Lesson 5: More Use Area Models and Partial Products to Multiply

Property

An area model represents a multiplication problem, such as $a \times N$, as the area of a rectangle.
The multi-digit number $N$ is decomposed into its expanded form (e.g., $123 = 100 + 20 + 3$).
The total area (product) is the sum of the smaller rectangular areas, which are the partial products.
For $a \times (b + c + d)$, the total product is $(a \times b) + (a \times c) + (a \times d)$.

Examples

• To model $6 \times 347$, you draw a rectangle with side lengths $6$ and $347$. Decompose $347$ into $300 + 40 + 7$. The partial products are the areas of the smaller rectangles: $6 \times 300 = 1800$, $6 \times 40 = 240$, and $6 \times 7 = 42$.

• To model $9 \times 4,582$, you draw a rectangle with side lengths $9$ and $4,582$. Decompose $4,582$ into $4000 + 500 + 80 + 2$. The partial products are the areas of the smaller rectangles: $9 \times 4000 = 36,000$, $9 \times 500 = 4,500$, $9 \times 80 = 720$, and $9 \times 2 = 18$.

Property

The partial products algorithm uses the distributive property to solve multiplication.
A multi-digit number is broken into the sum of its place values (expanded form), and each part is multiplied separately before adding the results.

Examples