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STUDY GUIDE

GRADE 8 | UNIT 9

Logical Reasoning
Table of Contents
Introduction
...
4
Objectives
...
6

-

Learn about It!
...
8

-

Check Your Understanding!
...
13

-

Learn about It!
...
16

-

Check Your Understanding!
...
21

-

Learn about It!
...
24

-

Check Your Understanding!
...
30

-

Learn about It!
...
32

-

Check Your Understanding!
...
38
Performance Task
...
40
Key to Let’s Practice!
...
45

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STUDY GUIDE

GRADE 8 | MATHEMATICS

Click Home icon to go back to
Table of Contents

UNIT 9

Logical Reasoning
We make decisions in our everyday life
...

Also, when we find ourselves in a situation where we have to convince someone of an
idea, that person may not easily accept it as truth unless we are able to provide evidence
or proof for our claims
...

Logical reasoning may also be needed by lawyers when they
defend their clients
...
It may also be used by marketers when they try
to persuade consumers to buy a product
...
This may be used in predicting behaviors of
individuals or phenomena
...

This unit will help us reason logically and support it with proofs which can be applied both
to mathematical and real-life situations
...
This will
help you assess your prior knowledge and practice some skills that you will need in
studying the lessons in this unit
...

1
...

a
...

b
...

c
...

d
...

e
...

f
...

g
...

h
...

i
...

j
...


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2
...

a
...

b
...

c
...

d
...

e
...

3
...

a
...

b
...

c
...

d
...

e
...


Objectives
At the end of this unit, you should be able to
•

use inductive reasoning or deductive reasoning in an argument; and

•

write a proof (both direct and indirect)
...
This activity may be done in pairs
...
Shown on the right are some figures

Figures:
a
...
You are
to guess what figure comes next
...
Draw the figures on a sheet of paper
...


4
...

5
...


compare your answers and discuss
the patterns you have observed
...
Did you get the same answers?

d
...
For the items on which you did not get
the same answers, are the patterns
you have observed in the figures

e
...
Doing this, you have somehow
applied what is called inductive reasoning
...


James leaves for school at 7:00 in the morning
...
James assumes then
that he will always be on time if he leaves at 7:00 a
...


Definition 1
...


Definition 1
...


There are two kinds of reasoning: inductive and deductive
...
3: Inductive Reasoning is a kind of reasoning
based on repeated observations and is
characterized by drawing a conclusion called
conjecture from observations of specific
examples
...

Definition 1
...


In this unit, it is important to know the difference among the following terms:
•

Statement – an expression or argument having a complete thought

•

Hypothesis – a phrase or statement where a conclusion is drawn

•

Conclusion – the relation from the given statements

•

Conjecture – an unproven statement which is based on observations

Let us consider the given problem earlier
...
It is most probable to be true because his conjecture is supported by his
observation that whenever he leaves at 7:00 a
...
, he always arrives on time for his
classes
...
Inductive reasoning does not
guarantee a true result or conclusion, but it provides a means of making a conjecture
...

The chair in the dining room is red
...

All chairs in the house are red
...


Try It Yourself!
Determine whether the following argument is a result of induction:
Every Friday, Sandie hangs out in the library
...

Therefore, Sandie will hang out in the library today
...

Solution:

2×4=8
8 × 4 = 32
32 × 4 = 128
By observing carefully, you will notice that each number in the pattern is
obtained by multiplying the preceding number by 4
...


Try It Yourself!
Find the next number in the sequence 3, 15, 75, 375, … using inductive reasoning
...

Solution:

Observe the pattern carefully
...

2+2=4
2+4=6
4 + 6 = 10
10 + 6 = 16
16 + 10 = 26
Thus, the probable next number in the list is 16 + 26 = 42
...


Real-World Problems
Example 4: Determine if the argument is a result of induction or
not
...
Alba’s children love to pizza
...
Alba’s son
...

Solution:

It is not a result of induction because it does not make observations and then
create a conclusion from the observations
...

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Example 5: Rick read a book for several days
...
On the second day, he read 15 pages
...
How many pages did
Rick read on the fifth day?
Solution:

We can use inductive reasoning to find the number of pages Rick read on the
fifth day by searching for a pattern in a sequence of numbers
...

We list the number of pages Rick read for the first few days
...


Notice that a constant is added to get the next number
...

Step 3:

Find the fifth number
...
Thus, the
fifth number is 26 + 7 = 33
...

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Try It Yourself!
A school’s mountaineering club started with 8 members on the first
week
...
How many members did it have on the
fifth week?

Check Your Understanding!
1
...

a
...
The eldest of that family has his
name that starts with X, so does the second
...

b
...
Today is Thursday, so we shall wear green
...
Glenda passed her first math quiz
...
She can be
confident she is going to pass the third
...
People from that party are not in favor of the school administration’s policies
...
So, she is not in favor of the policies, too
...
Make a conjecture for each of the following items
...

a
...
The sum of an even and an odd number (whether it is odd or even)
c
...
Answer the following questions
...
Emily writes a few numbers on her notepad
...
What do you think will be the next number that Emily writes?
b
...
How many cups of taho did
Mang Victor sell on Friday?
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Lesson 2: Deductive Reasoning
Warm Up!
B for Bridge
Materials Needed:

pen, paper

Instructions:
1
...

2
...

3
...
”
Member A may choose whatever clause he or she sees fit
...
Member B shall then complete member A’s statement by saying “…then
__________________________
...

5
...
”
6
...
” Member C may use any clause he or she sees fitting
to complete the statement in step 5
...
Finally, the three of you shall construct the statement “Therefore, if in step 3>, then ”
...
Whenever an if-then statement is formed, write them down on a sheet of paper
...
Do this for three rounds while switching roles
...

10
...


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Learn about It!
In the previous lesson, you have learned that there are two types of reasoning which
enable us to make a conclusion/conjecture in a certain situation
...

Another type of reasoning is the deduction or the deductive reasoning
...
1: Deductive reasoning is a type of
reasoning that uses general facts,
definitions, and accepted properties in a
logical order to write a logical argument
...
Sally has more than 145 credits
...


Definition 2
...

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Consider the following example:
Statements:

Angel knows that if she misses the class one day before the test, she
cannot make a good score in the test
...


Conclusion:

Angel will not get a good score in the test
...


Definition 2
...


Consider this another example:
Statements:

If you study hard, then you will have good grades
...


Conclusion:

If you study hard, then you will find a nice job
...

Statements:

If a student has at least 130 credits, then he or she must have
completed a bachelor’s degree at a state university
...


Conclusion:

Sally completed her bachelor’s degree at the state university
...
The
activity was titled that way for the reason that member B acts like a bridge that connects
the hypothesis made by member A with the conclusion made by member C
...
Can you observe how the transitivity
was applied to make the concluding statement for each round?

Let’s Practice!
Example 1: An obtuse angle measures more than 180°
...
Write
a conclusion using deductive reasoning
...

If an angle is obtuse, then it measures 180°
...

Since the second statement matches the hypothesis of the first statement,
the conclusion of the first statement is also true for the second statement
...


Try It Yourself!
The sum of the measures of the angles of a triangle is 180°
...
Write a conclusion using deductive reasoning
...

If a three-dimensional figure has six square faces, then it is a prism
...

The Law of Syllogism states that if this is so, the hypothesis of the first
statement implies that the conclusion of the second statement is true
...


Try It Yourself!
Draw a conclusion based on the following statements:
If I pass the entrance exam, I will enroll in that school
...


Example 3: Make a conclusion for the following:
If the sum of the squares of the two legs of a right triangle is equal to the
square of the longest side, then the triangle is a right triangle
...

Solution:
Step 1:

Determine first if the second statement matches the hypothesis of the first
statement
...

Using the Law of Detachment, since the hypothesis matches that of the
second statement, then the conclusion of the first statement also applies
...


Try It Yourself!
Make a conclusion for the following:
If a triangle is a 30°-60°-90° triangle, then its hypotenuse is twice as long as its
shortest side
...


Real-World Problems
Example 4: Write a conclusion for the following statements using
deductive reasoning
...

The sun came out
...

If the sun comes out, then I will hang the wet clothes outside
...

Since the second statement matches the hypothesis of the first statement,
the conclusion of the first statement is also true for the second statement
...


Example 5: Make a conclusion for the following:
If I cut my hair short, I will need less shampoo
...

Solution:
Step 1:

Determine what law may be used to draw the conclusion
...
Thus, we can use the Law of Syllogism
...

Using the Law of Syllogism, since the hypothesis of the second statement
matches conclusion of the first statement, then we can say that the
hypothesis of the first statement implies that the conclusion of the second
statement is true
...


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STUDY GUIDE

Try It Yourself!
Make a conclusion for the following:
If you can find a good graphic artist, we can have a poster
for the contest
...


Check Your Understanding!
1
...

a
...

If two lines intersect, then they intersect at a point
...

b
...

Lala is unmarried
...

c
...

Will has found real fulfillment
...

d
...

I will have to miss classes if I will have to go see the doctor
...

2
...
(6 points)
a
...

Kayla limits her hours on social media
...
The plant will not get the nutrients it needs if you do not water it
...

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STUDY GUIDE

c
...

If the classes are suspended, our teacher will speed up her discussion the next
time we meet
...
Solve the following problems
...
The arithmetic mean of two numbers 𝑥 and 𝑦 is

𝑥+𝑦


...
If a student is in grade 8, he or she is required to watch the school play
...

What is required of a grade 8 student? What law did you use to answer the
question?

Lesson 3: Direct Proof
Warm Up!
A Series of Indeterminate Events
Materials Needed:

paper, glue, strips of paper with sentences (to be provided by
the teacher)

Instructions:
1
...

2
...

3
...

4
...

5
...

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STUDY GUIDE

6
...

7
...

8
...

9
...


Learn about It!
There are so many statements we hear everyday
...
Some people prefer to verify ideas first or require evidences or proof before they
take it as truth
...
He wants to prove that
this is true for all even numbers
...
1: A proof is a logical argument which
establishes the truth of a statement
...
The most commonly used types of proofs are paragraph and two-column proof
...
2: A two-column proof, also called a formal
proof, consists of numbered statements
and reasons that show the logical order of
argument
...
3: A paragraph proof, also called an informal
proof, uses sentences to explain why a
conjecture in a given situation is true
...
4: A direct proof is a form of proof which
consists of a sequence of statements that
are either given or deduced from previous
statements, and whose last statement is
the conclusion to be proven
...

First, write its equivalent conditional statement to easily determine the conclusion that
needs to be proven
...

Then organize the statements and facts using a two-column proof
...
Make deductions using the given statement with
supporting facts or properties, and then end with the conclusion proven
...

Notice that in the two-column proof, you have a series of statements
...

This is similar to what you did in Warm Up! Recall that in the said activity, you ordered
sentences to form a logical narrative—starting with the event that happened first and
ending with event that happened last
...
This is also true for two-column
proofs
...


Let’s Practice!
Example 1: Given: 5𝑥 − 4 = 2𝑥 + 8
Prove: 𝑥 = 4

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STUDY GUIDE

Solution:
Statement
1 5𝑥 − 4 = 2𝑥 + 8
2 5𝑥 − 2𝑥 − 4 + 4 = 2𝑥 − 2𝑥 + 8 + 4

Reason
Given
Addition Property of Equality
(Add −2𝑥 and 4 to both sides of the
equation)

3 3𝑥 = 12

From statement 2

4 (3𝑥 = 4) (1)
3
𝑥
=
4
5

Multiplication Property of Equality
From statement 4

Try It Yourself!
Given 9𝑥 − 5 = 6𝑥 + 4, prove 𝑥 = 3
...

Solution:

Let 𝑎 and 𝑏 be two odd numbers
...


Given

2 𝑎 = 2𝑛 + 1 and 𝑏 = 2𝑚 + 1; where

Definition of odd numbers

𝑚 and 𝑛 are any integers
3 𝑎 + 𝑏 = 2𝑛 + 1 + 2𝑚 + 1
4 𝑎 + 𝑏 = 2𝑛 + 2𝑚 + 2

Substitution

5 𝑎 + 𝑏 = 2(𝑛 + 𝑚 + 1)

Factoring

From statement 3

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From statement 5, the sum of two odd numbers 𝑎 and 𝑏 is equal to a number
that has a factor of 2, therefore, the sum of two odd numbers is an even
number
...


Example 3: If 𝑀 is the midpoint of 𝐴𝐵
...


Solution:
Statement

Reason

1 𝑀 is the midpoint of 𝐴𝐵
...


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STUDY GUIDE

Real-World Problems
Example 4: Gauss Street cuts across Newton Street and Leibniz
Street, forming the angles shown in the picture
...

Solution:
Step 1:

To easily construct a two-column proof, we redraw the diagram using lines
and label some points
...
Also, it is given from the original figure that ∠𝐴𝐶𝐷 =
129° and ∠𝐻𝐹𝐺 = 51°
...


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STUDY GUIDE

Step 2:

Construct a two-column proof
...


Definition of Linear Pair

4 ∠𝐸𝐹𝐻 = 129°
5

supplementary
...


angles
...


Since we have shown that lines 𝐵𝐷 and 𝐸𝐺 are parallel, and since these lines
correspond to Newton Street and Leibniz Street, respectively, we can say that
Newton Street is parallel to Leibniz Street
...
If the sum of
their ages is 53, prove that Jojo’s age is 14
...

1
...

2
...

3
...
Euclid Street cuts across Pythagoras Street and Archimedes Street, forming the
angles shown in the picture
...


5
...
Now he has 11
...

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STUDY GUIDE

Lesson 4: Indirect Proof
Warm Up!
Prove Me Wrong
Materials Needed:

optional: paper, pen

Instructions:
1
...

2
...

3
...

4
...
What idea did you learn the hard way or what idea did you doubt?
b
...
What were your realizations eventually?
d
...
Your teacher shall choose a group that will share to the class what they have
shared to their group
...
You doubted it because
you were afraid you were going to fail your subject if you did not stay up late to study
and do your homework
...
You realized you were better off rested
...
However, eventually, you were
proven wrong and realized the idea is actually true or it actually makes sense
...
Read the following situation and the proof
that follows and find how it parallels what you shared in Warm Up!
...
However, she cannot directly prove this statement;
instead, she has proven that the sum of two even numbers is never
an odd number
...
4: An indirect proof is a form of proof where one
assumes that the opposite of a statement to be
proven is true, until the assumption leads to an
impossibility

or

contradiction,

which

then

proves that the given statement is correct
...
In this proof, assume that the
opposite of the statement is true, and then find a statement which contradicts the
assumed statement or a known fact
...

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STUDY GUIDE

Statement
1
2

𝑎 and 𝑏 are even numbers
𝑎 = 2𝑚 and 𝑏 = 2𝑛; where 𝑚 and 𝑛
are any integers

Reason
Given
Definition of even numbers

3

𝑎 + 𝑏 is an odd number

Given

4

𝑎 + 𝑏 = 2𝑘 + 1; where 𝑘 is any integer

Definition of odd numbers

5

𝑎 + 𝑏 = 2𝑚 + 2𝑛

Substitution (statement 2 to statement 4)

6

𝑎 + 𝑏 = 2(𝑚 + 𝑛); where 𝑚 + 𝑛 is any
integer

7

Factoring

Therefore, the sum of two even

Due to the contradiction between

numbers is also an even number
...


Statement 6 contradicts the definition of odd numbers expressed on statement 4; an odd
number is an integer in the form of 2𝑘 + 1, where 𝑘 is any integer
...

In Warm Up!, doubting the idea may be likened to negating the conclusion of a given
statement
...

This may be likened to arriving at a contradiction
...
This may be likened to stating that the conclusion in the given
statement is true
...

Solution:

Another way to prove the given statement is to use contraposition
...

Assume the negation of the conclusion to be true; that is, 𝑥 ≥ 5
...


1
2

Statement
𝑥≥5

Reason
Assumption (−𝑞)
Multiplication Property of Inequality

(𝑥 ≥ 5)(5)
5𝑥 ≥ 25

(−𝑝)

Statement 2 is true by the assumption in statement 1 using contraposition
...


Try It Yourself!
Prove that if 𝑥 + 7 > 9, then 𝑥 > 2 by indirect proof
...

1

< 0, then 𝑏 is not negative
...


Reason

Definition of nonnegative integers
Multiplication Property of Inequality
1
(Since is negative, the inequality sign is
𝑏
reversed
...

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STUDY GUIDE

Try It Yourself!
Prove that if

1

> 0, then 𝑎 is positive
...

Solution:

Assumption: If 𝑛 is odd, then 𝑛2 is an even number
...


Due to contradiction made by
statement 7, we conclude that the
original statement is true
...


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STUDY GUIDE

Try It Yourself!
Use an indirect proof to prove that if 𝑛 is even, then 𝑛2 is also an even number
...
Prove that
the triangular lot has no more than one right
angle using an indirect proof
...


Step 2:

Make an assumption
...
Since what we want to prove is
that the triangular lot represented by ∆𝐴𝐵𝐶 in the diagram has no more than
one right angle, the assumption we make is that it has more than one right
angle, say, angles 𝐴 and 𝐵
...

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STUDY GUIDE

Step 3:

Construct the two-column indirect proof
...


4 𝑚∠𝐶 = 0°

Subtraction Property of Inequality

5 ∆𝐴𝐵𝐶 has no more than

Statement 4 contradicts a known fact (an

one right angle
...


Since statement 4 contradicts the fact that angles in a triangle cannot have a
measure of 0°, then we can conclude that the original statement to be
proven is indeed true
...
June claims that
the base angles of the roof are congruent
...


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STUDY GUIDE

Check Your Understanding!
Construct a two-column indirect proof for each of the following
...
Given that 3𝑥 > 12, prove that 𝑥 > 4
...
Prove that the sum of two even numbers is an even number
...
Prove that if 𝑛2 divisible by 4, then 𝑛 is even
...
In a restaurant offering an eat-all-you-can promo, Jeff was able to eat 𝑥 cups of rice
...
If his
wife ate 3 cups of rice, prove that Jeff ate 4 cups of rice
...
In a school campus, the elementary building and the high school building are of the
same distance from the gym
...
Prove using an indirect proof
that the triangle formed by gym, canteen, and elementary building is congruent to
the triangle formed by the gym, canteen, and high school building
...
Write a rule to find the nth number in the following sequence:
2, –4, 8, –16, 32, …
2
...

3
...

A says B lies; B says C lies; C says A and B lie
...
To gather
suggestions and ideas, they decided to hold a debate on certain issues
...
The class
shall be divided equally among three issues and for each issue, there shall be two groups
with opposing positions, i
...
affirmative (pro) and negative (anti)
...
Should students be permitted to use cell phones in class?
2
...
Are school detentions effective in the long run or are its effectivity in correcting
student behavior short-lived?

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STUDY GUIDE

Coin toss or drawing of lots may be used to determine which groups are assigned to
which topics and to which stances (affirmative or negative)
...
Back your claims by using direct and indirect proofs
...

Performance Task Rubric
Below

Needs

Successful

Exemplary

Expectation

Improvement

Performance

Performance

(0–49%)

(50–74%)

(75–99%)

(99+%)

Only 1 of the

Only 2 of the

Only 3 of the

Four of the

concepts

concepts learned

concepts

concepts

learned in the

in the unit was

learned in the

learned in the

unit was used

used in the

unit was used in

unit was used in

in the debate
...


the debate
...


Arguments

Arguments

Arguments

Arguments

raised in the

raised in the

raised in the

raised in the

debate are

debate are

debate are

Persuasiveness

debate are do

somewhat

satisfactorily

outstandingly

of Arguments

not support

persuasive in

persuasive in

persuasive in

the assigned

supporting

supporting

supporting

position
...


position
...


Criteria

Use of the
Concepts
Learned in the
Unit

Written report
Usefulness of
written report

lacks a lot of
salient points
raised in the
debate
...


Written report
includes all
salient points
raised in the
debate
...

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STUDY GUIDE

Wrap-up

Logical
Reasoning

Methods of
Reasoning

Inductive
Reasoning

Kinds of Proofs

Deductive
Reasoning

Direct Proof

Indirect Proof

Key Terms/Formulas
Key Term

Description
It is characterized by drawing a general conclusion

Inductive Reasoning

from observations (information and patterns) of
specific examples
...
The two laws involved are:
•

Law of Detachment

•

Law of Syllogism

It is a form of proof consist of sequenced statements
Direct Proof

supported by facts, definitions, properties, theorems,
and postulates
...


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STUDY GUIDE

Key to Let’s Practice!
Lesson 1
1
...
1875
3
...
128 members
Lesson 2
1
...

2
...

3
...

4
...

Lesson 3
1
...

Statement

Reason

1 𝑎 and 𝑏 are even numbers
...


2(𝑛 + 𝑚) is even since 2 is one of

Factoring
its factors and the other is an
integer
...

Statement

Reason

1 𝐶 is the midpoint of 𝐴𝑌 and of 𝐵𝑋
2 𝐴𝐶 = 𝐶𝑌

Given

3 𝐵𝐶 = 𝐶𝑋
4 ∠𝐵𝐶𝐴 ≅ ∠𝑋𝐶𝑌

Definition of Midpoint

5 ∆𝐴𝐶𝐵 ≅ ∆𝑌𝐶𝑋
...


4
...
Since Jojo’s father is 11 years more than twice the age of Jojo,
Jojo’s father’s age is 2𝑥 + 11
Statement
1 𝑥 + 2𝑥 + 11 = 53

Reason
Given (The sum of Jojo and his
father’s age is 53
...
)

4 3𝑥 = 42

From Statement 3

5 (3𝑥 = 42) (1)
3

Multiplication Property of Equality
1

(multiply 3 to both sides of the
equation
...

Statement
𝑥≤2

1
2

Reason
Assumption (−𝑞)
Multiplication Property of Inequality

(𝑥 + 7 ≤ 2 + 7)
𝑥+7≤9

(−𝑝)

Statement 2 is true by the assumption in statement 1 using contraposition
...

2
...


Definition of nonpositive integers
Multiplication Property of Inequality
1
(Since is positive, the inequality sign is NOT
𝑎
reversed
...


3
...


statement 7, we conclude that the
original statement is true
...
Let the roof in Charles’ drawing be represented by isosceles triangle 𝐴𝐵𝐶 with ∠𝐴
and ∠𝐶 as its base angles
...


Given
...


Given (from the diagram)
...


5 ∠𝐴 ≅ ∠𝐶

Angles opposite congruent sides of a
triangle are congruent
...
)

Since the last statement is a contradiction to Statement 3, we can say that the
conclusion in the original statement is true, i
...


44

STUDY GUIDE

References
Aufmann, Richard N
...
Mathematical Excursions
...

CK-12 Foundation
...
” Accessed June 19, 2018
...
ck12
...
“Side Angle Side Postulate
...

http://www
...
com/geometry/congruent_triangles/side-angle-sidepostulate

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Title: Mathematics Study Guide
Description: Math----Statistics (probability and logical reasoning)

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