(The following lesson was created for use with students in grades six through nine. It first appeared in the June 1996 issue of the CMC Communicator, a publication of the California Mathematics Council. It is reprinted with permission of the CMC.)
Background for Activity:
When an artist friend tried to mail handmade invitations to a birthday celebration, she was told that the envelopes were not regulation size. The invitations could not be sent through the U.S. Mail unless they were placed in larger envelopes. Her dilemma inspired this measurement activity.
Directions:
Collect envelopes of various sizes, enough for each group of students to have at least four envelopes. Post on the wall a large sign listing the U.S. Postal regulations, effective January 1, 1995, which includes first class domestic mail at $0.32:
An envelope must be a rectangle which is at least 0.007 inches thick, but it must not be more than 1/4 inch thick. It must weigh one ounce or less. The length of the envelope must be at least 5 inches and its height at least 3 1/2 inches. The ratio of length to height must be between 1.3 and 2.5. Envelopes with length greater than 11 1/2 inches, or height greater than 6 1/8 inches, or thickness greater than 1/4 inch cost more to mail.
First, students trace on paper each envelope that has been distributed to their group, then measure and record the length and height of each envelope.
Next each group (or the whole class) will fold paper to see how thick 1/4 inch would be. Students discuss how they would measure 0.007 inch with the paper they have on hand. Our class decided that, since 2000 sheets made a stack much taller than seven inches, there was no danger that the paper was too thin (since 2000 sheets would represent 1000 envelopes).
Then students calculate the ratio of length to height to see if the dimensions are within the U.S. postal limits of 1.3 to 2.5.
Finally, using all the information, students determine which envelopes could be sent first class at $0.32.
On the second day, students discuss the postal regulations for $0.20 postcards:
A postcard must be a rectangle which is not larger than 4 1/4 inches by 6 inches by 0.0095 inches thick and not smaller than 3 1/2 inches by 5 inches by 0.007 inches thick.
One student designed a simple template by marking the key points (see Figure 1). Then it was easy to determine which postcards and envelopes could be mailed first class. The shaded area that was within regulations for postcards and letters had to fit between the too small and too large boundaries.
This measurement activity is particularly appropriate for holiday times when cards are exchanged.
Extension:
Postal regulations for first class mail can be expressed by algebra students in terms of linear inequalities. Students are surprised to see that the shaded area of acceptable envelopes is so small.