One of the luxuries of a setting like the Problems of the Week, at least for those of us who get to read the student submissions, is that you can think about each student’s work, individually, without the pressure of 23 other kids needing your help *right now*, as in a classroom setting. That extra time doesn’t mean we always reach profound conclusions or can even begin to think about what is necessarily inside the students’ heads, but at least we have time to try. With this in mind, I’d like to share four solutions to Congruent Rectangles Help that I’ve been pondering. The problem is about five congruent rectangles, arranged three vertically across the top and two horizontally across the bottom, that form a larger rectangle. Students are asked to find possible areas for the large rectangle that could result from integer dimensions for the small rectangles. (You might pause here and think about the problem for a minute if you’re not already familiar with it.)

I encourage you to jot down things that you *notice* and *wonder* about each submission as you read them. I’ll share some of my thoughts at the bottom.

Submission 1i acually estimated

width times height so 3 times 5 =15 or 4 time 5=20 or 5 times 5 =25

Submission 2the length is twelve and the width is twelve and the area is 12×12=144 and the perimeter is 48

12×12=144 4+4+4+4+4+4+6+6+6+6+6+6=48

Submission 3One area is 30. another is 63. I cant find a last one

I figured out that the width has to be a divisible of 3 because of the 3 sides of rectangles there. but i couldnt find anything else i got stuck

Submission 4One area is (5*4 for each small) for the large rectangle, one area is 90 (3*6 for each small), and one is 30 (2*3 for each small). EXTRA 5(x+y)=Area of the large rectangle

- 5*4 equals 20 for 1 small rectangle. 20*5 equals 100 for all small rectangles, or 1 large rectangle.
- 3*6=18 for 1 small rectangle. 18*5 equals 90, the area of 1 large rectangle.
- 2*3= 6 for one small rectangle. 6*5=30 and this is the are for 1 large rectangle.

EXTRAThe formula for solving any of these is 5(x+y)=Area of the large Rectangle. For example, 5(6*2)=40 or the area of the large rectangle.

Pause here if you’re still taking notes. Otherwise, read on.