Lesson plan Grade 8 Science
Cindermere's Combined Sewer: Sizing a Storm Storage Tunnel
A 50-minute grade 8 science lesson in which students build a linear function from storm data for a fictional city and decide how large a sewer storage tunnel must be.
Objectives
- Students will determine a rate of change from a table of rainfall and sewer volume data.
- Students will write and solve a linear function that predicts storm flow into a storage tunnel.
- Students will explain how a storage tunnel keeps untreated sewage out of a river during storms.
Vocabulary
Key words: combined sewer, overflow, interceptor, storage tunnel, rate of change, linear function
Teacher overview
Many older cities were built with one pipe that carries both sewage and street runoff, which works until a storm sends more water down that pipe than the treatment plant can handle. The extra flow spills out of a relief point into a river, carrying untreated sewage with it, so cities dig storage tunnels that hold the surge until the plant can catch up. This lesson uses storm records from the fictional city of Cindermere to build a linear function relating rainfall to the volume arriving at the tunnel, then asks students to size the tunnel. Students often think a storage tunnel treats the water it holds; it only stores the flow, and every gallon still goes to the plant afterward.
Procedure
Hook (5 min): Sketch two cross-sections on the board, one city block with separate storm and sewer pipes and one with a single combined pipe. Ask students to predict what happens in each system during a heavy storm, and record their predictions without correcting them yet.
Direct instruction (12 min): Explain that Cindermere's older neighborhoods have combined sewers, that the interceptor pipe carrying flow to the plant has a fixed capacity, and that anything above that capacity used to spill into the river. Introduce the storage tunnel as the fix. Then model the mathematics on a smaller neighboring district, not on the handout data: a 1-inch storm sends 3 million gallons to that district's tunnel and a 2-inch storm sends 7 million gallons, a change of 4 million gallons per inch of rain.
Guided practice (10 min): Stay with the neighboring district and write its function as a class. Substitute the 1-inch storm into V = 4r + b, so 3 = 4(1) + b and b = -1, giving V = 4r - 1. Check it against a 3-inch storm, which the district measured at 11 million gallons: 4(3) - 1 = 11. Ask what the 4 and the -1 each mean in this situation, then tell students the handout uses Cindermere's own storms and a different rate.
Independent practice (15 min): Students complete the handout on their own or in pairs, predicting volumes, solving for the storm size that fills the tunnel, and finding the overflow from a large storm.
Closure and exit ticket (8 min): Students answer handout item 7 on an index card. Collect the cards and confirm that at least 80 percent of students describe small storms as staying inside the interceptor rather than reaching the tunnel before moving to the next lesson.
Student Handout
The fictional city of Cindermere recorded four storms and measured how many million gallons reached its new storage tunnel during each one. Use r for rainfall in inches and V for volume in million gallons.
| Rainfall r (inches) | Volume V reaching the tunnel (million gallons) |
|---|---|
| 1.0 | 5 |
| 2.0 | 14 |
| 3.0 | 23 |
| 4.0 | 32 |
- Find the rate of change in million gallons per inch of rain.
- Write the linear function V = mr + b that fits the table.
- Use your function to predict the volume reaching the tunnel during a 2.5-inch storm.
- Cindermere's tunnel holds 20 million gallons. Solve 9r - 4 = 20 to find the storm size that exactly fills it. Round to the nearest hundredth of an inch.
- A 3.5-inch storm arrives. Find the volume that reaches the tunnel, then subtract the tunnel's 20-million-gallon capacity to find how many million gallons overflow.
- Solve 9r - 4 = 0 for r, rounded to the nearest hundredth of an inch.
- Using your answer to item 6, explain what happens during storms smaller than that rainfall, and what the -4 in the function represents.
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: 9 million gallons per inch.
From the 1-inch row to the 2-inch row, the volume rises from 5 to 14, a change of 9 million gallons for 1 inch of rain. The same rise appears between every pair of rows. - Answer: V = 9r - 4.
Substituting the first row into V = 9r + b gives 5 = 9(1) + b, so b = -4. Checking the 4-inch row: 9(4) - 4 = 32, which matches the table. - Answer: 18.5 million gallons.
V = 9(2.5) - 4 = 22.5 - 4 = 18.5 million gallons. - Answer: 2.67 inches.
9r - 4 = 20 gives 9r = 24 and r = 24/9 = 2.666..., which rounds to 2.67 inches of rain. - Answer: 7.5 million gallons.
V = 9(3.5) - 4 = 31.5 - 4 = 27.5 million gallons reaching the tunnel. Subtracting the 20-million-gallon capacity leaves 7.5 million gallons with nowhere to go. - Answer: 0.44 inches.
9r - 4 = 0 gives 9r = 4 and r = 4/9 = 0.444..., which rounds to 0.44 inches. - Answer: Storms below about 0.44 inches send nothing to the tunnel.
In a small storm the interceptor still has room, so all of the flow goes straight to the treatment plant. The -4 represents that headroom in the interceptor, which the rain has to fill before any water is diverted to the tunnel.
Differentiation
Support
Give students the function V = 9r - 4 already written, so items 3 through 6 are substitution and solving rather than model building. Provide a two-column organizer that separates "what the storm gives me" from "what I need to find" for each item.
Extension
Ask students to graph V = 9r - 4 on graph paper, mark the tunnel's 20-million-gallon capacity as a horizontal line, and shade the region where the fictional city of Cindermere would still overflow. Then have them find what tunnel size would handle a 5-inch storm.
ELL supports
Pre-teach "combined sewer," "overflow," and "storage tunnel" with a labeled cross-section showing one pipe splitting into a plant line and a tunnel line. Provide sentence frames such as "When the rain is more than ___ inches, the tunnel ___." Allow students to explain item 7 orally using the diagram before writing.
Extensions
Have students find out whether their own community has combined or separate sewers by checking the utility's website or a public works report, then write a paragraph comparing what they find with the fictional city of Cindermere's situation.