1. Warm-up - Sept 22 (Tuesday) 8. Which conditional and its converse form a true biconditional? a. Write the two conditional statements that make up this biconditional. b. Illustrate the first conditional from part (a) with a Venn diagram. c. Illustrate the second conditional from part (a) with a Venn diagram. d. Combine your two Venn diagrams from parts (b) and (c) to form a Venn diagram representing the biconditional statement. e. What must be true of the Venn diagram for any true biconditional statement? f. How does your conclusion in part (e) help to explain why you can write a good definition as a biconditional? 24. Consider the following statement. “An integer is divisible by 10 if and only if its last digit is 0.” page 57-HH page 59-HH

2. 8. Which conditional and its converse form a true biconditional? a. Write the two conditional statements that make up this biconditional. b.Illustrate the first conditional from part (a) with a Venn diagram. c. Illustrate the second conditional from part (a) with a Venn diagram. d.Combine your two Venn diagrams from parts (b) and (c) to form a Venn diagram representing the biconditional statement. e. What must be true of the Venn diagram for any true biconditional statement? f. How does your conclusion in part (e) help to explain why you can write a good definition as a biconditional? 24. Consider the following statement. “An integer is divisible by 10 if and only if its last digit is 0.”  Integers with last digit of 0 Warm-up - Sept 22 (Tuesday) Integers divisible by 10 1)If the integer is divisible by 10, then its last digit is 0. 2) If the last digit of an integer is 0, then it is divisible by 10. Integers divisible by 10 Integers With last digit of 0 Integers with last digit of 0 Integers divisible by 10 b b c c d d The two circles coincide. A good definition can be written as a biconditional because either of the coinciding circles of its Venn diagram can be the hypothesis and the other the conclusion. page 57-HH page 59-HH

3. 2-4 DEDUCTIVE REASONING Math Journal and Student Companion Pages 49-54 

4. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Understanding: Given true statements, deductive reasoning can be used to make a valid or true conclusion. Deductive reasoning often involves the Laws of syllogism and Detachment. BIG IDEA: Reasoning & Proof page 49- SC  If I spend as little as possible, I use the coupon. Explain how you know your solution is reasonable.  If I use the coupon. then I will get the lowest priced pair of jeans.  If I use the coupon. then I will get the lowest priced pair of jeans.

5. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Understanding: Given true statements, deductive reasoning can be used to make a valid or true conclusion. Deductive reasoning often involves the Laws of syllogism and Detachment. BIG IDEA: Reasoning & Proof page 60-HH A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make?

6. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Understanding: Given true statements, deductive reasoning can be used to make a valid or true conclusion. Deductive reasoning often involves the Laws of syllogism and Detachment. BIG IDEA: Reasoning & Proof A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make? The garden will be watered. page 60-HH

7. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Understanding: Given true statements, deductive reasoning can be used to make a valid or true conclusion. Deductive reasoning often involves the Laws of syllogism and Detachment. BIG IDEA: Reasoning & Proof page 50-SC What can you conclude from the given true statements? a. If there is lightning, then it is not safe to be out in the open. Marla sees lightning from the soccer field. b. If a figure is a square, then its sides have equal length. Figure ABCD has sides of equal length. p  q p  q p  q p  q Marla is not safe out in the open. No valid conclusion is possible.

8. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Given true statements, deductive reasoning can be used to make a Understanding: valid or true conclusion. BIG IDEA: Reasoning & Proof page 60- HH If it is July, then you are on summer vacation. If you are on summer vacation, then you work at a smoothie shop. If it is July, then you are on summer vacation. If you are on summer vacation, then you work at a smoothie shop. You conclude: ? ? ? Example If it is July, then you work at a smoothie shop.

9. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Given true statements, deductive reasoning can be used to make a Understanding: valid or true conclusion. BIG IDEA: Reasoning & Proof page 50- SC p  q q  r  p  r If a whole number ends in 0, then it is divisible by 5. No valid conclusion is possible.

10. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Given true statements, deductive reasoning can be used to make a Understanding: valid or true conclusion. BIG IDEA: Reasoning & Proof page 51- SC p  q q  r  p  r p  r p  r p  r p  r The Nile is the longest river in the world ; by Law of Syllogism and Law of Detachment.

11. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Given true statements, deductive reasoning can be used to make a Understanding: valid or true conclusion. BIG IDEA: Reasoning & Proof page 49- SC p  q p  q p  q p  q 1st conclusion  The Nile is the longer than the Amazon. q  r q  r q  r q  r The Nile is the longest river in the world. same conclusion  Yes; if Law of Detachment is used first, then it must be used again to reach the same conclusion. The Law of syllogism is not used. Yes; if Law of Detachment is used first, then it must be used again to reach the same conclusion. The Law of syllogism is not used.

12. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Given true statements, deductive reasoning can be used to make a Understanding: valid or true conclusion. BIG IDEA: Reasoning & Proof page 52- SC p  q q  r  p  r If it is Saturday, then you wear sneakers.

13. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Given true statements, deductive reasoning can be used to make a Understanding: valid or true conclusion. BIG IDEA: Reasoning & Proof page 52- SC Zachary catches the 8:05 bus. p  q p  q p  q p  q

14. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Given true statements, deductive reasoning can be used to make a Understanding: valid or true conclusion. BIG IDEA: Reasoning & Proof page 52- SC 3. Write the contrapositive of each conditional statement. If possible, make a conclusion from the two contrapositives and state the law of deductive reasoning you used. If an animal is a poodle, then the animal is a dog. If an animal is a dog, then the animal is a mammal. If an animal is not a dog, then the animal is not a poodle; If an animal is not a mammal, then the animal is not a dog. If an animal is not a mammal, then the animal is not a poodle;

15. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Given true statements, deductive reasoning can be used to make a Understanding: valid or true conclusion. BIG IDEA: Reasoning & Proof page 53 - SC The detective uses logical reasoning to go from given statements and clues to a final conclusion. This process is called deductive reasoning. The detective uses logical reasoning to go from given statements and clues to a final conclusion. This process is called deductive reasoning.

16. 2-4 Deductive Reasoning Objective: Use the Law of Detachment and the Law of Syllogism. Essential Given true statements, deductive reasoning can be used to make a Understanding: valid or true conclusion. BIG IDEA: Reasoning & Proof page 53 - SC Deductive reasoning uses logic (laws, postulates, theorems, or definitions) to reach conclusions and so conclusions are always true. Deductive reasoning uses logic (laws, postulates, theorems, or definitions) to reach conclusions and so conclusions are always true. Inductive reasoning uses observations to make conjectures that can be true or false (unless proven true). Inductive reasoning uses observations to make conjectures that can be true or false (unless proven true).

17. page 54 -- SC 

18. page 54 - SC 

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20. page 54 SC If a blub is screaming, then a greep is flinging. If a blub is screaming, then a greep is flinging.