Math problem set Grade 11 Math

Merrick Point Jar Test: Coagulant Dose and Turbidity Regression

A grade 11 problem set fitting a line to a fictional jar test's dose and settled turbidity, then predicting, finding residuals, and reading a correlation coefficient.

Grade: Grade 11
Subject: Math
Material: Math problem set
Time: 45 min
Prep: No prep
Materials: Calculator, Graph paper

Standards

HSS-ID.B.6 Common Core Math

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related, including fitting a function to the data, using functions fitted to data to solve problems, and informally assessing the fit of a function by plotting and analyzing residuals.

How this material aligns: Problems 3 through 5 and 7 use a fitted line to predict settled turbidity, compute residuals from observed jar readings, and judge fit, the two-variable modeling this standard targets.

All materials for HSS-ID.B.6 →

HSS-ID.C.8 Common Core Math

Compute (using technology) and interpret the correlation coefficient of a linear fit.

How this material aligns: Problem 6 interprets the correlation coefficient reported for the dose-turbidity fit, reading its sign and magnitude as the strength and direction of the linear relationship.

All materials for HSS-ID.C.8 →

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

Objectives

Students will use a fitted regression line to predict settled turbidity and compute residuals from observed data.
Students will interpret a correlation coefficient as the strength and direction of a linear relationship.

Vocabulary

Turbidity: Turbidity is a measure of how cloudy or murky water looks because of tiny particles floating in it. Clear water has low turbidity, and muddy water has high turbidity.

Also: coagulant, residual, correlation coefficient

Teacher overview

Before a treatment plant sets its chemical dose, operators run a jar test: several beakers of the same raw water get different doses of coagulant, and the settled turbidity is measured after mixing. The paired results form a scatter of dose against turbidity that a line can model. This set uses a fictional jar test at the Merrick Point plant so students can fit, predict, and check residuals on realistic data. A common misconception is that the linear trend continues without limit; in a real jar test, adding coagulant past the optimum eventually stops helping and can even raise turbidity, so a line only models the tested range.

Merrick Point Jar Test: Coagulant Dose and Turbidity Regression — Grade 11 Math

Name: ____________________________ Date: ______________

Problems

The Merrick Point Water Treatment Plant (a fictional plant used for this set) ran a jar test on one batch of raw water. Each jar received a different alum dose, and the settled turbidity was measured after mixing and settling.

Alum dose (mg/L)	Settled turbidity (NTU)
10	8.2
20	6.5
30	5.1
40	3.9
50	2.4
60	1.3

The least-squares line for this data is turbidity = 9.37 - 0.137 x dose, and its correlation coefficient is r = -0.998.

Find the mean alum dose across the six jars.
Find the mean settled turbidity across the six jars. Round to the nearest hundredth.
Use the least-squares line to predict the settled turbidity at an alum dose of 45 mg/L.
The jar at 40 mg/L measured 3.9 NTU. Find the residual (observed minus predicted) at that dose.
Find the residual at the 50 mg/L jar, which measured 2.4 NTU.
The correlation coefficient is r = -0.998. Interpret what its sign and its size say about the relationship between dose and turbidity.
The plant wants settled turbidity at or below 2.0 NTU before filtration. Use the line to find the minimum alum dose that meets that target. Round up to the nearest whole mg/L.
The residuals in items 4 and 5 are both small, and they have opposite signs. Explain why that pattern supports using a line to model this jar test.

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: 35 mg/L. (10 + 20 + 30 + 40 + 50 + 60) / 6 = 210 / 6 = 35 mg/L.
Answer: 4.57 NTU. (8.2 + 6.5 + 5.1 + 3.9 + 2.4 + 1.3) / 6 = 27.4 / 6 = 4.567, which rounds to 4.57 NTU.
Answer: about 3.21 NTU. 9.37 - 0.137 x 45 = 9.37 - 6.165 = 3.205, which rounds to 3.21 NTU.
Answer: about +0.01 NTU. Predicted at 40 mg/L: 9.37 - 0.137 x 40 = 9.37 - 5.48 = 3.89 NTU. Residual = 3.9 - 3.89 = +0.01 NTU.
Answer: about -0.12 NTU. Predicted at 50 mg/L: 9.37 - 0.137 x 50 = 9.37 - 6.85 = 2.52 NTU. Residual = 2.4 - 2.52 = -0.12 NTU.
Answer: A strong negative linear relationship. The negative sign means turbidity falls as dose rises, and the size, very close to -1, means the points lie almost exactly on a straight line, so the line is a dependable summary within the tested range.
Answer: about 54 mg/L. Solve 9.37 - 0.137 x dose = 2.0, so 0.137 x dose = 7.37 and dose = 53.8 mg/L. Rounding up to be at or below the target gives 54 mg/L, which predicts 1.97 NTU.
Answer: Small residuals mean each prediction is close to the measured value, and residuals with mixed signs mean the points scatter above and below the line rather than curving away from it. Together those signal that a straight line, not a curve, fits this data well.

Differentiation

Support

Provide the six points already plotted on graph paper with the least-squares line drawn, and give a worked residual for a seventh made-up point so students can copy the observed-minus-predicted procedure before doing items 4 and 5.

Extension

Ask students to predict turbidity at a dose of 80 milligrams per liter, then explain why the plant should not trust that prediction, connecting their answer to the overview's note about coagulant past the optimum.

ELL supports

Pre-teach "coagulant," "settled," "predict," and "residual" with a labeled sketch of a jar with cloudy water clearing as particles settle. Provide a sentence frame for item 6: "The sign is ___, so turbidity ___ as dose rises, and the size means the fit is ___."