Worksheet Grade 9 Math
Quarrylea Dam Spillway: Reading a Head and Discharge Table
A grade 9 worksheet where students read a spillway rating table, compute rates of change across intervals, and convert discharge between cfs, MGD, and gpm.
Objectives
- Students will read discharge and head values from a spillway rating table in both directions.
- Students will compute average rates of change over two intervals and judge whether the table is linear.
- Students will convert a discharge among cubic feet per second, million gallons per day, and gallons per minute.
Vocabulary
- Spillway
- A dam's overflow channel, giving extra water a safe path around or over the dam when the reservoir gets too full. It protects the dam from being damaged in floods.
- Acre-foot
- The amount of water it takes to cover one acre of land, about the size of a football field, one foot deep. Western states use acre-feet to measure large amounts of water.
Also: head, discharge, rating table
Teacher overview
A dam's spillway is described by a rating table that pairs the height of water above the spillway crest, called the head, with the flow the spillway passes at that height. Operators read the table in both directions during a storm: forward to find how much water is leaving, and backward to find how high the reservoir must rise to pass an expected inflow. This worksheet uses one fictional table for the fictional Quarrylea Dam and asks students to read it, measure how fast discharge changes across two different intervals, and convert the results into the units a water utility actually reports. The misconception this worksheet is built to surface is that a table of paired values must describe a straight line. Students who assume that will interpolate confidently at any head, and questions 4 through 6 show them the rate of change nearly doubling from one end of the table to the other.
Student Worksheet
The table below is the fictional rating table for the fictional Quarrylea Dam, which holds back the fictional Dunmoss Reservoir. Head is the depth of water above the spillway crest.
| Head (ft) | Discharge (cfs) |
|---|---|
| 0.5 | 60 |
| 1.0 | 170 |
| 1.5 | 310 |
| 2.0 | 480 |
| 2.5 | 670 |
| 3.0 | 880 |
- Find the discharge when the head is 2.0 feet.
- Find the head at which the spillway passes 670 cfs.
- Estimate the discharge at a head of 1.25 feet. Use the midpoint of the discharges listed at 1.0 foot and 1.5 feet.
- Find the average rate of change in discharge from a head of 0.5 feet to a head of 1.0 feet. Give your answer in cfs per foot of head.
- Find the average rate of change in discharge from a head of 2.5 feet to a head of 3.0 feet. Give your answer in cfs per foot of head.
- Compare your answers to questions 4 and 5. Is this rating table linear? Answer yes or no.
- Convert the discharge at a head of 2.0 feet to million gallons per day. One MGD equals 1.547 cfs.
- Convert the discharge at a head of 1.0 foot to gallons per minute. One cfs equals 448.8 gpm.
- Suppose a storm brings 600 cfs into the reservoir while the spillway passes 480 cfs. Find the extra water stored over 6 hours, in cubic feet. There are 3,600 seconds in an hour.
- Convert your answer to question 9 into acre-feet. One acre-foot equals 43,560 cubic feet.
- Explain in two or three sentences why an operator should not use this table to predict the discharge at a head of 6 feet.
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: 480 cfs.
The row for a head of 2.0 feet lists a discharge of 480 cfs.Answer: 2.5 feet.
Reading the table backward, 670 cfs appears in the row for a head of 2.5 feet.Answer: 240 cfs.
The listed values are 170 cfs at 1.0 foot and 310 cfs at 1.5 feet. Their midpoint is (170 + 310) divided by 2 = 240 cfs. This is an estimate, because the table is not a straight line.Answer: 220 cfs per foot.
The change is 170 - 60 = 110 cfs across a head change of 1.0 - 0.5 = 0.5 feet, and 110 divided by 0.5 = 220 cfs per foot.Answer: 420 cfs per foot.
The change is 880 - 670 = 210 cfs across a head change of 3.0 - 2.5 = 0.5 feet, and 210 divided by 0.5 = 420 cfs per foot.Answer: No.
A linear table would give the same rate of change over every interval of equal width. Here the rate nearly doubles from 220 cfs per foot at the low end to 420 cfs per foot at the high end, so the relationship curves upward.Answer: 310 MGD.
Dividing gives 480 cfs divided by 1.547 cfs per MGD = 310.3 MGD, which rounds to about 310 MGD.Answer: 76,296 gpm.
Multiplying gives 170 cfs x 448.8 gpm per cfs = 76,296 gpm.Answer: 2,592,000 cubic feet.
The net gain is 600 - 480 = 120 cfs. Over 6 hours there are 6 x 3,600 = 21,600 seconds, and 120 x 21,600 = 2,592,000 cubic feet.Answer: 59.5 acre-feet.
Dividing gives 2,592,000 divided by 43,560 = 59.50 acre-feet.Answer: A head of 6 feet is far outside the range the table covers.
The table stops at 3.0 feet, and the rate of change is still climbing at that point, so there is no way to tell from these rows how the spillway behaves twice that high. At high heads the approach channel and the crest shape can change the relationship in ways the listed rows do not show.
Differentiation
Support
Give these students the rate-of-change formula printed at the top of the page and a worked example using the fictional rows at 1.5 and 2.0 feet, so questions 4 and 5 repeat a pattern they have already seen. Provide the two conversion factors on a separate strip, each with a short sentence saying which unit is larger.
Extension
Ask students to add a row to the fictional Quarrylea table for a head of 3.5 feet, using the pattern in how the rate of change grows, and to state clearly what assumption their added row depends on. A second task is to find the head at which the spillway would pass 1,000 cfs and explain how confident they are in that estimate.
ELL supports
Provide a labeled side-view sketch of a dam showing the crest, the head above it, and the water leaving, with the words "head" and "discharge" placed on the picture. Give a sentence frame for question 6, such as "The table is not linear because the rate changes from ___ to ___."
Extensions
Students can look up a real spillway rating curve published by a state dam safety office or a federal reservoir operations page and compare its shape with the fictional table used here. A follow-up task is to explain, using the rates of change they computed, why a reservoir's level rises quickly at first during a storm and then more slowly as the head over the spillway grows.