Illuminations: Pi Line

Pi Line
Students measure the diameter and circumference of various circular objects, plot the measurements on a graph, and relate the slope of the line to π, the ratio of circumference to diameter.

Learning Objectives
 
Students will:
  • Select appropriate scales to plot data collected.
  • Write an equation of a line of the form y = ax.
  • Interpret π as a ratio and as a slope.

Materials
 
Circular objects of various sizes
Masking tape (preferably, of bright color)
Scissors
Graph paper
Ruler or measuring tape
Graphing calculator (optional)

Instructional Plan
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 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:

  1. Wrap masking tape around the circle, overlapping the tape at the ends.
  2. Cut the tape and put it on the whiteboard to display the circumference. Write the word circumference on the piece of tape.
  3. 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.
  4. 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 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 predictions.

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 questions:

  • What is the y-coordinate of the y-intercept for any line? [0]
  • 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? [0]
  • 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 line.

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
 

The following questions appear on the second page of the Slope, Pi, and Lines overhead.

Slope, Pi, and Lines Overhead Slope, Pi, and Lines Overhead

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.]
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.]
Does the ratio of circumference to diameter vary depending on the size of the circle or the type of measurement (in., cm)? Explain.
[No. The ratio of circumference to diameter is constant, because all circles are similar. What measurements are used has no impact on the ratio.]
How does your equation relating circumference and diameter relate to the slope intercept equation y =  mx +  b? What are the values of m and b in your equation?
[Written in slope-intercept form, the circumference formula would be y =  πx + 0, meaning that m = π, and b = 0.]
Why are x and y considered variables, and why are m and b considered constants?
[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.]

Assessment Options
 
  1. Use a "think-pair-share" strategy to have students discuss whether the ratio of circumference to diameter varies depending on the size of the circle. First, ask students to decide individually whether the ratio varies, and have the class vote. (You might want to use "two-finger voting" so that all students vote at the same time. Students raise one finger for the first choice or two fingers for the second choice.) If the voting reveals that some students think the ratio changes, pair those students with other students who think the ratio is constant. After discussion, have students revote. If some students still think the ratio varies, ask others to suggest ways of convincing the student that the ratio is constant. Suggestions might include calculating the ratio of circumference to diameter and calculating the slope of the line using various combinations of data points.

  2. In their journals, allow students to summarize what it means that slope is a ratio and that π is a ratio.

Extensions
 
  1. Allow students to consider the following situation:
    As a sports agent for athletes, June gets 15% of a player’s earnings. Determine at least three different ordered pairs of the form (athlete’s earnings, June’s commission). Plot these points; find the equation of the line through the points; determine the slope of the line; and discuss the meaning of the slope of the line. Of what two quantities is the slope a ratio? How is this problem similar to the circle problem?

  2. Give each group a sheet of centimeter graph paper with circles of different sizes drawn over the grid. Each group then estimates the radius of their circles as well as the area by counting squares. Students record the data for each circle as a point of the form (r,A) where r is the radius and A is the area. Students can then create a scatterplot of the points, but before they do so, have them speculate as to the shape of the graph; is it likely to be linear or quadratic? Students should then use the regression feature to find the equation of the graph and consider the coefficient of the variable. You might want to ask, "What would be a more accurate equation? How do you know?" [The area of a circle is given by the formula, A = πr2, so the coefficient should be approximately π.] Use the formula to form at least six data points of the form (r, A). Plot the points and discuss why π is not the slope of a line in this situation. For a given area, have students use their graphs to estimate the radius of the associated circle.

Teacher Reflection
 
  • 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?

NCTM Standards and Expectations
 
Algebra 9-12
  1. Approximate and interpret rates of change from graphical and numerical data.
Geometry 9-12
  1. Analyze properties and determine attributes of two- and three-dimensional objects.
  2. Investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.
Measurement 9-12
  1. Make decisions about units and scales that are appropriate for problem situations involving measurement.
This lesson prepared by Martha Haehl.
  
1 period   

NCTM Resources

Principles and Standards for School Mathematics (Book and E-Standards CD)
National Council of Teachers of Mathematics Thinkfinity
More and Better Mathematics for All Students
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