1. Argumentation Day 4 Math Bridging Practices Monday August 18, 2014

2. We have been wrestling with…

3. Monte Python – The Flying Circus

4. A Mathematical Argument It is… – A sequence of statements and reasons given with the aim of demonstrating that a claim is true or false – - “an argument is a collective series of statements to establish a definite proposition” (Monte Python) 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 Adrianne said it, and she’s always, always right.) Take-away 1! What is an argument?

5. 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 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 big or small. Take-away 1! What is an argument?

6. THINK! What is 16 x 25? Take-away 2! There’s a difference between an explanation and an argument.

7. Explanation of Steps Providing an Argument I took 16 and split it into 10 and 6. I multiplied 10 by 25, and I multiplied 6 by 25. And then I added those 2 numbers together. I took 16 and split it into 10 and 6 because I need to find 16 groups of 25, and so I can find 10 groups of 25 and then add to it 6 more groups of 25. So I multiplied 10 by 25 and 6 by 25. I added them together to give me 16 25s total, which is what I need. Explanation of Steps vs Providing an Argument Student Work 16 = 10 + 6 10 x 25 = 250 6 x 25 = 150 250 + 150 = 400 Take-away 2! There’s a difference between an explanation and an argument.

8. Explanation of Steps Providing an Argument I took half of 16 to get 8 and doubled the 25 to get 50. I did it again- so half of 8 was 4 and double 50 was 100. 4 times 100 is 400. I took half of 16 to get 8 and doubled the 25 to get 50. 16 x 25 and 8 x 50 are the same, because if you take ½ of one number (8), and double the other number (25), the product is the same. Explanation of Steps vs Providing an Argument Student Work 16 x 25 = 8 x 50 = 4 x 100 = 400 16 25 8 8 I did it again- so half of 8 was 4 and double 50 was 100. 4 times 100 is 400. 8 8 Take-away 2! There’s a difference between an explanation and an argument.

9. Toulmin’s Model of Argumentation Claim Data/Evidence Warrant Take-away 3! Argumentation involves claims, warrants and evidence.

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

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. 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 Take-Away 4! Not all arguments are valid (viable).

12. Is it a viable argument? sdf Reasoning and Proof, NCTM (2000), p 189 I think they have different areas because the triangle looks a lot bigger. I think they’re the same because they are both half of the bigger rectangles. They’re both half, so they have to be the same.

13. Take-away 5! (in progress) What “counts” as an acceptable (complete) argument varies by grade (age-appropriate) and by what is taken-as-shared in a class (established already as true). Regardless of this variation, it should be mathematically sound.

14. Example Consecutive numbers go even, odd, even, odd, and so on. So if you take any two consecutive numbers, you will always get one even and one odd number. And we know that when you add any even number with any odd number the answer is always odd. That’s how I know that no matter what two consecutive numbers you add, the answer will always be an odd number. Claim Angel’s Response Data/Evidence 2 consec #s are always one odd and one even Warrant Because we’ve shown before odd+even is odd It is an argument. It is viable. Depending on class, may or may not be complete.

15. STUDENT WORK – HEXAGON TASK Purposes To help us get clearer about strengths and weaknesses of different arguments To help us each develop a vision for what we value and will count as a strong argument in our classrooms

16. THE HEXAGON TASK Each figure in the pattern below is made of hexagons that measure 1 centimeter on each side. a)Draw Figure 5. Find the perimeter of Figure 5. b)If the pattern of adding one hexagon to each figure is continued, what will be the perimeter of the 25th figure in the pattern? Justify your answer.

17. Analyzing Student Work PTT -Just think. (~3 mins) Group time! (~15 mins) What is the student’s claim? In your own words, summarize the student’s argument. Take turns summarizing. (Column 2) Has this student constructed a viable argument to show the perimeter of the 25 th figure? [does it prove the claim?] (Column 3) Commentary – Explain why or why not. Perhaps address completeness -- what else would you want to see?

18. Is the response an argument that shows the perimeter of the 25 th figure? The result is justified! Valid, complete argument! Result is not justified [not a valid argument, or not complete –too many gaps] Place a sticky note on the poster according to the following. Put your Table # on the sticky note. We’re not sure Be prepared to share your thinking

19. Argumentation – solidifying What will count in your classroom for a valid argument? (What qualities or criteria are important to you?) How are these criteria communicated to students? What do you expect at the beginning of the year? Where will growth be?