Problem Solving and

Reasoning
Definition of terms
Approaches to Problem Solving
Problem Solving with Patterns
Polya’s problem solving strategy
What is a/an ...
Problem? - a task that requires the learner to
reason through a situation that will
be challenging but not impossible
- provides practice in using
Exercise? algorithm and maintaining the
basic facts

- encompasses exploring,
Problem reasoning, strategizing, estimating,
Solving? conjecturing, testing, explaining,
and proving.
Problem
Solving Goal Obstacle Solution
Process
Approaches to Problem Solving

Inductive Reasoning

Deductive Reasoning
Approaches to Problem Solving
Inductive Reasoning

• The type of reasoning that comes up to a conclusion by

examining specific examples.
• A conclusion that is made by applying inductive
reasoning is called a conjecture.
• Conjectures may or may not be correct.
Approaches to Problem Solving
Deductive Reasoning

• a valid form used in proving observations and arriving at

conclusions.
• a process by which one makes conclusions based on
previously accepted general assumptions, procedures,
and principles.
Inductive Reasoning
Examples of Inductive Reasoning
1. Determine the next term of this sequence.
13
1, 3, 5, 7, 9, 11, ____

2. Determing the next shape.

____________
Examples of Inductive Reasoning
3. Determine the next term of this sequence.
26
2, 2, 4, 6, 10, 16, ____

4. Determing the next shape.

1 2 3 ???
Examples of Inductive Reasoning
5. Determine the next term of this sequence.
?
1, 2, 4, 8, 16, ___
32

6. Using circles, dots, and lines, determine the next term

of this sequence.
?
1, 2, 4, 8, 16, ____
 3
6. Determine the next 8
term of this sequence. 2 6 7
12
?
1, 2, 4, 8, 16, ____ 10 11 13 4 ?
14
1 9 16
15
5
2
2

3 7
1 3
1 6
2 1 4 4
1 5
8

Using circles, dots, … a b c d e f

Number of Dots 1 2 3 4 5 6
Number of Region 1 2 4 8 16 ?
Making Conjectures
 Making Conjectures
1) Write a conjecture that describes the pattern 2, 4, 12,
48, 240. Then use your conjecture to find the next item in
the sequence.
Step 1: Look for a pattern 2 ∙ 2 = 4 12 ∙ 4 = 48
2, 4, 12, 48, 240 4 ∙ 3 = 12 48 ∙ 5 = 240
Step 2: Make a conjecture
The numbers are multiplied by 2, 3, 4, and then 5. The next
number will be the product of 240 ∙ 6 = 1440.

Answer: 1440
 Making Conjectures
2) Write a conjecture that describes the pattern shown.
Then use your conjecture to find the next item in the
sequence.

3 +6 9 +9 18 +12 30

Step 1: Look for a pattern 6=3∙2 9=3∙3 12 = 3 ∙ 4

Step 2: Make a conjecture
The figure will increase by the next multiple of 3. If we add 12 + 18 = 30. That is, the
next figure is made of 30 segments.
Answer: 30 𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠 Check: Draw the next figure
 Making Conjectures
3) Write a conjecture that describes the given. Then use
your conjecture to find the next item in the sequence.
1 1 1 1 1 1 1 1 1 𝟏
1, , , , → , , , , ,
1 4 9 16 25 𝟑𝟔
4 9 16 25
Step 1: Look for a pattern
12 22 32 42 52
Step 2: Make a conjecture
The next term has a denominator that is the next perfect square
(or next integer squared).
Predict the next number in the pattern: 𝟔𝟐 = 𝟑𝟔
Making Conjectures
4) Pick a number, multiply the number by 4, add 2 to the
product, divide the sum by 2, and subtract 1 from the
quotient. Repeat this procedure for several different
numbers and then make a conjecture about the
relationship between the original number and the final
number.
𝑪𝒐𝒏𝒋𝒆𝒄𝒕𝒖𝒓𝒆:
𝟑 𝟒 𝟓 The output is always
3 ∙ 4 = 12 4 ∙ 4 = 16 5 ∙ 4 = 20 twice the original
12 + 2 = 14 16 + 2 = 18 20 + 2 = 22 number.
14 18 22 6=3+3=𝟑∙𝟐
=7 =9 = 11
2 2 2 8=4+4=𝟒∙𝟐
7−1=𝟔 9−1= 𝟖 11 − 1 = 𝟏𝟎 10 = 5 + 5 = 𝟓 ∙ 𝟐
Counterexample
Counterexample (instance for which the
statement becomes false)
Direction: Find a number that provides a counterexample
to show that the given statement is false.
1) ∀ 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑥, 𝑥 3 ≥ 𝑥.
This is a counterexample.
Let 𝑥 = 3. Let 𝑥 = −3.
Therefore, the given
𝑥3 ≥ 𝑥 𝑥3 ≥ 𝑥 statement is false.
33 ≥ 3 −33 ≥ 3
27 ≥ 3 −27 ≥ 3
∴ −27 < 3
Counterexample (instance for which the
statement becomes false)
Direction: Find a number that provides a counterexample
to show that the given statement is false.
2) ∀ 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑥, 𝑥 + 3 = 𝑥 + 3.
This is a
Let 𝑥 = 1. Let 𝑥 = −1.
counterexample.
𝑥+3 = 𝑥 +3 𝑥+3 = 𝑥 +3 Therefore, the
given statement is
1+3 = 1 +3 −1 + 3 = −1 + 3 false.
4 =1+3 2 =1+3
4=4 2≠4
Deductive Reasoning
Deductive Reasoning
1. Using deductive reasoning, prove that the given
procedure below will always result to twice the original
number.
Pick a number, multiply the number by 4, add 2 to the
product, divide the sum by 2, and subtract 1 from the quotient.

𝒏
𝑛 ∙ 𝟒 = 4𝑛
∴ 𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑛𝑢𝑚𝑏𝑒𝑟 𝑛,
4𝑛 + 𝟐
𝑡ℎ𝑒 𝑟𝑒𝑠𝑢𝑙𝑡 𝑖𝑠 𝑎𝑙𝑤𝑎𝑦𝑠 𝟐𝒏 𝑜𝑟
4𝑛 + 2
= 2𝑛 + 1 𝑡𝑤𝑖𝑐𝑒 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟.
𝟐
2𝑛 + 1 − 𝟏 =
= 𝟐𝒏
𝟐𝒏
 Deductive Reasoning
2) Each of four neighbors, Sean, Maria, Sarah, and Brian, has a
different occupation (editor, banker, chef, or dentist). From the
following clues, determine the occupation of each neighbor.
a) Maria gets home from work after the banker but before the
dentist.
b) Sarah, who is the last to get home from work, is not the editor.
c) The dentist and Sarah leave for work at the same time.
d) The banker lives next door to Brian.
Editor Banker Chef Dentist
Sean X / X X Therefore,
Sean – Banker
Maria / X X X
Maria – Editor
Sarah X X / X Sarah – Chef
Brian X X X / Brian - Dentist
Problem Solving
with Patterns
Terms of a Sequence
 Terms of a Sequence
A sequence is an ordered list of numbers.
5, 14, 27, 44, 65, …

𝑎1 represents the first term in the sequence

𝑎2 represents the second term in the sequence
𝑎3 represents the third term in the sequence
⋮
𝑎𝑛 represents the 𝑛𝑡ℎ term in the sequence
 Terms of a Sequence
When we examine a sequence, the following questions are
frequently asked:
• What is the next term?
• What formula or rule can be used to generate the terms?

We will study the basic sequences and find the next term
of a sequence using a difference table.
A difference table is often used to show differences
between successive terms of the sequence.
 Terms of a Sequence
The following table is the difference table for the
sequence: 2, 5, 8, 11, 14, …

Sequence 2 5 8 11 14 17
First
Difference 3 3 3 3 3
 Terms of a Sequence
Consider the given sequence with its difference table.
5, 14, 27, 44, 65, …
Sequence 5 14 27 44 65 90
First Dif 9 13 17 21 25
Second
4 4 4 4
Dif
 Terms of a Sequence
Use the difference table to predict the next term in the
sequence.
2, 7, 24, 59, 118, 207, …

Squence
2 7 24 59 118 207 332
First Difference
5 17 35 59 89 125
Second
Difference
12 18 24 30 36
Third Difference
6 6 6 6
Nth Term Formula for a Sequence
Nth term Formula for a Sequence
Consider the formula 𝑎𝑛 = 3𝑛2 + 𝑛. This formula defines a
sequence and provides a method for finding any term of
the sequence.
𝑛=1 𝑛=3
𝑎𝑛 = 3𝑛2 + 𝑛 𝑎𝑛 = 3𝑛2 + 𝑛
𝑎1 = 3(12 ) + 1 = 3 + 1 = 𝟒 𝑎2 = 3(32 ) + 3 = 27 + 3 = 𝟑𝟎
𝑛=2 𝑛=4
𝑎𝑛 = 3𝑛2 + 𝑛 𝑎𝑛 = 3𝑛2 + 𝑛
𝑎2 = 3(22 ) + 2 = 12 + 2 = 𝟏𝟒 𝑎2 = 3(42 ) + 4 = 48 + 4 = 𝟓𝟐

𝟒, 𝟏𝟒, 𝟑𝟎, 𝟓𝟐, …

 Nth Term of a Sequence
Assume the pattern shown by the square tiles in the following figure
continues.
a) What is the 𝑛𝑡ℎ-term formula for the number of tiles in the 𝑛𝑡ℎ figure of
the sequence? 𝟑𝒏 − 𝟏
b) How many tiles are in the eighth figure of the sequence? 𝟐𝟑
c) Which figure will consist exactly 320 tiles? 𝟏𝟎𝟕𝒕𝒉 𝒇𝒊𝒈𝒖𝒓𝒆
𝒂)
𝒃) 𝟑𝒏 − 𝟏
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑙𝑒𝑠: 2, 5, 8, 11, …
𝐶𝑜𝑚𝑚𝑜𝑛 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒: 3 𝟑 𝟖 − 𝟏 = 𝟐𝟑
𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 𝑓𝑜𝑟𝑚𝑢𝑙𝑎: 3𝑛 −1 𝒄) 𝟑𝒏 − 𝟏 = 𝟑𝟐𝟎
𝟑𝒏 = 𝟑𝟐𝟎 + 𝟏 = 𝟑𝟐𝟏
𝟑𝟐𝟏
𝒏=
𝟑
𝑎1 𝑎2 𝑎3 𝑎4 𝒏 = 𝟏𝟎𝟕
Types of a Sequence
 Types of Sequence
1. Arithmetic Sequence (common difference)
1, 5, 9, 13,17, 21, 25, 29, 33, . ..
2. Geometric Sequence
2, 4, 8, 16, 32, 64, …
2. Triangular Number Sequence
1, 3, 6, 10, 15, 21, 28, 36, 45, ...
Polya‘s Problem Solving Strategy
Polya’s Problem
Solving Strategy
George Pólya: The Father of
Problem Solving

• George Pólya (1887-1985) was a Hungarian

mathematician.
• He made fundamental contributions to combinatorics,
number theory, numerical analysis and probability
theory. He is also noted for his work in heuristics and
mathematics education.
Pólya’s Four-Step Problem Solving
Strategy
• In 1945 George Pólya published the book How To Solve It
which quickly became his most prized publication.
• It sold over one million copies and has been translated
into 17 languages.
• In this book he identifies four basic principles of problem
solving.
• Understand the problem (UP)
• Devise a Plan (DP)
• Carry out the plan (COP)
• Look back/review the solution (RS)
 Understand 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?
 Devise a Plan
• Make a list of the known information.
• Make a list of information that is needed.
• Draw a diagram.
• Make an organized list that shows all the possibilities.
• Make a table or a chart.
• Work backwards.
• Try to solve a similar but simpler problem.
• Look for a pattern.
• Write an equation. If necessary, define what each variable represents.
• Perform an experiment.
• Guess at a solution and then check your result.
• Use indirect reasoning.
 Carry Out the Plan
• Work carefully.
• Keep an accurate and neat record of all your attempts.
• Realize that some of your initial plans will not work and
that you may have to devise another plan or modify your
existing plan.
 Review the Solution
• Ensure that the solution is consistent with the facts of
the problem.
• Interpret the solution in the context of the problem.
• Ask yourself whether there are generalizations of the
solution that could apply to other problems.
 Polya’s 4-Step Strategy
Problem 1: The product of the ages, in years, of three
teenagers is 4590. None of the teens are the same age.
What are the ages of the teenagers?

Understand the Problem:

There is a need to determine three distinct whole
numbers, from the list 13, 14, 15, 16, 17, 18, and 19,
that have a product of 4590.
 Polya’s 4-Step Strategy
The product of the ages, in years, of three teenagers is
4590. None of the teens are the same age. What are the
ages of the teenagers?
Devise a Plan:
If we represent the ages by 𝑥, 𝑦, and 𝑧, then 𝑥𝑦𝑧 = 4590. It is
not possible to solve this equation, but notice that 4590 ends
in a zero. Hence, 4590 has a factor of 2 and a factor of 5,
which means that at least one of the numbers we seek must
be an even number and at least one must have 5 as a factor.
The only number in the list that has 5 as a factor is 15. Thus,
15 is one of the numbers and at least one of the other
numbers must be an even number. At this point, try to solve
this problem by guessing and checking.
 Polya’s 4-Step Strategy
The product of the ages, in years, of three teenagers is
4590. None of the teens are the same age. What are the
ages of the teenagers?

Carry out the Plan:

15 ∙ 16 ∙ 18 = 4320
15 ∙ 16 ∙ 19 = 4560
15 ∙ 17 ∙ 18 = 4590
∴The ages of the teenagers are 15, 17, and 18.
 Polya’s 4-Step Strategy
The product of the ages, in years, of three teenagers is
4590. None of the teens are the same age. What are the
ages of the teenagers?

Review the Solution:

• Because 15 ∙ 17 ∙ 18 = 4590 and each of the ages
represents the age of a teenager, we note that the
solution is correct.
• Also, the numbers 13, 14, 16, and 19 are not factors
(divisors) of 4590. Thus, there are no other solutions.
UP:
Polya‘s 4-Step Strategy
Names of Children
Number of children
Problem 2: A pregnant lady named her children:
Dominique, Regis, Michelle, Fawn, Sophie and Lara.
DP: What will she name her next child?
Look for a pattern Jessica, Katie, Abby or Tilly?
COP:
Do, Re, Mi, Fa, La, Ti Answer:
∴ Name of next child: Tilly Tilly
She seems to follow the scale
RS:
Do, Re,
The lady seems to follow theMe,
scaleFa, So, La, and then Ti.
Do, Re, Me, Fa, So, La, and then Ti.
 Polya’s 4-Step Strategy
Problem 3: A volleyball team won two out of their last four
games. In how many different orders could they have two
wins and two losses in four games?
UP: There are many different orders. The team may have won two
straight games and lost the last two (WWLL). Or they lost the first
two games and won the last two (LLWW). Consider other
probabilities such as WLWL.

DP: Make an organized list. An organized list is a list that is

[produced using a system that ensures listing different orders to
be listed once and only once.
 Polya’s 4-Step Strategy
A volleyball team won two out of their last four games. In
how many different orders could they have two wins and
two losses in four games?
COP: Each entry in the list must contain two W’s and two LL’s.
The list must be in order and no duplications.

WWLL WLWL WLLW LWWL LWLW LLWW

RS: An organized list is made. It has no duplicates and the list

considers all possibilities. Therefore, there are six different orders
in which a volleyball team can win exactly two out of four games.
 Polya’s 4-Step Strategy
Problem 4: The figure shows 12 toothpicks arranged to form 3
squares. How can you form 5 squares by moving only 3
toothpicks?
UP: Form 5 squares by moving only 3
toothpicks.
DP: Experiment or Trial & Error
COP: Move each toothpick with hand.
RS: One of the squares is formed by the
outer boundary of the arrangement.
There was no requirement that each of
the five squares must be congruent to
each of the others.
 Polya’s 4-Step Strategy
Problem 4: The figure shows 12 toothpicks arranged to form 3
squares. How can you form 5 squares by moving only 3
toothpicks?
UP: Form 5 squares by moving only 3
toothpicks.
DP: Experiment or Trial & Error
COP: Move each toothpick with hand.
RS: One of the squares is formed by the
outer boundary of the arrangement.
There was no requirement that each of
the five squares must be congruent to
each of the others.
 Polya’s 4-Step Strategy
Problem 5: Two apples (A) weigh the same as a banana (B)
and a cherry (C). A banana weighs the same as nine
cherries. How many cherries weigh the same as one apple?
UP: COP:
2𝑨 = 𝑩 + 𝑪 2𝑨 = 𝑩 + 𝑪
𝑩 = 9𝑪 Substitute 9𝑪 for 𝑩
𝑨 = 𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑐ℎ𝑒𝑟𝑟𝑖𝑒𝑠? 2𝑨 = 9𝑪 + 𝑪 = 10𝑪
10𝑪
DP: 𝑨= = 𝟓𝑪
𝟐
Introduce 3 variables
A = weight of an apple Therefore, five cherries
B = weight of a banana weigh the same as one
C = weight of a cherry apple.
 Polya’s 4-Step Strategy
Problem 5: Two apples (A) weigh the same as a banana (B)
and a cherry (C). A banana weighs the same as nine
cherries. How many cherries weigh the same as one apple?
RS:
UP: COP:
= 𝑩 + 𝑪an equation with2𝑨
•2𝑨Solving = 𝑩 + 𝑪 is hard.
3 variables
𝑩 = 9𝑪 Substitute 9𝑪 for 𝑩
That
•𝑨 = 𝐻𝑜𝑤is𝑚𝑎𝑛𝑦
why,𝑐ℎ𝑒𝑟𝑟𝑖𝑒𝑠?
in place of 𝑩,
2𝑨we placed
= 9𝑪 +𝑪= 9𝑪.
10𝑪
• This means that we just 𝑨 have10𝑪
DP: = to deal
= 𝟓𝑪 with just 2
𝟐
Introduce 3 variables
variables: 𝑨 and 𝑪.
A = weight an apple
• Thus, solving for the unknown becomes easier.
B = weight of a banana,
C = weight of a cherry