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.

Busy Days

My last post was about a professional development conference I attended and presented at. Last week I went to another and presented again! In between, we had grandparents’ day at our school, so there are a couple of things I want to share today.

Last Friday I attended the Biennial Conference of the Pennsylvania Association of Independent Schools. It was hosted at the lovely campus of The Episcopal Academy in a Philadelphia suburb. I attended two sessions and presented my MTBoS love song at a third session of the day. When I presented at ECET^2NJPA I had a small, but engaged, crowd. At PAIS I was fortunate enough to have a full room with some people sitting on the counters. We had a lively conversation and one person in particular had some great questions. Recently, NCTM issued an editorial statement about the importance of curricular coherence. It can be read, in part, as a warning against using open source curriculum without deep and careful thought about how it all fits with what you are trying to accomplish in your classroom. My presentation focuses on my journey through the MTBoS and how the resources shared by our community helped inspire me to take on the task of writing a text for our school. It is a text that I hope represents some important values in our department. The text challenge I tackled was for our Geometry course and I have to admit that I felt a certain amount of freedom that I might not have if I was writing for Algebra I, Algebra II, or Precalculus. Those courses that are more in a direct vertical relationship with each other feel like they bear more weight in terms of coherence with each other. There is also more of a feeling that these courses depend directly on each other. I mention this because of a great question that came my way about this. One of the members of the conversation directly asked me about the decisions I made regarding the course content and I had to admit that I probably would have been more intimidated if I had tackled one of these more ‘core’ courses in the high school curricular stream. I felt that we had a really good conversation in the room about incorporating different activities into the classroom. Since the audience were all members of independent schools they probably have a little more flexibility than many of our public school friends have in terms of deciding what resources to incorporate into their classrooms. I have now made this presentation three times for three different organizations and I will probably put it to sleep now, but I am glad that I have had the opportunity to engage in this conversation and to spread the word about the deep well of resources that is the MTBoS.


Last Wednesday, two days before the conference, our school hosted our annual grandparents’ day. I have often been teaching Calculus in the afternoon and this rarely brings many grandparents into the room. This year I end my day with my Geometry class and we had about a dozen guests in class. I found a lovely activity at the Nrich site and my students and our guests had a terrific conversation tying together Cartesian coordinate plane ideas, transformations by vectors, and the idea of being able to project ahead in a sequence. I have been including problems from the Visual Patterns website and I think that Wednesday’s activity might have been a bit of a breakthrough. The conversation we had, and the inclination to want to show off for our guests, was more lively and engaging than I anticipated and on last Thursday’s test I saw better performance on the pattern recognition problem than I had previously seen. I cannot recommend Nrich too strongly. There is a wealth of great problems there and I am using another one for this Friday’s parents’ day visit when I expect to have another crowded room. One of the grandparents was here from California to visit her grandson and she stayed after class to chat and ask for a photo with me and her grandson joining her. I was flattered by her words and by the fact that she wanted to have this memory.