Reasoning & Logic

1. Reasoning & Logic Geometry Standard 1: Objective 1

2. You will use inductive and deductivereasoning to develop mathematical arguments. • You will formulate conjectures using inductive reasoning. • You will prove a statement false by using a counterexample. • You will write conditional statements, converses, and inverses, and determine the truth value of these statements. Core Objectives – The BIG Picture

3. Inductive Reasoning • Conditional Statements • Conjectures • Counterexample • Deductive Reasoning • Converses • Inverses • Truth Value Key Vocabulary

4. You and your partner will receive two small pieces of paper. • Follow the directions on the paper. • These are examples of Inductive Reasoning. • When you finish, pair-up with another partnership sitting next to you. • As a group of four, answer this question: What is Inductive Reasoning? Pair Exploratation Activity

5. When you make a conclusion based on a pattern of examples or past events, you are using Inductive Reasoning. Inductive Reasoning - Defined

6. If you see dark, towering clouds approaching, what might you do? • Why? • Even though you haven’t heard a weather forecast, your past experience tells you that a thunderstorm is likely to happen. • What are some examples of decisions you made today based on past experiences or patterns that you observed? Inductive Reasoning – REAL LIFE http://images.iop.org/objects/phw/news/13/7/5/cloud.jpg

7. Originally, mathematicians used inductive reasoning to develop geometry and other mathematical systems to solve problems in their everyday lives. Euclid http://www.math.buffalo.edu/mad/Ancient-Africa/gizaplateau.gif Inductive Reasoning – Math Background http://www.mathninja.net/history/euclid/images/euclid.jpg

8. Without drawing a picture, use Inductive Reasoning to fill in the table. Inductive Reasoning - Practice

9. You have just made a Conjecture. • A Conjecture is a conclusion that you reach based on Inductive Reasoning. • A Conjecture is an educated guess. • Sometimes a Conjecture is true, and other times it may be false. • How do you know whether your Conjecture, about the regions formed by connecting points on a circle, is true or false? Conjecture

10. One way to test the truth of a Conjecture is to try out different examples. • Test your Conjecture: • Draw a circle with 6 points on it. • Connect the points with segments. • Count the regions formed by the pattern. • Is your Conjecture true or false? Conjecture – Test it!

11. A false example is called a Counterexample. Counterexample - Defined

12. Suppose you went out with friends for the past 5 Friday evenings. Each time, your parents told you to be home by 11:00 PM. • On the first Friday, you came home at 10:55 PM. The second Friday, you came home at 10:49 PM. The next three Fridays, you came home at 10:59 PM, 10:51 PM, and 10:42 PM respectively. • You plan to hang out again this Friday. Using Inductive Reasoning, what might be your parents’ Conjecture? • Remember that a Conjecture is a general statement using Inductive Reasoning…not a specific example. • What would be a Counterexample? Counterexample – REAL LIFE

13. Write a Conjecture using Inductive Reasoning about a REAL LIFE example. Then give a Counterexample. • Share in groups of 4. Be sure to use the correct terms when discussing your example. Your Turn – REAL LIFE example

14. Guided Practice Akira studied the data in the table at the left and made the following conjecture. “The product of two positive numbers is always greater than either factor.” How did Akira use Inductive Reasoning in order to form his conjecture? Find a counterexample for his conjecture.

15. Guided Practice The graph shows the revenue from the sale of waste management equipment in billions of dollars. Find a pattern in the graph and then make a conjecture about the revenue for 2005. Make a conjecture about whether the rate of increase will continue forever. Explain your reasoning. What might prove your reasoning true or false?

16. REFLECT Pattern A Inductive Reasoning Conjecture Counterexample Inductive Reasoning A A Conjecture (rule, conclusion) A A A B Counterexample

17. Even though patterns can help make a conjecture, patterns alone do not guarantee that the conjecture will be true. • In logic you can prove that a statement is true for all cases by using Deductive Reasoning. • Deductive Reasoning is the process of using facts, rules, definitions, or properties in a logical order. Deductive Reasoning

18. If Marita obeys the speed limit, then she will not get a speeding ticket. • Marita obeyed the speed limit. • Therefore, Marita did not get a speeding ticket. • What was the fact, rule, definition, or property? • Notice the “If…then…” statement. Deductive Reasoning - Example

19. If Marita obeys the speed limit, then she will not get a speeding ticket. • Marita obeyed the speed limit. • Therefore, Marita did not get a speeding ticket. • The second sentence says that that the “If”-statement is true. • If Marita obeys the speed limit… • Marita obeyed the speed limit. • Marita met the first condition. Deductive Reasoning - Example

20. If Marita obeys the speed limit, then she will not get a speeding ticket. • Marita obeyed the speed limit. • Therefore, Marita did not get a speeding ticket. • Therefore, by Deductive Reasoning, the “then”-statement must be true. Deductive Reasoning - Example

21. In other words… • Deductive Reasoning starts with an If…then… statement. • This is a rule, fact, definition, or property. • “If…then…” statements are called Conditional Statements. • Then, you look at an example. See whether or not it makes the “If…” statement true. • If so, we logically conclude that the “then…” statement is also true. Deductive Reasoning - Restated

22. All students in Mr. Jackson’s Geometry class are enrolled at Bonneville High. • This is a fact. • NOTE: If it is not a fact (rule, definition, or property), then we can not use deductive reasoning. Deductive Reasoning - Example

23. All students in Mr. Jackson’s Geometry class are enrolled at Bonneville High. • Write a Conditional Statement (if…then…) using the rule. • “If you are a student in Mr. Jackson’s Geometry class, then you are enrolled at Bonneville High.” Deductive Reasoning - Example

24. “If you are a student in Mr. Jackson’s Geometry class, then you are enrolled at Bonneville High.” • Now comes the Deductive Reasoning (the logic) using a specific example. • Insert your name is a student in Mr. Jackson’s class. • This statement is a true statement. • NOTE: If it were NOT true, we could not use Deductive Reasoning. It would fail the “IF”-test. Deductive Reasoning - Example

25. “If you are a student in Mr. Jackson’s Geometry class, then you are enrolled at Bonneville High.” • Insert your name is a student in Mr. Jackson’s class. • Therefore, insert your name is enrolled at Bonneville High. • We conclude, by Deductive Reasoning that the “then” statement is also true. Deductive Reasoning - Example

26. Students that attend class every day, do their homework, and pass the tests will pass Geometry. • Write a Conditional Statement(if…then…). • Choose an example that would make the first condition (if-statement) true. • Use Deductive Reasoning to draw a logical conclusion. Deductive Reasoning – Guided Practice

27. Write a rule. Remember it must be true. • Exchange “rules” with a partner. • Write a Conditional Statementfor your partner’s rule. • Exchange with another partner. • Choose an example that makes the first condition (if-statement) true. • Pass it back to the owner. • Use Deductive Reasoning to write a logical conclusion. Deductive Reasoning – Paired Practice

28. In order for something to be logical (or “TRUE” as we say it in Geometry), both conditions (the “If”-statement and the “Then”-statement) must be true. Deductive Reasoning - Restated

29. Let’s look at the logic of math. • RULE: “If you multiply two positive numbers, then their product is also positive.” • That rule is already a conditional statement. • Deductive Reasoning: The two numbers (5) and (8) are positive; therefore, their product is also positive. • Notice the words “IF,” “THEN,” and “THEREFORE.” These are commonly used in deductive reasoning. Deductive Reasoning – Math Example

30. Write a conditional statement about a rule in math. • Now choose an example that makes the first condition true. • Use deductive reasoning to state the conclusion. Deductive Reasoning – Independent Practice

31. Geometry Textbook pages 786-793 (Glencoe, 2006). • Many careers and hobbies involve Geometry. • They all rely on Postulatesand Theorems. • Postulates are rules that we accept to be true. • Theorems are rules that have been proven to be true using postulates and other theorems. • They are both RULES; therefore we can use them to logically explain our Geometric World. What’s the point of all of this?

32. In mathematics, you will come across many if-then statements. • Another example, “If a number is even, then it is divisible by 2.” • If-then statements join two statements based on a condition. • Conditional statements have two parts. The part following If is the hypothesis. The part following Then is the conclusion. What’s the point of all of this?

33. “If a number is even, then it is divisible by 2.” • Hypothesis: A number is even. • Conclusion: The number is divisible by 2. • In geometry, postulates are often written as if-then or conditional statements. • You can easily identify the hypothesis and conclusion in a conditional statement. What’s the point of all of this?

34. Find a postulate in the textbook (pages 786-793) and read it to yourself. • What is the hypothesis? • What is the conclusion? What’s the point of all of this?

35. You may have noticed that not all postulates are clearly stated as if-then statements. • There are different ways to express a conditional statement. • The following statements all have the same meaning. • If you are a member of Congress, then you are a U.S. citizen. • All members of Congress are U.S. citizens. • You are a U.S. citizen if you are a member of Congress. Extension

36. What is another form of the statement, “All collinear points lie on the same line”? Extension – Guided Practice

37. With a partner, try this one: • What is another form of the statement, “If two lines are parallel, then they never intersect”? Extension – Paired Practice

38. The converse of a conditional statement is formed by exchanging the hypothesis and the conclusion. • Let’s form the converse of some previous examples. • “If you multiply two positive numbers, then their product is also positive.” • Hypothesis: Two numbers are positive. • Conclusion: The product of the numbers is positive. • Converse: “If the product of two numbers is positive, then the numbers are positive. • Can you think of a counterexample? Converse

39. “If you are a student in Mr. Jackson’s Geometry class, then you are enrolled at Bonneville High.” • Hypothesis: A student is in Mr. Jackson’s Geometry class. • Conclusion: The student is enrolled at Bonneville High. • Converse: If you are a student enrolled at Bonneville High, then you are in Mr. Jackson’s Geometry class. • Counterexample? Converse – more examples

40. “If Marita obeys the speed limit, then she will not get a speeding ticket.” • Hypothesis: • Conclusion: • Converse: • Counterexample? Converse – Guided Practice

41. With a partner, form the Converse of this statement: • “If a number is even, then it is divisible by 2.” • Converse: • Is there a counterexample? Converse – Paired Practice

42. On your own, write the Converse of these conditional statements and write a counterexample if there is one: • If a figure is a triangle, then it has three angles. • If you are at least 16 years old, then you can get a driver’s license. • If a figure is a square, then it has four sides. Converse – Independent Practice

43. With a partner, form the Converse of this statement: • “If today is Saturday, then there is no school.” • Converse: • Is there a counterexample? Converse – Paired Practice

44. You have formed the Converse of several Conditional Statements. • If a conditional statement is true, is its converse always true? • What logical REASONING did you use to draw that conclusion? Critical Thinking

45. The inverse of a conditional statement is formed by negating both the hypothesis and conclusion of the conditional. • Example: • Conditional: If it is raining, then it is cloudy. • Inverse: If it is not raining, then it is not cloudy. • Can you think of a counterexample? Inverses

46. What is the inverse of this conditional? • If a figure has five sides, then it is a pentagon. • Can you think of a counterexample? Inverses

47. Form the inverse of these conditional statements we have already seen. Determine if there is a counterexample. • If Marita obeys the speed limit, then she will not get a speeding ticket. • If a number is even, then it is divisible by 2. • If a figure is a triangle, then it has three angles. • If you are at least 16 years old, then you can get a driver’s license. • If a figure is a square, then it has four sides. • If you are a student in Mr. Jackson’s Geometry class, then you are enrolled at Bonneville High. • If today is Saturday, then there is no school. Converse – Practice

48. Mathematicians use symbols to represent words and phrases. • This is called “notation.” • The purpose of mathematical notation is to simplify the writing. • Without mathematical notation, 2x2+3x-5 would need to be written, “Two times a number squared plus three times that number minus five.” • …and that is an easy one! Truth Value – Notation

49. In math, we sometimes use “Truth Tables” in order to analyze the truth of a statement. • A statement is any sentence that is either true or false, but not both. • Therefore, every statement has a Truth Value, true (T) or false (F). Truth Value - Notation

50. Use the handout of the 7 Conditional Statements that we discussed. • Consider the Conditional and its Converse and Inverse. • Use Deductive Reasoning and write T if it is true for all cases (no counterexamples) and F if it is not true for all cases (you can think of at least one counterexample). • Item number one is done for you. Truth Value - Analysis