At this point in the year, my students have not yet solved systems of linear equations. But, they have enough knowledge to be able to graph two lines and see the point of intersection graphically, or find the common (x, y) values in a table. Nonetheless, my students are usually not proficient at setting the proper window on their TI-83 graphing calculators when they model real world problems. Because of this, I launch this lesson by having students practice finding an appropriate window to locate the point of intersection for each pair of linear equations.
To explore and practice this capability on the graphing calculator, I give the students three pairs of equations to graph for today's Launch:
1) Y1 = 5x + 400; Y2 = −3x + 80
2) Y1 = (-1/2)x + 100 Y2 = (3/4)x - 10
3) Y1 = 0.7x + 111.2; Y2 = −1.3x - 89.8
I walk around the class making sure each student is on task and help out as needed. I like to let students reach an appropriate window by trial and error instead of me giving them any values at all. I may ask more advanced students to help out by guiding students that are struggling. I keep an eye out so that the students helping don't actually perform the tasks for them.
Once a student finds the appropriate window, I will show them how to use the CALC command, followed by the INTERSECT command to find the point of intersection. Then continue to hit ENTER until they see the coordinates of the intersection on their screen.
I make sure I see everyone’s window for at least one graph, with the coordinates of the corresponding intersection on the screen. When I have doubts about someone’s work, I ask that he/she repeat the steps to me, just to make sure they understood and performed the tasks themselves.
The problem posed to the students in this section of this lesson is:
In general, it is more economical to buy a hybrid car instead of the regular model of an automobile.
Because the Hybrid Costs more than the standard model, the question that students end up trying to answer is:
After how long will the hybrid buyer actually begin to save money?
I begin the activity by projecting the image of the 2014 Toyota Avalon. I then pose the question:
If I wanted to buy this fancy car, should I buy a Hybrid model to save money? Here's some data that I want you to look at and help me make a decision:
I'll give students a few minutes to look at the data, then I'll ask:
Can you use the price of both vehicles and the estimated miles per gallon statistic for each car to predict which car will be cheaper. What else would you need to know?
After we discuss these questions we'll make some simplifying assumptions for today's lesson. I tell the class that we'll use some average numbers to help us make the comparisons:
Then, I will ask my students to plot the total cost (owning and operating) each car on the same graph, with years of ownership along the x axis. The model that they are creating is simplified, of course, but I find that my students are excited to apply their knowledge on this task.
I like to have my students graph by hand first, before using their graphing calculator, despite the practice at the start of the lesson. Graphing by hand provides an opportunity to think about an appropriate scale for the vertical axis. I think that this helps students to drill deeper into the problem. If time is an issue, I may skip this and go straight to using the graphing calculator. When we get to using the graphing calculator, I tell the class that it is very important to set smart range values when choosing the window. I find it helps to have students graph Y1 and Y2 consistently: they should set Y1 equal to the cost of owning and operating the regular car, and Y2 equal to the cost of owning and operating the hybrid car. This avoids confusion later in the lesson.
In Part II of the activity, I ask that each pair of students use both their equations to determine how long it will take before the hybrid becomes a money saver, (break-even point). Students should solve for x after setting both equations equal to each other. They then should check their answer with their graph.
To close the lesson, I ask a group to come up with their calculator and show their graph on the screen for their classmates to see. I ask that they explain what the x and y coordinates of the intersecting point represents. I also randomly ask a student from another group to show their solved equation on the board. This answer should coincide with intersection of the graph being shown.
I end by asking...."So is being the Hybrid Avalon better?" I hope that students say something with respect to the length of time that one expects to own the car. I also expect that some students will argue that it is better for the environment, so it is a good investment. There are, of course, lots of interesting mathematical conversations that can be had with this problem. Depending on time available and students' willingness to think in terms of quantities, patterns, and questions, we may explore them.
The answer to this problem is x = 6.78 (approximately), which means that it would take a bit over 6 years 9 months for the hybrid car to start becoming the more economical purchase.