Worksheet Grade 10 Math
Ledmoor Hollow Storms: A Rainfall and Peak Flow Line of Fit
A grade 10 worksheet where students fit a line to eight fictional storms at a creek gauge, compute residuals, and judge a prediction far outside the data range.
Objectives
- Students will find the slope and intercept of a line through two points of a paired data set.
- Students will use a fitted line to predict peak flow and compute residuals from observed values.
- Students will explain why a prediction far outside the data range is less reliable.
Vocabulary
Key words: peak flow, line of fit, residual, correlation coefficient, extrapolation
Teacher overview
Ledmoor Hollow, Starling Brook, and the eight storm records on this worksheet are fictional, and the paired values were set so that a line through the first and last points is easy to write by hand. A stream gauge reports peak flow in cubic feet per second, and pairing that peak with the rainfall that caused it is the first step in most flood studies. Students fit the line, use it, and then test it against the storms it did not pass through.
Student Worksheet
The town of Ledmoor Hollow keeps a gauge on Starling Brook. These eight storm records are fictional. Each pair gives the rainfall in inches and the peak flow in cfs.
Storm records: 0.5 inches and 75 cfs; 0.9 inches and 118 cfs; 1.3 inches and 180 cfs; 1.7 inches and 212 cfs; 2.1 inches and 275 cfs; 2.4 inches and 298 cfs; 2.7 inches and 345 cfs; 3.0 inches and 375 cfs.
Let x stand for rainfall in inches and y stand for peak flow in cfs.
Draw a line through the points (0.5, 75) and (3.0, 375). Find the slope of that line.
Find the y-intercept of the same line.
Write the equation of the line in the form y = mx + b.
Explain what the slope means for this creek. Use one or two sentences.
Use your equation to predict the peak flow for a storm that drops 1.3 inches of rain.
The gauge recorded 180 cfs for that 1.3-inch storm. Find the residual, which is the observed value minus the predicted value.
Use your equation to predict the peak flow for a storm that drops 2.1 inches of rain.
The gauge recorded 275 cfs for that 2.1-inch storm. Find the residual.
Use your equation to predict the peak flow for a storm that drops 1.7 inches of rain.
The gauge recorded 212 cfs for that 1.7-inch storm. Find the residual.
Use your equation to predict the peak flow for a storm that drops 4.5 inches of rain.
The largest storm in the record dropped 3.0 inches. Explain why the item 11 prediction is less reliable than the item 5 prediction.
A calculator gives a correlation coefficient of 0.998 for these eight fictional storms. Is the linear relationship strong or weak?
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: 120 cfs per inch. The slope is (375 − 75) ÷ (3.0 − 0.5) = 300 ÷ 2.5 = 120 cfs per inch of rain.
Answer: 15 cfs. Substituting the point (0.5, 75) gives 75 = 120(0.5) + b, so b = 75 − 60 = 15 cfs.
Answer: y = 120x + 15. The slope from item 1 and the intercept from item 2 give this equation, with x in inches and y in cfs.
Answer: Each additional inch of rain adds about 120 cfs to the peak flow. The intercept of 15 cfs is the line's prediction for zero rain, which lies outside the 0.5 to 3.0 inch data range, so it should not be read as a measured flow.
Answer: 171 cfs. Substituting gives 120(1.3) + 15 = 156 + 15 = 171 cfs.
Answer: 9 cfs. The residual is 180 − 171 = 9 cfs, so the line predicted less than the gauge recorded.
Answer: 267 cfs. Substituting gives 120(2.1) + 15 = 252 + 15 = 267 cfs.
Answer: 8 cfs. The residual is 275 − 267 = 8 cfs.
Answer: 219 cfs. Substituting gives 120(1.7) + 15 = 204 + 15 = 219 cfs.
Answer: −7 cfs. The residual is 212 − 219 = −7 cfs, so the line predicted more than the gauge recorded.
Answer: 555 cfs. Substituting gives 120(4.5) + 15 = 540 + 15 = 555 cfs.
Answer: The 4.5-inch storm sits outside the data range. Every fictional storm used to build the line fell between 0.5 and 3.0 inches, so the line has been checked only inside that band. A much larger storm can spill onto ground that stayed dry in the smaller storms, which changes how fast water reaches the brook, and the line has no record of that behavior. The 1.3-inch prediction sits inside the range and is supported by nearby points.
Answer: Strong. A correlation coefficient of 0.998 is very close to 1, so the points lie near a straight line and the relationship is strong and positive.
Differentiation
Support
Give students the eight fictional pairs already plotted on labeled axes with the line through (0.5, 75) and (3.0, 375) drawn in, so items 1 and 2 become a matter of reading the graph. Provide a residual table with the observed values filled in and only the predicted column blank.
Extension
Ask students to fit a second line through the points (0.9, 118) and (2.7, 345) of the fictional record, compute the same three residuals, and argue in writing which of the two lines fits better. A further challenge is to sum the residuals for each line and explain what a sum near zero does and does not tell them.
ELL supports
Provide a labeled picture of a residual, showing the observed point, the point on the line, and a short vertical arrow between them marked "residual." Offer a sentence frame for item 4: "For every one inch of rain, the peak flow goes up by about ___ cfs."
Extensions
Have students pull rainfall and peak flow pairs from a public gauge their teacher selects, plot them, and fit a line by the same two-point method used on the fictional Ledmoor Hollow data. A short written comparison of the two slopes makes a useful bridge to a later lesson on watershed size and runoff response.