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Wednesday, November 2, 2011

Completing the Square

I made some final tweaks to Completing the Square in Algebra 2, and I find it just amazing the difference between this year and previous years, in that so much more often now, I just know what to do.

It doesn't feel like I changed all that much, but the kids just get it. I don't think it's a difference in delivery or anything. Here are the important bits.

First, I took two days instead of one. Go to hell, pacing calendar. The first day is just to see the pattern and get the idea with easy easy problems. a = 1 and b is even. The second day we work with a != 1 and odd values of b (fractions. eep. but the kids are even dealing with fractions okay.)

Tee it up: why would we want to do this? It saves us time.

Look for patterns. The kids fill out this whole table all on their own. I don't say a thing. I convince them to try and focus by telling them that if they really get how this table works, their lives will be a million times easier for the next six months. It's an exaggeration but you need them to engage here.

The bottom three rows were new this year. Hardly any students needed an assist with the * rows. I was surprised. The important part - the mathematics - was the ** row. Again I was surprised that they mostly worked this out on their own. There were some kids, I had to point at numbers, and say "Look at the 10, the 25, and the 5. Look at the 14, the 49, and the 7. How are those related? How can you write that relationship but use b?"

Once we're all on board with the table, we put the pattern together with "the genius method" from before to solve a simple quadratic in standard form:

And that is basically that. We practice a bunch of easy ones. The next day, we come back and practice a bunch of really hard ones.
Here are the smartboard files: Day 1, Day 2.

I just find it stunning that you can plan out a lesson 95% correctly and it will miss most of your kids. And you can change one little thing - add three rows to a table - and now all the kids basically get completing the square, think it's easy, prefer it to other methods of solving quadratics, and tell you why they don't get why this is such a big deal. I feel a little like I have super powers.


  1. Old fashioned I guess, But if a kid can't explain what square he is completing, in some geometric sense, they just don't get it yet....

  2. I disagree. We don't in other contexts have any problem calling a number or an expression to be a square, thinking of a square as "something times itself". We call 4 and 9 and 121 "perfect squares" all the time and don't stop to imagine the quadrilateral. So (x-3)^2 is a square, and x^2 - 6x + 9 is a square.

    A geometric model just aggravates and confuses them and isn't worth the huge time investment.

  3. Also, the square might well be negative, in which case a geometric interpretation hardly makes sense but this whole procedure works just as well.

  4. I take a more moderate view on the geometric interpretation of completing the square. I wouldn't pick one OR the other, but would share both representations (and yes, screw the pacing guide). I know the geometric representation (which I didn't see until I was teaching) allowed me to make sense of concepts that were previously nothing more than procedures.

  5. If you ever feel like hitting them with the geometric way at some point I do have a simple dissection puzzle equivalent to completing the square.

  6. I like the square, but I also like the algebra. I will keep both.

    Occasionally a kid gets square-dependent. There's a reason. And occasionally strong kids like being able to handle both. And that's cool too.

    But if a kid chooses not to stay with the square, no biggie. They've seen it.


  7. I don't like the pictures/geometry, so this method suits me just fine and makes more sense to me than trying to draw out squares. Certainly Kate knows both methods and I'm sure we'd all teach to what our kids need/understand, so maybe this crew is more like me than the visually minded students in other classes.

  8. I took the steps of your "long way" and "genius way" and arranged them in a circle. They are all reversible, after all! Then I said "If you're given a quadratic in this form, do you want to go this way around the circle, in one step, or go that way and do seven steps?"

    I think arranging it in a circle makes the point that we have all these transformations we can do to a quadratic, and we can choose which ones we use and which direction we want to go.

    Also it allows me, a few days later, to draw a diameter of the circle and call it the quadratic formula.

    And of course, they have to sing the song from the famous Disney movie, The Algebra King: "The Circle of Quadratics". Or something like that.

  9. As a horrible math student, turned physics guy, turned physics teacher, turned math teacher, I loved this. I always saw completing the square as such a painful process and have always struggled to find a good reason or method to teach it.

    This gives me some traction as a teacher as well as a better personal understanding of the "why" in "why do we do this?" As a physics guy I so often feel like I missed out on the subtly of why certain math is done. Seeing math as only a tool and not as something elegant and potentially pretty darn clever...

  10. I LOVE this! I just finished teaching this topic, and my students do pretty well with it until they encounter a fraction. I spent one day on the square root property (too easy for a whole day) and another day on complete the square. The second day felt rushed and students freaked out when fractions showed up.

    Next time, I am going to use this and do the non-fraction problems one day and extend the pattern to tackle the fraction ones the next day. Thank you!!

  11. There are quite a few indirect benefits of completing the square.

    For instance, say we have f(x)=x^2-2x+7 which can be rewritten as f(x)=(x-1)^2+6; by inspecting the completed the square form we can deduce that f(x) is always greater than zero.

    Secondly, using the above f(x) description, we can also surmise that it depicts a quadratic curve with minimum point being (1,6)and hence the line of symmetry of the curve is given by x=1; in addition we realize that since the smallest y value is 6, the curve will NEVER cut the x-axis, ie there will be no x-intercepts (which could also be verified by computing the determinant b^2-4ac to be less than 0 or that the curve is absolutely positive as shown in the previous paragraph). Armed with all these information, you can quickly sketch the graph of y=x^2-2x+7.

    Just sharing, hope it helps. Peace.

  12. Joshua - I really like your circle idea. It is a great way to visualize the equivalence of different methods. I have found that just showing the two methods result in the same solution doesn't always convince students that the methods are equivalent. Another plus with your idea is that it provides an easy opportunity to talk about going backward. "What is an equation that has solutions of x = 1 and x = 4."

    Mr. Koh - I agree, it is always a good idea to reinforce the connections to other ideas. My experience is that students are usually open to these connections, but not necessarily on the days that we are practicing the skill of completing the square. I like to separate conceptual days from procedural days. Just what works for me.

    Kate and other geometry model detractors - I agree that the model can be time consuming. However, it helps if the model has been used for more than just that topic. The geometric model can help students understand factoring (or if used properly even linear equations). It also helps if instead of trying to draw the picture out students use algebra tiles to get the hang of the ideas.

  13. Good stuff Kate, I've always liked how you use recognizing patterns as a means for your kids to develop understanding. That's a fantastic life skill that will transfer for them.

    In this instance I'm in favor of both the algebra and the geometric representation, and I disagree with, "...just aggravates and confuses them..."

    Wearing my school leader shoes, I'd say that if a method can be successfully used to clarify something for one group of students but aggravates and confuses another group (students of the same level) then it's important to examine the presentation of the method. If a strategy can not only help explain a topic for a student but also lead to greater depth of understanding I have a hard time believing it's not worth the time investment.

    Which leads to my math teacher shoes. We don't use completing the square because it's easier to solve quadratics than factoring. We use it as a means of taking the square root of a quadratic that wasn't at first glance a perfect square. We also use it as proof of the quadratic formula. In both instances it is well worth the time investment of using a geometric model, which in my experience has always led to greater understanding.

  14. Nice approach - makes it very clear how simple the idea really is.

    I think a geometric sqaure approach or rectangle is great for an Algebra I intro to factoring. I prefer Kate's approach for the Algebra II classes I have seen.

    We are often looking at things in a functional way, height of a ball, etc., so introducing a geometric interpretation really muddies the water.


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