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 In Topic D, students learn that radical expressions naturally arise in geometry, such as the height of an isosceles triangle or the lateral length of a cone. The Pythagorean Theorem is applied to...

 Topic C revisits the Pythagorean Theorem and its applications, now in a context that includes the use of square roots and irrational numbers. Students learn another proof of the Pythagorean Theorem...

 In Topic B, students apply their knowledge of volume from previous grade levels (5.MD.C.3, 5.MD.C.5) to the learning of the volume formulas for cones, cylinders, and spheres (8.G.C.9). First,...

 Optional Topic E is an application of systems of linear equations (8.EE.C.8b). Specifically, a system that generates Pythagorean triples. First, students learn that a Pythagorean triple can be...

 It is recommended that students have some experience with the lessons in Topic D from Module 2 before beginning these lessons. In Lesson 13 of Topic C, students are presented with a general proof...

 Topic B begins with the definition of similarity and the properties of similarities. In Lesson 8, students learn that similarities map lines to lines, change the length of segments by factor r, and...

 Topic A begins by demonstrating the need for a precise definition of dilation instead of “same shape, different size” because dilation will be applied to geometric shapes that are not polygons. ...

 Optional Topic D begins the learning of Pythagorean Theorem. Students are shown the “square within a square” proof of the Pythagorean Theorem. The proof uses concepts learned in previous topics of...

 In Topic C, on the definition and properties of congruence, students learn that congruence is just a sequence of basic rigid motions. The fundamental properties shared by all the basic rigid motions...

 Topic B is a critical foundation to the understanding of congruence. All the lessons of Topic B demonstrate to students the ability to sequence various combinations of rigid motions while...

 Throughout Topic A, on the definitions and properties of the basic rigid motions, students verify experimentally their basic properties and, when feasible, deepen their understanding of these...

 Similar to Topic A, the subjects of area, surface area, and volume in Topics D and E are not new to students, but provide opportunities for students to expand their knowledge by working with...

 Similar to Topic A, the subjects of area, surface area, and volume in Topics D and E are not new to students, but provide opportunities for students to expand their knowledge by working with...

 Topic C introduces the idea of a slice (or cross section) of a threedimensional figure. Students explore the twodimensional figures that result from taking slices of right rectangular prisms and...

 In Topic B, students work extensively with a ruler, compass, and protractor to construct geometric shapes, mainly triangles (7.G.A.2). The use of a compass is new (e.g., how to hold it, and to how...

 In Topic A, students solve for unknown angles. The supporting work for unknown angles began in Grade 4, Module 4 (4.MD.C.5–7), where all of the key terms in this Topic were first defined, including...

 Students revisit scale drawings in Topic C to solve problems in which the scale factor is represented by a percent (7.RP.A.2b, 7.G.A.1). They understand from their work in Module 1, for example,...

 In Topic C, Students continue work with geometry as they use equations and expressions to study area, perimeter, surface area, and volume. This final topic begins by modeling a circle with a bicycle...

 In Topic B, students use linear equations and inequalities to solve problems. They continue to use bar diagrams from earlier grades where they see fit but will quickly discover that some problems...

 In Topic D, students bring the sum of their experience with proportional relationships to the context of scale drawings (7.RP.2b, 7.G.1). Given a scale drawing, students rely on their background in...

 Module 5 concludes with deconstructing the faces of solid figures to determine surface area. Students note the difference between finding the volume of right rectangular prisms and finding the...

 In Grade 5, students recognized volume as an attribute of solid figures. They measured volume by packing right rectangular prisms with unit cubes and found that determining volume was the same as...

 In Module 3, students used coordinates and absolute value to find distances between points on a coordinate plane (6.NS.C.8). In Topic B, students extend this learning to Lessons 7 and 8 where they...

 In Topic A, students use composition and decomposition to determine the area of triangles, quadrilaterals, and other polygons. They determine that area is additive. Students learn through...