Worksheet Grade 8 Math
Blencathra Dam Release: A Function From Two Readings
A grade 8 worksheet where students build linear functions for a fictional dam's reservoir and tailwater from only two gage readings each.
Objectives
- Students will find the rate of change of a linear function from two given ordered pairs.
- Students will find the initial value of a linear function by working backward from a known point.
- Students will use a constructed linear function to find an output and to solve for an input.
Vocabulary
Key words: rate of change, initial value, linear function, tailwater, reservoir elevation
Teacher overview
During a controlled release, the water stored behind a dam drops at a steady rate while the water just downstream rises at a steady rate, so both surfaces are linear functions of time. The gages at the fictional Blencathra Dam on the Stannel River log only a few readings per shift, which means an operator often has two points and needs the whole function. This worksheet asks students to build that function from two ordered pairs rather than from a full table. The usual error is dividing the change in time by the change in elevation instead of the other way around, which produces a rate with the units upside down.
Student Worksheet
In our scenario, a release begins at hour 0 at the fictional Blencathra Dam. Let t stand for hours since the release began.
The reservoir gage recorded an elevation of 1,042.4 feet at hour 2 and 1,038.8 feet at hour 8.
- Find the rate of change of the reservoir elevation, in feet per hour. Use a negative number for a falling level.
- Find the initial value, which is the reservoir elevation at hour 0.
- Write a linear function for the reservoir elevation E after t hours.
- What is the reservoir elevation at hour 12?
- The release must stop when the reservoir elevation reaches 1,034.0 feet. At what hour is that?
The tailwater gage below the dam recorded an elevation of 612.5 feet at hour 1 and 615.3 feet at hour 5.
- Find the rate of change of the tailwater elevation, in feet per hour.
- Find the tailwater elevation at hour 0.
- Write a linear function for the tailwater elevation T after t hours.
- What is the tailwater elevation at hour 10?
The outflow through the gates starts at 300 cubic feet per second and rises by 45 cubic feet per second each hour.
- Find the rate of change of the outflow, in cubic feet per second per hour.
- What is the outflow at hour 6?
- Which surface changes faster in feet per hour, the reservoir or the tailwater? Explain how you can tell from the two rates of change.
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: -0.6 feet per hour. The elevation changed from 1,042.4 feet to 1,038.8 feet, a change of -3.6 feet, over 8 minus 2, or 6 hours. Dividing -3.6 by 6 gives -0.6 feet per hour.
- Answer: 1,043.6 feet. Hour 2 is two hours after the start, and the level falls 0.6 feet each hour, so at hour 0 it was 1,042.4 plus 2 times 0.6, which is 1,042.4 plus 1.2, or 1,043.6 feet.
- Answer: E = 1,043.6 - 0.6t. The initial value 1,043.6 feet is the constant term and the rate of change -0.6 feet per hour is the coefficient of t.
- Answer: 1,036.4 feet. Substitute t = 12 to get E = 1,043.6 - 0.6(12) = 1,043.6 - 7.2, which is 1,036.4 feet.
- Answer: 16 hours. Set 1,043.6 - 0.6t = 1,034.0. Subtract 1,043.6 from both sides to get -0.6t = -9.6, then divide both sides by -0.6 to get t = 16 hours.
- Answer: 0.7 feet per hour. The elevation changed from 612.5 feet to 615.3 feet, a change of 2.8 feet, over 5 minus 1, or 4 hours. Dividing 2.8 by 4 gives 0.7 feet per hour.
- Answer: 611.8 feet. Hour 1 is one hour after the start, and the level rises 0.7 feet each hour, so at hour 0 it was 612.5 minus 0.7, which is 611.8 feet.
- Answer: T = 611.8 + 0.7t. The initial value 611.8 feet is the constant term and the rate of change 0.7 feet per hour is the coefficient of t.
- Answer: 618.8 feet. Substitute t = 10 to get T = 611.8 + 0.7(10) = 611.8 + 7, which is 618.8 feet.
- Answer: 45 cubic feet per second per hour. The description gives the rate directly, since the outflow rises by the same 45 cubic feet per second in every hour of the release.
- Answer: 570 cubic feet per second. The function is Q = 300 + 45t, so at t = 6 the outflow is 300 plus 45 times 6, which is 300 plus 270, or 570 cubic feet per second.
- Answer: The tailwater changes faster. Its rate of change is 0.7 feet per hour while the reservoir changes by 0.6 feet per hour, and comparing the sizes of the two rates, without their signs, shows which surface moves more in one hour.
Differentiation
Support
Give these students a two-point recording box for the fictional Blencathra gages, with the subtraction of the two elevations and the subtraction of the two hours already set up as a fraction. Assign items 1 through 5 and item 12, and post the reminder that the rate of change is feet divided by hours, not hours divided by feet.
Extension
Ask students to find the hour when the reservoir surface and the tailwater surface are exactly 424 feet apart, using both functions, and to explain what happens to that gap as the release continues. They should show the subtraction of the two functions before solving.
ELL supports
Pre-teach reservoir, tailwater, and release with a labeled side view of a dam showing the higher water behind it and the lower water below it, with an arrow for each direction of change. Post the pair "rate of change" with "per hour" and "initial value" with "at hour 0." Allow students to say each fictional Blencathra reading aloud as an ordered pair before computing.
Extensions
Give students a gage at a different fictional dam that reads 998.0 feet at hour 3 and 996.5 feet at hour 9 of that dam's own release, and ask them to build its function, state its rate of change and initial value, and decide whether that gage is more likely upstream or downstream of that dam's powerhouse. Have them justify the choice using the sign of the rate of change.