Adding and multiplying follow three simple, dependable laws. They are the rules that say which moves are always safe — when you can reorder numbers, regroup them, or split one apart without ever changing the answer. Learn these three and you can rearrange almost any sum into an easier one.

1. Commutative: order doesn't matter

You can swap the two numbers around and the answer stays the same. This is true for both adding and multiplying:

So 3 + 5 is the same as 5 + 3 (both 8), and 2 \times 4 is the same as 4 \times 2 (both 8). Think of two amounts being poured into one pile — it makes no difference which you pour first.

2. Associative: grouping doesn't matter

When you add (or multiply) three numbers, it doesn't matter which pair you combine first. The brackets just say "do this bit first", and you can move them freely:

Try (2 + 3) + 4: that is 5 + 4 = 9. Now 2 + (3 + 4): that is 2 + 7 = 9. Same total. This lets you pick the friendly pair first — for 7 + 8 + 2 it is easier to do 8 + 2 = 10 first, then 7 + 10 = 17.

3. Distributive: the star of the show

This is the most powerful law. Multiplying a number by a sum is the same as multiplying it by each part and then adding:

The picture below makes it obvious. A rectangle is a tall and b + c wide, so its area is a \times (b + c). Slice it down the line between b and c and you get two rectangles — one a \times b and one a \times c. The squares didn't move, so the two areas must add up to the whole. Step through it.

This is the trick that turns a hard multiplication into easy ones. To work out 6 \times 13, split 13 into 10 + 3:

Two easy products instead of one tricky one. The distributive law is exactly the area model at work, and it is the engine behind almost every multiplication shortcut you will ever learn.

See it explained

Khan Academy uses the distributive property to break a product into easier pieces: