The Unit Circle (Lesson 9.6)
Unit 1: Sequences and Linear Functions
Day 1: Recursive Sequences
Day 2: Applications of Arithmetic Sequences
Day 3: Sum of an Arithmetic Sequence
Day 4: Applications of Geometric Sequences
Day 5: Sequences Review
Day 6: Quiz 1.1 to 1.4
Day 7: Linear Relationships
Day 8: Point-Slope Form of a Line
Day 9: Standard Form of a Linear Equation
Day 10: Quiz 1.5 to 1.7
Day 11: Unit 1 Review
Day 12: Unit 1 Test
Unit 2: Linear Systems
Day 1: Linear Systems
Day 2: Number of Solutions
Day 3: Elimination
Day 4: Larger Systems of Equations
Day 5: Quiz 2.1 to 2.4
Day 6: Systems of Inequalities
Day 7: Optimization Using Systems of Inequalities
Day 8: Quiz 2.5 to 2.6
Day 9: Unit 2 Review
Day 10: Unit 2 Test
Unit 3: Function Families and Transformations
Day 1: Interpreting Graphs
Day 2: What is a function?
Day 3: Translating Functions
Day 4: Quiz 3.1 to 3.3
Day 5: Quadratic Functions and Translations
Day 6: Square Root Functions and Reflections
Day 7: Absolute Value Functions and Dilations
Day 8: Equations of Circles
Day 9: Quiz 3.4 to 3.7
Day 10: Unit 3 Review
Day 11: Unit 3 Test
Unit 4: Working with Functions
Day 1: Using Multiple Strategies to Solve Equations
Day 2: Solving Equations
Day 3: Solving Nonlinear Systems
Day 4: Quiz 4.1 to 4.3
Day 5: Combining Functions
Day 6: Composition of Functions
Day 7: Inverse Relationships
Day 8: Graphs of Inverses
Day 9: Quiz 4.4 to 4.7
Day 10: Unit 4 Review
Day 11: Unit 4 Test
Unit 5: Exponential Functions and Logarithms
Day 1: Writing Exponential Functions
Day 2: Graphs of Exponential Functions
Day 3: Applications of Exponential Functions
Day 4: Quiz 5.1 to 5.3
Day 5: Building Exponential Models
Day 6: Logarithms
Day 7: Graphs of Logarithmic Functions
Day 8: Quiz 5.4 to 5.6
Day 9: Unit 5 Review
Day 10: Unit 5 Test
Unit 6: Quadratics
Day 1: Forms of Quadratic Equations
Day 2: Writing Equations for Quadratic Functions
Day 3: Factoring Quadratics
Day 4: Factoring Quadratics. Part 2.
Day 5: Solving Using the Zero Product Property
Day 6: Quiz 6.1 to 6.4
Day 7: Completing the Square
Day 8: Completing the Square for Circles
Day 9: Quadratic Formula
Day 10: Complex Numbers
Day 11: The Discriminant and Types of Solutions
Day 12: Quiz 6.5 to 6.9
Day 13: Unit 6 Review
Day 14: Unit 6 Test
Unit 7: Higher Degree Functions
Day 1: What is a Polynomial?
Day 2: Forms of Polynomial Equations
Day 3: Polynomial Function Behavior
Day 4: Repeating Zeros
Day 5: Quiz 7.1 to 7.4
Day 6: Multiplying and Dividing Polynomials
Day 7: Factoring Polynomials
Day 8: Solving Polynomials
Day 9: Quiz 7.5 to 7.7
Day 10: Unit 7 Review
Day 11: Unit 7 Test
Unit 8: Rational Functions
Day 1: Intro to Rational Functions
Day 2: Graphs of Rational Functions
Day 3: Key Features of Graphs of Rational Functions
Day 4: Quiz 8.1 to 8.3
Day 5: Adding and Subtracting Rational Functions
Day 6: Multiplying and Dividing Rational Functions
Day 7: Solving Rational Functions
Day 8: Quiz 8.4 to 8.6
Day 9: Unit 8 Review
Day 10: Unit 8 Test
Unit 9: Trigonometry
Day 1: Right Triangle Trigonometry
Day 2: Solving for Missing Sides Using Trig Ratios
Day 3: Inverse Trig Functions for Missing Angles
Day 4: Quiz 9.1 to 9.3
Day 5: Special Right Triangles
Day 6: Angles on the Coordinate Plane
Day 7: The Unit Circle
Day 8: Quiz 9.4 to 9.6
Day 9: Radians
Day 10: Radians and the Unit Circle
Day 11: Arc Length and Area of a Sector
Day 12: Quiz 9.7 to 9.9
Day 13: Unit 9 Review
Day 14: Unit 9 Test
Learning Targets
Use special right triangles to find coordinates on a unit circle.
Understand that on a unit circle, cos = x-coordinate and sin = y-coordinate.
Evaluate sin, cos, tan functions using the unit circle.
Tasks/Activity | Time |
---|---|
Activity | 25 minutes |
Debrief Activity with Margin Notes | 10 minutes |
QuickNotes | 5 minutes |
Check Your Understanding | 10 minutes |
Activity: The Unit Circle
Lesson Handouts
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Answer Key
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Homework
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Experience First
This is one of my all-time favorite lessons. Not because it’s super exciting and engaging and students think algebra is so fun, but because it connects right triangle trig to unit circle trig in such a simple way. I don’t know if you know this or not, BUT THE UNIT CIRCLE IS JUST A WHOLE BUNCH OF SPECIAL RIGHT TRIANGLES!!! This truly blew my mind the first time I realized this. I know, some of you are laughing at me because you knew that all along. But I know some of you just had your mind blown too! If we can get students to think of the unit circle with special right triangles instead of memorizing a whole bunch of crazy gimmicks, I can retire now because I will have nothing greater to accomplish in my teaching career. Alright, so in order for students to get the most out of this lesson, you’ve got to set it up right. Each student needs this unit circle and this set of triangles. It’s important that you use these ones because the hypotenuse of the triangles is equal to the radius of the circle. Students will start out the lesson by finding sides lengths for a 30-60-90 triangle and 45-45-90 triangle that both have a hypotenuse of 1. Once they’ve done that, they should cut out the set of triangles that you give them and label all sides and all angles on both sides of the paper. Students can then use these triangles to fill in the coordinates for all of the points marked on the unit circle by fitting the angles of the triangle into the reference angle of the circle and then using the side lengths of the triangle to determine the x and y coordinates. I would recommend modeling one of the coordinates using a Think Aloud. As students are completing their circles, walk around and check their work. Once a student has been checked, they can help check other students too. After the unit circle is complete, they can start investigating the relationship between sine and cosine and the coordinates. You could again model the first one. This is a long activity. Do your best to keep students moving. Guiding Questions
this unit circle and this set of triangles
Formalize Later
The majority of the formalization comes when investigating the relationship between the trig functions and the coordinates and in the QuickNotes. The best thing you can do to help students formalize their understanding is to check in with them often. Ask them to explain their thinking. When they’re stuck, use a Think Aloud to model the thought process they should be having. Consider having a practice day after this lesson, in which students find trig ratios for a variety of different angles. We don’t recommend having students race to see who can fill in the full unit circle the fastest.