Beginning Algebra
Tutorial 15:
Introduction to Problem
Solving
 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.
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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).
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. |
|
 Example
1: Twice the difference of a number and 1 is 4 more
than
that number. Find the number. |
| Make sure that you read the question carefully several
times.
Since we are looking for a number, we will let
x = a number
|
 |
*Remove ( ) by using dist. prop.
*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:
The number is 6.
|
| Example
2: One number is 3 less than another number. If
the sum of the two numbers is 177, find each number. |
| 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
|
 |
*Combine like terms
*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:
One number is 90.
Another number is 87.
|
| The following formula will come in handy for solving
example 3:
Perimeter of a rectangle = 2(length) + 2(width)
|
| Example
3: In a blueprint of a rectangular room, the length is
1 inch more than 3 times the width. Find the dimensions if the
perimeter
is to be 26 inches. |
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 + 3w = length
|
 |
*Remove ( ) by using dist. prop.
*Combine like terms
*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:
Width is 3 inches.
Length is 10 inches.
|
|
Supplementary and Complementary
Angles
|
| Supplementary angles sum up to be 180 degrees.
Complimentary angles sum up to be 90 degrees.
|
| Example
4: Find the measure of each angle in the figure
below.
Note that since the angles make up a straight line, they are
supplementary
to each other.
|
| Make sure that you read the question carefully several
times.
We are already given in the figure that
x = 1 angle
5x = other angle
|
 |
*Combine like terms
*Inv. of mult. by 6 is div. by 6
|
| If x is 30, then 5x = 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.
|
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) |
| 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.
(answer/discussion
to 1c)
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Need Extra Help on These Topics?
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All contents copyright (C) 2001 - 2008, WTAMU and Kim Seward. All rights reserved.
Last revised on June 22, 2003 by Kim Seward. |
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