Completing the Square
28 Jan
Yesterday I ahem-ed and winked to my Algebra II class about them needing to know how to complete the square for class today. Teaching this topic last year was a nightmare. A total trainwreck. Students were having difficulty all over the place — they couldn’t simplify radicals, they didn’t get why the procedure worked, they were wondering how imaginary numbers came into play here, they confused the steps, they didn’t *get* it. And it was my fault.
Part of the problem was that we were doing too much, too fast. We had brought in graphing quadratics early on, and we were emphasizing the relationship between the equations and the graphs from the start. We also — in the middle of the quadratic unit — taught complex numbers. That’s too many huge things to deal with. Quadratics bring too much together, and we needed to keep the ideas and skills organized so they make sense.
So the other Algebra II teacher and I decided we’d try something different. First, this year, we introduced complex numbers without talking about quadratics. We motivated these numbers, and then we had students practice working with them, getting really comfortable with them. Second, when we started quadratics, we did so without any graphing. Period. We were doing all algebraic work.
Here’s how we progressed.
PART I: Review
Regular, very simple equations with solutions involving square roots, imaginary numbers, and real numbers:
Quick review of factoring:
A brief discussion of solving equations with perfect square terms — with imaginary and real solutions:
Part II: Completing the Square
Perfect Squares:
We talked about what a perfect square is and noticed a relationship between the four terms — when you FOIL. Importantly, students are going to see that the second and third terms are the same.
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Creating Perfect Squares!:
The next step of creating perfect squares really has them grapple with the fact that the missing constant term is simply half of the coefficient of the 
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Completing the Square:
For me, it was this step, just a short distance from the last step, which made the entire unit a success. Because now my students had seen the relationship between all the terms in a perfect square and actually seemed to understand them. My favorite part was that most students were getting problem 10 right — and it involves fractions! We also talked about how important signs are for this process.
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The End Game: Completing the Square
Before we actually “completed the square” I had students look at the last section of the review sheet.
We talked about how if you can write a problem in this form, that you can ALWAYS solve it. And what we were going to be doing is finding a way to write any quadratic equation in that form, so we can solve it.
Then, I went through an example — step by step — to get a problem to that form:
Then I had them solve it, like they had done previously. Most of them had no trouble solving it.
They practiced doing a few problems on their own — some which gave “nice” answers, some which gave answers with radicals, and some which gave complex answers.
Part III: Reinforcement
I made them practice a few more times, with some harder problems, and then I threw them a curve ball — a coefficient in front of the 
Overall, they really rocked it. How do I know they got it?
I started off this post by saying that I ahem-ed about giving a pop quiz in my class today. Well, I followed through on that. gave a pop quiz to my class on completing the square. I gave them one easy problem and one much more difficult problem — with fractions and radicals — on completing the square.
I got a whole bunch of perfect scores.
If you want, the worksheets I created are below:
Factoring Quadratics
Completing the Square, Part I
Completing the Square, Part II
Completing the Square, Pop Quiz
This looks fantastic. I love when you can analyze where their problems are and have the time and resources to make the “next go round” much smoother and successful. Go you.
Ms. Cookie
Thanks for sharing your experience. Thanks for also providing the 4 worksheets. I will be teaching this topic to my 8th graders in quarter 4. I know it’ll be tough for them.
@Ms. Cookie: It’s hard to get the time to do this, and more importantly, muster the energy. At this point, I’m just trying to get the lesson plans ready the night before.
@Diana: Good luck! What course are your 8th graders learning completing the square in? (I’m doing it with my 10th graders.)
You are right to completely separate the topic from graphing quadratics. Saving confusion? A definite gain. But also you set them up for an Aha! later on. No need to spoil the surprise.
This is for us a late Algebra I topic. I like using the drawing of the square itself, (x+3) by (x+3) for example, and we can inspect the four rectangles within, noticing two with equal area.
Now, when we have
we can set up the same thing: a big square, label it 
, label the top and the left 2x and 2x. Add two rectangles, equal area, so 10x each. Clearly we need to label the sides 5. And k should be 25…
Now, some kids prefer the algebra. Most, in fact. But in any event, we practice, as you are doing now, completing squares just for the sake of completing squares. Then we solve some equations. And after a day or so, I solve one big one…
Do they remember all of this in Algebra II? No. But it is familiar, and comes back. I know that sometimes all I am doing is setting the stage for learning that will happen in the future. Not a terrible thing.
Jonathan
@Jonathan-
Not a terrible thing at all! I don’t think we have completing the square in the Alg I curriculum, sadly. And I’m glad you think that it’s okay to divorce graphing from this. I have to say that it has been really wonderful — kids that are weak in algebra really got to test their skill with adding fractions, reducing radicals, and simplifying radicals.
We’ve since moved onto graphing, and I’ve been surprised how not immediate seeing the connection between the graph and the solution to quadratics are. I’ve been having students solve
for example, and then I’m having them graph 
and talk about the relationship between the graph and the zeros. I’m glad we have time to work on this connection, separately.
I’m almost definitely going to ask them a question on their next assessment about that — having them write it down.
Sam,
I ask very few “explain” questions on tests, but this is my favorite group of topics for exactly that. Things that are simple, I don’t care, get a kid to express in words the relationship between an equation and a graph, and you have something they are far more likely to remember.
It was on a test a month ago. (Last week, final exam, I just asked for advantages and disadvantages of several methods of solving quadratics. Kids work hard on expressing themselves on that sort of question).
Jonathan
Hi,
I don’t think it’ll be of much use for you, but maybe… I have a (very concise) on factorizing quadratics (w/o complex numbers) here: http://mbork.pl/Math_Studies_SL_materials
Regards
@mbork, we’re now done with factoring quadratics… but hopefully someone else who is doing it might find it useful!
Wow, when did you teach this to your 6th graders, and is this in your Curriculum? We did not cover this with our 8th graders at my school. I have had previous students contact me that was covering this in high school.
@Lcedwards82: I’m teaching this in 10th grade (high school).
i’m putting together a lesson plan on completing the square and was hoping to cite you as a source in my lesson. i’m a math education student putting together a teacher work sample for a pre-student teaching class. would you be okay with this? i really like your ideas on connecting the second and third terms of FOIL to cutting the “x” coefficient in half. i tried to do this with algebra tiles, but it unfortunately didn’t go over very well, probably because the students had never used them before (it was a group project and my pull for using them in the factoring lesson was unsuccessful). it also could be that i have very little experience and am not that good yet….
i’m impressed with your website and i have a feeling i might be referring to it in the future:)
take whatever you want!
the teacher motto: “beg, borrow, steal”