Friday, September 29, 2017

Lesson 3-2: Types of Angles (Day 32)

This is what Theoni Pappas writes on page 272 of her Magic of Mathematics:

"Our eyes begin to jump with this puzzle. Determine how many 2 * 1 * 1 blocks make up this structure."

This is the first page of the section "Putting Your Logic to Work." This section contains many puzzles for the reader to solve, the first being a "Rectangular Volume Illusion."

Unfortunately, this puzzle is completely dependent on its picture, which you readers of this blog are unable to see. Indeed, many of these puzzles require pictures. And so I'll only mention some of these questions briefly on the blog.

I do notice that the entire structure fits in a box with dimensions 6 * 6 * 4. So the entire box has a volume of 144 cubic units, and since each block has a volume of two cubic units, exactly 72 blocks fit inside the box. The problem is that some of the blocks are supposed to be missing. The figure is drawn in isometric form, so each block is drawn as three "parallelograms." We distinguish between, say, a 5 * 5 * 5 cube and the same cube with a small 1 * 1 * 1 cube missing by drawing the latter with the parallelograms going "the other way" where the missing unit cube is. This is why Pappas tells us that "our eyes begin to jump" as we try to figure out how many blocks are missing.

Since you can't see the blocks, I might as well skip to the answer. Pappas tells us that there are 61 blocks in the structure. This means that eleven of them are "missing."

Hey, as I'm looking at the answer page anyway, let's give the solutions to yesterday's riddles:

(1) Tom and Jerry are professional baseball players, but play on different teams. This particular day their teams were playing each other. Jerry is a catcher who prevented Tom from reaching home plate.

(2) While walking down the street, Eric had a terrible attack of hiccups. He went into the bar to get a glass of water to see if it would help him. The bartender, realizing Eric's problem, thought he would try to scare the hiccups out of Eric. It worked!

(3) Mary's husband died a number of years ago. She kept his ashes in an urn on her kitchen table. Also on her table was a bowl with goldfish. Mary had left her kitchen window open this particular day. Her neighbor's cat had come in, and in trying to get the goldfish, the cat knocked over the goldfish bowl and the urn. Her husband's ashes were spilled onto the floor.

Lesson 3-2 of the U of Chicago text is called "Types of Angles." In this chapter, students learn about zero, acute, right, obtuse, straight, complementary, supplementary, adjacent, and vertical angles.

In the new Third Edition of the text, this actually corresponds to Lesson 3-3. But the definitions of acute, right, and obtuse are actually combined with yesterday's Lesson 3-1. Only the last four definitions (mainly adjacent and vertical angles) remain in the new Lesson 3-3.

In between these, in the new Lesson 3-2, are rotations. I've mentioned before how strange is this that both the old and new editions define a rotation as the composite of two reflections in intersecting lines, yet the new edition has a section on rotations before defining reflections! The U of Chicago most likely placed this section here so that in introducing rotations, students become more familiar with angles. (Again, I point out that Hung-Hsi Wu of Berkeley, in his recommendations for Common Core Geometry, teaches rotations before reflections, but he defines rotations differently. His lessons have nothing to do with the new Lesson 3-2.)

In fact, Jackie Stone -- the main Blaugust participant last month -- also introduces rotations when teaching her students about angles, just like the U of Chicago text:

What was intended to be a five minute “review” of these skills to launch into the real lesson activity of the day turned into a much more in depth “teaching” of how to use this tool.  Although they might NEVER use a protractor outside of my class again I do find the task of measuring something using a tool useful. The task also spoke to the CCSS Math Practice Standards of attending to precision and using tools strategically. It is so challenging (especially at the beginning of the year) to determine what are appropriate scaffolds to help students work on a task. Moving forward, I plan to assume less which is actually a good thing because then we can talk about refined meanings of things.  For instance, because of their lack of background we were able to really talk about that the measurement in degrees was actually a measurement of a rotation. I think next year my approach might be different.

Again, because I jumbled up the first two lessons of Chapter 3 two years ago, this is what I wrote three years ago on this lesson:

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.

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. [2017 update: yes, I'm skipping her #2 -- dw]

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 [...]

And let's stop right here, because today's an activity day, and I haven't posted that Daffynition game yet this year -- so let me post it today! This is what I wrote three years ago about the activity. (Oh, and if you thought we were done with Serra's text after finishing Chapter 0 last month, think again!):

It's tough trying to find activities that fit this chapter. One source that I like to use for activities is Michael Serra's textbook, Discovering Geometry. Just like most other geometry texts, in Chapter 2 he discusses the concept of definition (Section 2-3, "What is a Widget?") Then the text introduces a project, "the Daffynition Game," where the students take turns making up definitions to real words.

A few comments I'd like to make about the game as introduced by Serra. Step 3 reads:

3. To begin a round, the selector finds a strange new word in the dictionary. It must be a word that nobody in the group knows. (If you know the word, you should say so. The selector should then pick a new word.)

The problem is that this depends on the honor system -- how do we know that a student who knows the word will actually admit it? Rather than depend on the students' honesty, why not make knowing the word an actual strategic move? That student will then earn a point for knowing the word -- and the student can still make up a fake definition in order to earn even more points? This means that the selector must be very careful to choose a word that isn't in the dictionary.

Another question is, what affect would this project have on the English learners? I'd say that this would be a great project for them, since they can learn both English and math in this lesson. But English learners might be at a disadvantage in this game, since if they choose a word that a native English speaker knows, the English speaker would earn a point (since I'm not counting on the honor system here). One debate that always comes up in a group activity is whether to group homogeneously or heterogeneously. For this project, it may be best to group homogeneously, but by English, rather than mathematical, ability.

Finally, this project requires students to look up words in a dictionary -- but what dictionary? I threw out the problems in earlier sections that depended on the availability of a dictionary. Perhaps the night before this activity, part of the homework assignment could be to look up a word in the dictionary and write down its definition -- but that assumes that the students will actually do the homework, and besides, there's no guarantee that the students have access to a dictionary at home (or online) either.

My solution is for the teacher to have enough index cards with words and definitions on them. Therefore the selector chooses an index card, not a word from the dictionary. Indeed, the teacher can give an index card to each student even before dividing the class into groups! But the selector should still follow the other steps as originally written in the Serra text.

OK, so let me post the worksheets. I decided to post only the first page of Lesson 3-2 (Lisa Bejarano's lesson) and then go directly to the Daffynition Game.

In cutting Bejarano's second page, I'm dropping some terms that don't appear until later in Chapter 3, but I also dropped "vertical angles" and "angle bisector," which do appear in Lesson 3-2. Teachers can either make sure to write those two dropped terms on index cards in the Daffynition Game, or else go full Bejarano and use the Frayer models as a full group project, just as the Colorado teacher originally intended.

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