Completing the Square, Starting with Geometry

Contrary to my own experience learning to Complete the Square, which was an entirely algebraic affair, there is now a beautiful geometric analogy to the process.  This is done a lot online, and the videos seem to follow a bit of a formula: show the algebra first, and alongside, show the geometric analogy.  I quickly found three examples on youtube: One, Two, Three.   Seems like the instructors are starting with completing the square like they were taught, but then saying, “hey – here is also a visual way to understand this!”

With some freshmen, I took a slightly different approach.  Thinking that the geometry isn’t just a reasonable analogy, but that it could be a roadmap to the idea, I started with the geometry, then went to the algebra.    Sort of.

To be honest, I started the day invoking some algebra – this was for a geometry class where the students had already seen the equation for a circle, so I started with the algebra they had seen, and quickly took them to where they would get stuck.   Then I started the geometry.


Students quickly answered (3,5), and a radius of 36.  No!  6! 6! 6!

Slide02Students: Uhhh….

There was some casting about on their parts, but really not getting anything that they felt they could hold onto.  So we started into completing the square.


Me: How many squares?

Students: 2!

Me: Orange square is x on a side.  The leftover is 3 units long.  Capisce?


Students worked on this. I had groups verify with each other so that they came to a consensus, class-wide, about the area being (x+3)(x+3), or something equivalent.


Groups did this, checking with partners.  Now a moment of instant gratification – check your work!


High fives all around


5!  No!  25!   I check for understanding all around.


People are getting it.  49 came faster.

Me: you get the idea.  Now let’s do some analysis. (the blue equations are animated in after I prompt for simplifying the black)


In the next slide – students saw only the blue equation to begin with.  The black 25 and 49 come after they have stated what should be there.


Students could kind of remember the 25 and the 49, and did so much better by me going back to the previous slides.  Once they remembered, and we put those in, I reinforced the question: “How do you complete the square if you only have x2+10x ?  Or you only have x2 + 14x ?”    I encouraged them to discuss in partnerships.

And there it was. They created the rule: Take half of the coefficient of x, and square it. Add that to the expression.    Every single partnership.   To reinforce:


Me: Ok – does that work with the pictures?

We went back to the pictures.

One student: With our initial squares, we already have half of the x term.


Now to revisit the initial problem:Slide13.jpg

This took a while.  And no wonder: there are more skills involved.  Combine like terms. Complete the square(s).  Add them to BOTH sides of the equation.  Combine like terms again.   THEN convert to the form from which we can easily discern a center and radius.

What I didn’t anticipate was that last step – to reinforce the notion that x2 + 10x + 25 = (x+5)2

So the last part of the lesson was pretty didactic.

What stands out? The ease with which the students were able to navigate the slides with the visual geometric cues.  Completing the square proved to be really straightforward for them.  The following algebraic manipulation took the longest time.

So there is the next step, if I am to run this again: reduce the “cliff” that occurs at the last slide. More scaffolding to get to that final complicated equation.  I have a few thoughts, but would love to hear your ideas about how that might be a more gradual process that doesn’t require a bunch of me talking at the board, writing stuff down.


Ppt here: Completing the Square