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. 

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 ?”

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 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 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.

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.