Crossroads in Mathematics

Example 8

Problem Posing

It is a good idea to let students create their own problems from a given context once they have had some experience in problem solving. This gives students a great deal of insight into the problem-solving process. It is empowering for students to realize they can create problems. Theirs will be tougher than common textbook problems in many cases. When given the following problem situation, students often generate problems involving inequalities or questions about domain and range of functions.

Problem situation. Janet is making a rectangular end table for her living room. She has decided the tabletop should have a surface area of 625 sq. in.

Task. Write three different types of questions that could be asked about this problem situation. Solve two of them.

Some students responses:

What perimeters are possible?
What would the perimeter be if Janet makes it a square?
The distance between the end of the sofa and the wall where the end table will go is only 18 in. How long could the end table be?
If the table is at least 20 in. wide, how long could it be?
What range of dimensions (length and width) are reasonable?
What are the dimensions if the table is twice as long as it is wide?

Table 1 suggests an ongoing solution strategy for the last question that requires no algebra. Problem posing can be done early in an introductory college mathematics curriculum; it is not an activity appropriate for only the precalculus level. The method of successive approximations, or systematic guess-and-check, used in Table 1 is one that can be used by students at any level. Note the "Process" columns. They are valuable because, as is often the case, the arithmetic process can give insight into the algebraic structure. The "Process" columns can be used as an instructional bridge to algebra.

Table 1. Areas for tables of various dimensions, if the table is twice as long as it is wide

Length Width (in.) Area (in.2)
(in.) Process Result Process Result
40
30
34
36
35
35.4
35.3
40/2
30/2
34/2
36/2
35/2
35.4/2
35.3/2
20
15
17
18
17.5
17.7
----
40 x 20
30 x 15
34 x 20
36 x 20
35 x 17.5
35.4 x 17.7
----
800
450
578
648
612.5
626.58
----

Length	Width (in.)		Area (in.2)
(in.)	Process	Result	Process	Result
40 30 34 36 35 35.4 35.3	40/2 30/2 34/2 36/2 35/2 35.4/2 35.3/2	20 15 17 18 17.5 17.7 ----	40 x 20 30 x 15 34 x 20 36 x 20 35 x 17.5 35.4 x 17.7 ----	800 450 578 648 612.5 626.58 ----