Curriculum / Math / 10th Grade / Unit 6: Three-Dimensional Measurement and Application / Lesson 18
Math
Unit 6
10th Grade
Lesson 18 of 18
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Lesson Notes
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Apply constraints on volume, surface area, or cost to solve design problems with three-dimensional figures.
The core standards covered in this lesson
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.
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.
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).
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
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.
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..
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.
$$\left\{\begin{matrix} 3x+4y\leq 180\\ 100(.972\pi)x+100(2.304\pi)y\geq 30(252\pi) \end{matrix}\right.$$
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.
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.
Topic A: Area and Circumference of Circles
Describe and use the formulas for area and circumference of circles to solve problems.
Standards
A.SSE.A.1G.GMD.A.1N.Q.A.3
Calculate and justify composite area and circumference of circles.
N.Q.A.1N.Q.A.2N.Q.A.3
Solve multistep area and circumference of circles problems involving cost and other rates.
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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.1G.GMD.B.4
Use volume concepts and formulas to analyze and solve multistep problems with cylinders and prisms.
G.GMD.A.1G.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.
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.3G.GMD.B.4G.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.3G.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.1G.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.2G.GMD.A.3
Calculate the volume of a sphere and use this in the solution of problems.
G.GMD.A.1G.GMD.A.2G.GMD.A.3N.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.3G.GMD.B.4N.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.3G.GMD.B.4G.MG.A.1G.MG.A.3
Solve multistep volume and surface area problems with rates and unit conversions.
G.GMD.A.3N.Q.A.2N.Q.A.3
Apply density concepts to surface area and volume problems.
G.GMD.A.3G.MG.A.2N.Q.A.2N.Q.A.3
G.GMD.A.3G.MG.A.2G.MG.A.3N.Q.A.2N.Q.A.3
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