The Unit Radius and the Unit Hypotenuse

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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.

Big Idea

The unit circle provides a rich landscape in which students can find surprising patterns as they have the experience of creating real mathematics.

Opener - Finish up Figure #1 and Exit Slip Notes

15 minutes
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    Full Class Discussion - Debriefing the Work

    10 minutes

    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.

    1. They still need to get better at making use of structure and repeated reasoning. 
    2. They think that “show your work” always means “show a lot of work, even if something was easy,” which just isn’t true.  
    3. 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.

    Exit Slip - Index Card - Verbal Questions

    5 minutes

    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.