1.5 – proofs that are not valid

1. 1.5 – proofs that are not valid Chapter 1: Inductive and Deductive Reasoning

2. activity • What’s wrong with this picture? • What’s the area of the top figure? • What’s the area of the bottom figure? • Get out the graph paper! Trace the bottom shape out as accurately as you can. • Cut out the pieces as carefully as possible, and rearrange them into the top figure. • What do you notice?

3. Sometimes, things that seem like proofs may not be valid, because they use a mathematical operation that isn’t allowed, or just because they distort the information. It’s important to be able to spot these invalid proofs!

4. example Argument Tia’s Solution “I did some research and found out that 18 athletes have competed in both Games. This statement is not valid.” This statement is true. She has played on the national hockey team. The conclusion was false because the first statement was false. Hayley played for Canada in the softball competition in the 2000 Summer Olympics. • Athletes do not compete in both the Summer and Winter Olympics. • Hayley Wickenheiser has represented Canada four times at the Winter Olympics. • Therefore, Haley Wickenheiser has not participated in the Summer Olympics. Sometimes, the original premise of the argument is false!

5. Bev claims he can prove 3 = 4 Bev’s proof: What’s wrong with his proof? a + b = c is his premise, and since he set it at the start, it’s valid. His substitution and reorganization are valid! Can he divide by (a + b – c)? Whenever you see division, you should make sure that you are not dividing by zero, because it is an illegal operation. Since a + b = c, we can say that a + b – c = 0. So, we cannot divide by (a + b – c). Bev’s proof is invalid. • Suppose that a + b = c • This can be written as: • 4a – 3a + 4b – 3b = 4c – 3c • 4a + 4b – 4c = 3a + 3b – 3c • 4(a + b – c) = 3(a + b – c) • 4 = 3

6. Example: What’s wrong with this? • Liz claims she has proved that -5 = 5 Liz’s proof: • I assumed that -5 = 5. • Then I squared both sides: (-5)2 = 52 • I got a true statement: 25 = 25 • This means that my assumption, -5 = 5, must be correct. You cannot start a proof with a false assumption! Everything that comes after a false assumption doesn’t matter, because its reasoning is built on something untrue.

7. example Hossai’s Proof: Choose any number: n Add 3: n + 3 Double it: 2n + 6 Add 4: 2n + 10 Divide by 2: 2n + 5 Take away the number you started with: n + 5 Where is the error? Hossai is trying to prove the following number trick: Choose any number. Add 3. Double it. Add 4. Divide by 2. Take away the number you started with. Each time Hossai tries the trick, she ends up with 5. Her proof, however, does not give the same result.

8. Pg. 42 – 44, # 1, 2, 3, 5, 7, 9, 10 Independent practice

9. 1.6 – reasoning to solve problems Chapter 1: Inductive and Deductive Reasoning 1.7 – Analyzing puzzles and games

10. example The members of a recently selected varsity basketball team met each other at their first team meeting. Each person shook the hand of every other person. The team had 12 players and 2 coaches. How many handshakes were exchanged? Solution: 13 + 12 handshakes There were 14 people in total, so person 1 shook hands with each of the other 13 people. This pattern will continue until there are only two people left, and then the last handshake happens. Person 2 had already shaken hand with person 1. So, person 2 shook hands with each of the remaining 12 people 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 91 handshakes

11. Example: use reasoning Sue signed up for games at her school’s fun night. Seven other people were assigned to her group,making up four pairs of partners. The other members of her group were Dave, Angie, Josh, Tanya, Joy, Stu, and Linus. When the games started, Dave and his partner were to the left of Stu. Across from Dave was Sue, who was to the right of Josh. Dave’s brother’s partner, Tanya, was across from Stu. Joy was not on Stu’s right. Dave Angie Josh Tanya Joy Stu Linus Sue

12. challenge Each counter can leapfrog over one other counter. The goal is to get all of the blues/reds over to the opposite side! Use the counters that have been distributed to try it out. Feel free to share strategies. What is the fewest number of steps that you can succeed in?

13. Pg. 49 – 51, # 1, 2, 5, 6, 7, 8, 12, 13, 16, 18, 19. Independent practice

14. example Nadine and Alice are playing a toothpick game. They place a pile of 20 toothpicks on a desk and alternate turns. On each turn, the player can take one or two toothpicks from the pile. The player to remove the last toothpick is the winner. Nadine and Alice flip a coin to determine the starting player. Is there a strategy Alice can use to ensure that she wins the game? Hint: Try working backwards!

15. Solution • Alice needs to make sure that there are one or two toothpicks left after Nadine’s last turn. • So, Alice needs to make sure that there are three toothpicks on the desk for Nadine. • To make sure she can leave three toothpicks, she needs to leave 6 toothpicks the time before (either Nadine picks 1, and we have 5 and pick 2, or she picks 2 and we pick 1). • To make sure that she leaves 6, she will need to leave 9 before that. We see a pattern! • She will need to leave 12, 15 and 18 toothpicks for Nadine. • So, if Alice goes first she should pick 2, to leave 18. If Nadine goes first and she knows this strategy, then Alice can’t win. However, if Nadine doesn’t know the strategy, Alice can leave toothpicks in a multiply of three and win the game.

16. Pg. 55 – 57 # 3, 4, 5, 7, 9, 11, 12, 13 Independent practice