Lesson plan Grade 9 Math
Marrisford's Lead Sampling: Finding the 90th Percentile
A grade 9 lesson where students order twenty fictional tap-water lead results, find the 90th percentile the way the rule does, and test it against the limit.
Objectives
- Students will order a data set and locate the 90th percentile using the position rule stated in the handout.
- Students will compare a sample statistic with a regulatory limit and state the compliance result.
- Students will explain why a sample estimate does not describe every home in the population exactly.
Vocabulary
- Action level
- A trigger number in lead and copper testing. If too many homes test above this level, the utility must take steps like improving treatment and telling the public.
- First-draw sample
- A water sample taken from a faucet after the water has sat still in the pipes for hours, usually first thing in the morning. It shows how much lead can build up in water that sits in a home's pipes.
Also: percentile, population, skew
Teacher overview
Utilities do not judge lead compliance by an average. They collect first-draw samples from a set of higher-risk homes and compare the 90th percentile of those results against the action level, which is 15 parts per billion today and drops to 10 parts per billion in 2027. This lesson walks students through that calculation on a fictional city's twenty samples, then asks what the number does and does not say about every other home. The misconception the lesson addresses is that a 90th percentile below the action level means every home is safe; it means nine out of ten sampled homes fell below the value, and individual homes above it are expected.
Procedure
- Framing the question (8 min): Tell students that in this fictional scenario the Marrisford water utility must report a single number to the state each year, and that number is not the average. Ask what could go wrong if a city reported the mean of its lead samples. Collect two or three responses without correcting them, then state the rule the class will use: with n samples, the 90th percentile is the value in position n times 0.9 once the results are sorted from low to high.
- Modeling the rule (10 min): Work one fictional ten-home example on the board: 1.0, 1.4, 2.1, 2.5, 3.3, 4.0, 5.5, 6.2, 8.1, and 12.6 parts per billion. Because 10 times 0.9 is 9, the 90th percentile is the ninth value, 8.1. Ask the class where the highest reading sits and why it does not become the reported number. As a formative check, ask every student to write on a scrap of paper the position they would use for a set of 30 samples and hold it up. If fewer than about eight in ten have written 27, model a second set before moving on.
- Handout questions 1 through 5 (15 min): Students work individually or in pairs on the sorting, the percentile, the two comparisons against the limits, the median, and the mean. Circulate and check that students sorted the list before counting positions, since a miscount here propagates through the rest of the page.
- Discussion of shape, questions 6 through 8 (10 min): Bring the class together to compare the mean and the median. Ask what kind of data set makes the mean larger, and steer toward the two very high readings. Then have students finish the count and the percent items.
- Sample and population, questions 9 and 10 (8 min): Students draw the dot plot and answer the inference question. Emphasize that the utility sampled twenty homes out of thousands, and that those homes were chosen because they are higher risk rather than at random.
- Exit check (4 min): Each student writes one sentence stating what the fictional city's reported number would be and whether it meets the current limit and the 2027 limit. Collect on the way out.
Student Handout
The Marrisford water utility is a fictional city system. It collected one first-draw tap sample from each of twenty homes. All results are lead concentrations in parts per billion.
3.9, 0.9, 11.5, 2.2, 6.1, 1.3, 24.3, 3.0, 1.8, 9.0, 0.6, 4.5, 2.6, 16.8, 1.1, 5.2, 2.0, 7.4, 1.5, 3.4
Sort the results from low to high before you begin. With twenty samples, the 90th percentile is the value in position 20 times 0.9, which is the eighteenth value.
- Sort the results and state the eighteenth value, which is the city's reported 90th percentile.
- The action level today is 15 parts per billion. State whether the reported value from question 1 is above or below it.
- The action level drops to 10 parts per billion in 2027. State whether this same set of samples would meet that lower limit.
- Find the median of the twenty results.
- Find the mean of the twenty results. Round to the nearest hundredth.
- Explain why the mean is larger than the median for this data set.
- Count how many of the twenty homes returned a result above 15 parts per billion.
- Find your count from question 7 as a percent of the twenty homes sampled.
- Explain why the utility cannot conclude that exactly that percent of every home in the city is above the action level.
- Draw a dot plot of all twenty results on a number line scaled from 0 to 25 parts per billion, then describe the shape of the distribution in one sentence.
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: 11.5 parts per billion. Sorted, the results are 0.6, 0.9, 1.1, 1.3, 1.5, 1.8, 2.0, 2.2, 2.6, 3.0, 3.4, 3.9, 4.5, 5.2, 6.1, 7.4, 9.0, 11.5, 16.8, 24.3, and the eighteenth value is 11.5.
- Answer: Below the action level. The reported 11.5 is less than 15, so this fictional system would not exceed the current action level.
- Answer: No. The reported 11.5 is greater than 10, so the same set of samples would exceed the level that takes effect in 2027.
- Answer: 3.2 parts per billion. With twenty values the median is the average of the tenth and eleventh: (3.0 + 3.4) / 2 = 3.2.
- Answer: 5.46 parts per billion. The twenty results sum to 109.1, and 109.1 / 20 = 5.455, which rounds to 5.46.
- Answer: Two very high readings pull the mean upward. The results 16.8 and 24.3 sit far above the rest, and the mean adds their full size while the median depends only on the middle positions. That right-skewed shape is why the compliance rule uses a percentile rather than an average.
- Answer: 2 homes. Only 16.8 and 24.3 exceed 15 parts per billion.
- Answer: 10 percent. 2 out of 20 is 2 / 20 = 0.10, which is 10 percent of the sampled homes.
- Answer: The twenty homes are a sample, not the whole city. The utility deliberately chose higher-risk homes, so the sample is not a random draw from every household, and the true citywide share could be higher or lower. A different set of twenty homes would almost certainly produce a different percent, which is why the result is an estimate about the population rather than a count of it.
- Answer: A right-skewed plot with most dots below 5. The dots crowd between 0.6 and 5.2, thin out from 6 to 12, and leave two isolated dots near 16.8 and 24.3. The long tail on the right matches the gap between the mean and the median found in questions 4 and 5.
Differentiation
Support
Give students the fictional list already sorted and a printed number line with tick marks every one part per billion, so the work begins at counting positions rather than at ordering twenty decimals. Provide a worked copy of the ten-home board example for students to refer to while they answer question 1.
Extension
Ask students to determine what the twenty-first sample would have to be, in this fictional data set, for the reported 90th percentile to rise above the current action level. Have them explain why adding one sample changes which position the rule points to.
ELL supports
Pre-teach "sample," "percentile," and "action level" with a strip of twenty cards that students physically sort and then count to the eighteenth. Provide a sentence frame for question 9: "The utility tested ___ homes out of ___, so the result is an estimate of ___ rather than a count of ___."