Proof in High School Geometry

(This is not a comprehensive description of the Urban School's Math 2 course -- it is a quick overview, with an emphasis on how we handle proof.)

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 use Cabri). 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 "given" sides and angles the students copy to construct their triangles. The lengths and angles can of course be manipulated dynamically in the Cabri file. (For their constructions, students copy the lengths with the "compass" tool, and the angles by using parallel lines.)

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.