## Geometry Module 3, Topic B: Volume

Classroom Resource Information

Title:

Geometry Module 3, Topic B: Volume

URL:

https://www.engageny.org/resource/geometry-module-3-topic-b-overview

Content Source:

EngageNY
Type: Lesson/Unit Plan

Overview:

Students study the basic properties of two-dimensional and three-dimensional space, noting how ideas shift between the dimensions. They learn that general cylinders are the parent category for prisms, circular cylinders, right cylinders, and oblique cylinders, and study why the cross-section of a cylinder is congruent to its base. Next students study the explicit definition of a cone and learn what distinguishes pyramids from general cones, and see how dilations explain why a cross-section taken parallel to the base of a cone is similar to the base.  Students revisit the scaling principle as it applies to volume and then learn Cavalieri’s principle, which describes the relationship between cross-sections of two solids and their respective volumes. This knowledge is all applied to derive the volume formula for cones, and then extended to derive the volume formula for spheres. Module 3 is a natural place to see geometric concepts in modeling situations. Modeling-based problems are found throughout Topic B and include the modeling of real-world objects, the application of density, the occurrence of physical constraints, and issues regarding cost and profit.

Content Standard(s):
 Mathematics MA2015 (2016) Grade: 9-12 Geometry 35 ) Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. [G-GMD1] Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.HS.35- Make a prediction about the volume of a container, the area of a figure, or the perimeter of a figure. Ex: how many cubes will go in one figure vs. another. Limit to cylinder, circle. Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis 16. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.10.16 Given a cross section of a three-dimensional object, identify the shapes of two-dimensional cross sections (limited to sphere, rectangular prism, or triangular prism). Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis 17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed. a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments. b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.10.17 Compare and contrast the volume of real-world geometric figures.
Tags: area, circle, circumference, cone, cylinder, density, formula, problemsolving, pyramid, realworld, sphere, threedimensional, twodimensional, volume