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Lesson plan Grade 8 Math

Fernhaugh Digester: Why a Cone Holds One Third of a Cylinder

A grade 8 lesson where students compare a cone and a cylinder of equal size, then find the volume of a fictional digester with a cone bottom.

Objectives

  1. Students will compute the volume of a cylinder, a cone, and a sphere from a stated diameter and height.
  2. Students will show that a cone holds one third of a cylinder with the same radius and height.
  3. Students will explain why a change in radius affects volume more than an equal change in height.

Vocabulary

Key words: digester, cone, cylinder, sphere, radius, cubic foot

Teacher overview

A digester is a sealed tank where bacteria break down the solids removed from wastewater, producing a gas the plant can burn for heat. The fictional Fernhaugh Water Resource Recovery Facility runs one digester built as a cylinder with a cone-shaped bottom, topped by a spherical gas holder, so a single tank calls for all three of the grade 8 volume formulas. Students first check the one third relationship between a cone and a cylinder of the same radius and height, then use it on the tank itself. Two errors dominate this work: substituting the diameter where the formula calls for the radius, which makes every answer four times too large, and dropping the one third factor from the cone.

Procedure

Hook (5 min): Hold up a cone and a cylinder with the same base and the same height. Ask students to guess how many coneloads of rice it takes to fill the cylinder. Record the guesses, then pour and count. Three coneloads fill the cylinder, and classes usually guess two.

Direct instruction (13 min): On the board, work a demonstration tank that does not appear on the handout. Take a cylinder 8 feet across and 6 feet tall, and use 3.14 for pi. Write the radius step first and say it out loud: half of 8 is 4, and 4 squared is 16. Then 3.14 times 16 is 50.24, and 50.24 times 6 is 301.44 cubic feet. Now take a cone with the same 8-foot width and 6-foot height, and show that its volume is 301.44 divided by 3, or 100.48 cubic feet. Connect the pouring demonstration to the one third factor in the formula, then write all three formulas on the board, including the sphere.

Guided practice (12 min): As a class, sketch the digester at the fictional Fernhaugh plant as a cylinder sitting on a cone, labeling the width of each part as 12 feet. Ask a check-for-understanding question before students begin computing: "What number goes into the formula, 12 or 6?" Expect 6, the radius. If more than a few students say 12, work one cylinder together and compare the two results side by side so the factor of four is visible.

Independent practice (15 min): Students complete the Student Handout, which walks from the cone and cylinder comparison through the full tank volume and the gas holder. Circulate and check that the radius step is written down before the multiplication starts.

Closure and exit ticket (5 min): On an index card, students find the volume of a cone 10 feet across and 6 feet tall, using 3.14 for pi. Collect and check for 157 cubic feet, and sort the cards by whether the radius step appears.

Fernhaugh Digester: Why a Cone Holds One Third of a Cylinder — Grade 8 Math
Name: ____________________________ Date: ______________

Student Handout

All tanks below are at the fictional Fernhaugh Water Resource Recovery Facility. Use 3.14 for pi. Round each answer to the nearest hundredth unless the item says otherwise. One cubic foot holds 7.48 gallons.

The digester has a cylinder body 12 feet across and 10 feet tall, sitting on a cone bottom that is also 12 feet across and is 9 feet tall. A spherical gas holder with a radius of 7 feet sits above it.

  1. Find the volume, in cubic feet, of a cylinder that is 12 feet across and 9 feet tall.
  2. Find the volume, in cubic feet, of a cone that is 12 feet across and 9 feet tall.
  3. Your answer to item 2 is what fraction of your answer to item 1? Write the fraction in lowest terms.
  4. Find the volume, in cubic feet, of the digester's cylinder body, which is 12 feet across and 10 feet tall.
  5. Find the total volume of the digester, in cubic feet, by adding the cylinder body from item 4 to the cone bottom from item 2.
  6. Convert the total volume from item 5 to gallons. Round to the nearest gallon.
  7. Find the volume, in cubic feet, of the spherical gas holder, which has a radius of 7 feet.
  8. A second plant uses a cone bottom that is 12 feet across and 18 feet tall. Find its volume in cubic feet.
  9. How many times as much does the cone in item 8 hold as the cone in item 2?
  10. Suppose the cone in item 2 kept its 9-foot height but was made 24 feet across instead of 12. Explain why that change increases the volume more than doubling the height would.

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: 1,017.36 cubic feet. The radius is half of 12, or 6 feet, and 6 squared is 36. Then 3.14 times 36 is 113.04, and 113.04 times 9 equals 1,017.36 cubic feet.
  2. Answer: 339.12 cubic feet. The cone uses the same base and height as item 1, so divide by 3.
    One third of 1,017.36 is 339.12 cubic feet.
  3. Answer: 1/3. Divide 339.12 by 1,017.36 to get 0.3333, which is one third. A cone always holds one third of a cylinder with the same radius and the same height.
  4. Answer: 1,130.40 cubic feet. The radius is still 6 feet, so 3.14 times 36 is 113.04, and 113.04 times 10 equals 1,130.40 cubic feet.
  5. Answer: 1,469.52 cubic feet. Add the cylinder body and the cone bottom: 1,130.40 plus 339.12 equals 1,469.52 cubic feet.
  6. Answer: 10,992 gallons. Multiply 1,469.52 cubic feet by 7.48 gallons per cubic foot to get 10,992.0096, which rounds to 10,992 gallons.
  7. Answer: 1,436.03 cubic feet. The radius cubed is 7 times 7 times 7, or 343. Then 3.14 times 343 is 1,077.02, and 1,077.02 times 4 is 4,308.08. Divide by 3 to get 1,436.0267, which rounds to 1,436.03 cubic feet.
  8. Answer: 678.24 cubic feet. The radius is 6 feet, so 3.14 times 36 is 113.04, and 113.04 times 18 is 2,034.72. Divide by 3 to get 678.24 cubic feet.
  9. Answer: 2 times. Divide 678.24 by 339.12 to get exactly 2. Doubling only the height doubles the volume, because height appears to the first power in the formula.
  10. Answer: Volume depends on the radius squared but on the height only to the first power.
    Going from 12 feet across to 24 feet across doubles the radius from 6 to 12 feet, and 2 squared is 4, so the volume becomes four times as large. Doubling the height would only double it.

Differentiation

Support

Give these students a formula card with the three volume formulas and one worked example of each, plus the reminder "cut the width in half first" printed above the handout items for the fictional Fernhaugh tanks. Assign items 1, 2, 3, and 4 as the required set, and allow a calculator throughout.

Extension

Ask students to redesign the digester so the total volume stays at 1,469.52 cubic feet while the cone bottom is only 6 feet tall, finding the new height of the cylinder body and showing the algebra. Then have them state one practical reason a plant might prefer the taller cone.

ELL supports

Pre-teach digester, cone bottom, and gas holder with a labeled cross-section drawing of the tank, marking the radius, the diameter, and each height in different colors. Post the word pairs "across the top" with "diameter" and "half of across" with "radius." Allow students to label a diagram before writing any sentences.

Extensions

Have students find a cone-shaped and a cylinder-shaped container at home or in the classroom with roughly matching openings and heights, estimate each volume in cubic inches, and check whether the one third relationship holds. Ask them to explain any difference in terms of how well the two shapes actually matched in radius and height, rather than in terms of measurement luck.