10. A Methodical Approach to Problem Solving

Student:  Well, what about word problems? I can work it out once I know what to do, but how can I figure out the equation?

Dr. O:  You need to know and use Polya’s Method5 for problem solving.

Step 1: Understand the Problem

Step 2: Devise a Plan

Step 3: Carry Out the Plan

Step 4: Check

Step 1:  To understand what you are required to do in the problem, ask yourself some questions:

• What is my goal (the main questions being asked)?
• What information am I told in the item?
• What constraints or conditions are imposed by the problem?
• What will the results look like?
• How can I state the goal in mathematical terms and symbols?
• What additional information do I need, such as a formula, a definition, or a concept?
• Do I need to make a sketch?
• If I don’t know the additional information, where do I find it?
• What do I need to determine first to answer the question that is asked?
• How can I organize the information to reach my goal, that is, to answer the question?

Step 2: To devise a plan, you start by writing the English version of the answers to some of the questions from step 1, then translating them into mathematical expressions and equations. Once you have an equation that will satisfy the goal, you are ready for the next step.

Step 3: To carry out your plan, solve the equation or equations you devised to give you your final result. Make sure that you follow the order of operations and the correct processes needed.

Step 4: To check your result, make sure that you answer all the questions asked in the problem. Your result needs to satisfy the constraints established by the problem and be reasonable.

As with any practical method, practice makes perfect when you are applying it correctly. Let me demonstrate.

Suppose you were assigned the following application (a.k.a.: word problem):

The measure of the second angle of a triangle is three times the measure of the first and the third is nine degrees less than five times the first. Find the measures of the angles.

##### Step 1: Understand the Problem

Goal: Find the measures of the angles.

Information given: a triangle, which means three angles. I’m told about the measures of angles 2 and 3. What about the measure of angle 1? It is the basis for finding the measures of both angles 2 and 3.

What do I need to know about the measures of the angles of a triangle? The sum of the measures of the three angles is 180°.

Organizing: Because I need to know the measure of angle 1 to find the measures of angles 2 & 3, I represent the measure of angle 1 by the variable a.

Let the measure of angle 1 = a.

Now I must meet the constraints imposed by the problem to write the representations of the measures of angles 2 & 3.

The constraint for angle 2: The measure of the second angle of a triangle is three times the measure of the first, so I must state the measure of angle 2 as 3 times a.

Let the measure of angle 2 = 3a.

The constraint for angle 3: and the third is nine degrees less than five times the first. The key words require me to write the measure of angle 3 as 5a - 9.

Let the measure of angle 3 = 5a - 9.

Sketching and labeling a drawing of a triangle may help me see the situation more clearly.

Now I’m ready to devise a plan.

##### Step 2: Devise a Plan

I need to write an equation that will let me answer the question of the problem. Since the sum of the measures of the 3 angles in a triangle =180°, I can write:

(a)  + (3a)  + (5a - 9)  = 180°

##### Step 3: Carry Out the Plan

Begin by combining like terms in the left side expression (to the left of the equals sign).

9a – 9 = 180                          To get the term 9a by itself, I need to add 9 to both sides.

9a – 9 + 9 = 180 + 9             Simplifying produces

9a = 189                                My goal is to isolate a, so I’ll reduce the factor of 9 to a 1 by dividing both sides of the equation by 9.

Simplifying both sides gives

a = 21°                                   The measure of angle one is 21°. Have I reached the goal? No, I still need to find the measures of angles 2 & 3.

Angle 2 = 3a                          Substitute the value of angle one for a and evaluate.

Angle 2 = 3(21) = 63°            Now find angle three.

Angle 3 = 5a- 9                    Substitute the value of angle one for a and evaluate.

Angle 3 = 5(21) - 9 = 105- 9 = 96°

Angle 1 = 21°, Angle 2 = 63°, and Angle 3 = 96°

Step 4: Check

My angles need to meet the constraints of the problem and have a sum =180°.

First, review the problem and note the constraints.

The measure of the second angle of a triangle is three times the measure of the first and the third is nine degrees less than five times the first. Find the measures of the angles.

Triangle means three angles. No explicit constraint is placed on the first angle except that it is the basis for the other two.  Let the measure of angle 1 = a.  Since the measure of the second angle of a triangle is three times the measure of the first, I let the measure of angle 2 = 3a.  The measure of the third is nine degrees less than five times the first, so I let angle 3 = 5a- 9°.

I need to answer these three equations correctly and have the sum of the measures equal 180° to solve the problem. I calculated the measure of angle 1 as 20°.

Now I write my equations and check to see if my solutions meet the constraints.

Angle 1 = a = 21°

Angle 2 = 3a = 3(21) = 63°

Angle 3 = 5a - 9° = 5(21) - 9 = 105- 9 = 96°

The values I computed for measures of the three angles meet the constraints of the problem. To finish my check, I add the three measures together. If the sum is 180°, then I have the correct measures.

21° + 63° + 96° = 180°, so I have my solutions.

The textbook also introduces a five-step approach that is a variation of Polya’s in Chapter 2.

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