The Area Model and Partial Products

24 × 6 is easier if you split the 24 into 20 and 4.

Two easier multiplications

20 × 6 = 120, and 4 × 6 = 24. Together they make 144.

Those two answers have a name.

Partial products are the smaller multiplications that add up to the whole answer.

Why the rectangle helps

The pieces cover the whole rectangle with nothing left over, which is why their areas add to the total.

Split the rectangle by place value

To do 6 × 47, draw a rectangle 6 tall and 47 wide, and split the 47 into 40 and 7. You now have a 6 × 40 piece and a 6 × 7 piece: 240 + 42 = 282. The picture makes the split obvious and the arithmetic easy.

The pieces are called partial products

240 and 42 are partial products — parts of the final product. Writing them out is slower than the compact algorithm but far clearer, because each line corresponds to something you can point at in the picture.

This is the distributive property

6 × 47 = (6 × 40) + (6 × 7) is the distributive property, met as a picture in Grade 3 and now doing serious work. It is the single idea that makes all multi-digit multiplication possible, and it will still be in use when you multiply brackets in algebra.

Estimate to check the size

6 × 47 is roughly 6 × 50 = 300, so expect something a little under 300. The answer 282 fits. A dropped partial product usually produces an answer that badly fails this check, which is why it is worth forming the estimate first.