WORKSHOP 1·3
Handout
Four problem-solving steps — understand the problem, come up with a plan, carry out the plan, and reflect
on the answer — were outlined by Pólya in his 1945 book How to Solve It. In this workshop, you will work
on a problem using these four steps. Pólya listed a number of questions helpful in understanding a problem.
Here are a few of them.
1. What are we asked to find or show?
2. Can we restate the problem in our own words?
3. Can we draw a picture?
4. Do we understand all the words used in the problem?
Try to solve the following problems using Pólya’s four-step process. Remember that the process here is as
important as the answer. Each time you revisit the first step of understand the problem, ask one of the four
questions listed above.
Warm up question: A farmer has 240 m of fencing and wants to fence off a rectangular
field that borders a straight river. He needs no fence along the river. Let x be the length
of the rectangular field, find a function A(x) that represents the area of the field of length
x.
Work out this problem at the board. Make sure every member of your group is involved.
Problem: Amanda wants to print a poster-size picture of her dog. A printer tells her that
she can order a poster of any dimensions but its total area must be 1000 square centimetres.
Moreover, the printing machine requires the poster to have 5 cm margins on each side and 7
cm margins on top and bottom of the poster. Find a mathematical expression that represents
the printed area as a function of one of the dimensions of the poster.
Additional question: What is the domain of such function and why?
When you are confident with your answer, write out your full solution on a fresh piece of the board. This
should be a full solution, including things like sentences and punctuation. Then call over one of your TAs.
You will get full marks if your solution is correct, if you explain it well, and if you can answer the additional
question.