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Teaching Point-Slope

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Kent Haines asks this great question

and the threads are deep and many!

This question is important at a pedagogical level for a lot of reasons. As Kent observes, \(y = m(x-h) + k\) leads to a more natural introduction of the vertex form for parabolas \(y = a(x-h)^{2} + k\). Knowing how students are going to see math content moving forward is a great way to rationalize an approach to teaching certain content. In the case of linear equations, we have a bunch of options. Namely we have at least

  1. Standard form: \(Ax + By = C\) leads naturally to the format for conic sections once we normalize \(\frac{x}{a} + \frac{y}{b} = 1\). For example, ellipses are introduced as \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).
    • Or we could just go with the generalized standard form of \(Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0\)
    • And further, the form of \(Ax + By = C\) more easily develops the introduction to Linear Algebra via solutions of systems of equations. In particular Algebra students are taught some basic methods for solving a linear system of two equations and two unknowns. And if you’ve ever had the joy of teaching this, many students cannot cope with the fact that solving a 2-by-2 system is a matter of determining where two lines intersect. That difficulty may very well be tied to the form students have seen lines expressed.
  2. Point-slope form: \(y – y_{1} = m(x – x_{1})\) and Kent’s modification to “vertex form” \(y = m(x – h) + k\).
  3. Slope-intercept form: \(y = mx + b\) which leads to a natural segue to polynomials (in form), \(y = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + \underbrace{a_{1}x + a_{0}}_{\mbox{linear}}\)

Of course, all forms are equivalent and all the typical information can be teased out. I must interject myself with a small caveat: the form of \(\frac{x}{a} + \frac{y}{b} = 1\) is a bit restrictive than the other forms since lines passing through the origin cannot be properly represented, lest we divide by zero. But barring this, each form can be manipulated into the other form. So, is it better to teach one form over another?

I can understand the rationale underlying all the arguments above regarding natural segues into later topics. However, I am more aligned with David Radcliffe’s comment

And as a de facto starting point, I go with \(y = mx + b\). If we are to argue that there is a natural development of topics, then I would argue that slope-intercept would be a more natural choice than point-slope. Slope-intercept has an “input-output” feel and is far easier for plotting and more natural for programming. In that way, slope-intercept has broader general use than point-slope.

I can’t really be swayed by the vertex form argument for several reasons in no particular order:

  • I don’t want to confuse students that the “vertex of a line” is the point \((h,k)\). This gets murky because the vertex of a parabola has specific geometric interpretations.
  • While I can see that the idea here is to show that we have a line that is shifted horizontally \(h\) units and vertically \(k\) units, this becomes a tougher visual sell when comparing two parallel lines in “vertex form”. \(y = mx + b\) states explicitly that we have a shift of \(b\) units from the line \(y = mx\). Or equivalently in “vertex form”, \(y = mx + b\) becomes \(y = m(x – 0) + b\).
  • The continuity in content is limited with point-slope or “vertex form” in that while it may be easier to have a discussion about vertex form for parabolas, that is about the extent of its use. Maybe we see this again in a similar form in a Calculus course, but by then students have been taught more generally about translations of functions.
  • While it is true that for specific problem types point-slope may be an easier set up, one can argue that for specific other problem types other forms are easier. And I can’t see a certain problem type dominating the curriculum.

I don’t particularly care to introduce linear equations as \(Ax + By = C\) for a variety of different reasons. These reasons mostly center around the fact that at this stage in a student’s math education, they are just becoming familiar with the idea of input and output relationships (independent vs dependent) and they haven’t yet had enough exposure to algebraic manipulation of equations. The general form is more advanced and better saved for later. The part that is typically missing when introducing the general form is in getting students to see that this is a rewrite of \(y = mx + b\) and to reframe the conversation about independent vs dependent, especially in the context of systems of equations.

As such, I prefer to start with \(y = mx + b\) emphasizing the meaning of that particular form, how we can obtain other forms as needed, how we can arrive back at slope-intercept as needed, and what information we can glean from the form given. Ultimately, form shouldn’t matter, but if we had to pick something to start with first, I would go with slope-intercept.

Your thoughts?