Unit Circle Problem Solving, Synthesis, and Discovery
Lesson 6 of 9
Objective: SWBAT draw their own conclusions about the relationships between trig functions on the unit circle by working on today's problem set.
The purpose of this opener is to help students review some of the basics of unit circle:
Give at least three trigonometric ratios that are equivalent to sin(43π/36).
I say, “How many of you can imagine a 60 degree angle?” Everyone says they can - this is the angle in an equilateral triangle, for one thing. “How about a 90 degree angle?” I ask. Again no problem. I continue by asking about a 140 degree angle, then a 300 degree angle, and we see that even these angles are pretty easy to visualize now that we know about reference angle and central angles (see visualizing degrees vs radians).
After everyone is hyped up and feeling good about their angle instincts, I say, “Alright, so who can imagine a 43π/36 radian angle?” -- and of course, everyone catches my drift. As I’ve done previously, I might now compare the relationship between degrees and radians to the one between feet and meters or pounds and kilograms. “Everyone in this room can tell me their height, I say, but how many of you know your height in meters?” My point is that it’s a matter of our familiarity with a given unit of measurement.
In order help students see the structure (MP7) in the unit circle and to reason abstractly and quantitatively (MP2) as they think about what radians actually represent, I point out that we’re not completely unfamiliar with radians. We know where 0, π, and 2π are on the unit circle. Most students are ok with π/2 and 3π/2 -- so at the very least we can figure out which quadrant this angle will be in. I elicit from students the idea that 43π/36 is greater than 36π/36 but less than 54π/36, so it’s in the third quadrant. 43 is a little closer to 36 than to 54, so that helps us to get a basic visualization for what this angle will look like.
Of course, we can also just convert 43π/36 to 215 degrees, which most students can visualize, and this confirms the picture that we’ve already developed about the radian measure.
Completing the Opening Problem
At this point, it comes pretty naturally that the reference angle for 215 degrees is 35 degrees, and that sin(215) is negative, so we can find some other equivalent trig ratios. I try to get kids thinking about an equivalent cosine as well. As we find equivalent trig ratios, construct a unit circle diagram on the board that has representations of each of our solutions to this problem. This diagram will stay up as we get to work today, and we’ll able to use it for reference.
Return Check In Quiz from Previous Class
As students are getting to work and trying the opener on their own, I circulate and return the quiz. After the opener, I simple state the answer to the question, and I ask students to consider why this answer is what it is. I tell them they’ll have another chance to show me what they know on this learn target.
Problem Set #14
I hand out Problem Set 14, and to introduce it, I go through some brief introductory notes from today's Class_Agenda. The meat of today's class is open-ended work time, but as I explain below and in today's Explaing the Math resource, there are many possible directions that the lesson can take. At its heart, this lesson is an example of how I create a space for improvisation in my classroom by giving students some problems and seeing where they take us, but at the same time, I have some specific outcomes in mind.
This problem set serves two purposes: one as a summative assessment tool for our semester of problem solving, the other as a teaching tool that I’ll use to continue to help students develop their understanding of the content learning targets (Triangles 2 and Unit Circle 2) that are listed at the top of the problem set. (CCS F-TF.2)
Unlike all previous problem sets, which are to be completed as homework and are due a week after they are assigned, this problem set is due at the end of the next class and I give students the majority of two class periods to work on it.
Problem Set #14 as an assessment of problem solving skills
Problem solving has been a major focus of our work this semester, and as this class begins to wind down, I want students to show me how they have developed as problem solvers. Every problem set of the semester has been assessed on Habit 1: I can make sense of problems and persevere in solving them (MP1), and as this is the 14th problem set of the semester, I hope to see that students can both show growth and explain how they are demonstrating mastery of this learning target. Less frequently assessed, but not less important is Habit 7: I look for and can make use of structures and patterns (MP7). With this problem set, I want to see how well students are able to use trigonometric structures: triangles, the unit circle, the trig ratios, and the Pythagorean theorem, and use these structures to develop new understandings of trig concepts.
Side Note: Grading this Problem Set
Grading this problem set will necessarily be rather subjective. Because I know each of my students - where they started the semester as problem solvers, their relationships with mathematics, their ability to struggle - the way I grade each of them will be different depending on where they have come from. Instead of the rubric that most prior problem sets have included, this time I ask students to assess themselves on each of the four learning targets. (Please see page 2 of Problem Set 14, and note that I print this as one double-sided handout.) If a student can give a thoughtful reason why they should earn a 3 or 4 on Habit 1, and I know that they really are making as much sense as they’re able to make, I’m happy to give them those high grades on this assignment.
In general, I think it is important for all teachers to spend time teaching vocabulary and etymology, even for words that are not directly related to the content. The first "problem" is really just a question: "What is 'synthesis'?" This gets students talking immediately. When they ask me for help, I tell them that if they have a mobile device, they should look it up and see what they find. Some students immediately think of "photo-synthesis," so I might lead a conversation about that by eliciting the idea that photo-synthesis is the plant process of taking light and other inputs and turning them into something new. So synthesis is what we're going to do today. We're going to draw on various pieces of knowledge to construct some new knowledge.
In a very tangible way, students will be able to refer to their work from the previous lessons to do the work of synthesis. The Figure #1 and Figure #2 handouts from the classes that began Unit 3 will be useful, and I encourage students to refer to these handouts. I like the opportunity to show students that even though these handouts were not graded, they have value as tools that will help students succeed as they work on this assignment.
Problem Set #14 as a teaching tool for Unit 3 / content SLTs
By working on these problems, students will have the opportunity to better understand the definitions of the trigonometric functions on the unit circle, particularly the definition of tangent, which they haven't yet seen. They will also be able to discover more connections between the trigonometric functions: I am specifically laying the groundwork for the Pythagorean identities and further evidence that tan = sin/cos, but also hope that students are coming up with their own ideas about how the trig functions are related.
I expect students to take two class periods to complete this problem set. For today's lesson, I hope that most students will finish the first six problems. Then they will be able to complete the last two and the reflection in during the next class.
Please see today's Explaining the Math resource for detailed notes about each problem on Problem Set 14.
For today's closing, I give students 5 minutes to write an index card essay in response to the prompt:
How did you synthesize today?