Problem Solving
MATH 102
Contemporary Math
S. Rook
Overview
• Section 1.1 in the textbook:
– Polya’s four step method for problem solving
– Problem Solving Strategies
Polya’s Four-Step Method for
Problem Solving
Polya’s Four-Step Method for Problem
Solving
• George Polya was a famous mathematician (see
pg 14 in textbook) who developed a systematic
four step method for solving problems
• Step 1 – Understand the Problem
– Know what you are being asked to find
– List what you know from the problem
– Reread as many times as is necessary to fully
comprehend the problem
– Drawing pictures/diagrams often helps understanding
Polya’s Four-Step Method for Problem
Solving (Continued)
• Step 2 – Devise a plan
– May require a bit of ingenuity
• Step 3 – Carry out the plan
– Actually doing the math and answering the problem
• Step 4 – Check the answer
– Ensure that the answer makes sense in the context of
the problem
• Chances are that you will not think about EVERY
problem in these 4 steps
– You should at least think of EVERY problem in terms of
planning and executing
Problem Solving Strategies
Drawing a Tree Diagram
• Drawing a tree diagram is most helpful for a
problem that can be broken down in terms of
steps or stages
• If the number of options in each stage is
reasonable, a tree diagram can be used to list all
possibilities
– Suppose a soccer game can be won, lost, or drawn.
What are all the different outcomes if we play three
games?
• The number of options in each stage do not
necessarily have to be the same
Drawing a Tree Diagram (Example)
Ex 1: Suppose that we roll a 3-sided die with sides
numbered 1, 2, and 3 and then a 4-sided die with
sides numbered 1, 2, 3, 4.
a) List the possible outcomes of the rolls using
coordinate pairs. Make sure to indicate which
part of the coordinate represents the 3-sided die
and which part represents the 4-sided die.
b) How many possible outcomes are there?
c) How many of the outcomes was an odd
number rolled on the 3-sided die?
Problem Solving (Example)
Ex 2: Solve. Try to plan before calculating.
a) Approximately how long would it take a person to drive
375 miles if his vehicle has a rate of 60 miles per hour?
b) Michael has cut a 40-foot long sandwich into three
unequal pieces. The longest piece is three times as long
as the middle-length piece and the shortest piece is 5 feet
shorter than middle piece. What are the lengths of the
three pieces?
c) Alex Rodriguez of the NY Yankees and Magglio Ordonez
of the Detroit Tigers had a total of 295 runs batted in. If
Rodriguez had 7 more than Ordonez, how many did each
have?
Summary
• After studying these slides, you should know how
to do the following:
– Understand Polya’s four-step process for problem
solving
– Use a tree diagram to help solve problems
– Apply a variant of Polya’s four-step process to solve
different types of problems
• Additional Practice:
– See the list of suggested problems for 1.1
• Next Lesson:
– Inductive & Deductive Reasoning (Section 1.2)