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Quiz & test questions Grade 8 Math

Ospreyville Phosphorus Cleanup: A Grade 8 Unit Test

A grade 8 unit test on functions, unit rates, linear equations, and volume, set in a fictional town reducing phosphorus in the Sallowbrook River.

Objectives

  1. Students will decide whether a table of monitoring data represents a function.
  2. Students will compare two proportional relationships using unit rate and slope.
  3. Students will solve linear equations and find cylinder and cone volumes in a river cleanup setting.

Vocabulary

Key words: phosphorus, function, unit rate, settling basin, volume

Teacher overview

The town of Ospreyville is a fictional community on the Sallowbrook River. Too much phosphorus reaches the river from farm fields and from town runoff, so algae grow heavily there each summer and the town has started a cleanup program. This unit test draws every item from that one program, so students meet functions, unit rates, linear equations, and volume in a single setting rather than in four unrelated word problems. Teaching note: on items 1 through 3, expect students to call the first table a non-function because the reading 1.4 appears twice. Remind them before the test that a repeated output is allowed and only a repeated input with two different outputs breaks the definition.

Ospreyville Phosphorus Cleanup: A Grade 8 Unit Test — Grade 8 Math
Name: ____________________________ Date: ______________

Assessment

The town of Ospreyville is fictional, and so are the readings, farms, and tanks below. Use 3.14 for pi and 7.48 gallons for each cubic foot.

Table 1 lists four sampling sites on the Sallowbrook River.

Site number Phosphorus (mg/L)
1 0.8
2 1.4
3 0.9
4 1.4

Table 2 lists four samples with the water temperature at the time of each sample.

Temperature (degrees C) Phosphorus (mg/L)
14 0.8
16 1.1
14 1.3
18 1.5
  1. Is the phosphorus reading in Table 1 a function of the site number? Explain your answer.
  2. Is the phosphorus reading in Table 2 a function of the temperature? Answer yes or no.
  3. Name the input value in Table 2 that keeps the table from being a function.
  4. Fern Hollow Farm spreads 72 pounds of phosphorus across 18 acres. Find the unit rate in pounds per acre.
  5. Oatley Farm follows the rule p = 5a, where p is pounds of phosphorus and a is acres. Which farm spreads more phosphorus on each acre?
  6. Both farms are graphed on the same axes with acres on the horizontal axis. Which farm's line is steeper, and what does the steeper line mean?
  7. A settling basin holds 60 - 2.5d pounds of phosphorus after d days of treatment. On what day does 30 pounds remain?
  8. A second basin holds 45 - 1.0d pounds after d days. On what day do the two basins hold the same amount?
  9. How many pounds does each basin hold on the day you found in item 8?
  10. Ospreyville built a cylindrical settling tank 20 feet across and 12 feet deep. Find its volume in cubic feet.
  11. Convert the tank volume from item 10 to gallons.
  12. The cone-shaped hopper under that tank is 6 feet across and 4 feet deep. Find its volume in cubic feet.
  13. Explain why the rule in item 7 should not be used to predict the phosphorus left on day 30.

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: Yes.
    Each site number appears once and is paired with exactly one reading. The value 1.4 appears twice, but a repeated output does not break the definition of a function.
  2. Answer: No.
    The temperature 14 appears twice with two different readings, 0.8 and 1.3, so the reading is not a function of temperature.
  3. Answer: 14 degrees.
    That input is paired with 0.8 in one row and with 1.3 in another row.
  4. Answer: 4 pounds per acre.
    72 divided by 18 = 4 pounds of phosphorus for each acre.
  5. Answer: Oatley Farm.
    Oatley Farm spreads 5 pounds per acre, and Fern Hollow Farm spreads 4 pounds per acre, so Oatley Farm spreads more on each acre.
  6. Answer: Oatley Farm's line.
    Its slope is 5 while the other slope is 4, so its line rises faster. The steeper line means more pounds of phosphorus for every acre added.
  7. Answer: 12 days.
    60 - 2.5d = 30. Subtract 60 from both sides: -2.5d = -30. Divide both sides by -2.5: d = 12 days.
  8. Answer: 10 days.
    60 - 2.5d = 45 - 1.0d. Add 2.5d to both sides: 60 = 45 + 1.5d. Subtract 45: 15 = 1.5d. Divide by 1.5: d = 10 days.
  9. Answer: 35 pounds.
    First basin: 60 - 2.5(10) = 60 - 25 = 35 pounds. Second basin: 45 - 1.0(10) = 45 - 10 = 35 pounds. The two agree, which confirms item 8.
  10. Answer: 3,768 cubic feet.
    The radius is half of 20 feet, or 10 feet. V = 3.14 x 10^2 x 12 = 3.14 x 100 x 12 = 3,768 cubic feet.
  11. Answer: 28,184.6 gallons.
    3,768 x 7.48 = 28,184.64, which rounds to 28,184.6 gallons.
  12. Answer: 37.68 cubic feet.
    The radius is half of 6 feet, or 3 feet. V = one third x 3.14 x 3^2 x 4 = one third x 3.14 x 9 x 4 = one third x 113.04 = 37.68 cubic feet.
  13. Answer: The rule gives a negative amount.
    60 - 2.5(30) = 60 - 75 = -15 pounds, and a basin cannot hold less than nothing. The basin empties at day 24, so the rule only describes the first 24 days of treatment.

Differentiation

Support

Provide a formula sheet with the cylinder and cone volumes drawn and labeled, and with a blank line for the radius above each formula. Allow students to answer item 1 in a single sentence using the frame "each ___ has exactly one ___," and give partial credit on items 7 through 9 for a correct first step in this fictional scenario.

Extension

Ask students to find the day on which the second fictional basin holds exactly twice as much phosphorus as the first, and to write the equation they solved. Have them also decide whether any number of acres greater than zero would leave the two farms having spread the same total amount, and explain how the two rules justify their answer.

ELL supports

Pre-teach phosphorus, settling basin, hopper, and unit rate with a labeled cross-section of a tank and a short cloze paragraph about the fictional cleanup program. Provide a word bank for item 6 containing steeper, slope, per acre, and more.

Extensions

Have students find a local watershed report or a state water quality summary and locate one nutrient measurement in it. Ask them to compare the reporting units used there with the units in this fictional test and to write two sentences explaining what a reader would need to know before comparing two numbers from different reports.