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Math problem set Grade 8 Engineering & Design

Nettlebrook Reservoir Spillway: A Grade 8 Scatter Plot Set

A grade 8 problem set where students scatter-plot eight years of rainfall and reservoir inflow data, fit a trend line by eye, and set a spillway capacity constraint.

Objectives

  1. Students will construct a scatter plot of rainfall and reservoir inflow data and describe the association it shows.
  2. Students will informally fit a trend line to the data and estimate its slope using two representative points.
  3. Students will use the trend line to predict a design inflow value and evaluate it against an engineering capacity criterion.

Vocabulary

Spillway
A dam's overflow channel, giving extra water a safe path around or over the dam when the reservoir gets too full. It protects the dam from being damaged in floods.

Also: scatter plot, trend line, outlier, association

Teacher overview

The fictional Nettlebrook Reservoir supplies water to the town of Wrenfield, and its dam's spillway, the structure that safely releases extra water during heavy rain, was built decades ago. Before recommending whether the spillway needs to be enlarged, the town's engineering team wants students to study eight years of recorded April rainfall alongside the reservoir's peak inflow that same month, to see whether the two values rise together in a predictable way.

A common misconception in this math is treating every point on a scatter plot as equally reliable when fitting a trend line. One year in this data set sits well above the pattern the other seven years follow; students who let that single point drag their trend line off course will make a spillway prediction that no longer matches most of the recorded years, so identifying and discussing that point is part of the task.

Nettlebrook Reservoir Spillway: A Grade 8 Scatter Plot Set — Grade 8 Engineering & Design
Name: ____________________________ Date: ______________

Problems

Nettlebrook Reservoir's fictional engineering office recorded April rainfall and peak inflow for eight years:

Year April rainfall (inches) Peak inflow (cfs)
1 3 220
2 5 410
3 2 150
4 6 480
5 4 300
6 7 560
7 3 240
8 5 550
  1. Construct a scatter plot of this data, with rainfall (inches) on the horizontal axis and peak inflow (cfs) on the vertical axis. Plot all eight points.
  2. Describe the association shown in your scatter plot: is it positive, negative, or is there no clear pattern? Is the association roughly linear?
  3. Draw a trend line by eye through the cluster of points. Treat (2, 150) and (7, 560) as representative points on your line. Calculate the line's slope in cfs per inch of rainfall.
  4. Use your trend line's slope and the point (7, 560) to write an equation for the trend line in the form inflow = slope × rainfall + b. Then use your equation to predict the peak inflow for an 8-inch April rainfall.
  5. The engineering team's design criterion is that the spillway must safely handle the inflow predicted for an 8-inch rainfall, plus a safety margin. Use your prediction from problem 4. What minimum spillway capacity, in cfs, should the design target before the margin is added? Then explain in one sentence why engineers add a safety margin instead of designing for the predicted value exactly.
  6. One year's data point sits noticeably above the trend the other seven years follow. Identify that year and its point. Then explain in one or two sentences how including or excluding that point could change a spillway design decision.
  7. Use your trend line's equation from problem 4 again. Predict the peak inflow for a year with 4.5 inches of April rainfall, a value that falls between two years already recorded.
  8. Challenge. Fitting a trend line by eye is a reasonable first step for an early design estimate. Explain in two or three sentences why that is so, and why the engineering team would still want a more rigorous statistical method before setting the spillway's exact dimensions.

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: A correctly plotted scatter plot with all eight points, rainfall on the horizontal axis and inflow on the vertical axis.
    Seven of the eight points should cluster along a rising diagonal pattern; the point for year 8 (5, 550) sits noticeably above that cluster.

  2. Answer: Positive, and roughly linear.
    As rainfall increases, peak inflow also increases for nearly every year in the data, and the seven closely clustered points suggest a straight-line pattern rather than a curve.

  3. Answer: 82 cfs per inch.
    Slope = (560 − 150) ÷ (7 − 2) = 410 ÷ 5 = 82 cfs per inch of rainfall.

  4. Answer: About 642 cfs.
    Using point (7, 560) and slope 82: 560 = 82(7) + b, so 560 = 574 + b, and b = −14. The trend line is inflow = 82(rainfall) − 14. At 8 inches: inflow = 82(8) − 14 = 656 − 14 = 642 cfs.

  5. Answer: At least 642 cfs, before adding a margin.
    Engineers add a safety margin above a statistical prediction because a trend line is only an estimate based on eight recorded years; an unusually heavy storm could produce inflow higher than anything in the data used to build the prediction.

  6. Answer: Year 8, at (5, 550).
    The trend line predicts about 396 cfs for 5 inches of rainfall (82 × 5 − 14 = 396), but year 8 recorded 550 cfs, well above that. If engineers exclude this point as unusual, their trend line and spillway prediction stay lower; if they keep it, it pulls the prediction higher and suggests the spillway may need more capacity than the other seven years alone would indicate.

  7. Answer: About 355 cfs.
    Inflow = 82(4.5) − 14 = 369 − 14 = 355 cfs.

  8. Answer: Accept any well-reasoned response. A strong answer notes that fitting a line by eye is quick and gives a reasonable first estimate for early planning, letting the team see the general size of the problem, but it depends on which two points a person happens to pick and does not account for how far the outlier point sits from the rest of the data. A formal statistical method calculates the best-fitting line using every data point at once and can measure how well that line actually fits, giving the team a more defensible number before committing to a specific spillway size.

Differentiation

Support

Provide pre-printed graph paper with the axes already labeled and scaled for the fictional Nettlebrook data (rainfall 0-8 inches, inflow 0-600 cfs), and give students the two representative points for problem 3 already circled on a sample scatter plot.

Extension

Ask students to calculate the trend line's slope using a different pair of points (not (2, 150) and (7, 560)) and compare the resulting slope value, discussing why choosing different points to fit a line by eye can produce slightly different predictions.

ELL supports

Provide a labeled diagram of a dam and spillway showing where reservoir water rises and where the spillway releases it, using the words "rainfall," "inflow," and "capacity" next to the diagram's parts. Allow the descriptions in problems 2, 5, 6, and 8 to be written as short phrases or bullet points.

Extensions

Have students work out (using only the data given in this problem set) how the trend line's prediction would change if a ninth year of data, with 8 inches of rainfall and 610 cfs of inflow, were added to the table, and have them explain whether that new point falls close to or far from their original trend line's prediction from problem 4.

Every figure in this scenario is fictional, chosen so students work with realistic engineering numbers.