Lesson 2: Modeling Multiplication with Area Models and Partial Products

Property

An area model represents a multi-digit multiplication problem as the area of a big rectangle. First, break both multi-digit numbers into their place-value parts. The total area of the big rectangle (which is the final product) is the sum of the areas of all the small rectangles made from these parts. These small areas are called partial products.
When you multiply two broken-down numbers like $(a_1 + a_2) \times (b_1 + b_2 + b_3)$, you can find the total product by adding up all the partial products:

Examples

• To model $\boldsymbol{23 \times 347}$, you draw a rectangle with side lengths $23$ and $347$. Decompose $23$ into $20 + 3$, and decompose $347$ into $300 + 40 + 7$. The partial products are the areas of the smaller rectangles: $20 \times 300 = 6000$, $20 \times 40 = 800$, $20 \times 7 = 140$, $3 \times 300 = 900$, $3 \times 40 = 120$, and $3 \times 7 = 21$.

• To model $\boldsymbol{256 \times 347}$, you draw a rectangle with side lengths $256$ and $347$. Decompose $256$ into $200 + 50 + 6$, and decompose $347$ into $300 + 40 + 7$. The partial products are the areas of the smaller rectangles: $200 \times 300 = 60000$, $200 \times 40 = 8000$, $200 \times 7 = 1400$, $50 \times 300 = 15000$, $50 \times 40 = 2000$, $50 \times 7 = 350$, $6 \times 300 = 1800$, $6 \times 40 = 240$, and $6 \times 7 = 42$.

Property

The distributive property allows us to multiply a sum by multiplying each addend separately and then adding the products.
When we decompose factors by place value, the results of these smaller multiplications are called partial products.

Examples