Lesson plan Grade 10 Math
Penmarch Ridge Reservoir: Reading a Water-Year Storage Graph
A 50-minute grade 10 lesson in which students read key features from a fictional reservoir's twelve monthly storage readings and restrict the domain of its refill model.
Objectives
- Students will identify the maximum, minimum, and intervals of increase and decrease in a table of monthly reservoir storage readings.
- Students will calculate the average rate of change in storage over stated intervals of a water year.
- Students will restrict the domain of a linear refill model to the interval where the model still describes the reservoir.
Vocabulary
- 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: water year, average rate of change, domain, capacity
Teacher overview
Penmarch Ridge Reservoir is fictional, and the twelve monthly storage readings used here were set for this lesson so that the refill months rise by a constant amount. A water year runs from October through September, which is why the table begins in October rather than in January. The misconception this lesson addresses is that a function fitted to part of a data set keeps working outside that part; students who extend the refill line eight months forward get a storage value larger than the reservoir can physically hold, and the domain question exists to make that failure visible rather than abstract.
Procedure
Hook (5 min): Write two storage values on the board, 4,200 acre-feet in March and 2,300 acre-feet in September, and tell students both come from the same fictional reservoir in the same year. Ask what has to happen between those two months, and take three or four answers without confirming any of them.
Direct instruction (12 min): Define the acre-foot as the volume covering one acre to a depth of one foot, and define the water year. Model average rate of change on a short teacher example that is not on the handout: a fictional pond holding 900 acre-feet in April and 600 acre-feet in July changes by (600 − 900) ÷ 3, which equals −100 acre-feet per month. Then write a linear model on the board and ask what the largest sensible input would be if the pond held only 1,000 acre-feet. Do not work any handout value during this segment.
Guided practice (13 min): Distribute the handout and work items 1 through 4 together. Pause after item 3 so pairs can confirm that every consecutive difference from October through March is the same number before they write the refill model. Ask students to sketch the twelve readings on graph paper with months on the horizontal axis, choosing a vertical scale that fits values from 2,000 to 4,400 acre-feet.
Independent practice (12 min): Students complete items 5 through 11 alone or in pairs. Circulate and watch for two errors: subtracting in the wrong order when finding a rate of change during the drawdown, and reporting the answer to item 9 as though the reservoir could really hold that much.
Closure and formative check (8 min): Read item 5 aloud and ask for a show of hands on whether the refill model should be used at t = 8. If more than a quarter of the class votes yes, reteach the domain idea using the item 9 value of 4,890 acre-feet against the stated capacity before the next class.
Student Handout
Penmarch Ridge Reservoir is fictional. Its capacity is 4,200 acre-feet, and it cannot store more than that. The readings below were taken on the first day of each month of one water year. Let t be the number of months after the October reading, so October is t = 0.
Storage readings, in acre-feet: October 3,050; November 3,280; December 3,510; January 3,740; February 3,970; March 4,200; April 4,120; May 3,880; June 3,400; July 2,860; August 2,420; September 2,300.
What is the greatest storage value in the table?
Which month has the least storage?
Storage rises by the same amount each month from October through March. Find that constant monthly increase.
Using the increase from item 3, the refill model is S(t) = 3,050 + 230t. Use the model to find the value of t at which S(t) equals 4,200 acre-feet.
The reservoir cannot hold more than 4,200 acre-feet. Give the largest value of t for which the refill model still describes the reservoir.
Find the average rate of change in storage from the March reading of 4,200 acre-feet to the July reading of 2,860 acre-feet, a span of 4 months.
Find the average rate of change in storage from the May reading of 3,880 acre-feet to the June reading of 3,400 acre-feet.
Find the largest one-month drop in storage anywhere between March and September.
Extend the refill model S(t) = 3,050 + 230t to t = 8 and compute the value it predicts.
The fictional district sets a drought trigger at 2,500 acre-feet. Which is the first month in the table with storage below that trigger?
Explain in two or three sentences why the refill model has a restricted domain.
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: 4,200 acre-feet. March holds the greatest value in the table, and it equals the stated capacity of the fictional reservoir.
Answer: September. The September reading of 2,300 acre-feet is lower than every other value in the table.
Answer: 230 acre-feet per month. Each consecutive difference is the same: 3,280 − 3,050 = 230, 3,510 − 3,280 = 230, 3,740 − 3,510 = 230, 3,970 − 3,740 = 230, and 4,200 − 3,970 = 230.
Answer: 5 months. Solving 3,050 + 230t = 4,200 gives 230t = 1,150, so t = 5, which is the March reading.
Answer: 5 months. The model is valid on the interval from t = 0 to t = 5, because storage reaches capacity at t = 5 and the reservoir cannot rise further.
Answer: −335 acre-feet per month. The change is 2,860 − 4,200 = −1,340 acre-feet over 4 months, and −1,340 ÷ 4 = −335 acre-feet per month.
Answer: −480 acre-feet per month. The change is 3,400 − 3,880 = −480 acre-feet over 1 month.
Answer: 540 acre-feet. The month-to-month drops are 80, 240, 480, 540, 440, and 120 acre-feet, and the largest of these is the June-to-July drop of 540 acre-feet.
Answer: 4,890 acre-feet. Substituting gives 3,050 + 230(8) = 3,050 + 1,840 = 4,890 acre-feet, which is 690 acre-feet more than the fictional reservoir can hold.
Answer: August. The August reading of 2,420 acre-feet is the first value below the 2,500 acre-foot trigger, since July still held 2,860 acre-feet.
Answer: The model describes only the refill months. Storage rises at a constant 230 acre-feet per month from October until the reservoir fills in March, and after that the water spills or is released rather than accumulating, so the line no longer matches the readings. The physical capacity of 4,200 acre-feet caps the output, which limits the useful domain to values of t from 0 through 5.
Differentiation
Support
Give students the fictional table already plotted on labeled axes, with the six refill points circled, so items 3 and 4 become a matter of reading marked coordinates. Provide the average rate of change formula with the item 6 values already substituted, leaving only the subtraction and division to complete.
Extension
Ask students to write a second linear model for the June-through-August drawdown of the fictional reservoir, state its domain, and find the month when the two models would intersect if both were extended. A further challenge is to find the constant monthly release that would have kept storage at or above the 2,500 acre-foot trigger through September.
ELL supports
Provide a labeled diagram of one acre-foot beside the fictional reservoir's cross-section, with the words capacity, refill, and drawdown placed on the picture. Offer sentence frames for item 11: "The model works from t = ___ to t = ___ because ___," and "After that month the reservoir ___."
Extensions
Have students find published monthly storage records for a reservoir their teacher selects, then repeat items 1 through 8 with that record in place of the fictional Penmarch Ridge readings. A short written comparison of the two refill rates, one from the real record and one from the fictional table, makes a useful bridge to a later lesson on piecewise models.