In this section, students practice spatial visualization. First they do this by examining solids of rotation. Then they investigate cross-sections of a variety of solids. They create physical representations to show that cross-sections of a pyramid may be viewed as dilations of the base for scale factors that are between 0 and 1. Students study the effect of dilation on...

In this section, students extend their study of scaling to solids. They conclude that dilating a solid by a scale factor of multiplies all lengths by , surface areas by , and volumes by . They work backward from a scaled surface area or volume to find the scale factor involved, which requires the introduction of cube roots. Students create...

In this section, students combine concepts of dilations, cross-sections, and Cavalieri’s Principle with dissection to derive the formula for the volume of a pyramid or cone. First, they establish that any triangular pyramid whose base has area square units and whose height is units can be combined with two other triangular pyramids of equal volume to form a prism with...

In this section students revisit the volume of a cylinder from previous grade levels, and solids of rotation from previous lessons. Then students are introduced to Cavalieri’s Principle: Suppose two solids have equal heights. If, at all distances from the base, the cross-sections of the two solids have equal areas, then the solids have equal volumes. This leads to the...