Area Models and Completing the Square

I’m nearing the end of my time with my 9th graders, with this week dedicated to moving beyond factoring as the sole method for solving quadratic equations and towards more general methods like completing the square.

Late in May, David Wees shared materials which challenge students to investigate the relationships between “standard form” and “completing the square” form (aside – does anyone agree on proper terms for these?) using area models to build representations.  Given that I use area models often to introduce polynomial multiplication, I was eager to maintain consistency in the student understanding.


But before we dove into David’s lesson, I wanted students to revisit their understanding of area models.  In this Desmos Activity Builder lesson I created, students shared their interpretations of area models and worked in pairs to investigate non-square models.  In one of the final screens, students argued for the “correct” interpretation of a model.


Using the Desmos teacher dashboard, we could see clear visual arguments for both representations.  This was valuable as we ended the lesson for the day, and tucked that nugget away for Monday, when we would begin to formalize these equivalencies.


After the weekend, students worked independently through David’s Completing the Square lesson. Not only did students quickly move through the area models and the dual representations, the debates between students to explain how to move from one representation to the other were loud and pervasive.  I’m also loving how many of my students have started to use color as an effective tool in our OneNote-taking (below).


At the end of the sheet, all students completed problems which translate standard form to vertex form with no support from me (“no fuss…no muss”).  It dawned on me that something amazing had happened….my students had figured out completing the square without my ever talking about completing the square.

Tomorrow we’ll tackle those pesky odd-number “b” terms, but my students own this already!

7 responses to “Area Models and Completing the Square

  1. Nice work! I wonder if James Tanton’s quadatics course would offer any insight on tackling the odd b or non square a’s. This work with expressions and his work with equations would require different adjustments.

  2. I have just done a post on your stuff.

    As far as the non square a is concerned a change of variable is in order.
    3x^2 + 8x + 8
    multiply everything by 3
    3^2x^2 + 8*3x + 24
    replace 3x by y
    y^2 + 8y + 24
    rebuild as square
    (y + 4)^2 + 8
    replace y by 3x
    (3x + 4)^2 + 8
    divide everything by 3
    and don’t give up yet.
    Case for the formula !

    Or you could divide by 3 at step 1!

  3. Love this. I did something similar with area model for polynomials and rational functions. Combining the area model with algebra tiles really added a lot.

  4. What about sqr(x-3) + 4

  5. nice blog on mathematics

  6. Bob, the link to the student materials should be updated. Unfortunately my colleague wrote over the original version of my document but I managed to go into revision history and make a copy of it which is available here:

  7. Roxanne Paulsen

    How do you transition this to x^2 + 4x – 6? I have been using algebra tiles to explain this and love this handout. I am hoping to use it today. But, I struggle with how easily they will transition to negative units to start with instead of always having positive ones.

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