Authentic Lessons for 21st Century Learning

My Unit Is a Circle

The Unit Circle / Trigonometry

K20 Center, Alexandra Parsons | Published: September 18th, 2020 by K20 Center

  • Grade Level Grade Level 11th, 12th
  • Subject Subject Mathematics
  • Course Course Precalculus
  • Time Frame Time Frame 1-2 class period(s)
  • Duration More 90 minutes

Summary

This lesson is intended to introduce the unit circle, leading into the concept of radians as a unit rather than degrees. This lesson is intended to introduce the concept at the beginning of the unit, but it could also work as remediation if students are just not getting it.

Essential Question(s)

How are units connected but also separate?

Snapshot

Engage

Students will brainstorm prior knowledge related to circles and triangles.

Explore

Students will investigate the relationship between the radius and circumference of a circle.

Explain

Students will determine the relationship between pi and circumference.

Extend

Students will insert special right triangles into the circle and determine the relationships.

Evaluate

Students will reflect on their learning and feelings of learning.

Materials

  • Rulers

  • Protractors

  • String

  • Scissors

  • Colored pens or pencils

  • Copies of the Explore Circle handout (attached; one per student)

  • Copies of the Circle Knowledge handout (attached; one per student)

Engage

Put students into pairs (using the Elbow Partners strategy is super easy and quick, but any grouping strategy will do) and have them get a fresh piece of paper wherever they take notes.

Set a timer for 5 minutes, and tell students they need to Tell Me Everything about:

  • Circles

  • Special Right Triangles

Explore

Pass out a copy of the Explore Circle handout to each student. Students also need a ruler, string, scissors, and a protractor.

  • Tell students to carefully determine the exact center of the circle and draw a straight line representing the diameter.

  • Once the center and the diameter are denoted, tell students to cut a piece of string the length of the radius.

  • Starting from one point of the diameter, tell students to lay their string around the circumference of the circle, making a small mark where it ends. Have them repeat under they have gone around the entire circle.

At this point, the center has been determined, the diameter drawn, and the string has been cut to match the length of the radius. This images shows the first radius around the circumference mark.
This is what the circle should look like at the end, with six marks (seven including the starting point zero).

Explain

Once students finish marking their circle, pass out a copy of the Circle Knowledge handout to each student. Allow them time to answer the questions based on their marked circles from the Explore activity.

When they have finished answering, guide them through an Inverted Pyramid. Have them compare and edit their answers with a partner, and then have partners form groups of four, repeat the share and edit, and then determine a whole class consensus by having groups share out and see if all other groups have the same answers.

Extend

Once the whole group discussion is over, have students draw 30/60/90 and 45/45/90 triangles in each quadrant around the circle.

This is an example of the first quadrant done.
The finished circle.

Once students are done drawing their triangles, have them answer the questions of part two on their handout. If you would like, repeat the Inverted Pyramid strategy for these questions.

Evaluate

Have students (preferably in an interactive notebook, but wherever students take notes) complete a How am I Feeling/What am I Thinking? reflection. In this reflection, they will journal about their feelings about their learning and what they're thinking either about the content today or about how this content will be used in the future.

Resources