Lesson 18 | Three-Dimensional Measurement and Application | 10th Grade Mathematics

Objective

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

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.
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).

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 repeated in Lessons 16–18, which all focus on modeling. In lesson 18, we are focusing on applications of volume and surface area with constraints and design problems.
There is no Target Task and little Problem Set Guidance in this lesson due to the level of work and explanation required for the two Anchor Problems. During the lesson, be sure to mix in plenty of independent work time and feedback cycles to ensure adequate thinking and processing time.

Fishtank Plus

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Anchor Problems

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

Problem 1

The Fresha Drink Company is marketing a new soft drink. The drink will be sold in a can that holds 200 cubic centimeters. In order to keep costs low, the company wants to use the smallest amount of aluminum. Find the radius and height of a cylindrical can that holds 200 cubic centimeters and uses the smallest amount of aluminum. Explain your reasons and show all your calculations.

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

MARS Summative Assessment Tasks for High School Bestsize Cans

Bestsize Cans from the Summative Tasks is made available through the Mathematics Assessment Project under the CC BY-NC-ND 3.0 license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed June 2, 2017, 5:51 p.m..

Problem 2

Janine is planning on creating a water-based centerpiece for each of the 30 tables at a party she is throwing. She has already purchased a cylindrical vase for each table. The radius of the vase is 6 centimeters and the height is 28 centimeters. She will fill the vases halfway with water and then add a variety of colored marbles until the water line is approximately three-quarters of the way up the cylinder. The small marbles, 9 millimeters in radius, come in bags of 100 and cost $3. The large marbles, 12 millimeters in radius, come in bags of 100 and cost $4.

If Janine only bought 9-millimeter marbles, how much would she spend on marbles for the whole party? What if Janine only bought 12-millimeter marbles?
Janine’s parents, who are paying for the party, have told her she can spend, at most, $180 on the marbles. Below is a system of inequalities that she can use to determine how many marbles of each type she can buy, where $$x$$ represents the number of bags of 9-millimeter marbles and y represents the number of bags of 12-millimeter marbles. Annotate each inequality with what it represents in terms of the problem.

$$\left\{\begin{matrix} 3x+4y\leq 180\\ 100(.972\pi)x+100(2.304\pi)y\geq 30(252\pi) \end{matrix}\right.$$

Janine graphed the system of inequalities to determine her possibilities. Which inequality represents each portion of the graph? From the graph, what are some possibilities of combinations of marbles that Janine could buy?

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

Illustrative Mathematics Centerpiece

Centerpiece, accessed on June 3, 2017, 3:25 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.

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.

Include shorter probelms that focus on procedural fluency and review the concepts from the unit.

Open Middle Maximizing Volume of a Cylinder, Given Lateral Area

Lesson 17

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

Modeling with Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Additional Practice

Lesson Map

Request a Demo

Contact Information

School Information

What courses are you interested in?

ELA

Math

Are you interested in onboarding professional learning for your teachers and instructional leaders?

Any other information you would like to provide about your school?

Effective Instruction Made Easy