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Project Grade 12 Math

Jarrowdene Drought Reliability: A Simulated Ten-Year Study

A six-session grade 12 project in which students simulate reservoir storage under three drought policies and report a failure probability with its margin of error.

Objectives

  1. Students will build a recursive storage model bounded by a reservoir capacity and a critical level.
  2. Students will estimate a failure probability from repeated simulated decades and attach a margin of error.
  3. Students will compare three drought policies and defend a recommendation to a district board.

Vocabulary

Acre-foot
The amount of water it takes to cover one acre of land, about the size of a football field, one foot deep. Western states use acre-feet to measure large amounts of water.

Also: reliability, simulation trial, margin of error, trigger policy

Teacher overview

Water districts do not decide drought policy from what happens in an average year, because what empties a reservoir is a run of dry years rather than the long-run average. They ask instead how often a policy fails across many possible decades, and they answer with simulation. Over six sessions students build a spreadsheet model of a fictional reservoir, run repeated ten-year sequences under three candidate policies, and report the probability that storage falls to a critical level at least once, together with an honest margin of error. The deliverable is a written reliability report and a short presentation to a fictional district board, which must include a recommendation the board could act on.

Jarrowdene Drought Reliability: A Simulated Ten-Year Study — Grade 12 Math
Name: ____________________________ Date: ______________

Project Brief

Driving question. How often would each of three drought policies fail the fictional Jarrowdene Water District over a ten-year stretch, and how confident can the district be in that estimate?

Client. The board of the fictional Jarrowdene Water District, which draws its entire supply from one reservoir and must adopt a drought policy before the next irrigation season. Your team is presenting as the district's planning consultants.

The system. Every figure below describes the fictional district and is fixed for this project.

  • Reservoir capacity: 42,000 acre-feet. Storage can never exceed it; water above capacity spills and is lost.
  • Critical level: 9,000 acre-feet. Storage falling to or below this level in any year counts as a failure.
  • Starting storage: 30,000 acre-feet at the beginning of year 1.
  • Base annual demand: 15,000 acre-feet.
  • Annual evaporation loss: 1,200 acre-feet, taken every year regardless of policy.
  • Annual inflow model: 22,000 acre-feet with probability 0.20, 17,000 acre-feet with probability 0.35, 12,000 acre-feet with probability 0.30, and 7,000 acre-feet with probability 0.15.
  • The district's forty-year record: inflow was near 22,000 acre-feet in 7 years, near 17,000 in 15 years, near 12,000 in 12 years, and near 7,000 in 6 years.

Storage recursion. Next year's storage equals this year's storage plus inflow, minus demand under the policy in force, minus evaporation, with the result capped at capacity and reported as a failure if it reaches the critical level.

The three policies.

  • Policy A, no restriction: demand stays at 15,000 acre-feet every year.
  • Policy B, standing conservation: demand is cut 10 percent to 13,500 acre-feet in every year, drought or not.
  • Policy C, trigger-based restriction: demand stays at 15,000 acre-feet while storage at the start of the year is above 18,000 acre-feet, and drops to 12,000 acre-feet in any year that begins at or below that trigger.

Constraints on the work. Each policy must be run for at least 200 simulated ten-year sequences. Every reported probability must carry a margin of error your team can justify from your own repeated runs, not from a formula you cannot explain. Your recommendation must weigh the reliability figures against the fact that Policy B asks customers to conserve in wet years as well as dry ones.

Final deliverable. A written reliability report of four to six pages and a six-minute presentation to the board. The report must contain a labeled chart comparing the three policies, a statement of your simulation method detailed enough that another team could repeat it, the estimated failure probability and margin of error for each policy, and one recommendation with its trade-offs stated plainly.

Milestones

  1. Session 1, build and verify the model. Each team builds the storage recursion in a spreadsheet and hand-checks one ten-year sequence against a fixed inflow list supplied by the teacher, so that a single arithmetic path is confirmed before randomness is added. Output: one verified ten-year table with the cap and the critical-level test working.
  2. Session 2, test the inflow model against the record. Teams simulate forty draws from the stated inflow model many times over and compare the resulting spread of category counts with the district's forty-year record. Output: a written half-page judgment on whether the record is consistent with the model, with the simulated evidence attached.
  3. Session 3, run Policy A and Policy B. Teams attach the random inflow draw to the recursion and run at least 200 ten-year sequences for each of the two fixed-demand policies. Output: two failure counts, each with the number of trials that produced it, and a first histogram of ending storage.
  4. Session 4, run Policy C and estimate uncertainty. Teams implement the trigger rule, run at least 200 sequences, and then repeat every policy's 200-trial estimate at least ten times to see how much the estimate itself moves. Output: a table of three failure probabilities with a margin of error for each, justified by the spread of the repeated estimates.
  5. Session 5, draft the report and the chart. Teams write the method section and the results section and build the comparison chart with labeled, scaled axes. Output: a complete draft report exchanged with another team for a written review against the rubric.
  6. Session 6, revise and present to the board. Teams revise from the peer review and deliver the six-minute presentation to the class acting as the fictional Jarrowdene board, which asks at least one question per team about uncertainty. Output: the final report and the delivered presentation.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Rubric

Criterion 1: Model construction and constraints.

  • Level 4: The recursion is correct, the capacity cap and critical-level test both work, and the hand-checked sequence in milestone 1 matches the spreadsheet exactly. Every constraint from the brief appears in the model.
  • Level 3: The recursion and both bounds are implemented correctly, with a minor arithmetic or labeling slip that does not change any result.
  • Level 2: The recursion works but one bound is missing or misapplied, so some storage values exceed capacity or a failure goes uncounted.
  • Level 1: The model does not carry storage forward correctly from year to year, and results cannot be interpreted.

Criterion 2: Testing the inflow model against the record.

  • Level 4: The team simulates enough sets of forty draws to describe the ordinary spread of category counts, compares the record against that spread, and states a clear judgment with its reasoning and its limits.
  • Level 3: The team simulates the draws and reaches a defensible judgment, but describes the spread loosely rather than with a stated range.
  • Level 2: The team compares the record with the model's expected counts only, without simulating any spread.
  • Level 1: The record and the model are asserted to agree or disagree with no evidence offered.

Criterion 3: Reliability estimates and stated uncertainty.

  • Level 4: All three policies are run at the required number of trials, the failure probabilities are reported with a margin of error the team derived from its own repeated estimates, and the report explains what that margin does and does not cover.
  • Level 3: All three policies are run and reported with a margin of error, but the justification for the margin is thin or partly borrowed.
  • Level 2: Probabilities are reported without any usable statement of uncertainty, or one policy is run at fewer than the required trials.
  • Level 1: Figures are reported that the team cannot reproduce or explain.

Criterion 4: Recommendation and presentation to the board.

  • Level 4: The recommendation follows from the reliability figures, names the trade-off that Policy B imposes conservation in wet years, stays within six minutes, and answers the board's question about uncertainty directly. The chart is labeled and scaled correctly.
  • Level 3: The recommendation follows from the results and the presentation is clear, with a chart that has a minor labeling problem or a trade-off mentioned only briefly.
  • Level 2: The recommendation is stated but rests on one figure rather than the comparison, or the chart is unlabeled.
  • Level 1: No recommendation is given, or the recommendation contradicts the team's own results.

Differentiation

Support

Give these students a starter spreadsheet for the fictional Jarrowdene reservoir with the columns laid out and the capacity cap formula already written, so their construction work is the demand rule and the failure test. Reduce the requirement to two policies, A and C, and supply a worked example of a margin of error computed from ten repeated estimates on a different data set.

Extension

Ask these students to add a fourth policy for the fictional district in which the trigger level itself is the variable, then find the trigger storage that minimizes failure probability without cutting demand in more than a third of the simulated years. Have them present that result as a curve of failure probability against trigger level rather than as a single number.

ELL supports

Pre-teach "reliability," "trial," "trigger," "margin of error," and "acre-foot" with a labeled diagram of the fictional reservoir marked with capacity, the trigger level, and the critical level. Provide a report outline with headed sections and sentence frames for the results, such as "Across 200 simulated decades, Policy ___ failed in ___ percent of them, give or take ___." Allow the presentation to be delivered from written notes.

Extensions

Teams can extend the simulation for the fictional district to a twenty-year horizon and report how each policy's failure probability changes, then explain why a longer horizon raises the chance of at least one failure even when nothing about the annual model changes. A second follow-up replaces the four-category inflow model with a continuous distribution fitted to the same record and asks teams to describe what the change does to the tails, which are the years that drive the failures.