Illuminations: Rolling Into Radians

Rolling Into Radians

Student groups collect height versus distance data for rolling objects of different sizes. Each group produces two sinusoidal graphs of the data, one in which both axes are measured in units, and the other in which both axes are rescaled and free of units. Students note that the amplitude and period of the unit-free graph are the same for all groups, and then discuss how their measurement opens a different way of describing points on a unit circle.

Learning Objectives

Students will:

Measure distance and height for periodic phenomena
Graph periodic phenomena
Develop understanding of periodic phenomena vocabulary
Relate physical characteristics of a situation to components of a periodic function
Relate measurements on a circle to arc length

Materials

Vocabulary Overhead

Data Collection Activity Sheet

Paper measuring tapes

Rulers (starting with 0 at the very end of the ruler)

Cans of various sizes

Permanent markers

Masking tape

Graph Overhead

Overhead markers

Questions Activity Sheet

Questions Answer Key

Geometer's Sketchpad File

Instructional Plan

Begin the lesson by telling students that they will be studying a repetitive phenomenon, with a special name and associated vocabulary. Introduce the vocabulary of periodic functions using the Vocabulary overhead. When discussing "periodic," you may want to explicitly point out the portion of the pattern that repeats. Note that "phase shift" is not illustrated in the overhead; it is not needed for this lesson.

Vocabulary Overhead

Divide the class into groups of three or four. In each group, students should be assigned the roles of Ruler, Writer, Roller, and Graphic Artist. (Roles may be combined for group sizes less than four, but the Roller and Ruler should be different students.) While students read their job descriptions, distribute one can, one measuring tape, one ruler, and one Data Collection activity sheet to each group. Be sure pages are NOT printed back-to-back, as students will want to look at all three pages simultaneously. Each group should have a different-sized can, with rims at both ends so that the cans will roll straight. The rims should be in constant contact with the floor or table. Use a permanent marker or a small dab of paint or nail polish to mark a dot on one end of the can along the rim.

Rolling into Radians Data Collection Activity Sheet

Groups with smaller cans might use tables or desks, but groups with large cans (cookie tins work well) may need to use the floor. To approximate the radius of the can, students might take half the diameter or use their measuring tape to find the circumference and divide it by 2π. If the can has a plastic lid, there may be a dot molded into the center of the lid that students could use for measuring radius. Be sure that all measurements done by a group are in the same units, but different groups may use different units. Students should attach the measuring tape to the floor or tabletop using masking tape.

Data Collection

Have students read the instructions for data collection on their activity sheet. Respond to any questions. Measurement starts with the dot on the can on the ground at the zero mark on the measuring tape, and the 0s are already recorded on the Data Collection activity sheet. Instructions ask for eight measurements per rotation of the can; they may be made any convenient locations, not necessarily evenly spaced.

There are two important notes for data collection:

First, it is important that the can be rolled, not slid. If the table or floor allows for excess sliding, students might wrap one layer of masking tape around each rim of the can to increase the friction between the can and the rolling surface.
Second, the horizontal position (d on the data table) must always the point of contact between the can and the measuring tape as illustrated at the right. It should NOT be the location where the vertical ruler measuring the height of the dot touches the measuring tape. Students who make this error will graph a cycloid, which has sharp corners at the bottom, rather than a sinusoid.

Data Handling and Analysis

After data is collected, students will need to divide each number by the radius of the can. The most efficient way to do this is to enter the data into lists on a spreadsheet or graphing calculator, then produce new lists by entering a formula that divides the original lists by the radius. Have students produce their second graph using an overhead marker on the Graph overhead. Cut the overhead in half and give one to each group.

Rolling into Radians Graph Overhead

Students may recognize the graphs of the sine or cosine function. You might ask those students what sort of quantity x represents in the expression y = sin x; they will most likely respond that x is an angle. Point out that in this activity they did not measure angles. As groups finish the data collection, distribute the Questions activity sheet (one per student). Students should work within their groups, using their own graphs and data, to answer the questions, but they will need to compare their answers to other groups' for Question 8.

Rolling into Radians Questions Activity Sheet

Summary

Discuss student answers to Questions activity sheet. In the discussion for Questions 7 and 8, point out that the unit-less graphs are based on what is called radian measure, or measurements in terms of the radius length of the circles. Converting other (distance) measurements into radian measure is as simple as dividing by the radius. Relate radian measure to angle measure by asking students how many radians make up the circumference of a circle [2π] and how many degrees of arc are in a full circle [360°]. Put all groups' graphs up at once for a visual image of Question 8. Point out that the graphs produced by all groups are examples of functions known as "sinusoids," or "sinusoidal functions."

Questions for Students

In this activity, you measured distances the can rolled on the ground. How are those distances related to the can itself? In other words, how could you mark off those distances directly on the can?
[The distances on the ground correspond to lengths of arc on the can.]
What kind of ruler might you build so that the data you collected would be the same as the data you used for your group's second graph?
[A ruler in which each unit was one radius-length long. Some students might think of marking off radius lengths on a string or tape as a ruler.]
What physical characteristic of the can determined the periods and amplitudes of the first graphs produced?
[The circumference determined the period and the radius determined the amplitude. Some students may point out that the radius determined both.]
In some sense, the second graphs your groups produced were "natural," unaffected by the units of measure or the sizes of the cans. What are some advantages of having such a natural function?
[A natural function may be applied to many situations by scaling, shifting, or reflecting. Adding units is equivalent to scaling, and different units may be used for vertical scaling (amplitude, axis of oscillation) than for horizontal units (period). The original graph could serve the same purpose, but you would have to transform the units on the original graph to fit the new context, rather than just apply the units in the new context.]

Assessment Options

Give the students a copy of the "unit-less" graph they produced, stripped of scale, and ask them to draw scales on the axes corresponding to a carnival Ferris wheel with a radius of 14 feet that makes 5 revolutions per minute. The graph should have a period equal to the time for one revolution and an amplitude equal to the radius of the Ferris wheel. Some students might place the circumference of the Ferris wheel along the horizontal axis.
Ask students to determine the period and amplitude of the graph of the function ƒ(x) = sin x, where x is measured in radians. Ask them to compare the graph of ƒ(x) = sin x to their second graph. [The period is 2π and the amplitude is 1. The data graph has the same shape, but is shifted up one unit and to the right by π/2].
Ask students to go back to their graphs and label the data points with the approximate degree measure of the angles of rotation for the first revolution. This reinforces the notion that position on the circle may be measured both by arc length and by central angle and forces students to think about the relationship rather than simply applying a formula.

Extensions

Have students place their calculators in Radians mode and graph the function ƒ(x) = sin x over a window ‑2π ≤ x ≤ 2π, ‑2 ≤ y ≤ 3. Then have them try to modify the function so that the graph on the calculator matches their second graph. [y = sin(x – π/2) + 1, or y = 1 – cos x, or y = 1 + cos(x ± π) works.]
Students who are comfortable using geometry software might be asked to program an animation with a construction that would trace out the graph. It is a bit of a challenge to get the "rolling" aspect of the model to work out right. [One approach might be to construct a circle, then a point on the circle, then an arc terminating at the point. Mark the arc length, then translate a point on the "measuring tape" by the marked distance. Construct a circle congruent to the first circle, tangent to the "measuring tape" at the translated point, to simulate the rolling. A slightly more complicated construction using Geometer's Sketchpad is included in the lesson materials. Students might be given the Sketchpad file and asked to figure out both how it works and why the larger circle is used. The reason for the larger circle is because Geometer's Sketchpad won't measure arc angles larger than 2π radians.]
Name some phenomena that may be characterized by graphs of the shape you produced. With each phenomenon, describe in what units the horizontal and vertical measurements might be made. [Sample answers: Height of a rider on a Ferris wheel: vertical measurement of height, horizontal measurement of time. Wave in the water: vertical and horizontal measurement might both be distances, or, if they think of a buoy, the horizontal component of the graph might be time. Length of daylight, in hours, on the vertical axis might be graphed against date, in days, on the horizontal axis. Sound waves might have air pressure on the vertical axis and time on the horizontal axis.]

Teacher Reflection

Was assigning specific roles to students in the groups effective, allowing for efficient data collection?
Was there adequate time for discussion of the key points in the lesson?
Do your students understand that dividing measurements on a circle by the radius of the circle converts the measurements into radian measure?
Were your students comfortable describing points on a circle in terms of arc length instead of angle?
This lesson was careful to avoid introducing a formula for translating between radians and degrees. Are you comfortable with that approach, or would you rather have students learn the formulas first?

NCTM Standards and Expectations

Algebra 9-12

Generalize patterns using explicitly defined and recursively defined functions.
Understand relations and functions and select, convert flexibly among, and use various representations for them.
Draw reasonable conclusions about a situation being modeled.

Measurement 9-12

Analyze precision, accuracy, and approximate error in measurement situations.
Make decisions about units and scales that are appropriate for problem situations involving measurement.