1. Argumentation Day 3 Math Bridging Practices June 25, 2014

2. A Mathematical Argument It is… – A sequence of statements and reasons given with the aim of demonstrating that a claim is true or false It is not… – (Solely) an explanation of what you did (steps) – A recounting of your problem solving process – Explaining why you personally think it’s true for reasons that are not necessarily mathematical (e.g., popular consensus; external authority, etc. It’s true because my John said it, and he’s always always right.)

3. Point of Clarification What’s 7 + 11? (a) 7 + 11 is 18 because 7 + 1 is 8 and 8 plus10 is 18. This is more retelling steps. (b) 7 + 11 is 18 because 11 is 10 plus 1. I added the 1 onto the 7, to get 8, and then I did 10 plus 8 instead. This is a mathematical argument. It is given to support the claim that 7 + 11 is 18.

4. Argumentation Students offer a mathematical reason for why their method is correct Students offer a logical argument to show how they know that their result is correct Student work : When talking about calculations such as: “I multiplied the cost of one package by 7 because that’s how many packages are needed for 14 days.”

5. Point of Clarification Having students generate arguments can happen every day in your class! I would argue it should ha ha ha What can you make an argument for? Any well formulated claim about something in math that could be determined true or false – no matter how bit or small.

6. Point of Clarification “Arguments in math” – need a claim, need evidence, need to know how the evidence shows the claim true (or false). “Arguments in the courtroom” – need a claim (guilty or not?), need evidence, need to know how the evidence shows the claim true (or false) “Arguments among friends” “Debates”

7. Language to help us think about and talk about mathematical arguments

8. Toulmin’s Model of Argumentation Claim Data/Evidence Warrant

9. Toulmin’s Model of Argumentation Claim Data/Evidence Warrant THE ARGUMENT

10. Toulmin’s Model of Argumentation Claim 7 is an odd number Data/Evidence 2 does not divide 7 evenly Warrant Definition of odd/even An even number is a multiple of 2; An odd number is not a multiple of 2.

11. Example 5 and 6 are consecutive numbers, and 5 + 6 = 11 and 11 is an odd number. 12 and 13 are consecutive numbers, and 12 + 13 = 25 and 25 is an odd number. 1240 and 1241 are consecutive numbers, and 1240 +1241 = 2481 and 2481 is an odd number. That’s how I know that no matter what two consecutive numbers you add, the answer will always be an odd number. Micah ’ s Response

12. Example 5 and 6 are consecutive numbers, and 5 + 6 = 11 and 11 is an odd number. 12 and 13 are consecutive numbers, and 12 + 13 = 25 and 25 is an odd number. 1240 and 1241 are consecutive numbers, and 1240 +1241 = 2481 and 2481 is an odd number. That’s how I know that no matter what two consecutive numbers you add, the answer will always be an odd number. Claim Micah ’ s Response

13. Example 5 and 6 are consecutive numbers, and 5 + 6 = 11 and 11 is an odd number. 12 and 13 are consecutive numbers, and 12 + 13 = 25 and 25 is an odd number. 1240 and 1241 are consecutive numbers, and 1240 +1241 = 2481 and 2481 is an odd number. That’s how I know that no matter what two consecutive numbers you add, the answer will always be an odd number. Claim Micah ’ s Response Data/Evidence 3 examples that fit the criterion Warrant Because if it works for 3 of them, it will work for all

14. J: I am a British Citizen B: Prove it J: I was born in Bermuda ?

15. Toulmin’s Model of Argumentation Claim I am a British citizen Data/Evidence I was born in Bermuda Warrant A man born in Bermuda will legally be a British citizen

16. Note: What “counts” as a complete or convincing argument varies by grade (age- appropriateness) and by what is “taken-as- shared” in the class (what is understood without stating it and what needs to be explicitly stated). Regardless of this variation, it should be mathematically sound.

17. Applying Toulmin’s: Ex 1 Which is bigger: 73 – 26 or 76 – 26 – 3? a. 73 – 26 is the same as 76 – 26 – 3. I add 3 to 73 and then take 3 away at the end. b. 73 – 26 is the same as 76 – 26 – 3. If I add 3 to 73 and then take 3 away at the end, I’ve added nothing overall, so the answer is the same. c. 73 – 26 is the same as 76 – 26 – 3 because 73 – 26 is 47 and 76 – 26 – 3 is also 47.

18. Applying Toulmin’s: Ex 1 Which is bigger: 73 – 26 or 76 – 26 – 3? a. 73 – 26 is the same as 76 – 26 – 3. I can add 3 to 73 and then take 3 away at the end. b. 73 – 26 is the same as 76 – 26 – 3. If I add 3 to 73 and then take 3 away at the end, I’ve added nothing overall, so the answer is the same. c. 73 – 26 is the same as 76 – 26 – 3 because 73 – 26 is 47 and 76 – 26 – 3 is also 47. Data/evidence included; Missing warrant Warrant included too! Warrant – I did the math. Not “explanatory”

19. Applying Toulmin’s: Ex 2 Which is bigger? 4 +(x+3) 2 or π a. Pi, because you can’t figure out what 4+(x+3) 2 is b. 4+(x+3) 2 because 4 is bigger than pi and (x+3) 2 is always positive a. 4+(x+3) 2 because 4 is bigger than pi and (x+3) 2 is always positive, so you’re adding a positive value to 4.

21. GROUP PICTURE in Atrium LUNCH