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.
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