A Friendly Competition
Lesson 10 of 17
Objective: SWBAT interpret key characteristics of graph including rate of change and y-intercept. SWBAT write equations describing graphs. SWBAT begin to conceptualize a system of equations.
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:
- I make sure students graph both savings situations on the same graph. I might suggest they use colored pencils to show the different points.
- I find students often struggle with scale when making graphs. They should be able to determine that the months will go on the x-axis for the first graph and they will need to be able to show the number of months in a school year. They will have to think more about how to scale the y-axis. I might ask them if they could figure out the highest point they will have to place on the graph.
- Depending on the previous work I have done with students about discrete and continuous graphs, I might have conversations about whether or not students will draw a line through the points on the graph. I let students know that the savings situation would result in a discrete graph because the money is put in the account one time each month. We have a discussion about the lined paper scenario and see what students think. Likely, they would not use up the paper at a continuous rate in the intervals between the points. I discuss with them that although the real world situations they are using may not be continuous, drawing a line through the points on their graph will allow them to see the trend of the data. I remind students that the point of making a graph is to tell the story of the situation visually.
- Students may try to make the first questions about the graphs more complicated then they are. I like to tell students, "Don't make a mountain out of a molehill!" a quote I credit to my own high school geometry teacher. I ask students to be specific and clear in their response to the question: "How can you tell from the graph that on the first day of school, you had more money in your account than your friend did?" A good follow up question might be, "Where in the graph do you see that?" or "How does the graph show you that?" The same line of questioning is useful for the rate of change questions. I might guide students by asking, "Where does the graph show you that your friend is saving money at a faster rate than you are?"
- If students are still unclear on the form y= ax + b or y = mx + b of linear equations, they may struggle to write the equations for each graph. I help guide them here by using SMP 8: Look for and express regularity in repeated reasoning. I ask them, how would you figure out how much money you had in your account after 3 months? What about 6 months? What about 9 months? After three examples, students should recognize the pattern of multiplying the number of months by 30 and then adding the initial amount they put in the account (100). I then ask them, what about m months? How can you generalize these calculations when you're using a letter instead of a number? In addition, if I guide students for the first two equations and they ask similar questions for the lined paper equations, I try referring them back to the savings scenario rather than starting a fresh line of questioning. For example, if a student seems stuck writing equations for the lined paper problem, I might say, "Well, let's look back and see what we did in Question #4 for the savings account problem. What did you do here? Can you apply that work to this new situation? What's different about this situation? How can you account for that difference?"
Discussion + Closing
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.