Title: 
Geometry Module 3, Topic B: Volume 
URL: 
https://www.engageny.org/resource/geometrymodule3topicboverview 
Content Source: 
EngageNY

Type: 
Lesson/Unit Plan

Overview: 
Students study the basic properties of twodimensional and threedimensional 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 crosssection 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 crosssection 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 crosssections 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. Modelingbased problems are found throughout Topic B and include the modeling of realworld objects, the application of density, the occurrence of physical constraints, and issues regarding cost and profit. 
Content Standard(s): 
Mathematics MA2015 (2016) Grade: 912 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. [GGMD1]
Alabama Alternate Achievement Standards
 Mathematics MA2019 (2019) Grade: 912 Geometry with Data Analysis  16. Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects.
Alabama Alternate Achievement Standards
 Mathematics MA2019 (2019) Grade: 912 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



Tags:

area, circle, circumference, cone, cylinder, density, formula, problemsolving, pyramid, realworld, sphere, threedimensional, twodimensional, volume 
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Accessibility  Text Resources: Content is organized under headings and subheadings 
Comments  There are nine lessons on this topic.
This resource is free for teachers to access and use. All resources required for the lessons are available to print from the site. 