Chapter 8: Roots and Radicals 
Numbers in the Real World

Polya Process
 

George Polya (1887-1985) was born in Hungary before moving to the US to teach at Brown University and Stanford University.  Polya wrote many books and articles, including his well-known book: How to Solve It.  You may have already heard of his problem-solving process in your math classes.

His process involves four steps as listed below.  He outlines numerous strategies for each step; some of which are also included below.

  1. Understand the problem.
    1. Most people need to read the problem several times.
    2. You may need to rewrite the problem in your own words.
    3. What is the missing information that is needed?
    4. What information given is not needed?
  2. Devise a plan.
    1. Make a list of information provided.
    2. Make a list of information needed.
    3. Draw diagrams.
    4. Make a table.
    5. Guess and check.
    6. Look for a pattern.
    7. Work a similar, simpler problem.
    8. Compare this problem to known problem types.
  3. Carry out the plan.
    1. Work carefully.
    2. Do not skip steps.  Show all your work.
  4. Review your solution.  [Look back.]
    1. Is the solution feasible in terms of the problem?
    2. Are there other solutions?
    3. Are there generalizations from this problem that could apply to other similar problems?
    4. Are there other ways to arrive at the same solution?
We have added the last strategy to those of Polya because analyzing your approach to the problem is as important as the solution. 


Problem 1:  There are 40 ducks and 60 goats in a field.  If you count 106 legs, how many of each animal are there?
    (Hint:  Make a table and use the guess and check strategy.)
 
Problem 2: What is the units digit for 73134 ?
    (Hint:  Work simpler problems to look for a pattern.)

Problem 3:  A rectangular garden is 4 feet longer than it is wide.  Along the edge of the garden on all sides, there is a 2-foot gravel path.  How wide is the garden if the perimeter of the garden is 28 feet?
    (Hint:  Draw a diagram and use the guess and check strategy.)


Reconsider the above problems.
Check Your Answers