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Problem Solving
POLYA’S FOUR STEP IN WORD PROBLEM
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OBJECTIVE
After going through this module, you are
expected to:
• Use Polya's four step process to solve word
problems
• 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
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Solve the magic square. In order to make each
row and column equal to sum of 15, fill it with
appropriate number ranges from 1-9 and there
will be no repetition.
MAGIC SQUARE 101
ACTIVITY
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What is Problem Solving
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Problem
Solving
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• Problem Solving is a mathematical process.
PROBLEM
SOLVING
METHOD
ANSWER
SOLUTION
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IMPORTANCE?
• The ability to think creatively, critically, and
logically
• The ability to structure and organize
• The ability to process information
• Enjoyment of an intellectual challenge
• The skills to solve problems that help them to
investigate and understand the world
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Mathematical
Problem
title
defined
style as …
• a problem that is amenable to being
represented, analyzed, and possibly solved, with
the methods of mathematics. This can be a
real-world problem, such as computing the orbits
of the planets in the solar system, or a problem
of a more abstract nature.
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Types of Mathematical
Problem
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Routine Problem
• is a type of problem in which there is an immediate solution.
The problem solver knows a solution method and only needs
to carry it out.
Example:
"589 × 45 = ___"
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Non-routine Problem
• is a problem which requires analysis and insights into known principles of Mathematics. It
involves difficult problem solving.
Example:
Water lilies double in area every twenty-four hours. At the beginning of the
summer, there is one water lily on the lake. It takes sixty days for the lake to be
completely covered with water lilies. On what day is the lake half covered?"
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Mathematical
title style Statement
Expression to Mathematical Statement
Y+2
two is greater than y
p - 11
eleven less than p
6(n+4)
six times the sum of
the number n and four
Statement to Mathematical Expression
Four more than six
times a number is
equal to forty
6x + 4 = 40
A number divided by
four increased by
three is equal to
twenty seven
Seventeen is equal to
two reduced by three
times a number
2 – 3x = 17
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Convert the following Mathematical statement into expression
Two-thirds of a number minus ten is equal to
negative twenty-three
The product of twelve and a number is forty-eight
Convert the following expression into Mathematical statement
7x + 9
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POLYA’S TECHNIQUE
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• father of modern problem solving
• was a great champion in the field of
teaching effective problem-solving skills.
born in Hungary in 1887, received his
Ph.D. at the University of Budapest, and
was a professor at Stanford University
(among other universities).
• “How to Solve It.”
GOERGE POLYA
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POLYA’S PROBLEM
SOLVING TECHNIQUES
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Understand
Master
problem:
title style
❑ Read the problem over carefully and ask yourself:
• Do I know the meaning of all the words?
• What is being asked for?
• What is given in the problem?
• Is the given information sufficient (for the solution to be unique)?
• Is there some inconsistent or superfluous information which is
given?
❑ By way of checking your understanding, try restating the
problem in a different way.
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Design
planMaster
for solving
title the
styleproblem:
❑ In essence, decide how you are going to work on the
problem. This involves making some choices about
what strategies to use. Some possible strategies
are:
• Draw a picture or diagram -- making a picture which
relates the information given to what is asked for can
often lead to a solution.
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Design
planMaster
for solving
title the
styleproblem:
❑ In essence, decide how you are going to work on the
problem. This involves making some choices about
what strategies to use. Some possible strategies
are:
• Make a list -- this is a strategy which is especially
useful in problems where you need to count the
members of a set.
A = {1, 2, 3, 4, 5, 6, 7, 8}
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Design
planMaster
for solving
title the
styleproblem:
❑ In essence, decide how you are going to work on the
problem. This involves making some choices about
what strategies to use. Some possible strategies
are:
• Solve smaller versions of the problem and look for a
pattern -- almost any problem can be made simpler in
some way. By working out simpler versions, you can
often see patterns which help solve the original
problem.
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Design
planMaster
for solving
title the
styleproblem:
❑ In essence, decide how you are going to work on the
problem. This involves making some choices about
what strategies to use. Some possible strategies
are:
• Decompose the problem -- Many problems can be
broken into a series of smaller problems. This
strategy can turn a problem which on first glance
seems intractable into something more doable.
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Design
planMaster
for solving
title the
styleproblem:
❑ In essence, decide how you are going to work on the
problem. This involves making some choices about
what strategies to use. Some possible strategies
are:
• Use variables and write an equation -- the method of
algebra. Very useful in a lot of problems.
Let x be the
first…
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Carry
out
edit
theMaster
plan: title style
❑ Once you have an idea for a new approach, jot it down
immediately. When you have time, try it out and see if it leads
to a solution
❑ If the plan does not seem to be working, then start over and
try another approach. Often the first approach does not work.
Do not worry, just because an approach does not work, it
does not mean you did it wrong. You actually accomplished
something, knowing a way does not work is part of the
process of elimination.
❑ Once you have thought about a problem or returned to it
enough times, you will often have a flash of insight: a new
idea to try or a new perspective on how to approach solving
the problem.
❑ The key is to keep trying until something works.
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❑ After you have a proposed solution, check your solution
out.
• Is it reasonable?
• Is it unique?
• Can you see an easier way to solve the problem?
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NUMERIC WORD PROBLEM
LET’S TRY THIS!
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Example1:
Click
to editTwice
Master
thetitle
difference
style of a number and 1 is 4
more than that number. Find the number.
Step 1: Understand the problem
• let x = a number
Step 2: Devise a plan (translate)
• Twice the difference of a number and 1 is 4 more than that number.
2(x-1)
=
x+4
Step 3: Carry out the plan (solve)
• Remove () Using Distribution Property
2(x-1) = x + 4
2x – 2 = x + 4
• Inverse of 2 is add 2
x–2+2=4+2
x=6
• Get all “x” term in one side
2x – 2 – x = 4
x–2=4
Step 4: Look back (check and interpret)
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Try This!
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When 6 times a number is increased by 4, the result is 40. Find the number.
Step 1: Understand the problem
• Let the number be x
Step 2: Devise a plan (translate)
6x + 4 = 40
Step 4: Look back (check and interpret)
• Substitute
6(6) + 4 = 40
36 + 4 = 40
40 = 40
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Age Problem
LET’S TRY THIS!
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Example
2: Phil
Master
is Tom’s
title
father.
stylePhil is 35 years old. Three years
ago, Phil was four times as old as his son was then. How old is Tom
•
now?
Now
3 years ago
Phil
35
35 – 3
Tom
x
x-3
• Inverse of negative 3 add up 3
8+3=x–3+3
11 = x
x = 11
Step 4: Look back (check and interpret)
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Lisa is 16 years younger than Kathy. If the sum of their ages is 30. How old is
Lisa?
Step 1: Understand the problem
• Let x be the age of Lisa
Step 2: Devise a plan (translate)
• Lisa is 16 years younger than Kathy
Lisa
Kathy
x
x + 16
• the sum of their ages is 30
x + (x + 16) = 30
Step 3: Carry out the plan (solve)
Step 4: Look back (check and interpret)
• Remove () and Combine like terms
• Substitute
x + x + 16 = 30
(7) + (7) + 16 = 30
2x + 16 = 30
14 + 16 = 30
30 = 30
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RECTANGLE PROBLEM
LET’S TRY THIS!
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Example
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3: In
Master
a blueprint
title style
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
Step 1: Understand the problem
w = width
and
= 2l +2w
Step 2: Devise a plan (translate)
Length is 1 inch more than 3 times the width:
length = 1 + 3w
Step 3: Carry out the plan (solve)
• Remove () using Distribution Property
26 = 2(1+3w) + 2(w)
26 = 2 + 6w + 2w
• Combine like terms
26 = 2 + 8w
Step 4: Look back (check and interpret)
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The length of a rectangle is 3 less than 4 times the width. The perimeter is 34.
Find the length and width.
Step 1: Understand the problem
• Let w be the width; and
• Let length be 3 less than 4 times the width
• And P for perimeter = 34
Step 2: Devise a plan (translate)
P = 2L + 2W
w=?
l = 4w – 3
Step 3: Carry out the plan (solve)
P = 2L + 2W
34 = 2(4w – 3) + 2W
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•
Step 4: Look back (check and interpret)
• Substitute
w=4
l = 4(4) – 3
34 = 2(4w – 3) + 2W
34 = 2(16 – 3) + 2 (4)
34 = 2(13) + 8
34 = 26 + 8
34 = 34
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SEATWORK.
I. Identify what method is being described in the following statements. (Polya’s 4 Methods of problem solving)
(1 point each). 15 minutes only
1.
Check your solution out.
2.
The key is to keep trying until something works.
3.
If the plan does not seem to be working, then start over and try another approach
4.
This involves making some choices about what strategies to use.
5.
Read the problem over carefully and ask yourself:
6.
Use variables and write an equation
7.
Try restating the problem in a different way.
8.
Decompose the problem
9.
Once you have an idea for a new approach, jot it down immediately.
10.
Try a new perspective on how to approach solving the problem.
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SEATWORK.
II. Use Polya’s four-step problem-solving strategy to solve each of the following exercises. Check your solution
out.
1.
The sum of a number and 2 is 6 less than twice that number.
2.
A rectangular garden has a width that is 8 feet less than twice the length. Find the dimensions if the perimeter is
20 feet.
3.
Place the digits 4, 6, 7, 8, and 9 in the circles to make the sum across and vertically equal 19. Is there more than
one answer? Explain briefly.
4.
A number is divided by 3. Then 14 is added to the quotient. The result is 33. What is the original number?
5.
Sally was having a party. She invited 20 women and 15 men. She made 1 dozen blue cupcakes and 3 dozen red
cupcakes. At the end of the party there were only 5 cupcakes left. How many cupcakes were eaten?
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Assignment
Solve the following using Polya’s Method in Solving Problem.
1. There are two numbers whose sum is 72. One number is twice of the
other. What are the numbers?
2. The sum of 5 times a number and 8 is 48. Find the number.
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Thank You
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