**Learning Objectives**

After completing this tutorial, you should be able to:

- Use Polya's four step process to solve word problems involving numbers, rectangles, supplementary angles, and complementary angles.

** Introduction**

Whether you like it or not, whether you are going to be a mother, father, teacher, computer programmer, scientist, researcher, business owner, coach, mathematician, manager, doctor, lawyer, banker (the list can go on and on). Some people think that you either can do it or you can't. Contrary to that belief, it can be a learned trade. Even the best athletes and musicians had some coaching along the way and lots of practice. That's what it also takes to be good at problem solving.

**George
Polya**,
known as the father of modern problem solving, did extensive studies
and
wrote numerous mathematical papers and three books about problem
solving.
I'm going to show you his method of problem solving to help step you
through
these problems.

** Tutorial**

As mentioned above, I use Polya’s four steps to problem
solving to
show students how to solve word problems. **Just
note
that your math teacher or math book may word it a little differently,
but
you will see it all basically means the same thing.**

If you follow these steps, it will help you become more
successful in
the world of problem solving.

Polya created his famous **four-step process for
problem solving,** which is used all over to aid people in problem solving:

**Step 1: Understand the problem.**

Sometimes the problem lies in **understanding
the problem**.
If you are unclear as to what needs to be solved, then you are probably
going to get the wrong results. In order to show an understanding
of the problem, you, of course, need to read the problem
carefully.
Sounds simple enough, but some people jump the gun and try to start
solving
the problem before they have read the whole problem. Once the
problem
is read, you need to list all the components and data that are
involved.
This is where you will be assigning your variable.

**Step 2: Devise a plan (translate).**

When you **devise a plan (translate)**, you
come up with a way to
solve the problem. Setting up an equation, drawing a diagram, and
making a chart are all ways that you can go about solving your
problem.
In this tutorial, we will be setting up equations for each
problem.
You will translate them just like we did in **Tutorial
4: Introduction to Variable Expressions and Equations.**

**Step 3: Carry out the plan (solve).**

The next step, **carry out the plan (solve)**,
is big. This is where
you solve the equation you came up with in your 'devise a plan'
step.
The equations in this tutorial will all be linear equations. If
you
need help solving them, by all means, go back to **Tutorial
12: The Addition Property of Equality, Tutorial
13: The Multiplication Property of Equality, or Tutorial
14: Solving Linear Equations (Putting it all together)** and
review
that concept.

**Step 4: Look back (check
and interpret).**

You may be familiar with the expression 'don't
look back'. In
problem solving it is good to **look back (check and interpret).**.
Basically, check to see if you used all your information and that the
answer
makes sense. If your answer does check out, make sure that you
write
your final answer with the correct labeling.

A lot of numeric types of word problems revolve around
translating
English statements into mathematical ones. If you need a review
on
these translations, you can go back to **Tutorial
4: Introduction to Variable Expressions and Equations.**

**Just read and translate it left to right to set up
your equation**.

Make sure that you read the question carefully several
times.

Since we are looking for a number, we will let

*x* = a number

***Get all the x terms on one side**

***Inv. of sub. 2 is add 2**

If you take twice the difference of 6 and 1, that is
the same as 4
more than 6, so this does check.

**FINAL ANSWER: **

Make sure that you read the question carefully several
times.

We are looking for two numbers, and since we can write the one number in terms of another number, we will let

*x* = another number

one number is 3 less than another number:

*x* - 3 = one number

***Inv. of sub 3 is add 3**

***Inv. of mult. 2 is div. 2**

If we add 90 and 87 (a number 3 less than 90) we do get
177.

**FINAL ANSWER: **

**Another number is 87.**

The following formula will come in handy for solving
example 3:

**Perimeter of a rectangle = 2(length) + 2(width)**

Make sure that you read the question carefully several
times.

We are looking for the length and width of the rectangle. Since length can be written in terms of width, we will let

*w* = width

length is 1 inch more than 3 times the width:

**1 + 3 w = length**

***Inv. of add. 2 is sub. 2**

***Inv. of mult. by 8 is div. by 8**

If width is 3, then length, which is 1 inch more than 3
times the width
would have to be 10. The perimeter of a rectangle with width of 3
inches and length of 10 inches does come out to be 26.

**FINAL ANSWER:**

**Length is 10 inches.**

**Complimentary angles sum up to be 90 degrees.**

Make sure that you read the question carefully several
times.

We are already given in the figure that

*x* = 1 angle

**5 x = other angle**

***Inv. of mult. by 6 is div. by 6**

If *x* is 30, then 5*x* = 5(30) = 150. 150 and 30 do
add up to be
180, so they are supplementary angles.

**FINAL ANSWER:**

**The two angles are 30 degrees and 150 degrees.**

** Practice Problems**

These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. **Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

To get the most out of these, **you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem**. At the link you will find the answer
as well as any steps that went into finding that answer.

Practice Problems 1a - 1c:Solve the word problem.

1a. The sum of a number and 2 is 6 less than
twice that number.

(answer/discussion to 1a)

(answer/discussion to 1a)

1b. A rectangular garden has a width that is 8
feet less than
twice the length. Find the dimensions if the perimeter is 20
feet.

(answer/discussion to 1b)

(answer/discussion to 1b)

1c. Complimentary angles sum up to be 90
degrees. Find
the measure of each angle in the figure below. Note that since
the
angles make up a right angle, they are complementary to each other.

** Need Extra Help on these Topics?**

This webpage gives you the basics of problem solving and helps you with translating English into math.

**http://www.purplemath.com/modules/ageprobs.htm**

This webpage goes through examples of age problems, which are like
the numeric problems found on this page.

**Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.**

Last revised on July 26, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.