Lesson plan Grade 9 Math

Delford's Two Wells: A PFAS Blending Equation Lesson

A grade 9 lesson where students write and solve a weighted-average blending equation to keep two fictional wells at or below the 4.0 ppt PFOA limit.

Grade: Grade 9
Subject: Math
Material: Lesson plan
Time: 50 min
Prep: Low prep
Materials: Printed student handout, Calculators

Standards

HSA-CED.A.1 Common Core Math

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

How this material aligns: Handout questions 3 and 4 have students write a one-variable blending equation and solve it for the well rate that meets the limit, the create-and-solve reasoning this standard names.

All materials for HSA-CED.A.1 →

HSN-Q.A.1 Common Core Math

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

How this material aligns: Students carry parts per trillion, gallons per minute, and million gallons per day through the blend calculation in questions 6 and 7, choosing and interpreting units consistently.

All materials for HSN-Q.A.1 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-23.

Objectives

Students will write a weighted-average equation for the concentration of a blend of two water sources.
Students will solve the equation for the source rate that meets a stated contaminant limit.

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.

Also: weighted average, blending

Teacher overview

PFAS are a family of long-lasting chemicals, and the federal limit for two of them, PFOA and PFOS, is 4.0 parts per trillion each. When one well is high and another is low, the blended concentration is a weighted average of the two, based on how much water each well contributes. This lesson uses a fictional town to turn that weighted average into a one-variable equation students can solve. It is a modeling exercise: real utilities usually treat PFAS with granular activated carbon or other methods, and any blending plan needs state approval and continued monitoring, so students should read this as math practice rather than a recommended fix. A common misconception is averaging the two concentrations directly; the blend must be weighted by each well's flow.

Procedure

Hook (5 min): Ask students how they would average two test scores if one counted for more points than the other. Connect the idea of a weighted average to mixing two streams of water where one carries more of a chemical.

Direct instruction (15 min): Define the blending formula: blended concentration = (C1 x Q1 + C2 x Q2) / (Q1 + Q2), where C is concentration and Q is flow. Model one fictional warm-up example on the board: a well at 2 ppt running 500 gpm blended with a well at 8 ppt running 500 gpm gives (2 x 500 + 8 x 500) / 1,000 = 5 ppt. Point out that this example is instruction only, not a graded item, and that equal flows give the plain average while unequal flows shift the result.

Guided practice (15 min): Hand out the student page. In pairs, students complete questions 1 through 4, which build Delford's blend and solve for the well rate that meets the 4.0 ppt limit. Circulate and check that pairs weight by flow rather than averaging the two numbers.

Independent practice (10 min): Students complete questions 5 through 8 on their own, finding total flow, converting to MGD, and reasoning about a summer demand shortfall.

Closure / exit ticket (5 min): Students answer question 9 on a half sheet: if the high well rose to 12 ppt, what is the new maximum rate that keeps the blend at 4.0 ppt? Collect and check the setup of the equation. If many pairs still average the two concentrations, reteach weighting the next day.

Delford's Two Wells: A PFAS Blending Equation Lesson — Grade 9 Math

Name: ____________________________ Date: ______________

Student Handout

Delford (a fictional town) draws from two wells. Well 2 is clean at 1.0 ppt PFOA and can run up to 700 gpm. Well 4 is higher at 10 ppt PFOA and can run up to 500 gpm. The federal limit for PFOA is 4.0 ppt. Use blended concentration = (C1 x Q1 + C2 x Q2) / (Q1 + Q2). Round rates to the nearest whole gpm and concentrations to two decimals.

If Well 2 runs at 700 gpm and Well 4 runs at 500 gpm, find the blended PFOA concentration in ppt.
Does the blend from question 1 meet the 4.0 ppt limit? By how much does it miss or clear?
Keep Well 2 at 700 gpm and let Well 4 run at a rate q gpm. Write an equation for the blended concentration as a function of q.
Set your equation from question 3 equal to 4.0 ppt and solve for q.
State the maximum rate for Well 4, in gpm, that keeps the blend at or below 4.0 ppt.
With Well 2 at 700 gpm and Well 4 at the rate from question 5, find the total flow in gpm.
Convert the total flow from question 6 to million gallons per day (MGD), using 1 MGD = 694 gpm. Round to two decimals.
Summer demand needs 1,300 gpm total. Can Delford meet it with these two wells while staying at or below 4.0 ppt? Explain using your numbers.
Exit ticket: If Well 4 rose to 12 ppt, redo the limit calculation to find the new maximum rate for Well 4 that keeps the blend at 4.0 ppt.

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: 4.75 ppt. Blend = (1.0 x 700 + 10 x 500) / (700 + 500) = (700 + 5,000) / 1,200 = 5,700 / 1,200 = 4.75 ppt.
Answer: No, it misses by 0.75 ppt. The blend of 4.75 ppt is above the 4.0 ppt limit by 4.75 - 4.0 = 0.75 ppt.
Answer: C = (700 + 10q) / (700 + q). With Well 2 fixed at 1.0 ppt x 700 gpm = 700, and Well 4 adding 10q, over a total flow of 700 + q.
Answer: q = 350 gpm. Set (700 + 10q) / (700 + q) = 4.0; then 700 + 10q = 2,800 + 4q; 6q = 2,100; q = 350 gpm.
Answer: 350 gpm. That is the largest Well 4 rate that holds the blend at the 4.0 ppt limit.
Answer: 1,050 gpm. Total flow = 700 + 350 = 1,050 gpm.
Answer: 1.51 MGD. 1,050 gpm / 694 gpm per MGD = 1.513, which rounds to 1.51 MGD.
Answer: No. The largest compliant total is 1,050 gpm, which is below the 1,300 gpm summer need by 250 gpm, so Delford would fall short and would need treatment or another source to meet demand within the limit.
Answer: About 263 gpm. Set (700 + 12q) / (700 + q) = 4.0; then 700 + 12q = 2,800 + 4q; 8q = 2,100; q = 262.5, which rounds to 263 gpm.

Differentiation

Support

Provide the blending formula with C1, Q1, C2, and Q2 already labeled for Delford, and work question 1 as a class before pairs continue. Give a filled-in first algebra step for question 4 (cross-multiplying by the total flow) so students focus on solving for q.

Extension

Ask students to find the blended concentration if both wells ran at the same rate, and to explain why the answer equals the plain average of 1.0 and 10 ppt only when the two flows are equal. Then have them describe how the maximum Well 4 rate would change if Well 2 were cleaner.

ELL supports

Provide a two-jar diagram showing a clean well and a high well pouring into one tank, with arrows sized to their flows. Pre-teach "blend," "weighted average," and "limit," and give a sentence frame for question 8: "The most water Delford can supply within the limit is ___ gpm, which is ___ than the summer need."