What Is All the Fuss About Lines?

Yesterday, I led a webinar that demonstrated how Sketchpad can be a powerful tool for exploring Common Core algebra topics. My examples included solving for unknowns with a pan balance, exploring the slopes of lines, maximizing the area of a fixed-perimeter rectangle, and graphing trigonometric functions. I touched only briefly on each example during the webinar, and so here I’d like to return to the algebra of lines.

I’ve taught algebra and pre-calculus courses to college students, and without fail, they stumble when asked to write the equations of lines. In fact, I’ve seen students deal more successfully with the properties of trigonometric functions—their amplitude, period, and phase shift—than simple lines. What’s going on here?

I think that the problem stems in part from the terminology and algebraic machinery that accompanies students’ introduction to lines. First, students learn the formula for the slope of a line. Then, they are introduced to the point-slope form of a line and the slope-intercept form. Knowing when to use each form and doing the necessary algebra soon makes lines feel incredibly difficult. Does it really need to be so complex?

I think the answer is no, and one way to convince you is to share the following problem: Given a point A at (5, 3), how many lines can you name in under a minute that pass through it? Now wait, you might say, setting up those formulas takes some work! A minute is not much time.

But put aside what you know about formulas and procedures and think logically about what the problem is asking. In the interactive model below, you can change the three numerical boxed values a, b, and c. Give it a try and let us know your technique. I’ve seen calculus teachers think hard about this question before having an aha! insight, so don’t worry if you need to ponder for a while.

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