Copyright © 2011 Pearson Education, Inc. Approaches to Problem Solving

Copyright © 2011 Pearson Education, Inc.

Approaches to Problem Solving


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Unit 2C

Problem-Solving Guidelines and Hints


Problem Solving 101

Problem solving is one of the most important activities that we as humans face each and every day. It is our ability to salve problems in unique and creative ways that helps to set us apart from other forms of life that inhabit this planet. In order to solve problems, we must first have an understanding of what constitutes a problem.

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Problem Solving 101 (cont.)

The 1977 edition of the Webster’s New Collegiate Dictionary defines the word in the following manner:

problem noun - from Latin problema from Greek probl``ema

meaning literally “something thrown forward”

1a) a question raised for inquiry, consideration or solution

b) a proposition in mathematics or physics stating something to be done

2a) an intricate unsettled question

b) a source of perplexity, distress or vexation

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Many times in school, you have been told to do problems 1 through 10 or A through G and so on. We, in educational circles, tend to use the word “problem” to describe many different things. However, what is actually true is that many of the things assigned to students are not problems, in the sense that we want to consider here, but instead are actually “exercises”. Again to quote Webster;

exercise noun

3) something performed or practiced in order to develop, improve or display a specific skill

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Problem Solving 101 (cont.)


What is a PROBLEM?

In order to separate problems from mere exercises, we need a stricter definition of just what a problem is. Anything that you are asked to do that reinforces or refines knowledge or a skill that you already have is an exercise. Things like throwing a baseball/softball, kicking a soccer ball into the goal at practice, shooting a basketball, catching a football, finding the solution to x = – 4 and so on, are all examples of exercises. What should be obvious is that there are things that are problems for others, but may be just exercises for you, and vice versa. So, what is a problem?

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Problem – An Example

If you were asked to go shoot free throws, is that a problem or an exercise?

If you took a basketball and hoop to a stone age tribesman along the Amazon River, and ask him to go shoot free throws, is that a problem?

What is the difference between you and the tribesman?

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My definition of a problem

My definition of a problem: A problem is any situation that needs to be resolved such that the solution requires more skill and knowledge than the problem-solver may have at the time that the problem comes to the attention of the problem-solver. Inherent in my definition is the portion of Webster’s definition that deals with a problem being an intricate unsettled question and a source of perplexity, distress or vexation. I believe that a part of any true problem is being STUCK at some point in our attempt to arrive at a solution. Not STUCK, no PROBLEM ! ! Throughout this course, we will be working on the skill of problem solving. Note, in my definition of a problem, there is no mention of mathematics. My definition deals with any situation that qualifies under the definition! ! It could be mechanical problems, personal problems, problems at work as well as math problems!

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Step 1: Understand the problem.

Step 2: Devise a strategy for solving the problem.

Step 3: Carry out your strategy, and revise if necessary.

Step 4: Look back to check, interpret, and explain your result.

Four Step Problem-Solving Process

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Step 1: Understand the problem. Think about the context of the problem to gain

insight into its purpose. Make a list or table of the specific information

given in the problem. Draw a picture or diagram to help make sense of

the problem. Restate the problem in different ways to clarify the

question. Make a mental or written model of the solution.

Four Step Problem-Solving Process:Step 1

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Step 2: Devise a strategy for solving the problem.

Obtain needed information that is not provided in the problem statement.

Make a list of possible strategies and hints that will help you select your overall strategy.

Map out your strategy with a flow chart or diagram.


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Step 3: Carry out your strategy, and revise it if necessary.

Keep an organized, neat, and written record of your work.

Double-check each step so that you do not risk carrying errors through to the end of your solution.

Constantly reevaluate your strategy as you work. Return to step 2 if you find a flaw in your strategy.


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Step 4: Look back to check, interpret, and explain your result.

Be sure that your result makes sense. Recheck calculations or find an independent way

of checking the result. Identify and understand potential sources of

uncertainty in your result. Write your solution clearly and concisely. Consider and discuss any pertinent implications of

your result.


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Hint 1: There may be more than one answer.

Hint 2: There may be more than one strategy.

Hint 3: Use appropriate tools.

Hint 4: Consider simpler, similar problems.

Hint 5: Consider equivalent problems with simpler solutions.

Hint 6: Approximations can be useful.

Hint 7: Try alternative patterns of thought.

Hint 8: Do not spin your wheels.

Problem Solving Guidelines and Hints

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Find the total number of possible squares on the chessboard by looking for a pattern.

Solution

Start with the largest possible square:

There is only one way to make an

8 x 8 square.

Problem Solving Example #1

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Now, look for the number of ways to make a 7 x 7 square.


There are only four ways.

Problem Solving Example #1 (cont.)

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If you continue looking at 6 x 6, then 5 x 5 squares, and so on, you will see the perfect square pattern as indicated in the following table for this chessboard problem:



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n x n squares # 8 x 8 1 7 x 7 4 6 x 6 9 5 x 5 16 4 x 4 25 3 x 3 36 2 x 2 49 1 x 1 64

Total: 204



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There are three boxes, one contains only apples, one contains only oranges, and one contains both apples and oranges. The boxes have been incorrectly labeled such that no label identifies the actual contents of the box it labels. Opening just one box, and without looking in the box, you take out one piece of fruit. By looking at the fruit, how can you immediately label all of the boxes correctly.

Which box did you open and how can you be sure to label all boxes correctly?

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Choose one fruit from the box labeled both!

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You are in a classroom with 40 people of varying heights. The teacher/instructor of the room has asked you to exchange papers for the purpose of correcting them. However, nobody is allowed to change papers with anyone that is shorter than themselves. How many exchanges will occur?

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NONE

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A Mathematics test consists of 20 multiple-choice questions. A correct answer is awarded 3 points and one point is deducted for every wrong answer. No points are awarded or deducted for questions not attempted. A boy attempted a total of 19 questions and his total score for the test was above 32. Find the minimum number of correct answers he obtained.

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13 Correct

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In-Class Group Assignment

p. 123 7-15, 17 – 25 odd

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Copyright © 2011 Pearson Education, Inc. Approaches to Problem Solving