Curriculum / Math / 10th Grade / Unit 6: Three-Dimensional Measurement and Application / Lesson 11
Math
Unit 6
10th Grade
Lesson 11 of 18
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Identify cross-sections of pyramids and use the relationships between the cross-sections to determine the volume of truncated cones and pyramids.
The core standards covered in this lesson
G.GMD.A.2 — Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
G.GMD.A.3 — Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
The foundational standards covered in this lesson
G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
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
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Below is a diagram of a right rectangular pyramid that is intersected by a plane parallel to the base. The base dimensions are 4 inches by 6 inches. The height of the pyramid is 10 inches, and the plane intersects the pyramid at 5 inches from the base.
Cross Section of a Rectangular Pyramid by Chris Woo is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed Sept. 18, 2018, 11:54 a.m..
Determine the volume of the truncated cone shown below.
Grade 8 Mathematics > Module 7 > Topic D > Lesson 20 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..
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Below is a truncated cone with height and two radii marked.
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 10
Lesson 12
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.Â
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.Â
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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.Â
G.CO.A.1
Define a general cylinder and general cone. Identify two-dimensional shapes that when revolved will form a cylinder.Â
G.CO.A.1 G.GMD.B.4
Use volume concepts and formulas to analyze and solve multistep problems with cylinders and prisms.Â
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.Â
Use the Pythagorean Theorem to find missing measurements and calculate volume of pyramids, prisms, and compound shapes comprised of pyramids and prisms.Â
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.Â
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.Â
G.GMD.A.1 G.GMD.A.2
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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