What Will It Cost?

Jayakar James Ellen Kidwell Rose Mack John Ruebusch Nancy Stout

Class: Algebra II

Materials: Pencil, paper, straight edge, graph paper, graphing calculator and/or software with statistics graphing capability.

Goals: To: 1) reinforce graphing techniques (linear, step-function, quadratic, power, exponential); 2) support interpretations of graphs; 3) expand experience in curve-fitting (determining curve of best fit); and 4) strengthen ability to predict future behavior from graphs.

Background: Students should have 1) skills in using reference books and magazines (possible extension); 2) competency with graphing points and functions; 3) ability to input and statistically analyze data on a graphing calculator; 4) knowledge of the r factor in linear regression; and 5) facility in evaluating functions and solving equations.

Setting: In 1994, the United States Postal Service requested a price increase for mailing a one ounce first-class letter or package. Though the federal government did not grant the request, we can guess that a rate change is imminent. Based on postal rates in the past, can we predict what the postage charge for a one ounce mailing will be in the year 2000?

Problem: Based on the information from the 1993 Statistical Abstract on postal rates, you are to plot a graph of years versus postage rate, from 1958 to the present. Use statistical analysis of graphs to determine the curve of best fit. After identifying the corresponding function, find the probable cost of a stamp in the year 2000, the probable cost of a stamp in the year 2025, and the year in which postage will probably reach \$.50 for one ounce.

Evaluation: This project should take at most two days, depending on the level of proficiency on the calculator. We recommend that you evaluate this project for completeness and accuracy of the graphs. Our objective is to reinforce graphing techniques using a mathematical model. It is suggested that the explanation for the choice of curve be written as a narrative.

Extension: Students could record data from the beginning of the US Postal Service until the present. The rates have not always increased, and the data points are more scattered. Perhaps a different answer with other justifications would result. Additionally, students could work with data such as the winning times for the Olympic 400-meter women=s race or the men=s mile run, both of which would generate different graphs.

Teacher Notes: We found the quadratic equation to be the best fit because 7 of the 12 data points matched. However, we used integral values for the independent variable, except in the case of March 22 and November 1, 1981, for which we used values of 81.25 and 81.83. If all the independent variables are written as integers, the basic shape of the curve may change enough for a different curve to fit. In addition, we ignored the rate change on September 14, 1975, as that was simply a modification in the method rather than an actual variation.

References:

1. US Bureau of the Census, Statistical Abstract of the United States: 1993

3. UCSMP, Functions, Statistics, and Trigonometry, Scott Foresman, 1992

What Will It Cost?

Student Data Sheets

1. Fill in this table:

 Date Rate Date Rate

2. Graph these data points on graph paper using appropriate labels.

3. Graph these data points on a graphing calculator, using
mode.

a. What is the equation of the line that might fit these data?

b. What is the equation of the quadratic graph that might fit these data?

c. What is the equation of the power function that might fit these data?

d. What is the equation of the exponential function that might fit these data?

1. Which one of these is the best choice, and why?

f. What will a stamp cost in the year 2000?

g. What will a stamp cost in the year 2025?

h. In what year will a stamp cost \$.50?

5. Develop a step-function graph of the data points on the graphing calculator, then draw the graph on graph paper.

a. What was the longest period of time between rate changes?

b. What was the largest rate change within a single period? When did this occur?

Data from page 560, Statistical Abstract of the United States: 1993

 Date of Rate Change Letters (each oz) Letters (first oz) Letters (each added oz) Date of Rate Change Letters (each oz) Letters (first oz) Letters (each added oz) 1958 (Aug 1) \$0.04 ---- ---- 1978 (May 29) ---- \$0.15 \$0.13 1963 (Jan 7) \$0.05 ---- ---- 1981 (Mar 22) ---- \$0.18 \$0.17 1968 (Jan 7) \$0.06 ---- ---- 1981 (Nov 1) ---- \$0.20 \$0.17 1971 (May 16) \$0.08 ---- ---- 1985 (Feb 17) ---- \$0.22 \$0.17 1974 (Mar 2) \$0.10 ---- ---- 1988 (Apr 3) ---- \$0.25 \$0.20 1975 (Sep 14) ---- \$0.10 \$0.09 1991 (Feb 3) ---- \$0.29 \$0.23 1975 (Dec 31) ---- \$0.13 \$0.11

Funded in part by the National Science Foundation and Indiana University 1995

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