Unit 3 - Measurement in Congruent and Similar Figures Overview

To-Do Date: Jan 3 at 11:59pm

Informal transformations are the way we, as humans, compare two objects to see if they are congruent. We turn, twist, and flip objects to see if one can lay exactly on the other without bending, stretching, or breaking either object. When they match, we say the objects are congruent. If they do not match, but they have the same shape and the same scaled measurements, we say the objects are similar. Transformations in geometry give us language to describe these turns, twists, flips, and scaling precisely and systematically. This unit formalizes the concept of congruence and similarity of planar objects by identifying the essential components of rigid motion and similarity transformations. Students are expected to become proficient with transformations that involve coordinates as well as with transformations that do not involve coordinates. Throughout the course, transformations are presented as functions. This connection further develops students’ understanding of functions and connects the statistics and geometry units of the course. It also creates a bridge between Algebra 1 and Algebra 2 since concept of function permeates and links nearly all aspects of high school mathematics. Students develop further insights into congruence and similarity by exploring which transformations affect angle measures and distances between pairs of points and which do not. Students apply their understandings of transformations, congruence, and similarity to solve problems involving polygons and circles. Throughout Units 2–4, specific learning objectives require students to prove geometric concepts. Students’ proofs can be organized in a variety of formats, such as two-column tables, flowcharts, or paragraphs. The format of a student’s proof is not as important as their ability to justify a mathematical claim or provide a counterexample disproving one. They should develop an understanding that a mathematical proof establishes the truth of a statement by combining previously developed truths into a logically consistent argument. 

ENDURING UNDERSTANDINGS
Students will understand that …

  • Transformations are functions that can affect the measurements of a geometric figure. 
  • Congruent figures have equal corresponding angle measures and equal distances between corresponding pairs of points.
  • Similar figures have equal corresponding angle measurements, and the distances between corresponding pairs of points are proportional. 
  • The geometry of a circle is completely determined by its radius.

KEY CONCEPTS

  • 3.1: Transformations of points in a plane – Defining transformations to describe the movement of points and shapes
  • 3.2: Congruent and similar polygons – Using transformations to compare figures with the same size or same shape
  • 3.3: Measurement of lengths and angles in circles – Using measurements in circles to make sense of round flat objects in the physical world