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Math problem set Grade 8 Science

Dunmore Desalination Energy: Two Linear Functions Compared

A grade 8 science problem set where students model a fictional town's reverse-osmosis and conventional plant energy use as linear functions and compare the two slopes.

Objectives

  1. Students will construct a linear function relating water produced to energy used for each treatment method.
  2. Students will compare the two slopes to describe the energy trade-off of seawater desalination.

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.
Desalination
Turning salty water, like ocean water, into fresh water people can use. It works, but it takes a lot of energy and leaves behind very salty water that must be handled carefully.

Also: slope, linear function, kilowatt-hour

Teacher overview

Turning seawater into drinking water with reverse osmosis gives a coastal town a supply that a drought cannot dry up, but pushing water through the membranes takes energy. In this set, students compare the energy use of the fictional town of Dunmore's new seawater reverse-osmosis plant with its older conventional plant, writing each as a linear function of the water produced. Seawater reverse osmosis uses roughly 11 to 15 kWh per 1,000 gallons, while conventional treatment and delivery uses about 1 to 2 kWh per 1,000 gallons. A common misconception in this math is treating the difference between the two rates as a single fixed total; students learn that the gap grows with every thousand gallons because it is a difference in slopes, not a constant amount.

Dunmore Desalination Energy: Two Linear Functions Compared — Grade 8 Science
Name: ____________________________ Date: ______________

Problems

Suppose the fictional town of Dunmore's seawater reverse-osmosis plant uses 13 kWh of electricity for every 1,000 gallons it produces, and its older conventional plant used 1.5 kWh for every 1,000 gallons. Let g stand for thousands of gallons produced and E stand for energy in kWh.

  1. Write a linear function E = m × g for the reverse-osmosis plant. What is m, and what does it mean?
  2. Write the linear function E = m × g for the conventional plant. What is m?
  3. The reverse-osmosis plant produces 20,000 gallons (g = 20). How many kWh does it use?
  4. How many kWh would the conventional plant use to produce the same 20,000 gallons?
  5. How many more kWh does reverse osmosis use than conventional for those 20,000 gallons?
  6. Compare the slopes. For each additional 1,000 gallons, how many more kWh does reverse osmosis use than conventional?
  7. In one day the reverse-osmosis plant produces 50,000 gallons (g = 50). Find the energy the reverse-osmosis plant uses and the energy the conventional plant would use for the same amount.
  8. Explain the trade-off Dunmore accepts by choosing seawater reverse osmosis, using the idea that its energy line is steeper.

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

Answer Key
  1. Answer: m = 13 kWh per 1,000 gallons.
    The function is E = 13 × g; the slope 13 means each 1,000 gallons of reverse-osmosis water takes 13 kWh.
  2. Answer: m = 1.5 kWh per 1,000 gallons.
    The function is E = 1.5 × g; the slope 1.5 means each 1,000 gallons of conventional water takes 1.5 kWh.
  3. Answer: 260 kWh.
    E = 13 × 20 = 260 kWh.
  4. Answer: 30 kWh.
    E = 1.5 × 20 = 30 kWh.
  5. Answer: 230 kWh.
    260 − 30 = 230 kWh more for reverse osmosis over those 20,000 gallons.
  6. Answer: 11.5 kWh.
    13 − 1.5 = 11.5, so reverse osmosis uses 11.5 more kWh for each additional 1,000 gallons.
  7. Answer: 650 kWh and 75 kWh.
    Reverse osmosis: 13 × 50 = 650 kWh. Conventional: 1.5 × 50 = 75 kWh for the same 50,000 gallons.
  8. Answer: A steeper energy line for a drought-proof supply.
    Because the reverse-osmosis slope (13) is larger than the conventional slope (1.5), its energy climbs faster with every thousand gallons. Dunmore accepts higher and faster-growing energy use in exchange for a seawater supply that does not depend on rainfall.

Differentiation

Support

Provide both functions already written and a filled-in first row of a table so students substitute g values rather than build the equations. Allow a calculator and let students check problems 3 and 4 against each other before continuing.

Extension

Ask students to solve 13 × g = 390 kilowatt-hours to find how many thousands of gallons the reverse-osmosis plant makes on 390 kilowatt-hours, then find how much water the conventional plant could make on that same energy and explain what the comparison shows about the cost of desalination.

ELL supports

Pre-teach "reverse osmosis," "desalination," and "slope" with a simple sketch of salty water pushed through a membrane into fresh water. Provide sentence frames for item 8, such as "Reverse osmosis uses ___ energy because ___, but it gives ___."