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Math problem set Grade 7 Math

Rookery Point Desalination: An Energy Proportionality Set

A grade 7 problem set at the fictional Rookery Point desalination pilot, where students test a table for proportionality, find the constant, and write an equation.

Objectives

  1. Students will decide whether a table shows a proportional relationship by testing equal ratios.
  2. Students will identify the constant of proportionality and write it as an equation.

Vocabulary

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: proportional, constant of proportionality

Teacher overview

Turning seawater into drinking water takes a lot of electricity, and for a well-behaved plant the energy used grows in direct proportion to the water treated. This set uses the fictional Rookery Point desalination pilot, giving students an energy-versus-volume table to test for proportionality, a constant to find, and an equation to write and use. The set also includes a second pump's table that looks similar but is not proportional. The misconception this set is built to correct is treating any rising table as proportional: students must check that the ratio of energy to volume is the same in every row, because a fixed startup cost makes bigger volumes still use more energy without the relationship being proportional.

Rookery Point Desalination: An Energy Proportionality Set — Grade 7 Math
Name: ____________________________ Date: ______________

Problems

Rookery Point is a fictional desalination pilot. It records the energy in kWh used to treat seawater, with volume measured in thousands of gallons. Here is the log for its main pump:

Volume (thousands of gallons) Energy used (kWh)
2 7
4 14
6 21
10 35
  1. For each row, divide the energy by the volume. Are all four ratios equal?
  2. What is the constant of proportionality, in kWh per 1,000 gallons?
  3. Write an equation for the energy E in terms of the volume V, where V is in thousands of gallons.
  4. Use your equation to find the energy to treat 8,000 gallons, where V is 8.
  5. Use your equation to find the volume treated when the pump uses 49 kWh.
  6. A second pump logs 12 kWh at volume 2, 20 kWh at volume 4, and 28 kWh at volume 6. Is this pump's relationship proportional? Explain using the ratios.
  7. Back on the main pump, the pilot treats 20,000 gallons in one day. How many kWh does that take?
  8. The pilot's engineers expected the main pump to use between 3 and 4 kWh per 1,000 gallons. Does the main pump's rate fall in that range?

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: Yes; every ratio equals 3.5.
    7 ÷ 2 = 3.5, 14 ÷ 4 = 3.5, 21 ÷ 6 = 3.5, and 35 ÷ 10 = 3.5. All four ratios are equal, so the relationship is proportional.
  2. Answer: 3.5 kWh per 1,000 gallons.
    The equal ratio from problem 1 is the constant of proportionality, 3.5 kWh for every 1,000 gallons.
  3. Answer: E = 3.5V.
    Energy equals the constant times the volume, so E = 3.5V, with V in thousands of gallons.
  4. Answer: 28 kWh.
    E = 3.5 × 8 = 28 kWh.
  5. Answer: 14 thousand gallons.
    49 = 3.5 × V, so V = 49 ÷ 3.5 = 14, which is 14,000 gallons.
  6. Answer: No.
    The ratios are 12 ÷ 2 = 6, 20 ÷ 4 = 5, and 28 ÷ 6 = 4.67. They are not equal, so the second pump is not proportional; it has a fixed startup energy on top of a per-volume rate.
  7. Answer: 70 kWh.
    20,000 gallons is V = 20, so E = 3.5 × 20 = 70 kWh.
  8. Answer: Yes.
    The main pump uses 3.5 kWh per 1,000 gallons, which falls between the engineers' expected 3 and 4 kWh per 1,000 gallons.

Differentiation

Support

Add a third column to the table labeled "energy ÷ volume" and have students fill it in before answering problem 1. Seeing the same 3.5 appear in every row makes the constant of proportionality concrete before they are asked to name it or write the equation.

Extension

Ask students to write the equation for the second, non-proportional pump in the form E = startup + rate × V by finding its fixed startup energy and its per-volume rate from the log. Have them explain how this equation differs from a proportional one.

ELL supports

Pre-teach "proportional" with two side-by-side tables, one where the ratio stays the same and one where it does not, each marked "same ratio" or "changing ratio." Provide the sentence frame "the ratio is ___ in every row, so it is (is not) proportional" for problems 1 and 6.

Extensions

Have students compare the pilot's per-gallon energy rate for seawater with the roughly 1 to 2 kWh per 1,000 gallons that conventional treatment and delivery use, drawing both as lines on the same graph. The steeper line for the seawater plant shows, visually, why turning seawater into drinking water costs more energy per gallon. Full-scale seawater plants use roughly 11 to 15 kWh per 1,000 gallons, so a line for a real plant would be steeper still.