Module 2: Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2. To be able to discuss similarity, students must first have a clear understanding of how dilations behave. This is done in two parts, by studying how dilations yield scale drawings and reasoning why the properties of dilations must be true. Once dilations are clearly established, similarity transformations are defined and length and angle relationships are examined, yielding triangle similarity criteria. An in-depth look at similarity within right triangles follows, and finally the module ends with a study of right triangle trigonometry.
Unit 1: Scale Drawings (G.SRT1, 4, G.MG.3)
Students revisit what scale drawings are and discover two systematic methods of how to create them using dilations. The comparison of the two methods yield the Triangle Side Splitter Theorem and the Dilation Theorem.
Unit 2: Dilations (G.SRT1, 4)
Students study and prove the properties of dilations.
Unit 3: Similarity and Dilations (G.SRT1, 4)
Students learn what it means for two figures to be similar in general, and then focus on triangles and what criteria predict that two triangles will be similar. Length relationships within and between figures is studied closely and foreshadows work in Topic D. The topic closes with a look at how similarity has been used in real world application.
Unit 4: Applying Similarity to Right Triangles (G.SRT.4)
The focus in this unit is similarity within right triangles. Students examine how an altitude drawn from the vertex of a right triangle to the hypotenuse creates two similar sub-triangles. Students work with adding, subtracting, multiplying, and dividing radical expressions. Finally, students prove the Pythagorean Theorem using similiarity.
Unit 5: Trigonometry (G.CO.9, 10, 11)
Students link their understanding of similarity and relationships within similar right triangles formally to trigonometry. In addition to the terms sine, cosine, and tangent, students study the relationship between sine and cosine, how to prove the Pythagorean Theorem using trigonometry, and how to apply the trigonometric ratios to solve right triangle problems.
Module 3: Extending to Three Dimensions, builds on students’ understanding of congruence in Module 1 and similarity in Module 2 to prove volume formulas for solids.
Unit 1: Area (G.GMD.1)
Students begin their work with 3-dimensions by first developing a stronger sense of area in two dimensions. They find approximated areas of curved figures by “squeezing” them between inscribed and circumscribed polygons, and refine the sizes of the rectangles and triangles that make up those polygons such that the approximations approach the curved figure’s actual area. This informal limit argument prepares students for the development of volume formulas for cylinders and cones in Topic B and foreshadows ideas that students will formally explore in Calculus. Students study the basic properties of area using set notation, the effects of the scaling principle on area, and finally approximate the area of the disk, or circle, by inscribing a polygon within the circle, and consider how the area of the polygonal region changes as the number of sides increases and the polygon looks more and more like the disk it is inscribed within. Topic A provides students with a powerful and universal tool for geometric measurement in two dimensions and serves as an important bridge to understanding geometric measurement in three dimensions.
Unit 2: Volume (G.GMD.1, 3, 4)
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.