OMG! Great day today in Algebra. We started yesterday with the Leaves activity, thank you Brian Marks, and today I was wondering what to do with my classes, and how we could use the activity to further their understanding of lines and direct and indirect variation. Mind you, these kids are still struggling with what x and y intercepts are. A colleague of mine suggested an activity she observed in another teacher’s classroom, placing six graphs around the room and having students move around and monitor what they observe on each graph. I shuddered to think of these classes getting up and moving around. I have members from two rival gangs in these classes, and could see blood appear at the thought. After pondering this a few minutes, I decided on another type of the same activity. This one kept them in their seats, where they are safe, but allowed for discussion and interaction.
and put these questions on the board: How are these graphs similar? How do these graphs differ? What are the x and y intercepts? What are the equations?
At first, when I projected the graphs the students just stared at them. I could see their minds going, “This looks a lot like thinking, Mrs. Ryan”. I waited, walked around the room, kept turning and looking at the graphs myself, and finally said, “It’s ok to start with the obvious.” They continued to stare. I said, “One thing I’m noticing is that they both have arrows on the ends of the graphs.” This got something going. Several students began to look at them really, and realized they did know something. I had them write their own thoughts individually on paper, then pair share. Several good discussions started, and a couple of the pairs even noted that the bottom graph couldn’t have a linear equation because it was curved. I have to admit, I was impressed.
We then discussed this as a class. They were able to verbalize that both graphs appeared to be negative, the second one didn’t have any intercepts, etc. They were able to get the equation for the line, but balked at the idea of the second one having an equation, until they realized that because it was in the coordinate plane, it should have an equation. Even that was impressive. 🙂
I directed them back to the “Leaves” activity from yesterday. They pulled it out and looked at the equations they had made, and the graphs. They actually started to get excited and told me, “This graph looks just like the ones we made yesterday.” I asked them what they had been doing yesterday mathematically that created the graphs. They talked about the division, and what was divided by what and came up with y = 24/x. We talked about how that was specific to the situation in the activity, so they changed the 24 to n. Then I asked them to compare the equations and talk about what they noticed. They noticed that y was in the same place in both equations, but x was in the numerator in one and the denominator in the other. (We had to re-write the linear equation for that one, but it immediately became clear). Then we talked about how the difference in position might change what happens to the graph. That lead to a discussion about fractions, what we can’t divide by, and how the graph actually shows that x can never equal zero, and how that determines how the graph curves away from zero.
I was able to show them how the linear graph showed a direct variation between x and y, and the curved graph showed an opposite variation between x and y. The students left today feeling smarter than they ever have about math! We have never talked about direct and indirect variation in the class before, and now they actually know what it is.
I want to thank Justin Aion for his inspirational blogging which caused me to believe one more time these kids could do something valuable and were worth the effort.
Woohoo!! BTW I just learned how to link and add a picture with this blog. A very good day indeed!