The goal of this lesson is for students to begin their exploration of the unit circle, defined as a circle of radius 1 centered at the origin, which they continue in the following lesson and use throughout the remainder of the unit. They focus first on the symmetric nature of the \((x,y)\) coordinates of points on the unit circle and then learn that these points can also be defined by their angle of rotation, which leads to working with radian angle measurements.
This lesson builds on the geometry course in which students learned that all circles are similar and examined arcs intercepted by given angles. That work led to defining the radian measure of an angle as the ratio of the arc length traveled to the radius of the circle. This means that 1 radian is the angle when the length of the arc it intersects on a circle of radius \(r\) is \(r\). Students also learned that by this definition, and because \(\pi\) is the ratio of the circumference of the circle to its diameter, there are \(2\pi\) radians in a full circle. This lesson includes an optional activity if students need practice recalling the definition of radian measurement.
Students look for regularity in repeated reasoning as they apply radian measure to examine the distance a wheel travels as it rolls for several angles, reasoning that the measure of the angle of revolution corresponds to the distance traveled when the radius is 1 (MP8).
- Calculate the radian angle measurement a point on a wheel rotates through by relating it to the distance traveled by the wheel.
- Describe characteristics of points on a unit circle.
- Let’s learn about the unit circle.
Acquire 1 round object per student if using the optional activity Measuring Circles.
Be prepared to display applets for all to see during the activity syntheses of the activities “Measuring Circles” and “Around a Bike Wheel.”
Devices are required for the digital version of the extension in “Around a Bike Wheel,” ideally 1 per student.
- I understand that a radian angle measurement is the ratio of the arc length to the radius of the circle.
- I understand that points on a unit circle can be defined by their coordinates or by an angle of rotation.
The length of an interval at which a periodic function repeats. A function \(f\) has a period, \(p\), if \(f(x+p) = f(x)\) for all inputs \(x\).
A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).
The circle in the coordinate plane with radius 1 and center the origin.