Use transformations to explore definitions and theorems about congruence, similarity, and symmetry.

Typically, there are three different approaches from which instructors can choose in order to embed transformation geometry learning outcomes into a GeT course. Instructors may choose to teach a GeT course using a formal transformation approach; they may choose to include a dedicated transformation unit within a course that has a predominantly traditional Euclidean focus; or they may integrate transformation approaches and concepts throughout the course offering. 

There are two main types of transformations that arise in GeT courses: isometries, also known as congruence transformations (mappings that preserve both angle measure and segment length) and similarity transformations (mappings that preserve angle measures and proportionality of segment lengths). The set of isometries of the plane form an infinite non-Abelian group. Moreover, each element of the group can be classified as the identity transformation, a reflection, rotation, translation, or a glide reflection, and the composition of at most three reflections can be used to generate the other isometries. Depending on abstract algebra prerequisites, instructors may choose to highlight and formalize group properties and connections to finite symmetry groups. 

It is recommended that GeT instructors build on familiar function notation when introducing transformations. This may lead to a better understanding of sequences of isometries as compositions of isometries. This notation may also make it easier for students to use rigid motions to express symmetry. GeT instructors can take advantage of looking at proofs through multiple approaches (Euclidean, analytic, transformational) to deepen students’ understanding of specific theorems. For example, students can be prompted to compare other strategies after proving the base angles of an isosceles triangle are congruent by using the concept of symmetry. Some GeT instructors may begin with an informal approach to the understanding of reflections and rotations and extend the concepts of symmetry to study geometric shapes using transformations (translations, rotations, and reflections) and combinations of them. The definition of congruence is then conveyed in terms of rigid motions and the definition of similarity in terms of dilations and rigid motions. The notion of a translation typically will be introduced with directed line segments and a rotation with directed angles, though some instructors might take the opportunity to more deeply explore the concepts in terms of vectors, matrices, and coordinate geometry.

Instructors may instead choose to begin solely with sequences of reflections, which can be used to generate all other isometries of the plane. Each line of the plane is associated with a reflection that satisfies two properties: (1) every point on the line is fixed by the transformation and (2) the line is the perpendicular bisector of the segment connecting any point not on the line and the point’s image under the transformation. By exploring the images of points and figures resulting from sequences of reflections about parallel and intersecting lines, GeT students can discover and establish relationships with translations, rotations, and glide reflections. Instructors may choose to also have students explore, informally, orientation preserving/reversing properties and the aspects of the group structure (associativity of composition, existence of identity and inverses, and non-commutativity).

GeT instructors have reported that students sometimes struggle with understanding fixed point properties of transformations. A case in example is when a segment is rotated a specified number of degrees about a center of rotation when the center of rotation (the fixed point) is not on the segment. Even though the center of rotation is specified, some students often choose one end-point of the segment as a center of rotation and use it as the fixed point. Using activities and technology that allow students to experience multiple examples of such properties can be included in a GeT course (see SLO 6 on use of dynamic geometry software) to clarify such misunderstandings. Douglas and Picciotto (2018) provide a guide and activities for transformational proof in high school geometry for teachers and curriculum developers. They note that The Common Core State Standards for Mathematics (2010) only require using transformations to justify the triangle congruence and similarity criteria (e.g., SAS, AA) and do not specify transformational proofs beyond this. However, Douglas and Picciotto recommend a high school geometry course using both traditional Euclidean and transformational approaches, beyond congruence and similarity criteria. It is important that prospective teachers are exposed to multiple approaches so that they not only are prepared with a solid background in geometry content but also able to make informed decisions regarding the choices they make in their own classrooms.

To prepare GeT students for transformational proofs beyond triangle congruence and similarity criteria, the design principles of St. Goar and Lai (2021) may be useful. They suggest emphasizing the bidirectionality of the definition of congruence (or similarity), highlighting that transformations act on the entire figure (or plane), and encouraging GeT students to explain how and why a prescribed transformation necessarily maps one figure to another as part of their proving activity. Instructors may also choose to compare definitions based on properties of transformations with other definitions (see SLO 5 on role of definitions). For example, one could define a kite as a convex quadrilateral for which a diagonal is a line of symmetry and prove properties about its congruent sides and congruent angles that are used as definitions in a traditional approach. Other opportunities to learn transformational geometry include explorations of tessellations of the Euclidean plane as well as explorations of transformations in non-Euclidean geometries (see SLO 9 on non-Euclidean geometry).

References

Douglas, L. & Picciotto, H. (March 2018). Transformational proof in high school geometry: A guide for teachers and curriculum developers. Retrieved from https://www.mathedpage.org/transformations/proof/transformational-proof.pdf.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards. Washington, DC: Authors.

St. Goar, J., & Lai, Y. (2021) Designing activities to support prospective high school teachers’ proofs of congruence from a transformation perspective. PRIMUS. doi: 10.1080/10511970.2021.1940403