The Fundraiser Pattern on the Office Board
During the spring fundraiser, Riverside Middle School posts the daily collection totals in the front office. Monday's total is $3, Tuesday's is $7, Wednesday's is $15, and Thursday's is $31. The principal has promised a pizza party if Friday's total follows the same pattern and is at least $60. Three students study the board. Elena says the rule is 'double the previous total and add $1.' Dev says the differences are $4, $8, and $16, so the next difference should be $24. Jonah says the pattern is 'add $4, then $8, then $16,' so Friday must be $47. The office manager will only release the bonus if the predicted total is justified by a rule that fits every posted day. The students have four data points and no written rule, so they must decide which pattern is best supported rather than merely possible.
1. Using the rule that fits all four days, what should Friday's total be? Show the arithmetic and explain why Dev's and Jonah's predictions do not fit the evidence.
Read the worked answer
Friday should be $63. Elena's rule, 'double the previous total and add $1,' fits every posted day: 3 × 2 + 1 = 7; 7 × 2 + 1 = 15; 15 × 2 + 1 = 31; 31 × 2 + 1 = 63. Equivalently, the differences are 4, 8, 16, and then 32, so each difference doubles.
Dev's +$24 would give 31 + 24 = 55. That assumes the differences are 4, 8, 16, 24, but 24 does not continue the doubling already visible in 4 → 8 → 16; it abandons the rule after three terms. Jonah's +$16 would give 31 + 16 = 47, which simply repeats the last difference instead of continuing the pattern. Both predictions are possible inventions, but neither uses the same repeatable transformation that explains all four known totals. Since $63 is at least $60, the pizza-party condition is met.
2. A classmate says any number could be justified by inventing a rule after the fact. How should the team decide which rule is 'best supported' when more than one rule can fit four points?
Read the worked answer
The team should prefer the rule that is simple, uniform, and predictive. A good rule uses the same operation at every step, explains the already-visible structure (here, the differences double), and does not need a new exception for each day. Invented rules can always be made to fit past data, so the test is whether the rule was chosen before seeing the next value and whether it makes a single clear prediction. A useful check is to ask: does the rule also explain why the differences are 4, 8, 16 rather than just producing those numbers? Elena's rule does. Dev's and Jonah's rules require abandoning the doubling of differences. If an extra posted total were available, the team should test the rule against it before releasing the bonus.