Learning Activity

Unit Circle: Exploring Unit Circle Trigonometry With Desmos

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

Desmos is a free online graphing calculator that has several mathematical applications that can be used in the classroom. The Trigonometry: Unit Circle Application is a tool that can be used to study the Unit Circle and the different parts that make up the Unit Circle such as reference angles, radian measure, trigonometric ratios, and angle measurements.

This resource was created as a result of the ALEX Resource Development Summit.

Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.21

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle, building on work with non-right triangle trigonometry.

Unpacked Content

UP:MA19.A2.21

Vocabulary

  • Unit circle
  • Radian measure
  • Quadrantal
  • Traversed

Knowledge

Students know:
  • Trigonometric ratios for right triangles.
  • The appropriate sign for coordinate values in each quadrant of a coordinate graph.

Skills

Students are able to:
  • Accurately find relationships of trigonometric functions for an acute angle of a right triangle to measures within the unit circle.
  • Justify triangle similarity.
  • Find the reference angle for any angle found by a revolution on a ray in the coordinate plane.
  • Relate the trigonometric ratios for the reference angle to those of the original angle.
  • Determine the appropriate sign for trigonometric functions of angles of any given size.

Understanding

Students understand that:
  • Trigonometric functions may be extended to all real numbers from being defined only for acute angles in right triangles by using the unit circle, reflections, and logical reasoning.

Phase

Before/Engage
Learning Objectives

Learning Objectives

Students will use radian measure as a ratio of the arc length to the radius.

Students will be able to accurately find relationships of trigonometric functions for an acute angle of a right triangle to measure within the unit circle.

Students will be able to justify triangle similarity.

Students will be able to find the reference angle for any angle found by a revolution on a ray in the coordinate plane.

Students will be able to relate the trigonometric ratios for the reference angle to those of the original angle.

Students will be able to determine the appropriate sign for trigonometric functions of angles of any given size.

Activity Details

  1. Distribute the Chromebooks/laptops to the class under your normal distribution procedures.
  2. Pass out the Unit Circle with Desmos directions to each student.
  3. Model on the projector/interactive whiteboard/panel display how to get to the website using the directions that you distributed.
  4. Talk about each of the formula bars represented in the Desmos Trigonometry: Unit Circle Application.  
  5. Explain that the “formula 2” is a point. Talk about what the coordinates are.
  6. Ask the students, “Why would this point, (cos a, sin a),  be important to show on the Unit Circle?”
  7. Lead students to talk about the trigonometric ratios and reference angles. Be sure to demonstrate using the online application to model for the students.
  8. Discuss the triangle similarities around the Unit Circle. Be sure to demonstrate using the online application to model for the students.
  9. Talk about the ratio of the arc length to the radius of the circle. Be sure to demonstrate using the online application to model for the students.
  10. Talk about the signs in each quadrant.  Be sure to demonstrate using the online application to model for the students.
  11. Demonstrate the “a” value at different points around the circle.
  12. The teacher would want to give the students just enough knowledge to create curiosity about the subject matter.
  13. Allow the students to explore the activity for 2-3 minutes. Stress to the students that this time is for them to explore what the application demonstrates and come up with ideas of why the various functions work the way that they do.
  14. Go around the room watching the students as they are exploring the application.
  15. After the 2-3 minutes are up, instruct the students to do number 4 on their directions.
  16. Collect student responses on paper.
  17. Go through the answers and select answers that reflect ideas that would build student knowledge, display them on the board, and discuss with the class.
Assessment Strategies

Assessment Strategies

The teacher would assess the students by walking around the room observing what the students are doing with their graphs.

The teacher will also assess student understanding by reading and displaying the student answers to the question given.

Variation Tips

The teacher could assign the Unit Circle with Desmos Directions to the students via Google Classroom or another online learning management system instead of having paper copies. If the teacher chooses to do this, the teacher could pass out index cards for the students to answer the question.

This same activity can be used as an after activity. If doing this, I would suggest creating more questions to answer so that each objective in the lesson is covered.

Background and Preparation

Background / Preparation

  1. The teacher would need to make copies of the Unit Circle with Desmos to hand out to students.
  2. If the teacher has no prior knowledge of Desmos, the teacher would need to gain a working knowledge of the Desmos Graphing Calculator before this lesson.  
  3. The teacher would need to make sure a classroom set of Chromebooks or laptops were available for this lesson.
  4. A timer of some kind will be needed for this lesson.

Digital Tools / Resources

Desmos Trigonometry: Unit Circle Application
https://www.desmos.com/calculator/qmzx2skkzy

Approved Date

Owner (Author)

kristylacks
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