Connected Mathematics–a 6th through 8th grade mathematics curriculum-is designed with a teaching model in mind: Launch-Explore-Summary. In my professional development work, I help teachers to understand the teaching model in general, and I demonstrate effective techniques for particular problems.
For a few summers now, I have been demonstrating a Launch for a problem in the 8th grade unit Frogs, Fleas and Painted Cubes. I came up with the idea after leaving the middle school classroom, so I have not used it with students. But every summer, teachers ask where they can find written instructions for it and I have had to say that they do not exist.
My summer teaching partner has reported to me that it has been effective in her classroom, so I am now motivated to write it up and share it widely. In what follows, I will assume that the reader has access to the Student and Teacher Editions of the unit. The problem in question is Problem 2.1 of Frogs, Fleas and Painted Cubes.
The original launch
In the original problem, students consider the effects of a land swap in which a square piece of land is swapped for a rectangular one with the same perimeter. So a 5 unit by 5 unit square, might be swapped for a 3 unit by 7 unit one. Students record their information in a table and look for a pattern in the relationship between the areas of the two pieces of land.
In designing my alternate launch, I wanted to achieve two ends: (1) increasing access for visual learners, and (2) increasing the generality of the results.
(1) was important to me because I have been working in my own teaching on visual representation in mathematics, and because the problem asks us to represent something geometric in a table. It seemed like a natural place to increase the visual components of the problem.
(2) is important as a teaching principle. The original problem asks students to consider a variety of squares, but to always change each dimension of the square by 2 units. Thus a 5 by 5 square becomes 3 by 7, and a 6 by 6 square becomes 4 by 8. But there is nothing special about 2 units. A more general pattern emerges if we consider different numbers of units.
The new launch
For this problem, students will work in groups of 3 or 4. Each group receives a bunch of colored (say pink) inch grid paper and a bunch of white inch grid paper. After setting up the context as in the Student and Teacher Editions, have each student draw a square (with whole number side lengths) on a sheet of the white grid paper. Students should coordinate to make sure that each student has a square of a different size from their groupmates.
Each group is assigned a number from 1 to 6, repeating numbers if there are more than 6 groups.
Using the pink grid paper and the group’s assigned number, each student draws a new rectangle. Say my group’s assigned number is 3. Then I will transform my original square (say it was a 5 by 5 square) into a new rectangle by increasing one dimension by and decreasing the other by 3. My new rectangle would be 2 by 8. This preserves the perimeter. What does it do to the area?
To investigate this question, each student tries to cover his/her original white square with his/her pink rectangle. Students will need to cut these pink rectangles apart in order to cover as much as possible of the white square.
Then each group puts their covered squares onto a large poster paper and these are displayed around the room, in order of the assigned numbers (see below).
Now the class is ready to do the mathematical work of investigating the relationship between the assigned number, the size of the original square, and the area of the pink rectangles. Some possible observations that students might make and questions they might pose include:
- It does not matter what size square we start with; the difference between the area of the original square and the area of the pink rectangle is the same within each group.
- The group’s assigned number matters-as the assigned number increases, so does the difference between the areas of the original square and the pink rectangle.
- The area of the pink rectangle is always less than the area of the original square. We can see this because we never quite cover the white with the pink.
- Several of these images show a white square peeking out from behind the pink. Is it always possible to rearrange the pink so that a white square peeks out from behind?
- When will it be possible to arrange the pink so that it is in the shape of a square (with whole-number side lengths)?
After some work observing patterns and asking questions, now students should be ready to make the table in the text. However, students should alter the table to match their group’s assigned number, and they should conjecture how the tables of groups with other assigned numbers should look.