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Lesson plan Grade 11 Math

Sundale Reservoir PFAS: Making Sense of Parts Per Trillion

A 45-minute grade 11 math lesson using scientific notation to convert PFAS units, find the mass in a fictional reservoir, and compare readings to the federal limit.

Objectives

  1. Students will convert PFOA concentrations among parts per trillion, ng/L, and mg/L using scientific notation.
  2. Students will calculate the mass of PFOA in a reservoir from concentration and volume.
  3. Students will compare measured levels to the federal limit and compute percent reduction.

Vocabulary

PFAS
A large family of human-made chemicals used in things like nonstick pans, waterproof fabric, and firefighting foam. They are called forever chemicals because they take a very long time to break down.
Parts per trillion
A unit for measuring extremely small amounts of a chemical in water. PFAS limits are set in parts per trillion because these chemicals matter even in tiny doses.
Maximum contaminant level (MCL)
A maximum contaminant level, or MCL, is the highest amount of a certain substance that federal rules allow in drinking water. Water systems test regularly to prove their water stays under each limit.

Also: scientific notation

Teacher overview

PFAS limits are written in parts per trillion, a scale small enough that a reading is hard to interpret without first converting it into more familiar units. This lesson uses scientific notation to make that scale concrete for grade 11 students, working from a reading at Sundale reservoir, a fictional source. Students convert between units, then compute how few grams of a contaminant sit in an enormous reservoir, which explains why a "tiny" concentration can still exceed a health-based limit. A common student misconception is that a number as small as a few parts per trillion cannot matter; the mass and factor-over-limit calculations are built to test that belief with numbers.

Procedure

Hook (5 min): Ask students to guess how many drops of a contaminant in an Olympic pool would equal one part per trillion. Let a few guesses stand, then explain that the class will use scientific notation to reason about exactly these scales for a fictional reservoir.

Direct instruction (12 min): Establish the unit chain on the board: 1 part per trillion equals 1 nanogram per liter, which equals 1 times 10 to the negative sixth milligrams per liter. Note that EPA set the PFOA and PFOS limit at 4.0 parts per trillion in April 2024, with compliance due by April 2029. In May 2026 EPA proposed letting eligible systems apply for a two-year exemption, to April 2031; the 4.0 limit would not change, and the proposal was not yet final as of September 2026, so this rule is still in motion. Model converting 12 parts per trillion into milligrams per liter using scientific notation.

Guided practice (10 min): Together, compute the mass of contaminant in a small tank of stated volume, so students see concentration times volume produce a mass. Stress that concentration answers "how strong" while mass answers "how much total."

Independent practice (13 min): Students complete the Student Handout, converting units, computing the reservoir mass, comparing readings to the 4.0 parts-per-trillion limit, and finding the percent reduction that treatment achieves.

Closure and exit ticket (5 min): On an index card, students answer: "A source reads 8 parts per trillion of PFOA against a 4.0 limit. By what factor does it exceed the limit?" Collect the cards; the class is ready to proceed when at least 80% correctly answer 2 times the limit.

Sundale Reservoir PFAS: Making Sense of Parts Per Trillion — Grade 11 Math
Name: ____________________________ Date: ______________

Student Handout

Sundale reservoir is a fictional source. Use these values and conversions:

  • 1 part per trillion = 1 nanogram per liter = 1 times 10 to the negative sixth mg/L.
  • 1 gram = 1 times 10 to the ninth nanograms.
  • Reservoir volume: 500 million liters, or 5 times 10 to the eighth liters.
  • Raw water PFOA: 12 parts per trillion. Federal limit: 4.0 parts per trillion.
  • After granular activated carbon treatment, PFOA drops to 2 parts per trillion.
  1. Write the raw PFOA level of 12 parts per trillion in mg/L, using scientific notation.
  2. Convert 12 parts per trillion to nanograms per liter.
  3. The limit for PFOA is 4.0 parts per trillion. By what factor does the raw water at 12 parts per trillion exceed the limit?
  4. Find the mass of PFOA in the whole reservoir. Multiply the concentration of 12 nanograms per liter by the volume of 5 times 10 to the eighth liters. Give the answer in nanograms, in scientific notation.
  5. Convert your item 4 mass to grams.
  6. After treatment the PFOA is 2 parts per trillion. What percent reduction is that from 12 parts per trillion? Round to the nearest tenth.
  7. Is the treated water at 2 parts per trillion below the 4.0 limit, and by how much?
  8. A classmate says "12 parts per trillion is basically nothing, so treatment is a waste." Using your factor from question 3, give one quantitative reason the utility must still treat.

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: 1.2 times 10 to the negative fifth mg/L.
    12 parts per trillion x (1 times 10 to the negative sixth mg/L per part per trillion) = 1.2 times 10 to the negative fifth mg/L.
  2. Answer: 12 ng/L.
    Because 1 part per trillion equals 1 nanogram per liter, 12 parts per trillion equals 12 nanograms per liter.
  3. Answer: 3 times.
    12 parts per trillion divided by the 4.0 limit = 3, so the raw water is 3 times the limit.
  4. Answer: 6 times 10 to the ninth nanograms.
    12 nanograms per liter x 5 times 10 to the eighth liters = 60 times 10 to the eighth = 6 times 10 to the ninth nanograms.
  5. Answer: 6.0 grams.
    6 times 10 to the ninth nanograms divided by 1 times 10 to the ninth nanograms per gram = 6.0 grams in the entire reservoir.
  6. Answer: 83.3%.
    (12 minus 2) divided by 12 = 0.833, or 83.3% reduction.
  7. Answer: yes, by 2.0 parts per trillion.
    The treated water at 2 parts per trillion is below the 4.0 limit by 2.0 parts per trillion.
  8. Answer: The raw water is 3 times the enforceable limit.
    Even though 12 parts per trillion sounds small, it is 3 times the 4.0 federal limit, which is set that low because these compounds raise health concerns at trace levels, so the utility must treat.

Differentiation

Support

Provide a filled-in unit ladder showing parts per trillion, nanograms per liter, and milligrams per liter side by side with example values, and complete question 1 together before students continue. Allow calculators that display scientific notation directly.

Extension

Ask students to find how many grams of PFOA treatment removes from the reservoir when the level drops from 12 to 2 parts per trillion, and to compare that mass to the mass of a single grain of table salt they research from a reliable source.

ELL supports

Pre-teach "trace," "concentration," "mass," and "limit" with a visual contrasting a strong and weak dilution. Provide a sentence frame for question 8: "The utility must still treat because 12 parts per trillion is ___ the limit."

Extensions

Have students look up whether their own state reports PFAS results for local systems, then practice reading one published value and stating, in parts per trillion, how it compares to the 4.0 federal limit.