Chapter-3-PROBLEM-SOLVING.pdf hhhhhhhhhh

Generally, it is a situation you want
to change!
What is a problem?

A problem is a situation that
conforms the learner, that requires
resolution, and for which the path of the
answer is not immediately known.
There is an obstacle that prevents one
from setting a clear path to the answer.

Problem Solving has been defined
as higher-order cognitive process
that requires the modulation and
control of more routine or
fundamental skills" (Goldstein
& Levin, 1987).
What is a Problem Solving



A. Understanding Reasoning
Mathematical reasoning refers to the
ability of a person to analyze problem
situations and construct logical
arguments to justify his process or
hypothesis, to create both conceptual
foundations and connections, in order
for him to be able to process available
information.

People who can reasonand think analytically
tend
To note patterns, structure, or regularitiesin
both real-world situations and symbolic
objects;
To ask if those patterns are accidental or if
they occur for a reason
To conjecture and prove
NCTM pointed out that….



Students are expected to:
1. Define a statement
Reasoning
2. Identify the hypothesis and conclusion
in a statement
3. Write conditional statements
4. Write the Converse, Inverse,
Contrapositive of a given conditional
statement.


What kind of thinking isused
when solving problems?
Inductive or deductive?
B. Inductive and Deductive Reasoning

The type of reasoning that forms a conclusion
based on the examination of specific examples is called
inductive reasoning.
Inductive Reasoning

Conclusion
Specific
Examples
The conclusion formed by using inductive
reasoning is often called a conjecture, since it may or
may not be correct.


Example 1:
A baby cries, then cries, then cries to get a milk.
We conclude that if a baby cries, he/she gets amilk.
6, 9, 12,
Example 2:
Here is a sequence of numbers: 3,
____
What is the 5th number?
Examples
We can easily conclude that the next number is15.


Example 3:
7th
You are asked to find the 6th and
term in the sequence:
1, 3, 6, 10, 15, ______ , _____
The first two numbers differ by 2. The 2nd
and 3rd numbers differ by 3. The next difference
is 4, then 5. So, the next difference will be 6 and
Thus the 6th term is 15+ 6 = 21 while the 7th is
21 + 7 = 28.

Take note!

Inductive reasoning is not used just to
predict the next number in a list.
We use inductive reasoning to make a
conjecture about an arithmetic
procedure.
Make a conjecture about the example 2
and 3 in the previousslide…

Exercise
Use Inductive Reasoning
to Make a Conjecture
A. Consider the following procedure:
1. Pick a number.
2. Multiply the number by 8,
3. Add 6 to the product
4. Divide the sum by 2, and
5. 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.

Conjecture
Solution :
Let n represents the number
Multiply the number by 8 8n
Add 6 to the product 8n + 6
Divide the sum by 2
8𝑛+6
2
= 4n +3
Subtract 3 4n + 3 – 3 = 4n
we start with n and end with 4n . This means
that the number is 4 times the original
number.

Exercise
B. Consider the following proce
dure:
1. Pick a number.
2. Multiply the number by 9,
3. Add 15 to the product,
4. Divide the sum by 3, and
5. 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.


C. Consider the following procedure:
1. List 1 as the first odd number
2. Add the next odd number to 1.
3. Add the next odd number to the sum.
4. Repeat adding the next odd number to the previous sum.
Construct a table to summarize the result. Use inductive
reasoning to make a conjecture about the sum obtained.
Exercise


D. Observe the two sets of polygonsbelow:
What is the name of a polygon that can be used to
describe the polygons in column 2?
Use inductive reasoning to make a conjecture about the
polygons in column 2.
Exercise


Exercise
Scientists often use inductive reasoning. For instance, Galileo
Galilei (1564–1642) 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. Galileo did
not have a clock, so he measured the periods of pendulums in
“heartbeats.” The following table shows some results obtained for
pendulums of various lengths. For the sake of convenience, a length
of 10 inches has been designatedas 1 unit.
Use the data in the table and inductive
reasoning to answer each of the following
questions.
a. If a pendulum has a length of 49 units,
what is its period?
b. If the length of a pendulum is
quadrupled, what happens to its
period?


Take note:
Conclusions based on
inductive reasoning may be
incorrect.
As an illustration, consider
the circles shown. For each
circle, all possible line
segments have been drawn to
connect each dot on the circle
with all the other dots on the
circle. For each circle, count
the number of regions formed
by the line segments that
connect the dots on the circle.

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
Counterexamples


Verify that each of the following
statements is a false statement by finding
a counterexample.
For all numbers x:
a. 𝒙 > 𝟎
b. 𝒙𝟐 > 𝒙
c. 𝒙−𝟏 < 𝒙
Exercise 1

MMW by Joseph G. Taban , UNP

Verify that each of the following statements
is a false statement by finding a
counterexample.
For all numbers x:
Exercise 2


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:



Mathematics is essentially deductive
reasoning
Deductive reasoning is always valid
Deductive reasoning makes use of
undefined terms, formally defined
terms, axioms, theorems, and rules of
inference.


Example 1:
If a number is divisible by 2, then it must be even.
12 is divisible by2.
Therefore, 12 is an even number.
Example 2:
All math teachers know how to playsudoku.
Resty is a math teacher.
Therefore, Resty knows how to playsudoku.
Examples of Deductive Reasoning


Example 3:
If a student is a DOST scholar, he receives a
monthly allowance.
If a student receives a monthly allowance, his
parents will be happy.
Therefore, if a student is a DOST scholar, his
parents will be happy.
Example 4:
If ∠A and ∠B are supplementaryangles.
If m∠A = 100º, then m∠B =80º
Examples of Deductive Reasoning

The essence of deductive reasoning is
drawing a conclusion from a given
statement.
The deductive reasoning works best
when the statements used in the
argument are true and the statements
in the argument clearly follow from one
another.
Take note:



 Logic Puzzles can be solved by deductive reasoning and a chart
that enables us to display the given information in a visual
manner.
Example 1:
Each of four neighbors, Sean, Maria, Sarah, and Brian, has a
different occupation (editor, banker, chef, or dentist).
From the following clues, determinethe occupation of each
neighbor.
1. Maria gets home from work after the bankerbut 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.
Logic Puzzles

SOLUTION
CLUES:
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 Bria

n.
 From clue 1: Maria is not the
banker or the dentist.
 From clue 2, Sarah is not the
editor.
 We know from clue 1 that the
banker is not the last to get home,
and we know from clue 2 that
Sarah is the last to get home;
therefore, Sarah is not thebanker.
 From clue 3, Sarah is not the
dentist.
 As a result, Sarah is the Chef.
 Maria is the Editor.
 From clue 4, Brian is not the banker.
 Brian is the Dentist.
 Sean is the Banker.
Editor Banker Chef Dentist
Sean
X X X
Maria
X X X
Sarah X X X
Brian
X X X

 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 presidentbut 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 secretaryare next-door neighbors.
EXERCISE


Group Activity:

Distribute the Activity Sheet:
Watch the movie after 20 minutes.
Can you solve _Einstein’s Riddle -
Dan Van der Vieren.mp4


1. INTUITION
Intuition is the ability to acquire
knowledge without proof, evidence, or
conscious reasoning, or without understanding
how the knowledge was acquired.
“Intuition is a sense of knowing how to
act spontaneously, without needing to know
why” – Sylvia Clare
C. INTUITION, PROOF, AND CERTAINTY

Mina and Sarah are getting ready for
school. Mina said, “ I have a very strong
feeling that it will rain this afternoon. Let
us each bring a jacket”
Example of Intuition



By intuition, we know truth simply
by the process of introspection and
immediate awareness.

A proof is a sequence of statements that
form an argument.
There are two common methods of
proof:
1. Direct Proof
2. Indirect Proof
2.Proof


In a direct proof
You assume the hypothesis p
Give a direct series (sequence) of
implications using definitions,
axioms, theorems and rules of
inference
Show that the conclusion q holds.
Direct Proof



Show that the square of an even number is an
even number .
Rephrase: If n is even, then n2 is even.
Assume n is even
–Thus, n = 2k, for some k (definition of even
numbers) – n2 = (2k)2 = 4k2 = 2(2𝑘2)
– As n2 is 2 times an integer, n2 is thus even.
Direct proof example

The best way to improve proof skills is
PRACTICE.
Let them prove in Algebra
Ex. Prove that “If 8x – 5 = 19, then x =3.”
The sum of two odd integers is even.
For students



When we use
Indirect Proof
an indirect
proof to prove
a theory, we
follow three
steps.
An indirect proof is also called a proof by
contradiction, because we are literally looking
for a contradiction to a theory being false in order
to prove that the theory istrue.

If 𝒏𝟐 is an odd integer then n is an oddinteger.
Proof:
Assume the conclusion to be false. n is an even
integer
- n=2k for some integer k (definition of even
numbers)
- n2 = (2𝑘)2 = 4𝑘2 = 2(2 𝑘2)
- Since n2 is 2 times an integer, it is even.
Indirect proof example



There are three lines of inquiry to address
the problem of certainty in mathematics.
1. Look at the historical development of
mathematics
2. Sketch the individual cognitive
development in mathematics
3. Examine the foundations of certainty
for mathematics and investigate its
strengths and deficiencies
3. Certainty


Ancient mathematicians who were
interested in problem-solving are Euclid,
Rene Descartes, and Gottfried Wilhelm
Leibnitz.
One of the foremost recent mathematicians
to make a study of problem solving was
George Polya (1887–1985). He was born in
Hungary and moved to the United States
in 1940.
D. PROBLEM -SOLVINGSTRATEGIES


POLYA’S STEPS IN
PROBLEM SOLVING
Understandthe
Problem
Devise a Plan
Carry out the Plan
Look Back


 Do you understand all the words used in stating the
problem?
 What are you asked to find or show?
 Can you restate the problem in your own words?
 Can you think of a picture or diagram that might help
you understand the problem?
 Is there enough information to enable you to find a
solution?
Devise a
Plan
Carry out
the Plan
Look Back


Find the connection between the data and
the unknown. You may be obliged to consider
auxiliary problems if an immediate connection
cannot be found. You should obtain eventually a
plan of the solution.
Polya mentions that there are many
reasonable ways to solve problems. The skill at
choosing an appropriate strategy is best learned by
solving many problems. You will find choosing a
strategy increasingly easy.
Understand the
Problem
Carry out
the Plan
Look Back


A partial list of strategies is included:
j
Understand the
Problem
Carry out
the Plan
Look Back
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.


■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.
Understand the
Problem
Devise a
Plan
Look
Back


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 contextof the
problem.
■Ask yourself whether there are generalizations of
the solution that could apply to otherproblems.
Understand the
Problem
Devise a
Plan
Carry out
the Plan

 Discuss the 5 examples and give
comments/suggestions on how to improve the
strategies
 Apply Polya’s four steps in problem solving
Activity sheets - STRATEGY in PROBLEM SOLVING
.pdf
Group Activity



Predict the next term in a sequence
nth-term Formula for a Sequence
Word Problems which involves
numerical pattern
E. Mathematical Problems
Involving Patterns

An ordered list of numbers such as
5, 14, 27, 44, 65, ...
is called a sequence. The numbers in a sequence that are
separated by commas are the terms of the sequence. In the
above sequence, 5 is the first term, 14 is the second term, 27
is the third term, 44 is the fourth term, and 65 is the fifth
term. The three dots “...” indicate that the sequence
continues beyond 65, which is the last written term. It is
customary to use the subscript notation an to designate the
nth term of a sequence. That is,
TERMS OF A SEQUENCE


Give problems involving sequence of
numbers and worded problemsinvolving
numerical patterns
Ex. 1. Find the 10th term in the sequence
3, 7, 11, 15,…
2. Mark saves money from his allowance.
Each day he saves 12 pesos more than the
previous day. If he started saving 8 pesos in the
first day, how much will he set aside in the 5th
day?
Exercise:



Sudoku
Magic Squares
A magic square of order n is an arrangement of
numbers in a square such that the sum of the n numbers in
each row, column, and diagonal is the same number..
KenKen Puzzles
KenKen is an arithmetic-based logic puzzle that was
invented by the Japanese mathematics teacher Tetsuya
Miyamoto in 2004. The noun “ken” has “knowledge” and
“awareness” as synonyms. Hence, KenKen translates as
knowledge squared, or awareness squared.
KenKen puzzles are similar to Sudoku puzzles, but
they also require you to perform arithmetic to solve the
puzzle.
F. Recreational Problems using
Mathematics

Distribute Activity Sheets:
KENKEN PUZZLE.docx
Solve a KenKen Puzzle


 Activity Sheet - TOWER of HANOI.docx
Activity: Exploration


 Mathematical Excursions (Ch. 1) by R. Aufmann , et
al.
 Mathematical Excursions Ch. 2) by R. Aufmann et al.
References


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