Multi-digit whole-number multiplication can be decomposed by place value, such as expressing as , finding a partial product for each component, and adding those products. An area model makes these products visible, while the distributive property explains why their sum equals the total product; each partial product must retain its place value rather than be treated as an unrelated number. This understanding supports the standard multiplication algorithm and is limited here to whole-number factors.
To use partial products, break one factor into its place-value parts. Multiply each part by the other factor, then add the products.
Example: Find .
Decompose by place value:
Multiply by each part:
These are the partial products. Each one keeps the place value of its part.
Add the partial products:
Therefore,
The distributive property explains this process:
An area model would show three sections with areas , , and . The total area is , so the partial products add to the total product.
Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill
Earned ?