Lesson 17 | Three-Dimensional Measurement and Application | 10th Grade Mathematics

Objective

Apply density concepts to surface area and volume problems.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.

High School — Number and Quantity

N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.
N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

High School — Number and Quantity

N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

G.MG.A.2 — Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Modeling with Geometry

G.MG.A.2 — Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

The foundational standards covered in this lesson

A.CED.A.2

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
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

Define density as the amount of one quantity per square or cubic unit of another.
Identify a general plan for the solution of problems and revise this plan as necessary throughout.
Describe the level of precision necessary at different stages of the problem to determine the most accurate answer.
Identify information that is necessary to determine an accurate solution.
Describe how you will assess whether your solution is reasonable or not.
Justify the solution and process of solving the problem.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

The criteria for success are genereally repeated in Lessons 16–18, which all focus on modeling. In Lesson 17, we are focusing on applications of volume and surface area in relation to density.

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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

Below is a mockup of an aerial view of the proposed Apple headquarters. Twelve thousand people are proposed to work at the headquarters.

What is the population density when all the employees are at the headquarters?

Guiding Questions

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References

101Questions Apple Mothership

Apple Mothership by Dan Meyer is made available on 101Questions under the CC BY 3.0 license. Accessed June 2, 2017, 4:45 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

A cylindrical soda can is made of aluminum. It is approximately $$4\frac{3}{4}$$ inches high and the top and the bottom has a radius of approximately $$1\frac{3}{16}$$ inches.

Find the approximate surface area of the soda can. What assumptions do you use in your estimate?
The density of aluminum is approximately 2.70 grams per cubic centimeter. If the mass of the soda can is approximately 15 grams, how many cubic centimeters of aluminum does it contain?
Use the answers to parts (a) and (b) to estimate how thick the aluminum can is.

Guiding Questions

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References

Illustrative Mathematics How Thick Is a Soda Can? Variation I

How Thick Is a Soda Can? Variation I, accessed on June 2, 2017, 4:52 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Target Task

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

A gigantic key lime pie was made and served to Bostonians in South Station. The pie was cylindrical with a diameter of 9 feet, weighed about one thousand pounds, and was about 8 inches tall. If 2,000 people were served, how many pounds was each person given? What was the fraction of the gigantic pie that each person received?

References

Yummy Math Huge Key Lime Pie

Modified by Fishtank Learning, Inc.

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.

Illustrative Mathematics A Ton of Snow
Mathematics Vision Project: Secondary Mathematics Three Module 5: Modeling with Geometry — Lesson 4, "Set"
Illustrative Mathematics How Many Cells are in the Human Body?

Lesson 16

Lesson 18

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

High School — Number and Quantity

High School — Number and Quantity

Modeling with Geometry

Foundational Standards

Creating Equations

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Target Task

References

Additional Practice

Lesson Map

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