An Educator's Reference Desk Lesson Plan
Date: April 19, 1996
Grade Level(s): 9, 10, 11, 12
Subject(s):
Introduction:
This project can be used when students are studying or after they have studied quadratic functions, however, other concepts are involved. The instructor can decide whether to cover these concepts prior to the project or along with the project.
I designed the project to integrate various areas of mathematics.
Topics Addressed In Project
Description Of Project
To capture the student's attention on the day I introduce the project, I have a song playing as the students walk into the room. (I usually use Joy To The World by Three Dog Night.) I pose the following problem to the students:
Procedure
We generally have a discussion about why a higher selling price does not necessarily result in higher profit. Students are then asked to suppose that their favorite recording group had just released a new tape. To collect data, I ask how many of them would be willing to pay specific amounts for the tape. I keep increasing the amount until all students reject the price. The following table shows a sample of the data I have collected thus far. Projected purchases will be explained below. The example on this page is based on this data. When I do the project with a new group I add the new data to the data collected with previous groups.
| COST/TAPE | PURCHASES | PROJECTED PURCHASES |
| $0.00 | 82 | 200000 |
| $5.00 | 81 | 197561 |
| $6.95 | 80 | 195122 |
| $7.45 | 78 | 190244 |
| $7.95 | 77 | 187805 |
| $8.25 | 66 | 160976 |
| $8.75 | 61 | 148780 |
| $9.00 | 60 | 146341 |
| $9.25 | 50 | 121951 |
| $9.50 | 47 | 114634 |
| $10.00 | 35 | 85366 |
| $10.50 | 30 | 73171 |
| $11.00 | 24 | 58537 |
| $12.00 | 17 | 41463 |
| $13.00 | 11 | 26829 |
| $13.50 | 9 | 21951 |
| $14.00 | 1 | 2439 |
| $15.00 | 1 | 2439 |
| $16.00 | 0 | 0 |
Depending on the background of the students and the technology available, students can make a scatter plot of the data on paper or enter the data on a graphing calculator or into a data analysis program on a computer to create a scatter plot. The purpose of the scatter plot is to determine the equation for a best-fit line.
On paper, the students can either "eye-ball" the line or learn to determine the median-median best fit. Some graphing calculators and computer packages will determine the equation for the students, either using median-median or least-squares. The linear expression from the best-fit equation will represent the approximate number of tapes that could sold based on the selling price of a tape.
As in all modeling situations, certain constraints and parameters must be considered. For this example we assumed a setup and marketing cost of $20000, a per tape production cost of $1.50, and a projected market population of 200000 people. Students modify the best-fit equation to reflect the ratio of the number of people in the study to a market of 200000. Students use the linear expression from the modified best-fit equation to create functions for cost, income and profit dependent on the per tape selling price. As with the scatter plot, various methods can be used to graph and analyze the functions. Students can graph the functions by hand and use algebraic methods to determine points for the analysis or they can use graphing calculators or function plotting software on a computer.
The cost function is defined as the per tape production cost (1.50) times the number of tapes sold (linear expression) plus setup and marketing cost (20000).
The income function is defined as the per tape selling price (independent variable) times the number of tapes sold (linear expression).
The profit function is defined as the income function minus the cost function.
The coordinates of the vertex of the parabola representing the profit will represent the optimum solution with the dependent value indicating the maximum profit and the independent variable indicating the selling price that will produce the maximum profit. In addition to the meaning of the vertex, I ask the students to analyze the graphs by interpreting, in relation to this particular problem, points where the graphs intersect the vertical axis and the horizontal axis, the meaning of the slope of the cost function and the best-fit line, etc. Other questions often arise, such as, "If the per tape production cost increases, should the selling price increase by the same amount to maintain maximum profit?", "Are the assumed parameters realistic?", "Should other constraints and/or parameters be considered?". These and other questions make good discussion topics for the students.
Updated Data
The data used in the example above was collected from students at Loup City High School, the Upward Bound Program at Briar Cliff College, Sioux City, Iowa and the Math and Science Regional Center, Northwest Missouri State University, Maryville, Missouri.
One aspect of this project that keeps it interesting is that the results change each time that new data is added to the previous data. I would like to keep my data updated with groups from various areas. If you choose to use this project, please send me the specific data from your class including the number of students in your study, the number willing to buy the tapes at the various selling prices, the name of your school. etc. If you need to include prices not currently listed, that is certainly acceptable.
I will update the data list as more data is sent to me and will display the updated list on the Cassette Tape Project page at the Loup City Public School homepage web site (address above). I will indicate schools and classes that have contributed data.
You may submit your data by e-mail to emccartn@genie.esu10.k12.ne.us or by snail-mail to the following address:
I hope your students enjoy the project.