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, but that does not mean true understanding. 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. Using a paper plate and special right triangle printables, the lesson focuses 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! Therefore, there should be no need for memorizing!

## UNIT CIRCLE PAPER PLATE 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. I do not recommend teaching the special right triangles and unit circle at the same time.

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 onto a third colored paper (I used **blue**). Ensure matching markers/highlighters for each colored paper you use.

## UNIT CIRCLE SET-UP, DEGREES, & COORDINATES

**1.** First, 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.** Next, have a brief discussion reminding students about the total degrees in a circle (for example: 360° in a circle, a semi-circle has 180°, a line has a measure of 180°, etc.) Tell them the point (1,0) represents the 0° location. With or without the 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 the 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°. After that, 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 how to find the 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:

*π*radians.

*π*, how far is halfway around the circle?

*π*radians.

*π*/2 radians.

*π*/2 radians.

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To read more about discovery in math class, you can find my post about discovering the rules for rotations on a coordinate plane by clicking here.

## 3 thoughts on “Unit Circle Paper Plate”

Hi. We did this activity during the regular school some years ago. We would have them make it by hand using a protractor. I want to do this online and I wanted to print out the triangles for the students who do not have printers at home, so they can come by and pick them up from the school. I was wondering if you could please share the file where you have these triangles on a worksheet

Hi Sonia!

Yes, I am happy to share the printable with you! It is in my TeachersPayTeachers.com store, Kacie Travis. Click here to download it.

Thanks for working to make sure kids get access to it! You’re a great teacher!

Kacie

I love this activity! I recently started teaching PreCal and this helped me teach the Unit Circle and special right triangles. I know the Calc teachers after me like the students to know and understand special right triangles not just memorize the Unit Circle so this helped me teach both ways. Since this is a college class I was impressed to see several of them keep their plate AND the triangles and continue to refer to them. Thanks so much for these templates!