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Completing the Square on Equations of Circles

24 Mar

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?

Screen Shot 2014-03-23 at 8.00.10 PM

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?

03-23-2014 Image009Screen Shot 2014-03-23 at 7.59.12 PM

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?

Screen Shot 2014-03-23 at 7.57.58 PM

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.

03-23-2014 Image011

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

Screen Shot 2014-03-23 at 8.34.18 PM

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.

03-23-2014 Image010

Screen Shot 2014-03-23 at 8.09.51 PM Screen Shot 2014-03-23 at 8.09.43 PM

And so the formative assessment journey continues …

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1 Comment

Posted by on March 24, 2014 in Circles, Coordinate Geometry, Geometry

 

Tags: , , , , , ,

One response to “Completing the Square on Equations of Circles

  1. Alicelewis

    April 1, 2014 at 11:53 am

    This article which name is Completing the Square on Equations of Circles is helpful for me. You have shared great stuff. The way of describing this concept is interesting. I would like to say Thanks for sharing it.

     

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