Some of my students are taking an AVID class that happens to occur during my conference period, so occasionally I stop by.

Now, linear equations can be written in a lot of different forms, but we only focus on three. These students were working on equations in point slope form:

$$(y-y_0) = \frac{rise}{run}(x-x_0)$$

There was a heated debate between students in my class, and students from another teacher, on how best to find plottable points. The way our book teaches students to handle this form, and how the other teacher taught it, is to convert the equation into slope intercept form, and then proceed from there.

Of course, all of the problems they were working on had an $$x_0$$ that was an even multiple of the $$run$$. When I drew up a problem that didn’t meet that criteria, all of a sudden the kids from the other class were stumped.

When I teach it, I try to get the kids to perceive it as something like this:

$$(yblob) = \frac{rise}{run}(xblob)$$ where the blobs are treated kind of like variables. It then comes down to them being able to find a case where the xblob is 0 (or, IOW, $$x-x_0=0$$) as well as the yblob, or where the xblob is equal to $$run$$ while the yblob is equal to $$rise$$.

My hope is that being able to view the point slope form in this way will prepare them for the similar appearance of other curves under translation:

$$(y-y_0) = k(x-x_0)^2$$

or

$$(y-y_0)^2 + (x-x_0)^2 = r^2$$

I spent a bit of time feeling all superior about this. I’m a big fan of the setup, and I think in this instance I laid some decent groundwork for the future.

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Then the students went on to deal with an equation in standard form, and I realized that, in that particular case, I’d left them even less able to deal with the odd case than the other teacher had for point slope form.

I’d tried, and hoped, that they’d be able to develop connections between the different forms primarily by use of tables and graphs, to see the similarities there, and then be able to find the equivalences via algebraic manipulation.

I feel like I failed at that now. I’m not sure if it was a matter of time, or approach. I suppose that’s what happens for the first year when you teach a subject, but I’d really like to give it another shot *right now* to see if I could sneak in better ways of making those connections.