Curriculum / Math / 10th Grade / Unit 6: Three-Dimensional Measurement and Application / Lesson 17
Math
Unit 6
10th Grade
Lesson 17 of 18
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Lesson Notes
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Apply density concepts to surface area and volume problems.
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.
The foundational standards covered in this lesson
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 — Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
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?
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..
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.
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.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
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?
Huge Key Lime Pie by Brian Marks and Lesie Lewis is made available on YummyMath. Copyright © 2017 Yummy Math. All Rights Reserved. Accessed June 2, 2017, 4:20 p.m..
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.
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Apply constraints on volume, surface area, or cost to solve design problems with three-dimensional figures.
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
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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