Display the first page of the Slope, Pi, and Lines Overhead on the projector. Use these questions to conduct a brief
discussion. Note that these questions are merely to set the stage for
the activity; it is not necessary that each question be fully answered
during the discussion.
Slope, Pi, and Lines Overhead
After the discussion, draw a coordinate plane on a whiteboard
and label both axes with the same scale. (Actual measurements in
centimeters or inches would be good, if the scale can go high enough to
represent the circumference of the largest circle.) Points will only be
plotted in the first quadrant.
Demonstrate the following process, which will be used during the lesson:
- Wrap masking tape around the circle, overlapping the tape at the ends.
- Cut the tape and put it on the whiteboard to display the circumference. Write the word circumference on the piece of tape.
- Stretch another piece of masking tape across the widest part
of the circle (the diameter) through the center and cut off the ends.
Write the word diameter on the strip of tape.
- For each circle, stretch the tape for the diameter below the x‑axis and parallel to it. At its end, position the tape for the circumference of that circle so that one end rests on the x‑axis, and stretch the tape vertically. Plot and label the point at the top of the circumference strip. (See diagram below.)
Distribute the Slope, Pi, and Lines Activity Sheet. Students will answer the questions on this sheet as they proceed through the activity.
Slope, Pi, and Lines Activity Sheet
Divide students into groups of three students each. Each group
will need several circular objects of different sizes, a roll of
masking tape, a pair of scissors, and a whiteboard. Allow them to
measure and record the diameter and circumference of at least three
objects. More items can be used if time permits.
After all groups have plotted several points, reconvene the
entire class. Ask students to use their data to predict the
circumference and diameters of various circles if the other piece is
known. For instance, ask them to predict the circumference if the
diameter is 22 centimeters [approximately 69.1 centimeters], and ask
them to predict the diameter if the circumference is 12 centimeters
[approximately 3.8 centimeters]. Students should recognize that the
points form a straight line and that the line can be extended to make
Discuss where the y-intercept of the line is likely to occur.
Students should recognize that the points seem to be on a line that
will pass through the origin. To reinforce the idea, ask the following
- What is the y-coordinate of the y-intercept for any line? 
- In the context of this problem, what does an x-value of 0 mean? [In the graph, x-values represent the diameter, so an x-value of 0 indicates that the diameter is 0.]
- For a circle with a diameter of 0, what is the circumference? 
- So, where should the y-intercept occur for the line in your graph? [At the origin.]
Have students estimate a line of best fit for their scatterplot. Note that this is best done after the discussion about the y-intercept.
Although the masking tape measurements will give approximate points,
students can be certain that the point (0,0) occurs along the line of
best fit. Therefore, students can place a piece of uncooked spaghetti
with one end at the origin, and move the other end to approximate the
Allow students to generate an equation that represents their line of best fit.
You may wish to have students enter the data that they gather
into a graphing calculator and use the regression feature to find the
line of best fit. Alternatively, students can use the Spreadsheet and Graphing Tool as follows:
- Choose the Data tab. The diameters can be entered in Column A, and the circumferences can be entered in Column B.
- Select Y= or Plots and highlight Plot 1: Column A, Column B. A scatterplot of the data will appear when the Graph tab is selected. (The values in the Window tab may need to be adjusted to view all points in the scatterplot.)
- Return to the Y= or Plots tab. Students can estimate an equation for the line of best fit and return to the Graph to see how well their estimate approximates the data.
Spreadsheet and Graphing Tool
Discuss how the slope of the line relates to the circle. Ask, "What
formula does the above equation of a line approximate; that is, what
formula relates circumference to diameter?" Some students may know the
answer to this question because they know the formula C = πd,
and this question was discussed in the Pre-Activity Questions. Others
may not, and this is a good opportunity to discuss the concept of
"constant rate of change." If students have difficulty recognizing that
the slope of their line is approximately π, it might be helpful to have
them calculate the slope by hand using one of the data points and the y-intercept. Then, discuss what quantities are being compared.
Questions for Students
1. The following questions appear on the second page of the Slope, Pi, and Lines Overhead.
2. What does it mean to say that π is a ratio? What is being compared?
[Circumference is compared to diameter. Specifically, π is the ratio C:d.]
3. What does it mean to say that the slope of a line is a ratio? In this activity, what quantities were being compared?
[The slope of a line compares the ratio of change in y-values to change in x-values.
In this activity, the change in circumference was compared to the
change in diameter. Because this ratio is always equal to π, there is a
constant rate of change.]
4. Does the ratio of circumference to diameter vary
depending on the size of the circle or the type of measurement (in.,
[No. The ratio of circumference to diameter is constant,
because all circles are similar. What measurements are used has no
impact on the ratio.]
5. How does your equation relating circumference and diameter relate to the slope intercept equation y
? What are the values of m
in your equation?
[Written in slope-intercept form, the circumference formula would be y = πx + 0, meaning that m = π, and b = 0.]
6. Why are x
considered variables, and why are m
[The variables x and y represent quantities that change. Although also represented with lowercase letters, both m and b are not variables because their values do not change, so they are considered constants.]
- Did students develop a greater understanding of slope as a rate of change?
- Did students make the connection that pi is a ratio comparing circumference to diameter, no matter the size of the circle?
- How did you challenge the high-achievers in your class?
- Was your lesson appropriately adapted for the diverse learner?
- Did you set clear expectations so that students knew what was expected of them? If not, how can you make them clearer?