Curriculum  /  Math  /  10th Grade  /  Unit 6: Three-Dimensional Measurement and Application  /  Lesson 6

Three-Dimensional Measurement and Application

Lesson 6

Math

Unit 6

10th Grade

Lesson 6 of 18

Objective

Use volume concepts and formulas to analyze and solve multistep problems with cylinders and prisms.

Common Core Standards

Core Standards

  • G.GMD.A.1 — Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
  • G.GMD.A.3 — Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Foundational Standards

  • 7.G.B.6
  • 8.G.C.9

Criteria for Success

  1. Use the volume formula for a general prism as $$Bh$$, where $$B$$ represents the area of the base and $$h$$ represents the height. 
  2. Describe that base area depends on the shape of the base in the prism. If it is a cylinder, the base is a circle. 
  3. Apply constraints on dimensions provided in the context of the problem to modify general formulas to model the specific problem. 
  4. Given either the radius, height, or another dimension (like side length of the base of a square prism), find other measurements in the problem.
  5. Analyze the formulas to make conjectures about the volume of a prism. 

Tips for Teachers

  • Students may need to review finding the volume of cylinders and prisms before they can fully access this lesson. It is recommended to spend time outside of class building this specific skill. 
  • Have students use or develop a reference sheet as needed. A good example is the Massachusetts Comprehensive Assessment System Grade 10 Mathematics Reference Sheet.
  • A student may ask how the volume of an oblique cylinder compares to the volume of a right cylinder. This will be addressed in Lesson 10 when we discuss Cavalieri’s principle.
  • This lesson will be extended in Algebra 2, where the students will use the volume formulas to find combinations of heights and radius as a rational function. 
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Anchor Problems

25-30 minutes

Problem 1

Use a piece of 8.5" $$\times$$ 11" paper to create a baseless rectangular prism as shown below.

  1. Find the length, width, and height of the baseless prism.
  2. State the formula for finding the volume of the prism and calculate the volume.
  3. What do you notice about the length and width? Rewrite the formula with only two variables to reflect this observation.

Guiding Questions

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Problem 2

A container holds a maximum of 120 cubic inches of water. A cylindrical pitcher has a height of 4 inches. You want to transfer the water from the cylindrical pitcher to the container. What is the maximum radius of the pitcher, to the nearest tenth, that can be used to fill the container without it overflowing? 

Guiding Questions

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Problem 3

A can of soda is split between two glasses. If you want the most soda, which glass would you choose? Explain your reasoning.

Guiding Questions

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References

Dan Meyer's Three-Act Math You Pour, I Choose

You Pour, I Choose by Dan Meyer is licensed under the CC BY 3.0 license. Accessed June 8, 2017, 11:12 a.m..

Modified by Fishtank Learning, Inc.

Target Task

5-10 minutes

You are designing a square prism that has a height of 4 inches. What are two possible base side lengths if you want a minimum volume of 24 cubic inches and a maximum volume of 35 cubic inches?

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems that focus on circular cylinders and prisms with other base shapes. 
  • Include problems where students are given the height and volume and asked to find the possible base side lengths if the prism was a square prism? If the prism was a rectangular prism? 

Next

Define and calculate the volume of pyramids and cones. Describe the relationship between general cylinders and general cones with the same base area.

Lesson 7
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Area and Circumference of Circles

Topic B: Three-Dimensional Concepts and General Volume

Topic C: Cavalieri's Principle, Spheres, and Composite Volume

Topic D: Surface Area, Scaling, and Modeling with Geometry

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