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Comparing Products


Alignments to Content Standards: 5.OA.A.2

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.

Area_163854eb350dcf74ee7fe03e9d1942aa

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.

ShariV says:

about 6 years

This is a great task, but it seems to fit better with standard 5.OA.2 since it involves understanding expressions without manually completing the calculation. My understanding of 4.OA.1 is that it is more about being able to verbally describe multiplication situations and the meaning of multiplication.

Kristin says:

about 6 years

Thanks, ShariV, I agree with you. I changed the alignment to 5.OA.2 and rewrote the commentary to explain how this task builds on knowledge from earlier grades (including the original standard the task was aligned to).