Transcript Have A Problem???

Have A Problem???
“How To Solve It!”
 Polya’s Four Step Problem Solving
Process
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Today’s TEKS
Objective:
(8.14) Underlying processes
and mathematical tools.
The student applies Grade 8
mathematics to solve
problems connected to
everyday experiences,
investigations in other
disciplines, and activities in
and outside of school.
The student is expected to:
(B) use a problem-solving model that
incorporates understanding the
problem, making a plan, carrying
out the plan, and evaluating the
solution for reasonableness
(C) select or develop an appropriate
problem-solving strategy from a
variety of different types,
including drawing a picture,
looking for a pattern, systematic
guessing and checking, acting it
out, making a table, working a
simpler problem, or working
backwards to solve a problem
Why Problem Solving??
 Once, at an informal meeting, a social
scientist asked a math professor, “What’s
the main goal of teaching mathematics?”
The reply was, “Problem solving.”
 In return, the mathematician asked, “What
is the main goal of teaching the social
sciences?” Once more the answer was,
“Problem solving.”
Who Solves Problems??
– All successful engineers, scientists,
social scientists, lawyers, accountants,
doctors, business managers, and so on,
have to be good problem solvers.
– Although the problems that people
encounter may be very diverse, there
are common elements and an
underlying structure that can help to
facilitate problem solving.
Step 1: Understand the Problem
 Identify what you are trying




to find.
Summarize the information
that is available in your own
words.
Determine if the information
available is enough, ie. Do you
need a formula, etc.?
Strip the problem of irrelevant
details.
Don’t impose conditions that
do not exist.
Step 2: Devise a Plan
 Is this problem similar to another problem you have
solved?
 Can one of the Problem Solving Strategies be used?
 Often a considerable amount of creativity is required to
formulate a plan.
Strategies for Problem Solving
A Strategy is defined as an artful means to an end.
 Make a chart or
 Guess, test, and
table.
 Look for a pattern.
 Draw a picture or
diagram.
 Eliminate impossible
situations.
 Work Backwards.
revise.
 Use a variable.
 Design a model.
 Try a simpler
version of the
problem.
 Use reasoning.
Step 3: Carry Out the Plan
 Implement the strategy or
strategies that you have
chosen until the problem is
solved or until a new course
of action is suggested.
 Give yourself a reasonable
 Don’t expect to solve
amount of time in which to
correctly and immediately
solve the problem. If you
all problems. Problem
are not successful, seek
Solving takes time and
persistence.
hints from others or put the
problem aside for awhile.
 Don’t be afraid of starting
over. Often, a fresh start
and a new strategy will lead
to success.
Step 4: Look Back
 Interpret the results into a





sentence with your own words.
Check the results to be sure the
solution is correct.
Does your answer satisfy the
statement of the problem? Does
it make sense?
Ask if there is another way to
solve the problem.
Ask if there are other problems
that can be solved by using the
same techniques used in this
problem.
Make a point of thinking about
the strategy that finally worked
for this type of problem for future
reference.
A Sample of the Process in Action
– Problem:
– Find the sum of
1 1 1
1
 2  3  ...  10
2 2 2
2
Let’s try Step 1:
Understand the Problem
Step 1: Understand the Problem
 This problem will require getting a
common denominator, here 210,
converting each fraction, and finding
the sum of the numerators.
 This is obviously a long and tedious
process.
 Maybe there is a quicker way
to solve the problem...
Step 2: Devise a Plan
 Instead of doing a direct calculation, let’s combine some of the
suggested strategies.
 Namely, make a list of the first few sums and look for a pattern.
Step 3: Carry Out the Plan
1 1 1 3 1 1 1 7 1 1 1 1 15
,   ,    ,    
2 2 4 4 2 4 8 8 2 4 8 16 16
 The pattern of sums,
1 3 7 15
, , ,
2 4 8 16
 suggests that the sum of the ten fractions
is
210  1 1023
, or
.
10
2
1024
Step 4: Look Back
 This method of combining the
strategy of Solve a Simpler
Problem with Make a List and
Look for a Pattern is very
useful.
 For example, what is the sum
 Because of the large
1 1
1
 2  ...  100 ?
2 2
2
denominators, you wouldn’t
want to add these fractions
directly.
Problem Solving Recap...
 When presenting the problem
solving process and the sample
problem, great care was taken to
label and display each of the four
steps. Clearly, this is not
necessary every time you work a
problem.
 On the other hand, it is a good
idea to get into the habit of
recalling the four steps as you
plan and as you work through a
problem.
 The steps and strategies will be especially helpful when
you are making a plan.
 As you are planning to solve a problem, think of the
strategies as a collection of tools which you may select
from and utilize to successfully solve your problem.
Final Suggestions from
Successful Problem Solvers
 Accept the challenge of
solving a problem.
 Take time to explore,
reflect, think, …
 Talk to yourself. Ask
yourself lots of questions.
 Many problems require
an incubation period. If
you get frustrated, do not
hesitate to take a break your subconscious may
take over. But do return
to try again.
 Experience in problem
solving is very valuable.
Work lots of problems;
your confidence will grow.
 There is nothing like a
breakthrough, an “Aha!”,
as you solve your
problems.
 Always, always look back.
Try to see precisely what
the key step was in your
solution and make a
mental note for future
reference.
 Enjoy yourself! Solving a
problem is a positive
experience.
Credits:
 PowerPoint
Presentation by…
John P. Ashby
 Course: ED 488
 Instructor: Susan Pride
 Semester: Fall, 1999
 Date: November 11, 1999
 Title: How To Solve It
 Software Package: Microsoft
PowerPoint