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# Work Period: Extending the Definitions of Trig Ratios to the Unit Circle

Lesson 5 of 9

## Objective: SWBAT determine when the values of trig ratios are the same for two different central angles, by using reference angles and the unit circle.

## Big Idea: When class opens and closes with the same formative task, students have a clear idea of what to look for as they're practicing today.

*75 minutes*

#### Opener

*5 min*

There is just one question on this Check In Quiz, and I give students approximately five minutes to complete it. Students are not allowed to use calculators, and I also make it clear that all should be working on Side 1 only. As students work, I circulate and explain that they may know how to do this already, or they may still be developing their understanding. Either way, I direct their attention to the top of the page and ask them to notice that I'll assess them on **Mathematical Habit 1: I can make sense of problems and persevere in solving them (MP1)**, so whatever they know, they should try to demonstrate this habit on paper.

I collect these papers after about 5 minutes, and in a few minutes when students get into their individual work time, I'll flip through their work to get specific ideas about which students and which ideas I'll have to focus on today. This is a quick and easy, in-class **formative assessment** that provides me with immediate feedback.

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#### Practice With a Purpose

*60 min*

The opening quiz was just beyond what most students know so far, and I have done this purposefully as a way to motivate today's class work. In a brief full-class discussion, I try to elicit these two big ideas from students by asking what we need know in order to determine if trig ratios are equivalent for two angles.

I make sure to express what students are already thinking. With a calculator, this task would be easy: you could just calculate the sine of each of these four angles, and see which two are the same. And if I wanted to know the actual value of sin(55), of course that would require a calculator. "There is no need to memorize the values of *all* the trig ratios," I add. What the quiz prompt was getting at is that you can know if two trig ratios are *equivalent* without knowing their *exact value*.

**Delta Math - Practice Exercises**

With the two big ideas in mind, I show students what they'll practice today. **See today's narrative video** for details on what I show them. Also, here are screenshots of the modules I've used in each set of exercises.

I have yet to explicitly teach a definition for tangent on the unit circle, but the trig ratio exercises will have students working with tangent. I find that occasionally allowing students to first grapple with a sub-topic like this on Delta Math is a good strategy (though not to be overused). While they're working, I circulate and explain tangent in a few different ways to small groups of students, and I try to lay some groundwork for what we will more explicitly study in the next lesson.

One more note: the "Signs of Trig Functions" module uses inequalities to get students thinking about whether a trig function is positive or negative. I've found that a really nice **side lecture is to develop a diagram like this** to help students get a real feel for what's going on (see Work Period: Extending the Definitions of Trig Ratios for more) .

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#### Closing

*10 min*

With about 10 minutes left in class, I tell students to finish the problem they're working on, to shut down their computers, and return them to the laptop cart. "When all computers and calculators at your table are returned," I say, "I will return your Check In Quizzes to your table so you give it another go."

Students know this is coming. I show them that there's one key difference here: although I am assessing the same content target on both sides of this page, I am looking for a different mathematical practice. On Side 2, the Mathemical Habit is #6: **I attend to precision (MP6)**. I tell students that no matter what they were able to do on Side 1, I want them to use what they learned today to be more precise in their explanation of the work. How can they draw the most simple and precise diagram, based on what they've done today?

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- UNIT 1: Statistics: Data in One Variable
- UNIT 2: Statistics: Bivariate Data
- UNIT 3: Statistics: Modeling With Exponential Functions
- UNIT 4: Statistics: Using Probability to Make Decisions
- UNIT 5: Trigonometry: Triangles
- UNIT 6: Trigonometry: Circles
- UNIT 7: Trigonometry: The Unit Circle
- UNIT 8: Trigonometry: Periodic Functions

- LESSON 1: What's so Great About the Unit Circle?
- LESSON 2: The Unit Radius and the Unit Hypotenuse
- LESSON 3: Using Right Triangles to Make the Unit Circle
- LESSON 4: Sine and Cosine on the Unit Circle
- LESSON 5: Work Period: Extending the Definitions of Trig Ratios to the Unit Circle
- LESSON 6: Unit Circle Problem Solving, Synthesis, and Discovery
- LESSON 7: Tangent on the Unit Circle and Relationships between Trig Functions
- LESSON 8: Work Period: Unit 3 Review
- LESSON 9: Unit 3 Exam