Open Middle®
Directions: How can you tell which rectangle is bigger: a rectangle with a perimeter of 24 units or a rectangle with an area of 24 square units?
Hint
What does “bigger” mean? What information to we need to compare these two squares?
Answer
I have intentionally left the word “bigger” mathematically imprecise. Depending on whether “bigger” refers to a greater area or greater perimeter, you will get different answers. For example, a let’s assume that the rectangle with an area of 24 square units measures 6 units by 4 units. Now let’s consider two different rectangles with perimeters of 24 units. Rectangle A is 6 units by 6 units. That has an area of 36 square units and has a bigger area than the initial rectangle. Rectangle B is 11 units by 1 unit. That has an area of 11 square units and has a smaller area than the initial rectangle. Hopefully this will provide good opportunities to develop the need for precise mathematical language.
Source: Robert Kaplinsky
Hi, Robert! Minor thing here, but I think the hint should say “rectangles” and not “squares”…? Students could use any sort of rectangle to consider this problem, right?
i think the answer is the perimeter of 24 units because then the area is going to be bigger than 24 units.
i think area
Huh. I didn’t realize this was for 3rd grade so I solved for x(12-x)=24 and got 6 plus sqrt 12 for one side and 6 – sqrt 12 for the other. Then if the one with perimeter 24 is “skinnier” than that its area will be smaller, and if it’s “squarer” (like 6×6) it will be larger. I’m thinking that is not how you want a third grader to reason, though. 😉