Curriculum / Math / 10th Grade / Unit 6: Three-Dimensional Measurement and Application / Lesson 9
Math
Unit 6
10th Grade
Lesson 9 of 18
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Describe the cross-sections of prisms and cylinders and make conjectures about volume from the cross-sections.
The core standards covered in this lesson
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.
The foundational standards covered in this lesson
7.G.A.3 — Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
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
It is important to define a cross-section as the intersection of a plane with the solid that is parallel to the base, and a slice as an intersection of any plane with the solid.
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
The cylinder is full of water. The water flows out through a pipe at the bottom of the cylinder. Imagine looking down on the cylinder as the water flows out of it. Draw the shape of the surface of the water at five different levels.
Representing 3D Objects in 2D from the Classroom Challenges by the MARS Shell Center team at the University of Nottingham is made available by the Mathematics Assessment Project under the CC BY-NC-ND 3.0 license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed July 6, 2016, 12:01 p.m..
Sketch the cross sections of the coffee cup at each of the heights shown below.
Geometry > Module 3 > Topic B > Lesson 13 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
Given the following cross-sections, equilateral triangles from the same three-dimensional figure:
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
For each of the following cross-sections, sketch two possible figures from which the cross-sections were taken.
Geometry > Module 3 > Topic B > Lesson 6 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
Horizontal slices of a solid are shown at various levels arranged from highest to lowest. What could the solid be?
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.
Lesson 8
Lesson 10
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.
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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.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.
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
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
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