When you bring a letter or package to the post office, the clerk usually uses a computerized scale to determine the postage charge. Have you ever wondered how this amount is calculated? Or have you ever wondered how postage rates were calculated in the past? You can answer these questions and many other interesting questions about postal rates by accessing the post office’s web site.

This project is primarily designed for use in a mathematics class when discussing graphs and curve fitting. Part III would make a good integrative lesson with an economics class when studying inflation.

Previous Knowledge Needed.

For Part I, familiarity with the Internet, basic graphing knowledge, and equations of lines. For Part II, students should be familiar with linear, quadratic, power, exponential and piece-wise defined functions. They should be able to find "best-fitting" curves and be able to interpret the coefficient of determination.

Access the Internet to obtain current and historical postal rate information;

Define and graph piecewise-defined functions;

Determine a linear equation to relate weight and postal rate;

Find best-fit curves to describe how postal rates have changed over time;

Use the best-fit curves to predict the next postal rate change and when it will be;

In Part I, students will access the Postal Service web site and find the cost to mail a first-class letter. They will graph rate versus weight and find a linear equation to describe this relationship. In Part II, students will see how the cost of sending a first-class letter has varied over time. They will find a best-fit curve relating year and cost and use this curve to predict the cost to send a letter in future years. In Part III, students will look at inflation rates using the Consumer Price Index and try to determine if postal hikes matched the inflation rates. Some interesting economics questions regarding the causes of inflation can be studied at this time.

Completion of the Postal Math worksheets.

Internet access, TI-83 graphing calculator or other calculator or computer package that will compute best-fit curves.

Students may wish to study the formulas used to compute the CPI each year. They may want to research the causes and effects of inflation.

!!!! Warning !!!! The Web sites given in this lesson may have changed! Before using this lesson with your students, be sure to check if the sites are still working or if you must find another site. Sometimes the sites still have the relevant data but you may need to change the directions to access the data.

Students may recognize that the function they are working with in Part I, Questions 1 - 3 is an example of a transformation of the greatest integer function.

If you wish more background about the statistical concepts involved in the lesson, some good sites to check are:

http://davidmlane.com/hyperstat/index.html

http://www.anu.edu.au/nceph/surfstat/surfstat-home/surfstat.html

http://www.math.unb.ca/~maureen/SSCEdCom/basicstats/basicstats.html

http://www.math.unb.ca/~knight/BasicStat/$content.htm

http://www.bbns.org/us/math/ap_stats

http://www.grad.cgs.edu/wise/linksf.shtml

http://www.cvgs.k12.va.us/DIGSTATS

http://www.statsoft.com/textbook/stathome.html

http://www.stats.gla.ac.uk/steps/glossary/index.html

http://www.crpc.rice.edu/CRPC/GT/sboone/Lessons/lptitle.html

http://forum.swarthmore.edu/library/topics/statistics

http://www.psychstat.smsu.edu/introbook/skb00.htm

TI-83 instructions:

http://www.ti.com/calc/docs/act/koehler001.htm

http://www.wku.edu/~neal/manual/ti83.html

The Calculator website at the Mathematics Department of the University of Wisconsin-La Crosse will perform basic statistical calculations. If you do not have access to a simple statistical computer package or calculators with statistics options, your students may access http://www.compute.uwlax.edu/stats_htdocs/newmenu.html to perform statistical computations on-line.

In order to print out just a copy of the student worksheet, highlight this section, then copy and paste it into your word processor. You may then revise the worksheet if you wish.

What is the Consumer Price Index and how is it used?

The consumer price index is a measure which is used to track the change in prices for common household goods over time. The consumer price index is developed using a "market basket" approach. That is, researchers determine the cost of a particular set of goods and services every year. This cost is then compared against the cost of goods and services from other years.

To determine the percentage change in prices between two years, simply use the following formula:

Percentage Change between YEAR A and YEAR B = (CPI for Year B) / (CPI for Year A)

The CPI can also be used to compare dollar amounts between years using "constant dollars." Constant dollars are dollars which are adjusted for inflation. For example, if you know the price of an item in YEAR A, and would like to know what that item would have cost in YEAR B, adjusting for inflation, you would use the following formula:

Price in YEAR B dollars = (Price in Year A dollars) X (CPI for Year B) / (CPI for Year A)

IRP has developed a "CPI calculator" that automates this second calculation and illustrates the process of inflating and deflating dollar amounts from year to year. http://www.ssc.wisc.edu/irp/faqs/faq5dir/cpicalc.htm

The actual values of the CPI are listed below. There are actually several Consumer Price Indexes, depending on the particular "market basket" (set of consumer goods) included, the set of consumers involved, and geographical factors. Two of these CPIs are used most frequently. In the 1996 Green Book, the House Committee on Ways and Means explains the difference between the two indexes as follows:

You can read more details about the Consumer Price Index and how it is used at the Consumer Price Indexes Home Page maintained by the Bureau of Labor Statistics, at http://www.bls.gov/cpihome.htm

You can read more details about the Poverty Thresholds and how they differ from the Poverty Guidelines at a page maintained by IRP, at http://www.ssc.wisc.edu/irp/faqs/faq5.htm

Values of the CPI-U and CPI-U-X1

Notes:  
*Poverty Threshold is weighted average threshold, for a family of 4.
**CPI-U here is for base year of 1982-1984=100.
***1998 Poverty Threshold is a preliminary figure, as of January 19, 1999.


Year    Poverty Threshold*  CPI-U**   CPI-U-X1
          (family of 4)

1998      $16,655***          163.0
1997      $16,400             160.5
1996      $16,036             156.9
1995      $15,569             152.4
1994      $15,141             148.2
1993      $14,763             144.5     144.5
1992      $14,335             140.3     140.3
1991      $13,924             136.2     136.2
1990      $13,359             130.7     130.7
1989      $12,674             124.0     124.0
1988      $12,092             118.3     118.3
1987      $11,611             113.6     113.6
1986      $11,203             109.6     109.6
1985      $10,989             107.6     107.6
1984      $10,609             103.9     103.9
1983      $10,178             99.6      99.6
1982      $9,862              96.5      95.6
1981      $9,287              90.9      90.1
1980      $8,414              82.4      82.3
1979      $7,412              72.6      74.0
1978      $6,662              65.2      67.5
1977      $6,191              60.6      63.2
1976      $5,815              56.9      59.4
1975      $5,500              53.8      56.2
1974      $5,038              49.3      51.9
1973      $4,540              44.4      47.2
1972      $4,275              41.8      44.4
1971      $4,137              40.5      43.1
1970      $3,968              38.8      41.3
1969      $3,743              36.7      39.4
1968      $3,553              34.8      37.7
1967      $3,410              33.4      36.3
1966      $3,317              32.4      35.2
1965      $3,223              31.5      34.2
1964      $3,169              31.0      33.7
1963      $3,128              30.6      33.3
1962      $3,089              30.2      32.8
1961      $3,054              29.9      32.5
1960      $3,022              29.6      32.2
1959      $2,973              29.1      31.6
1958                          28.9      31.4
1957                          28.1      30.5
1956                          27.2      29.6
1955                          26.8      29.1
1954                          26.9      29.2
1953                          26.7      29.0
1952                          26.5      28.8
1951                          26.0      28.3
1950                          24.1      26.2
1949                          23.8      25.9
1948                          24.1      26.2
1947                          22.3      24.2
          

1. Find the current cost to mail a first class letter by weight. Access the Internet site http://www.usps.gov Choose Calculate Rates, then U.S. Domestic Rates, then First Class Mail. Fill in the postal rates to send a first class letter up to 10 oz.

2. What would it cost to send a letter that weighs

3.6 oz? __________

7 oz? __________

0.4 oz? __________

3. Plot points showing cost versus weight on the following grid.

4. Look at your table again in #1 above. How much does the cost increase when the weight increases

from 1 to 2 oz? __________

from 2 to 3 oz? __________

from 3 to 4 oz? __________

Is the cost increase the same for all one ounce increases in weight? __________

5. Let x denote the weight of a letter and y the cost to send it. Suppose x is given only in whole number of ounces, like 5 ounces, but not 3.2 ounces. Find a linear equation y = mx + b to calculate y in terms of x. Hint: what is the slope?

6. Draw the line you specified in #5 on the graph in #3.

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When the U. S. Post Office was first established, the cost to send a letter was based on the number of sheets in the letter and the distance it was traveling. Beginning in 1863, however, domestic letter rates became based solely on weight. Access the Postal Rate Commission site http://www.prc.gov to find out what the cost of a first-class stamp was from 1885 to the present. Choose Postal Rates and Fees, then History of Rates and First Class Stamp Rates.

1. Starting from 1958, enter the year and cost of a first-class stamp in two lists in your TI-83 calculator. Calculate a third list so that 1958 will be represented as year 1. If you have entered the years in list L₁, and the costs in L₂, then use the formula L₃ = L₁ - 1957. (If you use the actual years, the TI-83 will display the overflow error when fitting a power curve.)

2. Make a scatterplot of this data with the horizontal axis denoting year (where 1958 = Year 1) and the vertical axis denoting cost. What sort of equation do you think might describe these points?

3. Fit the following curves to this data:

Which curve has the "best" coefficient of determination? _____________________

4. Using the equations above, predict what the postage rate will be in the year 2000; in 2010.

5. Which predictions do you think are the best? Explain.

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After about 1950, postal rates started to increase very often and by large amounts. Was there similar inflation in other parts of the economy? One measure of the rate of inflation is the Consumer Price Index. The CPI tracks the change in prices over time for a particular set of basic consumer goods and services. Researchers report the cost of this set of goods and services each year. Hence one can calculate the rate of inflation from one year to another by comparing costs of this set of goods and services for these two years. You can read about the CPI, find out how it is measured and see a table of CPI values at the site http://www.ssc.wisc.edu/irp/faqs/faq5.htm

1. (3 points) Look at the CPI-U values from this site. Did it ever happen that the CPI-U decreased from one year to the next? If so, when?

2. (3 points) To see if postal rates increased at the same rate as the CPI, access the CPI Price Change Calculator site http://www.ssc.wisc.edu/irp/faqs/faq5dir/cpicalc.htm This site will convert a dollar amount from one year to its equivalent another year. In 1971, the cost of a first-class stamp was 7 cents. If postal rates followed the CPI inflation rates, what should a first-class stamp cost in 1988? Was the actual 1988 price more, less or the same as this value? (Enter .07 and 1971 in the first two boxes; then 1988 in the third box.)

3. (3 points each) Convert the following costs of a first-class stamp. Compare to the actual cost. The first one has been done for you.

4. Would you conclude that the Postal Service raised rates according to inflation as measured by the CPI-U? Explain.

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 1. Find the current cost to mail a first class letter by weight. Access the Internet site http://www.usps.gov Choose Calculate Rates, the U.S. Domestic Rates, then First Class Mail. Fill in the postal rates to send a first class letter up to 10 oz.

2. What would it cost to send a letter that weighs

3.6 oz? __ 0.99 __

7 oz? __ 1.65 __

0.4 oz? __ 0.33 __

3. Plot points showing rate versus cost on the following grid.

 The formula for the above graph is y = 0.22 Int(x) + 0.33

4. Look at your table again in #1 above. How much does the cost increase when the weight increases

from 1 to 2 oz? ___ 0.22 ___

from 2 to 3 oz?___ 0.22 ___

from 3 to 4 oz? ___ 0.22 ___

Is the cost increase the same for all one ounce increases in weight? Yes

5. Let x denote the weight of a letter and y the cost to send it. Suppose x is given only in whole number of ounces, like 5 ounces, but not 3.2 ounces. Find a linear equation y = mx + b to calculate y in terms of x. Hint: what is the slope?

y = 0.22x + 0.11

6. Draw the line you specified in #5 on the grid in #3.

When the U. S. Post Office was first established, the cost to send a letter was based on the number of sheets in the letter and the distance it was traveling. Beginning in 1863, however, domestic letter rates became based solely on weight. Access the Postal Rate Commission site http://www.prc.gov to find out what the cost of a first-class stamp was from 1885 to the present. Choose Postal Rates and Fees, then History of Rates and First Class Stamp Rates.

1. Starting from 1958, enter the year and cost of a first-class stamp in two lists in your TI-83 calculator. Calculate a third list so that 1958 will be represented as year 1. If you have entered the years in list L₁, and the costs in L₂, then use the formula L₃ = L₁ - 1957. (If you use the actual years, the TI-83 will display the overflow error when fitting a power curve.)

2. Make a scatterplot of this data with the horizontal axis denoting year and the vertical axis denoting cost. What sort of equation do you think might or might not describe these points?

The points seem to follow a straight line, especially after 1970.

3. Fit the following curves to this data:

Which curve has the "best" coefficient of determination? quadratic or linear

4. Using the equations above, predict what the postage rate will be in the year 2000; in 2010.

5. Which predictions do you think are the best? Explain.

The linear prediction. There is little inflation this year so it is unlikely that the post office will raise the rate to 37 cents (quadratic prediction). The power fit is not reasonable and the exponential much too high.

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After about 1950, postal rates started to increase very often and by large amounts. Was there similar inflation in other parts of the economy? One measure of the rate of inflation is the Consumer Price Index. The CPI tracks the change in prices over time for a particular set of basic consumer goods and services. Researchers report the cost of this set of goods and services each year. Hence one can calculate the rate of inflation from one year to another by comparing costs of this set of goods and services for these two years. You can read about the CPI, find out how it is measured and see a table of CPI values at the site http://www.ssc.wisc.edu/irp/faqs/faq5.htm

1. (3 points) Look at the CPI-U values from this site. Did it ever happen that the CPI-U decreased from one year to the next? If so, when?

Yes, once, from 1948 to 1949.

2. (3 points) To see if postal rates increased at the same rate as the CPI, access the CPI Price Change Calculator site http://www.ssc.wisc.edu/irp/faqs/faq5dir/cpicalc.htm This site will convert a dollar amount from one year to its equivalent another year. In 1971, the cost of a first-class stamp was 7 cents. If postal rates followed the CPI inflation rates, what should a first-class stamp cost in 1988? Was the actual 1988 price more, less or the same as this value? (Enter .07 and 1971 in the first two boxes; then 1988 in the third box.)

20.4 cents; in 1988 the actual cost was 25 cents, more than the CPI-U inflation rate

3. (3 points each) Convert the following costs of a first-class stamp. Compare to the actual cost. The first one has been done for you.

4. Would you conclude that the Postal Service raised rates according to inflation as measured by the CPI-U? Explain.

Generally yes, though it seems the Postal Service raised prices slightly more.