INTEGRATED MATH 3 First thing is first....Let’s check our homework before we begin Lesson 2.

1. INTEGRATED MATH 3 First thing is first....Let’s check our homework before we begin Lesson 2. I hope you understood your homework and didn’t have any problems!

2. Chapter 1: Reasoning and ProofLesson 2: If-Then Statements If-Then Statement: Frequently used in deductive arguments because if the hypothesis is satisfied then the conclusion follows. It is read as “if p, then q”

3. The p represents the hypothesis and the q represents the conclusion. If-then statements can be represented symbolically as: p Hypothesis: The condition that follows “if.” Conclusion: The condition that follows “then.”

4. Two goals to accomplish in this lesson: • Understand the need for careful analysis of the meaning of mathematical statements. • Make clear nature of true if-then statements.

5. Decide what is the hypothesis and conclusion. If a quadrilateral is a trapezoid, then two opposite sides are parallel. Hypothesis: a quadrilateral is a trapezoid. Conclusion: two opposite sides are parallel.

6. LET’S TRY ANOTHER ONE….. If a plant has water, then they will grow. Hypothesis: a plant has water. Conclusion: they will grow. ANY QUESTIONS?

7. Decide what can be concluded, if anything, from each of the following sets of statements. • Known fact: If a person has a Michigan driver’s license, then the person is 16 year of age or older. Given: Andy has a Michigan driver’s license. Conclusion: ?

8. Known fact: If a person in Michigan has a driver’s license, then the person is 16 years of age or older. Given: Janet is 18 years old. Conclusion: ? • Known fact: If two sides of a triangle are the same length, then the triangle is isosceles. Given: ABC has sides of length 2 cm, 5 cm, 5 cm. Conclusion: ?

9. ANSWERS: • Andy is 16 years of age or older. • No conclusion can be made. • ABC is isosceles.

10. Suppose it is true that “all sophomores at Calvin High School enroll in physical education.” • What is the hypothesis? • What is the conclusion? • If Tad is a sophomore at Calvin, what can you conclude? • If Rosa is enrolled in a physical education class at Calvin, what can you conclude?

11. Hypothesis: A student is a sophomore at Calvin High School. • Conclusion: The student enrolls in physical education. • Tad enrolls in physical education. • Nothing can be concluded about Rosa. We don’t know if she is a sophomore.

12. We already discussed in the first lesson that a converse exchanges the hypothesis and conclusion. Let’s practice with converses. Conditional Statement: If a polygon has four right angles, then the polygon is a rectangle.

13. Converse: If a polygon is a rectangle, then the polygon has four right angles. So all we did was exchange the hypothesis and conclusion. It is like flipping the two statements. Let’s do another one.

14. Conditional statement: If it is an acute angle, then it has a measure less than 90. Converse: If it has a measure of less than 90, then it is an acute angle.

15. New Definitions to add to your Vocabulary! • Inverse: The statement formed by negating both the hypothesis and conclusion of a conditional statement. • Contrapositive: The statement formed by negating and exchanging both the hypothesis and conclusion of a conditional statement.

16. Examples with Inverse and Contrapositive Conditional Statement: If two integers are even, their product is even. Inverse: If two integers are not even, their product is not even. Contrapositive: If their product is not even, then the two integers are not even.

17. Let’s try one more! Conditional Statement: If the drama class raises $2000, then they will go on tour. Inverse: If the drama class doesn’t raise $2000, then they will not go on tour. Contrapositive: If they will not go on tour, then the drama class will not raise $2000.

18. Now, let’s try all 3 with the conditional statement: • Converse • Inverse • Contrapositive We also have a short cut to remember them by: Flip, Not, Flip & Not

19. Flip (converse): Just flip the hypothesis and conclusion. • Not (inverse): Negate or put not in the hypothesis and conclusion. • Flip & Not (contrapositive): Flip the hypothesis and conclusion in addition to adding not or negating them.

20. EXAMPLES OF ALL 3: Conditional Statement: If it is three collinear points, then they lie on the same line. Converse: If they lie on the same line, then it is three collinear points. Inverse: If it is not three collinear points, then they don’t lie on the same line. Contrapositive: If they don’t lie on the same line, then it isn’t three collinear points.

21. Now comes the confusing part! What you have to do is figure if each statement is true or false after you change the conditional statement and give a counterexample if it is false. Let’s try an example to make sure you understand.

22. If a triangle has 3 equal sides, then it is equilateral. Converse: If it is equilateral, then a triangle has 3 equal sides. True Inverse: If a triangle doesn’t have 3 equal sides, then it isn’t equilateral. True Contrapositive: If it isn’t equilateral, then a triangle doesn’t have 3 equal sides. True

23. Last Example: If you live in Dallas, then you live in Texas. Converse: If you live in Texas, then you live in Dallas. False, you could live in Houston, Texas. Inverse: If you don’t live in Dallas, then you don’t live in Texas. False, you could live in Austin, Texas. Contrapositive: If you don’t live in Texas, then you don’t live in Dallas. True

24. This completes Lesson 1 on Reasoning Strategies. • Complete the homework assignment. • Complete the quiz with open notes.

25. Remember to complete your assignments! Have a great day…..