Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle (i.e. rigid motions) to those that do not (e.g. translation vs. horizontal stretch)
INTERPRETATION OF OBJECTIVE - G.CO.A.2
This objective has two parts to it:
(1) The objective speaks to performing and representing transformations in a variety of ways such as: transparencies, patty paper, constructions or dynamic geometry software. This is to become familiar with the impact that transformations have on the general motion and shape of the figures. Once we are able to perform these movements in the plane, then we transition to the coordinate plane to use coordinate rules to move/alter shapes, for example: (x,y) ----> (x + 2, 3y). These rules act like input/output function machines, where we input coordinates and they output the new coordinates. We link algebraic relationships such as functions & one to one functions to mappings and transformations in geometry.
(2) The objective also speaks about determining whether a transformation is isometric or not. Isometric transformations preserve distance and angles. We do not study the transformations in depth here, we are simply learning about the transformations that are isometric or rigid motion, and those that are not.
(1) The student will be able to distinguish between transformations that are isometric and those that are not isometric.
(2) The student will be able to recognize different types of transformations such as reflections, translations, rotations, dilations, and stretches.
(3) The student must know the definition of isometric transformations (rigid motion).
(4) The student will be able to explain the connection between the algebraic relationships of functions and one to one functions to the geometric relationships of mapping and transformations.
(5) The student will be able to use coordinate rules to move and/or alter a pre-image to determine its image or vice versa.
THE BIG IDEA
The big ideas of geometry, congruence and similarity, in the new core curriculum are developed from a transformational stand point. Congruence is connected directly to the isometric transformations of the plane, while similarity is connected to the non-isometric transformation of the plane, such as dilation. These early ideas form the foundation for later understanding of these major concepts of geometry. These ideas are generally quite easy for students to get but don’t pass over them lightly – they hold essential foundational concepts!!
TRAPS & PITFALLS
Poor algebra skills and limited understanding of functions will certainly be a pitfall for this objective. Students need to be able to substitute values, solve equations, and also demonstrate understanding of domain and range values. The more technical connections of a transformation being a one to one function can be confusing if these terms are new or misunderstood from algebra.
Notation can also be an area of difficultly in this objective. This is where we first introduce pre-image (A) and image (A’) and the prime notation ('). Students often confuse pre-images with images and struggle to work backwards if given the image first, and then asked to determine the pre-image. This is often true in algebra as well, when given the output value (y), and then asked to work backwards to determine the input value (x) always seems to be harder for students.
The student needs to come with an informal understanding of the motions in the plane such as flip (reflection), turn (rotate), slide (translate) and scale (dilate). This informal knowledge will help them connect to the new more formal definitions.
The student needs to come with an understanding of what a function is and how to input values (domain) and then solve for output values (range). One to one functions will also be revisited as we connect algebraic relationships to geometric relationships.
As mentioned in the Big Idea, these foundational ideas will lead to the much bigger concepts of congruence and similarity. Understanding isometric and non-isometric transformations is essential to success in this course.
MY REFLECTIONS (over line l)
Concerning the first part of the objective where we connect the algebra to the geometry with functions, I approached this is a very technical way and lost a few students whose algebra skills were weak. I was drawing domain and range charts and was trying to revive their knowledge of functions and it was difficult for many because what should have been review wasn't. Approaching it next time, I will build from the idea of coordinate rules. I want them to gain familiarity with input/output relationships in geometry just as they had done with functions in algebra. We will work on this simple concept first and then link in the deeper concept that transformations are one to one functions.
Concerning the second part of the objective where isometries are introduced I found great success here. Students picked it up quickly and they easily saw the patterns in the rules to determine if it was isometric or not. I had to pay close attention to problems that had them work backwards from knowing the image and then solving for the pre-image. Don’t focus on predicting which rules do what, focus on determining whether they produce isometries or not.