This document discusses inductive and deductive reasoning. It provides examples to illustrate the differences between inductive and deductive reasoning. Inductive reasoning involves making a generalization based on specific examples, while deductive reasoning uses stated assumptions or principles to logically deduce conclusions. The document cautions that conclusions based on inductive reasoning may be incorrect, as they are conjectures rather than proven statements, and a single counterexample can invalidate a conjecture.

1. Find the next term of the
following lists.
1) 5, 10, 15, 20, 25, ___
2) 3, 6, 9, 12, 15, ___
3) 1, 3, 6, 10, 15, ___
4) 7, 12, 17, 22, 27, 32, ___
5) 2, 5, 10, 17, 26, ___

2. Chapter 1
Lesson 3
PROBLEM
SOLVING
By:
Crysali Therese R. Dayaganon

3. Chapter 3. Problem Solving
3.1 Inductive and Deductive Reasoning
3.2 Problem Solving
3.3 Recreational Problems

4. 3.1 Inductive and Deductive
Reasoning
What is INDUCTIVE REASONING?
• A Type of reasoning that forms a conclusion
based on the examination of specific examples
• The conclusion formed is a conjecture, since it
may or may not be correct.

5. 3.1 Inductive and Deductive
Reasoning
Inductive Reasoning
Inductive reasoning is the process of reaching a
general conclusion by examining specific examples.

6. 3.1 Inductive and Deductive
Reasoning
Example 1. Use Inductive Reasoning to Predict a
Number
1) 1, 4, 9, 16, 25, 36, 49, ___
2) ½ , 2/3, ¾, 4/5, 5/6, ___
3) 1, 5, 12, 22, 35, ___

7. 3.1 Inductive and Deductive
Reasoning
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.

8. 3.1 Inductive and Deductive
Reasoning
Example 2. Use Inductive Reasoning to Make a Conjecture
Consider the following procedure:
Pick a number
Multiply the number by 8
Add 6 to the product
Divide the sum by 2
Subtract 3

9. 3.1 Inductive and Deductive
Reasoning
Solution:
Original
Number
2 3 4 5
Multiply by 8 16 24 32 40
Add 6 22 30 38 46
Divide by 2 11 15 19 23
Subtract 3 8 12 16 20
Conjecture: The procedure produces a number that is four times the original number.

10. 3.1 Inductive and Deductive
Reasoning
Check Your Progress.
Consider the following procedure:
Pick a number
Multiply the number by 9
Add 15 to the product
Divide the sum by 3
Subtract 5

11. 3.1 Inductive and Deductive
Reasoning
Scientists often use inductive reasoning.
Galileo Galilei used inductive reasoning to discover
that the time required for a pendulum to complete
one swing, called the period of the pendulum,
depends on the length of the pendulum.

12. 3.1 Inductive and Deductive
Reasoning
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 is the time it takes
for the pendulum to swing from left to right
and back to its original position.

13. 3.1 Inductive and Deductive
Reasoning
Example 3. Use Inductive Reasoning to Solve an Application
Use the data and inductive reasoning to answer each of the
following questions.
1) If a pendulum has a length of 49 units, what is its period?
2) If the length of a pendulum is quadrupled, what happens
to its period?
Length of
pendulum
(in units)
Period of
pendulum
(in
heartbeats)
1 1
4 2
9 3
16 4
25 5
36 6

14. 3.1 Inductive and Deductive
Reasoning
Check your Progress
A tsunami is a sea wave produces by an underwater earthquake.
The height of the tsunami as it approaches land depends on the
velocity of the tsunami. Use the table at the left and inductive
reasoning to answer the ff.:
 What happens to the height of a tsunami when its velocity is
doubled?
 What should be the height of a tsunami if its velocity is 30 ft/s?
Velocity of
the tsunami
(in ft/s)
Height of
tsunami
(in ft)
6 4
9 9
12 16
15 25
18 36
21 49
24 64

15. 3.1 Inductive and Deductive
Reasoning
Conclusions based on inductive reasoning may be
incorrect.
One can not be sure about a conjecture until a
general relationship has been proven.

16. 3.1 Inductive and Deductive
Reasoning
Pitfalls of Inductive Reasoning
We concluded that the probable next number in the list
2, 9, 16, 23, 30 is 37.
If the list 2, 9, 16, 23, 30 actually represents the dates
of Mondays in June, then the date of the Monday after
June 30 is July 7 (see the figure on the next slide). The
next number on the list would then be 7, not 37.

17. 3.1 Inductive and Deductive
Reasoning
Pitfalls of Inductive Reasoning
We concluded that the probable next number in the list
2, 9, 16, 23, 30 is 37.
If the list 2, 9, 16, 23, 30 actually represents the dates
of Mondays in June, then the date of the Monday after
June 30 is July 7 (see the figure on the next slide). The
next number on the list would then be 7, not 37.
One counterexample is sufficient to make the conjecture false.

18. 3.1 Inductive and Deductive
Reasoning
Counterexamples
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 (counterexample), then the statement is a
false statement.

19. 3.1 Inductive and Deductive
Reasoning
Example 4. Find a counterexample
Verify that each of the following statements is a false
statement by finding a counterexample.
For all numbers x:
1) 𝑥 > 0
2) 𝑥2
> 𝑥
3) 𝑥2 = 𝑥

20. 3.1 Inductive and Deductive
Reasoning
Deductive Reasoning
Deductive reasoning is the process of reaching a
conclusion by applying general assumptions,
procedures, or principles.

21. 3.1 Inductive and Deductive
Reasoning
Example 5. Use Deductive Reasoning to Establish a Conjecture
Show that the following procedure produces a number that is four times
the original number.
Procedure:
 Pick a number.
 Multiply the number by 8.
 Add 6 to the product.
 Divide the sum by 2.
 Subtract 3.
Solution:
Let n represent the original number.
 8n
8n + 6

8𝑛+6
2
= 4𝑛 + 3
4n + 3 – 3 = 4n
Note :
We started with n and
ended with 4n
This example produces
a number that is four
times the original
number.

22. 3.1 Inductive and Deductive
Reasoning
Inductive Reasoning vs. Deductive Reasoning
Example 6. Determine Types of Reasoning
Determine whether each of the following arguments is an example of inductive
reasoning or deductive reasoning.
1) During the past 10 years, a tree has produced plums every other year. Last
year the tree did not produce plums, so this year the tree will produce plums.
2) All home improvement cost more than the estimate. The contractor
estimated that my home improvement will cost Php150,000.00. Thus my
home improvement will cost more than Php 150,000.00.

23. 3.1 Inductive and Deductive
Reasoning
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.