One of the most challenging topics to teach in high school mathematics is “Completing the Square.” This is because educators do not always fully understand the topic and because it includes several algebraic steps that are incredibly challenging for students.

An eye-opening experience was when I first generated visuals for each step in completing the square using an area model.

Algebraic StepVisual Model
x² + 6x + 8A visual showing x² + 6x + 8 using a algebra tiles.
x² + 6x + 8 + 1 – 1A visual showing x² + 6x + 8 + 1 - 1 using a algebra tiles.
x² + 6x + 9 – 1A visual showing x² + 6x + 9 + 1 using a algebra tiles.
(x + 3)² – 1A visual showing (x+3)² -1 using a algebra tiles.

The name “Completing the Square” is not arbitrary! Visually, one can see that we are literally taking an incomplete square (at least in cases like the one above) and making it complete. This visualization makes the algorithm’s goal obvious and helps students see what they are trying to accomplish.

However, the steps above are not sufficient for students. The area model above is much easier to understand if you already know the mathematics it represents. Since children don’t, we need to introduce the area model with more straightforward examples before using it for more complex ones. Here’s a worksheet that aims to do this.

A short worksheet showing more examples of area models being connected to algebraic models.

Here’s the link to the full worksheet.

Once we have used the area model to establish the purpose of the steps when completing the square, we gradually remove the visuals. This is because we don’t want students to need to draw out the visuals each time to help them solve the equation. The goal is for students to understand conceptually what completing the square is and what steps are needed to complete it. The visuals are an aid for this goal.

There are other cases to consider (for example, expressions like x² + 6x + 10 and 2x² + 6x + 10). Once students have a handle on the simpler cases, these examples will be easier for them to manipulate algebraically.

The key idea here is that some ideas are much more obvious when represented visually than when we focus purely on a symbolic approach.