Definitions in Geometry update

I blogged a few years ago about starting geometry with developing definitions. I’ve made some changes since then, and I have additional ideas for next year that I want to remember.

I’ve found it more useful to start the school year introducing geometry as art, and developing a need for definitions as students struggle to describe the process to create their designs.

Once there is a need for developing agreed upon definitions for terms, I want to use this video to motivate how and why definitions develop:

To make a definition is to highlight and call attention to a feature or structural property. Historically this comes out of working on a problem, not as a prelude to it. The point is you don’t start with definitions, you start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction. Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion.

– Paul Lockhart, Lockharts Lament (p.22)


Then, students will develop their own terms and definitions (We ended up with of holes, tubes, and bubbles – you can see the fun thread here.) and we can see how complete it is by trying to classify different objects: a sock? a slice of Swiss cheese? a block of Swiss cheese? etc…

We may engage in some form of Attacks and Counterattacks to help students refine their definitions as the situation requires.

After this introduction of what definitions are and how they work, I will use examples and counterexamples for students to work in small groups and develop definitions of other geometry terms, as described here.

Starting Geometry with Definitions

Every year, geometry starts with students defining many key terms so that we can use this vocabulary as we work through the content. For some reason, this school year, I couldn’t remember what I’d done in the past and I took this to mean that it wasn’t as awesome as it could be. As I planned the first few days I had these ideas in mind:

  • I wanted students to know 16 key geometry terms.
  • I wanted to use the frayer models that fit in students interactive notes as Sarah Hagan describes here.
  • I wanted students to develop their own definitions and not simple copy from a book so that they owned the vocabulary and processed each term.
  • I wanted to create and foster a culture of collaboration in my class as the school year began.
  • I wanted students to depend on their peers and teacher provided resources for support of content and not rely on the teacher as the purveyor of information.
  • I wanted to be able to easily verify student’s definitions for accuracy outside of class time.
  • I wanted to pre-assess students ability to write effective definitions and writing in general.

I love the Kagan Geometry book  (but really wish it was available in a digital format) I don’t always follow the prescribed structures, but the resources can be very useful. There are pages in this book for developing definitions that contain only images of examples and non examples – which fit well with the frayer model that I planned to use. In searching the MTBoS for ideas, I found this post by Pam Wilson.

This what I ended up doing. I am satisfied with the way this went and will do it again next year:

The setup:

I copied the terms, blown up large onto different colored card stock  &  laminated them. Each color represented a group, so I make 5 colors with 3 words per color & I kept one to use as an example with the whole class. I also made a ton of copies of Sarah’s Frayer model for students to use.

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The implementation:

1. I used the widget example from Discovering Geometry (chapter 1). It shows strange blobs and says “these are widgets”, then there is another group of strange blobs and it says “these are not widgets”. I have students define widgets in their groups. Then they read their definition and we try to draw a counterexample. Then we discussed what makes a good definition and we were ready to go!

2 I projected the “perpendicular lines” examples and non examples. We completed a frayer model for the term.

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3. Students worked in small groups with their 3 terms copying the examples & non-examples, then writing good definitions for each term. I set a timer for 10 minutes.

4. Groups rotate to another station. repeat. 10 minute timer.

5. This continued until the end of class. Students turned in their completed Frayer model sheets.

<At this point each student had about half of the 16 words defined.>

6. That evening I read their definitions (every single one!) and wrote feedback in the margins. No grade.

7. The next class period, I gave all of the students back their definitions with my feedback and gave them time to correct or improve their work.

8. Give one, get one – Speed dating style! Students each got a blank Frayer definition sheet and sat across from a student who was not in their original group. The would talk, find a term that they needed and share. Each students would give one definition to their partner and get one from their partner. Then a timer would go off and they would rotate & repeat.

9. While the students speed dated, I listened and taped pickers to the back of their interactive notebooks.

10. As a quick check for understanding, the students used their plickers and answered multiple choice questions on the terms for the last 10 minutes of class.

11. Homework was to cut them out and put them on specified pages of their interactive notes.

12. I made a Word Wall by simply taping the laminated cards to the wall after the lesson. Easy Pezy!

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