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Habits of Mind: Equations as Point-Testers

December, 2010

Over years of classroom teaching, our lead developer Al Cuoco saw students who were very skilled at manipulating and graphing equations missing out on a critical relationship between equations and graphs. They didn't know that the equation is a point-tester for the graph. Without this understanding they were stumped by what should be simple questions, such as whether a certain point is on a particular graph.

For example, a student asked to graph the equation:

might be perfectly capable of doing the algebraic manipulation necessary to change the form of the original equation to this special form that makes it easier to graph:

They might be able to sketch a very accurate graph of the equation:

But then, when asked whether the point (7.5,3.75) is on this ellipse, most students wouldn't know how to proceed, except to say that it looks like it might be. In the CME Project curriculum, students learn early and often that the equation is a point-tester for the graph. This means that if the coordinates of the point satisfy the equation, the point is on the graph. If the coordinates don't satisfy the equation, the point is not on the graph. If you ask a CME Project student whether the point (7.5,3.75) is on this ellipse, they'd substitute the coordinates of the point for x and y in the equation of the ellipse:

Since the result is not equal to 1, this point is not on the graph of this ellipse. This robust connection of the equation with its graph is emphasized throughout the curriculum, and serves students well as they attempt to answer questions and solve problems with graphs.