What is the thinking that led this student to make this mistake?

I’ve been teaching geometry for six years, and I figure I must have seen this mistake dozens of times. It’s so common that I have a name for it in my class — it’s a **part-whole** issue. Students know that AD is to DB as AE is to EC, and I think DE gets (correctly) associated with AD and AE while BC gets (correctly) associated with DB and EC. The issue, though, is that AD, DE, AE are all *whole *sides whereas as DB and EC are *parts *of sides. So while this student is correct to associate these sides, the student is comparing whole side lengths to parts rather than finding the proportion between different whole side lengths.

I’d be pretty surprised if other geometry teachers haven’t seen this mistake too, and I’d be interested to hear their explanations of why this mistake is so common.

When I shared this on twitter, the main conversation was about the quality of the problem, and especially the fact that this diagram is not to scale.

@mpershan Well, it's not merely "not to scale" it is "horribly misleading".

— Enter your real name (@vickersty) June 20, 2016

@mpershan @Dsrussosusan @Seestur The side representing 2 is > twice as big as the side that's s 6 — and we wonder why they think math lies?

— geonz (@geonz) June 20, 2016

I was surprised by this response for two reasons:

- While I wouldn’t want my students to start studying this math with this task (they didn’t) I think the wildly out-of-scale diagram is a nice way to draw students’ attention to the underlying relationships between the sides. I often encourage students to make quick sketches to help guide their thinking, and these sketches don’t have to be to scale in order to be helpful.
- Most importantly: The student whose work we’re studying did not have an issue with the diagram! He had successfully solved the first four problems, and then he offered a reasonable (but incorrect) answer to the last one. The underlying issue this student had is easily explained without the diagram, and it’s one that I’ve seen often with accurate diagrams.

Then again, there were so many people on twitter suggesting that this problem has major issues, it’s making me pause and wonder if they have a point. I’ll have to think more about it.

In any event, I then started thinking about addressing and furthering the thinking that this student had. This wasn’t just an isolated mistake — a lot of students in class had similar issues. I wanted to start class with an activity that would help further their thinking on this type of problem. What activity could I do?

Because I wanted to help students see the subtle difference between part/whole and whole/whole comparisons, I decided to use a Matching Connecting Representations activity (see more of these here).

I came up with two different versions. Any ideas on how to improve them? Would they spur kids to think about different strategies?

**Featured Comments: **

Some dissent from **S Freedman**:

I really like the lack of scale in the drawings. It’s important to teach that diagrams can be misleading. The math isn’t lying, just their unconscious interpolating brains.

**Max **wants to tackle the ambiguity with the diagram head-on, and offers a “Which One Doesn’t Belong” activity for doing so.