Slide 1

Math 20: Foundations FM20.2 Demonstrate understanding of inductive and deductive reasoning including: analyzing conjectures analyzing spatial puzzles and games providing conjectures solving problems. A. What is Proof? Slide 2 Time to Get Started! The Mystery of the Mary Celeste p.4 Work through with a partner. You have 15 minutes. Slide 3 Slide 4 What DO YOU Think? p.5 Slide 5 1. What is Enough Proof? FM20.2 Demonstrate understanding of inductive and deductive reasoning including: analyzing conjectures analyzing spatial puzzles and games providing conjectures solving problems. Slide 6 1. What is Enough Proof? Conjecture A testable expression that is based on available evidence but is not yet proved. Inductive Reasoning - Drawing a general conclusion by observing patterns and identifying properties in specific examples. Slide 7 Investigate the Math p.6 Slide 8 Slide 9 Example 1 Slide 10 Example 2 Is this conjecture convincing? Why or why not? Slide 11 Example 3 Slide 12 Example 4 Slide 13 Slide 14 Summary p. 12 Slide 15 Practice Ex. 1.1 (p.12) #1-14 #3-19 Slide 16 2. How to Prove Conjectures FM20.2 Demonstrate understanding of inductive and deductive reasoning including: analyzing conjectures analyzing spatial puzzles and games providing conjectures solving problems. Slide 17 2. How to Prove Conjectures Turn to page 16. Choose 2 illusions and develop a conjecture by looking at the illusions Next prove the conjecture. Slide 18 Slide 19 Your brain can be deceived that is why Inductive Reasoning can lead you to a conjecture but can not prove it for all cases. Slide 20 Practice Ex. 1.2 (p.17) #1-3 Slide 21 3. What is a Counterexample? FM20.2 Demonstrate understanding of inductive and deductive reasoning including: analyzing conjectures analyzing spatial puzzles and games providing conjectures solving problems. Slide 22 3. What is a Counterexample? When points on the circumference of circle are joined by chords the circle is divided into sections Conjecture: As the number of connected points on the circumference of a circle increases by 1, the number of regions created within the circle increases by a factor of 2. How can we prove this? Slide 23 When we try 6 points on the circumference of a circle it does not increase by a factor of 2 This is called a counterexample because it disproves our conjecture. Why is only one counterexample enough to disprove a conjecture? Slide 24 Example 1 Previously we made 2 conjectures about the difference between consecutive perfect squares. The difference between two consecutive perfect squares is a prime number. The difference between two consecutive perfect squares is an odd number. Find a counterexample for each conjecture. Slide 25 Find another counterexample for the first conjecture. Can you find a counterexample for the second conjecture? Slide 26 Example 2 Slide 27 Summary p.22 Slide 28 Practice Ex. 1.3 (p.22) #1-16 #3-19 Slide 29 4. Proving for All Cases FM20.2 Demonstrate understanding of inductive and deductive reasoning including: analyzing conjectures analyzing spatial puzzles and games providing conjectures solving problems. Slide 30 4. Proving for All Cases Proving something to be true for all cases by drawing a specific conclusions through logical reasoning by starting with general assumptions that are known to be valid is called Deductive Reasoning So what does this mean? Lets look at the following conjecture. Slide 31 Slide 32 Slide 33 What type of reasoning did Jon use? What type of reasoning did we use? How does this differ from what Jon used? Slide 34 Example 1 We have been looking at the difference between 2 consecutive perfect squares. We last time came up with the conjecture that the difference between 2 consecutive perfect squares is an odd number. Lets prove this for all cases. Slide 35 Example 2 a) b) Slide 36 Transitive Property: If two quantities are equal to the same quantity, then they are equal to each other. If a = b and b = c, then a = c. Slide 37 Example 3 Slide 38 The process we just used to complete the previous example is referred to as a Two Column Proof Slide 39 Example 4 Slide 40 Slide 41 Summary p. 31 Slide 42 Practice Ex. 1.4 (p.31) #1-14 #4-18 Slide 43 5. Finding Holes in you Proofs! FM20.2 Demonstrate understanding of inductive and deductive reasoning including: analyzing conjectures analyzing spatial puzzles and games providing conjectures solving problems. Slide 44 5. Finding Holes in you Proofs! Invalid Proof A proof that contains an error in reasoning or that contains invalid assumptions. Premise A statement assumed to be true. Circular Reasoning An argument that is incorrect because it makes use of the conclusion to be proved. Slide 45 Investigate the Math p.36 Slide 46 Slide 47 Example 1 Slide 48 Slide 49 Example 2 Slide 50 Example 3 Slide 51 Example 4 Slide 52 Is there a number that will not work in Hossais number trick? Explain. Slide 53 Example 5 Slide 54 Summary p.41 Slide 55 Practice Ex. 1.5 (p.42) #1-7 #3-10 Slide 56 6. Solving Problems with Reasoning FM20.2 Demonstrate understanding of inductive and deductive reasoning including: analyzing conjectures analyzing spatial puzzles and games providing conjectures solving problems. Slide 57 6. Solving Problems with Reasoning Try the Trick for some different numbers and see if it is true. Slide 58 Use deductive reasoning to try and prove the number trick. Slide 59 Investigate the Math p.45 Slide 60 Slide 61 Example 1 Slide 62 Did we use inductive reasoning or deductive reasoning? Slide 63 Example 2 Slide 64 Did we use inductive reasoning or deductive reasoning? Slide 65 Summary p.48 Slide 66 Practice Ex. 1.6 (p.48) #1-14 #3-16 Slide 67 7. Some Puzzles and Games FM20.2 Demonstrate understanding of inductive and deductive reasoning including: analyzing conjectures analyzing spatial puzzles and games providing conjectures solving problems. Slide 68 7. Some Puzzles and Games Start with one marker each, then 2, then 3 and continue until 5 markers. Find the minimum number of moves for each situation and record the results in a table Slide 69 Investigate the Math p. 52 Slide 70 Example 1 Slide 71 Slide 72 Slide 73 Summary p. 55 Slide 74 Practice Ex. 1.7 (p. 55) #1-13 #4-15