Graphs from the Unit Circle

• Lesson
9-12
3

In this lesson, students use uncooked spaghetti to transfer lengths from the unit circle to a function graph on large butcher paper. In the process, they discover the key features of sine and cosine graphs. The activity is presented for students working in degrees, but another version of the handouts is provided for students working in radians.

This lesson is intended to be taught in degrees. However, alternative worksheets in radians are also provided. You may use either version, or even use both versions in the same class as a way to differentiate for students at various levels of development. Note: The radian version of this activity should not be used to introduce radian measure. It is intended for students who already understand the use of radians.

Pre-Assessment

This lesson is intended for students who have already explored unit circle trigonometry. In particular, students must understand the relationship between the coordinates of a point on the unit circle and the sine and cosine of its central angle in standard position. A quick review during the pre-assessment stage below is helpful for some students.

Display the first question on the Pre-Assessment Overhead or draw the diagram on the board. Assign students to label the lengths of the sides of the right triangle. Create similar examples if students need extra practice.

Display the second question on the overhead or draw the diagram on the board. Review and discuss the answers to both questions. For Question 1, the side lengths are 3, 4, and 5. For Question 2, cos θ should be written on the side on the x-axis and sin θ should be written on the side perpendicular to the x-axis.

Activity Set-Up

Form students into groups of 2 or 3; larger or smaller groups are not recommended for this activity. This group size will allow students to collaboratively manipulate the large-scale materials and promote discussion to solve any problems that may occur. Too many students in the group may leave some members with little or nothing to contribute.

Introduce the activity by telling students that it’s time to focus on the physical relationships in trigonometric functions. Students will use a protractor to measure specific angles around the circle at the beginning of the activity, but no numeric measurements are used after that. Students will explore the relationships between lengths by understanding how all the measurements are based only on the initial spaghetti, which is one unit (hence the unit circle). The focus for the majority of the lesson is on comparative physical lengths of spaghetti, not numeric measurements. Some students will need to be reminded of this throughout the activity.

Distribute the Graphs from the Unit Circle activity sheet to each student. Depending on the ability level of your students, you can choose to do this activity in degrees or radians. Students tend to be more comfortable working in degrees. However, if your students are fluent in using radians, you can choose to have them work in those units. Although they are working in groups, stress to students that each person should turn in a paper. That way each student will have a copy of the work to keep with his or her notes for future reference.

Do not distribute the Questions activity sheet until students have finished the Graphs one correctly. The beginning of the Questions activity sheet essentially answers the questions at the end of the Graphs activity sheet. It is best if students have a chance to discover, or remember, these answers.

Instructions for students are on the handout so groups may begin right away. Various check points are included throughout the activity. They look like this: OK _________. Students are instructed on the activity sheet to check with you at each check point to make sure their work to that point is on track. Initial or stamp the check point when you feel the student is ready to go on to the next section. This ensures that each group is interacting with you throughout the activity and is staying on track. Also, it means you do not have to read those sections on the students’ papers after you collect them, since you have already checked them.

The first challenge for students is to figure out how to draw a circle with a radius equal to one piece of spaghetti. Their usual compass isn’t big enough. Compasses made for chalkboards or whiteboards will work, but it is a nice challenge for students to figure out how to make it themselves. If time is an issue, you may want to give students a chalk board or whiteboard compass, since it could save 15 or 20 minutes of class time.

As students begin drawing their circles, circulate around the room to make sure they are well drawn. Be sure to remind students to mark the center of the circle before drawing it. Also, make sure the circle is not too sloppy. A lopsided circle will lead to a very lopsided sine curve and cosine curve.

The initial set-up of the unit circle and function graph on each group’s paper should look like this:

As students begin to measure and mark their angle measures, watch to be sure they lay the string around the circle in a counter-clockwise direction starting at (1,0). This reinforces angle measures in the unit circle for angles in standard position. The graph will work no matter which direction they put the string on the circle, but this will help to reinforce what students have already learned about unit circle trigonometry.

Activity Exploration

Question 1 on the Graphs activity sheet asks what the x-values represent on the function graph. At this stage of the process, the important thing is just for students to realize that they are angle measures. For students who don’t see it right away, remind them that they labeled each graph in degrees. Ask students:

• What do we usually measure in degrees?
[Students should recall that angles are measured in degrees.]
• Then, what is labeled on the x-axis?
[Students should conclude that since the x-axis is being labeled in degrees, it most likely represents the angle measures.]

Transferring the spaghetti heights from the unit circle to the function graph looks like this:

Question 2 on the Graphs activity sheet asks what the y-values represent on the function graph. Students need to recall that this piece of the triangle in the unit circle is the sine value, which they reviewed in Question 2 of the Pre-Assessment. Therefore, the y-values of the function graph are sine values. For a student who doesn’t see it right away, ask questions related to a previous lesson on unit circle trigonometry or the Pre-Assessment activity from this lesson. Ask, "How did you label that leg of the triangle in the pre-assessment activity?"

After initialing or stamping the checkpoint at the end of the Graphs activity sheet, give each student the Question activity sheet. Based on which Graphs activity sheet you used, each student should be given either the degrees or radians version of the activity sheet.

In Questions 7–10, students are challenged to determine what length in the unit circle is used to create y = cos x on the function graph. For students who struggle to pick the correct measure ask, "Where did you put the cos θ label in the pre-assessment activity?" Students should recall that cos θ represented the side of the triangle along the x-axis.

Students may need help to realize that the horizontal leg of the triangle from the unit circle has to be placed vertically on the function graph since cosine is the x-value in a unit circle but the y-value in the graph of y = cos x.

Closure

As students complete the activity collect the activity sheets and allow students to hang their graphs on the wall. You may wish to find one graph that is particularly neat and accurate to put at the front of the room as a resource during future lessons.

Post instructions for the closure activity so that groups finishing the activity at different times can begin this assignment as they finish:

Draw a graph of y = sin x in your notebook, this time including 2 complete periods, from x = –360° to x = 360°. Do the same for y = cos x on a different set of axes. Describe the domain, range, maxima, minima, and intercepts of each graph.

This assignment is designed to assess how much students understand these key features of the graphs. Note: If working in radians, the graphs should be drawn from x = –2π radians to x = 2π radians.

On the last day of the lesson, return the handouts to students for a discussion of the Questions from the Unit Circle activity sheet. Refer to the Questions from the Unit Circle Answer Key for sample answers. Randomly select students to answer each question, referring to the graphs on the wall as needed.

Reference

Peterson, Blake E., Patrick Averbeck, and Lynanna Baker. 1998. Sine curves and spaghetti. Mathematics Teacher 91: 564–67.

If working in degrees: If working in radians:

Assessment Options

1. Use the check points to facilitate ongoing assessment. Later, determine some number of points for each section in order to assign a grade for the assignment.
2. Show students graphs created by other students or graphs you've created with mistakes and ask them to correct them. For example, flip the sine graph vertically across the x-axis and ask students how they know it is wrong.
3. Assign a journal-writing task in which students explain the connection between the unit circle graph and the function graph.

Extensions

1. Similar activities can be done for tangent, cotangent, secant, and cosecant. You can introduce the other trigonometric functions using the same activity, or have students create the graphs on their own on a smaller scale.
2. Have students look at the graphs when the amplitude and/ or period have changed.  Discuss the transformation that occurs.

Questions for Students

1. What happens to the sine value on the unit circle at 180°? How will you show that on the function graph?

[The circle intersects the x-axis, so the sine is 0. Therefore, it should be a zero of the function or an x-intercept of the graph.]

2. What happens to the cosine value on the unit circle at 180°? How will you show that on the function graph?

[Cosine is the x-value in the unit circle, so that would be –1. At 180° on the function graph, the y-coordinate should be –1, so this is a minimum point on the function graph.]

3. What happens to the cosine value on the unit circle at 90°? How will you show that on the function graph?

[The x-value on the unit circle for a 90° angle in standard position is 0, so the cosine is 0 at 90°. It should be a 0 of the function or an x-intercept of the graph.]

Teacher Reflection

• Was students’ level of enthusiasm/involvement high or low? Explain why.
• Did you challenge the achievers to go beyond quick answers and pursue deeper explanations and understanding? How?
• Did all members of each group contribute to the project and remain fully engaged? How do you know? How could you improve that next time?
• Did you provide appropriate support for students who struggle with graphing by hand? How?
• How did students demonstrate understanding of the relationship between the unit circle and the function graph?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were the effective?

Learning Objectives

Students will:

• Explore the properties of the unit circle.
• Discover the graphs of sine and cosine curves.
• Discover the domain, range, maxima, minima, and intercepts of sine and cosine graphs.
• Compare the properties of sine and cosine curves.

NCTM Standards and Expectations

• Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions.
• Use symbolic algebra to represent and explain mathematical relationships.
• Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts.

Common Core State Standards – Practice

• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.