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Worksheet Grade 9 Math

Marren River Oxygen Log: A Dissolved Oxygen Data Worksheet

A grade 9 worksheet using a fictional river's dissolved oxygen readings to find average rate of change and build a linear model against the 5 mg/L fish threshold.

Objectives

  1. Students will calculate the average rate of change of dissolved oxygen from a data table.
  2. Students will build a linear model and use it to predict when oxygen reaches a threshold.

Vocabulary

Dissolved oxygen
Dissolved oxygen is the oxygen gas mixed into water that fish and other underwater animals breathe. Cold, moving water holds more oxygen than warm, still water.

Also: average rate of change, threshold, linear model

Teacher overview

Warm water holds less dissolved oxygen than cold water. In many rivers, daytime photosynthesis raises oxygen during the afternoon, but in a shaded or heavily loaded stretch, warming can pull the level down on a hot afternoon, as it does in this fictional log. A field crew recorded dissolved oxygen and temperature in the Marren River (a fictional river used for this worksheet) at four times during one summer day. This worksheet has students read that log, find average rates of change, and build a short linear model. Most fish need dissolved oxygen above 5 mg/L, so students also judge how close the river came to that limit. A common misconception is reading a rising evening number as an error; students should connect it to the falling temperature instead.

Marren River Oxygen Log: A Dissolved Oxygen Data Worksheet — Grade 9 Math
Name: ____________________________ Date: ______________

Student Worksheet

Use this monitoring log from the Marren River for every problem. All readings are from the same site on the same summer day.

Time Temperature (deg C) Dissolved oxygen (mg/L)
6 a.m. 18 8.4
12 p.m. 24 6.9
4 p.m. 27 5.2
10 p.m. 22 6.1

Most fish need dissolved oxygen to stay above 5 mg/L.

  1. Which reading is closest to the 5 mg/L level most fish need?
  2. By how many mg/L does the 4 p.m. reading sit above the 5 mg/L level?
  3. Find the average rate of change in dissolved oxygen, in mg/L per hour, between 6 a.m. and 4 p.m.
  4. Assume the oxygen keeps falling at the rate from item 3 after 4 p.m. Write a linear equation for dissolved oxygen D (in mg/L) as a function of h, the hours after 4 p.m.
  5. Use your equation from item 4 to predict how long after 4 p.m. the oxygen would reach 5.0 mg/L.
  6. The real 10 p.m. reading was 6.1 mg/L, higher than your model predicts. Give one physical reason the oxygen rose in the evening.
  7. Find the average rate of change in temperature, in degrees Celsius per hour, between 6 a.m. and 4 p.m.
  8. Describe the general relationship between the temperature readings and the oxygen readings in this log.
  9. Based only on the four readings, was the river ever below the 5 mg/L level during the day? Support your answer with the data.
  10. Suppose a warm discharge raised the 4 p.m. temperature by 3 degrees Celsius. Using the pattern in this log, would you expect the oxygen to be above or below 5.2 mg/L, and what concern does that raise?

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: 5.2 mg/L (the 4 p.m. reading). It is the lowest of the four readings and the closest to the 5 mg/L fish threshold.
  2. Answer: 0.2 mg/L. 5.2 - 5.0 = 0.2 mg/L above the threshold, a thin margin.
  3. Answer: -0.32 mg/L per hour. From 6 a.m. to 4 p.m. is 10 hours; (5.2 - 8.4) / 10 = -3.2 / 10 = -0.32 mg/L per hour.
  4. Answer: D = 5.2 - 0.32h. The starting oxygen at 4 p.m. is 5.2 mg/L, falling 0.32 mg/L for each hour h after 4 p.m.
  5. Answer: About 0.6 hour. Set 5.0 = 5.2 - 0.32h; then 0.32h = 0.2, so h = 0.625 hour, roughly 38 minutes after 4 p.m.
  6. Answer: The temperature fell from 27 to 22 degrees Celsius after 4 p.m., and cooler water holds more oxygen, so the level rose instead of continuing to drop. (Oxygen mixing in from the air, called reaeration, also helps.)
  7. Answer: 0.9 deg C per hour. (27 - 18) / 10 = 9 / 10 = 0.9 degrees Celsius per hour.
  8. Answer: They move in opposite directions. As the temperature rose through the afternoon, the dissolved oxygen fell; warmer water holds less oxygen, so the two readings show an inverse relationship.
  9. Answer: No. Every recorded value was at or above 5.2 mg/L, and the lowest, 5.2 mg/L at 4 p.m., is just above the 5 mg/L threshold, so no reading dropped below it.
  10. Answer: Below 5.2 mg/L. Warmer water holds less oxygen, so a 3-degree rise would likely push the oxygen under 5.2 and possibly below the 5 mg/L level fish need, raising a fish-kill concern.

Differentiation

Support

Provide the two interval lengths (10 hours for items 3 and 7) already marked, and model the average-rate-of-change formula, change in value over change in time, with a simple first example before students work the log. Allow calculators.

Extension

Ask students to compute the average rate of change in oxygen for the evening interval from 4 p.m. to 10 p.m., compare its sign and size to the daytime rate from item 3, and explain what the change in sign means about the river through the full day.

ELL supports

Pre-teach "dissolved oxygen," "temperature," and "threshold" with a labeled thermometer and a picture of a fish using its gills, with a note that dissolved oxygen is mixed invisibly into the water. Provide a sentence frame for item 8: "When temperature goes ___, dissolved oxygen goes ___, because warm water holds ___ oxygen."