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 …