# Project with transformation geometry

I am working on a unit plan dealing with tranformation geometry level secondary education.

Students do enjoy creating those visual transformation but

I am trying to figure out what could be a nice project with transformation geometry... that can answer a dilemma existing in the real life and that can be appealing to a teenager.

NEW EDIT: I have been reading/prospecting the very interesting link below. the central point is symmetry and transformation. The math of it seems to be focused on the creating groups (i.e. group symmetry, the "17-wallpapers"), how symmetry is fundamental in physics, defining or constructing a transformation or a composition of transformation.

So far, I envision as final project where students create an expository gallery of (let's say) 6 tessellations. Starting with some tessellations involving simple plane-figures and simple transformation up to having a final tessellation involving a composition of figure (maybe a pentagon as mandatory to create a constraint as it does not "tessellate by itself"). For each tessellation, 2 parts: The 1st part the image of the construction phases done with geogebra + the description of those of the transformation fitting with mathematics phrasing, 2nd part the creative result and a paragraph explaining the "beauty" of it. The "beauty of it" would essentially be about describing the symmetries and (eventually, if it is not too complex for students) the operation under this group. Students will then assemble those in a gallery or in an electronic portfolio.

So far this is what I heading to ...

Symmetry (and tessllation) is a first time encounter for me. What do you think about what I am proposing? What could the drawbacks be?

• I find your plan provides a good encounter with the subject. Especially the division into two parts seems useful. What I would think about is the "how and where your students learn what" within the project. Next to geogebra, there there is a bunch of more specialized tools. Two of them: iOrnament by Richter-Gebert (as iOS or Android App) or Tesselations and Symmetries, a HTML-5 Tool (with geogebra sheets) by Malin Christensson. Anyway I hope your project turns out as you intend. – SCS May 7 '18 at 15:16

Transformation geometry is a beautiful subject with a lot of possibilities for projects.

It gives a dynamic view on geometry - as opposed to a static view, e.g. on the notion of congruence. It is - in my opinion - a good starting point to experience typical mathematical ideas on the road to abstraction.

• optimization problems: Understanding the law of reflection, Fermat point, Napoleon's theorem and alike...

• structuring congruence transformations (isometries of the Euclidean plane): see how every rotation can be written as combination of two reflections; to see which transformations are orientation preserving and which ones are orientation reversing or understanding glide reflections reveals a base to understanding the Euclidean group of isometries eventually. this provides a nice entry point to group theory.

• understanding symmetry as a property of transformations leaving a figure invariant e.g. to understand frieze or wallpaper patterns as well as tessellations

• exploring subgroups as a strategy to structuring relations among figures, as e.g. in the house of quadrilaterals

Possibly, if you don't want to get too abstract (e.g. studying symmetry groups), comparing frieze and wallpaper patterns from your surrounding is a nice way to see the world around through "math glasses". I usually let my students take pictures from patterns to analyze, or we look at patterns from different cultures, or they are asked to create their own patterns with some properties given. If you have dynamical geometry software such as geogebra available, that naturally gives a nice tool for possible projects.

So far so good, I hope you find this answer helpful. I'm sure there are other uses, ideas for projects, maybe other people are willing to share their ideas within further answers or comments.

• Question regarding the paper you posted the link on the so-called "subgroup" of house of quadrilateral. The paper mentioned different "sub-groups". Is he using the term "group" such as it is defined in mathematics as a GROUP under an operation (i.e. closed, identity, inverse, associative )? I did not see anything going in that direction. For example, the sub group with the diagonals seems to be more a "tree organization " of possible quadrilaterals ... – gegu May 6 '18 at 3:55
• Yes; There are different notions what "the house of quadrilaterals" should be. It mainly depends on what construct one uses to distinguish different sorts of quadrilaterals (in itself a worthwhile task...). Many teachers in Europe discuss this way before students have an understanding of the notion of a group. If you draw the diagram of subgroups and the corresponding quadrilaterals with this subgroup as symmetry group, you get a version as well. Cheers – SCS May 6 '18 at 7:06
• ... and you're right, it rather should be called "tree of quadrilaterals". But then, to young students, you can put everything into a house - as you can see here. ;-) – SCS May 6 '18 at 7:13

Do teenagers still play minigolf or pool? Both of those activities may be analyzed using transformation geometry. Not exactly a single project, but an area to explore perhaps

• Hi, Mr. David.There is no minigolf in my Country (Belize). they are not allowed to step in places (bars) where there are pools. Could you elaborate a bit what you had in mind? maybe it can be done with another activity... – gegu May 3 '18 at 2:31
• – David Steinberg May 4 '18 at 1:41