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Logical Reasoning(An Introduction to Geometry)

MATHEMATICS Grade 8

“If a number is even,

then it is divisible by 2”

The statement above is written in

conditional form, or in if-then form.

A conditional statement has 2 parts:

◦ A hypothesis, denoted by p, and

◦ A conclusion, denoted by q.

◦ In symbols, “If p, then q.” is written

as p => q.

Remember!

A conditional statement may be true or

false.

To show that a conditional statement is

false, you need to find one example

(called a counterexample) in which the

hypothesis is fulfilled and the conclusion is

not fulfilled.

Remember!

To show that a conditional statement is

true, you must construct a logical

argument using reasons. The reasons can

be a definition, an axiom, a property, a

postulate, or a theorem.

Converse

The converse of the conditional

statement is formed by interchanging the

hypothesis and conclusion.

For instance, the converse of p => q

is q => p.

The converse may also be true or false.

Examples:

“If m∠A = 45, then ∠A is acute.”

◦ This statement is true because 45 <90.

Converse: “If ∠A is acute, then m∠A = 45.”

◦ The converse is false, because some

acute angles do not measure 45.

Examples:

“If m∠B = 90, then ∠B is right angle.”

◦ This statement is true because the

measure of the right angle is exactly 90.

Converse: “If ∠B is right angle, then m∠B =

90.”

◦ The converse is true. (Explanation same

as above)

Examples:

“If today is Sunday, then it is a weekend

day.”

◦ This statement is true because Sunday is

a weekend day.

Converse: “If today is a weekend day, then

it is Sunday.”

◦ The converse is false. Saturday (a

counterexample) is also a weekend day.

Other statements

◦ Conditional: p => q

“If p, then q”

◦ Inverse: ~p => ~q

“If not p, then not q.”

◦ Contrapositive: ~q => ~p

“If not q, then not p.”

The symbol (~) shows the negative of the hypothesis and conclusion.

Remember:

To form the inverse of the conditional

statement, take the negation of both the

hypothesis and the conclusion.

To form the contrapositive of the

conditional statement, interchange the

hypothesis and the conclusion of the

inverse statement.

Examples:

Conditional:

“If m∠A = 45, then ∠A is acute.”

Converse:

“If ∠A is acute, then m∠A = 45.”

Inverse:

“If m∠A is not 45, then ∠A is not acute.”

Contrapositive:

“If ∠A is not acute, then m∠A is not 45.”

Examples:

Conditional:

“If m∠B = 90, then ∠B is right angle.”

Converse:

“If ∠B is right angle, then m∠B = 90.”

Inverse:

“If m∠B is not 90, then ∠B is not a right angle.”

Contrapositive:

“If ∠B is not a right angle, then m∠B is not 90.”

Examples:

Conditional:

“If today is Sunday, then it is a weekend day.”

Converse:

“If today is a weekend day, then it is Sunday.”

Inverse:

“If today is not Sunday, then it is not a weekend day.”

Contrapositive:

“If today is not a weekend day, then it is not Sunday.”

For NOW:

◦ Conditional: p => q

“If p, then q”

◦ Converse: q => p

“If q, then p”

◦ Inverse: ~p => ~q

“If not p, then not q.”

◦ Contrapositive: ~q => ~p

“If not q, then not p.”

Remember:

If the statement is true, then the

contrapositive is also logically true.

If the converse is true, then the inverse is

also logically true.

In determining if the inverse, converse,

and contrapositive of the statement is

true or false, assume that the given

statement is true.

Seatwork (1 whole int. pad)

Write the converse, inverse, and

contrapositive of each conditional

statement. Determine the truth value of

each statement. If the statement is false,

give a counterexample.

Seatwork (1 whole int. pad)

1. If the degree measure of an angle is between 90 and 180, then the angle is obtuse.

2. If a quadrilateral has four congruent sides, then it is a square.

3. If a bird is an ostrich, then it cannot fly.

4. If today is Friday, then tomorrow is Saturday.

5. If there is no struggle, then there is no progress.

3.

Converse: If an angle is obtuse, then its

degree measure is between 90 and 180.

Inverse: If the degree measure of an angle

is not between 90 and 180, then the angle

is not obtuse.

Contrapositive: If an angle is not obtuse,

then its degree measure is not between

90 and 180.

All statements are TRUE.

4.

Converse: If a quadrilateral is a square,

then it has four congruent sides.

Inverse: If a quadrilateral has no

congruent sides, then it is not a square.

Contrapositive: If a quadrilateral is not a

square, then it has no congruent sides.

All statements are TRUE.

8.

Converse: If a bird cannot fly, then it is an

ostrich.

Inverse: If a bird is not an ostrich, then it

can fly.

Contrapositive: If a bird can fly, then it is

not an ostrich.

Converse and Inverse statements

are FALSE. Counterexample: Penguin

Contrapositive is TRUE.

1.

Converse: If tomorrow is Saturday, then

today is Friday.

Inverse: If today is not Friday, then

tomorrow is not Saturday.

Contrapositive: If tomorrow is not

Saturday, then today is not Friday.

Deductive Reasoning

Joash Caleb Z. Palivino

MATHEMATICS Grade 8

Deductive Reasoning

To deduce means to reason from the

known facts.

Deductive Reasoning is the process of

using facts, rules, definitions, or properties

to reach logical conclusions from given

statements.

Deductive Reasoning

In deductive reasoning, assume that the

hypothesis is true, and then write a series

of statements that lead to the conclusion.

Each statement is supported by a reason

that justifies it.

Deductive Reasoning

Law of Detachment

◦ Draws conclusion from a true

conditional statement p q and a

true statement p.

◦ If p q is a true statement and p is

true, then q is true.

Deductive Reasoning

Law of Detachment (Example)

Given:

If a car is out of gas , then it will not start.

Sarah’s car is out of gas.

Valid Conclusion:

Sarah’s car will not start.

Law of Detachment

Given:

If two numbers are odd, then their sum is

even.

The numbers 3 and 5 are odd numbers.

Conclusion: The sum of 3 and 5 is even.

Given:

If you want good health, then you should get

8 hours of sleep each day.

Aaron wants good health.

Conclusion: Aaron should get 8 hours of sleep each day.

Law of Detachment

Given:

If you are a good citizen, then you obey

traffic rules.

Aaron is a good citizen.

Conclusion: Aaron obeys traffic rules.

VALID CONCLUSION.

Law of Detachment

Given:

If a pet is a rabbit, then it eats carrots.

Jennie’s pet eats carrots.

Conclusion: Jennie’s pet is a rabbit.

INVALID CONCLUSION.

There are other animals that eat carrots

besides rabbit, like hamster.

Seatwork (Math NB – Ans. only)

Determine if the conclusion is valid or

invalid. If invalid, explain your reasoning by

giving a counterexample.

1.

Given: If students pass an entrance exam,

then they will be accepted into

college.

Latisha passed the entrance exam.

Conclusion: Latisha will be accepted to

college.

2. Given: Right angles are congruent.

∠1 and ∠2 are right angles.

Conclusion: ∠1 and ∠2 are congruent.

3. Given: An angle bisector divides an

angle into two congruent

angles.

Ray KM is an angle bisector of

∠JKL

Conclusion: ∠JKM and ∠MKL are

congruent.

4.

Given: If a game is rated E, then it has

content that may be suitable for

ages 6 and older.

Cesar buys a computer game that

he believes is suitable for his little

sister who is 7.

Conclusion: The game Cesar purchased has

a rating of E.

Rating Age

EC 3 and older

E 6 and older

E10+ 10 and older

T 13 and older

M 17 and older

5. Given: All vegetarians do not eat meat.

Theo is a vegetarian.

Conclusion: Theo does not eat meat.

6. Given: If a figure is a square, then it

has four right angles.

Figure ABCD has four right

angles.

Conclusion: Figure ABCD is a square.

7.

Given: If you leave your lights on while

your car is off, your battery will

die.

Your battery is dead.

Conclusion: You left your lights on while

the car was off.

8.

Given: If Dante obtains a part-time job,

he can afford a car payment.

Dante can afford a car payment.

Conclusion: Dante obtained a part-time job.

9.

Given: If the temperature drops below

32 degrees Fahrenheit, it may

snow.

The temperature did not drop

below 32 degrees Fahrenheit on

Monday.

Conclusion: It did not snow on Monday.

10.

Given: Some nurses wear blue uniforms.

Sabrina is a nurse.

Conclusion: Sabrina wears blue uniform.

Answers

1) The conclusion is valid.



4) The conclusion is invalid. The rating

can also be EC.


Answers

6) The conclusion is invalid. The figure

could be a rectangle.

7) The conclusion is invalid. The battery

could be dead for another reason.

8) The conclusion is invalid. Dante could

afford a car payment for another reason.


10) The conclusion is invalid. Not all

nurses wear blue uniform.

Deductive Reasoning

Law of Syllogism

◦ Draw conclusions from two true

statements when the conclusion of one

statement is the hypothesis of another.

◦ If p q is true and q r is true, then

p r is also true.

Deductive Reasoning

Law of Syllogism (Example)

Given:

If two angles of a triangle are congruent, then the sides opposite these angles are also congruent.

If two sides of triangle are congruent, then the triangle is isosceles.

Valid Conclusion:

If two angles of a triangle are congruent, then the triangle is isosceles.

Law of Syllogism

Given:

If a number is a whole number, then the

number is an integer.

If a number is an integer, then it is a

rational number.

Conclusion: If a number is a whole

number, then it is a rational number.

Determine if a valid conclusion can be

reached from the given statements

Given:

If an angle is supplementary to an obtuse

angle, then it is acute.

If an angle is acute, then its measure is

less than 90.

Conclusion: If an angle is supplementary

to an obtuse angle, then its measure is

less than 90.



Given:

If a parallelogram has a right angle, then

it is a rectangle.

If a parallelogram has a right angle, then

it is a square.

Conclusion: NO VALID CONCLUSION.



Given:

If an angle is a right angle, then the

measure of the angle is 90.

If two lines are perpendicular, then they

form a right angle.

Conclusion: If two lines are

perpendicular, then the measure of the

angle formed is 90.



Given:

If you are a good citizen, then you pay

your taxes.

If you are a good citizen, then you obey

traffic rules.

Conclusion: NO VALID CONCLUSION.


Use the Law of Syllogism to draw a valid

conclusion from each set of statements, if

possible. If no valid conclusion is possible,

write no valid conclusion.

1. If Tina has a grade of 90% or greater, she will

be on the honor roll.

If Tina is on the honor roll, then she will have

her name in the school paper.

2. If the measure of an angle is between 90 and

180, then the angle is obtuse.

If an angle is obtuse, then it is not acute.

3. If a number ends in 0, then it is divisible by 2.

If a number ends in 4, then it is divisible by 2.

4. If a triangle is a right triangle, then it has an

angle that measures 90.

If a triangle has an angle that measures 90,

then its acute angles are complementary.

5. If you interview for a job, then you wear a

suit.

If you interview for a job, then you will

update your resume.

6. If two lines in a plane are not parallel, then

they intersect.

If two lines intersect, then they intersect in a

point.

7. If it continues to rain, then the soccer field

will become wet and muddy.

If the soccer field becomes wet and muddy,

then the game will be canceled.

8. If the bank robber steals the money, then the sheriff will track him down.

If the bank robber steals the money, then the bank robber will be rich.

9. If the truck runs over some nails, then a tire will go flat.

If a tire goes flat, then the deliveries will not be made on time.

10. If Jane encounters a traffic jam today, she reports to work late.

If Jane reports to work late, her boss penalizes her.

Inductive Reasoning

Joash Caleb Z. Palivino

MATHEMATICS Grade 8

Identifying a Pattern

Monday, Wednesday, Friday, …

◦ Alternating days of the week make up

the pattern.

◦ The next day is Sunday.

3, 6, 9, 12, 15, …

◦Multiples of 3 make up the pattern.

◦ The next multiple is 18.

Inductive Reasoning

Inductive reasoning is a process of

observing data, recognizing patterns, and

making generalizations from observations.

Inductive reasoning is reasoning from

specific to general.

In using inductive reasoning to make a

generalization, the generalization is called

a conjecture.

More on Identifying a Pattern

1, 2, 4, 8, 16, …

◦ Each term is 2 times the previous term.

◦ The next two terms are 32 and 64.

1, 4, 9, 16, 25, …

◦ Each term is a square number.

◦ The next two terms are 36 and 49.

Making a Conjecture

The product of an even number and

an odd number is _____.

◦ List some examples and look for a

pattern.

(2)(3) = 6

(2)(5) = 10

(4)(3) = 12

(4)(5) = 20

The product of an even number and an

odd number is even.

Making a Conjecture

Study each number patterns:

12 + 28 = 40

-14 + 6 = -8

-10 + 30 = 20

0 + 22 = 22

18 + 16 = 34

8 + 38 = 46

Conjecture: The sum of two even numbers

is an even number.

Making a Conjecture

Study each number patterns:

4 (5) = 20

9 (8) = 72

11 (6) = 66

-12 (-3) = 36

-5 (8) = -40

-41(4) = -164

Conjecture: The product of an odd number

and an even number is an even number.

Remember!

Inductive reasoning may not always lead

to the right conclusion.

To show that a conjecture is always true,

you must prove it.

To show that a conjecture is false, you

have to find only one example in which

the conjecture is not true. This case is

called a counterexample .


Use inductive reasoning to find the next

two terms of each sequence. Justify your

answer.

Use inductive reasoning to find the next two

terms of each sequence. Justify your answer.

1. 1, 10, 100, 1000, ___, ___

2. 1, 3, 9, 27, 81, ___, ___

3. 1, 1, 2, 3, 5, 8, 13, ___, ___

4. 0, 2, 6, 12, 20, 30, 42, ___, ___

5. O, T, T, F, F, S, S, E, N, ___, ___

6. J, F, M, A, M, J, J, ___, ___

7. ½, ¼, 1/8, 1/16, ___, ___

8. ½, 9, 2/3, 10, ¾, 11, ___, ___

9. S, M, T, W, T, ___, ___

10. A, C, E, G, ___, ___

Logic Puzzle

Alice met a lion and a unicorn. Suppose that the lion lies on Monday, Tuesday, and Wednesday and the unicorn lies on Thursday, Friday, and Saturday. At all other times both animals tell the truth. Alice has forgotten the day of the week during her travels through the Forest of Forgetfulness.

Lion:Yesterday was one of my lying days.

Unicorn: Yesterday was one of my lying days, too!

Alice, who was very smart, was able to deduce the day. What day of the week was it? Explain.

Logic Puzzle

Tweedledum and Tweedledee are identical twins

who decided to entertain themselves by confusing

Alice.

One of the brothers – of course, we don’t know

which – says, “In this puzzle, each of us will pick

one of two cards, either an orange one or a blue

one. The one with the orange card will always tell

the truth. The one with the blue card will always

lie.”

Logic Puzzle

Alice picks out Tweedledee immediately!

Which one is it, and how did she figure it out?

I have the

blue card,

and I am

Tweedledee!

You are not!

I am

Tweedledee.

Logic Puzzle

Alice looked confused for a moment, then

thought as logically as she could and solved the

puzzle.

Who is Tweedledum? How can you tell?

Tweedledum is

now carrying a

blue card!

Logic Puzzle

Three sisters are identical triplets. The oldest

by minutes is Sarah, and Sarah always tells

anyone the truth. The next oldest is Sue, and

Sue always will tell anyone a lie. Sally is the

youngest of the three. She sometimes lies and

sometimes tells the truth.

Victor, an old friend of the family's, came over

one day and as usual he didn't know who was

who, so he asked each of them one question.

Logic PuzzleVictor asked the sister that was sitting on the left, "Which sister is in the middle of you three?" and the answer he received was, "Oh, that's Sarah."

Victor then asked the sister in the middle, "What is your name?" The response given was, "I'm Sally."

Victor turned to the sister on the right, then asked, "Who is that in the middle?" The sister then replied, "She is Sue."

This confused Victor; he had asked the same question three times and received three different answers.

Who was who?

Performance Task # 2: Logic Puzzle

Task:

You work for a company that publishes logic

puzzle booklets. Your task is to create an

original logic puzzle that requires the use of

inductive and/or deductive reasoning to

determine the solution.


Mechanics:

You must use at least 3 people/objects for

your puzzle.

You must provide list of statements (clues)

that will help solve the puzzle.

Test your puzzle on at least 2 people (not your

groupmates).


Mechanics:

Submit the following on March 15, 2017:

◦ One (1) blank puzzle sheet + One (1)

puzzle sheet solution key. [FINAL]

(Format: Short bond paper (8.5” x 11”),

computerized, font style and size of your

choice (but should be legible and

understandable))

◦ All copies of DRAFT and TESTED

puzzle sheets.


Mechanics:

◦ Reflection Journal (individual). (Format:

Computerized on short bond paper, Arial, 12,

1.5” spacing, 1” margin – all sides). The

reflection paper must address the following:

What geometry skills are used for the project?

Can I use these skills outside of class? How?

How did we get started? What were my first

thoughts?

How does our team work? How do each member

contribute to the group’s success?


Mechanics:

Late submission of Performance Task will have a demerit of 2 points each day.

Rubric:

GROUP (80%)

◦ Content (25%)

◦ Clues (25%)

◦ Solution (20%)

◦ Mechanics (10%)

INDIVIDUAL – Reflection Journal (20%)

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