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.
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.
Rolling into Radians Data Collection Activity Sheet
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
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.
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
Rolling into Radians Questions Activity Sheet
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."
This activity is adapted from Allan Bellman et al., Mathematics: Modeling Our World, Course 3 (New York: W. H. Freeman, 1999), 403-405.