Tuesday, September 22, 2015

Lesson 3-2: Types of Angles (Day 19)

Today we finally reach angles. With the changes I made to yesterday's lesson, today's lesson has been modified. Today I will cover Lesson 3-1, "Angles and Their Measures," and the first part of Lesson 3-2, "Types of Angles." As it turns out, splitting Lesson 3-2 makes perfect sense in light of the McDougal Littell text -- the text that caused me to modify the lesson in the first place. After all, McDougal Litttell's Lesson 1.4 contains all of the U of Chicago's 3-1 (degree measures) plus part of 3-2 (acute, right, obtuse angles), while the rest of 3-2 (the linear pair definition) plus 3-3 (that linear pairs are supplementary) appear in 1.5.

I appeal to the McDougal Littell text to justify squeezing three lessons into two. But of course, the lessons that I post here don't correspond exactly to those in that text. Basically, all I have to do is add acute, right, obtuse, straight, and angle bisector to 3-1 to obtain 1.4. In 1.5, McDougal Littell states that linear pairs are supplementary as well as gives a definition of vertical angles, but doesn't prove that vertical angles are congruent -- even though this appears in 3-3. The difference is that the U of Chicago's 3-3 appears after the logic Chapter 2, and so 3-3 can build on that logic. But McDougal Littell's 1.5 appears before the logic of its respective Chapter 2, and so the Vertical Angles Theorem can't be proved in that text until Chapter 2.

I decided to repost my entire worksheet for 3-1 plus one side of the worksheet for 3-2. Last year, I came up with using Frayer models to teach the various types of lessons. But once again, the way I divide 3-2 in half doesn't correspond exactly to McDougal Littell's 1.4-1.5 split. Four vocabulary terms (complementary, supplementary, adjacent, linear pair) that are in today's lesson are 1.5 terms, while tomorrow's angle bisector is a 1.4 term.

Furthermore, students will learn what a linear pair is today, but won't learn that the angles forming such a linear pair are supplementary until tomorrow. Then again, McDougal Littell students learn what vertical angles in Chapter 1, but not that they're congruent until Chapter 2. Since I believe that angles really do belong in Chapter 1, it's actually normal that students learn about various types of angles in the first chapter but not about their properties until the second.

I could have edited today's lesson, but I decided not to. I only edit lessons when I reorder them so wildly that students are asked to "review" things they haven't learn yet -- like the time they would've had to answer questions about parallelograms when they've hardly learned about parallel lines.

This is what I wrote last year about today's Lesson 3-2. This includes where I originally got the idea of using Frayer models, as well as the modifications I made to today's lesson:

Lesson 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 Lesson 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 Lesson 2.3, which corresponds roughly to Lesson 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 Lesson 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 Lesson 3-2 of the U of Chicago text -- and indeed, this year I'm waiting until tomorrow to post it, as part of the domino effect of my changing the lesson from yesterday. In last year's post I included a YouTube video from Square One TV on perpendicular lines -- but instead I'll post the "Angle Dance" video instead, as I had intended to post this yesterday:

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 last week 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 assignments in consecutive weeks 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.

I notice today that Lisa Bejarano is still an active math teacher blogger. Last week, she updated another one of her posts last year about how she taught constructions better this year. She had the students play "Euclid, the Game," and I recognize the post of Euclid's First Theorem -- yes, the equilateral triangle construction that appears in Lesson 4-4 of the U of Chicago text.

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