This is an initial draft for the fifth chapter of a course guide for a year-long 9th grade Algebra & Geometry course. I am requesting your assistance. For more information on the course and this project, please see this post.
Now that we’ve established the formal definitions of rotations, reflections, and translations, we’re ready to define a congruence as a finite composition of these rigid transformations. That forces us to define the concept of composition and it also leads us to want to develop a more convenient notation for transformations that can handle compositions well, namely parentheses notation. It would be rather upsetting if we had a figure S which was congruent to T, but T wasn’t congruent to S. That is we’d like to be assured that a congruence from S to T necessarily implies the existence of a congruence from T to S, but that we can handle if we introduce the notion of the inverse of a transformation.
We could introduce all of these concepts just motivated by the desire to define congruence, but I have found the notion of symmetry (which for now we consider to be a congruence from a figure to itself) to be more directly engaging. It also helps motivate better why we’d be interested in identifying the specific transformations that compose, instead of just being interested in whether such a sequence exits.
(A) Key Ideas: Students should understand parentheses notation for transformation, what it means to compose two transformations, and what is meant by the inverse of a transformation. Students should understand the formal definition of congruence in terms of rigid transformations. They should also understand the distinction between a congruence (noun) and congruent figures (adjective). Two figures are congruent if there exists a congruence from one to the other. Students should understand that we consider a symmetry to be a congruence from a figure to itself.
(B) Related standards from the common core:
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).
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
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.
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.
(C) Connections with what students have done: When I did a unit like this last year I was surprised to discover that students had only learned about symmetry in terms of reflectional symmetry. When shown a figure with only rotational symmetry there was some heated discussion about whether it should be considered to possess any symmetry and the consensus reached was it should not because there were no “lines of symmetry”.
(D) Connections with what students will do: The definition of congruence will extend once we introduce dilations to a definition of similarity (chapter 8). These allow us to establish criteria for triangle congruence or similarity (chapter 9). Transformations are nice examples of functions and the notation and concepts used here (such as composition and inverse) will be generalized next year (10th grade). Students are also laying the groundwork here for the concept of groups which they may encounter in college mathematics.
(E) Texts and videos to which students and parents could refer to help understand key ideas: James Tanton has a nice pamphlet on symmetry which explores the question of how symmetry is defined mathematically. By including dilations and hence self-similarity, he looks at an even broader notion of symmetry than just focusing on congruences. He also spends some time proving that the rigid motions are isometries using SAS and properties of parallelograms (as opposed to common core which starts with the assumption that rigid motions are isometries and uses that to prove SAS and properties of parallelograms).
(F) Questions and tasks that can generate discussion to explore key ideas in class: Last year I started by handing out a sheet with several figures on one side (and frieze and wallpaper patterns on the other side) and asking which figures possess symmetry and are some figures “more symmetric” than others. This led to some great discussion, although as I noted above, the students had in their minds that only reflectional symmetry should be considered symmetry. I had to state that mathematicians and artists were often interested in a broader notion of symmetry. (I shared a lot of work of M.C. Escher and other photos of art and architecture using symmetry). I also pointed out some benefit to being able to perform a sequence of symmetries and still consider that to be a symmetry (which would not be the case if we only allowed reflections).
Later we looked at a square and tried to list out all the symmetries and keep track of what happens when combine two symmetries of the square (essentially writing out the multiplication table for the dihedral group of symmetries of the square). This motivated composition which in turn motivated some improved notation (parentheses) as well as the concept of inverse.
(G) Questions and tasks that can be used to help assess student understanding of key ideas: One could give the students a figure and ask them to simply list the symmetries. One could also ask students to identify the composition of two symmetries as a single symmetry.
(H) Interesting and challenging ways to extend and apply the key ideas: By looking at only the orientation preserving symmetries of a regular polygon we encounter the cyclic groups which we can then connect with the notion of modular arithmetic (which some students have encountered before). One could also look at classifying all frieze patterns or all wallpaper patterns.
(I) Standard textbook problems related to the key ideas: I’ve seen a standard problem of given a figure draw all “lines of symmetry” We can extend this traditional exercise by also asking for point symmetries and asking what order they are, as well as considering translational symmetry.
(J) Misconceptions and “tricks” to avoid: Parentheses notation often gets confused with multiplication. That can still happen here, although it is not nearly as bad since the elements of the domain aren’t numbers but rather points in the plane.
If you have any questions, comments, and/or resources, please share them. Thanks for your help.