Completing the Square

Completing the Square always seems to be one of the toughest concepts for Algebra 2 kids to do and understand.  I always seem to be looking for a new way to tackle it with my kids.

A few years ago I started with the vertex form of the equation – teaching the transformations and showing my kids how to write the equation in vertex form by finding the vertex on the calculator.  While I hate teaching calculator “methods” before the algebraic methods this did have a benefit.  Pretty quickly my kids saw the pattern between the middle term (b) and the x-coordinate of the vertex and then they figured out that they needed to subtract the square of that from the constant to get the y-coordinate.  At least it worked pretty well when the leading coefficient was 1!  When we had an “a” value that was more than 1 it was a bit more problematic.  They saw the pattern but it was harder.  I showed them algebraically what was happening but to say they weren’t buying it was an understatement.  The idea of a perfect square trinomial just seemed hard to grasp even though we had done factoring and looked at the pattern before.  With no other way present I stuck with this one.

Then, a few months ago I posted about using boxes to help students with Polynomial Long Division.  At the time I asked the question – “Could we use this for completing the square?”  Bob (@bobloch) replied with a great post showing how to use the boxes for completing the square.  I just recently taught this concept in my Algebra 2 Honors class and my kids loved the boxes.  I started by showing them how that might work with a = 1.  They got it!  Made sense to them.  Then we went to a = 2 and I actually had them make 2 boxes.  Then I went to 3.


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That’s when one of my kids brilliantly asked – “What happens when a = 40?  I don’t want to make 40 boxes.”  So we had a great discussion about how to make one box but multiply it by our coefficient and several kids actually commented on how that helped them see what was going on in the algebraic form. Whoo Hooo!  Win!

Now for the crazy part – as I was discussing how this lesson show go with my student teacher, she said, sorta jokingly… “Since they understand the ones where a = 1 its a shame we couldn’t do something like slide and divide with this. ”  (I just learned about slide and divide for factoring about two years ago – amazing! Found a proof online too!)  Well, this sounded like a challenge to me so I tried it.  With some weird tweeks it works and I can explain why I think it works but I can’t prove it yet.  See what you think.  Anyone will to try to tackle a proof?

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