The Free Month That Shifts a Line
Mr. Boateng's eighth-grade algebra class is comparing two gym memberships during a unit on linear functions. Gym A charges a $30 enrollment fee, $25 per month, and waives the monthly fee for the first month. Gym B charges $90 to enroll and $15 per month with no promotions. Students must determine when the two gyms cost the same and recommend a membership for someone staying a full year. During work time, Mr. Boateng notices that most students wrote 30 + 25m = 90 + 15m, solved it to get m = 6, and concluded that Gym B is cheaper after six months. Only two students accounted for the free month, and one of those subtracted $30 instead of $25. The class graphs lines and solves two-step equations reliably, but interpreting a non-integer solution in context was flagged as a weakness on the last district benchmark.
1. Write the cost functions for both gyms, find the break-even point, and interpret the result in context.
Read the worked answer
Let m be the number of months of membership, m ≥ 1.
Gym A: A(m) = 30 + 25(m − 1) = 25m + 5. The free month means m − 1 payments of $25 after the enrollment fee. Gym B: B(m) = 90 + 15m.
Set equal: 25m + 5 = 90 + 15m, so 10m = 85 and m = 8.5.
Interpretation: no whole number of months makes the costs exactly equal. Check adjacent months. At m = 8: A = 25(8) + 5 = $205; B = 90 + 15(8) = $210. At m = 9: A = $230; B = 225. So Gym A is cheaper through month 8, the lines cross partway through month 9, and Gym B is cheaper from month 9 onward.
Check the endpoints too: at m = 1, A costs $30 (enrollment only) while B costs $105, so A starts far ahead. The slopes tell the rest of the story: A grows $25 per month and B $15 per month, so B must eventually overtake A, and the only question is when. The non-integer break-even is a signal to test whole-month values, not a number to round silently to 8 or 9.
2. Diagnose the m = 6 answer. What assumption did students make, and what is the most efficient way to show them it is wrong?
Read the worked answer
Students modeled Gym A as 30 + 25m, which charges for all m months and ignores the waiver. Equivalently, they treated the promotion as affecting the enrollment fee rather than one month's dues. Setting 30 + 25m equal to 90 + 15m gives 10m = 60, so m = 6, with both sides equal to $180.
The fastest refutation is a substitution check at an easy value, m = 1. The students' model predicts A(1) = 30 + 25 = $55, but the actual cost is $30, since the first month is free. A model that misses a known data point cannot be right. A two-row table for m = 1, 2, 3 makes the gap concrete and shows the true A-values sitting $25 below the students' line at every month - a vertical translation, not a slope change.
The one student who subtracted $30 instead of $25 has conflated the fee with the monthly rate: dropping the intercept by $30 gives A(m) = 25m, which would make the first month cost $25 rather than $30 and shift the break-even again. Both errors are modeling errors about what the promotion attaches to, and both are caught by checking a single month against the real situation.
3. Make a recommendation for a 12-month membership, and explain whether it would change at 6 months. What does the non-integer break-even mean for the decision?
Read the worked answer
At 12 months: A = 25(12) + 5 = $305; B = 90 + 15(12) = $270. Gym B saves $35 over the year, about 11% of Gym A's cost.
At 6 months: A = 25(6) + 5 = $155; B = 90 + 15(6) = $180. Gym A saves $25. So the recommendation does flip, and the flip happens where the lines cross, between month 8 and month 9, not at month 6.
The non-integer solution means the decision is genuinely sensitive to the horizon. A member who says "a year" gets B; a member who says "the school year, about nine months" is nearly indifferent ($230 versus $225) and should weigh other factors - contract terms, cancellation fees, location. A common student error is to round 8.5 up and declare B cheaper "after 8 months," which is false for month 8. The defensible statement is comparative and piecewise: A for 8 months or fewer, B for 9 months or more, with the crossover occurring mid-month. Asking students to state the recommendation as a conditional sentence, rather than a single gym name, is the assessment move that reveals whether they understand the intersection.