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Worksheet Grade 8 Math

Tallowfield Digester Records: Function Tables and Rules

A grade 8 worksheet using a fictional water reclamation plant's digester records to decide which tables are functions and to write the rule for one of them.

Objectives

  1. Students will decide whether a table of plant records represents a function.
  2. Students will write a linear rule from a table and use it to find new outputs.
  3. Students will explain why a repeated input with two different outputs breaks the definition of a function.

Vocabulary

Key words: function, input, output, ordered pair, rate of change, digester

Teacher overview

The town of Tallowfield is a fictional community whose water reclamation facility runs an anaerobic digester. Bacteria inside the sealed tank break down sludge and release biogas, which the plant burns to heat the building. Operators log the tons of sludge fed in and the biogas that comes out, so the plant's records are ready-made function tables. Teaching note: many eighth graders decide that a table is not a function as soon as they see a repeated number in either column. Table B repeats an input on purpose, paired with two different outputs. Point out that a repeated output alone would not break the definition, and that only a repeated input paired with two different outputs does, and have students say the rule aloud as "each input gets exactly one output" before they answer item 4.

Tallowfield Digester Records: Function Tables and Rules — Grade 8 Math
Name: ____________________________ Date: ______________

Student Worksheet

Tallowfield Water Reclamation is a fictional plant. Table A shows one week of digester records. Table B shows four separate readings taken by an operator who wrote down the digester temperature and the gas flow at that moment.

Table A

Sludge fed (tons) Biogas made (cubic feet)
1 8,000
2 16,000
3 24,000
4 32,000

Table B

Digester temperature (degrees F) Gas flow (cubic feet per hour)
98 4,200
99 4,350
98 4,500
100 4,600
  1. Table A pairs each amount of sludge with the biogas it made. Is Table A a function? Use the meaning of the word function in your answer.
  2. Write a rule for Table A. Use g for the biogas in cubic feet and s for the tons of sludge.
  3. Use your rule to find the biogas made from 7 tons of sludge.
  4. Table B pairs each temperature reading with a gas flow. Is Table B a function? Answer yes or no.
  5. Name the input value in Table B that keeps the table from being a function.
  6. List the four ordered pairs shown in Table A.
  7. Do the four points from Table A lie on one straight line? Explain how the table shows this.
  8. The gas holder at the plant stores 60,000 cubic feet. What is the greatest whole number of tons of sludge whose biogas still fits in the holder?
  9. A second digester at the same plant follows the rule g = 6,500s. How much more biogas does the first digester make from each ton of sludge?
  10. Use the second digester's rule to find the biogas made from 5 tons of sludge.
  11. On Monday the plant fed 3 tons to the first digester and 4 tons to the second digester. How much biogas did the two digesters make in all?
  12. Explain why a table that lists the same input twice, paired with two different outputs, is not a function.

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 sludge amount in Table A appears once and is paired with exactly one biogas amount, which is what a function requires.
  2. Answer: g = 8,000s.
    The biogas rises by 8,000 cubic feet for every extra ton, and 1 ton pairs with 8,000, so the rule is g = 8,000s.
  3. Answer: 56,000 cubic feet. g = 8,000 x 7 = 56,000 cubic feet.
  4. Answer: No.
    The input 98 appears twice with two different outputs, so Table B is not a function.
  5. Answer: 98 degrees.
    The temperature 98 is paired with 4,200 in one row and with 4,500 in another row.
  6. Answer: (1, 8,000), (2, 16,000), (3, 24,000), (4, 32,000).
    Each ordered pair lists the input first and the matching output second.
  7. Answer: Yes.
    The output rises by the same 8,000 cubic feet for each extra ton, so the rate of change is constant and the points lie on one line.
  8. Answer: 7 tons.
    60,000 divided by 8,000 = 7.5, and the sludge is counted in whole tons, so 7 tons is the largest amount that fits. Seven tons make 56,000 cubic feet, while 8 tons would make 64,000 cubic feet.
  9. Answer: 1,500 cubic feet.
    8,000 minus 6,500 = 1,500 cubic feet more per ton in the first digester.
  10. Answer: 32,500 cubic feet. g = 6,500 x 5 = 32,500 cubic feet.
  11. Answer: 50,000 cubic feet.
    First digester: 8,000 x 3 = 24,000. Second digester: 6,500 x 4 = 26,000. Total: 24,000 + 26,000 = 50,000 cubic feet.
  12. Answer: The input has two outputs.
    A function must send each input to exactly one output. When the same input appears with two different outputs, there is no single answer to the question "what does this input produce," so the pairing is not a function.

Differentiation

Support

Give students a highlighted copy of Table B with the two rows that share the input 98 marked in the same color, and let them circle the repeat before they answer item 4. For item 2, supply the frame g = ____ x s and ask only for the missing number, which they can find by dividing any output by its input.

Extension

Ask students to add a fifth row to Table B that keeps the table from being a function for a second reason, then a different fifth row that would leave the original problem in place but add no new one. Have them explain in writing how the two rows differ.

ELL supports

Pre-teach input, output, ordered pair, and digester with a simple machine sketch that shows sludge going in on one side and biogas coming out on the other. Provide sentence frames for items 1 and 4: "Table ___ is a function because each input has ___ output," and "Table ___ is not a function because the input ___ has ___ outputs."

Extensions

Have students collect a real function table from the classroom, such as the number of people at a table and the number of cups needed, and write its rule in the same form used in item 2. Then ask them to design a table of digester records that is deliberately not a function and to trade with a partner, who must find and explain the row that breaks the rule.