Ohio Resource Center
Content Supports - Activities and rich problems
Products of Reflections
Discipline
Mathematics
8, 9, 10, 11, 12
Professional Commentary

Students investigate compositions (called "products") of two reflections and express the result as a single transformation. There are two cases to consider: (1) If the reflection lines are parallel, the product is a translation, and (2) If the reflection lines intersect, the product is a rotation. This mathematically rich problem was prepared by the Ohio Resource Center to accompany the Mathematics Program Models for Ohio High Schools developed by the Ohio Department of Education. (author/sw)

Common Core State Standards for Mathematics
Standards for Mathematical Practice
CCSS.Math.Practice.MP5
Use appropriate tools strategically.
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations:
1. Lines are taken to lines, and line segments to line segments of the same length.
2. Angles are taken to angles of the same measure.
3. Parallel lines are taken to parallel lines.
8.G.A.2
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
High School - Geometry
Congruence
Experiment with transformations in the plane
HSG-CO.A.2
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 to those that do not (e.g., translation versus horizontal stretch).
HSG-CO.A.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Understand congruence in terms of rigid motions
HSG-CO.B.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
Make geometric constructions
HSG-CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Ohio Mathematics Academic Content Standards (2001)
Geometry and Spatial Sense Standard
Benchmarks (8–10)
F.
Represent and model transformations in a coordinate plane and describe the results.
9.
Show and describe the results of combinations of translations, reflections and rotations (compositions); e.g., perform compositions and specify the result of a composition as the outcome of a single motion, when applicable.
Principles and Standards for School Mathematics
Geometry Standard
Apply transformations and use symmetry to analyze mathematical situations
Expectations (9–12)
use various representations to help understand the effects of simple transformations and their compositions.