SMP8: Look for and Express Regularity in Repeated Reasoning #LL2LU

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning.

CCSS.MATH.PRACTICE.MP8

But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4

I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3

I can look for and express regularity in repeated reasoning.

Level 2

I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1

I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

 

What do you notice? What changes? What stays the same?

We use a CAS (computer algebra system) to help our students practice look for and express regularity in repeated reasoning.

What do we need to factor for the result to be (x-4)(x+4)?

What do we need to factor for the result to be (x-9)(x+9)?

What will the result be if we factor x²-121?

What will the result be if we factor x²-a²?

We can also explore over what set of numbers we are factoring using the syntax we have been using. And what happens if we factor x²+1? (And then connect the result to the graph of y=x²+1.)

 

What happens if we factor over the set of real numbers?

Or over the set of complex numbers?

What about expanding the square of a binomial?

What changes? What stays the same? What will the result be if we expand (x+5)²?

Or (x+a)²?

Or (x-a)²?

What about expanding the cube of a binomial?

Or expanding (x+1)^n, or (x+y)^n?

What if we are looking at powers of i?

We can look for and express regularity in repeated reasoning when factoring the sum or difference of cubes. Or simplifying radicals. Or solving equations.

Through reflection and conversation, students make connections and begin to generalize results. What opportunities are you giving your students to look for and express regularity in repeated reasoning? What content are you teaching this week that you can #AskDontTell?

[Cross-posted on Experiments in Learning by Doing]

 

Completing the Square on Equations of Circles

I wrote about this lesson last year. So just a few updates for this year.

Our goal – the second part of the standard:

G-GPE.A.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

We started with a Quick Poll. I figure it’s going to be hard to complete the square if we don’t know what the square of a binomial actually is.

If someone has a counterexample, then the statement must be false. Who marked false that has a counterexample for this statement not always being true?

One student let x=2 to show that the statement wasn’t always true.

Did anyone else use a number?

Various other numbers had been used to show the statement was false.

Did anyone show it was false a different way?

One student expanded (x+1)² to show that it wasn’t always equal to x²+1.

We used CAS to look for regularity in repeated reasoning. What happens when you square a binomial?

We started with the familiar, the equation for the circle with its center and radius. What happens if we expand that equation – and instead start with the expanded form? How would we go backwards to get to the center and radius form of the equation?

More than one student couldn’t believe I made a big deal about what we needed to add to complete the square. It was so obvious to them that we needed to undo what we had done when we expanded: divide by 2 and then square.

We call this completing the square to find the center and radius of a circle.

And just in case someone needs another visual, we look at Completing the Square from Algebra 2 Nspired.

And then we tried a few where we didn’t know the center-radius form before we started.

And then we checked to see how well students were working on their own, finding out that we are not quite ready to move on.

And so the formative assessment journey continues …