Lesson 7

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

Display the 2 images for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the images. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” 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 \(ABC\) and \(DEF\) similar? How do you know?” (Yes, they are similar because they are both right triangles with the same angles, only triangle \(DEF\) is double the size of triangle \(ABC\).)
• “How can you use the coordinates of \(A\) to find the coordinates of \(D\)?” (Since triangle \(DEF\) has side lengths double that of triangle \(ABC\), point \(D\) has \((x,y)\) coordinates double those of point \(A\).)

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 \(\cos(\theta)\) and \(\sin(\theta)\)

If needed, provide students with copies of separate clock faces.

Select students to share their strategies for identifying the coordinates on the clocks with a minute hand of length 5 inches and 15 inches in the order listed in the Activity Narrative. If no students wrote their answers using exact values, such as\(\left(5\cos\left(\frac \pi 6\right), 5\sin\left(\frac \pi 6\right)\right)\) for the 5 inch minute hand 10 minutes after the hour, ask students to consider how they could write the exact coordinates. After a brief quiet think time, invite students to share their thinking, highlighting any callbacks to their previous work identifying exact values by using cosine and sine without approximating. If needed, here are some questions for discussion:

• “What angle does the minute hand make, with the positive \(x\)-axis, at 10 minutes after the hour?” (\(\frac \pi 6\) radians.)
• “If the minute hand was 1 foot long, could you use sine and cosine to express the coordinates of its tip?” (The coordinates, in feet, are \(\left(\cos\left(\frac \pi 6\right), \sin\left(\frac \pi 6\right)\right)\).)
• “What about when the minute hand is 5 inches long? 15 inches long?” (The coordinates, in inches, are \(\left(5\cos\left(\frac \pi 6\right), 5\sin\left(\frac \pi 6\right)\right)\) and \(\left(15\cos\left(\frac \pi 6\right),1 5\sin\left(\frac \pi 6\right)\right)\).)

Conclude the discussion by selecting students to share how they calculated the length of the minute hand when the coordinates of the end of the minute hand are known. Make sure to highlight these strategies by inviting students to consider each and share a reason why they prefer one over the other:

• use the Pythagorean Theorem
• recognize the angle as \(\frac{\pi}{4}\) radians due to the \(x\)- and \(y\)-coordinates matching

The purpose of this activity is for students to consider the height of points on a circle not centered at \((0,0)\). 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. 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.

For students having difficulty figuring out different rotations or heights on the Ferris wheel, suggest that they add more labels to the diagram, such as some of the known heights.

Begin the discussion by asking, “Would you ever be 75 feet off the ground? Explain your reasoning.” (No, the highest you would ever be is 70 feet, which is directly above the center of the Ferris wheel rotated \(\frac \pi 2\) radians from 0 radians.)

Invite students to share how they estimated the angles of rotation that put \(P\) 60 feet off the ground. Highlight students who:

• labeled the diagram of the Ferris wheel in order to identify that there are 2 angles, one in quadrant 1 and one in quadrant 2
• used the unit circle display to identify angles where the value of sine is close to \(\frac{2}{3}\), since 60 feet is 20 feet higher than the center and 20 feet is \(\frac{2}{3}\) the radius of the wheel
• wrote an equation to represent the situation, such as \(60=40 + 30 \sin(\theta)\)

For any of these methods that were not used, ask students to consider them now. A key point here is recognizing that the angle needed is one where \(\frac23\) the length of the radius is added to the height of the center of the wheel.