Inductive Reasoning and Conjectures

Lesson 2-1:
Patterns and
Inductive Reasoning

Inductive reasoning is reasoning based on patterns you
observe.

Look for a pattern. What are the next two terms in each sequence?

A. 3, 9, 27, 81, …

B.

A conjecture is a conclusion you reach using
inductive reasoning.

Look at the circles. What conjecture can you make about the number of
regions 20 diameters form?

It is important to gather enough data before you make a
conjecture.

What conjecture can you make about the sum of the first 30 even
numbers?

Sales of backpacks at a nationwide company decreased over a period
of six consecutive months. What conjecture can you make about the
number of backpacks the company will sell in May?

Not all conjectures turn out to be true. You should test
your conjecture multiple times. You can prove that a
conjecture is false by finding one counterexample . A
counterexample is an example that shows that a
conjecture is incorrect.

What is a counterexample for each conjecture?

A.

If the name of the month starts with the letter J, it is a summer
month.

B. You can connect any three points to form a triangle.

Multiple Select

Identify the counterexample(s) for the statement below.

"When you multiply a number by 2, the product is greater than the original number."

• 0(2) = 0

• 5(2) = 10

• 1/2 (2) = 1

• (-3)(2) = -6

Lesson 2-2: Conditional Statements

The study of if-then statements and their truth
values is a foundation of reasoning.

Drag and Drop

Example: If an angle measures 130, then the angle is obtuse.

Hypothesis:​

Placeholder

Conclusion: ​

Placeholder

Drag these tiles and drop them in the correct blank above

an angle measures 130

the angle is obtuse

If an angle measures 130

then the angle is obtuse

Draw

What are the hypothesis and the conclusion of the conditional? Underline the hypothesis once and the conclusion twice.

Multiple Choice

What is the hypothesis in this conditional statement?

"If 7y - 5 = 9, then y = 2."

• y = 2

• 7y - 5 = 9

Drag and Drop

What is the conclusion in this conditional statement?



"If this animal is a spider, then it has eight legs."



Conclusion: ​

Placeholder

Drag these tiles and drop them in the correct blank above

it has eight legs

this animal is a spider

If this animal  has eight legs

If this animal is a spider

How can you write the following statement as a conditional?

“Vertical angles share a vertex.”

Step 1: Underline the hypothesis once and the conclusion twice.


Vertical angles share a vertex.

Step 2: Write the conditional.

If ____________________________, then ____________________.

The truth value of a conditional is either true or false.
To show that a conditional is true, show that every time
the hypothesis is true, the conclusion is also true.

To show that the conditional is false, if you find one
counterexample for which the hypothesis is true and
the conclusion is false, then the truth value of the
conditional is false.

Is the conditional true or false? If it is false, find a
counterexample.

If a number is divisible by 3, then it is odd.

Multiple Choice

Which of the following is a counterexample for this conditional statement?

"If a girl has blonde hair, then she also has blue eyes."

• Beth, who has blonde hair and brown eyes

• Victoria, who has brown hair and brown eyes

• Haley, who has brown hair and blue eyes

• Isabelle, who has blonde hair and blue eyes

The negation of a statement p is the opposite of the
statement. The symbol is ~p and is read “not p.”

The negation of the statement “The sky is blue” is
“The sky is not blue.”

Every conditional has three related
conditional statements.

They are:
● Converse
● Inverse
● Contrapositive

Multiple Choice

What is the converse of the following conditional?

"If Josh gets a raise, then he takes a vacation."

• If Josh does not get a raise, then he does not take a vacation.

• If Josh takes a vacation, then he gets a raise.

• If Josh does not take a vacation, then he does not get a raise.

Multiple Choice

What is the contrapositive of the following conditional?

"If Arianna is shopping for celery, then she is at a farmers' market."

• If Arianna is not shopping for celery, then Arianna is not at a farmers' market.

• If Arianna is not at a farmers' market, then she is not shopping for celery.

• If Arianna is at a farmers' market, then she is shopping for celery.

Multiple Choice

Look at the following conditionals:

"If Susan is not tired, then she is going to go running."

"If Susan is going to go running, then she is not tired."

Is the second conditional the contrapositive, converse, or inverse of the first conditional?

• contrapositive

• converse

• inverse

Multiple Choice

What is the inverse of the following conditional?

"If Colton is not going to eat mocha ice cream, then it is not his birthday."

• If Colton is going to eat mocha ice cream, then it is his birthday.

• If it is not Colton's birthday, then he is not going to eat mocha ice cream.

• If it is Colton's birthday, then he is going to eat mocha ice cream.

What are the converse, inverse, and contrapositive of the
conditional statement below?

If a vegetable is a carrot, then it contains beta carotene.

Converse: __________________________________________________

Inverse: ____________________________________________________

Contrapositive: ______________________________________________

The End

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