Lesson 6 | Three-Dimensional Measurement and Application | 10th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Geometric Measurement and Dimension

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.

Geometric Measurement and Dimension

G.GMD.A.3 — Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Foundational Standards

The foundational standards covered in this lesson

7.G.B.6

Geometry

7.G.B.6 — Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
8.G.C.9

Geometry

8.G.C.9 — Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

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

Tips for Teachers

Suggestions for teachers to help them teach this lesson

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

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

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

Find the length, width, and height of the baseless prism.
State the formula for finding the volume of the prism and calculate the volume.
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

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

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?

Open Middle Cylinders
Illustrative Mathematics Volume Formulas for Cylinders and Prisms
EngageNY Mathematics Geometry > Module 3 > Topic B > Lesson 6 — Focus on questions about area and volume
Kuta Software Free Geometry Worksheets Volume of Prisms and Cylinders
MARS Summative Assessment Tasks for High School Fun Size Cans
Estimation 180 How Many Red Licorice Come in a Container? Day 37

Lesson 5

Lesson 7

Lesson Map

Topic A: Area and Circumference of Circles

Describe and use the formulas for area and circumference of circles to solve problems.

A.SSE.A.1 G.GMD.A.1 N.Q.A.3

Calculate and justify composite area and circumference of circles.

N.Q.A.1 N.Q.A.2 N.Q.A.3

Solve multistep area and circumference of circles problems involving cost and other rates.

N.Q.A.1 N.Q.A.2 N.Q.A.3

Topic B: Three-Dimensional Concepts and General Volume

Describe the terms point, line, and plane. Define and classify polyhedrons, specifically prisms and pyramids.

G.CO.A.1

Define a general cylinder and general cone. Identify two-dimensional shapes that when revolved will form a cylinder.

G.CO.A.1 G.GMD.B.4

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

G.GMD.A.1 G.GMD.A.3

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

G.GMD.A.1 G.GMD.A.3

Use the Pythagorean Theorem to find missing measurements and calculate volume of pyramids, prisms, and compound shapes comprised of pyramids and prisms.

G.GMD.A.3 G.GMD.B.4 G.SRT.C.8

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

Describe the cross-sections of prisms and cylinders and make conjectures about volume from the cross-sections.

G.GMD.A.3 G.GMD.B.4

Describe Cavalieri’s principle relating equal area cross-sections and volume, and how this relates to the formulas for volume. Derive the volume of a sphere using Cavalieri’s principle.

G.GMD.A.1 G.GMD.A.2

Identify cross-sections of pyramids and use the relationships between the cross-sections to determine the volume of truncated cones and pyramids.

G.GMD.A.2 G.GMD.A.3

Calculate the volume of a sphere and use this in the solution of problems.

G.GMD.A.1 G.GMD.A.2 G.GMD.A.3 N.Q.A.3

Calculate the volume of compound objects and those with subtracted solids. Determine how the volume will be affected by scaling one or more dimensions.

G.GMD.A.3 G.GMD.B.4 N.Q.A.3

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

Use lateral surface area formulas to solve problems.

N.Q.A.2

Use the surface area and volume to solve application problems.

G.GMD.A.3 G.GMD.B.4 G.MG.A.1 G.MG.A.3

Solve multistep volume and surface area problems with rates and unit conversions.

G.GMD.A.3 N.Q.A.2 N.Q.A.3

Apply density concepts to surface area and volume problems.

G.GMD.A.3 G.MG.A.2 N.Q.A.2 N.Q.A.3

Apply constraints on volume, surface area, or cost to solve design problems with three-dimensional figures.

G.GMD.A.3 G.MG.A.2 G.MG.A.3 N.Q.A.2 N.Q.A.3

Three-Dimensional Measurement and Application

Objective

Common Core Standards

Core Standards

Geometric Measurement and Dimension

Geometric Measurement and Dimension

Foundational Standards

Geometry

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Target Task

Additional Practice

Lesson Map

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