mathemathecs

1. MATM111
Our lady of Fatima university
COLLEGE OF ARTS AND SCIENCES
Pampanga Campus
Math and Physics Department
NATURE OF MATHEMATICS:
Problem Solving and Reasoning
Jerica Nicole R. Flores, MAEd
Lecturer

2. Learning Outline
III. Nature of Mathematics: Problem Solving and Reasoning
3.1.
Inductive
Reasoning
3.2.
Deductive
Reasoning
3.3.
Polya’s
Strategy

3. Apply inductive and deductive reasoning to solve
problems.
Learning Outcomes
III. Nature of Mathematics: Problem Solving and Reasoning
Solve problems involving patterns and recreational
problems following Polya’s Strategy.
At the end of the chapter the students are expected:
Organize one’s method and approaches fro proving
and solving problems.

4. Nature of Mathematics
Problem Solving and Reasoning
• Problem is a situation that confronts
the learner , that requires
resolution, and for which the path
to the answer is not immediately
known.
• Drill/Exercise a situation that
requires resolution but the method
is clear and the way to the answer is
easily seen.

5. Nature of Mathematics
Problem Solving and Reasoning

6. Nature of Mathematics
Problem Solving and Reasoning
INDUCTIVE REASONING

7. Nature of Mathematics
Problem Solving and Reasoning
EXAMPLES:

8. PROBLEM SOLVING
AND REASONING
INDUCTIVE REASONING
EXAMPLE

9. Nature of Mathematics
Problem Solving and Reasoning
DEDUCTIVE REASONING

10. Nature of Mathematics
Problem Solving and Reasoning
DEDUCTIVE REASONING
Key terms:
ARGUMENT
Is the reason or reasons
offered for or against
something
PREMISES
Minor or major
propositions or
assertions that serve
as the bases for an
argument. It can be an
assumption, law, rule,
idea, or observation.
SYLLOGISM
An argument
composed of two
statements or
premises followed by a
conclusion.
CONCLUSION
The last step in a reasoning
process.

11. Nature of Mathematics
Problem Solving and Reasoning
EXAMPLES:

12. Nature of Mathematics
Problem Solving and Reasoning
POLYA’S STRATEGY
Named after George Polya (1887-1985). It is
a four-step problem solving strategy that are
deceptively simple.
Polya’s Four-step:
I. Understand the Problem
II. Devise a plan
III.Carry out the plan
IV.Review the solution

13. PROBLEM SOLVING
AND REASONING
POLYA’S STRATEGY
PREPARATION:
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
problems?
 What is the goal?

14. PROBLEM SOLVING
AND REASONING
POLYA’S STRATEGY
THINKING
TIME
:
Devise a Plan
 Make a list of the known information
 Make the list of the information that is need
 Draw a diagram
 Make an organize list that shows all the possibilities
 Make a table or chart
 Try to solve a similar but simpler problem.
 Write an equation. If necessary define each variable
represents
 Perform an experiment
 Guess at a solution and then check your result

15. PROBLEM SOLVING
AND REASONING
POLYA’S STRATEGY
INSGIHT:
Carry out your 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

16. PROBLEM SOLVING
AND REASONING
POLYA’S STRATEGY
VERIFICATION
:
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 your self whether there are generalizations of the
solution that could apply to other problem.

17. PROBLEM SOLVING
AND REASONING
POLYA’S STRATEGY Sample Problems
Example 1. The sum of two numbers is 30. The first
number is twice as large as the second one, what are
the numbers?
•PREPARATION(UNDERSTAND THE PROBLEM)
•THINKING TIME (DEVISE A PLAN)
•INSIGHT (CARRY OUT PLAN)
•VERIFICATION (LOOK BACK AND REVIEW)

18. PROBLEM SOLVING
AND REASONING
POLYA’S STRATEGY
II. DEVISE A PLAN:
LET X = 2ND NUMBER
2X = FIRST NUMBER
Equation: 2X + X = 30
III.CARRY OUT PLAN:
2X + X = 30
3X= 30
THEREFORE X = 10 AND 2X=20
IV. REVIEW THE
SOLUTION:
2(10) + (10) = 30
20 + 10 = 30
I. UNDERSTAND THE
PROBLEM:
WE NEED TO DETERMINE TWO
DISTINCT NUMBERS.

19. PROBLEM SOLVING
AND REASONING
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.

20. PROBLEM SOLVING
AND REASONING
Example: A
sequence of four
figures is shown
below. If the figures
were continued,
how many circles
would be there in
figure 10?
Fig 1
Fig2 Fig 3
Fig 4

21. Nature of Mathematics
Problem Solving and Reasoning

22. PROBLEM SOLVING
AND REASONING
Example 4. In tossing two coins at a time, what is the
probability of having 2 heads in a single throw?
steps
Understand the
problem
Devise a plan
Carry out the plan
Review the
solution

23. RECITATION

24. RECITATION

25. THANK YOU!
GO
FLEX!