Writing the Equation of a Line

Today's opener is posted on the board as students arrive.  Before the late bell, I tell everyone in attendance that I'm going to teach a quick lesson when the bell rings, so they should be prepared.  I have graph paper ready for anyone who asks for it, but I also tell everyone that it's alright just to watch the lesson that's about to happen, because they'll have a chance to practice in just a few minutes.  Then I ask if anyone can figure out the slope of this line just by looking at the coordinates of the two points.  This gets my high-achievers excited, as they try to work it out quickly.

I call it an opener, but this is really today's mini-lesson.  When the bell rings, I get started.  I plot the two points on the blank graph (I've also included .png files of simple axes in the resource folder here - I made them and you can use them freely.).  Then I review the process that everyone followed yesterday on Delta Math, of finding the vertical and horizontal changes between two points, then using these to find the slope of the line.  

The key to this problem is that I want students to see how slope can be used to find any number of points on a line.  Moving from (-2,4) to (4, -5) on the graph, we have to move down 9 and right 6, which reduces to a slope of -3/2.  Once we have that, I tell everyone, "here is the most important thing I'm going to show you today," and starting from (-2, 4), I move down three and right two to get another point.  I repeat, drawing two new points, and then arriving at (4, -5).  Then I keep going.  Soon there are six points on the graph, and I show the class that it's not too difficult to connect these without a ruler.  "Using slope, you can draw a relatively straight line without needing a ruler," I say.  

Our next task is to finish writing the equation of the line.  I write y = mx + b below the second question on the board.  This is one of those structures that almost all of my students have seen before, but their collective understanding of what it is and what it means is usually all over the place.  Today's lesson serves to review the process of writing an equation of a line for all students, but also to get at the concept of what the equation of a line really represents.  "We know the slope," I say, pointing to the m before moving my finger to the b, "now we just have to figure out the y-intercept."

I love this mini-lesson because the y-intercept is already on the board.  It was one of the points I plotted when I showed everyone how to use slope to generate more points.  Some students recognize it right away.  I circle that point, (0,1), and beside I write "y-intercept" and "b = 1".  Then I just fill in the parameters of the slope-intercept form.  All of what I've just described happens within the first five minutes of class.  By design, I move as quickly as the kids will let me.  The resulting notes look like this.

Following the opener/mini-lesson, we transition to student work time.  It's a pretty straightforward workshop model here: I show kids how to do something, then they practice.  So what I'd really like to point out are the two little teaching moves that make this a very successful sort of skill and drill lesson.

First of all, I purposefully move at a fast pace.  I think it's important to vary the pace in a math classroom.  My default is to move slowly: giving students time to think through and grapple with the problems they're solving, and to construct their own understanding of each new idea.  But today, I move quickly through the mini-lesson, transition to work time, and then with ~15 minutes left, make another abrupt change to a quiz about arithmetic sequences.  I want students to squeeze in as much thinking as they can about this important review topic, and by moving from the practice with linear equations to the quiz about arithmetic sequences, I hope that they can - subconsciously even - make connections between the two.

My second tweak to the lesson is that I offer students a choice about the level of practice in which they'll engage.  To begin today's work time, I tell students that they're going to practice what I just showed them.  I say that I've prepared three versions of today's classwork: Easy, Medium, and Hard, and that everyone should come up and take a copy of whichever one they'd like.  Asking students to come up and make a choice goes a long way.  They take the initiative to choose what they'd like to work, which increases their ownership over the assignment.  Also, the opportunity to stand up and take a quick walk - even to the front of the room and back - signifies a transition in the lesson.  Both of these add up to increased engagement, and it's simple tricks like this that go so far in developing a productive classroom environment.

Once they get to work, students will have to unpack the notes I just gave, so I expect that they'll refer back to those frequently.  This work opens up a lot of space for me to have individual conversations with kids.  As I circulate, there are a wide variety of questions kids have, at all sorts of levels:

• Did I plot the points correctly?  Students have to make sure they're getting x and y right, both in the coordinate pair ("It's alphabetical order!" I say) and which axis is which.
• Does it matter if I count over or up first?  No, as long as you remember that change in y is the numerator and change in x is the denominator of slope.  We also address the idea that it doesn't matter which point I start from, as long as we pay attention to the direction we're moving.  I like for students to really understand that moving up and right or down and left results in a positive slope, either way.
• How do I find the y-intercept?  I carry a maroon pen today, so I can annotate student work, circling the y-intercept to help them develop a visual definition.  Using slope to count out to find additional points, and then seeing the y-intercept as a point helps build this concept.  We'll get to algebraic methods for computing the y-intercept tomorrow, but it has to have a context first.
• How do I write the equation?  This is my favorite question, because I get to tell kids that they've already done all the work!
• Am I right?  Classic.  I show students how to use a new point they've just discovered on the graph by substituting its coordinates for x and y, to see that they make the equation true.  I cannot stress enough the importance of kids having some grounding for what a linear equation actually is and what it means.  When I share this note with students, their most common reaction is, "that's cool!"

If I have time to ask my own questions, I'll ask students to seek out some arithmetic sequences in the new points they find on the graph. Both the x-coordinates and the y-coordinates of successive lattice points will make their own arithmetic sequence.  Observing this is yet another way to deepen conceptual understanding of slope.