Partial-products multiplication

Partial-products multiplication splits each factor into its place values and multiplies every piece of one by every piece of the other. Each of those small multiplications is a partial product. Adding the partial products gives the answer.

The method is slower to write than the standard algorithm and much easier to explain. Every line on the page corresponds to a real rectangle of the answer, which is why it pairs naturally with the area model.

How the method works

1. Write each factor in expanded form, for example 27 as 20 + 7.
2. Multiply each part of the first factor by each part of the second.
3. Write every partial product on its own line.
4. Add the partial products.

A two-digit by two-digit problem always produces exactly four partial products. If there are three, one has been missed.

A worked example

A second example

Why it works

Multiplication distributes over addition. Splitting both factors and multiplying every pair is exactly what the distributive property allows:

The area model turns that identity into a picture. The whole rectangle has to equal the sum of its parts, so no partial product can be dropped without the picture failing to add up.

Common mistakes

Missing a partial product

Most often the tens times ones pair. Count the lines: two digits by two digits gives four, three digits by two digits gives six.

Losing the zeros

Writing 2 x 4 = 8 instead of 20 x 40 = 800 shrinks the answer by a factor of a hundred. Multiply the place values, not the digits.

Check yourself