What's so Great About the Unit Circle?

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SWBAT understand why the radian measure of an angle is equivalent to the length of an arc on the unit circle.

Big Idea

The Unit Circle is so great because it makes our lives so easy! For example: on the unit circle, radians and arc length are the same thing.

Opener - Unit Shapes Warm-Up Problems & Learning Target

10 minutes

This lesson is a pretty direct interpretation of the Common Core Standard F-TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.  When I introduce a new learning target, I like to start by going through the vocabulary in the learning target, and because the word “unit” is so important, that’s where we begin. 

Teacher's note: Readers may want to review my Prezi before reading on.

In today’s opener, students create a set of what I informally call Unit_Shapes.  I recognize that the “unit equilateral triangle” does not have the same lore attached to it as the unit circle, but that’s not the point here.  My goal is just to really get students thinking about the word “unit”.  I am trying to establish the unit circle as a common-sense mathematical tool (MP5), and in order to do that, I first want students to feel very comfortable with the reasoning behind its naming.

I give students a minute or two to set up their paper by making four sections and drawing the shape indicated in each section.  Next I show them the problem for each slide.  I move quickly here, allowing no more than a minute of individual time on each problem before leading a brief class discussion.  I make the point to students that this should feel like this a review of special triangles, so we’re not going to spend too long on it right now.  

Over the next few classes, students will make a few documents that will come together to help them make sense of the unit circle.  “Unit Shapes” is the first one.  See today’s Explaining the Math Resource for a few extra notes on this opener.

After students find the circumference and area of the unit circle - a task whose satisfying simplicity never fails to capture their attention - I pose the title question of today’s lesson for the first time: “What’s so great about the Unit Circle?”  I can count on several of them to shout that it “makes things easy,” and this is what I’m going for.  I will ask this question several times today, in an effort to build excitement about the unit circle, and also to make sense (MP1) of this important mathematical structure.  Because we’re going to get back to the unit circle, I make sure to leave diagram #4 in a prominent place on the chalkboard.

Learning Target Review

Next we look at the learning target.  I put it up on the screen and just ask kids to read it to themselves.  Then I ask if anyone has any questions about any of the words in this SLT.  Because we just spent the first few minutes of class talking about how great and easy the unit circle is, they’re feeling comfortable with that.  Compared to other learning targets we’ve seen, the language here is straightforward and familiar.  It’s the learning target’s claim that radian measure and arc length are equivalent on the unit circle that should build on our budding excitement about this new structure.  

I ask students what this learning target is really saying.  (I don’t tell them this now, but at the end of class, they’ll be asked to put this in their own words.)  They catch on to the idea that on the unit circle, arc length and radian measure are the same thing.  It makes sense to them that this is a possibility, because they already have it in mind that the unit circle is so great.  Some ask why this is true, and I say that that’s what we’re going to look at today.

Review of Arc Length Formula & Practice Exercises

10 minutes

Because today’s new learning target is about arc length we’re going to spend a few minutes reviewing that.  Together, we look the the formula_for_arc_length, and I ask students what this formula means.  I refer back to SLT Circles 3 (CCS G-C.B.5), which defines radian measure as a constant of proportionality between radius and arc length, and I point out that S = r*theta and y=kx certainly have a similar structure (MP7).  I’m hoping that students remember that theta must be given in radians for this formula to work.  To check for their understanding, I point to theta and ask, “could I just plug in 60 degrees here?”  If they’re not sure, I don’t dwell on it - I just show them the notes.  The example exercises will give them a chance to make sure they’ve got it.

Practice Exercises

There are six practice problems on today’s Prezi.  The first three have a radius greater than 1, and for the last three r=1.  On four of the six problems, theta is given in radians, on two, it’s in degrees.  I’m really trying to get students to make connections between a few ideas at once, and to be flexible thinkers.  I’m also, once again, trying to help students see that the unit circle is great because it makes things easy.  Those last three exercises do the trick.  We put these up on the board, and for the second time this class period, students are excited to see how simple things can get when r=1.  They’re buying into the unit circle thing, and I’m building capital for when we dig a little deeper into the topic.

Return Unit 2 exams

10 minutes

I return the exams that students took in the previous class, and I give them some time to talk about it.  I do not worry too much about structure here.  There’s a culture in my class in which students want to have a little time to discuss their exams with each other, so I allow for that.  I also make a firm point that, no matter what their scores on this exam, they shouldn’t worry about this.  We’re going to keep digging into all these ideas.  Even “summative” assessments aren’t really summative in my implementation of mastery-based grading, because there are always more chances to demonstrate mastery on each learning target.  

As I often tell them, I say, “The best way to improve your grade is do your best on what we’re doing TODAY!”  The next activity we’ll do is directly related to the third learning target from Unit 2, so the evidence that I’m not just blowing smoke should be pretty clear to kids. 

The Unit Circle Figure 1

30 minutes

Homework and Exit Slip

15 minutes

Problem Set 12 consists of two problems that will get kids thinking about 45/45/90 triangles, two that require them to apply their knowledge of arc length, and three problems in which they start to extend their knowledge of sine and cosine beyond right triangles.  

The Figure #1 handout serves as today’s exit slip.  I tell students that it’s ok if they’re not done yet, but I’m going to collect their work on Figure #1.  (I emphasize that this is not for a grade, just to check in.)  “Before the bell rings,” I say, “take a few minutes to give Figure #1 a title, then fill in the SLT and answer the question ‘What’s happening here?’ at the bottom of the page.”  I clarify by saying it’s ok if the learning target is not perfect, word for word.  In fact, I would prefer students to write it in their own words and tell me what they think they were learning about today.  I explain that they can answer the question, “What’s happening here?” however they wish.

After the lesson, as I read through the titles, SLTs, and thoughts about what’s happening, I get great insight into what kids are thinking about the work.  There are very literal interpretations of what’s happening: “we are making a circle going up by 15 degrees and searching for radian measure,” there are attempts to talk about patterns: “every 90 degrees the denominator repeats itself forwards and in reverse,” and there are various rewordings of the learning target.  The titles are fun to read, as the span the spectrum from literal to silly.  What I like about this exit task is that it is a rich formative assessment that needs no grade.  Students have fun with the title, they try to impress me with the SLT, and they reveal their thinking by answering the question.