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.

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

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

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

— Cognitively_Accessible_Math (@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.

## 7 replies on “Comparing Parts of Sides Instead of Whole Sides”

Problem I feel is that we are taking a visual cue about the relative location of a segment and using that to write proportions, rather than grasping that the underlying mathematics has to do with the relationships of similar triangles which is established by the parallel segments. There’s a lot of background math that goes on here and students are jumping to the parts that involve writing in order to form a solution.

Is the brevity of the question and its lack of justification required part of why students don’t recognize (the need for) the full scope of the background relationships?

2 strategies… highlight the two similar triangles in different color highlighters… or sketch the overlapping triangles separately. That second one is a common strategy I’ve used for proofs involving overlapping triangles and seems equally beneficial here.

P.S. How are you generating the images for these comparing representation questions? Those seem time consuming to create? ( I realize once you have one, you’re probably changing a small text box to create the other two which isn’t AS hard… )

Not at all time-consuming for me! I use Geogebra and Google Slides.

I can’t promise that making digital materials like these won’t be time-consuming at first, but I’d estimate that it took me about 20-30 minutes to make both of these Connecting Representation activities.

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.

I’m working on a hypothesis that the question of “drawn to scale” vs. “not drawn to scale” gets mathematically deep and important if we attend to it.

Most students (and I’m guessing even plenty of math-literate adults like me) would have a very hard time articulating what information geometric figures are intended to convey in any sort of rigorous or precise way. The September assertions in Geometry classes across the country about what one ought or ought not to assume from a diagram are in fact probably covering up deep mathematical concepts about what geometric objects are, and which geometric relationships are important, and what geometers consider to be the same and different.

Seems to me like a case for Which One Doesn’t Belong! Here’s a super-based version to maybe start a conversation?

I am of two minds about the representations. I agree that students will see these on assessments and need to know them as a genre. But how will they learn to recognize when they are out of proportion? How do I recognize them?

It’s no mystery where the mistake comes from: given numbers, use them in a computation. 1/3 does show some idea of correspondence, maybe. A truly proportional picture gives them a visual cue – that most would probably still not catch. What if they always had to draw their own picture?

I like this structure of match, match, extend, though. Good routine. I think the 2nd works more towards creating a representation which is building more power than the first.

In fact, there are so many more triangles which might or might not belong. For example the set of two triangles from your recommended separated out diagram. Other sets of two triangles that are and aren’t similar in the same ways. Which are equivalent to the original case and which aren’t? I shall add more pictures when I’m not on my phone, if others remain interested in this avenue.

More triangles