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Worksheet Grade 9 Math

Fenwick PFAS Carbon Filter: Parts-per-Trillion Removal Worksheet

A grade 9 PFAS worksheet: students use percent removal and a linear breakthrough model to keep Fenwick's carbon filter below the parts-per-trillion limit for PFOA.

Objectives

  1. Students will calculate percent removal needed to bring PFOA below its federal limit.
  2. Students will use a linear model to predict when a carbon filter reaches breakthrough.
  3. Students will interpret PFAS results against a limit whose compliance timeline regulators are still revising.

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.
Activated carbon
Activated carbon is a special charcoal full of microscopic holes that grab and hold chemicals as water flows past. It is the same material used in many home water filter pitchers.

Also: percent removal, breakthrough

Teacher overview

The fictional town of Fenwick removes PFAS from its drinking water with a granular activated carbon filter. The federal limit for PFOA is 4.0 ppt. Fenwick's raw water carries more PFOA than that, so the filter must bring the level down and hold it down as the carbon ages. This worksheet uses percent removal and a simple linear model of "breakthrough" to decide when the carbon has to be replaced.

Fenwick PFAS Carbon Filter: Parts-per-Trillion Removal Worksheet — Grade 9 Math
Name: ____________________________ Date: ______________

Student Worksheet

Fenwick (a fictional town) measures PFOA in parts per trillion (ppt). Its raw water holds 20 ppt of PFOA. The federal limit, called the MCL, is 4.0 ppt. A fresh carbon filter removes 95 percent of the PFOA at startup.

  1. By how many ppt must treatment lower the PFOA to reach the 4.0 ppt limit, starting from 20 ppt?
  2. What percent removal brings 20 ppt down to exactly 4.0 ppt? Use (in - out) / in x 100.
  3. The fresh filter removes 95 percent of the PFOA. What is the treated level, in ppt?
  4. Is the startup treated level below the 4.0 ppt limit? By how many ppt?
  5. As the carbon ages, treated PFOA rises. Model it as C = 1.0 + 0.25m, where C is treated PFOA in ppt and m is months of use. Find C after 8 months.
  6. Breakthrough is the month when treated PFOA reaches the 4.0 ppt limit. Solve 1.0 + 0.25m = 4.0 for m.
  7. Fenwick replaces the carbon before breakthrough. Explain why running the filter past breakthrough would be a problem.
  8. A second carbon tank is added after the first. The first lowers 20 ppt by 95 percent to 1.0 ppt. The second removes another 90 percent of what is left. What is the final PFOA level?
  9. PFAS rules are still changing. EPA set the 4.0 ppt limit in 2024 and gave water systems until April 2029 to comply. In May 2026 EPA proposed letting eligible systems apply for a two-year exemption, to April 2031, but that proposal was not final as of September 2026. Give one reason Fenwick might act before the deadline anyway.
  10. Suppose the raw water rises to 30 ppt. What percent removal is now needed to reach the 4.0 ppt limit? Round to the nearest whole percent.

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: 16 ppt. 20 ppt - 4.0 ppt = 16 ppt of PFOA to remove.
  2. Answer: 80 percent. (20 - 4) / 20 x 100 = 16 / 20 x 100 = 80%.
  3. Answer: 1.0 ppt. 20 x (1 - 0.95) = 20 x 0.05 = 1.0 ppt.
  4. Answer: Yes, by 3.0 ppt. The 1.0 ppt treated level sits under the 4.0 ppt limit by 4.0 - 1.0 = 3.0 ppt.
  5. Answer: 3.0 ppt. C = 1.0 + 0.25(8) = 1.0 + 2.0 = 3.0 ppt.
  6. Answer: 12 months. 1.0 + 0.25m = 4.0, so 0.25m = 3.0 and m = 12 months.
  7. Answer: Past breakthrough the treated water rises above the 4.0 ppt limit, so the filter would send water over the MCL to customers.
  8. Answer: 0.10 ppt. The second tank leaves 1.0 x (1 - 0.90) = 1.0 x 0.10 = 0.10 ppt.
  9. Answer: Acting early keeps PFOA low for customers now and avoids a rushed, costly project if the exemption is never finalized or Fenwick does not qualify for it, which would leave the April 2029 deadline in place.
  10. Answer: 87 percent. (30 - 4) / 30 x 100 = 26 / 30 x 100 = 86.67%, which rounds to 87%.

Differentiation

Support

Give students the percent-removal formula (in - out) / in x 100 on a reference card, worked once with sample numbers that do not appear on the worksheet. Allow a calculator, and provide the aging model already written out for items 5 and 6 so students can focus on solving.

Extension

Ask students to rewrite the aging model so that treated PFOA starts at half the startup level and rises twice as fast. Have them find the new breakthrough month and compare it with the 12-month result from item 6.

ELL supports

Pre-teach "removal," "treated," "raw water," and "breakthrough" with a simple diagram showing dirty water going in, an arrow through a carbon tank, and clean water coming out. Provide a sentence frame for item 7: "Running past breakthrough is a problem because ___."