Water for Schools
Menu

Lesson plan Grade 7 Math

Brambleway Digester Biogas: A Grade 7 Proportionality Lesson

A 50-minute grade 7 math lesson where students test a fictional digester's biogas records for proportionality, find the constant, and write the equation.

Objectives

  1. Students will compute the unit rate of biogas produced per pound of volatile solids fed to a digester.
  2. Students will decide whether a table of paired values is proportional and identify the constant of proportionality.
  3. Students will write and solve an equation that predicts biogas output from solids fed.

Vocabulary

Key words: digester, biogas, volatile solids, unit rate, constant of proportionality

Teacher overview

The fictional town of Brambleway runs an anaerobic digester at its water resource recovery facility. Bacteria inside the sealed tank break down the solids collected from wastewater and give off biogas, which the plant burns to heat its buildings. Operators log two numbers every day, the pounds of volatile solids fed into the digester and the cubic feet of biogas the digester returns, and those paired records are the data set for this lesson. The most common student error here is writing the unit rate upside down, dividing pounds of solids by cubic feet of gas, which yields a number that cannot be used to predict output; watch for it while students work item 1.

Procedure

Hook (5 min): Ask students what a wastewater plant should do with the solids it removes from sewage. After a few responses, explain that the fictional Brambleway plant seals those solids in a heated tank where bacteria digest them, and that the tank gives back a fuel gas the plant uses instead of buying heat.

Direct instruction (12 min): Put a two-column table on the board using numbers that are not on the handout: a digester fed 200 pounds of volatile solids returned 2,400 cubic feet of biogas, and on another day 350 pounds returned 4,200 cubic feet. Model finding each unit rate, 2,400 divided by 200 and 4,200 divided by 350, and show that both give 12 cubic feet per pound. Name that repeated value the constant of proportionality, write the equation b equals 12v, and stress the order of the division by asking what 200 divided by 2,400 would represent.

Guided practice (13 min): Distribute the handout. Work item 1 together, then have pairs complete items 2 through 4 and compare their reasoning with a neighboring pair. Circulate and check that students are dividing cubic feet by pounds.

Independent practice (12 min): Students finish items 5 through 8 alone. Remind them that item 6 asks for an equation as well as a number, and that items 7 and 8 use a day when the digester did not perform as expected.

Closure and exit ticket (8 min): Each student answers one question on an index card: "A digester was fed 180 pounds of volatile solids and produced 2,340 cubic feet of biogas. What is the unit rate in cubic feet per pound?" Collect the cards and sort them into correct and reversed rates. The class is ready to move on when at least 80 percent of the cards show the division done in the correct order.

Brambleway Digester Biogas: A Grade 7 Proportionality Lesson — Grade 7 Math
Name: ____________________________ Date: ______________

Student Handout

The fictional Brambleway water resource recovery facility recorded four days of digester data.

Day Volatile solids fed (pounds) Biogas produced (cubic feet)
Monday 150 1,950
Tuesday 200 2,600
Wednesday 260 3,380
Thursday 320 4,160
  1. Find the unit rate for Monday in cubic feet of biogas per pound of volatile solids.
  2. Find the unit rate for Thursday in cubic feet of biogas per pound of volatile solids.
  3. Do all four days give the same unit rate? Answer yes or no, and name the value you compared.
  4. What is the constant of proportionality for this table, in cubic feet per pound?
  5. On Friday the digester was fed 285 pounds of volatile solids. Using the constant, how many cubic feet of biogas should it produce?
  6. On Saturday the operators want 5,200 cubic feet of biogas. Write an equation using v for pounds of volatile solids, then solve it for v.
  7. On Sunday the digester was fed 240 pounds but produced only 2,880 cubic feet. What was Sunday's unit rate in cubic feet per pound?
  8. How many cubic feet short of the expected Sunday output was that?

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: 13 cubic feet per pound. Divide biogas by solids: 1,950 divided by 150 equals 13.
  2. Answer: 13 cubic feet per pound. Divide biogas by solids: 4,160 divided by 320 equals 13.
  3. Answer: Yes. Every row gives 13 cubic feet per pound, because 1,950 divided by 150, 2,600 divided by 200, 3,380 divided by 260, and 4,160 divided by 320 all equal 13, so the table is proportional.
  4. Answer: 13 cubic feet per pound. The constant of proportionality is the repeated unit rate found in items 1 through 3.
  5. Answer: 3,705 cubic feet. Multiply the solids by the constant: 285 times 13 equals 3,705.
  6. Answer: 400 pounds. The equation is 13v equals 5,200, so v equals 5,200 divided by 13, which is 400 pounds of volatile solids.
  7. Answer: 12 cubic feet per pound. Divide 2,880 by 240 to get 12, which is below the usual 13.
  8. Answer: 240 cubic feet. The expected output was 240 times 13, or 3,120 cubic feet, and 3,120 minus 2,880 equals 240 cubic feet.

Exit ticket answer: 13 cubic feet per pound, because 2,340 divided by 180 equals 13.

Differentiation

Support

Give these students a handout with the Monday row's division already set up as 1,950 divided by 150 and a printed reminder that reads "unit rate equals cubic feet divided by pounds." Allow calculators for every item and accept a labeled arrow diagram in place of the written equation in item 6.

Extension

Ask students to decide what a drop from 13 to 12 cubic feet per pound would cost the plant over a 30-day month at 250 pounds fed per day, and to explain in writing which quantity in the equation changed and which stayed fixed.

ELL supports

Pre-teach "fed," "produced," and "per" with a labeled sketch of a tank showing solids going in on one side and gas leaving on the other. Provide the sentence frame "For every 1 pound of solids, the digester makes ___ cubic feet of biogas" for items 1 through 4.

Extensions

Have students graph the four rows as points with pounds on the horizontal axis and cubic feet on the vertical axis, draw the line through the origin, and explain why a proportional relationship must pass through the point where both values are zero. Then have them plot Sunday's point and describe where it falls relative to the line.