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Lesson plan Grade 9 Math

Hazelmere's Snowmelt Season: Where a Linear Model Stops

A grade 9 lesson on a fictional mountain reservoir, where students build two linear models from snowmelt data and restrict each domain to what makes sense.

Objectives

  1. Students will write linear functions for a melting snowpack and for reservoir storage from stated rates.
  2. Students will determine the practical domain of each function and explain what limits it.
  3. Students will interpret the slope and the vertical intercept of a storage function in context.

Vocabulary

Snowpack
The snow that piles up in the mountains over the winter. As it melts in spring and summer, it acts like a giant water tank feeding rivers and reservoirs.
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: snow water equivalent, practical domain, capacity

Teacher overview

In mountain watersheds, most of the year's water arrives during a few weeks of spring melt, and reservoir operators plan around two limits at once: the snow runs out, and the reservoir fills. Both quantities change at a steady rate, so both are modeled with linear functions, yet neither model is honest past the day its physical limit is reached. This lesson has students build both functions for a fictional reservoir and decide where each one stops. The misconception it addresses is that a linear model applies to every input the algebra will accept; students who extend the snowpack function past day 30 will report a negative snowpack, a result the physical situation rules out.

Procedure

  • Hook (6 min): Show students the two given rates for this fictional watershed, one falling and one rising, and ask which limit they think will be reached first: the snow running out or the reservoir filling. Take a quick show of hands and record the split on the board without resolving it, since the handout settles the question.
  • Building the two models (12 min): Work through the structure of a linear model with the class, using a different fictional example than the handout: a snowpack of 30 inches melting at 1 inch per day. Ask for the function, the day it reaches zero, and what the model would claim on day 40. Write the phrase "practical domain" on the board and define it as the set of inputs for which the model still describes the real situation. As a formative check, ask students to give a thumbs up if a domain of 0 to 30 days is reasonable for that example; if fewer than about eight in ten agree, work a second short example before releasing the handout.
  • Handout questions 1 through 3 (8 min): Students write the snowpack function, solve for the day it reaches zero, and state its practical domain. Circulate and look for students who write the melt rate as positive; the coefficient must be negative because the snowpack is decreasing.
  • Handout questions 4 through 6 (10 min): Students write the storage function, find the day the reservoir reaches capacity, and state that function's practical domain. Watch for students who use the capacity as the starting value rather than as the ending value.
  • Handout questions 7 through 11 (10 min): Students compare the two domains, compute the volume that arrives after the reservoir is full, convert it to gallons, sketch the storage graph, and interpret its intercept and slope. Pairs work well here.
  • Exit check (4 min): Each student writes one sentence naming which limit binds first in this fictional watershed and what an operator would have to do during the last two days of melt. Collect on the way out.
Hazelmere's Snowmelt Season: Where a Linear Model Stops — Grade 9 Math
Name: ____________________________ Date: ______________

Student Handout

Hazelmere Reservoir is a fictional mountain water supply. Let d be the number of days since melt began on April 1. Use these values.

  • Snowpack water content on April 1: 24 inches
  • Melt rate: 0.8 inches per day, steady until the snow is gone
  • Reservoir storage on April 1: 4,200 acre-feet
  • Meltwater adds 100 acre-feet per day while snow remains
  • Reservoir capacity: 7,000 acre-feet
  • 1 acre-foot = 325,851 gallons
  1. Write a function S(d) for the inches of snowpack water content remaining after d days of melt.
  2. Solve for the day the snowpack reaches zero.
  3. State the practical domain of S(d), and explain in one sentence what happens to the model past that day.
  4. Write a function V(d) for the reservoir storage in acre-feet after d days of melt.
  5. Solve for the day the reservoir reaches its capacity.
  6. State the practical domain of V(d).
  7. Explain why the practical domain of V(d) ends before the practical domain of S(d).
  8. Find the volume of meltwater, in acre-feet, that arrives after the reservoir is already full.
  9. Convert your answer to question 8 into gallons.
  10. Sketch the graph of V(d) over its practical domain on labeled, scaled axes, and mark the point where the reservoir reaches capacity.
  11. State what the vertical intercept of V(d) and the slope of V(d) each represent for this reservoir.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: S(d) = 24 - 0.8d. The snowpack starts at 24 inches and loses 0.8 inches each day, so the rate of change is negative.
  2. Answer: 30 days. Set 24 - 0.8d = 0, so 0.8d = 24 and d = 30.
  3. Answer: 0 to 30 days. Past day 30 the function returns negative inches of snowpack, which cannot describe a real watershed, so the model stops there.
  4. Answer: V(d) = 4,200 + 100d. Storage starts at 4,200 acre-feet and rises by 100 acre-feet per day of melt.
  5. Answer: 28 days. Set 4,200 + 100d = 7,000, so 100d = 2,800 and d = 28.
  6. Answer: 0 to 28 days. Past day 28 the function returns more than the 7,000 acre-feet the reservoir can hold, so the model no longer applies.
  7. Answer: The reservoir fills before the snow runs out. Storage reaches its 7,000 acre-foot limit on day 28, while the snowpack lasts until day 30, so the physical limit that binds first belongs to the reservoir.
  8. Answer: 200 acre-feet. Melt continues for 2 days after the reservoir is full, at 100 acre-feet per day: 2 x 100 = 200 acre-feet that must be released or spilled.
  9. Answer: 65,170,200 gallons. 200 acre-feet x 325,851 gallons per acre-foot = 65,170,200 gallons.
  10. Answer: A straight line from (0, 4200) to (28, 7000). The horizontal axis is days since April 1, scaled 0 to 30; the vertical axis is storage in acre-feet, scaled 4,000 to 7,200. The point (28, 7000) is marked as the day the reservoir fills, and the line stops there rather than continuing upward.
  11. Answer: The intercept is the April 1 storage of 4,200 acre-feet, and the slope is the daily inflow of 100 acre-feet per day. The intercept describes the reservoir before any melt has arrived, and the slope describes how fast melt raises the level while snow remains on the ground.

Differentiation

Support

Give students a two-column table for this fictional watershed with days 0, 5, 10, 15, 20, 25, and 30 down one side, and have them fill in the snowpack and storage values before writing either function. Seeing the storage column pass 7,000 makes the domain restriction concrete before the algebra begins.

Extension

Ask students to write a revised storage function for the fictional reservoir in a warm year where melt runs at 150 acre-feet per day, find the new fill date and the new practical domain, and explain why a faster melt gives operators less time to move water downstream.

ELL supports

Pre-teach "snowpack," "capacity," and "domain" with a two-panel drawing of a full snowfield in April and a bare slope in May, next to a tank filling to a marked line. Provide a sentence frame for question 7: "The reservoir is full on day ___, but the snow lasts until day ___, so the model for ___ stops first."