I begin class by letting students know they will be creating graphs to reflect some real world situations. I begin by reading aloud the Friendly Competition activity and questions together. I divide students into pairs or small groups and let them begin working on the problems.
I let students get to work in pairs or homogenous groups on today's activity. As students work, I circulate and check student progress. Here are some issues I watch for:
When students are nearing completion on their work, I choose different groups to share out their graphs. If I think it will be helpful, I lead a discussion here about using different scales for the graphs and comparing students representations.
I then lead a discussion about the two different graphs, side by side, drawing connections between their similar questions. I make sure students are able to articulate how the graph shows them where the starting values are. I remind students that we call this point the y-intercept. I also ask them to clearly articulate how the graph shows them how quickly they are saving money or how quickly they are using up paper. I like to draw a comparison here between the savings graph, which has two different positive rates of change and the lined paper graph which has the same negative rate of change. I try to be sure students see that the savings graphs are growing as time goes on and the lined paper supply is decreasing as time goes on. I ask students how those differences relate to the problem situations.
Next, I ask students what's happening in the graph that has the lines that don't cross. I elicit from students that when two situations have the same rate of change, the resulting graphs of the equations will be parallel lines. I ask them why this makes sense.
I ask different students to share out how they came up with their equations for each situation. I might highlight the various strategies that the students use. Again, I make sure they can point out where the starting values are in the equations and where the rates of change are in the equations. This is a good opportunity to draw a connection between the three different representations: the situations, the graphs, and the equations.
The final questions about when the savings accounts will have the same amount of money and if the students ever have the same amount of lined paper are systems of equations questions. The point of including those questions is so that students begin to get an intuitive sense of what a system of equations is. I do not use any formal language or explanation here. I do bring out the connection between the graph where the lines cross and the point at which they cross. I want to elicit from students that the point where the lines cross represents the month when both accounts will have the same amount of money. Then I ask students why the lined paper lines will never cross.
Reflection: At the close of the whole group discussion with students, I give them some time to reflect on today's activity. I like to use an exit ticket to give some structure to the end of class and allow students some quiet time to think about what they will take away from today's class. A prompt for today's class might be:
How did you determine the equations that fit the savings situation? What tips would you have for other students who were taking on this task?
This material is adapted from the IMP Teacher’s Guide, © 2010 Interactive Mathematics Program. Some rights reserved.