Your guardian offers you a one-year allowance contract and one chance to sign. Option A: twenty dollars a week, every week, all fifty-two weeks. Option B: one cent the first week — then the payment doubles every week after that. Choose now; no switching later.
The task
Commit before computing: every group member writes A or B on the board and signs it. Then find out what your signature cost you. Track both options two ways — the payment arriving each week, and the total collected so far — until your group can answer with evidence: in which week does B’s payment first beat A’s? In which week does B’s total catch A’s total? And the contract-lawyer question: your guardian, panicking, offers to cap the contract early — after which week does ending it stop being worth their while? End with a rule for B’s total after weeks that does not require adding numbers, and a sentence on why the two catch-up weeks are different.
Facilitation notes — for the teacher
The payment-versus-total distinction is the whole exploration; most groups conflate them at first, and the two different catch-up weeks force the separation. Let the doubling column overflow — the moment a weekly payment exceeds a million dollars is worth pausing on out loud. The shortcut for B’s total (each total is one cent less than the next payment) tends to be discovered numerically; ask groups to say why before showing any formula — it is the geometric series identity in its friendliest costume. A’s total, meanwhile, is arithmetic-series territory: pair the first and last weeks and watch the pairing trick reappear from Patterns day.
What mathematics tends to surface
Option A is an arithmetic story: constant payments, a total that grows linearly. Option B is geometric: constant ratio, a total that grows like its own payments. The shocking lateness of the crossover, followed by the violence of what comes after, is the felt difference between linear and exponential growth. B’s total-without-adding rule is the sum of a geometric series, met numerically before symbolically — Series does it properly, and Sequences and Their Rules holds both stories side by side.
Where it leads
Replace doubling weekly with multiplying by 1.05 yearly and Option B becomes a savings account — compound interest is this fable at a gentler ratio, as Money Over Time will show. Every annuity question in this unit is B’s total-column trick wearing a suit.
The answer is not on this page
Neither crossover week is printed here. Your signature is on the board; the evidence is your group’s to build.
Curriculum connection
C2.1
identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representation
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C2.4
solve problems involving arithmetic and geometric sequences and series, including those arising from real-world applications
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