Leo+DadMade for Leo
The Distributive Law (expanding)
Rung 4 of 4 · Mastery

Expanding Out in the Real World

This law isn't just a bracket trick — it's how clever people multiply in their heads. And run it backwards and you've started factorising, the next concept along.


BuildDo the head-multiply, then flip to “reverse: factorise” and pull the common factor out.
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Audio WalkthroughComing Soon
Video ExplainerComing Soon

Ask someone quick to work out 6 × 23 and they won't reach for a pen — they'll split the 23. That split is the distributive law, doing your mental arithmetic for you.

Splitting to Multiply in Your Head

6 × 23 is awkward, but 23 = 20 + 3, and you can do those bits easily: 6 × 20 = 120 and 6 × 3 = 18, then add: 120 + 18 = 138. That's exactly 6(20 + 3) = 6×20 + 6×3 — the same law from rung 1, only now the "brackets" are a number you chose to break up. The toy draws it as a rectangle split into a tens piece and a ones piece, so you can see the two products.

The move: break a hard number into friendly pieces, multiply each piece by the outside number, then add the answers. 7 × 41 = 7×40 + 7×1 = 280 + 7 = 287.

Running It Backwards

Here's the payoff. Expanding turns a(b + c) into ab + ac. Go the other way — start with 6x + 9, notice that 3 divides both terms, and pull it out the front: 6x + 9 = 3(2x + 3). That's factorising, and it's just expanding in reverse. Flip on the reverse challenge in the toy and you're already doing the very first move of the next concept.

Why This Is the Finish Line

The area model made it make sense. The two strokes made it quick. Naming the slips made it safe. But seeing the same law power your mental maths and open the door to factorising — that's when it stops being a bracket rule and becomes a tool you actually reach for. That's mastery, and it's the bridge straight into factorising algebraic expressions.

Us, Thinking Out Loud

What's a multiplication you could split to do in your head right now?

Factorising is "undo expanding" — how do we find the number to pull out the front?

Of the four rungs, which should we re-drag in a fortnight?