Worksheet Grade 11 Math
Talmadge Reservoir Elevation: Reading a Hydrograph's Key Features
A grade 11 worksheet reading domain, range, increasing and decreasing intervals, extrema, and rate of change from a fictional reservoir's monthly elevation table.
Objectives
- Students will state the domain and range of a reservoir-elevation function from a data table.
- Students will identify increasing and decreasing intervals, extrema, and average rate of change and interpret them in context.
Vocabulary
- Drawdown
- Drawdown is how far the water level in a well drops when a pump pulls water out. The harder and longer you pump, the bigger the drawdown gets.
Also: domain, range, relative maximum
Teacher overview
A reservoir rises and falls through the water year as inflow from rain and snowmelt competes with releases for supply and evaporation. Plotting the water-surface elevation month by month produces a function whose shape tells the story of the year. This worksheet uses a fictional record for Talmadge Reservoir so students can read a function's key features from a table: domain, range, where it increases and decreases, its high and low points, and its average rate of change. Students often report only the highest value as the range; the reminder here is that a range needs both the lowest and highest outputs.
Student Worksheet
Talmadge Reservoir (a fictional reservoir used for this worksheet) sits in a mountain watershed. The table gives its water-surface elevation, in feet above a fixed datum, at the start of each month. Treat the month number as the input and the elevation as the output.
| Month | Elevation (feet) |
|---|---|
| 1 (Jan) | 812 |
| 2 (Feb) | 818 |
| 3 (Mar) | 826 |
| 4 (Apr) | 831 |
| 5 (May) | 828 |
| 6 (Jun) | 819 |
| 7 (Jul) | 808 |
| 8 (Aug) | 798 |
| 9 (Sep) | 792 |
| 10 (Oct) | 795 |
| 11 (Nov) | 802 |
| 12 (Dec) | 809 |
- State the domain of this function as the set of month numbers in the table.
- State the range of this function using its lowest and highest elevation values.
- Over which months is the elevation increasing? List every interval.
- Over which months is the elevation decreasing?
- Identify the month of the relative maximum elevation and give its value.
- Identify the month of the minimum elevation and give its value.
- Find the average rate of change in elevation from April to September. Include the units and the sign.
- Interpret your answer to item 7 in the context of the reservoir's water year.
- Explain why the domain stops at month 12 and what an elevation reading "between" two listed months would represent.
- The reservoir's usable supply range is between 795 and 831 feet. During which listed month does the elevation fall below that usable range?
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: Months 1 through 12. The input values are the whole numbers 1 to 12, one for each month of the year.
- Answer: 792 feet to 831 feet. The lowest elevation is 792 feet (September) and the highest is 831 feet (April), so the range runs from 792 to 831 feet.
- Answer: January to April and September to December. Elevation rises from 812 up to 831 across the first four months and again from 792 up to 809 across the last four.
- Answer: April to September. Elevation falls from 831 down to 792 across those months.
- Answer: April, 831 feet. It is the highest point, with lower values on both sides.
- Answer: September, 792 feet. It is the lowest elevation in the table.
- Answer: about -7.8 feet per month. (792 - 831) divided by (9 - 4) = -39 / 5 = -7.8 feet per month.
- Answer: The negative rate means the reservoir is drawing down. During the dry season the water surface drops by roughly 7.8 feet each month as releases and evaporation exceed inflow.
- Answer: The domain stops at month 12 because a calendar year has only twelve months, so no month 13 exists in this record. A reading between two months would be an elevation on a day partway through, which the monthly table does not list but which the real reservoir passes through continuously.
- Answer: September. Its elevation of 792 feet is below the usable lower limit of 795 feet; every other month is at or above 795 feet.
Differentiation
Support
Provide a printed coordinate grid with the twelve points already plotted so students can trace increases and decreases visually before answering. Define "relative maximum" and "range" on a reference card with a small labeled example.
Extension
Ask students to find every month-to-month interval whose average rate of change is steeper than the drawdown rate found in item 7, and to explain what physical event in the watershed might cause the single steepest drop.
ELL supports
Pre-teach "elevation," "increasing," "decreasing," and "drawdown" with arrows on a simple rising-and-falling curve. Provide sentence frames: "The elevation increases from month ___ to month ___," and "The lowest value is ___ feet in ___."