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Mathematics - Secondary Curriculum Secondary Mathematics I
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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.4

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

  • 3D Transmographer
    This lesson contains an applet that allows students to explore translations, reflections, and rotations.
  • Constructions
    This site provides both a video and step-by-step directions on how to complete a variety of constructions.
  • Defining Reflections
    The goal of this task is to compare and contrast the visual intuition we have of reflections with their technical mathematical definition.
  • Defining Rotations
    The goal of this task is to encourage students to be precise in their use of language when making mathematical definitions.
  • 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 Rotations
    The purpose of this task is to use the definition of rotations in order to find the center and angle of rotation given a triangle and its image under a rotation.
  • Identifying Translations
    The purpose of this task is to study the impact of translations on triangles.
  • 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.
  • Symmetries of a circle
    This task asks students to examine lines of symmetry using the high school definition of reflections.
  • Taking a Spin (pdf)
    Although students are often asked to find the angles of rotational symmetry for given regular polygons, in this task they are asked to find the regular polygons for a given angle of rotational symmetry, a reversal that yields some surprising results. This task would be most appropriate with students who have at least some experience in exploring rotational symmetry.
  • 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.
  • Translations, Reflections, and Rotations
    Students are introduced to the concepts of translation, reflection and rotation in this lesson plan.
  • Unit Squares and Triangles
    This problem provides an opportunity for a rich application of coordinate geometry.

UEN logo - 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.