1. Argumentation as an Everyday event! Tuesday, February 3, 2015

2. Agenda How is solving a linear equation making an argument? –How does the language we use reveal the argument? –Possibly consider: quadratic & sys of eq Parabola problem and SBAC’s take With time –Always, Sometimes, Never –Graphic Design prompt

3. Please solve 3x + 2 = -2x + 5

4. Full argument: 3x + 2 = -2x + 5 3x + 2 = -2x + 5 + 2x What: add 2x to both sides Why: to try to get x alone, so we can ‘see’ what x value(s) makes the equation true. Warrant: because of the addition property of equality. So, if you add the same quantity to both sides of an equation, you do not change the solution set for that equation. Whatever x value makes 3x + 2 equal to -2x + 5 also makes 3x + 2x + 2 equal to -2x + 2x + 5. But why was adding 2x useful? Because 2x is the additive inverse of -2x, so that their sum is 0, and I’ll no longer have a x term on the right hand side. 5x + 2 = 5 -2 -2 What: added -2 to both sides Why: to try to get x alone so we can ‘see’ the solution. Warrant: same as above, except, whatever makes 5x + 2 = 5 also makes 5x + 2 – 2 = 5 – 2. But why -2? Because -2 is the additive inverse of 2, and we know a + -a is the same as a – a. 5x = 3 (1/5) * 5x = (1/5)*3 What: multiply both sides by 1/5 Why: to try to get x alone so we can ‘see’ the solution

5. 5x = 3 (1/5) * 5x = (1/5)*3 What: multiply both sides by 1/5 Why: to try to get x alone so we can ‘see’ the solution Warrant: because of the multiplication property of equality, which states that if two quantities are equal, then when you multiply them by the same quantity, the product is equal But why times? And why 1/5? Because 1/5 is the multiplicative inverse of 5, and so the product of 1/5 and 5 is 1, which gets us where we want to go, because any number times 1 is itself – we can soon ‘see’ what x is! x = 5/3What: x is 5/3! We have it!

6. How is solving a linear equation making an argument? –How does the language we use reveal the argument? –What are we doing mathematically? What language do we use to describe this? “keep both sides equal” “do the same thing to both sides” “move the 5 to the other side” “get x alone” or “isolate the variable” “cancel the 3s”

7. Similar, but a little different… Quadratic and square root functions 1.x 2 – 4x – 5 = 16 1. 2 + √(x – 1) = x - 5 2.2x + y = 8 3x – 2y = -2

8. WHAT DOES THIS LOOK LIKE IN THE CONTEXT OF SMARTER BALANCED?

9. Parabola problem - From Field Test Do the problem! Discuss your thoughts on strong arguments to justify which graph corresponds to which equation –Aim to use language of claim, warrant, evidence, adequate precision of language and computation (see grid) Look at student work – order; comment Consider SBAC’s take

10. The Parabola Problem (SBAC)

11. What does SmarterBalanced have to say? 2s - A, E, F 1s - B, C, H 0s – D, G Their scoring guide

12. IMPORTANT NOTE Smarter Balanced has to judge too! Smarter Balanced seems to be giving “the benefit of the doubt” to some degree – at least at the lower grades. In Cognitive Lab Tests, they found 1/3 of students would have gotten a higher score if all their work had been submitted (and not just what they entered online). My take away: It’s important to teach beyond the Smarter Balanced scoring guides.