Comparing Products

Task

Leo and Silvia are looking at the following problem:

How does the product of $60 \times 225$ compare to the product of $30 \times 225$?

Silvia says she can compare these products without multiplying the numbers out. Explain how she might do this. Draw pictures to illustrate your explanation.

IM Commentary

The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on

3.OA.5 Apply properties of operations as strategies to multiply and divide

and

4.OA.1 Interpret a multiplication equation as a comparison.

Solution

Since 60 is twice 30, the product $60 \times 225$ is twice the product $30 \times 225$. We can write this as an equation: $$60 \times 225 = (2 \times 30) \times 225 = 2 \times (30 \times 225).$$

The above explanation corresponds to the following picture.

The area of a 225 by 60 rectangle ($60 \times 225$) is double that of a 225 by 30 rectangle ($30 \times 225$). If we scale the width of the rectangle by a factor of 2, then the area of the resulting rectangle doubles. In other words, if one of the factors of the product $30 \times 225$ is scaled by a factor of 2 then the product is scaled by a factor of 2.