reasoning and proof

reasoning and proof

Quarter 2, Week 7

At the end of the session, I would be able to:

1.  M8GE-IIh-1: Use inductive or deductive reasoning in an argument.

2.  M8GE-IIi–j -1: Write a proof (both direct and indirect).

Quarter 2, Week 7

At the end of the session, I would be able to:

1.  M8GE-IIh-1: Use inductive or deductive reasoning in an argument.

a. differentiate the inductive and deductive reasoning; and

b. apply inductive and deductive reasoning in an argument.

Concepts:

Inductive Reasoning

-    A reasoning that uses specific examples or set of observations to arrive at a general rule, generalizations, or conclusions.

-    “from specific to general”

 

Deductive Reasoning

-    A type of reasoning that uses basic and/or general statements to arrive at a conclusion. 

-    It starts with a given condition called hypothesis followed by a series of statements with corresponding reasons that leads to the desired conclusion.

-    “from general to specific”

 

REVIEW

Directions: Rewrite each statement into if – then form, then identify the hypothesis by underlining it once and the conclusion twice.

1.   Two points determine a line.

2.   Ilonggos are from Iloilo.

3.   A quadrilateral has four sides.

Open Ended

Directions: Rewrite each statement into if – then form, then identify the hypothesis by underlining it once and the conclusion twice.

1. Two points determine a line.

Open Ended

Directions: Rewrite each statement into if – then form, then identify the hypothesis by underlining it once and the conclusion twice.

2. Ilonggos are from Iloilo.

Open Ended

Directions: Rewrite each statement into if – then form, then identify the hypothesis by underlining it once and the conclusion twice.

3. A quadrilateral has four sides..

In the previous lesson , you have learned about identifying the hypothesis and conclusion of conditional statements, the inverse, converse and contrapositive of an if-then statements.

Learning these concepts are important for you to understand how to deduce at conclusion based on logical reason.

Inductive Reasoning

 

EXAMPLE:Look at the figures carefully. What is next?

• Looking at the shaded region in the figures, the next figure is

Inductive Reasoning

 

EXAMPLE: I counted ten people with blond hair; therefore, all of the people have blond hair.

Inductive Reasoning

 

EXAMPLE: 5,10, 15, 20, 25, _____. The next number is 30.

Inductive Reasoning

 

EXAMPLE: Every time I eat shrimps; I suffer from an allergic attack. Therefore, it can be

concluded that I am allergic to shrimps.

Inductive Reasoning

EXAMPLE: My mathematics teacher is smart.

   My previous mathematics teacher was smart.

   My father’s mathematics teachers were smart too.Therefore, I conclude that mathematics teachers are smart.

Deductive Reasoning

 EXAMPLE

1. Students of Iloilo National High School are responsible.

Ann is a student of Iloilo National High School.

Therefore, Ann is responsible.

Deductive Reasoning

 EXAMPLE

2.  Even numbers are divisible by 2.

32 is an even number.

Therefore, 32 is divisible by 2.

Deductive Reasoning

 EXAMPLE

3.  Right angles are congruent.

Angle A and angle B are right angles.

Therefore, angle A  and angle B  are congruent.

Deductive Reasoning

 EXAMPLE

4. COVID-19 patients are quarantined.

Martha is a COVID-19 patient.

Therefore, Martha is quarantined.

Deductive Reasoning

 EXAMPLE

5. Filipinos are hospitable.

Alex is a Filipino.

Therefore Alex is hospitable.

Fill in the Blank

Activity 1: Complete Me!

Directions: Draw a conclusion from each given situation and identify the kind of reasoning used.

1. Complementary angles are two angles whose sum is 90°.∠𝐴 and ∠𝐵 are complementary. Therefore, ______________.

Fill in the Blank

Activity 1: Complete Me!

Directions: Draw a conclusion from each given situation and identify the kind of reasoning used.

2. In the sequence 3, 6, 9, 12, … . The next number is __________.

Fill in the Blank

Activity 1: Complete Me!

Directions: Draw a conclusion from each given situation and identify the kind of reasoning used.

3. All rectangles have congruent diagonals. Square is a rectangle. Therefore, _______________.

Fill in the Blank

Activity 1: Complete Me!

Directions: Draw a conclusion from each given situation and identify the kind of reasoning used.

4. 2, 4, 6, 8,... The sum of the first 5 even numbers is ___________.

Fill in the Blank

Activity 1: Complete Me!

Directions: Draw a conclusion from each given situation and identify the kind of reasoning used.

5. A pentagon is a polygon with five sides. Polygon ABCDE is a regular pentagon. Therefore ________________.

Multiple Select

Activity 2: Let’s Do this!

Directions: Identify the type of reasoning used in each of the following situations.

1. Today, Sam notices that Iloilo city alarm sounds at 9:00 P.M. The next day of the same time, she notices that the city alarm sound again. She remembers that the city alarm did sound at the same time the other day. She then concludes that the city alarm sounds every 9:00 P.M.

• inductive

• deductive

Multiple Select

Activity 2: Let’s Do this!

Directions: Identify the type of reasoning used in each of the following situations.

2. No foreigner can be elected senator in our country. Peter is a foreigner. Therefore, Peter cannot be elected senator in our country.

• inductive

• deductive

Multiple Select

Activity 2: Let’s Do this!

Directions: Identify the type of reasoning used in each of the following situations.

3. Grade 8 students at Iloilo NHS conducted an experiment on tomatoes by applying Ferrous Sulfate. After three weeks, they observed the number of tomatoes produced increased. They concluded that the used of Ferrous Sulfate as organic fertilizer can increase the production of tomatoes.

• inductive

• deductive

Multiple Select

Activity 2: Let’s Do this!

Directions: Identify the type of reasoning used in each of the following situations.

4. A student who gets a perfect score in mathematics 8 will be given extra credits. Ann got a perfect score in mathematics 8. Thus, Ann will be given extra credit.

• inductive

• deductive

Multiple Select

Activity 2: Let’s Do this!

Directions: Identify the type of reasoning used in each of the following situations.

5. All the residents of Barangay Magsaysay, Lapaz are exempted from taxes. The parents of Melanie reside in Barangay Magsaysay. Therefore, Melanie’s parents are exempted from taxes.

• inductive

• deductive

Quarter 2, Week 7

At the end of the session, I would be able to:

2.  M8GE-IIi–j -1: Write a proof (both direct and indirect).

a. define direct proof and indirect proof;

b. differentiate a direct proof from an indirect proof; and

c. write direct proof and indirect proof.

Proof

-     A logical argument in which each statement is supported or justified by given information, definition, axioms, postulates, theorems, and previously proven statements.

Ways of Writing a Proof

1.   Paragraph Form

A way of writing a proof where you write a paragraph to explain why a conjecture for a given situation is true.

 

2.   Flow Chart Form

A way of writing proof where a series of statements are organized in logical order using boxes and arrows. Each statement together with its justification is written in a box. Arrows are used to show how each statement leads to another.

 

3.   Two-Column Form

Proof in two-column form has statements and reasons. The first column is for the statements and the other column is for the reasons.

Ways of Writing a Proof

EXAMPLE: 
         Given:
    

 $4\left(x-2\right)=20$ 

Prove: $x=7$   

Ways of Writing a Proof

EXAMPLE: 
         Given:
    

 $4\left(x-2\right)=20$ 

Prove: $x=7$   

Ways of Writing a Proof

EXAMPLE: 
         Given:
    

 $4\left(x-2\right)=20$ 

Prove: $x=7$   

Direct Proof

-     One of the most familiar forms of proof. The method of proof is to take an original statement p (hypothesis), which we assume to be true, and use it to show directly that another statement q(conclusion) is true.

 

Writing a Direct Proof

1.   Assume that the hypothesis is true.

2.   Use what we know about the hypothesis and other facts as necessary to deduce that the conclusion is true.

 

Indirect proof

Indirect proof is a type of proof in which a statement to be proved is assumed false (by negation) and if the assumption leads to an impossibility, then the statement assumed false has been proved to be true.

Writing an Indirect Proof

1.   Assume the opposite of the conclusion of the statement.

2.   Proceed as if this assumption is true to find the contradiction.

3.   Once there is a contradiction, the original statement is true.

Fill in the Blank

Activity 1: I Need Proof!

Directions: Fill in the blanks by choosing your answers from the box.

Prove that if 2(𝑏+1) =−6, then 𝑏=−4.

Given:2(𝑏+1) =−6 Prove: 𝑏=−4

Fill in the Blank

Activity 2: Prove Me Indirectly!

Directions: Arrange the statements to its logical order using an indirect proof. Use letters A to F.

Prove that if 𝑦=−3, then 4(𝑦−3) ≠−20.

1. 4𝑦−12=−20 by distributive property of equality.

2. Hence our assumption is false and 4(𝑦−3) ≠−20is true

3. Using addition property of equality, 4𝑦−12+12=−20+12.So 4𝑦=−8 by simplifying.

4. Assume that 4(𝑦−3) =−20. Take this statement as true and solve for 𝑦.

5. Thus, division property of equality, 4𝑦/4=−8/4. So 𝑦=−2 by simplifying.

6. But 𝑦=−2, contradicts the given statement that 𝑦=−3.