Reasoning
and
Problem Solving
Reasoning
Inductive Reasoning – is the process of
reaching a general conclusion
by examining specific examples.
A conclusion based on inductive
reasoning is called a conjecture.
A conjecture may or may not be
correct.
3
PROBLEM SOLVING
EXAMPLE 1:
Use inductive reasoning to
predict the next number in
each of the following lists.
a)3, 6, 9, 12, 15, ?
b)1, 3, 6, 10, 15, ?
PROBLEM SOLVING
4
Your Turn
Use inductive reasoning to
predict the next number in
each of the following lists.
a. 5, 10, 15, 20, 25, ?
b. 2, 5, 10, 17, 26, ?
5
PROBLEM SOLVING
EXAMPLE 2:
Consider the following procedure: Pick
a number. Multiply the number by 8, add 6
to the product, divide the sum by 2, and
subtract 3.
Complete the above procedure for
several different numbers. Use inductive
reasoning to make a conjecture about the
relationship between the size of the
resulting number and the size of the
original number.
6
PROBLEM SOLVING
Solution:
Pick any number:
a
Multiply by 8:
8a
Add 6 to the product:
8a+6
Divide the sum by 2:
8𝑎+6
2
Subtract 3:
4a+3-3 = 4a
=
8𝑎
2
6
2
+ = 4𝑎 + 3
• Therefore, the given procedure produces a
number that is four times the original number.
7
Your Turn
• Consider the following procedure: Pick a
number. Multiply the number by 9, add
15 to the product, divide the sum by 3,
and subtract 5.
• Complete the above procedure for several
different numbers. Use inductive
reasoning to make a conjecture about the
relationship between the size of the
resulting number and the size of the
original number.
PROBLEM SOLVING
8
EXAMPLE 3:
Length of pendulum, in
units
Period of pendulum, in
heartbeats
1
1
4
2
9
3
16
4
25
5
36
6
The period of a pendulum
Use the data in the table and inductive is the time it takes for the
reasoning to answer each of the following pendulum to swing from
left to right and back to its
questions.
original position.
a) If a pendulum has a length of 49 units,
what is its period?
b) If the length of a pendulum is
quadrupled,
period?
what
happens
to
its
9
EXAMPLE 3:
Length of
pendulum, in
units
1
4
9
16
25
36
Period of
pendulum, in
heartbeats
1
2
3
4
5
6
a) If a pendulum has a length of 49 units, what is its period?
10
EXAMPLE 3:
Length of
pendulum, in
units
1
Period of
pendulum, in
heartbeats
4
2= 4
9
3= 9
16
4= 16
25
5= 25
1= 1
36
6= 36
a) If a pendulum has a length of 49 units, what is
its period?
Each pendulum has a period that is the square root of its
length. Thus, the pendulum with a length of 49 units will have
a period of 7 heartbeats.
11
EXAMPLE 3:
Length of
pendulum, in
units
1
4
9
16
25
36
Period of
pendulum, in
heartbeats
1
2
3
4
5
6
a) If the length of a pendulum is quadrupled, what happens
to its period?
12
EXAMPLE 3:
Length
Period
Length
Period
Length
Period
1
4
9
16
25
36
1
2
3
4
5
6
1
4
9
16
25
36
1
2
3
4
5
6
1
4
9
16
25
36
1
2
3
4
5
6
a) If the length of a pendulum is quadrupled, what happens to
its period?
Quadrupling the length of a pendulum doubles its period.
13
Your Turn
Velocity of
Height of Tsunami,
Tsunami, ft/sec
ft
6
4
9
9
12
16
15
25
18
36
21
49
24
64
• Use inductive reasoning to answer each of the following
questions.
• a. What happens to the height of a tsunami when its velocity
is doubled?
• b. What should be the height of a tsunami if its velocity is 30
14
feet per second?
NOTE
• Conclusions based on inductive reasoning may
be incorrect.
Example:
Ms. Dela Cruz wears white on Monday, Tuesday,
Wednesday, and Thursday. What color does she
wear on Friday?
PROBLEM SOLVING
15
Deductive Reasoning – is the process of
reaching
a
conclusion
by
applying general assumptions,
procedures, or principles.
16
PROBLEM SOLVING
EXAMPLE 1:
All birds can fly. Tweetie is a bird.
Therefore, Tweetie can fly
Every Filipino of age 18 and above can
vote. Juan is a 24 year old Filipino
man. Therefore, Juan can vote.
17
PROBLEM SOLVING
EXAMPLE 2: Solve a Logic Puzzle
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.
1) Maria gets home from work after the
banker but before the dentist.
2) Sarah, who is the last to get home from
work, is not the editor.
3) The dentist and Sarah leave for work at the
same time.
4) The banker lives next door to Brian.
PROBLEM SOLVING
18
Editor Banker
Chef
Dentist
Sean
Maria
Sarah
Brian
PROBLEM SOLVING
19
1) Maria gets home from work after the
banker but before the dentist.
Editor Banker
Sean
Maria
Sarah
Brian
x
Chef
Dentist
x
20
PROBLEM SOLVING
2.) Sarah, who is the last to get home
from work, is not the editor.
3.) The dentist and Sarah leave for work
at the same time.
Editor Banker
Sean
Maria
Sarah
Brian
x
x
Chef
Dentist
x
x
21
PROBLEM SOLVING
4. The banker lives next door to Brian.
Editor Banker
Sean
Maria
Sarah
Brian
x
x
Chef
Dentist
x
x
x
22
PROBLEM SOLVING
1) Maria gets home from work after the
banker but before the dentist.
2) Sarah, who is the last to get home
from work, is not the editor.
Editor Banker
Sean
Maria
Sarah
Brian
x
x
x
x
Chef
Dentist
x
x
23
PROBLEM SOLVING
Therefore, Sarah is the Chef
Editor Banker
Sean
Maria
Sarah
Brian
x
x
x
x
Chef
Dentist
/
x
x
24
PROBLEM SOLVING
Sean is the Banker
Sean
Maria
Sarah
Brian
Editor Banker
/
x
x
x
x
Chef
Dentist
/
x
x
25
PROBLEM SOLVING
Since there is only one chef, mark the
other cell in CHEF COLUMN with x
Sean
Maria
Sarah
Brian
Editor Banker
/
x
x
x
x
Chef
x
x
/
x
Dentist
x
x
26
PROBLEM SOLVING
Maria is the Editor
Sean
Maria
Sarah
Brian
Editor Banker
/
/
x
x
x
x
Chef
x
x
/
x
Dentist
x
x
27
PROBLEM SOLVING
Brian is the Dentist
Sean
Maria
Sarah
Brian
Editor Banker
x
/
/
x
x
x
x
x
Chef
x
x
/
x
Dentist
x
x
x
/
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PROBLEM SOLVING
Your Turn Solve a Logic Puzzle
Brianna, Ryan, Tyler, and Ashley were recently
elected as the new class officers (president,
vice president, secretary, treasurer) of the
sophomore class at Summit College. From the
following clues, determine which position each
holds.
1. Ashley is younger than the president but
older than the treasurer.
2. Brianna and the secretary are both the
same age, and they are the youngest members
of the group.
3. Tyler and the secretary are next-door
neighbors.
29
PROBLEM SOLVING
Exercise: INDUCTIVE OR DEDUCTIVE
• 1. I heard lots of barking last night. The neighbor’s dog must’ve
been pretty upset about something, since he rarely barks.
• 2. All dogs bark. Fido is a dog, so he barks.
• 3. No book in English begins numbering its pages on a left hand
page. This is a book in english, therefore it will begin its
numbering on a right hand page.
• 4. Based on a survey of 3300 randomly selected registered voters,
56.2% indicate that they will vote for the incumbent officials in the
upcoming election. Therefore, approximately 56% of the voles in
the upcoming election will be for the incumbent.
• 5. All mollusks are invertebrates. Snails are mollusks, so snails
must be invertebrates.
• 6. Jack is taller than jill. Jill is taller than Joey. Therefore, Jack
is taller than Joey.
• 7. Cats routinely kill birds and mice. There is a cat, so it
almost assuredly kills birds and mice.
• 8. No whales live in freshwater, and the lake is a fresh water,
so there are no whales living there.
• 9. In the sequence 3,6,9,12,15, the next term is going to be
18.
ANSWERS
• 1. I heard lots of barking last night. The neighbor’s dog
must’ve been pretty upset about something, since he rarely
barks.
• Answer: Inductive
• Reason: The speaker is relying on a collection of
experiences to draw an inference.
• 2. All dogs bark. Fido is a dog, so he barks.
• Answer: Deductive.
• Reason: The premises guarantee the conclusion
• 3. No book in English begins numbering its pages on a left
hand page. This is a book in english, therefore it will begin its
numbering on a right hand page.
• Answer: Deductive
• Reason: The conclusion follows necessarily from the
premises.
• 4. Based on a survey of 3300 randomly selected registered voters,
56.2% indicate that they will vote for the incumbent officials in the
upcoming election. Therefore, approximately 56% of the voles in the
upcoming election will be for the incumbent.
• Answer: Inductive
• Reason: The conclusion does not follow necessarily follow from the
premise. The conclusion follow with some probability…
• 5. All mollusks are invertebrates. Snails are mollusks, so
snails must be invertebrates.
• Answer: Deductive
• Reason: The premise guarantee the conclusion
• 6. Jack is taller than jill. Jill is taller than Joey. Therefore, Jack
is taller than Joey.
• Answer: Deductive
• Reason: The premise guarantee the conclusion.
• 7. Cats routinely kill birds and mice. There is a cat, so it
almost assuredly kills birds and mice.
• Answer: Inductive
• Reason: Words such as “routinely” and “almost assuredly”
indicate the inductive character of the argument.
• 8. No whales live in freshwater, and the lake is a fresh water,
so there are no whales living there.
• Answer: Deductive
• Reason: The premises guarantee the conclusion.
• 9. In the sequence 3,6,9,12,15, the next term is going to be
18.
• Answer: Inductive
• Reason: The conclusion is based on pattern.
Problem Solving
1. Problem Solving with Patterns
Sequences
A sequence is an ordered list of
numbers. Each number in a sequence is
called a term of the sequence. The 𝑎𝑛 is
used to designate the 𝑛𝑡ℎ term of a
sequence.
A formula that can be used to
generate all the terms of a sequence is
called an 𝑛𝑡ℎ − 𝑡𝑒𝑟𝑚 formula.
PROBLEM SOLVING
42
EXAMPLE 1:
Predict the Next Term
Use a difference table to predict the
next term in the following sequences.
a. 5, 14, 27, 44, 65
b. 2, 7, 24, 59, 118, 207, …
43
PROBLEM SOLVING
Solution:
5
d1
14
9
d2
27
13
4
44
17
4
65
21
4
44
PROBLEM SOLVING
Solution:
5
d1
14
9
d2
27
13
4
44
17
4
65
21
4
90
25
4
Answer: The next term is 90.
45
PROBLEM SOLVING
Solution:
2
d1
d2
7
5
24
17
12
d3
59
35
18
6
118
59
24
6
207
89
30
6
46
PROBLEM SOLVING
Solution:
2
d1
d2
7
5
24
17
12
d3
59
35
18
6
118
59
24
6
207 332
89
30
6
125
36
6
ANSWER: The next term is 332.
47
PROBLEM SOLVING
Your Turn:
Use a difference table to predict the
next term in the sequence.
1, 14, 51, 124, 245, 426
48
PROBLEM SOLVING
2. Problem-Solving Strategies
One
of
the
foremost
recent
mathematicians to make a study of
problem solving was George Polya
(1877-1985). He was born in Hungary
and moved to the United States in 1940.
the basic problem-solving strategy that
Polya advocated consisted of the
following four steps.
49
PROBLEM SOLVING
Polya’s Four-Step Problem-Solving
Strategy
1) Understand the problem.
2) Devise a plan.
3) Carry out the plan.
4) Review the solution.
50
PROBLEM SOLVING
Polya’s four steps are deceptively
simple. To become a good problem
solver, it helps to examine each of these
steps and determine what is involved.
51
PROBLEM SOLVING
Once you have found a
solution, check the solution.
Ensure that the solution is
consistent with the facts of
the problem.
• Work carefully.
• Keep an accurate and neat
record of all your attempts.
• Realize that some of your
initial plans will not work
and modify your plan.
PROBLEM SOLVING
You must have a clear
understanding of the problem.
“Can you restate the problem
in your own words?”
Successful problem solvers
use a variety of techniques
when they attempt to solve a
problem.
52
1) Understand the problem.
You must have a clear understanding of the
problem. Consider the following questions.
• 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?
PROBLEM SOLVING
53
2. Devise a plan.
• Successful problem solvers use a variety of
techniques when they attempt to solve a problem.
• ■ Make a list of the known information.
• ■ Draw a diagram.
• ■ Make an organized list that shows all the
possibilities.
• ■ Make a table or a chart.
• ■ Work backwards.
• ■ Look for a pattern.
• ■ Write an equation. If necessary, define what each
variable represents.
PROBLEM SOLVING
54
3. Carry out the plan.
Once you have devised a plan, you must carry
it out.
• 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.
55
PROBLEM SOLVING
4. Review the solution.
• Once you have found a solution, check 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.
56
PROBLEM SOLVING
EXAMPLE 1:
Apply Polya’s Strategy
A baseball 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?
57
PROBLEM SOLVING
PROF. ANNABELLE Q. SOLLANO
Solution
Understand the problem:
A baseball 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?
• There are many different orders. The team may
have won two straight games and lost the last
two (WWLL). Or maybe they lost the first two
games and won the last two (LLWW). Of
course there are other possibilities, such as
WLWL.
58
PROBLEM SOLVING
PROF. ANNABELLE Q. SOLLANO
Solution
Device a plan
A baseball 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?
• Since it is about “order”, we will make an
organized list of all the possible orders. Each
entry in our list must contain two W’s and
two L’s.
PROBLEM SOLVING
PROF. ANNABELLE Q. SOLLANO
59
Solution
Carry Out the Plan
A baseball 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?
• 1. WWLL
• 2. WLWL
• 3. WLLW
• 4. LWWL
• 5. LWLW
• 6. LLWW
PROBLEM SOLVING
PROF. ANNABELLE Q. SOLLANO
60
Solution
Review
A baseball 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?
• The list has no duplicates and the list considers all
possibilities, so we are confident that there are six
different orders in which a baseball team can win
exactly two out of four games.
61
PROBLEM SOLVING
PROF. ANNABELLE Q. SOLLANO
Solution
Answer:
Therefore, there are 6 different
orders in which a baseball team can
win exactly two out of four games.
62
PROBLEM SOLVING
PROF. ANNABELLE Q. SOLLANO
EXAMPLE 2: Apply Polya’s Strategy
In consecutive turns of a
Monopoly game, Stacy first paid
$800 for a hotel. She then lost half
her money when she landed on
Boardwalk. Next, she collected $200
for passing GO. She then lost half
her remaining money when she
landed on Illionois Avenue. Stacy
now has $2,500. How much did she
have just before she purchased the
hotel?
63
PROBLEM SOLVING
Understand the Problem
In consecutive turns of a Monopoly game,
Stacy first paid $800 for a hotel. She then lost
half her money when she landed on
Boardwalk. Next, she collected $200 for
passing GO. She then lost half her remaining
money when she landed on Illionois Avenue.
Stacy now has $2,500. How much did she
have just before she purchased the hotel?
We need to determine the number of dollars that
Stacy had just prior to her $800 hotel purchase.
PROBLEM SOLVING
64
Device a plan
In consecutive turns of a Monopoly
game, Stacy first paid $800 for a hotel.
She then lost half her money when she
landed on Boardwalk. Next, she collected
$200 for passing GO. She then lost half
her remaining money when she landed on
Illionois Avenue. Stacy now has $2,500.
How much did she have just before she
purchased the hotel?
Since we know the end result, let’s try the
method of working backwards.
PROBLEM SOLVING
65
Carry out the plan
In consecutive turns of a Monopoly game, Stacy
first paid $800 for a hotel. She then lost half her money
when she landed on Boardwalk. Next, she collected
$200 for passing GO. She then lost half her remaining
money when she landed on Illionois Avenue. Stacy now
has $2,500. How much did she have just before she
purchased the hotel?
• Stacy must have had $2500x 2 = $5000 just before she
landed on Illinois Avenue;
• $5000-$200= $4800 just before she passed GO;
• $4800x2= $9600 prior to landing on Boardwalk.
• This means she had $9600+800= $10,400 just before she
purchased the hotel.
66
PROBLEM SOLVING
Review
In consecutive turns of a Monopoly game, Stacy
first paid $800 for a hotel. She then lost half her money
when she landed on Boardwalk. Next, she collected
$200 for passing GO. She then lost half her remaining
money when she landed on Illionois Avenue. Stacy now
has $2,500. How much did she have just before she
purchased the hotel?
• To check our solution we start with $10,400 and proceed
through each of the transactions.
$10,400 - $800 = $9600.
Half of $9600 = $4800.
collected $200 (4800+200= $5000).
Half of $5000 = $2500.
67
PROBLEM SOLVING
Answer:
She have $10, 400 just before
she paid for a hotel.
68
PROBLEM SOLVING
God bless
69
PROBLEM SOLVING
Reference:
Aufmann, R. N., Lockwood, J. S., Nation, R.
D. & Clegg, D. K. (2013).
Mathematical Excursions,
Third Edition. CA: Brooks/Cole,
Cengage Learning.
70
PROBLEM SOLVING