James Dunseith NORTH HIGH, WORCESTER, MA
Algebra I : Unit #7 - Lines : Lesson #5

A Graph is a Set of Points

Objective: SWBAT understand that the graph of a linear equation in two variables is the set of all its solutions plotted in the coordinate plane.
Standards: 8.F.A.3 HSA-CED.A.2 HSA-REI.D.10 HSF-BF.A.1 HSF-LE.A.2 MP2 MP5 MP7
Subject(s): Math
60 minutes
1 Opener / Mini Lesson: How to write the equation of a line through two points, without graphing. - 8 minutes

For the second consecutive day, I start with a fast-paced mini-lesson.  During yesterday's lesson, students practiced finding the slope and y-intercept of a line by first plotting points on the coordinate plane.  Today's lesson starts by leaving that for the more abstract task of creating an equation of a line connecting two points, without graphing, before brining it back to the graph later in the class.  I want students to develop the algebraic chops for completing this key task, but I also want them to keep coming back to the idea that an equation in two variables represents a set of points on the coordinate plane, and this lesson is designed to help them get there.

Here is what is posted on the front board as students arrive.  I tell everyone to take a minute to write what they see here in their notebooks, and then I go through these two steps.  For the second straight day, I move pretty quickly.  I want to demonstrate this and then get it into the hands of my students as quickly as possible.  When I'm done, the notes look like this.  I've followed the pretty traditional approach of first calculating slope, then placing that parameter into slope-intercept form, and finally using the x and y coordinates from one point to solve for b.

As I've done all week, I teach students to find slope without explicitly using the formula.  Instead, I tell them to simply look at how much the y and x values change from one point to another.  I find that eliminating the formula - at least for now - allows students the chance to really understand what slope is, rather than worrying about whether or not they plugged in x1 and x2 in exactly the right places.

Similarly, after I show students how to use algebraic moves to determine the y-intercept, I also show them a tabular approach.  Some students prefer this method, and some prefer the algebra; I like giving kids options for how to work a problem like this.  This approach of constructing a table and then counting by the slope allows them to see a connection to yesterday's work, where we counted out the slope on a graph, and it allows them to see why we can define the y-intercept as the point where x = 0.  

What is a "Formula," Anyway?
Developing a Conceptual Understanding

There is always some subset of students who recall the notation-heavy version of the slope formula.  There is a larger subset of students who can almost recall the formula before getting bogged down in where "x-one" and "y-two" go.  When the inevitable conversation is raised about why we're not using the formula, I ask, "What does that formula mean?"  I give students a few moments to chew on that, and maybe we make progress, but I'm also wary of moving quickly here, because talk of formulas is precisely what causes another subset of my kids to tune out.

Instead, as I've outlined above, I reiterate what we're doing when we calculate slope.  We are using a ratio to compare the amount that y changes to the amount that x changes.  I provide the visual.  Here, there's the inevitable exclamation - this way is so much easier! - at which point it's my job to swing the pendulum back again and show students that whether you use the formula or draw arrows, you're really doing the same thing!

So while I'm thinking of this, here are some side notes that came to mind as I taught the lesson today:

Note #1: I'm not doing away with mathematical notation or vocabulary.  I want students to understand the "delta" symbol for change, and at the same time that I replace the subscripted formula with labeled arrows, I'm using that symbol accordingly.  I do not shy away from using the word ratio to talk about slope, because that's what slope is.

Note #2: In general, why does subtraction feel so hard for so many kids, but telling the distance between two numbers feels so much easier?  Does the way that students get tripped up on the formula make it feel harder to subtract?

Note #3: The graph that results from today's activity - and the "steps" that many students will draw on it - is similar in flavor to the arrows I use to denote slope.  It's fertile ground for an extension to talking about reducing fractions.  I don't specifically set out to do that here, but to do so is useful!

2 Individual Practice - 20 minutes

After moving through the opener-mini-lesson as quickly as possible, I distribute today's practice handout.  There are five versions of the front of the handout and one version of the back.  I make double-sided copies, and an equal number of each version.

As I describe in this video, the five practice problems are different on each version of the handout, but all of the solutions are the same.  Of course, as kids get to work, I make sure to point out that everyone at each table has a different version of the handout.  I save the punchline for later.

Using the notes from the start of class, which I leave on the board but expect to see in student notebooks, everyone has about 20 minutes to practice writing the equation of a line through two points.  Some students use the algebraic method to find the y-intercept and others employ the tabular approach that I tacked on to the mini-lesson.  

After three days of work on this topic, students are increasingly confident in their skills here.  Of course, I make sure to point out the role that hard work has played in their growth.

3 Group Practice (Formative Assessment) - 15 minutes

As I've described in the previous section, the key to this lesson is that, even though the problems on each version of the practice assignment are different, the solutions are the same.  The key to this final part of today's lesson is making the transition from individual work time to group work.

I tell students to stop wherever they are on the front of the assignment, and flip it over.  On the back, they'll find that I've written the number of one of the problems at the top of the paper.  I tell everyone to find all of their classmates who have the same number written there.  In each of these new groups, each student will have a different version of the assignment.  (The ideal class size for this assignment is 25 students, which would mean that five students have each version of the assignment, and each version has a different problem number written on the back.  Adjustments are made accordingly for larger and smaller class sizes.)

Once the new groups are made, I remind everyone once more that everyone has a different version of the assignment, and then I get to enjoy making the big reveal that the solutions are the same on each assignment.  The expressions kids make when I tell them are teaching gold.  Then it's time for the instructions.  "On the back of this assignment," I say, "you and your new group-mates are going to focus on one problem - the one whose number is on the back of your assignment.  Each of you must plot your pair of points for this problem on the graph, then you must also plot the points from every other member of your group."

I want students to see - first looking at the points on the graph, and then by fully understanding the concept - that all the points in their group make a straight line, and that the equation of this line must be the same, no matter which two points any individual started with.  Students work together to plot those points, then they make sure they all agree on the equation.  If they finish early, then they can go back and check that they share the other four solutions.  

I plan on collecting this handout at the end of class, just so I can check in what everyone knows.  If the class goes especially smoothly, and I can see that everyone is pretty confident, I might skip the collection part.