*By Linda Richard, Curriculum Writer*

I used to teach my high school students a catchy song to memorize the distance formula. We all had fun goofily singing this song. My students hummed it to themselves during tests and successfully calculated distances. I was pleased with this outcome—but what did my students actually *understand *about distance in the coordinate plane? In retrospect, very little.

Why did I need a song? Well, to my students, the distance formula was an arbitrary conglomeration of operations, variables, and subscripts. Their success on distance problems hinged entirely on their ability to memorize this conglomeration. If they forgot any part of the formula, they had no way to recover.

Even in years when I provided the formula for my students rather than requiring memorization, the skill they acquired was that of substituting numbers into a formula, which wasn’t the goal I had for my learners. The formula didn’t deepen my students’ understanding of the coordinate plane. It didn’t add to their conceptual framework, or leave them with lasting understanding. If anything, it deepened their suspicion that math is a set of disconnected formulas to memorize, churn through, and forget.

In *Illustrative Mathematics’ Geometry *course, distance calculation is presented as an application of the Pythagorean Theorem. To find the distance between two points, they draw, or imagine, a right triangle whose hypotenuse is the distance between the points. The legs of the triangle are vertical and horizontal lines extending from the two points.

Students begin with integer-coordinate points for which they can find the lengths of the legs by counting squares on the grid. When presented with more challenging coordinate values, they conclude they can subtract coordinates to find the leg lengths. Then, the values are substituted into the Pythagorean Theorem.

$a^2 + b^2 = c^2$

$(x_2 – x_1)^2 + (y_2 – y_1)^2 = d^2$

No formula is necessary, because the triangle can always be used as a tool. As students build fluency through repeated calculations, they may themselves develop a formula to provide efficiency in their work, but can always rely on their conceptual understanding to see them through.

This framing of distance lends itself naturally to further extensions. For example, a circle is defined by distance: it’s the set of points $(x, y)$ that are $r$ units from a given center, $(h,k)$. To develop the equation for a circle, we can draw a triangle, label the sides, and apply the Pythagorean Theorem.

$a^2 + b^2 = c^2$

$(x-h)^2 + (y-k)^2 = r^2$

A similar process can be used to write equations for simple parabolas given a focus and a directrix.

This is an example of the IM curriculum’s work on *extension* and *connection*. Students see that the Pythagorean Theorem is a simple, powerful idea threading through many grade levels, connecting geometry and algebra. Calculating distances and writing equations for circles and parabolas aren’t *new *topics; rather they are extensions of a concept students already know. Students are invited to see math as a set of connected ideas that *make sense*.

**Next Steps**

- Where else can we deemphasize memorization in favor of sense making?

##### Linda Richard

Linda Richard teaches mathematics at Cascadia College in the Pacific Northwest. She has taught high school mathematics, written math curriculum, and run teacher professional development programs. Linda helped pilot and roll out Washington State’s Bridge to College Mathematics course, designed to help students transition successfully to post-secondary math. She loves finding new ways to connect all students to the beauty and usefulness of mathematics.