Worksheet Grade 9 Math
Wickcombe's Ivyshaw Row Flush Test: Lead Against Liters Drawn
A grade 9 worksheet using a sequential flush profile from one home's lead service line to practice average rate of change and one-variable equations.
Objectives
- Students will calculate average rates of change from a sequential sampling profile.
- Students will solve a linear equation for the volume drawn at a given lead concentration.
- Students will compare the mean and median of a small data set and explain the difference.
Vocabulary
- Lead service line
- A lead service line is an old water pipe made of lead metal that connects a house to the water main. Lead can get into the drinking water, so these pipes are being dug up and replaced across the country.
- 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.
Also: sequential sampling, flush, parts per billion
Teacher overview
When a home is served by a lead service line, the water that has sat in that pipe overnight carries the most lead, and the concentration drops as fresh water from the main replaces it. Technicians measure this by sequential sampling: they open a tap and collect a series of samples, one liter at a time, recording the lead in each. In many real homes the water that stood in the service line reaches the tap a few liters into the flush rather than in the first liter, so the highest reading often comes later than the first sample. This worksheet uses one such fictional profile, simplified so the highest reading comes first, from a home in the fictional village of Wickcombe. Students measure how fast the concentration falls over two stretches of the flush, solve for the volume at which the water crosses two regulatory levels, and compare the mean and median of the readings. The misconception to correct here is that the average of a flush series describes what the household drinks. The samples are a deliberate sequence rather than a random draw, so the mean of the seven values describes the flush itself, not a typical glass of water, and questions 7 through 9 are written to make that difference visible.
Student Worksheet
The readings below come from a fictional sequential flush test at a home on Ivyshaw Row in the fictional village of Wickcombe. The home is served by a lead service line. A technician opened the kitchen tap and collected one-liter samples in order, recording the lead in parts per billion.
| Liters drawn | Lead (ppb) |
|---|---|
| 0 | 32 |
| 1 | 26 |
| 2 | 20 |
| 3 | 14 |
| 4 | 8 |
| 5 | 5 |
| 6 | 4 |
- Find the average rate of change in lead from 0 liters to 4 liters. Give your answer in ppb per liter.
- Find the average rate of change in lead from 4 liters to 6 liters. Give your answer in ppb per liter.
- The first five readings fall on the line L = 32 - 6v, where v is liters drawn and L is lead in ppb. Use that line to predict the lead after 2.5 liters.
- The lead action level is 15 ppb. Using L = 32 - 6v, find the number of liters at which the lead reaches 15 ppb.
- The action level drops to 10 ppb in 2027. Using L = 32 - 6v, find the number of liters at which the lead reaches 10 ppb.
- Does the line L = 32 - 6v still fit the readings taken after 4 liters? Answer yes or no.
- Find the mean of the seven readings in the table.
- Find the median of the seven readings in the table.
- Explain in one or two sentences why the mean of these readings is larger than the median.
- Find the percent decrease in lead from the first draw at 32 ppb to the reading at 4 liters.
- Explain in two or three sentences why the water drawn first in the morning holds the most lead.
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: -6 ppb per liter.
The change is 8 - 32 = -24 ppb across 4 - 0 = 4 liters, and -24 divided by 4 = -6 ppb per liter. The negative sign means the concentration is falling.Answer: -2 ppb per liter.
The change is 4 - 8 = -4 ppb across 6 - 4 = 2 liters, and -4 divided by 2 = -2 ppb per liter. The lead is still falling, but only a third as fast as over the first stretch.Answer: 17 ppb.
Substituting gives L = 32 - 6(2.5) = 32 - 15 = 17 ppb.Answer: 2.8 liters.
Solving 32 - 6v = 15 gives 6v = 17, so v = 2.83 liters, which rounds to about 2.8 liters.Answer: 3.7 liters.
Solving 32 - 6v = 10 gives 6v = 22, so v = 3.67 liters, which rounds to about 3.7 liters. Reaching the lower level takes almost a liter more of flushing.Answer: No.
The line predicts 32 - 6(5) = 2 ppb at 5 liters and 32 - 6(6) = -4 ppb at 6 liters, while the measured values are 5 ppb and 4 ppb. A negative concentration is impossible, so the straight-line model has stopped describing the flush.Answer: 15.6 ppb.
The seven readings total 32 + 26 + 20 + 14 + 8 + 5 + 4 = 109 ppb, and 109 divided by 7 = 15.57 ppb, which rounds to 15.6 ppb.Answer: 14 ppb.
Sorted in order the readings are 4, 5, 8, 14, 20, 26, 32, and the fourth of seven values is 14 ppb.Answer: The two highest first-draw readings pull the mean upward.
The values are bunched at the low end and stretch out toward 32 ppb at the high end, and a mean responds to how far those high values sit from the rest while a median only counts their position in the order.Answer: 75 percent.
The drop is 32 - 8 = 24 ppb, and 24 divided by 32 = 0.75, which is 75 percent of the first-draw value.Answer: That water has been standing still overnight in the pipes and fixtures nearest the tap.
Water standing in contact with lead pipe, solder, or brass picks up lead over hours, so the first water out in the morning has had all night to do so. Once the tap runs long enough to pull fresh water from the main into the house, the concentration falls toward whatever the main itself carries.
Differentiation
Support
Give these students the rate-of-change formula and one worked interval from the fictional Wickcombe profile, using the stretch from 0 to 2 liters, before they attempt questions 1 and 2. For questions 4 and 5, provide the equation already set up as 32 - 6v = 15, so the work begins at the solving step rather than the translation step.
Extension
Ask students to fit a second line to the fictional readings from 4 to 6 liters, write it in slope-intercept form, and state the volume at which the two lines cross. A further question is how a technician's conclusion would change if only the first sample had been collected instead of the full series.
ELL supports
Provide a labeled picture showing a service line running from a street main into a house, with the first sample bottle drawn at the tap, so "service line," "flush," and "draw" are anchored to an image. Give sentence frames for questions 9 and 11, such as "The first water has more lead because ___."
Extensions
Students can read their own water utility's Consumer Confidence Report to see how lead results are summarized, then explain why a utility reports a percentile of many homes rather than a single home's flush profile. A second task is to describe what a household could do while waiting for a lead service line to be replaced, based on the pattern the fictional readings show.