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, 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 attending to structure can take advantage of that symmetry 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.
Lesson overview
 4.1 Warmup: Notice and Wonder: Angles Around the Unit Circle (5 minutes)
 4.2 Activity: Angles Everywhere (15 minutes)

4.3 Activity: Angle Coordinates Galore (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 4.4 Cooldown: What are the Circle Coordinates? (5 minutes)
Learning goals:
 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.
Learning goals (student facing):
 Let’s look at angles and points on the unit circle.
Learning targets (student facing):
 I can find different angles on the unit circle and estimate their coordinates.
Required materials:
 Copies of blackline master, if desired
 Index cards
 Rulers
 Straightedges
 Tools for creating a visual display
 Tracing paper
Required preparation:
 At all times in this lesson students should ideally have access to index cards, 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.
Standards:
 This lesson builds on the standards: CCSS.HSFTF.A.1MS.FTF.1CCSS.HSGSRT.CMO.G.SRT.C
 This lesson builds towards the standard: CCSS.HSFTF.A.2MS.FTF.2
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