The Unit Radius and the Unit Hypotenuse
Lesson 2 of 9
Objective: SWBAT understand why radian measure is equal to arc length on the unit circle, and to make generalizations about sine and cosine when the length of the hypotenuse is 1.
Finish Figure #1
A few students still need a little time to finish this handout, and if they do, I’ll have a few quiet words with them about urgency and the focus it’s going to take to get the most out of this unit.
I also circulate, and table by table, ask students to think about the arc length formula, S = r * theta, and what happens to this formula when r = 1. This is an explicit and straightforward explanation of why arc length and radian measure are equivalent on the unit circle, and I check to see that students are getting the gist of this idea.
Unit Right Triangles
I distribute this second handout to all students whether they’re ready or not, and I tell them to look for the connections between Figure 1 and Unit Right Triangles. They should be looking for and making use of structure (MP7), building on the relationship between right triangles and circles, and noticing how these structures are expressed within the unit circle. Over the next few days, I tell students that they will synthesize (video - as always, I ask about what this word means, and I explain it a little) the ideas from each of these handouts, so even if it’s not obvious right away how these are related, sleuthing glasses should be kept handy.
This is a pretty straightforward handout that gives students the background they’ll need to understand the behavior of sine and cosine on the unit circle. I don’t tell them any of this, however, because the whole purpose of my approach to introducing the unit circle is to allow students to figure things out for themselves. Just as I did with the opener today, I provide students with this handout, and I wait for them to start making observations and asking questions. The only thing I emphasize is that when they once again see the question, “What’s happening here?” students should try to think deeply about the patterns they see and the generalizations they can make. I want to see how quickly they can look for and express regularity in repeated reasoning (MP8); this task is quite repetitive, and can be done quite quickly, it’s just a matter of students seeing that.
As they work, some students finally notice a relationship between sine and cosine. When they make the connection that sin(x) and cos(90-x) are equivalent, I try to get them to explain why this is happening. I also refer them back to SLT Triangles 2 from our first unit. I show them that if they’re unhappy with their current mastery grade on that SLT, now is a chance to think about what it means, and how they can achieve a higher grade.
Figure #2 - Not everyone gets to this today.
The differentiation in this lesson lies in the fact that everyone can move at their own pace. I can count on around half of my students finishing “Unit Right Triangle” with a lot of time to spare, so I have the handouts ready:
Figure #2 is at the center of my lesson, Sine and Cosine on the Unit Circle. Please refer to that lesson for the narrative that describes how I introduce this handout and the questions that follow.
Unit Right Triangles could be completed within about 5 minutes by someone who knows both a) the definition of sine and cosine on the unit circle and b) understands the connection between the hypotenuse of a right triangle and the radius of a circle on the coordinate plane. These are precisely the ideas I’m trying to get kids to understand by doing this work.
This handout takes a good deal longer than 5 minutes for most of my students for a few reasons.
- They still need to get better at making use of structure and repeated reasoning.
- They think that “show your work” always means “show a lot of work, even if something was easy,” which just isn’t true.
- Many students have the idea that they have to apply the Pythagorean Theorem to solving these Unit Right Triangles, which although not the case, is a delightfully provocative proposition.
Depending on how class is progressing, I call the class to attention with between 10 and 15 minutes left. I have been paying attention to how much work students are showing on the Unit Right Triangles handout. I might select a few students with different approaches to put their work on the board, but I want to try to move students back to the idea that “1” really makes things easy by writing sin(x) = a/1 on the board. I ask them how to solve that equation for a, and it always surprises me how many say that we have to “multiply both sides by 1.” It’s true of course, but now I put on the show: “What’s seven divided by 1?” I say, and I start to pace and raise my voice. “What’s a million divided by 1?” You get the idea. I throw out some nutty 9 digit numbers, before, finally, “So, what’s a divided by 1?” And we see that it’s ok if “show your work” is a simple line or two. After all, the ease of having a 1 in a calculation part of what makes the unit circle so great?
If there’s time and it feels like students will be into it, I might throw in a little preview of the Pythagorean Identity by asking kids to type sin(40)^2 + cos(40)^2 on their calculators. But I don’t say too much about this, I’m just getting the idea out there. We’ll have plenty of time for that in the next few lessons.
I hand out index cards and tell students that I’d like them to answer a few questions on this card. They should write their name at the top, and they should write the date, because that’s a record of when this thinking happened. I ask the following questions, one at a time, and I give students a minute or two to answer each one.
1. If I graph it on the coordinate plane, what is the equation of the unit circle?
2.How is the pythagorean theorem connected to the equation of a circle?
3. Did you notice any new patterns, connections, similarities or equalities today?
This Exit Slip is check-in for me on what students understand; it also helps give students a preview of what’s coming next. Student answers to the first two questions will help me figure out what I have to emphasize as we move on to the next lesson. The third question will help me see the kind of depth students are applying to today’s thinking.