The following exploration would be a great activity to use in a high school Algebra class. It explores a real-life function in which students must create an equation to describe the function and then use it to predict future values.
The data below is based on the first class letter postage for the United States from 1919 to 1998. Using Microsoft Excel, plot the data and develop a prediction function.
| year | rate(in cents) |
| 1919 | 2 |
| 1932 | 3 |
| 1958 | 4 |
| 1963 | 5 |
| 1968 | 6 |
| 1971 | 8 |
| 1974 | 10 |
| 1975 | 13 |
| 1978 | 15 |
| 1981 | 20 |
| 1985 | 22 |
| 1988 | 25 |
| 1991 | 29 |
| 1994 | 32 |
| 1998 | 33 |
In Excel, you can create a graph that represents this data.
Looking at this graph, it is clear that the function to describe the data will be some type of exponential function. In order to find a prediction function, I used another type of technology, the TI-83 Plus. On the TI-83, go to STAT. This brings up the EDIT menu in STAT. You have to edit your list by entering the data, so hit the "ENTER" key. Now you are at the data table. The top row is L1, L2, etc. The L1 data to go in the first column is the list of x-values (the list of years). So enter the years into the first column. (It may be easier to use 19 for the year 1919, 33 for the year 1933, etc.) The data for the L2 column is the list of postage stamp prices ( the y-values). After all of the data is entered, hit the STAT button again. Under STAT, go to the CALC (calculate) menu and down to ExpReg (exponential regression) and hit ENTER. Now you have a screen that says"ExpReg". You have to tell the calculator what data to use in creating an equation for the data. The data comes from L1 and L2 with the L1 list being the x-values. So hit "2nd" "1" to get the L1 data listed first and then the "," key. Now the y-values have to be entered, so hit "2nd" "2" to the the L2 data listed. Finally, hit ENTER and the calculator will create the function to describe your data. The calculator comes back with the following information:
So the estimated regression equation for the price of stamps is y = .0065*1.0401^x. With this estimated equation, we can use the program Graphing Calculator to create a graph of the function.
Note that the years are along the x-axis and the prices are along the y-axis. The x-value 20 corresponds to 1920 and the x-value of 100 corresponds to the year 2000. The y-value of 0.2 corresponds to 20 cents. With the Graphing Calculator software, you can move around to different points on the function to find their values. This is a very valuable tool for students. It allows them to find values of the function without a lot of pencil-and-paper computation. By moving along the function, it is easy to answer the questions posed in this problem.
Question 1: When will the cost of a first class postage stamp reach $1.00?
By moving the little blue box seen above in Graphing Calculator, a student could manipulate its position until it was at or very close to 100 (representing $1.00) on the y-axis to find the x-value at $1.00. At the top of the grid, you can see the x and y values of whatever point you are on in the plane. When the price is about $1.00, the corresponding year is x = 129 meaning the actual year of 2029.
Question 2: When will the cost be 64 cents?
Again, moving the blue box to a cost of about 64 cents, you can find that the corresponding year is 2018.
Question 3: How soon should we expect the next 3 cent increase?
A three cent increase would mean that the price of stamps be 36 cents. So, following our estimated function, we can predict that stamps will cost 36 cents in the year 2003.
The three questions above could also be solved without using the software after an estimated function is obtained. By pencil- and-paper computation or with a TI-89 calculator, you would do the following:
Question 1: Set up the equation 1.00 = 0.0062*1.0401^x.
Solving for x, you get x = 129.2882 which corresponds to the answer obtained when using Graphing Calculator.
Question 2: Set up the equation 0.64 = 0.0062*1.0401^x.
Again, solving by hand or by letting the TI-89 solve the equation, you will get x = 117.9372.
Question 3: Set up the equation 0.36 = 0.0062*1.0401^x. Solving by hand or with the calculator, x = 103.30312.
EXTENSIONS:
The main goal of this problem was to find the year in which the postage prices would change. As a teacher, you could extend this problem by asking what month of the year calculated would the price change. An even further example would be to ask the students to calculate the particular day that the postage prices would change. It would also be important to discuss that the work being done here is to make predictions about the future. Why are predictions important? What other types of predictions could be made?
Allowing students to use software like Graphing Calculator (or even graphing calculators) helps to enhance their learning and get them excited and involved in what they are doing. The technology also helps students make connections between mathematics and the real-world.