Categories
Similar Figures Similarity, Right Triangles and Trigonometry

Comparing Parts of Sides Instead of Whole Sides

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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.

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?

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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. When talking abouth math, or any other problem solving situation, which part of the brain is responsible for problem solving?

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

wodb

Categories
Experiments Feedback Similar Figures Similarity, Right Triangles and Trigonometry

Children and Teachers on Feedback

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How do you predict that a group of students (9th Graders, Geometry, nearly all are comfortable with scaling) would respond to this prompt? Do you think they’ll disagree? Converge on one option? What reasons do you think they will bring to support their answers? Do you think that their responses will differ significantly from the responses that a group of teachers would give? If so, how?

Sheesh, that’s a lot of prompts. Let’s condense that:

  1. What do you predict students will respond?
  2. How do you predict that a group of teachers will respond?
  3. How would you respond?
Categories
Decimals Geometry Similar Figures Similarity, Right Triangles and Trigonometry

Decimal Misconceptions? Meet similar triangles.

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A note from the submitter:

Along the lines of one I sent you awhile back. This is one of my best students, and several other students gave answers with similar misconceptions. I pretty much ignored it last time it came up, thinking that it was an anomaly, but I think it’s a significant hole in my students’ understanding. Students were using calculators today.

What’s going on here in the student work? What’s the connection to the earlier post?

Categories
Similar Figures Similarity, Right Triangles and Trigonometry

“They are not similar because you have to add different numbers…”

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Another interesting instance where additive and multiplicative reasoning get entangled when working with similar figures.

Categories
Similar Figures Similarity, Right Triangles and Trigonometry

Scaling by 1/2

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I wonder if the student would’ve made the same mistake with “scaling by 2.” In any event, this isn’t necessarily a nasty misconception to fix, but I’m always interested by the circumstances when addition and multiplication get tangled up.

Categories
Geometric Measurement and Dimension Pythagorean Theorem Right Triangles Similar Figures Similarity, Right Triangles and Trigonometry

A whole bunch of questions about right triangles

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What other mistakes would you expect to see from this problem? How do you teach so as to help students avoid these pitfalls?

Thanks to Tina Cardone for the submission.

Categories
Similar Figures Similarity, Right Triangles and Trigonometry

Similar Figures

Nico

What’s the fastest way to help this student?

Thanks to Nico for the submission.