1. M1 Lesson 4.4
January 21, 2009
Deductive Reasoning

2. Purpose
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.

3. What do we need then?
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…

4. Example: Solve for x. 3(2x+1) + 2(2x+1) + 7 = 42 – 5x
This is the work you are used to showing when solving algebraic equations.

5. Example: Solve for x, give reasons for each step.
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
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.*

6. More fun with deductive reasoning…
Law of Syllogism
If hypothesis p, then conclusion q,
If hypothesis q, then conclusion r.
If hypothesis p, then conclusion r.
If these statements are true,
then the following statement is true.

7. Example using Law of Syllogism:
If Manuel has a healthy diet, then he will have a healthy body.
If Manuel has a healthy body, then he will feel good.
So… If Manuel has a healthy diet, then he will feel good.

8. More examples… your turn!
Directions: Use the Law of Syllogism to write the conditional statement from the pair of true statements.
If the season is Fall, then the leaves will change colors.
If the leaves change colors, then they will fall from the trees.

9. then the leaves will fall from the trees!
Conclusion:
If the season is Fall,
then the leaves will fall from the trees!

10. If 2x > 10, then 2x > 7. If x > 5, then 2x > 10.
Example 2: Try this one…
Directions: Use the Law of Syllogism to write the conditional statement from the pair of true statements.
If 2x > 10, then 2x > 7.
If x > 5, then 2x > 10.
Notice the conclusion of the second statement is the hypothesis of the first statement…

11. Conclusion:
If x > 5, then 2x > 7.

12. Homework
Come up with two of your own examples using the Law of Syllogism
Credits:
Slides 2-5 taken from Discovering Geometry: An Investigative Approach by Michael Serra. Published by Key Curriculum Press. 2003