Ch1-L3.2-Problem-Solving.pptx

CHAPTER 1
LESSON 3.2 PROBLEM
SOLVING
Prepared by:
Crysali Therese R. Dayaganon

Chapter 1 Lesson 3.2 Problem Solving
3.2 Problem Solving
George Polya (1887 – 1985) was
a mathematics educator who
strongly believed that the skill
of problem solving can be
taught.
He developed a framework
known as Polya’s Four-Steps in
Problem Solving. This process
addressed the difficulty of
students in problem solving.

3.2 Problem Solving
Step 1: Understanding the
Problem
Needless to say that if you do
not understand the problem
you can never solve it.
It is also often true that if you
really understand the problem,
you can see a solution.

3.2 Problem Solving
Step 1: Understanding the Problem
• Can you restate the problem in your own words?
• Can you determine what is known about these types of
problems?
• Is there missing information that, if known, would allow
you to solve the problem?
• Is there extraneous information that is not needed to solve
the problem?
• What is the goal?

3.2 Problem Solving
Step 2: Devising a Plan
Polya mentioned that there are
many reasonable ways to solve
problems. The skill at choosing
an appropriate strategy is best
learned by solving many
problems. You will find choosing
a strategy increasingly easy.

3.2 Problem Solving
Step 2: Devising a Plan
• Guess and check
• Look for a pattern
• Make an orderly list
• Draw a picture
• Eliminate possibilities
• Solve a simpler problem
• Use symmetry
• Use a model
• Consider special cases
• Work backwards
• Use direct reasoning
• Use a formula
• Solve an equation
• Be ingenious

3.2 Problem Solving
Step 3: Carrying Out the Plan
This step is usually easier than
devising the plan. In general,
all you need is care and
patience, given that you have
the necessary skills. Persist
with the plan that you have
chosen. If it continues not to
work discard it and choose
another.

3.2 Problem Solving
Step 3: Carrying Out the Plan
• Be patient
• Work carefully
• Modify the plan or try a new plan
• Keep trying until something works
• Implement the strategy or strategies in Step 2
• Try another strategy if the first one isn’t working
• Keep a complete and accurate record of your work
• Be determined and don’t get discouraged if the plan does not work
immediately

3.2 Problem Solving
Step 4: Looking Back
Polya mentioned that much can
be gained by taking the time to
reflect and look back at what you
have done, what worked, and
what didn’t. Doing this will
enable you to predict what
strategy to use to solve future
problems.

3.2 Problem Solving
Step 4: Looking Back
• Look for an easier solution
• Does the answer make sense?
• Check the results in the original problem
• Interpret the solution with the facts of the problem
• Recheck any computations involved in the solution
• Can the solution be extended to a more general case?
• Ensure that all the conditions related to the problem are met
• Determine whether there is another method of finding the solution
• Ensure the consistency of the solution in the context of the problem

3.2 Problem Solving
Example 1
Suppose the NCAA
basketball championships
is decided on a best of five
series game. In how
different ways can a team
win the championships?

3.2 Problem Solving
Solution
Step 1: Understand the Problem.
There are many different orders to win the
championships. The team may have won three
straight games (WWW) or maybe they could lose the
first two games and won the last three games
(LLWWW). There are also other possibilities such as
WWLW, WLWW, or WLWLW.

3.2 Problem Solving
Solution
Step 2: Devise a Plan
Make an organized list of all possible orders and
ensure that each of the different orders is accounted
for only once.

3.2 Problem Solving
Solution
Step 3: Carry Out the Plan
Each entry in the list must contain
three Ws and may contain one or
two losses. Use a strategy to each
other. One strategy is to start to
write Ws, then write L if it is not
possible to write W. This strategy
produces ten (10) different orders
shown at the right.
WWW (Start with three wins)
WWLW (Start with two wins)
WWLLW (Start with two wins)
WLWW (Start with one win)
WLLWW (Start with one win)
WLWLW (Start with one win)
LWWW (Start with one loss)
LWWLW (Start with one loss)
LWLWW (Start with one loss)
LLWWW (Start with two losses)

3.2 Problem Solving
Solution
Step 4: Look Back
The list above is organized and contains no
duplications. It includes all
possibilities, we can conclude that there are ten (10)
different ways in which a basketball team can win the
NCAA championships in the best of 5 games.

3.2 Problem Solving
Example 2
Determine the digit
100 places to the
right of the decimal
point in the decimal
representation
7
27
.

3.2 Problem Solving
Solution
Express the fraction
7
27
as a decimal and look for a
pattern that will enable us to determine the digit 100
places to the right of the decimal point.

3.2 Problem Solving
Solution
Step 2: Devise a Plan.
Dividing 27 into 7 by long division or by using a calculator produces
the decimal 0.259259259... . Since the decimal representation repeats
the digits 259 over and over forever, we know that the digit located
100 places to the right of the decimal point is either a 2, a 5, or a 9. A
table may help us to see a pattern and enable us to determine which
one of these digits is in the 100th place. Since the decimal digits
repeat every three digits, we use a table with three columns.

3.2 Problem Solving
Solution
The First 15 Decimal Digits of
7
27

3.2 Problem Solving
Solution
Step 3: Carry Out the Plan.
Only in column 3 is each of the decimal digit locations
evenly divisible by 3. From this pattern we can tell
that the 99th decimal digit (because 99 is evenly
divisible by 3) must be a 9. Since a 2 always follows a
9 in the pattern, the 100th decimal digit must be a 2.

3.2 Problem Solving
Solution
Step 4: Look Back
The above table illustrates additional patterns. For instance, if each
of the location numbers in column 1 is divided by 3, a remainder of 1
is produced. If each of the location numbers in column 2 is divided by
3, a remainder of 2 is produced. Thus we can find the decimal digit in
any location by dividing the location number by 3 and examining the
remainder. For instance, to find the digit in the 3200th decimal place
of
7
27
, merely divide 3200 by 3 and examine the remainder, which is 2.
Thus, the digit 3200 places to the right of the decimal point is a 5.

3.2 Problem Solving
Example 3
Two times the sum of
a number and 3 is
equal to thrice the
number plus 4. Find
the number.

3.2 Problem Solving
Solution
We need to make sure that we have read the question
carefully several times. Since we are looking for a
number, we will let 𝑥 be a number.

3.2 Problem Solving
Solution
We will translate the problem mathematically. Two times the
sum of a number and 3 is equal to thrice the number plus 4.
2 𝑥 + 3 = 3𝑥 + 4

3.2 Problem Solving
Solution
We solve for the value of 𝑥, algebraically.
2 𝑥 + 3 = 3𝑥 + 4
2𝑥 + 6 = 3𝑥 + 4
3𝑥 − 2𝑥 = 6 − 4
𝑥 = 2

3.2 Problem Solving
Solution
Step 4: Look Back
If we take two times the sum of 2 and 3, that is the same as
thrice the number 2 plus 4 which is 10, so this does check.
Thus, the number is 2.

3.2 Problem Solving
Example 4
Two times the sum of a
number and 3 is equal to
thrice the number plus 4.
Find the number.

3.2 Problem Solving
Example 5
Three siblings Ivan, Ilya, and Iara. Ivan gave Ilya
and Iara as much money as each had. Then Ilya gave
Ivan and Iara as much money as each had. Then Iara
gave Ivan and Ilya as much money as each had. Then
each of the three had Php 128. How much money did
each have originally?

3.2 Problem Solving
Solution
The problem is a little bit confusing and needs to be
carefully analyzed.

3.2 Problem Solving
Solution
We will be working backwards

3.2 Problem Solving
Solution
Fourth: Each has Php 128.
Third: Iara gave Ivan and Ilya as much money as each has.
Second: Ilya gave Ivan and Iara as much money as each has.
First: Ivan gave Ilya and Iara as much money as each has.
Stages Ivan Ilya Iara
Fourth 128 128 128
Third 64 64 256
Second 32 224 128
First 208 112 64

3.2 Problem Solving
Solution
Step 4: Look Back
Thus, Ivan, Ilya and Iara’s initial money are Php208, Php112 and Php 64, respectively.
Stages Ivan Ilya Iara
First 208 112 64
Second 208-112-64=32 112+112=224 64+64=128
Third 32+32=64 224-32-128=64 128+128=256
Fourth 64+64=128 64+64=128 256-64-64=128

3.2 Problem Solving
What is a sequence?
• An ordered list of objects (or events)
• The numbers in a sequence that are separated by commas are the
terms of the sequence.
Example:
5, 14, 27, 44, 65, …
• The subscript notation 𝑎𝑛 is used to designate the nth term of a
sequence.

3.2 Problem Solving
What is the next term of the sequence?
5, 14, 27, 44, 65, …

3.2 Problem Solving
We can use difference table to show the differences between
successive terms of the sequence.
Sequence: 5, 14, 27, 44, 65, …
First difference 9 13 17 21 …
Second difference 4 4 4

3.2 Problem Solving
We can use difference table to show the differences between
successive terms of the sequence.
Sequence: 5, 14, 27, 44, 65, 90, …
First difference 9 13 17 21 25 …
Second difference 4 4 4 4 …

3.2 Problem Solving
What is the next term of each sequence?
• 3, 7, 11, 15, 19, ____
• 4, 7, 11, 16, 22, ____
• 6, 9, 14, 26, 50, ____

3.2 Problem Solving
References:
Aufmann, Richard N. et al. 2013. Mathematical Excursions. Third Edition. USA:
Brooks/Cole, Cengage Learning.
Sirug, Winston S. 2018.Mathematics in the Modern World. Manila: Mindshapers Co.,
Inc.

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