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?