GE 1 - Mathematics in The Modern World: Author: Rose Lyn M. Rapsing

GE 1 – Mathematics in the Modern World
Module 5
Author: Rose Lyn M. Rapsing Section 1: The Nature of Mathematics
1.3 Problem Solving and Reasoning
Inductive and Deductive Reasoning
Overview
There are different types of reasoning most of which are explained in
psychology books and texts. This module discusses two types of reasoning –
deductive and inductive reasoning.
Inductive reasoning (“bottom-up logic”) is distinct from deductive reasoning
(“top-down logic). While the conclusion of a deductive argument is certain, the truth
of the conclusion of an inductive argument is probable, based upon the evidence
given. Also in deductive reasoning, a conclusion is reached reductively by applying
general rules while in inductive reasoning, the conclusion is reached by generalizing
or extrapolating from specific cases to general rules.
The module is divided into three parts. The first part is an introduction to
inductive and deductive reasoning. Lessons include inductive VS deductive
arguments. Second part is the lessons on inductive reasoning. Whereas, lessons on
the operations on modular arithmetic is discussed in the third part.
Learning Outcomes
After working on this module, you will be able to:

1. apply inductive and deductive reasoning to solve problems;
2. perform inductive and deductive reasoning; and
3. use inductive and deductive reasoning to create conclusion.
Activities To Do
1. Imagine that you ate a dish of strawberries and soon afterward your lips swelled.
Now imagine that a few weeks later you ate strawberries and soon afterwards your
lips again became swollen. The following month, you ate yet another dish of
strawberries, and you had the same reaction as formerly. You are aware that
swollen lips can be a sign of an allergy to strawberries. Using the given data, what
can you conclude?
2. Consider the following list of natural numbers: 5,10,15,20. What is the next number
of this list?
Questions To Ponder
The given activity simply ask for your
 In making conclusions, what do you need to consider?
 If you are asked to find the 50th term of the same series of numbers, will you still use
the method or technique you used in the activity? How effective your method will
be?
Department of Mathematics, College of Science, University of Eastern Philippines 94

Module 5
Introduction to Inductive and Deductive Reasoning
An inductive argument differs from a deductive argument. Inductive argument reaches a

conclusion based on specific examples, whereas deductive argument reaches a conclusion by
applying general assumption.
We have two basic approaches for how we come to believe something is true.
The first way is that we are exposed to several different examples of a situation, and from
those examples, we conclude a general truth. For instance, you visit your local grocery store daily
to pick up necessary items. You notice that on Friday, two weeks ago, all the clerks in the store
were wearing football jerseys. Again, last Friday, the clerks wore their football jerseys. Today, also
a Friday, they’re wearing them again. From these observations, we can conclude that on all
Fridays, these supermarket employees will wear football jerseys to support their local team.
This type of pattern recognition, leading to a conclusion, is known as inductive reasoning.
Knowledge can also move the opposite direction. Say that you read in the news about a
tradition in a local grocery store, where employees wore football jerseys on Fridays to support the
home team. This time, you’re starting from the overall rule, and you would expect individual
evidence to support this rule. Each time you visited the store on a Friday, you would expect the
employees to wear jerseys.
Such a case, of starting with the overall statement and then identifying examples that
support it, is known as deductive reasoning. See image below:
https://courses.lumenlearning.com/engcomp1-wmopen/chapter/text-inductive-reasoning/

Module 5
1 Determine whether each of the following arguments is an example of inductive reasoning

or deductive reasoning.
1. All plants are photosynthetic. Algae is a plant. Therefore, algae is photosynthetic.
Example
2. The chair in the living room is red. The chair in the dining room is red. The chair in the
bedroom is red. Therefore, all the chairs in the house are red.
3. All numbers ending in 0 or 5 are divisible by 5. The number 35 ends with a 5, so it must
be divisible by 5.
4. Samantha got an A on each of her first four math tests, so she will get an A on the next
math test.
5. The first lipstick I pulled from my bag is red. The second lipstick I pulled from my bag is
red. Therefore, all lipsticks in my bag are red.
Solution
1. The conclusion is a specific case of a general assumption. Thus, this argument is an
example of deductive reasoning.
2. The argument is from specific examples to general conclusion, so the argument is an
example of inductive reasoning.
3. The argument is from general assumption to specific conclusion, so the argument is
an example of deductive reasoning.
4. The argument reaches a conclusion based on specific examples, so it is an example
of inductive reasoning.
5. The argument reaches a conclusion based on specific examples, so it is an example
of inductive reasoning.
Self-Assessment Activity 1
Determine whether each of the following arguments is an example of inductive or deductive
reasoning.
1. We had rain each day for the last five days, so it will rain today.
2. Acute angles are less than 90 degrees. This angle is 40 degrees, so it must be an acute
angle.
3. If a figure is a rectangle, then it is a parallelogram. Figure A is a rectangle. Therefore,
Figure A is a parallelogram.
4. Every time you eat peanuts, you start to cough. You are allergic to peanuts.
5. All birds have feathers. All robins are birds. Therefore, robins have feathers.
reasoning.
Because the argument is from general assumption to specific conclusion. This argument is an example of deductive 5.
The argument reaches a conclusion based on specific examples, so it is an example of inductive reasoning. 4.
reasoning.
Because the argument is from general assumption to specific conclusion, so the argument is an example of deductive 3.
2.
Because the conclusion is a specific case of a general assumption, this argument is an example of deductive reasoning.
The argument reaches a conclusion based on specific examples, so it is an example of Inductive Reasoning. 1.
Answers to SAA 1

Module 5
Inductive Reasoning
Inductive reasoning is a method of logical thinking in which we use observations combined

with experiential information we already know to be true to reach a conclusion. When we are able
to look at a specific set of data and form general conclusion based on existing knowledge from
past experiences, we are using inductive reasoning.
We have been employing inductive reasoning for a very long time. Inductive reasoning is
based on our ability to recognize meaningful patterns and connections. By taking into account
both examples and our understanding of how the world works, induction allows us to conclude
that something is likely to be true. By using induction, we move from specific data to a
generalization that tries to capture what the data “means”.
Inductive reasoning is often used to generate predictions or to make forecasts.
When we examine a list of numbers and predict the next number in the list according to
some pattern we have observed, we are using inductive reasoning.
2 Use inductive reasoning to predict the next number in each of the following lists.
1. 4, 8, 12, 16, 20, 24, ?
Example
2. 1, 2, 4, 8, 16, ?
3. 3, 5, 9, 15, 23, 33, ?
Solution
1. Each successive number is 4 larger than the preceding number. Thus, we predict
that the next number in the list is 4 larger than 24, which is 28.
2. It appears here that in order to obtain each number after the first, we must double
the previous number. Therefore, the most probable next number is 16x2= 32.
3. The first two numbers differ by 2. The second and the third numbers differ by 4. It
appears that the difference between any two numbers is always 2 more than the
preceding difference. Since 23 and 33 differ by 10, we predict that the next number
in the list will be 12 larger than 33, which is 45.
When you are making a general conclusion of something through examining specific
examples then we are using inductive reasoning. The conclusion that is formed using inductive
reasoning is called a conjecture. This conjecture is an idea that may or may not be correct.
Inductive reasoning is not used just to predict the next number in a list. It is also used to
make a conjecture about an arithmetic procedure. In the following examples we must pick at least
3 different numbers to compare the size of the original numbers and the resulting numbers, and
then we make a conjecture about it.

Module 5
3 Complete the procedure below 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.
1. Consider the following procedure: Pick a number. Multiply the number by 12. Add
Example
12 to the product. Divide the sum by 4, and subtract 3 from the quotient.
2. Consider the following procedure: Pick a number. Multiply the number by 6 and
add 8. Divide the sum by 2, subtract twice the original number, and subtract 4.
Solution
1. Suppose we pick 3 as our original number. Then the procedure would produce the
following results:
Original number: 3
Multiply by 12: 12 x 3 = 36
Add 12: 36 + 12 = 48
Divide by 4: 48 ÷ 4 = 12
Subtract 3: 12 - 3 = 9
We started with 3 and followed the procedure to produce 9.
Starting with 4 as our original number produces a final result of 12.

In each of these cases the resulting number is three times the original number.
We conjecture that following the given procedure produces a number that is three
times the original number.
(5 ∙ 6)+8
2. If the original number is 5, then 2
– (5 ∙ 2) – 4 = 5, which is equal to the
original number.
(7 ∙ 6)+8
If the original number is 7, then 2
original number.
(10 ∙6)+8
If the original number is 10, then 2
original number.
It appears, by inductive reasoning, that the procedure produces a number that
is equal to the original number.

Module 5
A. Use inductive reasoning to predict the next number in each series
1. 5, 10, 15, 20, 25, ?
2. 1, 8, 27, 64, 125, ?
3. 5, 11, 17, 23, 29, 35, ?
4. 3, 7, 11, 15, 19, 23, ?
5. 1, 5, 12, 22, 35, ?
B. Use inductive reasoning to decide whether each statement is correct.
1. Pick any counting number. Multiply the number by 8. Subtract 4 from the product. Divide
the difference by 2. Add 2 to the quotient. The resulting number is four times the original
number.
2. Pick a number. Add 4 to the number and multiply the sum by 3. Subtract 7 and then
decrease this difference by the triple of the original number. The following procedure
always produces the number 5.
A. Make a conjecture.
Pick any counting number. Multiply the number by 6. Add 8 to the product. Divide the sum
by 2. Subtract 4 from the quotient.
To show that a conjecture is always true, you must prove it.
To show that a conjecture is false, you have to find only one example in which the
conjecture is not true. This case is called a counterexample.
Formally, we have the following definition.
1 Counterexamples
Definition
A statement is a true statement provided that it is true in all cases. If you can find one case
for which a statement is not true, called a counterexample, then the statement is a false
statement.
In the following example, we will verify that each statement is false by finding a
counterexample for each.

C. The procedure produces a number which is three times the original number.
B. (1) correct, (2) correct
A. (1) 30, (2) 216, (3) 41, (4) 27, (5) 51
Answers to SAA 2
Module 5
4 Verify that each of the following statements is a false statement by finding a

counterexample.
𝑥
1. For all numbers 𝑥, = 1.
Example
𝑥
1
2. For all numbers 𝑥, 𝑥 > .
𝑥
3. For all numbers 𝑥, |𝑥| > 0.
Solution
0 0
1. Let 𝑥 = 0. Then, 0
= 0. Because 0
is not equal to 1, we have found a
𝑥
counterexample. Thus, “For all numbers 𝑥, 𝑥 = 1.” is a false statement.
1
2. For 𝑥 = 1 we have 1 > 1. Since 1 is not greater than 1, we have found a
1
counterexample. Thus “For all numbers 𝑥, 𝑥 > 𝑥.” is a false statement.
3. Consider 𝑥 = 0. Then |0| = 0. Because 0 is not greater than 0, we have found a
counterexample. Thus “for all numbers 𝑥, |𝑥| > 0.” is a false statement.
Verify that each of the following statements is a false statement by finding a counterexample.
1. For all numbers 𝑥, 𝑥2 > 𝑥.
2. For all numbers 𝑥, √𝑥2 = 𝑥.
3. For all numbers 𝑥, – 𝑥 < 𝑥.
4. For all numbers 𝑥, 𝑥3 ≥ 𝑥.
5. For all numbers 𝑥, 𝑥 + 𝑥 > 𝑥.
numbers x, x + x > x” is a false statement.

Let x = 0. Then 0 + 0 > 0. Since 0 is not greater than 0, we have found a counterexample. Thus, “For all 5.
Thus, “For all numbers x, 𝑥 3 ≥ 𝑥” is a false statement.
Consider x = −2. Then −23 ≥ −2. because −8 is not greater than −2, we have found a counterexample. 4.
all numbers x, –x < x” is a false statement.
For x=−1, we have –(−1) < −1. Since 1 is not less than −1, we have found a counterexample. Thus, “For 3.
all numbers x, √𝑥 2 = 𝑥” is a false statement.
Let x = −2. Then (−2)2 = 2. Because −2 is not equal to 2, we have found a counterexample. Thus, “For 2.
numbers x, 𝑥 2 > 𝑥” is a false statement.
Let x = 1. Then 12 > 1. because 1 is not greater than 1, we have found a counterexample. Thus, “For all 1.
Answers to SAA 3
Module 5
Deductive Reasoning
Another type of reasoning is called deductive reasoning. Deductive reasoning is

distinguished from inductive reasoning in that it is the process of reaching a conclusion by
applying general principles and procedures.
Deductive reasoning is a logical process where conclusions are made from general cases.
General cases are studied after which conclusions are made as it applies to a certain case (Rips,
1994). Argument from analogy is one of the examples under deductive reasoning.
Inductive reasoning and deductive reasoning are both used to create conjecture about an
arithmetic procedure. In inductive reasoning we create a conjecture by completing a certain
procedure for several different numbers (example 3). Whereas, in deductive reasoning we will
use 𝑛 to represent the number that we pick, where 𝑛 is a natural number.
5 Use deductive reasoning to create a conjecture on the following procedures.

1. Consider the following procedure: Pick a number. Multiply the number by 12. Add
Example
12 to the product. Divide the sum by 4, and subtract 3 from the quotient.
2. Consider the following procedure: Pick a number. Multiply the number by 6 and
add 8. Divide the sum by 2, subtract twice the original number, and subtract 4.
Solution
1. Let 𝑛 represents the original number.
Multiply the number by 12: 𝑛 ∙12 = 12𝑛
Add 12 to the product: 12𝑛 + 12
12𝑛+12
Divide the sum by 4: 4 = 3𝑛 + 3
Subtract 5: (3𝑛+3) – 3 = 3𝑛
We started with 𝑛 and obtain 3𝒏. The procedure given in this example produces a
number that is three times the original number.
2. Let 𝑛 represent the original number.

Multiply the number by 6: 6 ∙ 𝑛 = 6𝑛
Add 8 to the product: 6𝑛 + 8
6𝑛+8
Divide the sum by 2: 2 = 3𝑛 + 4
Subtract twice the original number: (3𝑛 + 4) – 2𝑛 = 𝑛 + 4
(The original number here is 𝑛, twice the original number is 2𝑛.)
Subtract 4: (𝑛 + 4) – 4 = 𝑛
We started with n and obtain 𝒏. The procedure given in this example produces a
number that is exactly the same as the original number.

Module 5
A. Use deductive reasoning to create a conjecture on the following procedures:
1. Procedure: Pick a number. Multiply the number by 6, add 10 to the product, divide the
sum by 2, and subtract 5.
2. Pick any counting number. Multiply the number by 8. Subtract 4 from the product. Divide
the difference by 2. Add 2 to the quotient.
B. Use deductive reasoning to show that the following procedure always produces the
number 5. Procedure: Pick a number. Add 4 to the number and multiply the sum by 3.
Subtract 7 and then decrease this difference by the triple of the original number.
(Use 𝑛 to represent the original number.)
The procedure always produces a number 5.

And decrease by triple the original number: (3𝑛 + 5) – 3𝑛 = 5.
Subtract 7: (3𝑛 + 12) – 7 = 3𝑛 + 5
Multiply the sum by 3: (𝑛 + 4) ∙ 3 = 3𝑛 + 12
Add 4 to the number: 𝑛 + 4
Let n represent the original number
B.
2. The procedure produces a number that is four times the original number.
number.
A. 1. The procedure produces a number that is three times the original
Answers to SAA 4

Module 5
Logic Puzzle
Logic puzzles can be solved by using deductive reasoning. A chart that enables us to display
the given information in a visual manner enables us to solve the following problems:
6
Each of four friends Donna, Sarah, Nikkie, and Xhanelle, had a different pet (fish, cat, dog,
Example
rabbit). From the following clues, determine the pet of each individual:
1. Sarah is older than her friend who owns the cat and younger than her friend
who owns the dog.
2. Nikkie and her friend who owns the rabbit are both of the same age and are
the youngest members of their group.
3. Donna is older than her friend who owns the fish.
Solution
fish cat dog rabbit
From clue 1, Sarah does not own Donna
a cat or a dog. In the chart (on Sarah 1 1
the right), write 1 (which Nikkie
stands for “ruled out by clue 1”) Xhanelle
in the cat and dog column for
Sarah.
From clue 2, Nikkie does not fish cat dog rabbit

own a rabbit and a dog being the
Donna 2
youngest. Since Sarah is not the
Sarah  1 1 2
youngest from clue 1, then
Sarah does not own a rabbit as Nikkie 2 2 2
well. Write 2 (ruled out by Xhanelle 2
clue 2) in rabbit column for Nikkie
and in rabbit column for Sarah’s row, therefore Sarah owns the fish. Put a which means
Sarah’s pet is a fish. So Donna, Nikkie and Xhanelle do not own the fish.
From Clue 3, Donna is older than

Sarah; hence, Donna owns the fish cat dog rabbit
dog. Write 3(ruled out by clue Donna 2 3  3
3) in cat and rabbit columns for Sarah  1 1 1
Donna. There are now ’s in Nikkie 2  2 2
rabbit column for Donna, Sarah Xhanelle 2 3 3 
and Nikkie; therefore Xhanelle
owns the rabbit. Write 3 in the cat column for Xhanelle; hence, Nikkie owns the cat. Put a
check in that box.
Thus, Sarah owns the fish, Donna owns the dog, Xhanelle owns the rabbit, and Nikkie owns
the cat.

Module 5
7
Colin, Maxwell, Eesh, Matthew and Nathan discovered that they have birthdays on the
Example
same day but during different months. Not one person is the same age. Their friends are
trying to guess their ages (7, 9, 10, 12, 14). From the following clues find out each person’s
age.
1. Eesh is older than Matthew.
2. Maxwell was seven 3 years ago.
3. Nathan’s age is not an odd number.
4. Matthew is ½ of Colin’s age.
5. Matthew and Eesh are 2 years apart.
Solution:
From clue 1, Eesh is older than 7 9 10 12 14
Matthew. It means that Eesh is not the Colin
youngest and Matthew is not the Maxwell
oldest. In the following chart, write Eesh 1
1(means rules out by Clue 1) in the Matthew 1
7 column for Eesh and 1 in the 14
Nathan
column for Matthew.
7 9 10 12 14
Clue 2, says Maxwell was 7, 3 years
Colin 2
ago. He must be 10 now. So, put a  in
Maxwell 2 2  2 2
the 10 column for Maxwell. Then put
2 in the 10 column for all other Eesh 1 2
children, because they cannot be 10 if Matthew 2 1
Maxwell is. It also means that we can Nathan 2
put 2 in all other ages for Maxwell.
Clue 3 says that Nathan’s age is not an

odd number. So we can put 3 in 7 and 7 9 10 12 14
9 column for Nathan. Colin 4 4 2 4 
Maxwell 2 2  2 2
Clue 4 says that Matthew is ½ Colin’s Eesh 1 2 4
age. 7 is the only number that is ½ of Matthew 4  2
4 1
another number in the age chart. So, Nathan 3 3 2 4
Matthew must be 7 and Colin must be
14. Put 4 in all other ages for Matthew and Colin, then put 4 in 7 and 14 column for all
other children.
The clue 5 says that Matthew and Eesh

are 2 years apart. Since Matthew is 7, 7 9 10 12 14
then Eesh is probably 9. Put  in the 9 Colin 4 4 2 4 
column for Eesh and 5 in the 12 Maxwell 2 2  2 2
column. And since 12 is the only age Eesh 1  2 5 4
left, means that Nathan is 12. Matthew  4 2 4 1
Nathan 3 3 2  4
Thus, Colin is 14, Maxwell is 10, Eesh is 9, Matthew is 7, and Nathan is 12.

Module 5
Use deductive reasoning to solve the following logic puzzles.
A. Match Students with Their Major: Michael, Clarissa, Reggie, and Ellen are attending
University of Eastern Philippines (UEP). One student is an information technology major,
one is a chemistry major, one is a math major, and one is a biology major. From the
following clues, determine which major each student is pursuing.
1. Michael and the information technology major are next door neighbors.
2. Clarissa and the chemistry major have attended UEP for 2 years. Reggie has
attended UEP for 3 years, and the biology major has attended UEP for 4 years.
3. Ellen has attended UEP for fewer years than Michael.
4. The math major has attended UEP for 2 years.
B. Little League Baseball: Each of the Little League teams in a small rural community is
sponsored by a different local business. The names of the teams are the Dodgers, the
Pirates, the Tigers, and the Giants. The businesses that sponsor the teams are the bank,
the supermarket, the service station, and the drugstore. From the following clues,
determine which business sponsors each team.
1. The Tigers and the team sponsored by the service station have winning records
this season.
2. The Pirates and the team sponsored by the bank are coached by parents of the
players, whereas the Giants and the team sponsored by the drugstore are coached
by the director of the community center.
3. Jake is the pitcher for the team sponsored by the supermarket and coached by his
father.
4. The game between the Tigers and the team sponsored by the drugstore was
rained out yesterday.
sponsored by the service station.

the supermarket, the Tigers was sponsored by the bank, and the Giants was
B. The Dodgers was sponsored by the drugstore, the Pirates was sponsored
information technology major, and Ellen is the chemistry major.
A. Michael is the biology major, Clarissa is the math major, Reggie is the
Answers to SAA 5

Module 5
KenKen Puzzle
KenKen and KenDoKu are trademark names for a style of arithmetic and logic puzzle
invented in 2004 by Japanese math teacher Tetsuya Miyamoto, who intended the puzzle to be an
instruction-free method of training the brain.
As in Sudoku, the goal of each puzzle is to fill the grid with digits (1 through 4 for a 4x4
grid, 1 through 5 for a 5x5 grid, 1 through 6 for a 6x6 grid, etc. ) so that no digits appears more
than one in any row or any column. Grids ranges from 3x3 to 9x9. Additionally, KenKen grids are
divided into heavily outlined groups of cells(often called “cages”) and the numbers in the c ells of
each cage must produce a certain “target” number when combined using a specified mathematical
operation (one of addition, subtraction, multiplication, or division)
KenKen’s rules:
1. Fill in each square cell in the puzzle with a number between 1 and the size of the
grid. For example, in a 4×4 grid, use the numbers 1, 2, 3, & 4.
2. Use each number exactly once in each row and each column.
3. The numbers in each “Cage” (indicated by the heavy lines) must combine — in any
order — to produce the cage’s target number using the indicated math
operation. Numbers may be repeated within a cage as long as rule 2 isn’t violated.
4. No guessing is required. Each puzzle can be solved completely using only logical
deduction. Harder puzzles require more complex deductions.
8 Solve the following KenKen puzzle.

Example
http://www.puzzazz.com/how-to/kenken

Module 5
8
Solution:
Example
Each cage in a KenKen contains a target number and

most contain an operator. If you see a single-cell cage
with just a number and no operator, it means that the
value in that cell is the target number. Such single-cell
cages work like givens in Sudoku puzzles. You won’t
see these in every puzzle, but when you do see one,
you should start there. In this puzzle, we can
immediately place a 4 in the upper right cell:
Whenever we place a number, this narrows down the

possibilities for other cells, so we want to look for that.
In this puzzle, we know that the 7+ cage in the third
column must contain 3 & 4, since that is the only
possibility that adds to 7. Given the 4 that we just
placed, combined with the rule that we must use each
number exactly once in each row and each column, we
can now tell which of the cage’s cell contains a 3 and
which contains a 4:
We can now tell that the two empty cells in the third
column (in the 4× cage) contain 1 & 2, but we don’t
know the order. However, given that information, we
can place a 2 in the lower right cell to make the cage’s
product be 4:
Remember that a cage can repeat numbers in an ir-

regularly shaped cage as long as no number is
duplicated within a single row or column. In this
puzzle, before we knew the two values in the 7+ cage,
we didn’t know if the 4× cage contained the numbers
1, 1, & 4 or 1, 2, & 2. Now that we know about the 2,
we can immediately finish the 4× cage because we
know the second 2 must be in the third row:

Module 5
In the 2÷ cage in the upper left, there are only two

possibilities: 1 & 2 or 2 & 4. The latter is excluded by
the 3 & 4 already in the row, so we can deduce that
two cells contain 1 & 2 (in an unknown order). Since
knowing this doesn’t let us place any additional
numbers immediately, we can use notes to help us
remember it for future use.
Next, we can look at the 1- cage in the first column.

Without knowing any constraints, the cells can
contain 1 & 2, or 2 & 3, or 3 & 4. But our notes show
us that the first column will already contain either a 1
or 2, which means the 1- cage cannot contain 1 & 2.
This means it must have either 2 & 3 or 3 & 4.
Whichever it is, it means the 1- cage will definitely
contain a 3. And that means the bottom left cell
cannot be a 3, which means it must be a 4:
That also lets us place the 3 in the bottom row and

then the 4 in the second row tells us how to place the
1 and the 4 above it:
Now we can place the 2 and then the 1 in the top row:

Module 5
Next, we finish up the first column, and we can tell the order because the third row
already has a 2 in it:
Finally, we wrap up the puzzle by placing the last two numbers in the fourth column:
Note that this is just one way to solve this puzzle. Because this is an easy puzzle, there is more than one
deductive path for solving. With harder puzzles, this is not always the case.
See: http://www.puzzazz.com/how-to/kenken
Solve the following KenKen Puzzles:
8x 4+ 2- 2÷ 1-
3 2÷ 1 1 2- 9+
2- 6+ 6x 4 3+
3- 15x
5+ 1-
9+ 1-

Module 5
Magic Square
A magic square of square of order n is a square grid (where n is the number of cells on each
side) filled with distinct positive integers in the range such that each cell contains a different
integer and the sum of the integers in each row, column, and diagonal is equal. The sum is called
the magic constant or magic sum of the magic square. A square grid with n cells on each side is
said to have order n.
A magic square of order 3:
9 Use deductive reasoning to determine the missing numbers in each magic square.
2 16
Example
11 10
7 12
14 1 15
Solution
We have a magic square of order 4:
The sum of the numbers in horizontal way is 34.
2 16
11 10
7 12
14 1 15
34
So, the sum of the numbers in each row, column, and diagonal must be 34.
Thus, we have
2 16 13 3
11 5 8 10
7 9 12 6
14 4 1 15
Use deductive reasoning to determine the missing numbers in each magic square.
1 16 21
15 18 11
17 13
20 8 19 12 6
5 3 22 25

Module 5
Summary
Inductive reasoning is the process of reaching a general conclusion by examining specific
examples. A conclusion based on inductive r easoning is called a conjecture. A conjecture may or
may not be correct.
Deductive reasoning is the process of reaching a conclusion by applying general
assumptions, procedures, or principles.
A statement is a true statement provided it is true in all cases. If you can find one case in
which a statement is not true, called a counterexample, then the statement is a false statement.
A statement may have many counterexamples, but we need only find one counterexample
to verify that the statement is false.
Responses To Consider
After working with this module, were you able to grasp the concept of using inductive and
deductive reasoning in making conclusions, predicting possible outcomes and finding
counterexamples?
The lesson on logic puzzles and KenKen puzzle requires critical thinking skills. Did you
encounter problems while solving these puzzles?
The lesson on finding counterexamples and making conjectures based on a procedure

require skills on the ordinary arithmetic. Did you encounter problems in performing these
operations?
As you go along this module, what lesson/s did you feel the most difficult for you? Try to list
them down and give time to consult your teacher for further discussion.
References
Aufmann, R., Lockwood, J., et.al, Mathematics in the Modern World, Rex Bookstore, Inc.,
2018.
Gonzales, J.O. , et.al. (2015). Essential Statistics. Manila: MaxCor Publishing House, Inc.
Nocon, R., Nocon, E., Essential Mathematics for the Modern World, C & E Publishing, Inc.
2018.
Other Materials
https://link.springer.com/referenceworkentry/10.1007%2F978-3-319-47829-6_1045-1
https://ivypanda.com/essays/deductive-and-inductive-reasoning-essay/#conclusion
http://www.puzzazz.com/how-to/kenken
Note To Students
Deadline of submission of Worksheet and Reflection Paper to the Municipal Link:
____________________________

Module 5
Note to the Municipal Link: WORKSHEET 5

1. The worksheet and reflection should
be forwarded to: 1.3 Inductive and Deductive Reasoning
MARY JANE B. CALPA
Chair, Department of Mathematics To the Students:
College of Science, 1. Fill out the “Student’s Information” completely.
University of Eastern Philippines 2. Write all your solutions/answers on the space
Catarman, Northern Samar provided for each item.
3. Write legibly. Use blue- or black-ink ball pen only.
2. Please check the date of submission 4. Submit on or before the indicated deadline.
indicated on the student’s information
below. 5. For queries, please contact your respective GE 1
Thank you. instructor/professor.
Student’s Information:
Student Number: Last Name, First Name M.I.: Course – Year:
Class ID Number: Professor/Instructor’s Name: Date of Submission:
I. Determine whether the argument is an example of inductive reasoning or deductive

reasoning. Write your answers before the number.
1. The left-handed people I know use left-handed scissors; therefore, all left-handed
people use left-handed scissors.
2. Elephants have cells in their bodies, and all cells have DNA. Therefore, elephants
have DNA.
3. Every year we get a thunderstorm in May. Since it is May, we will get a
thunderstorm.
4. Jennifer always leaves for school at 7:00 a.m. Jennifer is always on time. Jennifer
assumes, then, that if she leaves at 7:00 a.m. for school today, she will be on time.
5. Every chicken we've seen has been brown. All chickens in this area must be brown.
6. All noble gases are stable. Helium is a noble gas, so helium is stable.
7. Every cat that you've observed purrs. Therefore, all cats must purr.
8. All cats have a keen sense of smell. Fluffy is a cat, so Fluffy has a keen sense of smell.
9. All horses have manes. The Arabian is a horse; therefore, Arabians have manes.
10. Cats don’t eat tomatoes. Tiger is a cat. Therefore, Tiger does not eat tomatoes.
II. Use Inductive reasoning to predict the next number in each list.
1. 1, 1, 2, 3, 5, 8, 13, 21, ?
2. 5, 7, 11, 17, 25, 35, ?
3. 1, 4, 9, 16, 25, 36, ?
4. 0, 3, 8, 15, 24, 35, ?
1 2 3 4 5 6
5. , , , , ,
2 3 4 5 6 7
,?

Module 5
III. Use inductive reasoning to decide whether each statement is correct.
1. Pick a number, multiply the number by 2. Add 10 to the product. Divide the sum by
2. Subtract 5 from the quotient. And multiply the result by 7. The resulting number
is seven times the original number.
2. The sum of any two even counting numbers is always an even counting number.
3. Pick a number. Multiply the number by 6. Add 3 to the product. Divide the sum by 3.
And subtract 1 from the quotient. The resulting number is always equal to te
original number.
4. The product of two odd counting numbers is always an odd counting number.
5. Pick a number. Multiply the number by 6 and add 8. Divide the sum by 2, subtract
twice the original number, and subtract 4. The resulting number is twice te original
number.
IV. Verify that each of the following statement is a false statement by finding a
counterexample.
1. For all numbers x, 𝑥 4 > 𝑥.
2. For all numbers x, ∣x + 3∣ = ∣x∣ + 3.
3. For all numbers x, (𝑥 + 4)2 = 𝑥 2 + 16.
4. For all numbers x, ∣x + x ∣ > x
5. For all numbers x, x < 𝑥 2 .

Module 5
V. Use Deductive reasoning to solve the following logic puzzle.
1. Four kids went to an unusual pet store. Each child picked out a different animal to
take home. Can you match the child with their new friend?
Clues:
1. No child has a pet that starts with the same letter as their name.
2. Dave does not have a pet that lives in the water.
3. Molly is allergy to smoke.
4. Wendy loves to fly.
Troll Water horse Mermaid Dragon

Dave
Wendy
Molly
Tracy
Conclusion: _________________________________________________________________________________________
2. Five pirates buried their secret treasures on an island. Each treasure is buried near
a different landmark. Where did each pirate bury his or her treasure?
Clues:
1. There are no tall trees near Bart or Bonnie's treasure.
2. Jean and Bonnie did not bury their treasure near the rock.
3. Sam buried his treasure near water.
4. Bart did not bury his treasure near the hill.
5. The water near Roger’s treasure is calm and quiet.
Wild Green hill

Giant rock Tall tree Jolly pond
waterfalls
Roger
Bonnie
Bart
Sam
Jean
Conclusion: _________________________________________________________________________________________

Module 5
VI. Solve each of the following KenKen puzzle.
1. 10+ 10+
10+ 10x
10+
10+ 10+ 10x
2. 11+ 2÷ 20x 6x
3- 3÷
240x 6x
6x 7+ 30x
6x 9+
8+ 2÷
VII. Use deductive reasoning to determine the missing numbers in each magic square.
13 18 27 20
31 4 36 2
21 14 16 25
30 3 5 32
17 26 15 24
8 35 1 6 33

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GE 1 – Mathematics in the Modern World

After working on this module, you will be able to:

Department of Mathematics, College of Science, University of Eastern Philippines 94

An inductive argument differs from a deductive argument. Inductive argument reaches a

Department of Mathematics, College of Science, University of Eastern Philippines 95

1 Determine whether each of the following arguments is an example of inductive reasoning

Department of Mathematics, College of Science, University of Eastern Philippines 96

Inductive reasoning is a method of logical thinking in which we use observations combined

Inductive reasoning is often used to generate predictions or to make forecasts.

Department of Mathematics, College of Science, University of Eastern Philippines 97

We started with 3 and followed the procedure to produce 9.

Starting with 4 as our original number produces a final result of 12.

Department of Mathematics, College of Science, University of Eastern Philippines 98

To show that a conjecture is always true, you must prove it.

Formally, we have the following definition.

Department of Mathematics, College of Science, University of Eastern Philippines 99

4 Verify that each of the following statements is a false statement by finding a

numbers x, x + x > x” is a false statement.

Another type of reasoning is called deductive reasoning. Deductive reasoning is

5 Use deductive reasoning to create a conjecture on the following procedures.

2. Let 𝑛 represent the original number.

Department of Mathematics, College of Science, University of Eastern Philippines 101

The procedure always produces a number 5.

Department of Mathematics, College of Science, University of Eastern Philippines 102

From clue 2, Nikkie does not fish cat dog rabbit

From Clue 3, Donna is older than

Department of Mathematics, College of Science, University of Eastern Philippines 103

Clue 3 says that Nathan’s age is not an

The clue 5 says that Matthew and Eesh

Department of Mathematics, College of Science, University of Eastern Philippines 104

sponsored by the service station.

Department of Mathematics, College of Science, University of Eastern Philippines 105

8 Solve the following KenKen puzzle.

Department of Mathematics, College of Science, University of Eastern Philippines 106

Each cage in a KenKen contains a target number and

Whenever we place a number, this narrows down the

Remember that a cage can repeat numbers in an ir-

Department of Mathematics, College of Science, University of Eastern Philippines 107

In the 2÷ cage in the upper left, there are only two

Next, we can look at the 1- cage in the first column.

That also lets us place the 3 in the bottom row and

Department of Mathematics, College of Science, University of Eastern Philippines 108

Department of Mathematics, College of Science, University of Eastern Philippines 109

Department of Mathematics, College of Science, University of Eastern Philippines 110

The lesson on finding counterexamples and making conjectures based on a procedure

Department of Mathematics, College of Science, University of Eastern Philippines 111

Note to the Municipal Link: WORKSHEET 5

Class ID Number: Professor/Instructor’s Name: Date of Submission:

I. Determine whether the argument is an example of inductive reasoning or deductive

Department of Mathematics, College of Science, University of Eastern Philippines 112

1. For all numbers x, 𝑥 4 > 𝑥.

2. For all numbers x, ∣x + 3∣ = ∣x∣ + 3.

3. For all numbers x, (𝑥 + 4)2 = 𝑥 2 + 16.

4. For all numbers x, ∣x + x ∣ > x

5. For all numbers x, x < 𝑥 2 .

Department of Mathematics, College of Science, University of Eastern Philippines 113

Troll Water horse Mermaid Dragon

Wild Green hill