Sunday, September 29, 2013

When Is the Right Answer the Right Answer?

This week, my students have been working on determining equations of a line based on properties of parallel and perpendicular lines (GRE 604 from the ACT College Readiness Standards for Mathematics), which involves problems like this one:

Several concepts popped up throughout the week while working on this skill: determining slope, slope-intercept form, point-slope form, and the relationships of slopes between lines that are either parallel or perpendicular to each other.

Throughout the week, I have been insisting that my students give their solutions to these problems in slope-intercept form, as shown in this student's work:

(There are some other things going on here that would also be interesting to talk about, but that will have to wait for another day.) 

Perfectly reasonable solution method, isn't it? Put the original line equation in slope-intercept form, determine the slope, use point-slope form to get the equation of the parallel line, and then solve for y to put that equation in slope-intercept form.

This morning, I found myself wondering why I was insisting on having my students put their answer in slope-intercept form.

Is it really necessary? I mean, couldn't the student have just stopped at point-slope form and still been correct? I mean, plug a few things into Desmos and it's hard to argue otherwise:

I've been thinking about this and struggling with this all morning. The focus of this particular ACT skill isn't necessarily for students to determine the equation of line and put it in slope-intercept form; the skill is just to determine the equation of a line based on properties of parallel and perpendicular lines.

In the problem above, the student is given the equation of a line and a point on another line that is parallel. The student knew to look for the slope of the original line, knowing that the parallel line they were looking for would have the same slope. After determining the slope, the student created the equation of the parallel line using point-slope form.

Should it stop there? After all, the student correctly applied the properties of parallel lines and determined a correct equation. That's what the skill is all about, right? Why was I insisting that the student put their answer in slope-intercept form? I'm not sure it's necessary, and I think it also creates a situation where the student can make a simple algebra mistake and come up with an equation that is no longer "correct." On the other hand, expecting students to be able to put the equation in slope-intercept form isn't all that unreasonable, is it? After all, the student did just that with the equation of the original line in the problem, in order to determine the slope of the parallel line. Is that a good enough reason to insist on it, though?

This is just one specific case. I know this isn't the only instance in mathematics where something like this happens. When is the right answer the right answer?


  1. I try not to request a particular form unless there is a need. If you ask many students how to write the equation of a line they only know y=mx+b since that's the one teachers ask for. Occasionally having students graph in a calculator that says y= or asking for the intercept are fines ways to motivating writing in slope intercept, but don't insist on it unless you need it.

    1. Thanks for the comments, Tina! Yeah, when I thought about it this weekend, I didn't see a need for my students to put their answers into slope-intercept form. Converting to slope-intercept form isn't really the point of this skill. I had the same thoughts about needing to convert to slope-intercept form on a graphing utility, but we aren't doing that with this particular assignment. My students seemed pretty happy when I told them today that they could just leave their answers in point-slope form. I think it makes sense.

  2. I teach college, so that may affect my answer. I don't think we should force this. I tell my students I like slope intercept form because it tells me so much. (But maybe other forms would speak to me more if I gave them a chance.) I don't take points off if they decide to leave their answer in point slope form. By the time they get to me, most of them are used to slope intercept form, and do put it that way. I worry that they're too used to it, and think it *is* the (only) equation of a line.

    1. Sue, I agree! It's so funny how slope-intercept form gets regarded as "the" equation of a line. I think maybe that's part of the reason why I was having students leave their answers in slope-intercept form: I had that bias ingrained into me when I was a student.

      Oh, my: slope-intercept bias. Is that a form of math discrimination or something?

      (insert joke about discriminant here)

      Anyway, thanks for your comments!

  3. For me I always had to remember that Math is a different language. And like with all languages there are ways to get the point across without perfection, but sometimes perfection is necessary. If my students could recognize the correct equation in slope intercept then I would see that as being evidence of knowledge. I think being able to create equations in slope intercept form fluently show a mastery of the particulars. But, I also taught middle school students. At various levels it applies differently.

    1. I can certainly see what you're saying about recognizing the correct equation in slope-intercept form and being able to demonstrate a certain level of mastery. There are times when it's useful to have students use that form specifically. But like you said, it applies differently at various levels. To me, the focus of this particular skill seemed to be just finding "an equation." After thinking on it, I took that to mean the equation could be given in whatever form the student felt was most appropriate or convenient to use. (I kind of like that, because the student then has to make decisions about what they think is the best way to come up with an equation. That's kinda CCSSMP.1-ish, isn't it?)

      Thanks for your comments!