Paper Plate Unit Circle

Teaching the unit circle should be so much more than memorizing. Unfortunately, that was how I learned it in Trig class. I don't remember all the details about it how it was presented, but I remember studying it and memorizing all the parts. I have seen tricks on Pinterest to help students memorize it and while there is some value in ways to help students remember things, this lesson will hopefully provide your students with a much deeper understanding of the parts of the unit circle, focusing on how the degrees, the coordinates, the special right triangles, and the circumference of the circle all relate...and it's hands-on and color-coded! There should be no need for memorizing!

HAVE FUN!

MATERIALS: plain paper plates, highlighters/colored markers/pencils, scissors, protractor, three different colors of copy paper
PREPARATION: (For full understanding, students should have already learned the special right triangles. They should know the relationship between the angle measures and side lengths. They should be easily able to find the leg lengths with a hypotenuse of one.) Copy the 45-45-90 triangles onto one colored paper (I used pink), the 30-60-90-A triangles onto another colored paper (I used yellow), and the 30-60-90-B triangles on to a third colored paper (I used blue). Ensure matching markers/highlighters for each colored paper you use.

Set-up, Degrees, and Coordinates

1. Distribute paper plates. Have students fold them in half and in half again. These creases represent the x- and y-axes.
2. With a black sharpie, trace the folds. Label one x-axis and the other y-axis. Label the origin (0,0).
3. Tell students, for the sake of this activity, the paper plate has a radius of one unit. Keep this reminder handy throughout the activity. Use the radius of one to discuss the coordinates created by the intersection of the axes with the edge of the plate. Label (1,0), (0,1), (-1,0), and (0,-1).
4. Have a brief discussion reminding students about the total degrees in a circle (for example: 360° in a circle, semi-circle has 180°, line has a measure of 180°, etc.) Tell them the point (1,0) represents the 0° location. With or without use of a protractor, have students discuss in groups or as a class, the degrees of each of the other coordinate points. Label all degrees: 0°, 90°, 180°, 270°, and 360°.

5. Have students fold their plates along the diagonal so that the 0° line touches the 90° and 180° line touches 270°. Then, fold along the opposite diagonal so the 90° line touches 180° and 0° touches 270°. Make creases. Trace these creases with a new color (preferably, the same color as the paper used to copy the 45-45-90 triangles.) Discuss the degrees of the new lines and label each using the same color. (45°, 135°, 225°, 315°)
6. Distribute one 45-45-90 triangle to each student. Label the right angle and the 45° angles. Using the hypotenuse length of one unit, have students determine the leg lengths and label the lengths in the boxes. (√2/2) Use those side lengths to investigate the coordinate points of the intersection of the lines that were just made with the paper plate. (For example, move horizontally along the x-axis √2/2 units and vertically along the y-axis √2/2 units to arrive at the intersection (√2/2, √2/2)). Use this triangle to find the coordinate points of all the new colored lines (45°, 135°, 225°, 315°)

7. Use a protractor to measure the 30° angle and make a tiny mark.  Do the same for 210°, which students can use the 30° from the 180° line for help. Make a fold on those marks. Label 30° and 210°. Use the protractor to make marks at 150° and 330°. Make a fold on those marks. Label 150° and 330°. Trace these creases with a unique color (preferably the same colored paper used to copy the 30-60-90-A triangles).

8. Distribute one 30-60-90-A triangle to each student. Label the 30°, the 60°, and 90° angles. Using the hypotenuse length of one unit, have students determine the leg lengths and label the lengths in the boxes. (1/2 and √3/2) Use those side lengths to investigate the coordinate points of the intersection of the lines that were just made with the paper plate. (For example, move horizontally along the x-axis √3/2 units and vertically along the y-axis 1/2 unit to arrive at the intersection (√3/2,1/2)). Use this triangle to find the coordinate points of all the new colored lines (30°, 150°, 210°, 330°).

9. Use a protractor to measure the 60° angle and make a tiny mark.  Do the same for 240°, which students can use the 60° from the 180° line for help. Make a fold on those marks. Label 60° and 240°. Use the protractor to make marks at 120° and 300°. Make a fold on those marks. Label 120° and 300°. Trace these creases with a unique color (preferably the same colored paper used to copy the 30-60-90-B triangles).

10. Distribute one 30-60-90-B triangle to each student. Label the 30°, the 60°, and 90° angles. Using the hypotenuse length of one unit, have students determine the leg lengths and label the lengths in the boxes. (1/2 and √3/2) Use those side lengths to investigate the coordinate points of the intersection of the lines that were just made with the paper plate. (For example, move horizontally along the x-axis 1/2 unit and vertically along the y-axis √3/2 units to arrive at the intersection (1/2,√3/2)). Use this triangle to find the coordinate points of all the new colored lines (60°, 120°, 240°, 300°).

Radians

1. Remind students of how to find circumference of a circle and connect that formula to the unit of radians. 

2. Use guided questioning to help them discover and label the remaining radians. For example:

     a. At 0°, how many radians have we traveled? 0 radians.

     b. After one full trip around the circle, how far have we traveled? 2 radians.

     c. If one full circumference around the circle is 2, how far is halfway around the circle?  radians.

     d. How many radians have you traveled to the 90° line? /2 radians.

     e. How many radians have you traveled to the 270° line? 1 and ½ radians or 3/2 radians.

 Continue with questioning until students catch on to how the fractions relate to the radians and they are all labeled.

I have created a document that includes the directions listed above, a template for the three types of triangles, and a blank unit circle to use as a quiz. You can download it for free from my Teachers Pay Teachers store. Enjoy!