Interpreting an Intersection: When Is Plan B Cheaper?
An Algebra I class is comparing two streaming plans. Plan A charges $12 per month with no start-up fee. Plan B charges a $30 start-up fee plus $7 per month. The teacher asks students to determine when Plan B becomes cheaper and to justify the answer with an equation or graph. Student 1 writes 12x = 30 + 7x, solves x = 6, and concludes, 'Plan B is cheaper after 6 months.' Student 2 graphs both lines, sees they cross at (6, 72), and writes, 'At 6 months Plan B is cheaper because the line is lower.' Student 3 uses a table: at 5 months A costs $60 and B costs $65; at 6 months both cost $72; at 7 months A costs $84 and B costs $79. Student 3 concludes, 'Plan B is never cheaper at first, then it is cheaper after the point where they meet.' A fourth student says, 'Plan B has a smaller monthly rate, so it is always cheaper in the long run, but the start-up fee means it is not cheaper at the beginning.' The teacher must decide which responses are mathematically acceptable and what to emphasize next.
1. Evaluate Student 1's and Student 2's conclusions. Are they mathematically correct as written? How should the teacher respond?
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Student 1's algebra is correct: 12x = 30 + 7x gives 5x = 30, so x = 6. The conclusion 'cheaper after 6 months' is correct only if 'after' means strictly more than 6 months. At x = 6 both plans cost $72, so Plan B is not cheaper at 6 months. Student 2's conclusion is wrong as written: at 6 months the graphs intersect, so neither line is lower; Plan B is cheaper at month 7, when A costs $84 and B costs $79. The teacher should press on the words 'at' and 'after' and ask for the cost at x = 6 and x = 7. Student 2 may be reading the intersection as a point where one line is already below the other, a common graph-reading error.
2. Student 3's table is correct. What does 'after the point where they meet' need to mean in this discrete context, and how can the teacher use the table to sharpen the conclusion?
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Because the plans are billed monthly, 'after the point where they meet' must mean the next whole month, month 7, not month 6. At month 6 the costs are equal; at month 7 Plan B is cheaper. The table makes the discrete domain visible: there is no month 6.5 in the billing context. The teacher can ask: 'Is there any whole month before month 7 when B costs less than A?' and 'If the service prorated charges by the day, would the answer change?' This distinguishes a continuous linear model from a discrete payment model. Student 3's statement is acceptable if the teacher accepts 'after' as excluding the intersection; it should be sharpened to 'from month 7 onward.'
3. Student 4 claims Plan B is 'always cheaper in the long run.' Is that precise? Write a corrected statement and explain how the teacher could use the phrase 'in the long run' to connect to rate of change.
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Student 4's phrase is imprecise. Plan B is not 'always cheaper'; it is cheaper for x > 6 (or for whole months x ≥ 7) and more expensive for x < 6. 'In the long run' can be made precise by comparing slopes: Plan A increases by $12 per month, Plan B by $7 per month, so B closes the $30 start-up gap at $5 per month; 30 ÷ 5 = 6. A corrected statement is: 'Plan B has the lower monthly rate, so it becomes cheaper after the crossover at 6 months; it is not cheaper before that.' The teacher can use this to connect rate of change, initial value, and the meaning of the intersection to the long-run behavior of linear functions.