Case Overview: Relating Perimeter and Area
Background on the Classroom Lesson
This video comes from a secondyear high school math class using an integrated curriculum. For homework, the students explored how increasing the perimeter of a figure affects its area using figures with a given perimeter.
Student Activity
The activity is part of the "Do Bees Build It Best?" unit from Year 2 of the Interactive Mathematics Program.
Citation: Fendel, D. M., Resek, D., Alper, L., & Fraser, S. (2010). Interactive mathematics program: Integrated high school mathematics, Year 2. Key Curriculum Press.
Students in the class you will see have been discussing answers to the following sets of problems:
Suppose that a farmer went to the supply store to buy fencing for a square corral and found that they were having a halfprice sale. Instead of buying 300 feet of fencing, she could now afford 600 feet of fencing. This will allow her to make a bigger space for her animals.

What would be the area of the square space if she used 600 feet of fencing? How does this compare to the area of a square space made with 300 feet of fencing? In other words, what would doubling the perimeter do to the area?

What would be the area of an equilateral triangle space using 600 feet of fencing? How does this compare to the area of an equilateral triangle made with 300 feet of fencing?

What would the area of the square and triangular spaces be if she used 900 feet of fencing?
The video clip focuses on the problem below:

What generalizations can you make from your results in Questions 1 through 3?
Here is what was pictured on the overhead:
Overview of the Video
In the video, the class focuses on generalizing and justifying their results about the area when doubling and tripling equilateral triangles and squares with a perimeter of 300. When watching (or rewatching), focus on the different explanations students use to justify their generalizations. Consider the different ways that students think about the problem and what each explanation or representation reveals about the student's understanding.
Questions to Consider about Student Thinking
We think some of the richest student thinking in this video includes Jake's idea and his conversation with Monica, Riaz's comment (and whether it is the same or different from Jake's), and Coleman's drawing and what it reveals about the pattern. In what follows, we provide sets of questions about these ideas to scaffold analysis of students' ideas.

Discussing Jake's Idea:

What is Jake's idea?

What role do the "dimensions" of a figure have in Jake's explanation?

What do you think Monica understands about Jake's idea?


Discussing Riaz's Idea:

What is Riaz's idea? How does he explain where the squaring comes from?

Is Riaz's explanation the same as Jake's or different?

 Discussing Coleman's Idea:

What is Coleman's idea? How does he explain where the squaring comes from?

How is Coleman's idea similar to or different from Riaz's and Jake's ideas?

Sherin, M. G., Russ, R. R., Walkoe, J., & Dyer, E. (2023). Algebra classroom video cases. Freezing Time Research Group. https://www.freezingtime.sesp.northwestern.edu/videocases. © 2023. Licensed under Creative Commons AttributionNonCommercialShareAlike 4.0 International.