Lesson plan Grade 9 Math

Feldmere's Pennyquick Lane Main: Building an Exponential Decay Model

A 50-minute grade 9 lesson where students build an exponential model of chlorine residual along a dead-end water main and compare it against a linear guess.

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

Standards

HSF-LE.A.2 Common Core Math

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs.

How this material aligns: Students construct an exponential function from the two input-output pairs measured at the pump station and at the four-hour sampling point on this main.

All materials for HSF-LE.A.2 →

HSF-IF.B.4 Common Core Math

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

How this material aligns: Students interpret the model's initial value, its decreasing behavior, and why its graph approaches zero residual without ever reaching it.

All materials for HSF-IF.B.4 →

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 milligrams per liter, million gallons per day, and pounds per gallon through the chlorine feed calculation in question 9.

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 2026-09-04.

Objectives

Students will construct an exponential function from two input-output pairs of residual readings.
Students will compare exponential and linear predictions for the same measured data.
Students will calculate a chlorine feed rate in pounds per day from a dose and a flow.

Vocabulary

Chlorine residual: Chlorine residual is the small amount of chlorine still left in the water after it leaves the treatment plant. It keeps protecting the water from germs all the way to your faucet.
Dead-end main: A dead-end main is a section of water pipe that stops instead of connecting to another pipe on the far end, so water can only flow in and out from one direction. Water can sit still longer in a dead-end main, so utilities flush them regularly to keep the water fresh.

Also: decay factor, exponential model, chlorine demand

Teacher overview

Chlorine added at a treatment plant does not stay put. It reacts with pipe walls and with whatever it meets along the way, so the residual measured far out in a distribution system is always lower than the residual leaving the plant, and it falls fastest where water sits longest. This lesson follows a dead-end main on the fictional Pennyquick Lane in the fictional Feldmere Water District, where the residual leaving the pump station is 1.20 mg/L and a sample taken four hours of travel time later reads 0.60 mg/L. The misconception the lesson targets is the assumption that the residual falls by the same amount each hour. Students who make that assumption predict a negative residual at twelve hours, which is a useful result precisely because it is impossible, and it opens the argument for a model that falls by equal factors instead of equal differences.

Procedure

Launch (6 min): Give the class the two readings from the fictional Feldmere main, 1.20 mg/L leaving the pump station and 0.60 mg/L four hours later, and ask for a prediction at eight hours. Record every distinct prediction on the board, which usually yields both 0.00 and 0.30 among the answers.

Model construction (14 min): Work with the class to find the factor connecting the two readings, dividing 0.60 by 1.20 to get 0.5 per four-hour period. Write the model as C = 1.20 times 0.5 raised to the power n, where n counts four-hour periods. Then build the competing linear model from the same two points, a loss of 0.15 mg/L per hour, and put both on the board side by side.

Guided practice (10 min): Have students extend both models to eight and twelve hours in a two-column table and compare. When the linear column produces a negative residual at twelve hours, ask what a negative concentration would mean physically. Sketch both curves on one set of axes labeled in hours and mg/L.

Independent practice (15 min): Students complete questions 1 through 11 on the handout for the same fictional main, including the chlorine feed calculation in question 9.

Closure and formative check (5 min): Students answer handout question 11 on a slip of paper, describing what the exponential model does as travel time keeps growing. Collect and sort the slips for the phrase "reaches zero," which signals a student is still reading the model as a countdown rather than a decay, and address those responses first next session.

Feldmere's Pennyquick Lane Main: Building an Exponential Decay Model — Grade 9 Math

Name: ____________________________ Date: ______________

Student Handout

Every reading below comes from the fictional Feldmere Water District's dead-end main on Pennyquick Lane. Water leaves the pump station with a chlorine residual of 1.20 mg/L. A sample taken where the water has traveled for four hours reads 0.60 mg/L. Let n count four-hour periods of travel time.

Divide the four-hour reading by the reading at the pump station to find the decay factor for one four-hour period.
Write the exponential model for the residual C after n four-hour periods, using your factor from question 1.
Use your model to find the residual after 8 hours of travel time.
Use your model to find the residual after 12 hours of travel time.
Over the first four hours the residual fell from 1.20 mg/L to 0.60 mg/L. Find the average loss per hour over that stretch.
Suppose an operator assumed the residual keeps falling by that same amount every hour. Find what that linear model predicts at 12 hours.
Explain in one sentence why the value you found in question 6 cannot describe a real sample.
Systems generally hold a residual of at least 0.20 mg/L out in the distribution system. Using your exponential model and sampling only at 4, 8, and 12 hours, give the first of those sampling times at which the modeled residual falls below 0.20 mg/L.
The pump station feeds chlorine at a dose of 1.4 mg/L into a flow of 0.75 MGD. Find the chlorine feed rate in pounds per day, using pounds per day = dose in mg/L x flow in MGD x 8.34.
The maximum residual disinfectant level for chlorine is 4.0 mg/L, a limit on the residual in the pipes. The residual here can never be higher than the dose. Find how far below that level the 1.4 mg/L dose sits.
Describe in one sentence what your exponential model does to the residual as the travel time keeps increasing.

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: 0.5.
Dividing gives 0.60 divided by 1.20 = 0.5, so half the residual remains after each four-hour period.
Answer: C = 1.20(0.5) raised to the power n.
The initial value 1.20 mg/L is the residual leaving the pump station, and 0.5 is the factor found in question 1.
Answer: 0.30 mg/L.
Eight hours is two four-hour periods, so C = 1.20(0.5) squared = 1.20(0.25) = 0.30 mg/L.
Answer: 0.15 mg/L.
Twelve hours is three four-hour periods, so C = 1.20(0.5) cubed = 1.20(0.125) = 0.15 mg/L.
Answer: 0.15 mg/L per hour.
The residual dropped 1.20 - 0.60 = 0.60 mg/L across 4 hours, and 0.60 divided by 4 = 0.15 mg/L per hour.
Answer: -0.6 mg/L.
The linear model gives 1.20 - 0.15(12) = 1.20 - 1.80 = -0.6 mg/L.
Answer: A concentration cannot be negative.
A sample can hold no chlorine at all, giving zero, but there is no such thing as less than none, so the linear model has run past the range where it could describe anything real.
Answer: 12 hours.
The modeled residual is 0.60 mg/L at 4 hours and 0.30 mg/L at 8 hours, both at or above 0.20 mg/L, while at 12 hours it is 0.15 mg/L, which is below 0.20 mg/L.
Answer: 8.76 lb/day.
Multiplying gives 1.4 mg/L x 0.75 MGD x 8.34 = 8.757 pounds per day, which rounds to 8.76 lb/day.
Answer: 2.6 mg/L.
Subtracting gives 4.0 - 1.4 = 2.6 mg/L, so even the full dose, and therefore any residual, sits well below the maximum residual disinfectant level.
Answer: It keeps falling by half each period and approaches zero without reaching it.
Each additional four-hour period multiplies the residual by 0.5, which makes the value smaller and smaller yet never exactly zero, so the model has no moment at which the chlorine is entirely gone.

Differentiation

Support

Give these students a partially completed table for the fictional Feldmere main with the 4-hour row filled in for both models, so their first task is extending a pattern rather than writing a function from scratch. Provide the exponential form with the initial value already in place and only the factor left blank, and allow calculators for every power.

Extension

Ask students to find, to the nearest tenth of an hour, when the modeled residual on the fictional Pennyquick Lane main first drops below 0.20 mg/L, using a table of values at tenth-of-an-hour steps between 10 and 11 hours, or a graph, rather than a formula. Then have them explain why flushing a dead-end main raises the residual there, in terms of what flushing does to travel time.

ELL supports

Pre-teach "residual," "decay factor," "dose," and "dead-end main" with a labeled sketch of a pipe running from a pump station to a single house at the end of a street. Provide sentence frames for questions 7 and 11, such as "The residual gets smaller each period because ___."

Extensions

Students can compare the fictional Feldmere model against the way real utilities report residuals in a Consumer Confidence Report, which lists a range across many sampling points rather than a single value. A follow-up task is to explain, using the model, why two homes on the same street can post different residuals when one sits on a busy main and the other on a dead end.