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

SLO 10 Summary

Two main types of transformations arise in GeT courses: isometries (also known as congruence transformations) and similarity transformations. Isometries include reflections, rotations, translations, and glide reflections; a similarity transformation is the composition of an isometry and a dilation. A GeT course can contain a dedicated unit on transformations, or transformational concepts can be integrated throughout the course (even to the extent of a purely axiomatic treatment). In order to enhance and facilitate prospective teacher learning, familiar function notation can be incorporated in an introduction to transformations. This may lead to a better understanding of a sequence of transformations as a composition. While some GeT courses 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, others may instead begin solely with sequences of reflections, which can be used to generate all other isometries of the plane.

SLO 10 Narrative

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: two figures, A and B, are considered congruent if, and only if, there exists a sequence of rigid motions, r, that superimposes figure A onto figure B, that is r(A)=B. Instructors may consider pointing out advantages of this definition of congruence, most notably that the definition applies to all congruent figures, not just congruent triangles; they may also highlight that transformations act on the entire plane. Similarity is defined analogously 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, 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. 

It is important that GeT students move beyond recognizing the properties of transformations and learn to reason with them. The Common Core State Standards for Mathematics (2010) require using transformations to justify the triangle congruence and similarity criteria (e.g., SAS, AA). Although the Common Core does not specify transformational proofs beyond this, Douglas and Picciotto (2018) provide a guide and activities for transformational proof in high school geometry for teachers and curriculum developers; the guide recommends a high school geometry course using both traditional Euclidean and transformational approaches, beyond congruence and similarity criteria for triangles. St. Goar and Lai (2021) note ways to move beyond triangle congruence criteria proofs and to incorporate the bidirectionality of the definition of congruence (or similarity). They state that in order to prove figure A is congruent to figure B based on a transformation approach, a student must use two key elements: (1) they must specify a sequence of rigid motions, r, from figure A to figure B (NCTM, 2018) and (2) they must justify deductively that the image of figure A under the rigid motions actually is figure B, that is r(A)=B. Similarity proofs are constructed analogously. 

Instructors may also choose to compare definitions based on properties of transformations with other definitions [see SLO 5]. 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. The transformational context could be an opportunity to compare different axiomatic contexts [see SLO 4]. For example, without basic assumptions about transformations, traditionally one of the triangle congruence criteria—ASA, SAS, or SSS—must be assumed as an axiom without proof. By contrast, in a transformation context, ASA, SAS, and SSS may all be proved deductively without such assumptions (Venema, 2006). 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].

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 so that they are able to make informed decisions regarding the choices they make in their own classrooms. Additional resources that may assist instructors with this effort include Venema (2006), which has a robust mathematical discussion of transformations, and Henderson & Taimina (2005), which offers an inquiry-based learning approach to transformations and a way to include rigid motions in other geometric contexts such as spherical and hyperbolic 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.

Henderson, D. W., & Taimin̦a, D. (2005). Experiencing geometry : in Euclidean, spherical, and hyperbolic spaces (2nd ed.). Prentice Hall.

National Council of Teachers of Mathematics (2018). Catalyzing change in high school mathematics: Initiating critical conversations. Author.

National Governors Association Center for Best Practices [NGA], & Council of Chief State School Officers [CCSSO]. (2010). Common Core State Standards for Mathematics. Available at http://www.corestandards.org/Math/

St. Goar, J., & Lai, Y. (2021) Designing activities to support prospective high school teachers’ proofs of congruence from a transformation perspective. PRIMUS. doi: 10.1080/10511970.2021.1940

Venema, G. A. 2006. Foundations of Geometry. Pearson.