A.4.5.2 Data Plans

A college student is choosing between two data plans for her new cell phone. Both plans[br]include an allowance of 2 gigabytes of data per month. The monthly cost of each option can[br]be seen as a function and represented with an equation:[br][list][*]Option A: A(x)=60[/*][*]Option B: B(x)=10x+25[/*][/list]In each function, the input, x, represents the gigabytes of data used over the monthly[br]allowance.
1. The student decides to find the values of A(1) and B(1) and compare them. What are those values?
2. After looking at some of her past phone bills, she decided to compare A(7.5) and B(7.5). What are those values?
3. Describe each data plan in words.[br]
4. Graph each function on the same coordinate plane.
Then, explain which plan you think she should choose.
5. The student only budgeted $50 a month for her cell phone. She thought, “I wonder how many gigabytes of data I would have for $50 if I go with Option B?” and wrote B(x)=50. What is the answer to her question? Explain or show how you know.
Select students to share their interpretations of the two data plans. Make sure students see that:[br][list][*]The equation A(x)=60 tells us that, regardless of the extra gigabytes of data used, x, the cost, A(x) is always 60.[/*][*]The 10x in the rule of B(x) tells us that each extra gigabyte of data used costs $10, and that there is a $25 fixed fee.[/*][/list]Explain to students that the two functions here are linear functions because the output of each function changes at a constant rate relative to the input. Option B involves a rate of change of $10 per gigabyte of data over the monthly allowance, while Option A has a rate of change of $0 per gigabyte over the allowance.[br]Then, ask students how they went about graphing the functions. Students are likely to have plotted some input-output pairs of each function. If no students mention identifying the slope and vertical intercept of each graph, ask them about it.[br]Next, focus the discussion on students’ response to the last question and how they found out the gigabytes of data that could be bought with $50 under Option B. Select previously identified students to share their strategies, in the order listed in the Activity Narrative. If no one mentions solving using a graph or solving , bring these up.[br]Explain the following points to help students connect some key ideas: [br]We can graph functions like and without plotting individual coordinate pairs.[br][list][*]A(x) is the output of function A and is represented by vertical values on a coordinate[/*][*]plane. The vertical values are typically labeled with the variable y, so we can write y=A(x) and graph y=60 to represent function A.[/*][*]Likewise, B(x) is the output of function B and is represented by vertical values on a plane. We can write y=B(x) and graph y=10x+25 to represent function B.[/*][/list]To solve equations like B(x)=50 means to find one or more values of x that make the equation true. We can do this, among other ways, by using the graph of B and by solving an equation algebraically.[br][list][*]On the graph of B, we can look for one or more values of x that correspond to the vertical value of 50. This might involve some estimating.[/*][*]Because B(x) is equal to 10x+25 and B(x) is also equal to 50, we can write 10x+25=50 and solve the equation.[/*][/list]If time permits, invite students to share which option they believe the college student should choose and why.[br]
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– LeonMilewski

Information: A.4.5.2 Data Plans