Secondary Mathematics I

Strand: GEOMETRY - Congruence (G.CO)

Experiment with transformations in the plane. Build on student experience with rigid motions from earlier grades (Standards G.CO.1-5). Understand congruence in terms of rigid motions. Rigid motions are at the foundation of the definition of congruence. Reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems (Standards G.CO.6-8). Make geometric constructions (Standards G.CO.12-13).

Standard G.CO.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Point out the basis of rigid motions in geometric concepts, for example, translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle.

• Constructions
This site provides both a video and step-by-step directions on how to complete a variety of constructions.
• GEOMETRY - Congruence (G.CO) - Sec Math I Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Congruence (G.CO).
• Identifying Unknown Transformations
This applet allows the student to drag a shape and then observe the changes to its behavior. They then determine whether the alteration is due to reflection, a rotation, or a translation/slide transformation.
• Introduction to the Materials (Math 1)
Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
• Module 6: Transformations & Symmetry - Student Edition (Math 1)
The Mathematics Vision Project, Secondary Math One Module 6, Transformations and Symmetry, builds on students experiences with rigid motion in earlier grades to formalize the definitions of translation, rotation, and reflection.
• Module 6: Transformations & Symmetry - Teacher Notes (Math 1)
The Mathematics Vision Project, Secondary Math One Module 6 Teacher Notes, Transformations and Symmetry, builds on students experiences with rigid motion in earlier grades to formalize the definitions of translation, rotation, and reflection.
• Module 7: Congruence, Construction & Proof - Student Edition (Math 1)
The Mathematics Vision Project, Secondary Math One Module 7, Congruence, Construction, and Proof, begins by developing constructions as another tool to be used to reason about figures and to justify properties of shapes. Individual constructions are not taught for the sake of memorizing a series of steps, but rather to reason using known properties of shapes such as circles.
• Module 7: Congruence, Construction & Proof - Teacher Notes (Math 1)
The Mathematics Vision Project, Secondary Math One Module 7 Teacher Notes, Congruence, Construction, and Proof, begins by developing constructions as another tool to be used to reason about figures and to justify properties of shapes. Individual constructions are not taught for the sake of memorizing a series of steps, but rather to reason using known properties of shapes such as circles.
• Reflected Triangles
This task is a reasonably straight-forward application of rigid motion geometry, with emphasis on ruler and straightedge constructions, and would be suitable for assessment purposes.
• Showing a triangle congruence: a particular case
This task provides experience working with transformations of the plane and also an abstract component analyzing the effects of the different transformations.
• Showing a triangle congruence: the general case
The purpose of this task is to work with transformations to exhibit triangle congruences in a general setting.
• Symmetries of a circle
This task asks students to examine lines of symmetry using the high school definition of reflections.
• Tangent Lines and the Radius of a Circle
This task presents a foundational result in geometry, presented with deliberately sparse guidance in order to allow a wide variety of approaches.
• Tessellations: Geometry and Symmetry
Students can explore polygons, symmetry, and the geometric properties of tessellations in this lesson.
• Translations, Reflections, and Rotations
Students are introduced to the concepts of translation, reflection and rotation in this lesson plan.
• Trigonometric Identities and Rigid Motions
The purpose of this task is to apply translations and reflections to the graphs of the equations f(x)=cosx and g(x)=sinx in order to derive some trigonometric identities.
• Unit Squares and Triangles
This problem provides an opportunity for a rich application of coordinate geometry.

http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.