What is distinctive about the approach to geometry laid out in the Common Core State Standards for Mathematics (CCSSM)? Let’s look to the Introduction to HS Geometry to find out:
The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes generally). Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes—as when the reflective symmetry of an isosceles triangle assures that its base angles are congruent. In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition.
Symmetry. Rigid motions. Translations, rotations, reflections. Superposition. Most recent approaches to the study of geometry tinker with these ideas, but the CCSSM places them squarely at the center of the discussion. In this series of posts, let’s explore the potential advantages of this approach, while also owning up to the challenges it poses.
Benefit #1: Hands-on Exploration
The use of rigid motions allows students to explore figures in a concrete, tactile way. Students can trace a figure onto a transparency or patty paper, then spin it, slide it, or flip it over.
They can also manipulate figures dynamically in a virtual environment like GeoGebra. For instance, this sketch allows students to engage with corresponding angles on parallel lines.
Benefit #2: Visual Intuition
One of the key skills involved in thinking geometrically is the ability to engage in visual reasoning. Students are well-positioned to master the subject when they can see the reflection symmetry of a rectangle, which explains why it has congruent diagonals, or the rotation symmetry of a parallelogram, which explains why its diagonals bisect each other (and many other things as well!).
Benefit #3: Direction for Proofs
Proofs involving rigid motions fall into a consistent pattern: 1) determine which rigid motions take one figure onto another, 2) set about showing that the rigid motions do what you say they’ll do. This is not to say that there is anything easy about developing the proofs, but this approach does provide a clear sense of direction for students about the goal towards which they are supposed to be working.
For instance, to show that the isosceles triangle above has reflection symmetry (which proves that the base angles are congruent), we just need to prove that the reflection over line exchanges and while keeping in place. The burden in developing the proof falls on linking these claims to the definition of a reflection and the things we assume are true of reflections (namely, that they preserve distance and angles).
Are you convinced that rigid motions provide a powerful and coherent framework for the study of congruence? Even if you are, it still remains to see how the program plays out. In this series of posts, my aim is to show how students can live up to the vision outlined in the Introduction to Geometry:
During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs.
I hope you find this series valuable! Please feel free to add your thoughts to the comments.