Over the years you have made conjectures, using inductive reasoning, based on patterns you have observed.

When you make a conjecture, the process of discovery may not always help explain why the conjecture works.

You need another kind of reasoning to help answer this question.

Deductive Reasoning is the process of showing that certain statements follow logically from agreed-upon assumptions and proven facts. When you use deductive reasoning, you try to

reason in an orderly way to convince others that your conclusion is valid.

Examples: Lawyers use deductive arguments to prove their case to a jury. Every time you show your work in a math problem you are using deductive reasoning…. Let’s see an example…

3(2x+1) + 2(2x+1) + 7 = 42 – 5x 5(2x+1) + 7 = 42 – 5x 5(2x+1) = 35 – 5x 10x + 5 = 35 – 5x 10x = 30 – 5x 15x = 30 x = 2

This is the work you

are used to showing

when solving

algebraic equations.

3(2x+1) + 2(2x+1) + 7 = 42 – 5x

Original Equation

5(2x+1) + 7 = 42 – 5x

Combining Like Terms

5(2x+1) = 35 – 5x

Subtraction Property of Equality

10x + 5 = 35 – 5x

Distributive Property

10x = 30 – 5x

Subtraction Property of Equality

15x = 30

Addition Property of Equality

x = 2

Division Property of Equality

Here we have given specific reasons for each step and have formed a logical

argument that x=2.

*This is deductive reasoning.*

If Manuel has a healthy diet, then he will have a healthy body.

If Manuel has a healthy body, then he will feel good.

Syllogism: If Manuel has a healthy diet, then he will feel good.

If the season is Fall, then the leaves will change colors.

If the leaves change colors, then they will fall from the trees.

Syllogism: If the season is Fall, then the leaves will fall from the trees!

If you are at least 16 years old, you can get a drivers license.

Ricky is 16 years old.

Detachment: Ricky can get a drivers license.

If 2x > 10, then 2x > 7. If x > 5, then 2x > 10. Syllogism: If x > 5, then 2x > 7.

If you study hard, you will pass all of your classes. If you pass all of your classes, you will graduate.

Syllogism: If you study hard, then you will graduate.

If x2 > 9, then x2 > 8. If x > 4, then x2 > 9.

Syllogism: If x > 4 , then x2 > 8.

If you like pizza, you love Chicago style deep dish. Emily loves Chicago style deep dish.

Detachment: Emily loves pizza.

If Ron eats lunch today, then he will eat a sandwich. If Ron eats a sandwich, then he will drink a glass of milk.

Syllogism: If Ron eats lunch today, then he will drink a glass of milk.

If you want an “A” in Geometry, you will have to study hard. Sal is studying hard.

Detachment: Sal will get an “A” in Geometry.

If you do your homework, then you can watch TV. If you watch TV, then you can watch your favorite show

Syllogism: If you do your homework, then you can watch your favorite TV show.

If Kendra pops a bag of popcorn, then she will eat the popcorn. If Kendra eats the popcorn, then she will need something to drink.

Syllogism: If Kendra pops a bag of popcorn, then she will need something to drink.

If you give Kfed money, then he makes an awesome album.

Hypothesis- Conclusion-

The ______ of a conditional statement is formed by exchanging the hypothesis and the conclusion.

Conditional- If it is sunny outside, then it is hot.

Converse- If it is hot, then it is sunny outside.

What are the key words included in a biconditional statement?

What is the inverse of the following statement.

Example:

Cond. Stmt: If is sunny outside, then it is hot.

Inverse: If it is NOT sunny outside, then it is NOT hot.

Example:

Cond. Stmt: If it is sunny outside, then it is hot.

Converse: If it is hot, then it is sunny outside.

Contrapositive:If it is NOT hot, then it is NOT sunny.

What is the contrapositive for the following statement?

If it is sunny outside, then it is hot.