Classroom activity Grade 11 Math
Garrowmere Dam Data Stations: Linear, Power, or Exponential?
A 55-minute grade 11 station rotation where teams test four dam data sets with differences and ratios, then name and write the model family each one follows.
Objectives
- Students will test a data table for equal differences and for equal ratios over equal intervals.
- Students will identify a power relationship by dividing by the square of the input.
- Students will write a function for each data set and use it to predict beyond the table.
Vocabulary
Key words: first difference, common ratio, power model, seepage, trash rack
Teacher overview
Students who have met linear, exponential, and power models separately often cannot tell which one a new table belongs to, because each was introduced with its family name already attached. This rotation removes the label. Teams of three or four visit four stations, each holding one table of readings from the fictional Garrowmere Dam on the fictional Cadleigh River, and at each station they run the same three tests before deciding: first differences over equal steps, ratios of successive values, and the quotient of the output by the square of the input. The rotation mechanic maps onto the mathematics directly, since the four stations were chosen so that no single test settles all of them and a team that runs only one test will misclassify at least one station.
Activity Directions
Setup (5 min): Print the four fictional Garrowmere station cards and place them at four separated tables, each with a copy of the handout section for that station. Form teams of three or four and send one team to each station. Each team needs a calculator and graph paper. Announce that every station gets exactly ten minutes and that teams rotate on your signal.
Station 1, the reservoir during a steady release (10 min): The card gives reservoir elevation at five-day steps while the gates hold a fixed opening. Teams compute first differences, decide the family, write the function, and answer handout questions 1 through 3.
Station 2, turbine output against discharge (10 min): The card gives generator output at four discharges with the head held constant. Teams compute the quotient of output by discharge, decide the family, and answer handout questions 4 and 5.
Station 3, head loss across the trash rack (10 min): The card gives head loss at the same four discharges. Teams first compute the quotient of head loss by discharge, find that it is not constant, then divide by the square of the discharge instead, and answer handout questions 6 through 8.
Station 4, seepage after the gates close (10 min): The card gives measured seepage on five consecutive days. Teams compute ratios of successive readings, decide the family, and answer handout questions 9 through 11.
Debrief (10 min): Once every team has visited all four stations, ask each team to report the family it assigned to one station and the test that settled it. Post the four verdicts on the board and ask handout question 12 aloud. Close with two questions: which test would have misclassified station 3 if used alone, and why two of the four stations are both linear despite looking different on a graph.
Student Handout
All readings below come from the fictional Garrowmere Dam on the fictional Cadleigh River.
Station 1. Reservoir elevation during a steady release.
| Day | Elevation (ft) |
|---|---|
| 0 | 812.0 |
| 5 | 810.5 |
| 10 | 809.0 |
| 15 | 807.5 |
| 20 | 806.0 |
- Find the common first difference in elevation across one five-day step.
- Write the elevation E as a function of the number of days d.
- Use your function to predict the elevation on day 35.
Station 2. Generator output against discharge, head held constant.
| Discharge (cfs) | Output (MW) |
|---|---|
| 400 | 3.6 |
| 800 | 7.2 |
| 1,200 | 10.8 |
| 1,600 | 14.4 |
- Divide output by discharge for the 800 cubic feet per second row.
- Predict the output at a discharge of 2,000 cubic feet per second.
Station 3. Head loss across the trash rack.
| Discharge (cfs) | Head loss (ft) |
|---|---|
| 400 | 0.20 |
| 800 | 0.80 |
| 1,200 | 1.80 |
| 1,600 | 3.20 |
- Divide head loss by discharge for the 400 cubic feet per second row.
- Divide head loss by the square of the discharge for the 400 cubic feet per second row.
- Use the constant from question 7 to predict the head loss at 2,000 cubic feet per second.
Station 4. Seepage past the closed gates.
| Day | Seepage (cfs) |
|---|---|
| 0 | 160 |
| 1 | 120 |
| 2 | 90 |
| 3 | 67.5 |
| 4 | 50.625 |
- Divide each seepage reading by the one before it. Give the common ratio.
- Write the seepage S as a function of the number of days d.
- Predict the seepage on day 6, to one decimal place.
- Sort the four stations into linear, power, and exponential families.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
Answer: -1.5 ft.
Every five-day step drops the elevation by the same 1.5 feet, which is a constant first difference.Answer: E = 812.0 - 0.3d.
The drop is 1.5 feet across 5 days, or 0.3 feet per day, starting from 812.0 feet on day 0.Answer: 801.5 ft.
Substituting gives 812.0 - 0.3(35) = 812.0 - 10.5 = 801.5 feet.Answer: 0.009 MW per cfs.
Dividing gives 7.2 divided by 800 = 0.009, and the same quotient appears in every row, so the output is proportional to discharge.Answer: 18.0 MW.
Multiplying gives 0.009 x 2,000 = 18.0 megawatts.Answer: 0.0005 ft per cfs.
Dividing gives 0.20 divided by 400 = 0.0005. This quotient does not hold in the other rows, since 3.20 divided by 1,600 is 0.002, so head loss is not proportional to discharge.Answer: 1.25 x 10^-6 ft per cfs squared.
Dividing gives 0.20 divided by 160,000 = 0.00000125, and the same constant appears in every row, so head loss follows a power model with exponent 2.Answer: 5.0 ft.
Squaring gives 2,000 squared = 4,000,000, and 0.00000125 x 4,000,000 = 5.0 feet.Answer: 0.75.
Dividing gives 120 divided by 160 = 0.75, and every later pair returns the same ratio.Answer: S = 160(0.75) raised to the power d.
The reading on day 0 is 160 cubic feet per second and each following day keeps three-quarters of the day before.Answer: 28.5 cfs.
Raising 0.75 to the sixth power gives 0.177979, and 160 x 0.177979 = 28.5 cubic feet per second.Answer: Stations 1 and 2 are linear, station 3 is a power model, and station 4 is exponential.
Stations 1 and 2 have constant first differences over equal steps, station 3 has a constant quotient only after dividing by the square of the input, and station 4 has a constant ratio between successive readings.
Differentiation
Support
Give these teams station cards with the difference or ratio column already drawn and the first entry filled in, so the arithmetic pattern is visible before any decision is asked for. Provide a one-page decision chart naming the three tests in order, and let these teams answer only questions 1, 4, 9, and 12 if the ten minutes run short.
Extension
Ask these teams to take base-ten logarithms of both columns at the fictional Garrowmere stations 2 and 3 and to explain what the two slopes reveal about the exponents. Then have them design a fifth station whose data would defeat all three tests, and state what family that data would belong to.
ELL supports
Pre-teach "discharge," "seepage," and "head loss" with a labeled cross-section of a dam showing the reservoir, the intake screen, the turbine, and water leaking past a closed gate. Provide a station card glossary and sentence frames for question 12, such as "Station ___ is ___ because the ___ stayed the same."
Extensions
Students can graph all four fictional Garrowmere data sets on the same set of axes with a shared scale and describe which pairs are hardest to tell apart by eye. A second task is to explain why an operator would care that trash-rack head loss follows a squared relationship, using the day the reservoir must pass an unusually large flow as the example.