1. CONFIDENTIAL 1 Grade 9 Algebra1 Using Deductive Reasoning to Verify Conjectures

2. CONFIDENTIAL 2 Warm Up Identify the hypothesis and conclusion for each conditional statement: 1) If the fire alarm rings, then every body should exit the building. 2) If two different lines intersect, then they intersect at exactly one point. 3) The statement AB = CD implies that AB = CD.

3. CONFIDENTIAL 3 1) Hypothesis: The fire alarm rings. Conclusion: Every body should exit the building. 2) Hypothesis: Two different lines intersect. Conclusion: Two different lines intersect at exactly one point. 3) Hypothesis: AB = CD Conclusion: AB = CD

4. CONFIDENTIAL 4 You have learnt that one counterexample is enough to disprove a conjecture. But to prove that a conjecture is true, you must use deductive reasoning. Deductive reasoning is the process of using logic to draw conclusions from given facts, definitions, and properties. Using Deductive Reasoning to Verify Conjectures

5. CONFIDENTIAL 5 Media Application Urban legends and modern myths spread quickly through the media. Many web sites and television shows are dedicated to confirming or disproving such myths. Is each conclusion a result of inductive or deductive reasoning? A) There is a myth that toilets and sinks drain in opposite directions in Southern and Northern Hemisphere. However, if you were to observe sinks, draining in two hemispheres, you would see that this myth is false. Since the conclusion is based on a pattern of observation, it is a result of inductive reasoning.

6. CONFIDENTIAL 6 B) There is a myth that you should not touch a baby bird that has fallen from its nest because the mother bird will disown the baby if she detects human scent. However, Biologists have shown that birds cannot detect human scent. Therefore, the myth cannot be true. Since the conclusion is based logical reasoning from scientific research. It is a result of deductive reasoning.

7. CONFIDENTIAL 7 Now you try! 1) There is a myth that an eelskin wallet will demagnetize credit cards because the skin of eels used to make the wallet holds electric charge. However, eelskin products are not made from electric eels. Therefore, the myth cannot be true. Is this conclusion a result of inductive or deductive reasoning? 1) Since the conclusion is based logical reasoning from scientific research. It is a result of deductive reasoning.

8. CONFIDENTIAL 8 In the deductive reasoning, if the given facts are true and you apply the correct logic, then the conclusion must be true. The law of detachment is one valid form of deductive reasoning. Law of detachment If p  q is a true statement and p is true, then q is true.

9. CONFIDENTIAL 9 Verifying conjectures by the law of detachment Verify if each conjecture is true by the law of detachment: A) Given: If two segments are congruent, then they have same length. AB = XY Conjecture: AB = XY Identify the hypothesis and conclusion in the given conditional. If two segments are congruent, then they have same length. The given statement AB = XY matches the hypothesis of a true conditional. By the law of detachment, AB = XY. The conjecture is valid.

10. CONFIDENTIAL 10 B) Given: If you are tardy three times, you must go to detention. Shea is in detention. Conjecture: Shea was tardy at least three times. Identify the hypothesis and conclusion in the given conditional. If two segments are congruent, then they have same length. The given statement “Shea is in detention” matches the conclusion of a true conditional. But this does not mean the hypothesis is true. Shea could be in detention for another reason. The conjecture is not valid.

11. CONFIDENTIAL 11 Now you try! 2) Determine if the conjecture is valid by the law of detachment. Given: If a student passes his class, the student is eligible to play sports. Ramon passed his class. Conjecture: Ramon is eligible to play sports. 2) By the law of detachment, the conjecture is valid.

12. CONFIDENTIAL 12 Another valid form of deductive reasoning is law of syllogism. It allows you to draw conclusions from two conditional statements when the conclusion of one is the hypothesis of the other. Law of syllogism If p  q and q  r are true statements, then p  r is a true statement.

13. CONFIDENTIAL 13 Verifying conjectures by the law of syllogism Verify if each conjecture is true by the law of syllogism: A) Given: If m/A < 90 0, then /A is acute. If /A is acute, then it is not a right angle. Conjecture: If m/A < 90 0, then it is not a right angle. Let p, q and r represent the following. p: The measure of an angle is less than 90 0. q: The angle is acute. r: The angle is not a right angle. You are given that p  q and q  r. Since q is the conclusion of the first conditional and the hypothesis of the second conditional, you can conclude that p  r. The conjecture is valid by the law of syllogism.

14. CONFIDENTIAL 14 B) Given: If a number is divisible by 4, then it is divisible by 2. If a number is even, then it is divisible by 2. Conjecture: If a number is divisible by 4, then it is even. Let x, y and z represent the following. x: A number is divisible by 4. y: A number is divisible by 2. z: A number is even. You are given that x  y and z  y. The law of syllogism cannot be used to draw a conclusion since y is the conclusion of both the conditionals. Even though the conjecture x  y is true, the logic used to draw the conclusion is not true.

15. CONFIDENTIAL 15 Now you try! 3) Determine if the conjecture is valid by the law of syllogism. Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal. Conjecture: If an animal is a dog, then it has hair. 3) The conjecture is valid by the law of syllogism.

16. CONFIDENTIAL 16 Applying the law of Deductive reasoning Draw a conclusion from the given information: A) Given: If a team wins 10 games, then they play in finals. If a team plays in the finals, then they travel to Boston. The Ravens won 10 games. Conclusion: The Ravens will travel to Boston. B) Given: If two angles form a linear pair, then they are adjacent. If two angles are adjacent, then they share a side. /1 and /2 form a linear pair. Conclusion: /1 and /2 share a side.

17. CONFIDENTIAL 17 4) Draw a conclusion from the given information. Given: If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is not a quadrilateral. Polygon P is a triangle. 4) Polygon P is not a quadrilateral. Now you try!

18. CONFIDENTIAL 18 BREAK

19. CONFIDENTIAL 19 GAME Click on the link below for some exciting puzzle http://www.thekidzpage.com/onlinejigsawpuzzles/kids- jigsaw-puzzles/12-piece-jigsaw/12-27-06-piggy.html

20. CONFIDENTIAL 20 Assessment Does each conclusion use inductive or deductive reasoning? 1)inductive reasoning 2)deductive reasoning 1)At Bell High School, students must take Biology before they take Chemistry. Sam is in Chemistry, So Marcia concludes that he has taken Biology. 2)A detective learns that his main suspect was out of town on the day of crime. He concludes that the suspect is innocent.

21. CONFIDENTIAL 21 3) Given: If you want to go on a field trip, you must have a signed permission slip. Zola has signed the permission slip. Conjecture: Zola wants to go on a field trip. 4) Given: If the side of lengths of a rectangle are 3 ft and 4 ft, then its area is 12 ft 2. A rectangle has side length of 3 ft and 4 ft. Conjecture: The area of the rectangle is 12 ft 2. 3) By the law of detachment, the conjecture is valid. 4) By the law of detachment, the conjecture is not valid. Verify if each conjecture is true by the law of detachment:

22. CONFIDENTIAL 22 Verify if each conjecture is true by the law of syllogism: 5) By the law of syllogism, the conjecture is valid. 6) By the law of syllogism, the conjecture is not valid. 5) Given: If you fly from Texas to California, you travel from the Central to the Pacific time zone. If you travel from the Central to the Pacific time zone, then you gain two hours. Conjecture: If you fly from Texas to California, you gain two hours. 6) Given: If the figure is a square, then the figure is a rectangle. If the figure is a square, then it is a parallelogram. Conjecture: If the figure is a parallelogram, then it is a rectangle.

23. CONFIDENTIAL 23 7) Draw a conclusion from the given information. Given: If you leave your car lights on overnight, then your car battery will drain. If your battery is drained, your car might not start. Alex left his car lights on last night. 7) Battery of Alex’s car will drain.

24. CONFIDENTIAL 24 Media Application Urban legends and modern myths spread quickly through the media. Many web sites and television shows are dedicated to confirming or disproving such myths. Is each conclusion a result of inductive or deductive reasoning? A) There is a myth that toilets and sinks drain in opposite directions in Southern and Northern Hemisphere. However, if you were to observe sinks, draining in two hemispheres, you would see that this myth is false. Since the conclusion is based on a pattern of observation, it is a result of inductive reasoning. Let’s review

25. CONFIDENTIAL 25 B) There is a myth that you should not touch a baby bird that has fallen from its nest because the mother bird will disown the baby if she detects human scent. However, Biologists have shown that birds cannot detect human scent. Therefore, the myth cannot be true. Since the conclusion is based logical reasoning from scientific research. It is a result of deductive reasoning.

26. CONFIDENTIAL 26 In the deductive reasoning, if the given facts are true and you apply the correct logic, then the conclusion must be true. The law of detachment is one valid form of deductive reasoning. Law of detachment If p  q is a true statement and p is true, then q is true.

27. CONFIDENTIAL 27 Verifying conjectures by the law of detachment Verify if each conjecture is true by the law of detachment: A) Given: If two segments are congruent, then they have same length. AB = XY Conjecture: AB = XY Identify the hypothesis and conclusion in the given conditional. If two segments are congruent, then they have same length. The given statement AB = XY matches the hypothesis of a true conditional. By the law of detachment, AB = XY. The conjecture is valid.

28. CONFIDENTIAL 28 Another valid form of deductive reasoning is law of syllogism. It allows you to draw conclusions from two conditional statements when the conclusion of one is the hypothesis of the other. Law of syllogism If p  q and q  r are true statements, then p  r is a true statement.

29. CONFIDENTIAL 29 Verifying conjectures by the law of syllogism Verify if each conjecture is true by the law of syllogism: A) Given: If m/A < 90 0, then /A is acute. If /A is acute, then it is not a right angle. Conjecture: If m/A < 90 0, then it is not a right angle. Let p, q and r represent the following. p: The measure of an angle is less than 90 0. q: The angle is acute. r: The angle is not a right angle. You are given that p  q and q  r. Since q is the conclusion of the first conditional and the hypothesis of the second conditional, you can conclude that p  r. The conjecture is valid by the law of syllogism.

30. CONFIDENTIAL 30 Applying the law of Deductive reasoning Draw a conclusion from the given information: A) Given: If a team wins 10 games, then they play in finals. If a team plays in the finals, then they travel to Boston. The Ravens won 10 games. Conclusion: The Ravens will travel to Boston. B) Given: If two angles form a linear pair, then they are adjacent. If two angles are adjacent, then they share a side. /1 and /2 form a linear pair. Conclusion: /1 and /2 share a side.

31. CONFIDENTIAL 31 You did a great job today!