Unit 8 Modeling with Geometry

Lesson 1

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 2

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 3

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 4

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.

Lesson 5

Learning Focus

Find missing sides of special right triangles without using trigonometry.

Lesson Summary

In this lesson, we learned there are some special right triangles for which missing sides of the triangle can be found when only one side is known, without using trigonometry! This happens when the right triangle is the result of decomposing a familiar shape, such as a square or an equilateral triangle, into two congruent right triangles.

Lesson 6

Learning Focus

Find missing sides and angles in non-right triangles.

Lesson Summary

In this lesson, we learned how to find missing sides of oblique triangles by decomposing them into non-congruent right triangles and applying trigonometry and the Pythagorean theorem to the right triangles.

Lesson 7

Learning Focus

Derive the Law of Cosines and the Law of Sines.

Lesson Summary

In this lesson and the previous lesson, we examined two important relationships that exist between the sides and angles of triangles, the Law of Cosines and the Law of Sines. Because of these relationships, we can solve for missing sides and angles in any triangle, not just right triangles.

Lesson 8

Learning Focus

Apply the Law of Cosines and the Law of Sines to solve problems.

Lesson Summary

In this lesson, we practice using the Law of Sines and the Law of Cosines and applied them to a practical situation. We learned that when we are given SSA information about an oblique triangle there are two possible triangles that satisfy these conditions. The Law of Sines doesn’t give us both solutions, but the Law of Cosines will. We also developed a new formula for finding the area of a triangle when we are given SAS information about the triangle.