Project Grade 7 Math
Marchbank Township's Private Well Survey Project
A four-session grade 7 project where students judge a sampling plan, estimate from a fictional township's 30 well tests, and report to its board.
Objectives
- Students will judge whether a proposed sampling plan represents the whole population of wells.
- Students will estimate a population share from a sample and state it as a probability.
- Students will compare estimates from repeated sub-samples and describe the variation.
Vocabulary
- Nitrate
- Nitrate is a form of nitrogen that plants use as food. It gets into water from fertilizer, animal waste, and septic systems, and too much of it is unhealthy for both people and streams.
Also: private well, population, random sample, representative
Teacher overview
Households on private wells test their own water, because a private well is not covered by the sampling a public utility must do. That leaves rural counties without a reliable count of how many wells carry more nitrate than the county would like to see, because only a handful have ever been tested. Over four sessions, students working for a fictional township board decide what a fair sample looks like, estimate from 30 tested wells, and hand the board a written recommendation with a probability statement in it. Teachers should tell students plainly that the review level in this project was invented for the scenario, and that real drinking-water limits are set by state and federal rules.
Project Brief
The fictional township of Marchbank has 240 homes on private wells. The board wants to know how many of those wells carry high nitrate. Nitrate reaches groundwater from fertilizer, manure, and failing septic systems. The board cannot pay to test all 240 wells. It paid for 30 tests instead.
For this project only, the board set a review level of 5.0 mg/L. A well at or above that level goes on a follow-up list. This review level is invented for the scenario and is not a legal standard.
Your team works as the board's advisers. Your final product is a four-slide report or a poster, delivered to the board in the last session. It must include all four of these:
- A short paragraph saying which sampling plan is fair, and why.
- The share of the 30 tested wells that are at or above the review level, written as a fraction, a decimal, and a percent.
- An estimate of how many of the 240 wells are at or above the review level, with your reasoning shown.
- A probability statement for one randomly chosen Marchbank well, written as a number between 0 and 1.
Here are the 30 test results the board paid for.
| Well | Nitrate (mg/L) | Well | Nitrate (mg/L) | Well | Nitrate (mg/L) |
|---|---|---|---|---|---|
| W01 | 0.4 | W11 | 4.9 | W21 | 0.3 |
| W02 | 1.2 | W12 | 8.1 | W22 | 4.2 |
| W03 | 6.8 | W13 | 2.7 | W23 | 9.4 |
| W04 | 2.1 | W14 | 1.1 | W24 | 2.9 |
| W05 | 0.9 | W15 | 5.9 | W25 | 1.7 |
| W06 | 5.4 | W16 | 0.8 | W26 | 5.2 |
| W07 | 3.3 | W17 | 3.6 | W27 | 3.1 |
| W08 | 7.2 | W18 | 2.4 | W28 | 0.7 |
| W09 | 1.8 | W19 | 6.1 | W29 | 6.5 |
| W10 | 0.6 | W20 | 1.5 | W30 | 2.2 |
Milestones
Session 1, choose a sampling plan. Two plans are on the table. Plan A tests the 30 wells nearest the township office, because a crew can reach them in one day. Plan B numbers all 240 wells and draws 30 numbers at random. Your team writes one paragraph naming the fairer plan and giving two reasons. Say what could go wrong with the other plan. Turn in the paragraph at the end of the session.
Session 2, estimate from the sample. Using the table of 30 results, count how many wells are at or above the review level of 5.0 mg/L. Write that count as a fraction of 30, then as a decimal, then as a percent. Scale the share up to all 240 wells and show the multiplication. Also find the mean and the median of the 30 readings, and write one sentence on why they differ. Turn in a worked page.
Session 3, test how steady the estimate is. Draw three separate random sub-samples of 10 wells from the 30, using slips or a spinner. Draw without replacement within a sub-sample, and return all 30 slips to the pool before starting the next sub-sample. For each sub-sample, find the share at or above the review level and the estimated number of wells out of 240. Record all three estimates in a table beside your session 2 estimate. Write two sentences on how much the estimates moved, and on which estimate you trust most.
Session 4, report to the board. Build the four-slide report or poster listed in the brief. Include your probability statement as a number between 0 and 1 and say in plain words what it means for one homeowner. Add one sentence recommending what Marchbank should do next, such as testing more wells or offering free tests in one part of the township. Present for four minutes and take questions.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Rubric
Score each criterion on the four levels below.
| Criterion | 4, exceeds | 3, meets | 2, approaching | 1, beginning |
|---|---|---|---|---|
| Sampling plan reasoning | Names Plan B as fairer, gives two sound reasons, and describes a specific way Plan A could skew the result. | Names Plan B as fairer and gives a sound reason tied to representing all 240 wells. | Names Plan B but the reason restates the plan rather than explaining fairness. | Names a plan with no reason, or chooses Plan A. |
| Share and estimate | Fraction, decimal, and percent all agree, and the scale-up to 240 wells is shown with correct multiplication and labeled units. | Share and scale-up are correct with minor labeling gaps. | One of the three forms is wrong, or the scale-up uses the wrong total. | Count of wells at or above the review level is wrong, so later work does not follow. |
| Variation across sub-samples | Reports three sub-sample estimates, compares them with the full-sample estimate, and explains why the larger sample is steadier. | Reports three sub-sample estimates and describes how much they moved. | Reports fewer than three sub-samples, or reports them without comparison. | Sub-samples are missing or not drawn at random. |
| Report and probability statement | Probability is between 0 and 1, matches the sample share, and is explained in plain words for a homeowner, with a specific next step for the board. | Probability is between 0 and 1 and matches the sample share, with a next step named. | Probability is stated but as a count or a percent above 1, or the meaning is unclear. | No probability statement, or it contradicts the team's own data. |
Teacher note on the provided data: 9 of the 30 wells are at or above 5.0 mg/L, which is 9/30, or 0.30, or 30 percent, and scales to about 72 of the 240 wells. The mean of the 30 readings is about 3.43 mg/L and the median is 2.8 mg/L, so the mean sits higher because a few large readings pull it up. Sub-sample results in milestone 3 will vary by team, which is the point of that session.
Differentiation
Support
Give these students a counting sheet with the 30 readings already sorted from smallest to largest, so the count at or above 5.0 mg/L becomes a matter of finding the cut point rather than scanning the whole table. Provide the scale-up as a filled proportion with one blank, and let them use a calculator for the mean.
Extension
Ask these students to work out how many additional wells the fictional board would need to test for its estimate to cover a quarter of the township, and to write a paragraph on why an estimate from 60 wells would be steadier than one from 30 without being twice as good.
ELL supports
Pre-teach "sample," "population," "estimate," and "at or above" using the 30-reading table with the review level marked by a highlighted line. Provide sentence frames for the report: "Out of ___ wells tested, ___ were at or above the review level," and "The chance that one well is at or above the review level is about ___."
Extensions
Have teams design a follow-up sampling plan that splits the fictional township into four areas and tests an equal number of wells in each, then explain in writing what that plan would show that a single random draw of 30 would miss.