Completing the Literal Square

An interesting post comparing polynomial division methods on the More Than a Geek blog reminded me of my own experience teaching completing the sqaure.  There were times in my career where I absolutely dreaded teaching this method, and tried my best to dance around it.  Now, my attitudes have changed, as mastering this method and looking at it in different ways provides so many interesting avenues for analyzing a quadratic function.

The post asks if the “box” method for teaching muliplication of binomials could be extended to completing the square, and I am happy to report that it can!  In fact, using binomial boxes may reach more visual learners and let them complete a square in a more literal sense.

First, some reminder of the “box” method for multiplying binomials (Note, don’t EVER call it FOIL!!!).  This method allows a more visual approach of the double-distributive property, and a visual organizer for students who require this level of structure:

Mult Binomials


We start with a quadratic, which we are interested in converting to vertex form:

STEP 1: shove that -9 out of the way, and set up a binomal multiplication box:

Step 1

STEP 2:  Fill in the x-squared box, and put half of the b term in each of the x-term boxes:

Step 2

STEP 3:  Now, quite literally, we need to complete the square by filling in the last box.  Also, since we add 25 to right side, we must also subtract 25 from the same side.

Step 3

STEP 4: A little housecleaning, and we have our quadratic in vertex form.

Step 4

My ideas for this are hardly unique. Check out these great blogs for more completing the square action:

Finally, enjoy a quick video where I walk through the box method, with a few stickier examples.


By Bob Lochel

HS Math Teacher. Hatboro-Horsham School District, Horsham, PA.

4 replies on “Completing the Literal Square”

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