Introduction to Trigonometry
Trigonometry Unit: Introduction to Radians (from
http://illuminations.nctm.org/LessonDetail.aspx?id=L844)
Day ____ of ____
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
Outcomes/Assessment:
Formative:
Self/Peer
Materials/Need to write on the board
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
Starter
5-10 min
Introduction
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," 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.
Stage 1: Experimenting
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.
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
10-15 Min
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.
Stage 2: Reflect & Discuss
10-15 Min
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.
Stage 3: Hypothesizing & Articulating
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
10 Min
graphs and data, to answer the questions, but they will need to compare their
answers to other groups' for Question 8.
Stage 4: Verifying & Refining
Using a large group discussion, have students present their results and verify that all
students have reached the same, correct result. 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
10 Min
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.
Homework
Distribute Homework.
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?
Extension/Remediation
1. 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.]
2. 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.]
3. 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.]