Argumentation Day 3

Math Bridging PracticesJune 25, 2014

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.)

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.

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.”

03.16$729.2$

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.

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”

Language to help us think about and talk about mathematical arguments

Toulmin’s Model of Argumentation

Claim

Data/Evidence

Warrant

Toulmin’s Model of Argumentation

Claim

Data/Evidence

Warrant

THE ARGUMENT

Toulmin’s Model of Argumentation

Claim7 is an odd number

Data/Evidence2 does not divide 7 evenly

WarrantDefinition of odd/even

An even number is a multiple of 2;

An odd number is not a multiple of 2.

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

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

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/Evidence3 examples that fit

the criterion

WarrantBecause if it works

for 3 of them, it will work for all

NOTE: this has the structure of an argument, but this does not show the claim to be true. (not a viable argument)

J: I am a British CitizenB: Prove it

J: I was born in Bermuda

?

Toulmin’s Model of Argumentation

Claim I am a British citizen

Data/EvidenceI was born in Bermuda

WarrantA man born in Bermuda will

legally be a British citizen

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.

Applying Toulmin’s: Ex 1Which 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.

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”

Applying Toulmin’s: Ex 2Which 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

c. 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.

GROUP PICTURE in AtriumLUNCH