Escape! meeting notes 2 — Miras

Notes from the Fall Bay Area meeting of Escape from the Textbook! (continued)

back to Part 1

Carlos Cabana challenged us to think of uses for Miras, and for tangrams.

Having done a lot with tangrams over the years, I chose to work on Miras. I have a class set of those at school but have not used them hardly at all. Miras are pieces of red plexiglas that work as mirrors, but also allow you to see through them. Thus you can see both an object, and its image in a reflection. Moreover, the mirror is beveled at the bottom, so you can use it as a straightedge to draw a line that is pretty much exactly the reflection line.

As it turns out, Miras can be used to do all these constructions:

• a perpendicular to a line
• an angle bisector
• the perpendicular bisector of a line segment
• copying an angle (perhaps not in every circumstance?)
• a parallel line

In other words, like patty paper, they can be used to provide an alternative to straightedge and compass construction. There are two things you cannot do with them (or with patty paper, for that matter): draw a circle, and use arcs of circles to copy segment lengths, as in the construction of an SSS triangle. In other words, compasses are still needed, but using these alternatives when possible makes geometric construction accessible to a much broader range of students.

We gave ourselves these challenges, which we met successfully:

• given one side, construct a square, using only a Mira
• given a circle, construct an inscribed equilateral triangle

(The latter construction leads easily to the inscribed regular hexagon as well.)

At the end of the day, the Mira does require a little more introductory overhead than patty paper, so I don’t think I’ll switch to Miras in the near future in my own geometry (Math 2) classes at Urban. Our current combination of patty paper, straightedge, and compass works well as an intro to construction on a laptop. On the other hand, I may well do some Mira work in my Space class, as the class includes a huge unit on isometries, including reflection in a line.

More notes from that meeting in a future post!

–Henri

PS: see my Geometric Puzzles in the Classroom page for material related to tangrams. For actual tangram activities, and activities involving mirrors, see my book Geometry Labs, if you have it. If you don’t, watch this space: I will soon make it available as a free download on my Web site.

Update: Geometry Labs is now available here!

on to Part 3