Lesson plan Grade 10 Math
Farrowdean Combined Sewer: Storm Storage as a Constraint
A 55-minute grade 10 lesson in which students model a fictional town's storm storage tank as an inequality in rate and duration, then judge four storms as viable or not.
Objectives
- Students will compute the excess storm volume that reaches a storage tank from a given inflow rate and duration.
- Students will write an inequality in two variables describing the storms a storage tank can hold.
- Students will classify specific storms as viable or non-viable against stated storage and dewatering constraints.
Vocabulary
Key words: combined sewer, overflow, constraint, viable solution, dewatering
Teacher overview
The town of Farrowdean, its treatment plant, its storage tank, and Ostley Brook are all fictional, and every rate and volume in this lesson was set for the modeling work rather than drawn from a real system. A combined sewer carries both sewage and street runoff in one pipe, so a storm raises the flow arriving at the plant far above its dry-weather level. The misconception this lesson addresses is that a storm is dangerous only when it is intense; the tank fills on volume, so a mild storm lasting many hours can fill the tank exactly while a sharper but shorter storm uses only half of it, and items 1 through 8 are built to surface that.
Procedure
Hook (5 min): Put two fictional storms on the board, one at 14.0 MGD lasting 8 hours and one at 8.0 MGD lasting 18 hours, and ask which is more likely to overflow the town's tank. Record the vote and move on without resolving it.
Direct instruction (13 min): Explain that the fictional plant can treat 6.0 MGD and that anything above that goes to a tank holding 1.5 million gallons. Model the excess-volume calculation on a teacher example that is not on the handout: a fictional storm at 10.0 MGD for 12 hours sends (10.0 − 6.0) × 12 ÷ 24, which equals 2.0 million gallons, to the tank. Then set up the general relationship with r for rate and h for hours, and sketch the boundary curve on labeled axes so students see that the same tank corresponds to many pairs of rate and duration.
Guided practice (14 min): Distribute the handout and work items 1 through 4 together. Pause after item 3 so pairs can confirm that they converted hours to a fraction of a day before multiplying. Have each pair mark Storms A and B as points on their graph of the boundary and check that the marks land on the side their arithmetic predicted.
Independent practice (15 min): Students complete items 5 through 12 alone or in pairs. Circulate and watch for two errors: forgetting to subtract the 6.0 MGD plant capacity before multiplying by the duration, and reporting the total excess volume when the item asks for the overflow beyond the tank.
Closure and formative check (8 min): Return to the hook and read items 6 and 5 aloud, in that order. Ask students to name which of those two storms sends more excess water toward the tank, and whether either one overflows, and to state the volumes that settle it, then ask the same question about Storm A and Storm C, where the milder storm is the one that fills the tank. If more than one-third answer either question from the inflow rate alone without computing a volume, rework items 1 and 5 on the board before the next class.
Student Handout
Farrowdean is a fictional town. Its plant can treat 6.0 MGD during a storm, and everything above that rate is sent to a tank holding 1.5 million gallons. Once the tank is full, the excess overflows to Ostley Brook. Assume each storm holds a steady inflow rate for its whole duration.
Storm A arrives at 9.0 MGD for 6 hours. How many million gallons reach the tank above the 6.0 MGD plant capacity?
Does Storm A overflow to the brook? Answer yes or no.
Storm B arrives at 12.0 MGD for 10 hours. Find the excess volume in million gallons.
How many million gallons overflow to the brook during Storm B?
Storm C arrives at 8.0 MGD for 18 hours. Find the excess volume in million gallons.
Storm D arrives at 14.0 MGD for 8 hours. Find the excess volume in million gallons, rounded to the nearest hundredth.
Find the overflow volume for Storm D, rounded to the nearest hundredth.
A storm holds steady at 10.0 MGD. Find the longest duration in hours that produces no overflow.
Using r for the inflow rate in MGD and h for the duration in hours, write the inequality that must hold for the fictional tank to contain the whole storm.
The town considers enlarging the tank to 3.0 million gallons. Would Storm D then be contained? Answer yes or no.
After a storm the 1.5 million gallon tank is full and must be emptied back to the plant within 12 hours. Find the required dewatering rate in MGD.
A slow storm runs at 7.2 MGD for 30 hours. Find its excess volume in million gallons.
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: 0.75 million gallons. The excess rate is 9.0 − 6.0 = 3.0 MGD, and 3.0 × 6 ÷ 24 = 3.0 × 0.25 = 0.75 million gallons.
Answer: No. The 0.75 million gallons fits inside the fictional tank's 1.5 million gallon capacity, so nothing reaches the brook.
Answer: 2.5 million gallons. The excess rate is 12.0 − 6.0 = 6.0 MGD, and 6.0 × 10 ÷ 24 = 6.0 × 0.4167 = 2.5 million gallons.
Answer: 1.0 million gallons. The tank holds 1.5 million gallons, so 2.5 − 1.5 = 1.0 million gallons overflows.
Answer: 1.5 million gallons. The excess rate is 8.0 − 6.0 = 2.0 MGD, and 2.0 × 18 ÷ 24 = 2.0 × 0.75 = 1.5 million gallons, which exactly fills the fictional tank.
Answer: 2.67 million gallons. The excess rate is 14.0 − 6.0 = 8.0 MGD, and 8.0 × 8 ÷ 24 = 8.0 × 0.3333 = 2.6667, which rounds to 2.67 million gallons.
Answer: 1.17 million gallons. Subtracting the tank capacity gives 2.6667 − 1.5 = 1.1667, which rounds to 1.17 million gallons.
Answer: 9 hours. The excess rate is 10.0 − 6.0 = 4.0 MGD, so 4.0 × h ÷ 24 = 1.5 gives 4.0h = 36 and h = 9 hours.
Answer: (r − 6.0)(h ÷ 24) ≤ 1.5. The first factor is the flow above the fictional plant's capacity in MGD, the second converts the storm duration to a fraction of a day, and the product must stay at or below the tank's 1.5 million gallon volume.
Answer: Yes. Storm D delivers 2.67 million gallons, which fits inside a 3.0 million gallon tank with about 0.33 million gallons to spare.
Answer: 3.0 MGD. Twelve hours is half a day, so the rate must satisfy 0.5 × rate = 1.5, giving 1.5 ÷ 0.5 = 3.0 MGD.
Answer: 1.5 million gallons. The excess rate is 7.2 − 6.0 = 1.2 MGD, and 1.2 × 30 ÷ 24 = 1.2 × 1.25 = 1.5 million gallons, which sits exactly on the boundary from item 9 with no margin left.
Differentiation
Support
Give students a filled-in table with columns for inflow rate, excess rate, duration in hours, duration in days, and excess volume, with the first fictional storm already completed, so items 1 through 7 become a matter of extending a pattern. Provide the item 9 inequality with only the 1.5 missing.
Extension
Ask students to graph the boundary from item 9 on labeled axes with r from 6 to 20 MGD and h from 0 to 40 hours, shade the region of storms the fictional tank contains, and describe the shape of the boundary in a sentence. A second challenge is to find the tank size that would contain every storm listed on the handout and justify the choice in writing.
ELL supports
Provide a labeled cross-section of the fictional plant, tank, and brook, with arrows marked "treated," "stored," and "overflow," so the three destinations of storm water stay visually separate. Offer sentence frames for items 2 and 10: "This storm sends ___ million gallons, which is ___ than 1.5, so the tank ___."
Extensions
Have students research how a real utility their teacher selects reports its combined sewer overflow events, then compare the reporting units with the million-gallon figures used for the fictional Farrowdean tank. A short written note on which units make overflow volumes easiest to compare across towns makes a useful bridge to a later lesson on choosing quantities for public reporting.