Break-Even Analysis with Two Membership Plans
Mr. Chen’s algebra class is comparing two fitness-club plans. Club A charges a $25 sign-up fee plus $8 per month; Club B charges a $45 sign-up fee plus $5 per month. Students graph C = 25 + 8m and C = 45 + 5m. One group solves 25 + 8m = 45 + 5m, gets m = 6.67, and concludes that Club B is cheaper after 6 months because 6.67 rounds to 6. Another student says the y-intercept is the monthly fee. A third says Club B is always cheaper because its slope is smaller. The class has 30 students and is about to write a recommendation for a family choosing a one-year membership. Mr. Chen wants the discussion to address domain, rounding, and interpretation of slope and intercept rather than only the algebraic solution.
1. Evaluate the three student claims. Which are mathematically defensible, which are not, and what precise language should replace each incorrect claim?
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The claim that Club B is always cheaper is false. A smaller slope means Club B’s cost grows more slowly, but its higher intercept makes it more expensive at the start. The intercepts are sign-up fees, not monthly fees, so the second student is misreading the parameters: slope is the monthly rate and intercept is the one-time charge. The first group’s algebra is correct but the contextual rounding is not. At m = 6, Club A costs 25 + 48 = 73 and Club B costs 45 + 30 = 75, so A is cheaper. At m = 7, A costs 81 and B costs 80, so B is cheaper. Since monthly billing is discrete, B is cheaper for whole-month memberships from month 7 onward, not from month 6. Precise wording: ‘The plans break even at about 6.67 months; for 7 or more months, Club B costs less.’
2. Write a follow-up question that requires students to use the context to interpret the break-even value and the intercepts. Explain what a complete response would include.
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Follow-up: ‘A family plans to join for exactly 12 months. Find each total cost, the savings from the cheaper plan, and explain what the break-even value means. State a sensible domain for m.’ A complete response computes Club A: 25 + 8(12) = 121; Club B: 45 + 5(12) = 105; savings = 16 dollars. It explains that the break-even value near 6.67 months is where the costs are equal, but because memberships are paid monthly, the practical switch occurs at 7 months. It identifies the intercepts as upfront fees and the slopes as monthly rates. It restricts the domain to whole nonnegative months for this billing context, or at least m ≥ 0 if continuous time is allowed. A response that only solves the equation without interpreting the parameters is incomplete.
3. Mr. Chen has one class period before the unit test. He can either assign 10 more equations or use two contrasting graphs. Which is more defensible, and how would he check whether the misconception persists?
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The contrasting graphs are more defensible because the class’s weakness is interpretation, not equation solving. Most students already solved the equation, but they misread the intercept, rounded the break-even value without checking costs, or treated slope alone as a complete comparison. Two graphs on the same axes, with intercepts and intersection labeled, let students see why the lower-slope line can start higher and become cheaper later. Mr. Chen should pair the graphs with a short table at m = 0, 6, 7, and 12 and ask for a contextual recommendation. To check whether the misconception persists, use an exit prompt with a new pair of plans and ask for the first whole month at which the second plan is cheaper. A strong response checks actual costs at the neighboring whole months rather than rounding the intersection automatically.