Lesson 4

The Unit Circle (Part 2)

Lesson Narrative

The goal of this lesson is for students to identify radian angles and the coordinates associated with those angles on the unit circle. In doing so, this draws special attention to the symmetry of the \(x\)- and \(y\)-coordinates of points on the unit circle. This work also calls back to the repeating nature of periodic functions highlighted at the start of the unit and will be built on when students study cosine, sine, and tangent as functions in future lessons.

Students start the lesson noticing and wondering about a specific set of angles around the unit circle. Next, they identify and label a unit circle with 24 angles each \(\frac{\pi}{12}\) apart from just a few starting angles. Students are given freedom to identify all 24 angles in a variety of ways, such as by taking advantage of symmetry, using an index card to find right angles, using patty paper to copy angles, or by folding. In the following activity, students shift their thinking from angles to the \((x,y)\) coordinates associated with those angles. The work here also leverages the symmetry inherent in the unit circle and students can take advantage of repeated reasoning as they label the coordinates of all 24 points (MP8).

During the Lesson Synthesis, students create visual displays of the unit circle. Post these in the classroom for student reference throughout the remaining lessons of the unit. In later lessons, students use the work here to extend their understanding of the domain of cosine, sine, and tangent to all real numbers.

Learning Goals

Teacher Facing

  • Describe the structure of angles expressed in terms of radians on the unit circle.
  • Describe the structure of the coordinates associated with specific radian measurements on the unit circle.

Student Facing

  • Let’s look at angles and points on the unit circle.

Required Preparation

At all times in this lesson, students should ideally have access to index cards, tracing paper, straightedges, and scientific calculators which they may wish to use in approaching tasks which are open to multiple approaches.

An optional blackline master of a unit circle is included for use in the activity “Angles Everywhere.”

Provide access to tools for creating a visual display of the unit circle during the Lesson Synthesis.

Learning Targets

Student Facing

  • I can find different angles on the unit circle and estimate their coordinates.

CCSS Standards

Building On


Building Towards

Glossary Entries

  • period

    The length of an interval at which a periodic function repeats. A function \(f\) has a period, \(p\), if \(f(x+p) = f(x)\) for all inputs \(x\).

  • periodic function

    A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).

  • unit circle

    The circle in the coordinate plane with radius 1 and center the origin.