Linear Functions

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.

8 thoughts on “Linear Functions”

  1. Hmm. Let me think “aloud” here.

    You could make the way of finding the eqn in point-slope form the same as slope-intercept. The eqn is y-g=m(x-h) for some g,m,h. Find m and then sub in both points for x,y to find g,h. The problem is, of course that there is no one answer for g,h. in fact, what they’ll get from both is g=mh + b, which may just be more confusing!

    Personally, I can never remember what the point-slope form looks like and I always figure it out geometrically. That is, I always draw a line with (to use your example) the points (3,1),(5,4) and (x,y) on it and say to myself “the slope between any (x,y) and (5,4) is the same as the slope between (3,1) and (5,4) so (4-y)/(5-x)=(4-1)/(5-3)”. Then I rearrange from there. Perhaps always focussing on this geometry might help?

    I wonder what would happen if you went through doing the slope-intercept form geometrically? That is, you find the slope. Then figure out how far it is to the y-axis by reasoning like this: “To get to the y-axis from (3,1), we have to move -3 in the x-direction. Since the slope is 3/2 that means going -3*3/2 in the y-direction, so the y-intercept must be 1-3*3/2=1-9/2=-7/2.”

    Perhaps a big table comparing geometric and slope+sub-in approaches for both might make it all a bit clearer.

    I also wonder what happens when you ask for eqn of line through (0,5) and (4,3)? Can they notice that they know the y-intercept already and just go straight to slope-intercept? Perhaps a discussion of when you can “go straight to” and when you need more work would help? The point-slope form you can go straight-to more often than the slope-intercept.

    Incidentally, with the “two-intercept” form x/a+y/b=1 (where a is the x-intercept and b is the y-intercept) it’s REALLY rare to be able to go straight-to and hard to do by subbing in points, and could provide a really interesting comparison.

    So there’s some thinking aloud. Don’t know how useful it is, but thanks for listening.

    1. David
      This approach reminds me of what I told my students during the few times that I taught Algebra I. I would tell them that I wanted them to think of the slope as the number representing how much y changes overtime x changes by 1. I have tried to avoid the whole ‘rise over run’ business in large part because I do not know what ‘run’ is. I also think that tying my idea together with yours of the physical reference to the change in coordinates would be super helpful. This also addresses a concern shared with me by a colleague that he wonders just how deeply ingrained is the idea that the graph of a function is that set of points that makes the equation of the function true. I think that I want to make more of a big deal of this and your suggestion of going back to fundamental principles in building a table might be a helpful way to do this. So, I think that when I cycle back around to this – and I am certain that I will – I will try to shore up the foundations with more representations (tables, sketches, equations, physical slope walk offs on the graph) and I will also try to point out the fundamental similarities – rather than the differences – between the equation forms. I worry about introducing the two-intercept form for fear of shaking up some already shaky foundations, but I need to think more about this.

      Thanks for jumping in on this conversation!

      1. I think the idea that the graph of an equation being the set of points making the equation true is not very well ingrained at all. The inability of my university students to answer questions like “is (1,3) on this line?” is testament to that! I think there could be a few more of that sort of question all the way through, as well as “give three points on this line”.

        The table I was referring to was more like this:

        Style of equation: slope-intercept, point-slope, general form, two-intercept

        Slope appears directly in equation: yes, yes, no, no

        Y-intercept appears directly in equation: no, yes, no, yes

        X-intercept appears directly in equation: no, no, no, yes

        Lines missing: vertical lines missing , vertical lines missing, no lines missing, vertical and horizontal lines missing

        Finding from two points: …

        Finding from point and slope: …

        1. Aha – a more sophisticated version of a table than the good old table of values. This set up reminds me a bit of a calculus approach to talking about clues regarding the graph of a function based on derivative info. This would be a sophisticated way to discuss this with my Geometry gang.

          Thanks for the idea.

  2. I have always tended to use point-slope form to find the slope-intercept form, when given two points, or a point and a slope. Perhaps helping students become more comfortable with using and manipulating the point-slope form would help.

    As for intercepts and specific points, I have understood that intercepts are simply the points where the other coordinate in 0. That is to say, the y-intercept is the point at which x=0, and the x-intercept is the point at which y=0. Knowing this, and having an equation of the line, in any form, allows for substitution of the known coordinate, (in the case of an intercept it is either x=0 or y=0) to find the value of the corresponding coordinate.

    Of course, this approach depends upon understanding all the parts of a linear equation, and what it means to be a function. I have worked with other students who struggle to visualize slope, intercepts, and specific points on a line. I believe more practice in identifying an equation based on a graph, or a graph based on an equation, or a graph or equation based upon two given points, will lead to more comfort in that regards, and confidence in the ability to pick out specific details of a line.

    Hope that makes sense and addresses some of the issue.

    1. Roy – I think that part of the battle is that for too many people the difference in notation between the two equation forms equals a difference in meaning between them. Where I see that they are the same equation in slightly different form – and I know you can see it this way – too many people are intimidated by the abstractions involved in jumping from equation to graph and back. Certainly more practice tying together these different representations will always help. I also like the idea you present that, instead of suggesting that we leave the point-slope form alone, we take the extra step or two to convert it into the more comfortable slope-intercept form. I may win more converts this way instead of leaving my line equations in a form that looks incomplete.

      Thanks for dropping by to join the conversation!

  3. I used to teach finding the slope intercept for the way you described. However, I have found that if they default to using point-slope, and then transform the equation into slope-intercept form, they are more successful.

    Another issue I run into is the fact that standard form for a linear function is not the same as standard form for any other polynomial function. My algebra students have a hard time wrapping their heads around this. This year I began by teaching the linear function y=x as the parent function and then used transformations. This way I can follow the same format for quadratic functions, and in Algebra II for cubics, etc. I was trying to find a way to show them that linear functions follow the same rules and are not inherently unique.

    1. Meg

      Thanks for jumping in the conversation and my apologies for being so slow to respond.

      I like your idea of starting in one place but encouraging the conversion to a more familiar, or more ‘complete’ form of the equation. Another thought I had, probably way too picky for my Geometry students, is converting y – h = m(x – g) simply into y = m(x – g) + h as this form matches with vertex form and other forms they will encounter in Precalculus. I don’t think that the possible gains outweigh the additional confusion that this is likely to cause.

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