Unit 3 Modeling with Geometry

Lesson 1

Learning Focus

Use similarity to scale area and volume.

Lesson Summary

In this lesson, we learned how area and volume scale if we scale the linear measures in a 3-D shape. Consequently, if we scale up a 3-D figure, we do not need to recalculate its surface area and volume using formulas for volume and area.

Lesson 2

Learning Focus

Derive and use formulas for right prisms and pyramids.

Lesson Summary

In this lesson, we derived a formula for the volume of right prisms with non-rectangular bases, and a formula for the volume of pyramids. These formulas were derived by decomposing rectangular prisms in various ways and considering how the sum of the volumes of the resulting prisms or pyramids had to add up to the volume of the original rectangular prism.

Lesson 3

Learning Focus

Identify shapes formed by slicing a solid with a plane.

Lesson Summary

In this lesson, we identified cross-sections, or slices, of various 3-D shapes, such as cubes and cylinders. Some of the cross-sections we found were obvious, but some were surprising. We also learned how to draw the cross-section on a 2-D representation of the three-dimensional shape.

Lesson 4

Learning Focus

Develop a strategy for drawing solids of revolution.

Lesson Summary

In this lesson, we learned how to create solids of revolution by rotating a 2-D shape around an axis of rotation and we examined the cross-sections that are formed when solids of revolution are sliced perpendicularly to the plane that contains the axes.

Lesson 5

Learning Focus

Calculate the volume of solids of revolution that can be approximated by cylinders and portions of cones.

Lesson Summary

In this lesson, we learned how to approximate the volume of a solid of revolution like a vase whose silhouette contained curves, rather than straight lines. By decomposing the shape into smaller pieces, we could approximate the volume of each piece using formulas for cylinders, cones, and frustums.

Lesson 6

Learning Focus

Apply geometric modeling to solve a real-world problem.

Lesson Summary

In this lesson, we calculated the weight of a solid of revolution by knowing the cross-sectional region that defined the solid and the density of the material from which the solid would be made. This is an example of geometric modeling.