Modeling in Mathematics: When is 99 Billion to Big

Modeling in Mathematics: When is 99 Billion Too Big?

Passing a McDonald Restaurant, you may still remember billboards which displayed counts of hamburgers the giant chain restaurant sold since the franchise opened. In early 1998 the count was 99 billion or 99 Giga burgers. The billboards were designed to display only two digits and the giant chain was fast approaching 100 billion, a three digit count. When the number was reached, how did McDonalds resolve the challenge of too many digits?

This project examines the growth of the franchise as seen in the hamburger counts or total number of McDonald chain restaurants. Can a mathematical model be found to model the growth? If graphed, what shape will the curve be? What type of a function is the best-fit?

The project is most suitable for students with access to TI-83⁺ calculator who enjoy a mathematical challenge. Counts or data on the total number of hamburgers (or restaurants) must be collected, a scatterplot with a best-fit graphing model found, and an analysis given.

If interested in the challenge, you need first research the counts of total burgers sold and/or the number of existing chain stores within last half of century. You may consider counts for each 5 year period; example end of … 1960, 1965, 1970, .... Organize data into table as shown below. For mathematical processing of data, click on "Calculator" link and "Scatterplot." Posted is a step by step tutorial on entering data in TI-83⁺ and finding the best fit line.

Beginnings to now…

McDonald Fast Food Chain began in early 1940's with the first restaurant was opened by McDonald's brothers. A decade later, McDonald's franchise was born. By mid 50's, approximately 50 million burgers were sold in what then was "a few" restaurants. Use the Web and other resources, to trace the total number of burgers sold through the past half a century. Use suggested guides to present a scholarly mathematical paper on your findings:

1. Present the data in a table. Information need not be found for each year, but should be given in even intervals, approximately twice for each decade. Title the table and give headings for each column, as example below shows:

Year	Number of hamburgers sold	(x , y); y in billions
…	…
…	…
1956	50 million	(6, 50)

2. To complete (x, y) coordinate column in the table, use suggested guide:

If your data begins with year 1950, let x = 0, represents year 1951; x = 5 represent year 1955; x = 10 represents year 1960. As shown in the table, (6, 50) is (x, y) coordinate which mathematically expresses the count of 50 billion for 1956.

3. Enter the data in your graphing calculator. Refer to step by step tutorial posted under "Calculator" link. Once you have obtained a scatterplot on the calculator, observe the pattern formed by the plotted points. Can a straight line be drawn that closely approximates most of the points? If so, use linear equation model. As explained in the TI-83⁺ tutorial, linear regression model is under STATS, CALC, LinReg(ax+b) . Do you observe the plotted pattern of points as non-linear? Is rapid growth indicated? If so, consider exponential model as a possible fit. On TI-83⁺ tutorial, linear regression model is under STATS, CALC, ExpReg.

Feel free to contact instructor for assistance with use of calculator. If on MCC campus, SAGE learning center can offer assistance.

5. To graphically represent found information, plot the data on XY-coordinate system.

Label X-axis as years and let number of hamburgers (or chain stores) be located on

Y-axis. Draw the best fit graph of the equation. Title your graph.

6. Describe the nature of the curve. Your discussion should give the best-fit model

(linear, exponential equation), discuss the model, and the credibility of predicted

short and long term future behavior.

Optional: You may also include statistics for total number of McDonald's restaurants for the same time period. However, if both are presented, a clear difference in table titles and graph colors is a must to make the distinctions understandable.

Project Format:

1. Write a typed scholarly discussion of your findings. No part of studied material may be "cut and paste." Any rewording of the references must be given full credit. Include a neatly presented graph. Interpret your findings. Note interesting observations, conclusions your research produced. Did anything surprise you?

2. Final page should be titled References. List main sources of information, using an accepted format for references. For Web sources, a title, author, URL address and posting date are required.

3. The general format of title page, short introduction, body, and conclusion is required. The introduction can simply state the purpose or intent of your work. Tables, graphs, and mathematical development with discussion are the body of your paper. Interpreting your findings can be the conclusion.