Wednesday, January 28, 2015

Equations of Lines (Day 101)

Equations of lines don't appear in the U of Chicago geometry text at all. Instead, we go back to U of Chicago's Algebra (that is, Algebra I) text. There, equations appear in the following sections in Chapter 8, "Slopes and Lines":

Section 8-4: Slope-Intercept Equations for Lines
Section 8-5: Equations for Lines with a Given Point and Slope
Section 8-6: Equations for Lines Through Two Points
Section 8-7: Fitting a Line to Data
Section 8-8: Equations for All Lines

Obviously, in Lesson 8-4, the equation y = mx + b appears. But as it turns out, the point-slope form doesn't appear in the U of Chicago at all. Instead, the slope-intercept form is used to find equations in both 8-5 and 8-6. Then again, I notice that most students have trouble remembering point-slope, so we might as well teach only the slope-intercept form. In Lesson 8-8, the standard form Ax + By = C of a linear equation appears.

A question that is often asked during this lesson is, why so we use m to denote slope? One urban legend is that it refers to the French word monter, meaning to climb. An English cognate of this word is "mountain," and of course mountains have slopes. The problem with this theory is that there is no evidence that the mathematician Rene Descartes ever used the letter m. One would think that if m had a French origin, Descartes -- you know, the French creator of the Cartesian plane on which slope is usually measured -- would have been the first to use it. A discussion appears at this thread:

Notice that John Conway -- the mathematician I previously mentioned as an advocate of the inclusive definition of trapezoid -- is a participant in this 20-year old thread. (That's right -- when Conway wrote in this thread, I was myself a young geometry student!) Conway suggests that m may stand for "modulus of slope." One teacher tells his students that m stands for "move" and b stands for "begin," since this is how students learn to graph lines in slope-intercept form.

The origin of b is little more well-known. It refers to the idea that a is the x-intercept of a graph, and b is the y-intercept. The equation of an ellipse centered at the origin uses a and b in this way -- but if the ellipse is translated so that its center isn't at the origin, a and b no longer stand for the intercepts.

Finally, back when I was student teaching, my students came up with their own mnemonic for the slope-intercept formula. The letters y = mx + b stand for "your mom's ex-boyfriend."

Slope-Intercept Property:
The line with equation y = mx + b has slope m and y-intercept b.

Once we have the slope-intercept formula, we can algebraically prove that its slope is m and that its y-intercept is b. The y-intercept is the point that lies on the y-axis -- and as we saw yesterday, this is the point whose x-intercept is 0. Setting x = 0 in the slope-intercept formula gives y = b -- that is, the y-intercept is b. To find the slope, we let (x_1, y_1) and (x_2, y_2) be two arbitrary points on the line:

(y_2 - y_1) / (x_2 - x_1)
= (mx_2 + b - (mx_1 + b)) / (x_2 - x_1)
= (mx_2 + b - mx_1 - b) / (x_2 - x_1)
= (mx_2 - mx_1) / (x_2 - x_1)
= m(x_2 - x_1) / (x_2 - x_1)
= m. QED

Finally, we can show how the standard form Ax + By = C is also the equation of a line. We notice that if B = 0, we have the horizontal line x = C / A. Otherwise, we may divide by B:

Ax + By = C
        By = -Ax + C
          y = (-A/ B)x + C / B

and so we have a line with slope -A / B and y-intercept C / B. QED

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