Classroom activity Grade 10 Math
Gallowbrook Jar Test: A Randomization Card Simulation
A 50-minute grade 10 activity where students shuffle result cards from a fictional two-dose jar test to judge whether a 1.00 NTU difference could come from chance alone.
Objectives
- Students will compute the difference in mean settled turbidity between two coagulant doses.
- Students will explain what a re-randomization simulates about a two-treatment experiment.
- Students will use a set of simulated differences to judge whether an observed difference is likely to be chance.
Vocabulary
- Jar test
- A jar test is a small-scale experiment operators run in the lab before treating a whole tank of water, testing different amounts of chemical in separate jars to see which dose clears the water best. It helps operators avoid wasting chemicals or under-treating the water.
- 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, randomization, simulated difference
Teacher overview
This is a card-shuffling simulation for groups of three, built on a fictional jar test at the Gallowbrook treatment plant in which twelve jars of the same raw water received one of two alum doses. The card work models the statistics directly, because writing each settled turbidity on a card and reshuffling the cards into two piles of six acts out the assumption that the dose made no difference at all. Groups run their own shuffles to feel how much the difference moves, then analyze a printed record of twenty re-randomizations so that every student works from the same numbers.
Activity Directions
Setup (8 min): Arrange groups of three and give each group twelve index cards, a handout, and a calculator. Have groups write one settled turbidity value on each card, using the twelve fictional readings on the handout, and mark nothing else on the cards. Explain the design of the fictional test: one batch of raw water was split into twelve jars, six jars were assigned at random to a 20 mg/L alum dose and six to a 30 mg/L dose, and every jar was stirred and settled the same way. State the question the plant wants answered, which is whether the higher dose really settles the water clearer.
Running the simulation (20 min): Groups first compute the two dose means and the observed difference from the handout, then set the cards face down, shuffle all twelve, and deal them into two piles of six. They compute the mean of each pile and record the difference, first pile minus second pile, on a tally sheet. Ask each group to complete at least four shuffles and to post their differences on the board. Stop when the board holds at least twenty posted values. Point out that the shuffling deliberately ignores which dose a card came from, which is what makes it a picture of "no effect."
Analysis (14 min): Direct groups to the printed record of twenty re-randomizations on the handout, produced by an earlier class, and have them complete items 6 through 11 from that record rather than from the board. Ask each group to compare the spread of the printed record with the values their own class posted and to say in one sentence whether the two sets look similar.
Debrief (8 min): Ask groups to report their answer to item 7, then ask the closing question: if the dose truly made no difference, how often would shuffling alone produce a gap as large as the one the plant measured? Confirm that none of the twenty simulated gaps reached it. Close by asking what the plant should do next, and steer the answer toward repeating the fictional test on a second batch of raw water rather than toward treating one result as settled.
Student Handout
The Gallowbrook plant and this jar test are fictional. One batch of raw water was split into twelve jars. Six jars were assigned at random to a 20 mg/L alum dose and six to a 30 mg/L dose. Settled turbidity was measured in NTU. Round every answer to the nearest hundredth.
Settled turbidity at the 20 mg/L dose, in NTU: 3.4, 2.9, 3.8, 3.1, 3.6, 3.6
Settled turbidity at the 30 mg/L dose, in NTU: 2.5, 2.1, 2.8, 2.4, 2.0, 2.6
Twenty simulated differences in NTU, recorded by an earlier class that shuffled the same twelve values into two piles of six: 0.13, −0.27, 0.40, 0.07, −0.53, 0.33, −0.13, 0.20, 0.60, −0.40, 0.00, 0.27, −0.20, 0.47, −0.07, 0.13, −0.33, 0.53, −0.60, 0.20
Find the mean settled turbidity for the six jars at the 20 mg/L dose.
Find the mean settled turbidity for the six jars at the 30 mg/L dose.
Find the observed difference in means, taking the 20 mg/L mean minus the 30 mg/L mean.
Is this fictional jar test a sample survey, an observational study, or an experiment?
Explain what the card shuffling represents about the two doses. Use one or two sentences.
How many of the twenty simulated differences are 1.00 NTU or greater?
What proportion of the twenty is that? Give a decimal.
How many of the twenty simulated differences are 0.50 NTU or greater?
What is the largest of the twenty simulated differences?
Find the mean of the twenty simulated differences.
Based on item 7, is the observed difference easily explained by chance alone? Answer yes or no.
Explain why the two alum doses were assigned to the twelve jars at random. Use one or two sentences.
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: 3.40 NTU. The six readings total 3.4 + 2.9 + 3.8 + 3.1 + 3.6 + 3.6 = 20.4 NTU, and 20.4 ÷ 6 = 3.40 NTU.
Answer: 2.40 NTU. The six readings total 2.5 + 2.1 + 2.8 + 2.4 + 2.0 + 2.6 = 14.4 NTU, and 14.4 ÷ 6 = 2.40 NTU.
Answer: 1.00 NTU. The difference is 3.40 − 2.40 = 1.00 NTU, with the lower dose leaving the cloudier water.
Answer: An experiment. The fictional plant assigned the two doses to the jars rather than observing doses that were already in place, which is what separates an experiment from a survey or an observational study.
Answer: The shuffling acts out the assumption that the dose changed nothing. If the dose had no effect, each turbidity value would have landed in whichever pile it fell into by chance, so reshuffling the same twelve values and splitting them again shows the range of gaps chance alone can produce.
Answer: 0 simulated differences. The largest value in the printed record is 0.60 NTU, so none reaches 1.00 NTU.
Answer: 0.00. The proportion is 0 ÷ 20 = 0.00, which is none of the twenty simulated differences.
Answer: 2 simulated differences. The values 0.60 and 0.53 are at or above 0.50 NTU, and the next largest is 0.47 NTU.
Answer: 0.60 NTU. No other value in the printed record of twenty exceeds it.
Answer: 0.04 NTU. The twenty values sum to 3.33 − 2.53 = 0.80 NTU, and 0.80 ÷ 20 = 0.04 NTU, close to zero as expected when the piles are formed at random.
Answer: No. The largest of the twenty simulated differences was 0.60 NTU, which still falls 0.40 NTU short of the observed 1.00 NTU gap, so chance alone is an unlikely explanation and the higher dose appears to have settled the fictional raw water clearer.
Answer: Random assignment spreads unmeasured differences evenly. Jars can differ in stirring position, temperature, or the exact split of the raw water, and assigning doses at random keeps those nuisances from lining up with one dose, so a gap in the results is more fairly credited to the dose itself.
Differentiation
Support
Give groups the two dose means already computed for the fictional test and the twenty simulated differences already sorted from least to greatest, so items 6 through 9 become a matter of reading a sorted list. Provide a dot plot of the twenty values with a marked line at 1.00 NTU so the comparison in item 11 is visual.
Extension
Ask groups to find how large the observed difference would have to be for exactly two of the twenty simulated values to match or exceed it, then state what that would mean about the strength of the evidence. A second challenge is to repeat the analysis using only the first four readings at each dose from the fictional test and to explain why a smaller experiment gives weaker evidence.
ELL supports
Provide a card labeled "real result" beside a card labeled "chance result" and have students place the observed 1.00 NTU value and one simulated value under the matching label before beginning the analysis items. Offer a sentence frame for item 5: "Shuffling the cards shows what would happen if the dose ___, because each value could land in ___ pile."
Extensions
Have groups design a second fictional jar test comparing three alum doses rather than two, describe how they would assign jars at random, and explain what would change about the shuffling procedure. A short written protocol naming the doses, the number of jars, and the recording plan makes a useful bridge to a later lesson on multiple comparisons.