 5 = 45`, which is odd (ends in 5). You can tell because both factors are odd, and odd × odd is odd; also 5-table answers ending in 5 are always odd.
Art hook 9 groups of 5 dots try to pair up two-by-two; one dot is always left unpaired and glows, showing the product is odd.
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Rani says `3 × 8` and `8 × 3` need two different amounts of counting. Is she right? Explain using an array.
Answer
No — both equal `24`. An array of 3 rows of 8 turned on its side is 8 rows of 3: same dots, same total. This is commutativity, so you only learn the fact once.
Art hook A 3x8 dot-array rotates 90 degrees into an 8x3 array; the dot count stays pinned at 24 the whole time, showing the flip is free.
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Mark 10 points evenly around a circle and number them 0 to 9. From each point `n`, draw a line to point `2 × n` (if you go past 9, keep counting from 0 again). Which two points do points `6` and `7` connect toward, using the 2 table?
Answer
`2 × 6 = 12`, and going past 9 lands on point `2` (12 is 10 + 2). `2 × 7 = 14` lands on point `4`. So the 2-table wraps around the ring, and doing this from every point weaves a heart-shaped (cardioid) pattern.
Art hook The classic times-table cardioid: 10 dots on a circle, each connected to its (2n) neighbour with wrap-around, producing a woven heart. Let the child change the multiplier to see the curve morph.