Math problem set Grade 8 Science

Marrow Valley Reservoir: Fitting a Line to Rainfall and Storage Data

A grade 8 math problem set where students scatter-plot a fictional reservoir's rainfall and storage data, use a given fitted line, and evaluate its predictions.

Grade: Grade 8
Subject: Science
Material: Math problem set
Time: 40 min
Prep: Low prep
Materials: graph paper, ruler, calculator (optional)

Standards

8.SP.A.1 Common Core Math

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities, describing clustering, outliers, and positive, negative, linear, or nonlinear association.

How this material aligns: Problem 1 requires students to construct and interpret a scatter plot of Marrow Valley's eight years of rainfall and storage data, describing the association.

All materials for 8.SP.A.1 →

8.SP.A.2 Common Core Math

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line and assess the model fit by judging the closeness of the data points to the line.

How this material aligns: Problems 3 through 7 require students to use an informally fitted line to make predictions and judge how closely the line matches Year 4's recorded storage.

All materials for 8.SP.A.2 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-16.

Objectives

Students will construct and interpret a scatter plot to describe the association between rainfall and reservoir storage.
Students will use an informally fitted line to predict reservoir storage from a given rainfall amount.
Students will evaluate how well a fitted line matches recorded data and identify when a prediction is an unreliable extrapolation.

Vocabulary

Key words: scatter plot, line of best fit, extrapolation, association

Teacher overview

The fictional Marrow Valley Reservoir has eight years of annual rainfall and year-end storage data, numbers created for this problem set. This problem set has students plot that rainfall-storage data as a scatter plot, informally fit a line to the pattern, and use the line to make and evaluate predictions. A common misconception is treating a fitted line as equally reliable at every rainfall value; the problems below are built to show that predictions made well beyond the recorded data range, an extrapolation, deserve more caution than predictions made within it.

Marrow Valley Reservoir: Fitting a Line to Rainfall and Storage Data — Grade 8 Science

Name: ____________________________ Date: ______________

Problems

Marrow Valley's reservoir manager (fictional, created for this problem set) recorded the following eight years of data. Use this table for every problem.

Year	Annual rainfall (inches)	Year-end storage (percent of capacity)
1	18	52
2	22	61
3	15	45
4	27	74
5	20	58
6	24	66
7	16	50
8	25	68

Plot the eight years of data as a scatter plot, with rainfall on the x-axis and year-end storage on the y-axis. Describe whether the association between rainfall and storage looks positive, negative, or roughly linear.
Draw the line from problem 3 (storage percent = 2 × rainfall + 16) on your scatter plot. Which year's point sits farthest from that line?
A reasonable line of best fit for this data is storage percent = 2 × rainfall (in inches) + 16. Use this line to predict the storage percent for a year with 21 inches of rainfall.
Predict the storage percent for a year with 30 inches of rainfall using the same line. Do you think this prediction is reliable? Explain why or why not, given the range of rainfall values in the original data.
Using the line, about how many inches of rainfall would be needed for the reservoir to reach 100 percent storage?
Compare your prediction in problem 5 to the actual range of rainfall in the eight-year data set (15 to 27 inches). What does this comparison suggest about how likely a year with that much rainfall would be for Marrow Valley?
Find the difference between the line's prediction and the actual recorded storage value for Year 4 (27 inches of rainfall, 74 percent storage). What does this difference tell you about how well the line fits Year 4 compared to the other years?
Based on the overall pattern in the scatter plot, does the reservoir manager have good evidence that more rainfall leads to higher year-end storage? Explain your reasoning using the data.

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: positive, roughly linear.
Plotting the eight points shows storage percent generally rising as rainfall increases, with points clustering fairly close to a straight line rather than curving or showing no pattern, so the association is positive and roughly linear.
Answer: Year 4.
The line predicts 2 × 27 + 16 = 70 percent for Year 4, but Year 4 recorded 74 percent, so its point sits 4 percentage points above the line, while every other year is within 2 points of it.
Answer: 58 percent.
Storage percent = 2 × 21 + 16 = 42 + 16 = 58 percent.
Answer: 76 percent.
Storage percent = 2 × 30 + 16 = 60 + 16 = 76 percent. This prediction extends beyond the original data's rainfall range of 15 to 27 inches, so it is an extrapolation and less reliable than a prediction made within the recorded range.
Answer: 42 inches.
Set 100 = 2x + 16. Subtract 16 from both sides: 84 = 2x. Divide by 2: x = 42 inches.
Answer: A 42-inch rainfall year is well outside the recorded range.
A 42-inch year would be about 15 inches more than the wettest year (27 inches) in the eight-year data set, suggesting such a year would be unusual, or possibly outside anything Marrow Valley has experienced in this record.
Answer: 4 percentage points.
The line predicts 2 × 27 + 16 = 70 percent, but the actual recorded value was 74 percent, a difference of 74 − 70 = 4 percentage points, the largest gap of any year in the data set, showing the line fits Year 4 less closely than it fits the other years.
Answer: yes.
Seven of the eight years fall within about 2 percentage points of the fitted line, and storage percent consistently rises as rainfall increases across the data set, which is reasonable evidence of a positive association between rainfall and year-end storage.

Differentiation

Support (below grade level)

Provide pre-drawn axes with gridlines already labeled in increments of 2 inches on the x-axis and 5 percent on the y-axis, and help students draw the fitted line from problem 3 on those axes before they work problems 3 through 7.

Extension (above grade level)

Ask students to calculate the difference between the fitted line's prediction and the actual recorded value for all eight years, not just Year 4, then identify which year's recorded storage comes closest to the line's prediction and which comes farthest from it.

ELL supports

Pre-teach "scatter plot," "line of best fit," and "extrapolation" with a simple visual example unrelated to reservoirs, such as plant height measured over several weeks, before applying the same idea to Marrow Valley's data. Allow problem 1's description to be given as a checklist of prepared phrases (positive or negative, strong or weak, linear or not linear) that students circle instead of writing a paragraph.

Extensions

Have students research, with teacher guidance, their own region's approximate annual rainfall and discuss, using Marrow Valley's fitted line only as a rough example rather than a real prediction for their own reservoir, why a line fitted to one reservoir's data should not be assumed to apply to a different location.