Friday, October 3, 2014

Activity: The Parallel Tests (Day 41)

As we already know, many students confuse the concepts of "corresponding angles," "alternate interior angles," and "same-side interior angles." Indeed, this is one reason I began with the Two Perpendiculars Theorem as our first Parallel Test -- determining whether two lines are perpendicular doesn't require a student to know the difference between corresponding and alternate interior angles.

But now we really want students to know the difference. And so there are several ways to make an activity out of learning the difference.

We begin with a diagram of two lines cut by a transversal, forming eight angles, just as in the diagrams of the previous lesson. But now we have students choose numbers from 1-8 at random and then name the types of angles given by the numbers.

Let's assume the diagram from the previous lesson. If one of the chosen angles is 3, and the other is...

-- 1, then we have a linear pair.
-- 2, then we have vertical angles.
-- 4, then we have a linear pair.
-- 5, then we have same-side interior angles.
-- 6, then we have alternate interior angles.
-- 7, then we have corresponding angles.

But what about 8 -- what types of angles are 3 and 8? As it turns out, these angles have no standard given name in geometry.

Notice that angles 1 and 8 are occasionally called "alternate exterior angles." The U of Chicago mentions this in Section 5-6, Question 10. Similarly, we note that angles 1 and 8 could be called "same-side exterior angles."

But no one would even care about alternate interior angles at all, except for the fact that we have Parallel Tests (and Consequences) that mention them. So we can extend the activity so that it refers to the Parallel Tests. In particular, we have the students choose two angles at random, and we ask the question, if these two angles had the same measure, would it necessarily make the lines parallel? If one of the chosen angles is 3, then choosing angles 6 or 7 as the other angle would produce a win. But we don't win if we choose angles 3 and 5 -- same-side interior angles need to be supplementary, not equal, to produce parallel lines. Neither do 3 and 2 produce a winning combination -- these vertical angles are of equal measure regardless of whether the lines are parallel. But 1 and 8 -- alternate exterior angles -- are a winning combination.

A more advanced game would be to make pairs that, if supplementary, would make the lines parallel be the winning combinations. This is tricky because, as it turns out, the pair with no name, 3 and 8, is a winning combination. Students might try proving this if this is disputed.

Truly advanced students might try a game where a student chooses three angles rather than two, assumes that all three have equal measure, then earns a point for each pair of lines that can be proved parallel. Some combinations, such as 1, 5, and 9, are two-pointers -- but these are rare.

Sorry, but you're on your own for this one! The diagram is so simple that it's not worth scanning and posting to the blog.

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