Completing the square completes me

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My favorite topic to teach is completing the square. Weird, right? I just love it. I start by having kids solve radical expressions. I say “what cha gonna do?” and then say “square both sides!” We then move to solving equations where we need to square root both sides and I catch them in the chant! “what cha gonna do?” “Square both sides.” Really? Then they think. And then they think some more. “Oooo,” they say. X squared = plus or minus the square root of x. Major deal of course. So now the chant becomes, “what cha gonna do?’ and they say, “square root both sides.” Many forget the plus and minus no matter how many we do it. I do this before solving quadratics, which may be unconventional, but it works.

We solve equations with a squared term, a linear term and a constant or two and we discover how to complete the square in order to be able to take the square root of both side after some massaging. It’s awesome. It’s magical. It has absolutely nothing to do with quadratics as far as students know. We are merely balancing equations.

Then we do go into quadratics. There is always some smart alec that wants to dive into the quadratic formula. I hate this, but soldier on. They may NOT use the quadratic formula until they can prove it. End of story. We go into vertex form of a quadratic. They dig the structure and can assemble the vertex form of a quadric if they know the vertex and the multiplier. We then look at discovering coming up with a quadratic equation with far less information. We know the structure of vertex form because that is taught in Math 1 (supposedly). Of course the structure is re-taught. But, now we have a reason for knowing how to complete the square. Yeah! It’s so similar to coming up with the slope intercept form of a linear equation. Plug in what you know (what you are given) and chug away. That there must be a perfect square trinomial to make the square root of both side s of the equation is the secret weapon. All must balance. No illegal moves. Taking the multiplier into consideration is a challenge, but we get there.

Because I introduce completing the square multiple times and early, it sticks by the time we prove the quadratic formula. It gets revisited when we do geometry and prove the Pythagorean theorem. The sparks and light bulbs that go off are so fun!

I tell my kids when we first complete the square that this is a skill that could make them a hero some day. When I first started teaching algebra 7 years ago, my daughter was in a college chemistry class. Her class had a problem that looked unsolvable until a student suggested that they complete the square. That student was a hero. I tell my students that some day they too can be a hero with completing the square. They are instructed to notify me the day this happens. Sadly, no notifications so far.

I think that success in teaching completing the square comes from students being exposed to it over a period of time, even before they really need it. If you wait until they need it, the mind is too clogged with other ways to solve quadratics. I tell my kids, too frequently solving a quadratic via the quadratic formula is like taking a barge down a river. Completing the square is a kayak. Have fun! Use it. Be a hero!


Making It Real

So, I went TMC16 and for whatever reason, I kept running into Denis Sheeran @MathDenisNJ. It was almost annoying. For both of us. I did get his book Instant Relevance this fall and he told me he would love to hear what I think. So here it goes.

When I ordered the book, I thought, “crap that’s a lot of money at this time of year.” See, I just set up my classroom. Then the mail came and I thought, “Are you kidding me? This is the shortest book since Jonathan Livingston Seagull!”

So now I think this. ALL BOOKS THAT ARE MEANT TO HELP TEACHERS BE BETTER SHOULD BE THIS LENGTH!! I am so serious. I’m beat at night. I can’t do much more than 5 pages. I want to learn more and become a better me. I really do. But some books are so boring and so long and I have the attention span of a gnat. I teach middle school for crying out loud.

So, thank you Denis. I never felt bad or distracted or loserish as I read your book. And this happened.

I took pictures. I had a bag if rolled up coins in the mud room that have never made their way to the bank. I wondered what I was missing out on so I figured my kids could help me.


I showed them a pic of the sac of coins. They guessed high and low. Then the best thing ever happened. One girl who I just moved from math 8 to math 1 (thank you county for being so great at giving us quality tools to place kids properly when they move in—sarcasm intended) said, “hey that’s like one of these things we’re working on. Can I show you?” She walks up to the board and does this.


I am blown away. I should be smart enough to see all the connections, but I don’t. I need my students to do that. I love this gal. I love my job.


My 1st class, math 8, worked as a class to figure out how much $ there was. I introduced the problem to my math 1 kids (periods 2 & 3) and then they pursued it to the end in small groups after they finished another task in class. They totally dug it.


Math 2 kids are a bit harder to impress, but they finished the entire task as an opener as I progressed through fotes.

All of this is just my way of saying, thanks Denis. It works. I get it. I dig it. I’m doing it.

Practicing on real kids!

You have to love a PD where you get to watch other teachers as they hone their craft as well as get to practice yourself…on REAL live kids!!! The kids were on fire as they made models of houses. I wanted to take one home, but I didn’t. (I mean a kid, not their house. The house was swell and all, but talking with these kids was a treat!)

This set-up the need for area and scale and unit conversion in order to make cost estimates of building materials.

The problem created the need for the content. The content did not set the stage for some hokey, convoluted, boring application.

Kind of getting excited for Aug 29!!!

Oh linear functions–please don’t be boring!


This month I was going to change the world, or at least change the climate in my 2 math 8 classes. It’s been miserable lately. I took responsibility for that and decided to make major changes in class as the second semester began. I changed the daily organization, the look of the room and most importantly, the lessons. Of course, I wanted everything to become perfect all at once, but that didn’t happen. Here’s how change began lesson-wise.

Two weeks ago I started by using a hundreds board and chips to revisit linear equations. We looked for patterns using 5 transparent plastic chips placed in the shape of a plus sign on the numbers on a hundreds board. We added the numbers under the chips; we looked at how we could get the greatest sum in a given pattern, the smallest sum, etc. We moved the plus sign around. We added by decomposing numbers. It was good.


(I modified an old Illuminations lesson that I found in an old notebook and I forgot it at school. Sorry. I’ll copy and edit into here Monday or so.) The lesson was good. I was happier, the kids were happier. Students were noticing patterns. We made a table of ordered pairs by using the middle number in the plus sign as the input and the sum of all five squares as the output. Students graphed individually. Then I gave each student a smiley face sticker to place on a big graph locating one of the ordered pairs they found. The goal was for students to see that even though they each did the activity alone, their points all made a pattern that graphed as a straight line. Since some of the points weren’t too close to the straight line, we also were able to notice parts of a pattern that don’t come out the way we expect. This was a time for self-correction. By giving each student the exact same type sticker, there was no way to single out a particular kid as being the one who didn’t get it. This activity allowed us to work with linear equations –again– without the students realizing it.

The next day we extended the 100s board activity onto a calendar. We did some calendar math using four by four grids. We saw patterns in diagonal sums, four corner sums, and center four sums. This was then extended to the abstract as I had students choose any day in the four by four region to be n. All other days were then defined in terms of that n by adding or subtracting.

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(My students struggle with turning concrete math into abstract math so this was a good exercise.) Sums were then found again. Setting these sums equal to one another or setting a sum equal to the original concrete sum yielded the original date on which the n was placed. I preferred this to choosing the first box as n and simply adding to get subsequent dates in terms of n. Students who persevered were delighted when the solved equation yielded the original n value. It was cool that n was different for various students, but n stilled tied to the original n-date the student picked. That was much more fun when n wasn’t 1! When the students would each get their respective n values, I said to them individually, “where have you seen that number before?” Their eyes lit up when they realized it was their original n value. Any opportunity to link the concrete to the abstract is a win.

Students then worked with squares with only one number and completed the rest of the squares relative to the given square’s value according to patterns that would occur on a calendar.Screenshot 2016-02-06 16.32.17

For the final assessment activity, I had the students solve missing squares for a calendar that was from the fictional planet Crayon…they only have 5 days in a week on Crayon! (I didn’t even steal that part from anywhere, btw!)


That was probably the best two consecutive days if school so far this year in math 8.

I have tear-stained pages of lessons that I tried this week using geo boards to create linear equations and simple systems of equations. That seemed like a great idea. I extended that to equations with one solution, no solution, and infinitely many solutions. The students weren’t nearly as impressed as I was, so I’ll try to polish that up and post geo board-graphing at a later date.