Math problem set Grade 11 Math
Cobalt Bay Desalination: Correlating Production and Energy Use
A grade 11 problem set where students compute and interpret a correlation coefficient between a fictional desalination plant's production and energy use.
Objectives
- Students will calculate a correlation coefficient from raw data using the standard formula.
- Students will interpret the strength and direction of a correlation coefficient in a real water-treatment context.
Vocabulary
- Reverse osmosis
- Reverse osmosis cleans water by pushing it through a membrane with holes so tiny that salt and most other dissolved things cannot fit through. It can even turn seawater into fresh water.
Also: correlation coefficient, positive correlation, causation
Teacher overview
Cobalt Bay Desalination Facility, a fictional reverse-osmosis plant used for this problem set, tracks its daily production against how much energy it uses, since forcing seawater through a membrane takes far more energy than conventional treatment. A common misconception in this set is assuming that a strong correlation coefficient proves production directly causes higher energy use; correlation measures how tightly two variables move together, not what causes what, so this set ends by asking students to name a factor that could weaken that relationship without changing the underlying cause.
Problems
The fictional Cobalt Bay facility recorded its production, in millions of gallons per day (MGD), and its energy use, in thousand kilowatt-hours (kWh), over five sampled days:
| Day | Production, x (MGD) | Energy use, y (thousand kWh) |
|---|---|---|
| 1 | 2 | 32 |
| 2 | 4 | 48 |
| 3 | 6 | 88 |
| 4 | 8 | 104 |
| 5 | 10 | 144 |
Use the correlation coefficient formula: r = (nΣxy − ΣxΣy) ÷ the square root of [(nΣx² − (Σx)²)(nΣy² − (Σy)²)], where n is the number of data points.
- Calculate Σx and Σy for the five days of data.
- Calculate Σxy, Σx², and Σy² for the five days of data.
- Use your sums from problems 1 and 2 in the correlation coefficient formula to calculate r. Round to the nearest hundredth.
- Based on your r-value, describe the relationship between production and energy use as weak, moderate, or strong, and as positive or negative.
- Reverse osmosis typically uses about 11 to 15 thousand kWh per MGD produced. Estimate Cobalt Bay's rate using its first and last data points (day 1 and day 5): what is the average change in energy use per MGD of added production? Does it fall within the typical range?
- A conventional treatment plant producing the same 10 MGD as Cobalt Bay's day 5 would use only about 1 to 2 thousand kWh per MGD, or 10 to 20 thousand kWh total. Using Cobalt Bay's actual day 5 energy use (144 thousand kWh), calculate how many more thousand kWh Cobalt Bay used compared to the top of that conventional range.
- Name one factor, other than production volume itself, that could weaken the correlation you calculated, causing energy use to not track production as tightly in a real facility.
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: Σx = 30, Σy = 416. Σx = 2 + 4 + 6 + 8 + 10 = 30 (MGD). Σy = 32 + 48 + 88 + 104 + 144 = 416 (thousand kWh).
Answer: Σxy = 3,056, Σx² = 220, Σy² = 42,624. Σxy = (2×32) + (4×48) + (6×88) + (8×104) + (10×144) = 64 + 192 + 528 + 832 + 1,440 = 3,056. Σx² = 4 + 16 + 36 + 64 + 100 = 220. Σy² = 1,024 + 2,304 + 7,744 + 10,816 + 20,736 = 42,624.
Answer: r ≈ 0.99.
Numerator: (5 × 3,056) − (30 × 416) = 15,280 − 12,480 = 2,800. First bracket: (5 × 220) − 30² = 1,100 − 900 = 200. Second bracket: (5 × 42,624) − 416² = 213,120 − 173,056 = 40,064. Denominator: the square root of (200 × 40,064) = the square root of 8,012,800 ≈ 2,830.69. r = 2,800 ÷ 2,830.69 ≈ 0.99.Answer: strong positive correlation.
An r-value of about 0.99 is very close to 1, meaning production and energy use rise together almost perfectly consistently across the five sampled days.Answer: about 14 thousand kWh per MGD.
From day 1 (2 MGD, 32 thousand kWh) to day 5 (10 MGD, 144 thousand kWh): change in y is 144 − 32 = 112; change in x is 10 − 2 = 8; 112 ÷ 8 = 14 thousand kWh per MGD, which falls within the typical 11 to 15 thousand kWh per MGD range for reverse osmosis.Answer: 124 thousand kWh.
144 (Cobalt Bay's actual day 5 energy use) − 20 (top of the conventional range for 10 MGD) = 124 thousand kWh more.Answer: short answer.
Factors such as changes in feedwater salinity or temperature, membrane fouling, or equipment maintenance could raise or lower energy use on a given day without a matching change in production volume, which would pull the data points away from a tight line and lower the correlation coefficient.
Differentiation
Support (below grade level)
Provide a table with the xy, x², and y² columns already labeled and partially filled in, so students focus on completing the pattern and summing the columns rather than setting up the table from scratch. Allow a calculator for the square root step in problem 3.
Extension (above grade level)
Ask students to add a sixth day with production of 12 MGD and an energy use value of their own choosing, then recalculate r with the new point included and explain whether their chosen value strengthened or weakened the correlation.
ELL supports
Pre-teach "correlation," "strong," and "weak" with a simple visual: three scatter plots side by side showing points tightly clustered along a line, loosely scattered, and randomly placed, labeled strong, moderate, and weak. Allow students to state their answer to problem 4 as a short phrase paired with the calculated r-value rather than a full sentence.