Lesson 15 | Three-Dimensional Measurement and Application | 10th Grade Mathematics

Objective

Use the surface area and volume to solve application 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.
G.GMD.B.4 — Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Geometric Measurement and Dimension

G.GMD.B.4 — Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

G.MG.A.1 — Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 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.1 — Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 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.
G.MG.A.3 — Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Modeling with Geometry

G.MG.A.3 — Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

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

Describe how to find the surface area of cylinders and other prisms using the principle that surface area is just the sum of the area of all the surfaces.
Use the formulas for the surface area of a cone, $$\pi r^2+\pi r$$ℓ, where $$r$$ is the radius and ℓ is the slant height of the cone, and the surface area of a sphere, $$4\pi r^2$$, to solve problems.
Identify necessary information to solve a problem, and use known quantities to gather that information.
Choose general formulas appropriately to develop a specific formula for finding the surface area or volume, and apply appropriate dimensions.
Compare surface area of figures by analyzing dimensions and formulas.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Have students use or develop a reference sheet as needed. A good example is the Massachusetts Comprehensive Assessment System Grade 10 Mathematics Reference Sheet.
Students may need to review the surface area formulas before they can fully access this lesson. It is recommended to spend time outside of class building this specific skill.

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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

Three tennis balls are in a can, each with a diameter of 2.6 inches.

How much felt is used to cover the three tennis balls?
What is the minimum surface area of the can, including the top and bottom?

Guiding Questions

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Problem 2

The following four packages all contain the same amount of candy. Rank the packages based on the least amount of packaging used to the most amount of packaging used.

Guiding Questions

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References

101Questions Dandy Candies

Dandy Candies by is made available on 101Questions under the CC BY 3.0 license. Accessed March 20, 2018, 2:37 p.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

What is the surface area and volume of the solid formed by revolving the following two-dimensional figure around the $$y$$-axis?

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 such as “an orange that has a diameter of 3 inches is peeled, and the peel is placed on a piece of paper that is 8.5 inches by 11 inches. How much of the paper is not covered by the orange peel?”
Include problems such as “a tube of wrapping paper has a diameter of 2 inches and a height of 2 feet. The paper goes around the tube five times. Will you have enough paper to wrap four rectangular prisms that have lengths of 8 inches, heights of 5 inches, and widths of 3 inches? If so, how much wrapping paper will you have left over? If not, how much more will you need?”
Include problems where students need to find the partial surface area with one surface missing.
Include problems where one dimension is increased and students are asked to examine the effect on the surface area and the volume.
Include problems where a two-dimensional shape is spun around an axis and students are asked to find the surface area and volume.

Kuta Software Free Geometry Worksheets Surface Area of Pyramids and Cones — Focus on surface area only.
Mathematics Vision Project: Secondary Mathematics Three Module 5: Modeling with Geometry — Lesson 3, "Set" (pp. 17-18)
Illustrative Mathematics Shipping Rolled Oats
MARS Summative Assessment Tasks for High School Evaluating Statements about Enlargements — Focus on surface area questions
Open Middle Maximizing Rectangular Prism Surface Area
Kuta Software Free Geometry Worksheets Surface Area of Prisms and Cylinders — Focus on surface area only

Lesson 14

Lesson 16

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

Modeling with Geometry

Modeling with Geometry

Foundational Standards

Geometry

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

References

Target Task

Additional Practice

Lesson Map

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