At first, not all geometry students know what we're talking about when we talk about angles and their measures. This is probably why it is so difficult to teach some of them how to use a protractor: they don't know what they're measuring. Here are two kinesthetic activities that help a little.

*** Arm angles**

Ask the students to stand up, and stick an arm straight out so that it is neither horizontal nor vertical. (This helps separate the idea of "straight" from the idea of "horizontal or vertical". Students often confuse those.) Then ask them to stick the other arm out, so that their arms make an acute angle, a right angle, an obtuse angle. It is surprising how difficult it is for many students to do this.

A payoff down the line is that arm angles can be used to introduce the idea of the intercepted arc. The arc intercepted by my angle is the arc I can see between my two arms. This probably seems pointless to some of you, but a number of students have told me that this is what helped them see what was meant by the intercepted arc.

*** Walking polygons**

This is based on turtle geometry as initially introduced by the Logo programming language, but it does not require a computer. A few "walking polygons" lessons are available in my Geometry Labs book. Here are two ideas:

- I tell students I will walk along the edge of an invisible parallelogram. Then I do the following, twice: two steps forward, a 135° turn to the left, one step forward, a 45° turn to the left. Of course it's not really possible for them to guess my turn angles, but it is entirely possible to figure out their sum, since after doing the two turns I'm facing in the opposite direction.

- Likewise, I tell them I'm going to walk a regular pentagon (or some other regular polygon -- preferably not a square.) At the end of the walk, I'm facing the way I faced at the beginning. This leads to a discussion of a formula for the size of the turn (or exterior) angle in a regular n-gon, and thus the size of the corresponding interior angle.

It is fun (and educational) to follow this up by programming a turtle in Logo or one of its descendants. (I use BYOB, a Logo descendant by way of Scratch.)

--Henri

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