In this module, students learn about translations, reflections, and rotations in the plane and, more importantly, how to use them to precisely define the concept of congruence. Up to this point, congruence has been taken to mean, intuitively, same size and same shape. Because this module begins with a serious study of geometry, this intuitive definition must be replaced by a precise definition. This module is a first step; its goal is to provide the needed intuitive background for the precise definitions that are introduced in this module for the first time.
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In Module 3, students learn about dilation and similarity and apply that knowledge to a proof of the Pythagorean theorem based on the angle-angle criterion for similar triangles. The module begins with the definition of dilation, properties of dilations, and compositions of dilations. The instruction regarding dilation in Module 3 is structured similarly to the instruction regarding concepts of basic rigid motions in Module 2. One overarching goal of this module is to replace the common idea of "same shape, different sizes" with a definition of similarity that can be applied to geometric shapes that are not polygons, such as ellipses and circles.
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The module begins with work related to the Pythagorean theorem and right triangles. Before the lessons of this module are presented to students, it is important that the lessons in Modules 2 and 3 related to the Pythagorean theorem are taught (i.e., Module 2 Lessons 15 and 16 and Module 3 Lessons 13 and 14). In Modules 2 and 3, students used the Pythagorean theorem to determine the unknown side length of a right triangle. The solutions from those modules are revisited and are the motivation for learning about square roots and irrational numbers in general.
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Working with the Pythagorean TheoremMathematics Instructional Plans (MIPs) help teachers align instruction with the Mathematics Standards of Learning (SOL) by providing examples of how the knowledge, skills and processes found in the SOL and curriculum framework can be presented to students in the classroom.