I have been doing math with kids for a while. Sixteen years paid. And more before that.

Why did this system of linear equations take kids more than triple the time I anticipated?

Writing good questions for the classroom, designing homework, making tests and quizzes. And when I offer a question, I expect a response. Sometimes I set up a mistake, prepared to talk about it. Sometimes I introduce a small twist, to take something that is easy for the kids, and make it challenging. I chose problems to drive conversations, knowing in advance where we are going. Sometimes I set up compare and contrasts (but I never call them that). Why is number three so much harder than number two, even though they look the same?

For tests, I know which skills I am looking for. I can avoid complications from tangents that I don’t mean to assess. I can create five of “the same” where the difficulty increases gradually, pulling them in. Or, for strong students, I can thoroughly mix easy and hard questions, and hector them to choose what to do first. “It’s you and the test, but which one of you is in charge?”

Which is all to say, I know where things are going. Usually.

Last week I gave a quiz.  Solve graphically for x and y, and check:   (in your check, indicate if your solution is “close” or “exact”).  Then solve the same system algebraically, and check. I figured fifteen minutes. It took them about 45. Can you look at the equations, and offer ideas of what went wrong?

$3x-4y = 8$
$y - 4 = \frac{6}{5}(x-3)$

Aftermath

The grades were about what I expected – before I watched the unplanned marathon unfold. And I did share my concerns with both classes the next day, and I solicited their opinions of what went wrong, as well. But I’d like yours.

1. December 9, 2012 pm31 9:50 pm 9:50 pm

Wondering: solving graphically, do students have to put into y= form or can they graph from equations as given? solving symbolically were most eliminating or substituting or something else? would any/many of the students multiply the 2nd equation by 5 to simplify the symbolic manipulation?

• December 9, 2012 pm31 10:11 pm 10:11 pm

Many students insist on putting the equation into slope/y-intercept form – though I encourage them to vary their method based upon the equation. In this case, many students used the equation in point slope form to directly graph (3,4) then see that (8,10) was awkward, and fall back on (-2,-2). The other equation I thought more would find two points, and graph, but many more moved it into “y= ” first.

We have spent equal amounts of time using “elimination” or “linear combination” and using substitution. I don’t think we have indicated any preference, instead looking at each example for what is easiest or most immediate.

December 10, 2012 am31 1:31 am 1:31 am

I graphed the first equation using the intercepts. The second one, I drew (3, 4) and wrote next to it m = 6/5. Then I noticed the slope of the first equation was 3/4. This implied an intersection in quadrant 3, but I didn’t bother guessing, because the fractions already seemed pretty terrible.

An algebraic solution seemed more or less straightforward, but there are plenty of ways for kids to screw up if they don’t start out by multiplying equation 2 by 5, for instance, and then noticing you can get 6x by doubling equation 1. I can imagine kids who are not that comfortable with algebra trying to apply some algorithm for solving linear systems by multiplying eq. 1 by 1/4 and adding, and struggling with the fractional coefficients.

Is this an Algebra 1 class? I don’t see why there is such an emphasis on putting the equation into slope/y-intercept form unless they have much more experience with it.

• December 10, 2012 am31 7:27 am 7:27 am

This is an algebra I class (albeit, one that is ending in two months). The emphasis on slope/y-intercept is a habit the students bring to the class, and not one I encourage.

Not multiplying by 5 may have been a significant problem here – leaving the algebra full of fractions.

• December 10, 2012 pm31 4:28 pm 4:28 pm

Ah, I didn’t understand why kids would instinctively go for slope/y-intercept form in Algebra I class; I think I learned it in Algebra I, many many years ago rather than coming in with that knowledge.

I think teaching flexibility is an admirable goal, but maybe not something that is ultimately the best use of time? I mostly work with math circle kids, who are strongly incentivized to be flexible for math competitions, because it gives faster answers / reduces the possibility of computation errors / reveals hints for solutions, but there’s a lot of practice involved in looking at a problem, identifying multiple solution paths, and choosing the easiest one (computationally).

If converting to slope/y-intercept form for graphing is familiar to the students and that’s what they do, then I think that sounds fine. Maybe you just need to (re)emphasize that the different representations are equivalent, and can be easily graphed if you understand what the coefficients represent.

• December 10, 2012 pm31 5:41 pm 5:41 pm

The purpose of this course, in a way, is to take kids who already have a good chunk of algebra 1, ands to put them in control of the algebra they are using.

I am teaching bright kids who can already solve a system. But they do not have great flexibility. I teach them point-slope, and make them graph without converting form. (it is a fight). I give them systems with fractional solutions, so that running into a solution with single-digit denominators represents unusual mercy. I teach them that graphing will produce, in many cases, a good approximation of a solution, rather than an exact solution, and that that can be okay.

If a system is easy enough, any approach is ok. But I want the kids to have multiple approaches available. And I want the problems to be tough enough that the kid actually needs to choose which tool is best. I want them to take executive control of their algebra, and not perform on auto-pilot.

This class is a luxury. In many schools these kids would have tested out. But we want this one term algebra to sharpen the tools they will use in Algebra II and later in Calculus. I want these kids to control the algebra they are working on. Remember, they were successful here, just slow.

I’ll write about the (pretty good) follow up lesson later this week.

December 10, 2012 pm31 2:31 pm 2:31 pm

What did they spend more time on, the graphing or the algebraic solution? I see difficulties in each stage. My calculus students (being married to the procedures they’re comfortable with) would have trouble with this.

It seems you are trying to teach them to be more flexible, and that’s great.

What steps of it did you see them struggling with? Had they seen plenty of equations in the form of the second one?

• December 10, 2012 pm31 3:07 pm 3:07 pm

I have some graphing-unable students, and they did struggle on both. But for most of them, it was the algebra. I am trying to teach flexibility… but something went wrong here. They’d seen plenty of equations in point-slope form, but I don’t think they’d had to do much manipulation. In fact, I’ve spent quite a bit of time trying to convince the kids not to manipulate them, as many kids instinctively and unnecessarily move every equation into slope/y-intercept form before graphing…