Rates of Change for Algebra One Students

Pat Mauch

I. Introduction

One of the topics that is often addressed in a first year calculus course is the idea of related rates of change. The graphing calculator has now made it possible to study these kinds of change by collecting discrete data and fitting a curve to that data. This curve can then be used to approximate instantaneous rates of change. The following experiment and related activities could be used with first year algebra students.

II. The Balloon Problem

The following problem on related rates was taken from Calculus Volume One by Tom M. Apostol. "Suppose a gas is pumped into a spherical balloon at a constant rate of 50 cubic centimeters per second. Assume that the gas pressure remains constant and that the balloon always has a spherical shape. How fast is the radius of the balloon changing when the radius is 5 centimeters?" The solution involves derivatives and the chain rule with an answer of 1/2pi (approximately .159).

III. Collecting Data in Lab Situation

Equipment:

Graphing calculators (TI-82 was used for the example).

Spherical balloons.

Outside calipers.

Small hand pump of the type used to inflate air mattress (optional).

Data collection:

Divide the students into groups of three. One student will inflate the balloon by pumping or blowing uniform breaths into the balloon at five second intervals. One student will measure the diameter of the balloon at five second intervals. The third student will record the data of time and diameter for each interval. The diameter values should be converted to radius values before they are interred in the calculator.

Calculator modeling: (General instructions for the TI-82 calculator will be given. These can be modified to other graphic calculators.)

1. Enter the time units as the first set of data points in list one.

2. Enter the measured radius values in list two.

3. Use a scatter plot to plot the discrete values from the experiment.

4. Find the best fit regression line. (The students should find this to be the power regression.)

5. Trace the regression line to find the values on either side of y = 5.

6. Determine the rate of change by comparing the change in y with the change in x.

Example data:

L(1) L(2)

(time) (radius measured)

_____ _____

5 4

10 5

15 5.5

20 6

25 6.75

30 7

35 7.5

40 7.75

45 8

50 8.5

Using the power regression y = ax^b a = 2.34201560 and b = .3243611769. Near y values yield y = 4.9848088 with x= 10.265957 and y = 5.0736281 with x = 10.840426. The resulting rate of change is approximately .155. If you zoom in on the curve, the result can be improved to approximately .157.

This particular experiment is hands on and should be fun for the students. The resulting curve is very "friendly" and small errors in the lab should still produce results that are reasonably good.

IV. Revisiting the Problem

The students should keep the results of this experiment in their portfolios. After the class has covered isolation of variables in an equation, the problem can be approached in a different way.

1. Enter the values from 1 to 10 in list 1.

2. Enter the values ( 50*L(1)) in list 2.

3. Enter the radius values () in list 3.

4. Plot L(1) and L(3).

5. Plot the regression line.

6. Trace the regression line to find the rate of change.

Example data:

L(1) L(2) L(3)

(time) (volume) (radius)

_____ _______ ______

1 50 2.2854

2 100 2.8794

3 150 3.2961

4 200 3.6278

5 250 3.908

6 300 4.1528

7 350 4.3718

8 400 4.5708

9 450 4.7538

10 500 4.9237

Since the regression line is a perfect fit, the answer can be determined to whatever degree of accuracy is needed by zooming in on the graph. I feel it is important to plot and find the regression line rather than simply graph the equation. This reinforces the idea of working from discrete values to the continuous and keeps a strong link with the previous lab work. This may be a good time to discuss the relationship between the a and b values in the regression equations (lab and revisit ) and the value of .75pi and 1/3. The idea of inverse functions, curve fitting, and residuals can also be explored at this time.

V. Extensions of the Problem

Perhaps the original problem is incomplete or doesn't address some of the more interesting situations involved with balloon inflating. Here is a short list of other ideas that can be explored.

1. How fast is the surface area of the balloon changing at a particular time?

2. How fast is the radius and surface area of the balloon changing at the time when it bursts?

3. How long will it take the balloon to burst?

4. If we experiment with spherical balloons of different sizes (masses), can we predict when it will burst? ( Determine a constant that can be used, in conjunction with volume or radius, to predict bursting.)

5. Find the mass of the balloon and, assuming that it is filled with He rather than air, determine the size of the balloon required for it to rise.

6. How high will the balloon rise? How fast will it rise? Will it burst?

7. Explore all of these ideas with a balloon that is in the shape of a cylinder.

8. Others?

VI. Conclusion

The concept of change and rates of change is one that is very important to the study of mathematics, and its application to real life situations. In the past these ideas were not explored, to any great extent, with students in junior high or beginning high school. Now, with the graphing calculator and computer spreadsheets available to most classrooms, the instructor has the opportunity to explore these topics with students much earlier in their mathematical development. The work in this paper is simply an example of the type of work that can be done to explore change.

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