Tuesday, December 15, 2009

Algebra 1: Situation Graphing


I learned a few years back that jumping right into graphing slope-intercept equations never worked. This is one of those concepts that, before I became a math teacher, I never would have guessed would be so hard for students to master. Start at the y-intercept, use the rate of change to plot the next point, and you're done - right? Yeah, not really. So after a couple of years of teaching, reteaching, re-reteaching, and tearing my hair out, I decided to try some other things. Eventually, I realized that a ton of scaffolding of the concept of slope was needed, along with firmly rooting linear functions in situated contexts.

One of the constant problem areas is deciding which way to draw the line for a negative slope. To graph something like y = -(2/3)x + 5, students would often move down 2 and left 3. My old attempts at correcting this focused only on the mathematical explanation: -(2/3) = -2/3 = 2/-3. So, you either go down 2 and right 3, or up 2 and left 3. If you go down 2 and left 3, that means -2/-3, which is 2/3. This is a perfectly reasonable way to explain it, but it didn't really provide much of a lifeline to my lower-skilled students, as it hinges on mastery of the division rules of signs, as well as remembering that a fraction also represents a division problem.

The other common problem was for students to correctly identify the starting point number, but to plot it on the x-axis instead of the y-axis.

The way I run the unit now is to provide numerous opportunities to graph and describe situations, both with and without numbers, in just the first quadrant of the coordinate plane. Distance, income, height, and so on. The quantity being measured is always on the vertical axis, and the horizontal axis always represents time. When we eventually generalize to y = mx + b equations on the full coordinate plane, I use the situated contexts as memory anchors. If a student doesn't remember where to start, I say something like, "Where do we show that the Hare got a two foot head start? On the feet axis or on the seconds axis?" In these situations, a positive rate of change always means "moving up" and a negative rate of change always means "moving down", while time is always passing to the right. This is a much more helpful way for my students to think about how to graph their decontextualized lines. Suddenly, there is a reason for the direction the line is moving in, instead of just a sign rule.

Another benefit to this approach is that my students are now a lot more flexible with the form of the equations. My situated equations typically are in the form y = b + mx, which seems like a more natural connection to the preferred method for graphing. Once they grasp that the number without the variable is always the starting point, then they can handle both y = b + mx and y = mx + b relatively interchangeably. Also, it really helps them to understand the difference between equations like y = 2 and y = 2x. The first shows a starting point of 2, with zero rate of change. What does it look like on a graph if someone is not moving, but time is still passing? Exactly - a straight line! (I'm still working on that one - even my highest skilled students still say straight when they mean horizontal. My "all lines are straight" response doesn't usually clarify the way I'd like it to.) And in the second, the rate of change is 2. Ahh, it's like a graph of someone running 2 feet per second... but where did he start from? Zero? Where is that?

This approach takes a ton more time, of course, but I can't see any way around it for my students. I hope that I am providing them with a long-lasting ability to understand and graph linear functions. The semester is ending this week (final exams start tomorrow!), so the test will be to see how much they recall in January, when we move into the next unit. We'll be doing those oh-so-fun problems where you give them a point and a slope (or a parallel line and a point, or two points) and they have to give you the equation of the line. I'm going to experiment with doing every problem both graphically and algebraically (both in instruction and on assessments) to see if the focus on multiple representations helps them master these problems. I'll post more about that in late January (after I develop and teach it!).

My Slope and Graphing Linear Functions unit (Unit 4) is pretty much up-to-date in my box.com widget in the side bar. Here are a couple of examples (in pdf format) of the kinds of activities that they were doing. The Word and Keynote files are all in my box. I'd love to get feedback on any of this.

Lesson 10: Practice graphing with tables
Lesson 11: Learning to graph without tables

4 comments:

Eyawn said...

Wow, this is excellent stuff. Even more helpful since I'm at the exact same spot with my own algebra students. It's my second year teaching algebra and I can already see my own slow evolution of approach developing.

This year, I prefaced graphing with real world situations (Dan Meyer's graphing stories, plus practice with labeling http://exponentialcurve.blogspot.com/2009/11/xkcd-based-lesson-coordinate-plane.html).

Next, I've split up graphing into three separate concepts: graphing by t-table, graphing by intercepts, and graphing by slope-intercept. So far, I can already see the difference in understanding.

Transitioning into slope-intercept will likely be the toughest jump, but this post has got my math teacher juices flowing big time. Thanks for sharing and keep up the good work!

Dan Greene said...

Hi Eyawn. I'm always happy to get others' juices flowing. :) I wish I had the edublog world to draw on when I was just starting - my evolution as a teacher would have been much faster, I imagine. And maybe less painful?

In reworking the curriculum for this year, I decided to actually hold off on graphing by intercepts until later. I wanted to build a solid bridge between situations, tables, and the slope-intercept form of the equation. So for now, I only taught them how to convert a standard form equation into slope-intercept form.

I realized that intercepts are a deep concept also, just like slope, and I want to take the time to do them justice. So much depends on solid understanding of x-intercepts, and yet they really have nothing to do (necessarily) with the skill of graphing a linear function, so I'm deciding to wait.

Students can usually plot points without much trouble - except when one of the coordinates is zero. Why this is such a killer, I'm not sure yet. I think I will get to this after completing this unit and the systems of equation unit.

I tend to save the linear combination method of solving for Algebra 2, and only do substitution in Algebra 1, so most systems will be in y = mx + b form (or easily convertible). Though I may change this up as I think about it more...

Anyway, I look forward to reading about what's going on in your class, and how all this ends up going for your students.

jd2718 said...

Dan,

I think there is special power that belongs to the displacement vs time graph. Maybe the y per x where "per" is the fraction bar? And where "per" is already part of the vocabulary? (inches per month, miles per hour?)

I take numbered streets in Manhattan - I pretend there is a 0 - and plot blocks vs minutes. Before I even get to lines, I tell stories and get the kids to plot the points (ooh, when he stays still the line is flat!) Uptown and downtown translate easily.

Now, when I started, those students needed a ton of time. Now, I could dispense with the connections, the situations, except, you know what? I think even kids who don't 'need' them, benefit from them. So I keep them and will keep them.

I remember, first time I did this, new teacher, winter 98, principal comes in to watch - kids were saying speed instead of slope, or slope instead of speed - sometimes they said starting point, sometimes they said y-intercept... and I got positive feedback (I think it was already way into the unit, when we did parallel lines. In the situation, it's just so obvious if they are going the same speed but have different starting points... but you've clearly planned out your own version!)

Jonathan

Mrs. A said...

This is great stuff! Thanks for sharing.