Tuesday, September 25, 2012

A nasty shock

We're starting to work on coordinate geometry, which should be a review of lines: gradients, intercepts, equations, graphing. Students are expected to know how to work with lines since they've been doing it every year for at least four years or so. Yet I always find that students struggle to use points to find the equation of a line.
This time, I decided to first make sure that students were able to see whether a line passes through two points. I designed a small matching activity: given some cards with equations on them, and some cards with pairs of points on them, match the equations with the pairs of points. After they matched everything that it was possible to match, some odd and some empty cards would remain. I was hoping students would use the understanding they developed/formalized during the matching activity to come up with suitable matches for the odd cards.

You can find the cards here.
Easy as pie? No. It turns out that not one of the students in my class (11th grade) were able to match the equations with the points. They had simply no idea of how the x and y in the equation related to the x- and y-coordinates of the points. What the hell have they been learning for four years?

Some approaches that students tried were:
  1. Getting the gradient by using two points, then comparing this gradient to the one in the equation. Fine, as long as there is just one equation with that gradient.
  2. Graphing the equations ("but we don't remember how to graph lines from equations") and see if they pass through the pair of points. Fine, if they understood how to graph the lines and if the scale of the graph was appropriate.
  3. Making a table of values, to see if the pair of points would come up as a pair of values in the table. Fine, as long as the points have integer coordinates and the person has a lot of time and patience.
All these approaches show students struggling to find a method that works, without really aiming to understand the information given in the question. I blame the students' past textbooks. The one they had last year, for example, presents lines and equations and parallelism and perpendicular lines and how to find gradient and intercept from 2 points - and only then presents how to check if two points are on a line or not. I'm thinking of writing those authors a very nasty letter.


  1. #1) I always love it when you Europeans call it "gradient." It's great. As an IB teacher myself it's weirdly rewarding to see that this actually happens.

    #2) This is super. I am going to use this TOMORROW. Thank you!!!

    #3) I am in love with Sweden. Max Martin and Shellback are my bros and I've had a crush on Nina Persson since I was 16. My wife is part Swede too...sounds like beautiful country!

    Thanks again!

  2. Marshall - well we have to call it "gradient" if we teach IB, don't we? I liked this activity, too, even though (or because) it so clearly showed the complete lack of student understanding of how equations and graphs of functions relate to each other. Let me know how it goes in your class!
    Me, I'm starting from scratch next class. I think I can assume that students know what graphing points means and how to do it, but beyond that I'll try to assume nothing. Of course, finishing the syllabus will be a bit of a challenge (to put it mildly) but I couldn't bear to just ignore such huge gaps in conceptual understanding.
    I love Sweden too... mostly for the wonderful woods and mountains, though, not Max Martin. :)

  3. Arg, I cannot believe students can get to Algebra 2 and still be completely mystified by this activity!!!! Remind me not to do this one anymore, it's making me depressed.

    (Just kidding I'll still do it - it's very good - but I will have to make sure it's on a day I can go to happy hour afterward.)

  4. Marshall, it's strangely both comforting and frightening that my students are not the only ones struggling with this. :)

  5. Thanks for an interesting blog! I tried to follow your link to the cards you had used, but the cards on the page were all empty. Do I have to click anywhere to see the activitycards?

  6. Emelie, unfortunately google docs doesn't show equations, which is what's on that page. If you download the file and open it in a word document you should see everything alright. In the future I'll try to remember to convert to pdf before sharing.