Lesson plan Grade 9 Math
Duskerton Road Salt and Skelbrook Chloride: Residuals and Fit
A 55-minute grade 9 lesson where students fit a least-squares line to six winters of road-salt and stream-chloride data, then read the residuals and the correlation.
Objectives
- Students will fit a least-squares line to paired data using a calculator or spreadsheet.
- Students will compute residuals and use their pattern to judge the fit of the line.
- Students will interpret a correlation coefficient as the strength of a linear association.
Vocabulary
Key words: residual, least-squares line, correlation coefficient, chloride, extrapolation
Teacher overview
Road salt applied to winter pavement does not disappear when the snow melts, and much of the chloride it carries runs off into nearby streams. This lesson uses six winters of paired records from the fictional town of Duskerton and the fictional Skelbrook that drains it: tons of salt spread, and the chloride concentration measured in the stream each spring. Students fit a least-squares line, compute residuals, and interpret the correlation coefficient. Two errors are common here. The first is computing a residual as predicted minus observed, which reverses every sign; the residual is observed minus predicted. The second is treating a correlation near 1 as proof that the salt caused the chloride, when the statistic by itself only reports how tightly the two columns track each other.
Procedure
Launch (8 min): Show the six pairs of numbers from the fictional Duskerton records without a graph and ask students whether the two columns look related. Take a show of hands on whether a straight line would describe the relationship well, and record the tally to revisit at the end.
Model the fit (12 min): Demonstrate entering the six pairs into a spreadsheet or graphing calculator and producing the least-squares line. Work through what each part of the output means, naming the slope as chloride per ton of salt and the intercept as the modeled chloride when no salt is applied. Plot the line over the scatter plot on the projector.
Guided practice with residuals (12 min): With the class, compute the residual for the first data pair as observed minus predicted, and write it on the board with its sign. Have pairs of students compute two more residuals each, then collect all six on a residual plot. Ask whether the residuals scatter above and below zero without an obvious curve, which is the informal check that a line is a reasonable model here.
Independent practice (18 min): Students complete questions 1 through 10 on the handout, which uses the same fictional Duskerton data. Remind students to attach units to the slope, since a slope of 2 with no units communicates nothing.
Closure and formative check (5 min): Return to the launch tally and ask students to write one sentence on an exit slip answering whether the six winters show that road salt raises stream chloride. Collect the slips and count how many students claim proof of cause. If more than a third do, open the next session with a short counterexample before continuing.
Student Handout
The table below shows six winters of records kept by the fictional town of Duskerton, where x is the tons of road salt spread in the Skelbrook watershed and y is the chloride concentration measured in the stream the following spring.
| Salt applied, x (tons) | Spring chloride, y (mg/L) |
|---|---|
| 20 | 51 |
| 30 | 68 |
| 40 | 91 |
| 50 | 109 |
| 60 | 132 |
| 70 | 149 |
- Enter the six pairs into a calculator or spreadsheet and find the least-squares line. Report its slope, in mg/L per ton.
- Report the y-intercept of that same least-squares line, in mg/L.
- The least-squares line from questions 1 and 2 is y = 2x + 10. Use it to predict the spring chloride for a winter in which 45 tons of salt are spread.
- Using the line y = 2x + 10, find the residual for the winter with 30 tons of salt and a measured chloride of 68 mg/L. Residual means observed minus predicted.
- Using the line y = 2x + 10, find the residual for the winter with 60 tons of salt and a measured chloride of 132 mg/L.
- Explain in one sentence what the slope of the line means for this fictional watershed.
- Use a calculator or spreadsheet to compute the correlation coefficient r for the six pairs. Round to three decimal places.
- Describe in one sentence what your value of r says about the strength and direction of the association.
- Using the line y = 2x + 10, predict the spring chloride for a winter in which 100 tons of salt are spread.
- Explain in one sentence whether you would trust the prediction from question 9, given that the recorded winters run only from 20 to 70 tons.
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: 2 mg/L per ton.
The least-squares fit gives a slope of exactly 2, meaning the modeled chloride rises 2 mg/L for each additional ton of salt spread.Answer: 10 mg/L.
The fitted line is y = 2x + 10, so the y-intercept is 10 mg/L, the chloride the model predicts for a winter with no salt applied.Answer: 100 mg/L.
Substituting gives y = 2(45) + 10 = 90 + 10 = 100 mg/L.Answer: -2 mg/L.
The line predicts 2(30) + 10 = 70 mg/L, and the observed value was 68 mg/L, so the residual is 68 - 70 = -2 mg/L. The negative sign means the measured chloride came in below the line.Answer: 2 mg/L.
The line predicts 2(60) + 10 = 130 mg/L, and the observed value was 132 mg/L, so the residual is 132 - 130 = 2 mg/L, this time above the line.Answer: Each additional ton of salt is associated with about 2 mg/L more chloride.
The slope is a rate of change: chloride per ton of salt spread in this fictional watershed.Answer: 0.999.
A spreadsheet returns about 0.9991 for these six pairs, which rounds to 0.999.Answer: A very strong positive linear association.
The value sits just below 1, so the points fall very close to a rising straight line, and higher salt totals come with higher chloride readings.Answer: 210 mg/L.
Substituting gives y = 2(100) + 10 = 200 + 10 = 210 mg/L.Answer: No.
A winter with 100 tons lies well outside the 20 to 70 ton range the model was built from, so the prediction is an extrapolation, and nothing in this data set shows that the same straight-line relationship continues that far out.
Differentiation
Support
Provide these students with the predicted values already computed for all six winters in the fictional Duskerton data, so their work on residuals is one subtraction per row rather than a substitution followed by a subtraction. Give a printed reminder card reading "residual = observed - predicted" and allow spreadsheet use for every step.
Extension
Have students remove the winter with 40 tons from the fictional data set, refit the line, and report how much the slope and the correlation coefficient change. Ask them to explain why removing a point near the center of the x values moves the slope less than removing an end point would.
ELL supports
Pre-teach "residual," "observed," "predicted," and "association" using a simple picture of one data point sitting above a line with a vertical segment drawn between them and labeled. Provide sentence frames for questions 6, 8, and 10, such as "For every one more ton of salt, the model predicts ___ more chloride."
Extensions
Students can look up their own state's transportation department winter maintenance reports and compare the salt totals reported there with the range in the fictional Duskerton table. A follow-up discussion can consider what other variables, such as total snowfall or stream flow during the spring sampling, would need to be recorded before anyone could argue that the salt caused the chloride change.