Lesson 7

Finding Unknown Coordinates on a Circle

7.1: Notice and Wonder: Big and Small (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

What do you notice? What do you wonder?

Two circles with centers at the origin of the x y plane. 

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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 \(A\).)

7.2: Clock Coordinates (15 minutes)

Activity

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 \(\sin(\theta)\)

Launch

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

Student Facing

Here is a clock face.

A blank clock on a grid. 
  1. The length of the minute hand on a clock is 5 inches and the center of the clock is at \((0,0)\) on a coordinate plane. Determine the coordinates of the end of the minute hand at the following times. Explain or show your reasoning.
    1. 45 minutes after the hour
    2. 10 minutes after the hour
    3. 40 minutes after the hour
  2. The minute hand on another clock, also centered at \((0,0)\), has a length of 15 inches. Determine the coordinates of the end of the minute hand at the following times. Explain or show your reasoning.
    1. 45 minutes after the hour
    2. 10 minutes after the hour
    3. 40 minutes after the hour
  3. At a certain time, the end of the minute hand of a third clock centered at \((0,0)\) has coordinates approximately \((7.5,7.5)\). How long is the minute hand of the clock if each grid square is one inch by one inch? Explain or show your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

The center of a clock is \((0,0)\) on a coordinate grid. Its hour hand is half the length of its minute hand. The coordinates of the end of the hour hand are about \((3,0.5)\). What are the approximate coordinates of the end of the minute hand? Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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
Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students share their strategies for identifying the coordinates on the clocks, scribe their thinking on a visual display. Highlight the exact coordinates (expressions using cosine and sine) and the approximate coordinates in the same color. Use annotation to draw attention to the scale factors of 5 and 15 in the expressions.
Supports accessibility for: Visual-spatial processing; Conceptual processing

7.3: Around a Ferris Wheel (15 minutes)

Activity

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.

Launch

If students have access to devices that can run a GeoGebra applet, suggest that the Unit Circle Slider applet might be a helpful tool in this activity. Consider demonstrating how to use the applet. The slider works to change the angle that the given segment creates with the positive \(x\)-axis. The displayed coordinates indicate the coordinates of the intersection point of the line segment with the unit circle.

Display the image of the Ferris wheel for all to see. Have students estimate the height of each position as the Ferris wheel moves counterclockwise. In the activity, they will precisely calculate the value.

  • \(\frac \pi {12}\) radians
  • \(\frac \pi 2\) radians
  • \(\frac {5\pi}6\) radians
  • \(\frac {5\pi}3\) radians
Reading: MLR6 Three Reads. Use this routine to support reading comprehension, without solving, for students. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (A Ferris wheel with point \(P\) on the circle). After the second read, students list any quantities that can be counted or measured, without focusing on specific values (the distance off of the ground, the radius, and the angle of rotation in radians). After the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem while keeping the cognitive demand of the task.
Design Principle: Support sense-making

Student Facing

The center of a Ferris wheel is 40 feet off the ground, and the radius of the Ferris wheel is 30 feet. Point \(P\) is shown at 0 radians.

A circle. 

​​​​​​

  1. Calculate how high off the ground point \(P\) is as the Ferris wheel rotates counterclockwise starting from 0 radians.
    1. \(\frac \pi {12}\) radians
    2. \(\frac \pi 2\)radians
    3. \(\frac {5\pi}6\) radians
    4. \(\frac {5\pi}3\) radians
  2. As you go around on the Ferris wheel, at which position(s) would you be 60 feet off the ground? Explain your reasoning.

You may use this applet if you choose.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

A Unit Circle Slider applet is available in the digital version of this activity. 

Arrange students in groups of 2. Display the image of the Ferris wheel for all to see. Tell them the center of the wheel is 40 feet high and it has a radius of 30 feet. Ask students to give a rough estimate for these questions:

  • “How high off the ground will point \(P\) be after the wheel rotates \(\frac \pi {12}\) radians?” (That is a small rotation, so around 50 feet.)
  • “How high off the ground will point \(P\) be after the wheel rotates \(\frac \pi {2}\) radians?” (That’s a quarter turn, so \(P\) will be at the top, which is a height of 70 feet.)

Give students quiet work time to complete the actual calculations needed for the first question and then time to share their work with a partner before completing the rest of the activity.

Reading: MLR6 Three Reads. Use this routine to support reading comprehension, without solving, for students. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (A Ferris wheel with point \(P\) on the circle). After the second read, students list any quantities that can be counted or measured, without focusing on specific values (the distance off of the ground, the radius, and the angle of rotation in radians). After the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem while keeping the cognitive demand of the task.
Design Principle: Support sense-making
Action and Expression: Internalize Executive Functions. Provide students with a two-column table to organize the given angles of rotation (in radians) and corresponding height off the ground (in feet). The table will also provide a visual support for students to write out exact values using sine.
Supports accessibility for: Language; Organization

Student Facing

The center of a Ferris wheel is 40 feet off of the ground, and the radius of the Ferris wheel is 30 feet. Point \(P\) is shown at 0 radians.

A circle. 
  1. Calculate how high off the ground point \(P\) is as the Ferris wheel rotates counterclockwise starting from 0 radians.
    1. \(\frac \pi {12}\) radians
    2. \(\frac \pi 2\)radians
    3. \(\frac {5\pi}6\) radians
    4. \(\frac {5\pi}3\) radians
  2. As \(P\) goes around on the Ferris wheel, estimate which angle(s) of rotation put \(P\) 60 feet off the ground. Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

Lesson Synthesis

Lesson Synthesis

Tell students to imagine an electricity-generating wind turbine whose base is 200 feet tall and has three blades that are each 120 feet long at the top of the base. Ask students to draw and label a picture of the wind turbine. After a brief work time to make their sketch, here are some questions for discussion:

  • “What is the highest and lowest (above the ground) the windmill blades reach?” (320 feet and 80 feet)
  • “What angle would a windmill blade make in the highest or lowest position assuming 0 radians is directly to the right of the center, as on a unit circle?” (\(\frac{\pi}{2}\) radians and \(\frac{3\pi}{2}\) radians)
  • “What does the expression \(120\sin\left(\frac{\pi}{4}\right)\) represent in this situation?” (the vertical distance between the center of the wind turbine to the tip of the blade when it is at an angle of \(\frac{\pi}{4}\).)
  • “How high is the tip of the blade when it is at an angle of \(\frac{11\pi}{6}\) radians?” (140 feet since \(200+120\sin\left(\frac{11\pi}{6}\right)=140\))

7.4: Cool-down - An Airplane Propeller (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Photo of Big Ben.

Sine is helpful for finding heights of things moving in a circular motion. The minute hand on the Elizabeth Tower, a famous clock in London whose bell is nicknamed Big Ben, extends 11.5 feet to the edge of the 23 foot diameter clock face. The center of the clock is 180 feet above the ground. At 12:00 the height of the end of the minute hand above the ground is 191.5 feet (\(180 + 11.5\)) while at 12:30 it is 168.5 feet high (\(180 - 11.5\)). At 12:15 and 12:45 the end of the minute hand is 180 feet high.

What can we say about the height of the end of the minute hand at other times? Let’s start by imagining a unit circle centered on the clock.

An analog clock showing 12 : 10. 

At 12:10 the minute hand makes a \(\frac{\pi}{6}\) radian angle together with the ray through the 3 on the clock. Using our unit circle, we know \(\sin\left(\frac{\pi}{6}\right) = 0.5\). For a clock that has a radius of 1, the height of the end of the minute hand above the middle of the circle would be 0.5 feet. But this clock has a radius of 11.5, so the end of the minute hand is 5.75 feet above the center of the clock, since \(5.75=0.5\boldcdot 11.5\). Taken together with the center of the clock being 180 feet off the ground, the end of the minute hand is 185.75 feet above ground.

Another question we can ask is: when is the end of the minute hand 174.25 feet above the ground? Since \(174.25 = 180 - 5.75\), this means that the tip of the minute hand is 5.75 feet below the center of the clock, and this is \(\frac{5.75}{11.5}\) or \(\frac{1}{2}\) times its length. Using our unit circle again, the two angles where \(\sin(\theta) = \text-\frac{1}{2}\) are \(\frac{7\pi}{6}\) and \(\frac{11\pi}{6}\). This means the end of the minute hand is 174.25 feet off the ground at 40 and 20 minutes after the hour.