This section begins with students getting a copy of Defining Radian Measure. The first part of the handout asks students to recall the definition of similarity in terms of transformations, one of the big ideas of the course.
I give the students a minute to think about how to fill in the blanks, talk to their neighbors, and even look at their notes if they want to. After that, I call on students randomly to share their responses and I provide feedback and clarification as needed to move us toward the actual definition.
Next, I'll have a student read the out loud to the completion of the second to last paragraph about postulates. Then I'll orchestrate a pair-share in which I ask students why the two postulates are self-evident...i.e., why we should be ok accepting them without proof.
After that, I'll have students read the last paragraph on the first page to themselves at least twice so that they can understand our plan for proving all circles are similar. Then I'll ask them to discuss with their partner what the proof plan is. Finally, I'll ask a few students to share their understanding of the proof plan with the class.
Next, I'll direct students to look at page 2 where they'll need to use the word bank to fill in the blanks. First, I want students to get an idea of how they think the blanks should be filled. Then I'll want them to discuss with their partners until they agree on what should go in the blanks. When this has happened, I show the correct way to fill in the blanks as I summarize the logic of our proof.
Then it's time for students to process what they've talked about and heard when they respond to the prompt at the end of page 2 of the handout. I give about 5 minutes for students to complete their responses.
In this section, things begin with me calling on students randomly to read a chunk of text at a time from page 3 of Defining Radian Measure. As we're reading page three together, one goal is for students to read for understanding, but it's really a time for me to get some concepts into the air and to do some direct instruction. So as students read, I will pause the action to elaborate on what has been said and clarify things that may be confusing.
For example, the part about setting up a proportion then writing an equivalent proportion would probably go in one ear and out the other without me slowing things down and making sure students understand what is happening and why it's significant. So when we read that part, I stop and write the proportion on the board and explain why its valid based on properties of similarity. Then I recall the properties of proportions that allow us to rewrite the proportion. Finally, I'll emphasize why the second version of the proportion tells us something that the first version didn't, namely that the ratio of the arc length to the radius is constant.
Next I'll talk about what a radian is and why it's called a radian (i.e., the connection to radius). My goal is to give students enough input that they feel comfortable writing about what a radian is, because that will be concluding activity of this section. I give them 5 minutes to write.
This section ramps of the depth of knowledge level as student are asked to apply what they've learned in the previous section about the meaning of radian measure. Page 4 of the handout starts with two fill in the blank items. It's important that students understand these items before moving forward, so I'll ask them to fill in the blanks on their own at first, and then compare with a partner then make sure that they come to an agreement on what should fill in the blanks.
Next, I'll discuss what should go in the blanks and why. I do this just in case there are any stragglers from the previous section who still don't quite understand the meaning of radians.
Once we've taken care of that piece it's time for students to work with their partners explaining how they'd find the radian measure of angles using string, ruler, and calculator. I walk around listening to the conversations, challenging misconception or engineering dialogue as needed. One thing, I tend to advise students about is pointing to the diagram as their talking to their partner and labeling the diagram so that they can refer to points by name rather than just saying "right there".
Next, I'll have students quiet down and write about what they discussed with their partner. Before having them write, I notify them that I will be calling students at random to the document camera show and read what they wrote so (I tell them) make sure it's legible and complete.
When the students are done writing, I'll call a few students up to read just to make sure that this small sample, at least, is thinking correctly. Otherwise, I'll need to spend some more time on this concept.
In this section, students will be working on page 5 of the handout. Basically, they'll have to use arc lengths and radius lengths to calculate radian measure. In some cases these are given, and in other cases they have to do some problem solving to find what they're looking for. At this point in the class, I allow things to settle into more of a workshop atmosphere in which students are working independently and collaboratively at their own pace as the situation dictates. I walk around checking students' process and listening to them tell me that there's not enough information. To this claim, I usually reply, "What extra information would you need?"... and when they tell me, I ask "Are you sure there's no other way to get that information based on what is given?"
When the time has elapsed I'll model the first couple of problems so that students see what I expect from a presenter. Then I'll ask for volunteers to come up and present other problems. One way or the other, I'll make sure that all problems are demonstrated correctly before student go on the the final activity of this lesson, the final reflection.
I give students 5-7 minutes to write their final reflections.