Using Right Triangles to Make the Unit Circle
Lesson 3 of 9
Objective: SWBAT elaborate on the connections between right triangles and the equation of a circle, laying groundwork for the extension of the definitions of trig ratios to the unit circle on the coordinate plane.
The previous class ended with an exit slip. In reading those exit slips, I saw how many students were able to write the equation of the unit circle. Enough of them need review that I’ll start with that today. Another question on that exit slip was about connections between the Pythagorean Theorem and the equation of a circle on the coordinate plane. At this point, most students understand that these have the same structure. I want to develop their ability to use these connections to develop new ideas: looking for and making use of structure is Mathematical Practice #7; seeing connections between structures is key to making sense of problems (MP1), abstract reasoning (MP2), and modeling with mathematics (MP4).
Today’s opener consists of four problems that will pick where the exit slip left off and that will set the stage for what we’re doing today, which is to begin to see how the definitions of the trig functions can be extended using the unit circle (CCS F-TF.A.2). Please take a look at the first eight slides of today’s Prezi to get an idea of how these problems will guide the lesson. I give students a few minutes to consider the four problems for themselves, before moving pretty quickly through a class discussion of each problem. The purpose of these problems is as much to guide students through some notes at the start of class as to engage them in rigorous problem solving.
With the first problem, I make sure to explicitly state the equation of the unit circle for anyone who didn’t know it, and to get students to say why the equation is what it is. The second problem is an example of a “unit right triangle” that students were working with in the previous class, and I direct the class conversation toward seeing how quickly this problem can be done. Part of understanding the unit circle is seeing the number 1 as a tool, and using it strategically (MP5): students must recognize that when a right triangle has a hypotenuse with length 1, the task of finding its side lengths is simple. This understanding is prerequisite to a strong grasp of the structure (MP7) of the unit circle. Students will have a little time to finish the Unit Right Triangles handout during today’s work time.
The third question is about lattice points on the unit circle. Most students are comfortable with the idea that there are four, and with the idea that all other points will have coordinate values between -1 and 1. As an extension question, I ask if anyone can think of any rational numbers that fit into the Pythagorean Theorem if c = 1. This is the sort of problem that I just leave out there for now; we’ll return to it in an upcoming class.
Impossible Right Triangles
Finally, the most fun problem of the opener is #4, in which students are asked to sketch a right triangle with c = 1 and another angle measuring 120 degrees. I act very nonchalant when I put this problem up, and I read it in my most serious, straightforward voice. I’m looking at faces to see who is confused by this. I give students a little time to mull it over. There are two ways I have seen this go. The one I hope for is that students will just take over the discussion here, and someone will try to sketch this triangle on the board, and there will be some spirited argument about whether or not this can really work. The other possibility is that students will just be too confused to know what do, and they will sit there assuming it’s their fault that this is confusing. If I see the latter happening, I’ll be quick to jump in and say, “if you’re confused right now, then that’s because you’re really paying attention, and you’re right!” before leading a conversation about why this is impossible.
Either way, it’s fun to see if anyone comes up with the idea of a triangle with the angles 90, 120 and -30. Some classes get there on their own, others need a little guidance. When I ask students to see if the relationship cos A = sin B holds for any two angles A and B that add up to 90 degrees, students check it on their calculators and their curiosity spikes.
In some classes, we also talk about the degenerate right triangle, with angles of 0, 90 and 90 degrees. I like to draw a line on the board and emphatically point to its “three sides”: the hypotenuse, the leg with the same length, and the leg with length zero.
Learning Target Review: UC1 and UC2
We look at both learning targets for Unit 3. I briefly review the first one (Unit Circle 1), asking students to identify the key word in this SLT. It’s “why” of course! I tell students that they should be able to explain why radian measure and arc length on the unit circle are the same thing in 140 characters or less (tomorrow’s class opens with a check in quiz along those lines). I show them Figure #1, and remind them that this is a tool that should be complete.
Next I introduce the second learning target for the first time:
- Unit Circle 2: I can use the unit circle to extend the definitions of trigonometric functions to all real numbers.
I put it up on the screen and ask students to identify the key words in this SLT. Students notice that we’re still talking about the unit circle, and that this learning target is once again about the trig ratios. (We notice, with some curiosity, that this SLT actually uses the word “functions” instead of “ratios,” and I tell students to consider this a coming attraction.)
When a students points out the word “extend,” I try to hook the rest of the class on that. I ask what it means to extend something, and we get at the idea of making something bigger, so then I ask why the definitions of the trig ratios would have to be extended. We look back to the right triangles from today’s opener, and how it was possible to make have an acute angle like 15 degrees in a right triangle, but not to have an angle measuring 120 degrees. I ask how we have defined sine and cosine, and I note that as soon as someone says “opposite over hypotenuse,” they’re assuming the use of a right triangle. Students are pretty quick to say that we can’t have an acute angle greater than 89 degrees in a right triangle. I like to note that, technically, you could have an 89.999 degree angle, which gets students thinking a little more precisely. We all agree that the domain of the trigonometric functions has been 0 < x < 90. So that’s why we’re going to need to extend these definitions, and we’re going to use the unit circle to do it.
Students are able to move at their own pace today. Differentiation is built into today’s work time because each task leads into the next, and my role is to provide support wherever students need it. Students must finish their Unit Right Triangles handout (see description in previous lesson) to get the instructions for Figure #2a. When they finish the first part of Figure #2a, they get a laptop and learn about reference angles by completing exercises on Delta Math.
In order to start to extend the definitions of the trigonometric functions, students will plot their “Unit Right Triangles” on the coordinate plane. Figure #2a is where students will plot these triangles. There is a staggered start to their work on this, because I make sure that students have completed the Unit Right Triangles handout before showing them how to get started on Figure #2a.
See the narrative video for this section to see how I introduce this task to students, and how I demonstrate the self-checking nature of this exercise.
Figure #2a follows the same format as Figure #1: it doesn’t yet have a title and it has space at the bottom for students to write the learning target and to answer the question, “What’s happening here?” I’ve titled it Figure #2a because Figure #2b is the same thing, with a unit circle already graphed. I keep a few copies of Figure #2b around as an accommodation for any students who really don’t get the idea that all the unit right triangles will make a circle.
If all (or at least a majority of) students are ready at the same time, you can show them the instructions as a whole class. I almost always end up sitting down next to a student or two and demonstrating the process, then I rely on my early starters to explain the process to their classmates. Think of it as playing telephone with the instructions - this is a great structure because it gives kids a chance to shore up their understanding by explaining what they know to someone else.
When students are done plotting their unit right triangles in Figure #2a, they have will have some time to grab a laptop and work onDelta Math. (Here is a screen shot of the practice modules on today's Delta Math assignment.) Note that at this point, students have only plotted triangles in the first quadrant of Figure #2a. Before we extend these plots to the other three quadrants, I want students to spend some time getting familiar with angles outside the domain of 0 to 90 degrees.
On this assignment, students will work with central angles and reference angles - ideas we haven’t explicitly studied yet. This gives them a chance to develop this knowledge independently before we apply it in the next lesson.
This exit task is repeat of what students did with Figure #1 two classes ago. Figure #2a needs a title in the blank at the top of the page that will help someone understand what's going on here. I also ask students to fill in the bottom of the page by writing the learning target in their own words, and answering the question, "What's happening here?" I'll collect this from students as they leave.
When I debriefed student explanations of what's happening here, I pointed out that a lot of them wrote very literally what they had been doing on Figure #1 (ie, "I'm converting from degrees to radians," or "We're labeling a circle every 15 degrees."), and I asked them to think of this question more as, "What's going on behind the scenes here?" or "What patterns are happening here?" I'll be looking to see if they take their responses to that next level.
Of course, it's good information to know how much each of student has completed on this figure, and I'll look for that, but I'm not grading them. I'm most curious to see their titles, their wording of the SLT, and their description of what's happening here. I'll take some pictures of their responses and use them to help form the lecture notes for the next class (see resulting notes in the Prezi for the next lesson).