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)

### Questions for Students

 The following questions appear on the second page of the 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

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

### Extensions

 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? 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 9-12Approximate and interpret rates of change from graphical and numerical data. Geometry 9-12Analyze properties and determine attributes of two- and three-dimensional objects. Investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates. Measurement 9-12Make 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)

 More and Better Mathematics for All Students
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