Some of my Geometry students are wrestling with being able to accurately write linear functions given information about points and slopes. I am struggling with how to help them overcome this and I have been doing quite a bit of thinking about how we teach this and what kind of sense it might make to my students. I know that I have these fundamental ideas in my head – an equation is a relationship between the variables stated in the equation (it tells me how to turn an x into a y or vice versa) AND the graph of an equation is the set of points that makes the equation true. I know that I say these things and I am fairly certain that previous teachers have said similar things. I know that my students have repeated some of these things and they can (periodically) carry out these operations. Where the mystery lies for me is why this skill can only be inconsistently displayed and I have a couple of thoughts. I am interested in any wisdom you have about this question.
Almost every single one of my students prefers the slope-intercept equation of a line to any other form. Partially because of my history as a Calculus teacher and partially because I favor more direct problem-solving approaches, I am an advocate for the point-slope equation. I consider it a minor success that most of my students answered the first question on their recent quiz in this form. Here is the question: Find an equation of the line that passes through the points (3 , 1) and (5 , 4). Now, I am careful to ask for an equation rather than the equation but I do not know how much of an impression this might leave on any of my students. Almost every Geometry student answered this correctly and most left it in point-slope form. I was pleased. The next question was this one: Find the coordinates of the following points on the line you found in problem #1.
- The x-intercept.
- The point with an x-coordinate of 1.
- The point with a y-coordinate of 1.
Here is where things started to fall apart for many of my students. I have been thinking about the mistakes I see in class and on assessments and it occurs to me that there might be a fundamental problem that I do not know how to solve. When a student wants to write the first problem in slope-intercept form I instruct them to first find the slope, then replace the x and y in y = mx + b with coordinates of either given point to find the b value. I tell them that this way is harder, but many want to hold on to that equation form. If they want to approach the first problem with the point-slope form I tell them to first calculate slope then replace x1 and y1 in the equation y – y1=m(x-x1) with the coordinates of one of the points they know while leaving the x and the y alone. I am embarrassed that this inconsistency has never jumped out to me before, but why is it that in one equation we leave the x and y while in the other we replace the x and the y with coordinates of a given point?!???!? I have to imagine that some of my students are absolutely baffled by this inconsistency. I wish that they could verbalize that sense of confusion, but I just now figured it out for myself, so why should they be able to lock in on this? So, dear readers I implore you – help me figure out a better way, a more logically consistent way, that I can help direct my students. This is not an intellectual task that is beyond any of them, but I have to guess that a handful of them are so tired of being asked to do this and have sort of given up on the idea that they will ever master this concept. It is way too easy to just write it off and hope it will go away. I do not want this to be their reaction and I want to see them reliably be able to answer these questions.