Why 3 × 7 = 7 × 3 Halves the Times Tables
By the TimesTable.co team · Updated · 3 min read
One of the most useful ideas in the whole times tables is also one of the simplest: you can multiply in either order. 3 × 7 is 21, and so is 7 × 3. Once a child truly understands this, the times tables roughly halve in size, and the dreaded 7s and 8s become mostly facts they already know.
Why is 3 × 7 the same as 7 × 3?
Imagine 21 counters laid out in a rectangle of 3 rows with 7 in each row. That's 3 × 7. Now walk round to the side of the table and look again: you see 7 rows with 3 in each. That's 7 × 3. Nothing was added or taken away, so the totals must match.
Our multiplication visualiser shows exactly this as a dot array, ideal for convincing a child who isn't sure.
What is the commutative property?
"Commutative" means the order doesn't matter. Multiplication is commutative (a × b = b × a) and so is addition (3 + 5 = 5 + 3). Subtraction and division are not: 10 − 4 isn't the same as 4 − 10. Children don't need the word straight away, but they do need the idea.
How does this halve the times tables?
The 1 to 12 times table grid has 12 × 12 = 144 squares. But every fact appears twice, once on each side of the diagonal: 3 × 7 in one place and 7 × 3 in another. The only facts that appear once are the squares on the diagonal (1 × 1, 2 × 2 … 12 × 12). So there are only 78 different facts: the 12 squares plus 66 pairs.
Open the times table grid and click a table header and the matching row and column light up together, showing that each fact lives in two places.
Why does it make the hard tables easier?
By the time a child reaches the 7 times table, they've already learnt 7 × 2 in the 2s, 7 × 5 in the 5s, and so on. If they recognise that 7 × 4 is the same as the 4 × 7 they already know, the 7s shrink to just a couple of new facts. That's why the order you learn them in matters, and why our 7 times table guide starts by counting what's already known.
How do you teach it at home?
- Build a rectangle from Lego bricks, coins or pasta, say 4 rows of 6.
- Count them together (24), and write "4 × 6 = 24".
- Turn the rectangle round. Ask: how many rows now? How many in each? Write "6 × 4 = 24".
- Try it with a few different rectangles until your child predicts the answer before counting.
Does it help with division too?
Yes, indirectly. The fact family 3, 7, 21 gives four facts: 3 × 7 = 21, 7 × 3 = 21, 21 ÷ 3 = 7 and 21 ÷ 7 = 3. Knowing one multiplication fact unlocks two divisions, which is useful when your child moves on to the bus stop method or long division.
How is this used in practice?
Our smart practice treats 3 × 7 and 7 × 3 as one fact, so progress on one counts for both, which is also why the progress map fills from both sides at once.
Frequently asked questions
Is 3 × 7 the same as 7 × 3?
Yes. Both equal 21. Three groups of seven and seven groups of three contain the same number of things, just arranged differently.
What is the commutative property of multiplication?
It is the rule that you can swap the order of the numbers in a multiplication without changing the answer: a × b = b × a.
How many times tables facts are there really?
The 1 to 12 grid has 144 facts, but because each pair appears twice, there are only 78 different facts to learn.
Does the commutative property work for division?
No. 12 ÷ 3 = 4, but 3 ÷ 12 is a quarter. Swapping the numbers only works for multiplication and addition.
How do I explain it to a young child?
Make a rectangle of counters, say 3 rows of 7, and count them. Then turn the rectangle a quarter turn: now it is 7 rows of 3, and there are still 21 counters.
Do schools teach this?
Yes. In England it is part of Year 2 (multiplication can be done in any order), and in the US it is part of Grade 3 properties of operations.