Experience First

In this lesson students think about the process of completing the square visually and conceptually. We do not emphasize traditional algorithms but instead have students make area model diagrams to represent the process of completing the square and what this truly means. We use visual algebra tiles in this lesson but if you have the concrete manipulatives, feel free to get them out!

​

The lesson starts with a gentle introduction to expanding binomials using the area model. At this point student should feel comfortable remembering the linear term in the expansion but we use the area model anyway to highlight how x^2 really represents  the area of a square with side length x, that x represents the area of a rectangle of dimensions 1 by x, and that the constant term represents individual unit squares. This understanding is critical for what we do later in completing the square. 

​

The front page should help students make connections between the dimensions of the big square and the terms in the standard form of the quadratic. Before moving on to page 2, make sure students understand how half of the x-bars are used in the length of the big square, and half the x-bars are used in the width of the big square (since a square must have equal sides). Also make sure that students connect perfect square trinomials with the idea of having a vertex on the x-axis (meaning it can be written as a binomial squared and there is no shift at the end). 

​

On the back page, students expand to looking at non-perfect square trinomials and consider how many unit squares are left-over from the perfect square or how many unit squares are missing from the perfect square. Finally they are left to puzzle over Bree’s strategy for visualizing a quadratic where the leading coefficient is not 1. Her approach will be generalized to other quadratics of this type in the Check Your Understanding.

​

Formalize Later

Although drawing out the area model might seem cumbersome, students need access to these rich ideas through concrete visuals first. Over time we expect students to switch over to a simplified diagram (where the x bars are not individually drawn but just represented) and eventually they will likely not have to draw the diagram at all. This process of moving from concrete to abstract takes time for students, and often we rush them to the algorithm before students are ready for it.  Consider how the availability of a concrete model can help your struggling students but also your advanced students in gaining deep understanding instead of replicating rote procedures.  Furthermore the option to move to more symbolic representations when a student is ready is a way to differentiate your instruction.