Tuesday, September 2, 2014

Section 3-2: Types of Angles (Day 18)

Section 3-2 of the U of Chicago text discusses the various types of angles. It covers both the classification of angles by their measures -- acute, right, and obtuse -- as well as related angles such as vertical angles and those that form a linear pair. Complementary and supplementary angles also appear in this lesson.

As usual, I like to look for other online resources by other geometry teachers. And I found the following blog post from Lisa Bejarano, a high school geometry teacher from Colorado who calls herself the "crazy math teacher lady":


Bejarano writes that in her geometry class, she "starts with students defining many key terms so that we can use this vocabulary as we work through the content." And this current lesson, Section 3-2, contains many vocabulary terms. So this lesson is the perfect time to follow Bejarano's suggestion.

Now two textbooks are mentioned at the above link. One is published by Kagan, and unfortunately I'm not familiar with this book. But I definitely know about the other one -- indeed, I actually mentioned it in my very last post -- Michael Serra's Discovering Geometry.

The cornerstone to Bejarano's lesson is the concept of a Frayer model. Named after the late 20th-century Wisconsin educator Dorothy Frayer, the model directs students to distinguish between the examples and the non-examples of a vocabulary word.

Strictly speaking, I included a Frayer-like model in last week's Section 2-7. This is because the U of Chicago text places a strong emphasis on what is and isn't a polygon. But Serra's text uses examples and non-examples for many terms in its Section 2-3, which corresponds roughly to Section 3-2 in the U of Chicago text.

Here are steps used in Bejarano's implementation of this lesson:

1. I used the widget example from Discovering Geometry (chapter 1). It shows strange blobs and says “these are widgets”, then there is another group of strange blobs and it says “these are not widgets”. I have students define widgets in their groups. Then they read their definition and we try to draw a counterexample. Then we discussed what makes a good definition and we were ready to go!

(Note: Bejarano writes that she found the "widget" example in Chapter 1 of Serra's text, but I found it in Section 2-3 of my copy. Of course, I don't know how old my text is compared to Bejarano's -- mine is the Second Edition of the text, dated 1997. Hers could be the Third Edition or later.)

Notice that "widget" here is a non-mathematical (and indeed, a hypothetical) example that Bejarano uses as an Anticipatory Set. As we've seen throughout Chapter 2, many texts use non-mathematical examples to motivate the students.

2. I projected the “perpendicular lines” examples and non examples. We completed a frayer model for the term.

Now actually, "perpendicular lines" isn't included in Section 3-2 of the U of Chicago text. But I decided that I'll include it in this section anyway. This is because my plan is to skip the second half of chapter 3 and move directly to Chapter 4 on reflections. This way, I can use the results of reflections to prove the results that appear late in Chapter 3 (as we should in a Common Core class). But "perpendicular lines" isn't defined until Section 3-5 and "perpendicular bisector" not until Section 3-6 -- and of course reflections are defined in terms of perpendicular bisector. In other words, I can, and must, define the terms in Sections 3-5 and 3-6 now, but can't prove the results in those sections until later. The solution is to define both "perpendicular lines" and "perpendicular bisector" today. Oh, and speaking of "perpendicular lines," here's yet another relevant video from Square One TV:

3. Students worked in small groups with their 3 terms copying the examples & non-examples, then writing good definitions for each term. I set a timer for 10 minutes.

Now I don't necessarily want to include this as a group activity the way Bejarano does here. After all, I included the Daffynition Game in my last post as a group project (which comes from the very same section of Serra that she found the widget example), and I don't wish to have two consecutive assignments that require groups. Teachers who wish to retain the group structure should get the remaining steps for doing so from Bejarano's site directly (at the link above).

Finally, I notice that Bejarano color-coded the Frayer models. I haven't figured out her color code -- for example, why are "acute angle" and "obtuse angle" green, yet "right angle" is red? It could be that the colors are simply to divide the definitions into threes for the groups, with no guarantee that words in the same color are necessarily related at all.

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