The purpose of this Warm-up is to elicit the idea that we can use the unit circle and similarity to determine facts about circles of other sizes, which will be useful when, in later activities, students calculate the coordinates of points on circles with radii other than 1 unit. While students may notice many relationships between these images, the similarity of triangles and and the scale factor of 2 needed to go from the former to the latter are the important discussion points.

Ask students to share the relationships they found between the images. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the images. Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the idea that the second image is a scaled copy or dilation of the first does not come up during the conversation, ask students to discuss this idea. Consider using the following questions as prompts if none of the students notice or wonder anything about dilations:

• “How do the sizes of the circles compare?” (The second circle is double the size of the first.)
• “Are triangles and similar? How do you know?” (Yes, they are similar because they are both right triangles with the same angles, but triangle is double the size of triangle .)
• “How can you use the coordinates of to find the coordinates of ?” (Since triangle has side lengths double that of triangle , point has coordinates double those of point .)

The purpose of this task is for students to use the coordinates of a unit circle to determine information about coordinates on circles of different sizes. Returning to the context of clock hands, this activity focuses on the location of the end of a minute hand relative to the center of the clock.

While there are 3 different clock sizes in this activity, students taking advantage of the similarity between the circles (MP7) can use their solutions for the first clock to identify the solutions for the second clock that is 3 times the size of the first.

Monitor for students who express their answers using:

• Approximate values based on the unit circle display.
• Exact values using  and .

The purpose of this activity is for students to consider the height of points on a circle that is not centered at . Working within the context of a Ferris wheel, students use their understanding of the unit circle to identify the height of a point on the wheel relative to the center of the wheel and then combine this value with the given height of the center of the Ferris wheel to calculate the actual height off the ground. In this way, students think abstractly and quantitatively about the height of a point on a circle that has been shifted (MP2). This work builds on the previous activity and builds toward transformations of trigonometric functions, which will be used in future lessons.

While approximate answers are acceptable and can be easier to think about than an expression with sine in it, encourage students to write out the exact value for the heights using sine. This practice will help in the future when students write and algebraically transform equations for trigonometric functions.