In the midst of Transformations -- By Axelle Faughn

Mathematical transformations are present throughout the school curriculum and are regarded as one successful approach to teaching geometry. We first encounter them in elementary and middle school when studying the transformation of shapes (see www.turnonccmath.net for associated learning trajectories), but we find them again in higher level math classes when exploring function transformations in pre-calculus and calculus, and studying matrices in linear algebra. Shifts, changes and mutations being so important in understanding the world around us, it is hardly surprising that so much emphasis is put on representing and understanding these natural phenomena mathematically. As evidenced by Escher's work these mathematical concepts are also very often used in arts and architecture for creating patterns. In this post we give a quick overview of various transformational selfies from a student's perspective.

Picture 1: Door reflection

Transformations are fun to teach in the geometry class, they provide a wealth of hands on activities that can be explored using geosticks or geometry software. They are also a key component in understanding geometric constructions and emphasizing shape properties, for instance using a mirra, while helping to formulate concrete ideas about shape congruency and similarity. Mirroring is at the core of many human activities. In the math class, usually starting with symmetry within a shape, one can play with mirror images across various lines of reflection. Reflections across other lines in order to duplicate a shape are also introduced as illustrated by the dorm door in Picture 1, or highlighted in our recent angle post. Man-made structures definitely offer a multitude of illustrations for visualizing reflections in our world, but so do natural phenomena such as leaves or insects for instance.

Picture 2: Jack O Lantern shift

Translations, or shifts, are also a common type of transformation surrounding us. We already talked about them in the number line post to exemplify repeated behavior. They are certainly used extensively in tessellations as shown in Picture 3, but can also be used to explain motion using vectors as the geometry student did in Picture 2. This vector representation of movement is further used in physics to express force and motion. It is therefore important to emphasize it early on as a tool to understanding some basic notions of mechanics.

Picture 3: Tessellation

Picture 4: Clock rotation

Picture 5: Wheel rotation

Rotations can also be studied within or across shapes, sometimes causing incredible phenomena such as Solar Eclipses and Planet revolutions. Rotational symmetry, as shown in Picture 4 both in the clock and the surrounding mirror, is the basis for many dialing or locking systems and allows us to easily connect modulo arithmetic to geometry. When found across shapes such as Picture 5 illustrates, one important question remains to find the center of rotation as being the only point of invariance on the plane. Such concerns and exercises open students to recognizing invariance as a key mathematical concept in a changing world, and as a mathematical characteristic that they should pay attention to in the future. In our world of impermanence, this may also be a good strategy to adopt in real life, by recognizing the patterns of change and consistency.

Picture 6: Sun similarity

With similarity students encounter transformations of a different kind, having to work with  proportions rather than same size copies of an image. Enlargements and reductions have common uses in the office for instance while making copies or editing photographs. In the 2017 Solar Eclipse event, it is fairly common to see an explanation of the smaller Moon hiding the much bigger Sun by using the shadow cone, an immediate consequence of shape similarity from various angles of observation. Talking about the sun, here is an interesting take on similarity by one pre-calculus students (Picture 6).

Picture 7: Are these flyers similar?

Likewise, perspective drawing will use similarity considerations in order to give a fairly credible representation of a room or object being depicted. However in order for a transformation to qualify as similarity, students should be able to prove that the corresponding dimensions of two shapes are related by following the same proportion (or scaling) factor.
For instance in Picture 7, are the two flyers truly similar? And are the arches of the church entrance in Picture 8 truly enlargements of one another?

Picture 8: Church similarity

Picture 10: concentric circles

Picture 9: Congruent angles and
proportional sides in various squares

At times, geometry students will also make remarkable observations about similarity, without necessarily knowing they do... a good opportunity to highlight some the need for awareness regarding the mathematics surrounding us. In Picture 9 and 10 for instance students found rather complicated ways of stating the obvious... indeed aren't all circles similar (not just concentric ones as represented in Picture 10)? And what do you think of the squares in Picture 9?

Picture 11: Pier reflection

Picture 12: Composition of transformations

In the pre-calculus class transformations are approached analytically, therefore we add a set of axes in order to describe them in the coordinate plane using function transformations in the expression of the function itself. Pictures 11 and 12 illustrate these manipulations nicely using reflections and shifts, at times combining them. However before getting to such concept mastery these students had to acquire strong knowledge of how transformations can be recognized and composed onto various shapes and curves. The concepts learnt in geometry come in handy when the added difficulty of symbolic expressions comes into play.

The world of transformations is a fascinating and easily accessible one for who gets used to observing and looking for patterns in the world around them. As we enter a new school year, many other kinds of transformations are about to happen in the schools and classrooms around the globe. Some of them one can learn to express using mathematics, others will continue to appear mysterious to a mathematician's mind, but are not less fascinating. Maybe one final point I'd like to emphasize is that there is invariance to be found in the midst of all these changes, some in the shape of lines, points and vectors, others in the form of angles and proportions, and others may see love as the only invariant there needs to be. So love your transformations! And make the most of every day this upcoming school year!