## Student Misconceptions about the Similarity of Rectangles

28 Nov

Have you used any of NCSM’s Illustrating the Standards for Mathematical Practice modules? The modules are written to use as professional development with teachers. We have been using some of the Congruence and Similarity module with our students for several years now.

In the task Hannah’s Rectangle, students are asked which rectangles are similar to rectangle A. Each student had a copy of the rectangles, a piece of wax paper, and a straightedge. They really had a protractor, but I asked them not to use it to measure length.

I sent a Quick Poll to find out which rectangles are similar to A, and this is what I saw, with responses separated:

And responses grouped together:

Ten students selected the correct similar rectangles.

Seven students selected every rectangle as similar to rectangle A.

What would you do next?

I showed the responses separated, without correct answers marked.

We looked at D. Is it similar to A? Why or why not?

Then a student offered his misconception: I selected them all because I thought all rectangles were similar.

(I’ve used this task several times, and that misconception didn’t surface until this year. LJ wasn’t alone in his thinking … 7 students had selected all rectangles – it just hadn’t occurred to me why they had done so until he gave his reason.)

Are all rectangles similar?

With what shapes can we say, “All ___ are similar.”?

Why aren’t A and F similar?

One student had already determined that C was similar to A. She eliminated F because it had the same base length as C.

Another student dilated E about its center to get A, showing that the diagonals were collinear.

Another student dilated B about its top left vertex to get A, showing that the red lengths were equal.

I showed my students the video of Randy sharing his thinking with his class. Several students had used a similar method, but they didn’t use the same wording as Randy in explaining their thinking.

And so the journey continues … learning more every year about student misconceptions and grateful for those who write tasks to expose those misconceptions.