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Having fun with linear equations

September 10, 2012

For our third class in Algebra II honors, I wanted to talk about all the different forms of lines that students know. Many of my students have already encountered both point-slope form and the general form of a line, but I suspected that they didn’t see the connections between the various forms, and so that’s where I tried to place some emphasis. What follows is my amalgamation of the lesson on this topic I presented to two different classes.

Introducing point slope form

I started by using the awesome graphing calculator Desmos to project a point on the board and ask students to write the equation of the line, given a point and a slope.

Screen Shot 2012 09 10 at 11 03 27 PM

My students approached this in two ways, which they shared by showing their work with the Hovercam document camera connected to a 30′ cord so that we can pass the camera around to show anyone’s work.

One student simply plugged the given point into slope intercept form and then solved for the y-intercept.

But a second student mentioned point slope form, and was able to instantly write the equation using point slope form, which most of the class had seen.

At that point, I asked them how the following two ideas were connected

y=mx+b
and
y=m(x-x_1)+y_1

Expanding the first term in the second equation gives us:

y=mx-mx_1+y_1

and this will be equivalent to the the slope intercept form if b=-mx_1+y_1.

I then asked students to think about how this could work—what do each of the symbols in this equation mean:
b: y-intercept
m: m
x_1: x coordinate of given point
y_1: y coordinate of given point

How could it be that the y-intercept is equal to the negative slope times the x-coordinate of the point plus the y coordinate of the point?

It helped to look at a graph:

Screen Shot 2012 09 10 at 11 24 35 PM

After some thinking, students were able to explain that all the y intercept is is the y coordinate of the given point plus how much the y coordinate changes when you move over a distance x_1 to the y-axis.

From there I showed students a second way of looking at point slope starting with the definition of slope:

m=\frac{y-y_1}{x-x_1}

and a simple rearrangement will give

y-y1=m(x-x_1) or y=m(x-x_1)+y_1

General and Standard form

Most students had send standard form before, and quickly spouted it off as

Ax+By=C and it wasn’t until I wikipedia’d it, that I found out there was a difference between that and Standard form Ax+By+C=0. Thankfully, I didn’t know this when I taught the lesson and therefore couldn’t bother my students with such details.

Instead I asked them why we had this additional form, and what it was good for. Some students remembered it made finding intercepts easy, so we tried that, and then I asked them to figure out the meaning of the A,\; B \textrm{and}\; C terms, which they were quickly able to do.

x-intercept form

At this point, I said that I think the x-intercept doesn’t get enough love, and we should try to write an equation for a line, given the x-intercept and the slope, and asked students to set out to do this.

This turned out to be a very difficult question. No students thought to use point slope. Some went back to the tried and true effort of plugging in the x-intercept into y=mx+b and solving for b, and one student tried the approach of swapping the x and the y, and then solving for y. When we graphed this effort, we saw pretty clearly that this line wash’t the line we were looking for, but that student continued to play with the calculation, and stumbled upon the discovery that if you inverted the slope and you then figured out what you needed to multiply the inverted slope by to get the x-intercept, you could write the equation. This was our first awkward foray into inverting equations.

Eventually though, we got back to point slope form

y=m(x-x1)+y1
where (x1,y1)=(x_{int},0)

so y=m(x-x_{int})

and if you expand the right hand side, you get:

y=mx-mx_{int}, which is equivalent to slope intercept form if $b=-mx_{int}$, and that brings us full circle back to the beginning to realize that slope tells us how much the y value increases for every unit of change in the x value, and if you are at a y value of 0 (the x-intercept) and move over x_{int} units, then you should also be changing your y-value by -mx_{int}, with the negative to remind us that the you are calculating a change in x position, and the final x position (the y-intercept), is 0, so \Delta x=x_f-x_i=0-x_{int}.

How this looks for real

Here is how this all looked this past Saturday. I filmed my class and posted the footage below:

I would greatly appreciate any feedback you have.

Closing thoughts

Overall, I think this lesson met a goal of helping students see how all of the various forms for writing linear equations are connected, and how each of them has a raison d’être. You can see this pretty clearly in the tweets I asked the second section to write today 20 minutes into class, responding to the question what are you learning now:

Learning the value of knowing different formats to write equations for lines

Expanding on linear equation formulas

Learning different uses of general, point slope, and y-intercept forms of lines

Equations are made not memorized.

We are learning about the various merits of point-slope, slope-intercept, and standard form

So many ways to make linear equations, each is better for something #prosandcons

All parts of the diff forms of equations relate back to one another #mathisntrandom

The different linear equations for lines allows us to find variables quickly

Linear equations—point slope form, slope intercept form, and standard/general form #coolstuff

By far my favorite is the one we tweeted as a class, “Equations are made, not memorized.”

At the same time, I’m concerned about a number of things in this lesson. All of this seems pretty abstract—I’m interested in how general form is related to point slope, and the meaning of the A’s and B’s, but I doubt my students are.

I also felt like this was a lesson where I did a lot of talking and guiding, and students didn’t do much in the way of practice. Maybe this is ok because this material is review; I’ll know better after tomorrow when I finish grading their first assessments, but I definitely am thinking I’m going to need to find a better way to incorporate more practice, as opposed to problem solving, into our routine.

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