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Proof in High School Geometry

if then diagram

by Henri Picciotto

I worked at the Urban School from 1981 to 2013, and helped develop much of the curriculum and pedagogy of the math department. For an overview of the Urban School math offerings, click here.

This article is a quick summary of Urban's Math 2 course a few years ago, with an emphasis on the approach to proof. The course evolves from year to year, and may change substantially with the introduction of transformational geometry in the coming period.

We use much material from my Geometry Labs book throughout the course. This work does not emphasize formal proof, but it does lay the foundation for it through discussion-generating activities.

We also use problems from a variety of books. My favorite by far is Geometry: A Guided Inquiry by Chakerian, Crabill and Stein. This is the book that had the greatest impact on my approach to teaching high school math. We also use a few problems from Michael Serra's Discovering Geometry and a few from Harold Jacobs' Geometry, Third Edition.

We start with much work on the inscribed angle theorem and the Pythagorean theorem, plus some algebra and an intro to computer programming. In the second quarter of the course, we introduce congruent triangles. A foundational activity is "Non-Congruent Triangles", from Geometry Labs. (However we use patty paper to copy angles, not compass and straightedge. The latter is too complicated and not worth the time it takes to teach it.) We follow that up by having students do the usual very basic proofs using those postulates, without requiring any particular format.

Around the same time, we introduce interactive geometry software (we have used Cabri, but the same can be accomplished with Geometer's Sketchpad or GeoGebra.) One early lesson in that environment asks students to create triangles using SSS, SAS, and so on. This is a complement to the "non-congruent triangles" lesson and it helps to develop the students' understanding of the congruence postulates, to get to the bottom of why AAA and SSA do not guarantee congruency, and to discover the triangle inequality. Here is a screen shot of the GeoGebra version of this activity -- you can download the file and the worksheet. The lengths and angles can of course be manipulated dynamically.

making

(Download the GeoGebra file, and the worksheet)
If you create a GSP sketch or Cabri figure of this,
please send it to me and I'll post it here.

In the second half of the course, among other things, we do these three units:
Proof
Quadrilaterals | Teacher Notes
Construction | Teachers' Notes

The first unit introduces the rules of the game: what is a proof? What can we assume? What are some ways to present a proof? We apply these ideas to proofs of the main theorems from the first trimester: the isosceles triangle theorem, the inscribed angle theorem, the Pythagorean theorem.

The second unit is the main event: an arena where students use proofs and counterexamples to establish the validity of their conjectures. The unit is explained in the teachers' notes. This is what you should look at if you're looking for a more authentic approach to proof in your geometry class, though it may be most meaningful in the context of rethinking the course as a whole.

The third unit is a motivating and challenging unit on construction. Interactive geometry software makes it possible to once again bring to our students the ancient and all-important tradition of geometric construction. The puzzle-like quality of construction challenges makes them motivating and is a much smarter use of the software than the banal "do this, do that, what do you notice?" routine. Moreover, doing this work on the computer is vastly more accessible to today's students than using straightedge and compass, even though the underlying mathematics is largely the same.

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Geometric Transformations

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