Approaches to Problem Solving Copyright 2011 Pearson Education Inc Slide 2 3 Unit 2C ProblemSolving 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 uniqu ID: 564112
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Copyright © 2011 Pearson Education, Inc.Approaches to Problem SolvingSlide3
Copyright © 2011 Pearson Education, Inc.Slide 2-3Unit 2C
Problem-Solving Guidelines and HintsSlide4
Problem Solving 101Problem 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.
Copyright © 2011 Pearson Education, Inc.Slide 2-4Slide5
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``emameaning 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
Copyright © 2011 Pearson Education, Inc.
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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 noun3) something performed or practiced in order to develop, improve or display
a specific skillCopyright © 2011 Pearson Education, Inc.Slide 2-6
Problem Solving 101 (cont.)Slide7
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 ExampleIf 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?
Copyright © 2011 Pearson Education, Inc.Slide 2-8Slide9
My definition of a problemMy 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!Copyright © 2011 Pearson Education, Inc.Slide 2-
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Copyright © 2011 Pearson Education, Inc.Slide 2-10
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 ProcessSlide11
Copyright © 2011 Pearson Education, Inc.Slide 2-11Step 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 1Slide12
Copyright © 2011 Pearson Education, Inc.Slide 2-12Step 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.Four Step Problem-Solving Process:Step 2Slide13
Copyright © 2011 Pearson Education, Inc.Slide 2-13Step 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.Four Step Problem-Solving Process:Step 3Slide14
Copyright © 2011 Pearson Education, Inc.Slide 2-14Step 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.Four Step Problem-Solving Process:Step 4Slide15
Copyright © 2011 Pearson Education, Inc.Slide 2-15
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 HintsSlide16
Copyright © 2011 Pearson Education, Inc.Slide 2-16
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 #1Slide17
Copyright © 2011 Pearson Education, Inc.Slide 2-17
Now, look for the number of ways to
make a 7 x 7 square.
Find the total number of possible squares on the chessboard by looking for a pattern.
There are only
four
ways.
Problem Solving
Example #1 (cont.)Slide18
Copyright © 2011 Pearson Education, Inc.Slide 2-18
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:
Find the total number of possible squares on the chessboard by looking for a pattern.
Problem Solving
Example #1 (cont.)Slide19
Copyright © 2011 Pearson Education, Inc.Slide 2-19
Find the total number of possible squares on the chessboard by looking for a pattern.
Problem Solving
Example #1 (cont.)Slide20
Problem Solving Example #2There 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?
Copyright © 2011 Pearson Education, Inc.Slide 2-20Choose one fruit from the box labeled both!Slide21
Problem Solving Example #3You 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?
Copyright © 2011 Pearson Education, Inc.Slide 2-21NONESlide22
Problem Solving Example #4A 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.
Copyright © 2011 Pearson Education, Inc.Slide 2-2213 CorrectSlide23
In-Class Group Assignmentp. 123 7-15, 17 – 25 oddCopyright © 2011 Pearson Education, Inc.
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