Many of us teach area and perimeter but I’m guessing that most of us have not spent a lot of time thinking about the relationship between the two. This meeting began a investigation that was totally new to me. Tyler started Friday’s meeting by showing us a coordinate plane with an x axis labeled Perimeter and y axis labeled Area with 2 plotted points. Tyler asked us: What do you notice? What questions do you have?
Some of our questions:
- What does the dot represent? Does it represent a rectangle?
- What is the relationship between area and perimeter?
- Are both of the figures represented by these dots possible? Which is possible and which is impossible?
- Could the point describe a triangle? Pentagon? Other shapes?
Tyler then gave us graph paper and the following prompt:
“Draw a shape on squared paper and plot a point to show its perimeter and area. Which points on the grid represents squares, rectangles, etc? Draw a shape that may be represented by the point (4, 12) or (12, 4). Find all the “impossible” points.”
Over the next hour, we drew figures and calculated area and perimeter for rectangles, triangles, circles, etc. Some of us plotted rectangles on the coordinate plane and noticed patterns about where squares showed, for example, and determined how to know whether a point was possible or impossible.
Questions to explore:
- Why are squares interesting when we think about the relationship between area and perimeter?
- What would the graph of all possible triangles look like?
- The graph of all possible circles?
At the end of the meeting, we did some algebraic work to determine the exact dimensions of a rectangle with an area of 4 and a perimeter of 12. We knew from trial and error that 5.3 x .7 was close. The perimeter (5.3 + 5.3 + .7 + .7) is 12, but the area (5.3 x .7) is 3.71, not 4.
We started with the following two equations and solved a system of equations:
x * y = 4
2x + 2y = 12
(Click to see a solution of this system. Sorry for the coffee stain.)
We came to 5.24 x .76, which is closer but not exact due to rounding. 5.24 + 5.24 + .76 + .76 = 12 & 5.24 x .76 = 3.9824 (not 4).
What do you notice about the solution to the following system of equations, from the other point (4, 12) on the coordinate plane)?
x * y = 12
2x * 2y = 4
On the way out, Tyler shared the Gold Rush problem with us and told us about Math Memos, the new section on CollectEdNY.org which features teacher write-ups of rich problems, along with samples of student work and discussion of multiple solution methods. Tyler’s write-up of the Gold Rush problem is highly recommended. If you missed this meeting, I recommend that you try the Gold Rush problem on your own, then think about connections to the area vs. perimeter puzzle above. Share your thoughts about the problems or teaching questions below.
(By the way, Tyler is looking for teachers who are interested in writing up problems for Math Memos. Is anyone interested in teaching a problem from a CAMI meeting, collecting student work and writing it up for the site?)
Question for the group: Would anyone use the Area vs. Perimeter prompt at the beginning with an HSE class? How could we scaffold and prepare students for the open-ended nature and abstraction of the task? Is the Gold Rush problem adequate preparation?
An integer-sided rectangle with area A is called a rectangular personality of A. Which integer from 1 through 100 has the most rectangular personalities?
In other words, which integer area value between 1 and 100 has the most possible integer-only perimeters?
In attendance: Avril, Chaim, Deneise, Eric, Jane, Nikko, Solange, Tyler
Programs represented: Bard Prison Initiative, BMCC, Brooklyn Public Library, CUNY Adult Literacy, Fifth Avenue Committee
Location: The Brooklyn Public Library, Pacific Street Branch at 25 4th Avenue