Math problem set Grade 7 Science
Blue Heron Creek: What a Five-Site pH Sample Can and Can't Tell You
A grade 7 math set using a fictional creek's pH sampling data to practice mean and range, then reasons about how well a small sample represents a waterway.
Objectives
- Students will calculate the mean and range of a set of water-quality measurements.
- Students will calculate what percent of sampled sites fall outside a stated standard range.
- Students will compare two samples to reason about how well a small sample represents an entire waterway.
Vocabulary
Key words: mean, range, sample, population
Teacher overview
A fictional watershed group tests pH at five sites along Blue Heron Creek, chosen from 40 possible monitoring points along the creek, on the same day. This problem set has students calculate the mean and range of that sample, check individual sites against the real secondary drinking-water pH standard of 6.5 to 8.5, and then compare a second week's sample to reason about how much a single day's data can really tell you about an entire creek. The secondary standard is an aesthetic guideline for drinking water, used here as a benchmark for comparing sites. A common misconception in this math is trusting the mean alone to judge water quality; a mean that falls comfortably inside the standard range can still hide individual sites that fall outside it, as problem 5 will show directly.
Problems
Blue Heron Creek's watershed group tests pH at five sites on the same day. Their readings are:
| Site | A | B | C | D | E |
|---|---|---|---|---|---|
| pH | 7.2 | 6.1 | 7.8 | 8.9 | 7.0 |
The secondary drinking-water standard for pH is 6.5 to 8.5.
- Calculate the mean pH across the five sites. Show your work.
- Calculate the range of the five pH readings (the highest value minus the lowest value).
- Which site(s) fall outside the 6.5–8.5 standard range, and by how many pH units does each miss the range?
- What percent of the five sampled sites fall outside the standard range?
- Is the mean pH you calculated in problem 1 within the standard range? Explain how a mean can be misleading when some individual sites, like the ones in problem 3, actually fall outside that range.
- Blue Heron Creek has 40 possible monitoring points in total, but the watershed group tested only 5 as a sample. If the sample's percent of out-of-range sites (from problem 4) held true for the whole creek, about how many of the 40 points would you expect to fall outside the standard range?
- The following week, the group tests five different sites: pH readings of 7.5, 7.6, 7.4, 7.9, and 7.3. Calculate this second sample's mean, rounded to the nearest tenth.
- Calculate the range of the second sample (problem 7) and compare it to the range of the first sample (problem 2). Which sample shows more variation between sites?
- Based on comparing the two weekly samples (problems 1–2 and 7–8), would you trust a single day's sample of five sites to represent Blue Heron Creek's pH condition on every day of the year? Explain your reasoning using evidence from both samples.
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: 7.4.
(7.2 + 6.1 + 7.8 + 8.9 + 7.0) ÷ 5 = 37.0 ÷ 5 = 7.4. - Answer: 2.8.
Highest reading (8.9) − lowest reading (6.1) = 2.8. - Answer: Site B and Site D.
Site B (6.1) falls 6.5 − 6.1 = 0.4 pH units below the range. Site D (8.9) falls 8.9 − 8.5 = 0.4 pH units above the range. - Answer: 40 percent.
2 of the 5 sites (B and D) fall outside the range. 2 ÷ 5 × 100 = 40%. - Answer: yes.
The mean, 7.4, is within the 6.5–8.5 range. But that single number hides the fact that Site B (6.1) and Site D (8.9) both fall outside the range: two out-of-range sites in opposite directions can still average out to a mean that looks perfectly normal. - Answer: 16 points.
40 points × 0.40 = 16 points, the estimated number of the creek's 40 monitoring points that would fall outside the standard range if this sample's rate held true across the whole creek. - Answer: 7.5.
(7.5 + 7.6 + 7.4 + 7.9 + 7.3) ÷ 5 = 37.7 ÷ 5 = 7.54, which rounds to 7.5. - Answer: 0.6.
The second sample's range is 7.9 − 7.3 = 0.6, much smaller than the first sample's range of 2.8. The first sample shows more site-to-site variation, which could happen if its five sites were spread across parts of the creek with different pollution sources or groundwater inputs, while the second week's sites happened to cluster closer together in condition. - Answer: Answers will vary but should note that the two samples produced very different ranges (2.8 versus 0.6) even though their means were similar, which suggests that a single five-site sample on a single day may not reliably represent the creek's full range of conditions; more samples across more days and locations would give a more trustworthy picture.
Differentiation
Support (below grade level)
Provide the mean formula written out as a labeled arrow rule ("add all the values, then divide by how many there are") and complete problem 1 together as a class before pairs attempt problem 7 independently. Give students a printed number line marked from 6.0 to 9.0 to plot each site's pH and visually spot which points fall outside the shaded 6.5–8.5 standard range.
Extension (above grade level)
Ask students to calculate the median of the first sample (7.2) and compare it to the mean (7.4), then explain which measure, in this particular dataset, gives a more accurate single-number picture of "typical" water quality along the creek.
ELL supports
Pre-teach "mean," "range," "sample," and "population" with a simple visual: five dots on a number line representing the sample, with an arrow labeled "population" pointing to a much longer line representing all 40 possible points. Allow problems 3 and 9 to be answered with labeled diagrams and short phrases rather than full sentences.