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Converting Square Units


Alignments to Content Standards: 6.RP.A.3

Task

Jada has a rectangular board that is 60 inches long and 48 inches wide.

  1. How long is the board measured in feet? How wide is the board measured in feet?
  2. Find the area of the board in square feet.
  3. Jada said,
    To convert inches to feet, I should divide by 12.
    The board has an area of 48 in $ \times $ 60 in = 2,880 in 2 .
    If I divide the area by 12, I can find out the area in square feet.
    So the area of the board is 2,880 $ \div $ 12 = 240 ft 2 .
    What went wrong with Jada's reasoning? Explain.

IM Commentary

Since this task asks students to critique Jada's reasoning, it provides an opportunity to work on Standard for Mathematical Practice 3 Construct Viable Arguments and Critique the Reasoning of Others.

Solution

  1. The board is 5 feet long and 4 feet wide.
  2. The area of the board is 20 ft 2 .
  3. While it is true that you convert inches to feet by dividing by 12, that doesn’t work for converting square inches to square feet. Because a square foot is 12 inches on each side, there are 12 2 = 144 square inches per square foot (see the picture). 1_c034d8ff713d1cc92e169ec397958050 Thus, $$2,880 \text{ in}^2 \times \frac{1 \text{ ft}^2}{144 \text{ in}^2} = 2,880 \div 144 \text{ ft}^2 = 20 \text{ ft}^2.$$

Jeremy says:

over 1 year

This looks an awful lot like unit analysis, which from what I have gathered is not a Grade 6 expectation. Wouldn't it be more appropriate to solve this using a ratio table after determining the 1:144 relationship?

drishor says:

over 4 years

I love the idea of this task as this is something that students struggle with. Knowing why the division doesn't work for square feet is not intuitive to a 6th grader, (at least in my experience). What if there were two students who each solved the problem, one correctly and the other incorrectly? Students could construct an argument about who did it incorrectly and why, or who did it right and what made it right. It seems to me like, if I am a student who has no idea about "square inches", I may not understand what Jada did wrong, (and may even think she did it right), and I would have nowhere to go in terms of trying to reason it out.

Kristin says:

over 4 years

I think this is a great suggestion--we'd lke to hear how it goes with students.