The purpose of this Warm-up is to introduce students to the unit circle and the convention of describing the location of a point on the circle by using its angle of rotation.
Launch
Display the image. Tell students that the dashed lines are 1 unit in each direction from the origin, so each grid mark represents 0.1 unit in length.
Arrange students in groups of 2. Give 1 minute of quiet think time for students to come up with a description of the location of point . In groups, have students share their descriptions and come up with even more different ways to describe the location of point .
Tell students that a circle that is centered at the origin and that has a radius length of 1 unit is called the unit circle and that it has several features we’re going to explore over the next few lessons.
Activity Synthesis
Invite students to share their descriptions. Record responses for all to see.
If no students suggest it, ask how to describe the location of point using just an angle.
Display this image.
Tell students that for any point on a unit circle, we can define it using just one feature: an angle. Specifically, an angle that starts at the positive -axis and rotates counterclockwise.
Often, a point on the unit circle is described simply as “a point at angle ” since there is only one point on the circle that could meet that description. If time allows, tell students that if we had picked a point in Quadrant III or IV, it would have a greater angle of rotation while a point in Quadrant I would have a smaller angle of rotation.
Students will continue to work closely with the unit circle and points on the circle for the next several lessons, so they do not need to be fluent with the vocabulary of unit circles at this time. The remainder of this lesson will focus on and how to use radians to think about the angle of rotation for a point on the unit circle.
The goal of this synthesis is to help students build their confidence working with radian angle measurements and consider what the sign of the - or -coordinates tell us about the angle of a point on the unit circle.
Display the applet:
Use the slider to first show the arc of the circle equal to 1 radius in length. This recalls the definition of a radian from a previous course: the angle made by wrapping a length of one radius around an arc of a circle of radius . Continue with the slider to show how the entire circumference is a bit more than 6 radius-lengths since is almost 6.3.
Ask students about the point rotating counterclockwise around the unit circle. Here are some questions for discussion:
“How far does the point have to rotate before the -values of start to repeat in the same order?” (1 complete rotation of radians.)
“How many radians can rotate and have a negative -value?” (Any between and .)
“How many radians can rotate and have a negative -value?” (Any between and .)
“If point rotates through radians, where is it? How do you know?” (Point would be halfway through Quadrant III. radians is radian larger than and radian smaller than , which is the same as .)
Student Lesson Summary
One way to define a circle that is centered at is by the equation , where is the radius of the circle. A unit circle has , so the equation for this unit circle with center must be . Points on the unit circle have several interesting properties, such as having matching points on opposite sides of the axes due to symmetry. Another feature of points on a unit circle is that they can be defined solely by an angle of rotation that is measured in radians.
Radians are a natural tool to use to measure the distance traveled on a circle. Let’s say that the wheels on a bike have a radius of 1 foot. When the bike starts to move to the left, rotating the wheel counterclockwise, let’s think about what happens to point .
The point will return to its starting location when the wheel has rotated through an angle of radians. During this rotation, the bike will move a length equal to the circumference of the wheel, which is feet. In general, the angle of rotation of the wheel with a radius of 1 foot, in radians, is the same as the number of feet this bike has traveled. So what do we do when a wheel doesn’t have a radius of 1 unit? Since all circles are similar, we can use the same type of thinking—scaled up or down—to match the size of the wheel, which is something we’ll do in future lessons.
Thinking about the wheel as a unit circle, as shown in this image, the arc length of the circle from to has length equal to 1 unit, the radius of the unit circle. Because of this, the angle is said to measure one radian. If we continue to measure off radian lengths around the circle, it takes a little more than 6 to measure the entire circumference.
This makes sense because the ratio of the circumference to the diameter for a circle is , and so the circumference is times the radius, or about 6.3 radii.
Let’s think about some other angles on the unit circle. Here, angle measures radians because its arc is of a full circle (counterclockwise) or of . Angle is three quarters of a full circle (counterclockwise), so that’s radians.
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If students have trouble getting started, consider asking:
“Can you label some values on the axis, such as 0.5.”
“In a previous lesson, we used right triangles to help us find coordinates. What triangle could you draw to help you describe the location of point ?”
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Activity Narrative
The purpose of this activity is to build on students’ understanding of the unit circle and help students establish language for talking about points on the unit circle (MP6).
Monitor for students who:
Use the Pythagorean theorem or symmetry to identify coordinates.
Use angles or fractions of a circle to identify rotation around the circle.
The goal is to connect strategies that focus on rectangular thinking, including gridlines, horizontal and vertical lines of symmetry, and the legs of a right triangle, to strategies that focus on angular thinking, such as fractions of a circle, angles, or amount around the circumference of the circle.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Launch
Select students who used each strategy described in the Activity Narrative to share later. Aim to elicit both key mathematical ideas and a variety of student responses, especially from students who haven't shared recently.
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy, such as:
For each description, which points could it be describing?
The point has a -coordinate of 0.
The point is rotated of the way around the circle from .
The - and -coordinates of the point are the same value.
The - and -coordinates of the point have opposite values.
The point is rotated less than of the way around the circle from .
The point is 1 unit away from the origin.
Point has a -coordinate of 0.5.
What is the -coordinate of point ?
What are the coordinates of point ?
Point is rotated of the way around the circle from point . Describe the location of point . Show or explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is to introduce the idea that for any point on a unit circle, we can define it using just one feature: an angle. Focus on the connection between location on the circle and how far around the circle the point is rotated, rather than explicit use of radians or degrees to measure angle of rotation, as students will get this opportunity in an upcoming activity.
After strategies have been presented, display 2–3 approaches from previously selected students. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:
“What do the approaches have in common? How are they different?”
“How does the location of the point show up in each method?”
“Are there any benefits or drawbacks to one representation compared to another?”
This activity is optional because it offers additional practice using a concrete representation of radian measure.
The goal of this activity is to measure circles using the radius as a unit of measure. In grade 7, students measure the circumference and diameter of different-sized circles and observe that the pairs of measurements appear to be proportional. In this task, students measure the circumference of different-sized circles using the radius of their circle as a unit of measurement. This allows students to observe that the number does not appear to depend on the size of the circle and recall that this number is since is the constant of proportionality relating the circumference and diameter of a circle.
Monitor for students who have clear explanations for the exact number of radii needed to fit around the circle, such as by reasoning about an equation for the circumference of a circle.
Launch
Arrange students in groups of 4. Provide each student in a group with a different circle to measure.
Engagement: Develop Effort and Persistence. Check in and provide each group with feedback that encourages collaboration and community. For example, identify elements from students' strategies for the group to discuss, or invite students to record their responses to the first questions and discuss their observations as a group. Supports accessibility for: Social-Emotional Functioning, Organization
Activity
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Student Task Statement
Your teacher will give you a circular object.
About how many radii does it take to go halfway around the circle?
About how many radii does it take to go all the way around the circle?
Compare your answers to the previous two questions with your partners.
What is the exact number of radii that fit around the circumference of the circle? Explain how you know.
Why doesn’t the number of radii that fit around the circumference of a circle depend on the radius of the circle? Explain how you know.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to ensure that all students understand why the number of radii that fit around the circumference of a circle does not depend on the size of the radius of the circle. Begin the discussion by inviting students to share how many radii it took them to go all the way around a circle, and record the responses for all to see.
Invite previously identified students to share their reasoning about the exact number of radii that fit around the circumference of a circle. If not mentioned, make sure that students recall that the circumference of a circle is proportional to the diameter, , and that the constant of proportionality is , which we can see in the equation , or .
Next, select students to share their explanation for why the number of radii that fit around the circumference of a circle doesn’t depend on the radius of the circle. Record, for all to see, student explanations and any diagrams used.
Display this image of a circle, where an angle of 1 radian is marked:
Use a piece of string, or other flexible material such as ribbon, to show how the arc of the circle intersected by the angle has a length that is equal to the radius.
MLR8 Discussion Supports. For each explanation that is shared, ask, “Who can restate what shared using mathematical language?” Advances: Listening, Speaking
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Activity Narrative
The purpose of this activity is to help students recall radian measurement of angles, focusing on some key understandings, such as the relationship between arc length, 1 radian, and the radius of a circle. Students use repeated reasoning as they consider how far a wheel has traveled for several angle measures (MP8).
As in the previous activities of this lesson, the circle is a unit circle, but now students are asked to think in feet, which can feel more like a “real” measurement to students than “1 unit” does, and connects to how odometers on cars and bikes work by measuring the number of revolutions of the wheel and relating this to the distance traveled. Another way to say this is that the odometer keeps track of the angle the wheel has rotated: When this angle is measured in radians and the radius of the wheel is 1 foot, as it is here, the angle of rotation in radians is the distance traveled in feet.
In the digital version of Are You Ready for More?, students use an applet to describe the shape made by a point on a moving bike wheel. After writing their own description, the applet allows students to trace out the shape while analyzing the rotation, distance, and height of the point. The digital version allows students to check their thinking about the shape made by the point on the moving wheel.
Launch
Arrange students in groups of 2. Tell students that they are now going to consider the relationship between the angle of rotation for a point on a unit circle and the arc length made by a point that is rotating through the angle.
Ask students to read the opening statement and to complete the first 2 questions individually. After quiet work time, ask students to compare their responses to their partner’s. Follow with a whole-class discussion about the location of point . Once students are in agreement on the location of (possibly by reasoning about the 12 equal parts that make up the circle and the circumference of the circle), remind students that one way to measure angles is by using radians. The radian measure of an angle is the ratio of arc length to radius. Ask, “What angle, in radians, does rotate through to get to ?” (Since the arc length from to is 1 foot and the radius is 1 foot, the angle is 1 radian.) Display this definition of a radian for all to see throughout the activity.
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color-code angles and corresponding parts of the circumference of the circle. Supports accessibility for: Visual-Spatial Processing
Activity Synthesis
The purpose of this discussion is for students to share with their classmates their reasoning about the points they plotted on the wheel. Pair groups and tell students to take turns sharing their reasoning for how they plotted the new points determined by the different counterclockwise rotations of . After 2–3 minutes of discussion time, select students to share the angle that they identified that brings to its lowest location in its rotation. If not mentioned by students, highlight how rotating of the way around the circle means the radian measurement is radians (the ratio of arc length to radius is since the arc length is ).
Here are some questions for discussion:
“How far does the bike travel in one half revolution of the wheel?” ( feet)
“What angle is that angle of revolution?” ( radians)
“How far does the bike travel in one full revolution of the wheel?” ( feet)
“What angle of revolution is that?” ( radians)
“What is the relationship between the distance the bike travels and the angle measure in radians?” (They are the same.)
Students will continue to develop their fluency with radian measurement in the following lessons, so there is no need to delve deeper in this activity.
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to “What is the relationship between the distance the bike travels and the angle measure in radians?” Invite listeners to ask questions and give feedback that will help clarify and strengthen their partner's ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
HSF-TF.C.8
Prove the Pythagorean identity and use it to find , , or given , , or and the quadrant of the angle.
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
A bicycle wheel has a 1-foot radius. The wheel rolls to the left (counterclockwise).
What is the circumference of this wheel?
Mark the point where will be after the wheel has rolled 1 foot to the left. Be prepared to explain your reasoning.
Mark the point where will be after the wheel has rolled 3 feet to the left. What angle, in radians, does rotate through to get to ? Explain your reasoning.
Where will point be after the bike has traveled feet to the left? What about feet? feet? Mark these points on the circle. Explain your reasoning.
After traveling some distance to the left, point is at the lowest location in its rotation. How far might the bike have traveled? Explain your reasoning.
Student Response
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Building on Student Thinking
Students may want to use approximations instead of leaving answers in terms of when working with radians. There is nothing incorrect with doing so, but encourage these students to use exact values as much as possible instead of approximations. Remind them that a full rotation is radians, and we can think of all other angles in a unit circle as some fraction of that measurement. For the purposes of this unit, most angles given are an integer multiple of radian.
