SWBAT graph lines that represent 2 cell phone plans and solve the system of equations to determine the best plan.

The purpose of this lesson is for students to understand how to analyze a system of equations to determine when a plan is cheaper, more expensive, or cost the same at certain points on the graph.

10 minutes

In this lesson, the following materials need to be prepared in advance:

- A copy of the Warm up worksheet
- A copy of the Choose the Best Cell Phone Plan problem.
- A copy of the Salaries problem.
- A ruler.

I plan for today's Warm up to take 5 minutes for the students to complete and 5 minutes to review and to introduce the lesson.

I model the warm up below in the video:

20 minutes

The Choose the best Plan problem is intended to demonstrate a real world application that is relevant to students. In the problem students model two cell phone plans using multiple representations. Students work with their assigned partner on this problem. I use the Think-Pair-Share Strategy on this problem.

The difficulty that students have with this problem is to graph the lines that represent the two different cell phone plans. When they attempt to graph using slope-intercept form, they realize it is not an effective method for this problem. This is a good time to bring up Mathematical Practice 6 (Attention to Precision). The y-intercepts are easy to plot, 20 and 0, but the slopes of .05 and .15 are too small to be precise when using slope-intercept form.

To prompt a student that is having difficulty, I question him/her on what other method could be used to graph the lines. Questioning (instead of telling) is the technique that I use to help students persevere (Math Practice 1). Instead of providing immediate answers, questions enable the student to continue to engage in a productive struggle.

After questioning students on the other methods to graph these equations, students usually start creating a t-table. A t-table is useful in showing more points on the line to further analyze the graph. In the demonstration below, the t-table shows that the 2 cell phone plans will both cost $30 at 200 minutes. It also answers the next question about which would be the better plan when 300 minutes are used. Plan A saves $10 when using 300 minutes and is the cheaper plan when more than 200 minutes are used. Plan B is cheaper when the minutes are less than 200 minutes.

By making the t-table first for each plan, a student is able to create a scale for x and y based off of the input and output values in the table. If the student attempts to create the scale from the equation, some students make the mistake of scaling it from the slope which is a small decimal, and the increase of rise over run does not give a clear picture of the line. The scale of the graph must be created to plot the data that x and y represent as points on the graph to provide an accurate graph of the problem (Math Practice 6). When reviewing the problem, I emphasize the importance of defining the variables at the beginning of the problem, and labeling the graph.

20 minutes

In this closing problem about salaries, I provide the students with a real world application to analyze.

The first question describes an employer requesting the student to analyze the graph and write a report. In the report, the student uses the graph to determine when the Radio Barn Company pays less, more, or the same than the competitor, Woofer. The students need to provide reasoning from the graph to support their decisions.

The second question has students identify the difficulties in analyzing the graph. Some difficulties that I expect students to list are the following:

- the variables are not defined
- the graph is not titled
- each axes is not labeled with a definition of each variable

The third question has students label each axes with the definition of the variables used to analyze the problem, and title the graph appropriately.

Finally when the report is written for the analysis, I have students work with their assigned partner. One student is the employer and the other student presents the report with reasoning. The employer critiques the report acknowledging the likes of the report first, and then any suggestions for improvement second.