Math problem set Grade 8 Science

Sarsen Basin Snowpack: A Linear Model of Spring Melt

A grade 8 science problem set on a fictional basin's snow course: rate of change in snow water equivalent, melt-out dates, and basin storage in scientific notation.

Grade: Grade 8
Subject: Science
Material: Math problem set
Time: 45 min
Prep: No prep
Materials: Printed problem set, Calculators

Standards

MS-ESS2-4 NGSS

Develop a model to describe the cycling of water through Earth's systems driven by energy from the sun and the force of gravity.

How this material aligns: Problem 10 has students explain the river's late-spring peak using the melt model from problems 1 through 7, which treats snowpack as storage that the sun releases and gravity carries downhill.

All materials for MS-ESS2-4 →

8.F.B.4 Common Core Math

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description, a table, or a graph, and interpret them in terms of the situation modeled.

How this material aligns: Problems 1, 2, and 5 have students find a rate of change from a table and write the linear function for each snow course.

All materials for 8.F.B.4 →

8.EE.A.3 Common Core Math

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

How this material aligns: Problem 9 has students convert the basin's stored water to gallons and write that very large quantity in scientific notation as a power of ten.

All materials for 8.EE.A.3 →

Download Word file

This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-23.

Objectives

Students will write a linear function for snow water equivalent from a table of snow course readings.
Students will solve for the day a snowpack reaches zero and compare two courses at different elevations.
Students will convert basin storage from acre-feet to gallons and write it in scientific notation.

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, snow course, melt-out, rate of change

Teacher overview

A mountain snowpack works as a natural reservoir, filling through the winter and releasing water through the spring. Crews measure it at fixed sites called snow courses, and they report snow water equivalent, the depth of water a column of snow would make if it melted where it stands. The problems below follow the fictional Sarsen Basin, which drains into the Tunstall River and supplies a town downstream, through one melt season at two elevations. Students often treat snow depth and snow water equivalent as the same measurement; because snow is mostly air, a snowpack several feet deep usually holds only a fraction of that depth in water.

Sarsen Basin Snowpack: A Linear Model of Spring Melt — Grade 8 Science

Name: ____________________________ Date: ______________

Problems

Every measurement below comes from the fictional Sarsen Basin snow survey. Let d be the number of days after April 1.

The upper snow course recorded these readings.

Days after April 1 (d)	Snow water equivalent S (inches)
0	24.0
10	16.0
20	8.0

Find the rate of change in inches of snow water equivalent per day at the upper course.
Write the linear function S = md + b for the upper course.
Use your function to predict the snow water equivalent at the upper course on day 25.
Solve your function for the day when S = 0, the melt-out day at the upper course.
A lower snow course starts April 1 at 15.0 inches and loses 1.5 inches per day. Write the linear function L = md + b for the lower course.
Solve the lower course function for the day when L = 0.
On that same day, how many inches of snow water equivalent remain at the upper course?
Each inch of snow water equivalent across the Sarsen Basin equals 900 acre-feet of water. How many acre-feet does the upper course's remaining snowpack from problem 7 represent?
One acre-foot equals 325,851 gallons. Convert your answer to problem 8 into gallons and write it in scientific notation, rounded to three significant figures.
The Tunstall River runs highest in late spring rather than in midwinter, even though most of the year's precipitation falls in winter. Use the snowpack model to explain why.

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: -0.8 inches per day.
From day 0 to day 10 the reading falls from 24.0 to 16.0 inches, a change of -8.0 inches over 10 days, which is -0.8 inches per day. The same rate appears from day 10 to day 20.
Answer: S = -0.8d + 24.
The rate of change is the slope, and the day 0 reading of 24.0 inches is the starting value, so S = -0.8d + 24. Checking day 20: -0.8(20) + 24 = 8, which matches the table.
Answer: 4 inches.
S = -0.8(25) + 24 = -20 + 24 = 4 inches of snow water equivalent on day 25, which is April 26.
Answer: Day 30.
Setting -0.8d + 24 = 0 gives 0.8d = 24 and d = 30, so the upper course melts out on day 30, which is May 1.
Answer: L = -1.5d + 15.
The course loses 1.5 inches each day, so the slope is -1.5, and the April 1 reading of 15.0 inches is the starting value.
Answer: Day 10.
Setting -1.5d + 15 = 0 gives 1.5d = 15 and d = 10, so the lower course melts out on day 10, twenty days before the upper course.
Answer: 16 inches.
Substituting d = 10 into the upper course function gives S = -0.8(10) + 24 = -8 + 24 = 16 inches still stored high in the basin.
Answer: 14,400 acre-feet.
16 inches x 900 acre-feet per inch = 14,400 acre-feet of water still held as snow.
Answer: 4.69 x 10^9 gallons.
14,400 x 325,851 = 4,692,254,400 gallons. Moving the decimal point nine places to the left gives 4.692254..., which rounds to 4.69 x 10^9 gallons.
Answer: The snowpack stores winter precipitation and releases it as the sun warms the basin.
Winter precipitation falls as snow and stays on the ground instead of running off, so the river stays low. When spring sunlight melts the pack, the stored water runs downhill over a few weeks, and the higher course keeps feeding the river for weeks after the lower one is bare.

Differentiation

Support

Give students the upper course function S = -0.8d + 24 already written so problems 3, 4, and 7 become substitution and solving. Provide a number line marked in days from April 1 through May 1 so students can label the two melt-out days as they find them.

Extension

Ask students to graph both snow courses on one coordinate grid, shade the days when only the upper course still holds snow, and then predict what would happen to the fictional Tunstall River's flow if a warm spring raised the lower course's melt rate to 2.5 inches per day.

ELL supports

Pre-teach "snowpack," "snow water equivalent," and "melt-out" with a picture pair showing a deep snow column beside the much shorter column of water it becomes. Provide sentence frames such as "The snow water equivalent falls ___ inches each day, so ___." Allow students to answer problem 10 as a labeled diagram with arrows before writing.

Extensions

Have students find a published snow survey report for a mountain range near their state, compare its April 1 reading with the fictional Sarsen Basin's 24.0 inches, and write two sentences about what a low April reading would mean for a town downstream.