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\memoto{TB Programme Organizers}
\memosubject{Course Proposal/Overview}
\memodate{\today}
\logo{\includegraphics[width=0.15\textwidth]{TB-Logo-pb-w-big.png}}
\memofrom{Stephanie A.}
\begin{document}
\maketitle
\tableofcontents
\section{Preliminaries}
\begin{itemize}
\item \textbf{Instructor:} Stephanie A. (3 Sessions)
\item \textbf{Suggested Audience:} Up to $\sim$20  "advanced" HS students, will be working in groups of $\sim$3 for projects.  These students should have potential to: 
\begin{itemize}
\item Work rigorously to expand their mathematical knowledge, skills, and understanding 
\item Use origami as an avenue for mathematical communication while demonstrating beginning technical understanding of group theory characteristic of the university level
\item Work in teams and take on leadership roles that exploit personal their strengths  
\item Identify their interests and strategize creatively as learners in designing insightful solutions
\end{itemize}
by end of the mincourse.  
\item \textbf{Suggested Pre-reqs:} Some familiarity with proofs, combinatorics, or NT is a plus
\item \textbf{Suggested Anti-reqs:} Completion of a first course in abstract algebra covering group theory
\item \textbf{Date/Duration} (tentative): timeframe between March - April, $\sim$ 2 weeks
\item \textbf{Online Tools:} Discord, MS Teams, Miro, Google forms, Padlet, Slido 
\end{itemize}
\section{Learning Goals, Objectives, and Standards}
\subsection{Goal:} Participating in this mincourse, advanced HS students will be introduced to, understand, and apply basic finite group theory and requisite algebraic concepts through the lens of favorite childhood pastimes.
\subsection{Objectives:}
\begin{itemize}
\item  Familiarize themselves with new concepts in algebra with crafting as a computational aid
\item Independently navigate and apply knowledge from resources provided
\item Connect new definitions of group theory to physical math-making
\item Appreciate the aspect of discovery in “toying” with and creating math
\item Move from school-level algebra to algebraic structures, becoming more comfortable with the abstractness involved
\item Develop soft skills to comfortably tackle hard problems, approach new concepts, and reflect on their reason for learning

\end{itemize}
\subsection{Standards}
Students will:
\begin{itemize}
\item Visualize and
hypothesize to generate
plans for ideas and
directions for prototyping/designing a toy, game, or play experience illustrating and/or applying their mathematical learning.  They will apply diverse methods in their innovation within the scope of algebraic themes. 
\item Make multiple origami pieces that explore group theory. They will demonstrate understanding of freedom and responsibility in choosing how to approach their culminating projects.  
\item Explore how physically crafted models can enhance and empower learning.  They will reflect on their learning towards the end of the course and re-engage with mathematical concepts in a novel way via the vision they develop for their culminating projects. 
\item Justify and critique their own and others' origami solutions via collective Miro galleries and logical/mathematical arguments. 
\item Investigate mathematical concepts via origami  and compare/contrast their process with others.
\item Collect a repository of notes/thoughts/learning to impact their understanding.  
\item Empathize with "the user" in developing their projects, analyze their audience's responses, and determine the commonalities of that group which may make their product successful. 
\item Critically analyze and question others' approaches to learning and making. 
\item Critically evaluate other teams' projects based on the criteria set forth by the "judging rubric".  
\item Make useful and/or meaningful things in synthesizing their knowledge.
\item Assess impact of culminating projects. 
\end{itemize}

\section{Recommended Student Prep Resources}
\begin{itemize}
\item \href{https://www.youtube.com/watch?v=EsBn7G2yhB8&list=PLDcSwjT2BF_VuNbn8HiHZKKy59SgnIAeO}{Essence of Group Theory by Mathemaniac}
\item \href{https://cocalc.com/share/public_paths/89e8db18f6cbfcdac13b409f2745c9e157b2ea9b/Rec%20Problems%20Column%2FChoosing%20Symmetry%20Draft-1%2Fmain.pdf}{Choosing Symmetry article by Stephanie A.} (a problems column I wrote - can revise for TB)
\item \href {https://nathancarter.github.io/group-explorer/}{Group Explorer} and \href {https://web.osu.cz/~Zusmanovich/teach/books/visual-group-theory.pdf}{Visual Group Theory} Book
\item \href{https://origami.me/easy-origami/}{Beginner Origami}
\item Student Materials Needed:  origami paper in at least 4 colors, writing utensil, stable internet connection, tape (optional, recommended), cotton balls (optional, recommended)
\item Further Resources:
\begin{itemize}
\item \href{https://math.umd.edu/~jcohen/402/Pinter%20Algebra.pdf}{A Book of Abstract Algebra} by Charles C. Pinter
\end{itemize}
\end{itemize}
\section{Course Plans}

\subsection{Symmetry and Squares}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\includegraphics[scale=1]{figures/gens.png}
\caption{We discover symmetries and determine the generators for $D_4$.}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.05]{figures/D4frnt.jpg}
\caption{We will define the presentation for $D_4$.}
\end{figure}

\begin{figure}
\centering
\includegraphics[scale=.05]{figures/D4table1}
\caption{As well as fold/write out the corresponding Cayley table.}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.05]{figures/D4table2}
\caption{Note how the elements axis flap over the table.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsubsection{Prep}
\begin{itemize}
\item \href{https://www.imaginary.org/film/group-theory-live-performance-mathlapse}{Group Theory Live Performance Video}: \emph{How would you describe symmetry? What do you think is happening in this video?}
\item Come to session with at least 2 sheets of origami paper, one prefolded into an $8 \times 8$ grid

\end{itemize}
\begin{abstract}
We will gently introduce groups via paper folding of an origami paper square - the symmetry group of the square being the dihedral group $D_4$.  We will also figure out its group presentation as well as fold its Cayley table.     Specifically, we wish to be guided by
\begin{itemize}
\item How can we view symmetry from a more mathematical standpoint?
\item What are some examples of transformations?
\item How can we use transformations to explore the symmetries of the square?
\item Are there compositions of transformations which can form the same symmetry?
\item Can we find all the symmetries of the square?
\item What is the order of this group?
\end{itemize}
Materials: 2 sheets origami paper, writing utensil\\
 Keywords/Vocab: Transformations, Invariance, Symmetry, Symmetry Group, Dihedral Group, Cayley Table, Group Presentation, Cayley Table, Group Presentation
\end{abstract}
\subsubsection{Consolidate/Assess:}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\includegraphics[scale=.05]{figures/bookmarks}
\caption{Origami corner bookmarks}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}


\item Fold 4 origami \href{https://www.gatheringbeauty.com/blog//2017/11/make-your-own-origami-corner-bookmarks.html}{origami corner bookmarks} and identify a \href{https://proofwiki.org/wiki/Dihedral_Group_D4/Subgroups#google_vignette}{subgroup of $D_4$} with each, writing out the generator(s) with the corresponding set of elements.  

\item Place 4 bookmark corners around stack of origami paper and rotate.  Attempt to describe the subgroup you are modeling and the permutation of 4 corners.  .  Place 4 bookmark corners around stack of origami paper and rotate.  Attempt to describe the subgroup you are modeling and the permutation of 4 corners.  
\item Post pics of completed bookmarks on TB \href{https://miro.com/?gclsrc=aw.ds&&utm_source=google&utm_medium=cpc&utm_campaign=S|GOO|BRN|US|EN-EN|Brand|Exact&utm_adgroup=&adgroupid=140324301985&utm_custom=18258206285&utm_content=668037264239&utm_term=miro%20board&matchtype=e&device=c&location=9030970&gad_source=1&gclid=Cj0KCQiAhbi8BhDIARIsAJLOludk3XQGZmcw09Mq--BkTa3nhFhJhLGtqePNzdD9S923z4Qt5FWfS_4aAuVlEALw_wcB}{Miro} gallery.

\end{enumerate}





\end{itemize}
\subsection{Finite (cyclic) Groups and the Infinity Cube}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
\centering
\includegraphics[scale=.05]{figures/homomorphism.jpg}
\caption{$\mathbb{Z}/4\mathbb{Z}^+$ and $\mathbb{Z}/5\mathbb{Z}^\times$} replaced with arbitrary operation * and the set of elements $\{e, a, b, c\}$
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.05]{figures/cayleyhomomorphsim.jpg}
\caption{Theire Cayley tables are the same!  We have a group homomorphism.}
\end{figure}

\subsubsection{Prep:}
\begin{itemize}
\item  \href{https://www.imaginary.org/snapshot/computing-with-symmetries}{Oberwolfach Snapshots: Computing w/Symmetries}
\item \href{https://www.youtube.com/watch?v=N11vOq3h9XQ}{Quotient Groups by Mathemaniac}: \emph{Begin to conjecture how we will study the Rubik's Cube  within the coming days of this course.  Consider how its group structure might influence possible configurations, as well as what its normal subgroup might be.}
\item Come to session with five $4 \times 4$ prefolded sheets of origmami paper and  a writing utensil
\end{itemize}


\begin{abstract}
We further explore the concept of a finite group via cyclic groups, proving   $\mathbb{Z}/4\mathbb{Z}^+ \cong \mathbb{Z}/5\mathbb{Z}^\times$ via folded Cayley tables in preparation for determining the group of the infinity cube.  We consider 
\begin{itemize}
\item What are the elements of  $\mathbb{Z}/4\mathbb{Z}^+$ ?
\item What are the elements of $\mathbb{Z}/5\mathbb{Z}^\times$?
\item How do we generalize operations and elements? 
\item What does it mean when 2 groups are "the same"?
\item What makes these groups "cyclic"?  
\item How is $\mathbb{Z}/\mathbb{Z}4$ similar or different to $D_4$?
\item How can can we "cycle through" (i.e. permute) elements in a group?  
\end{itemize}

Keywords/Vocab: cyclic group, quotient group, cyclic graph, cycle permutation/$k$-cycle, generators, group homomorphism  
\end{abstract}
\subsubsection{ Consolidate/Assess: Build an \href{https://jonakashima.com.br/2019/07/14/origami-infinity-cube/}{Infinity Cube}}

\begin{itemize}



\item Materials: 

\begin{itemize}
\item 8 sheets of origami paper in two different colors or patterns (4 of each color)
\item 8 cotton balls (optional, recommended)
\item tape (optional, recommended)


\end{itemize}
\item Consider: \emph{How does your infinity cube model a group?}
\item Post pics of completed origami infinity cubes on TB \href{https://miro.com/?gclsrc=aw.ds&&utm_source=google&utm_medium=cpc&utm_campaign=S|GOO|BRN|US|EN-EN|Brand|Exact&utm_adgroup=&adgroupid=140324301985&utm_custom=18258206285&utm_content=668037264239&utm_term=miro%20board&matchtype=e&device=c&location=9030970&gad_source=1&gclid=Cj0KCQiAhbi8BhDIARIsAJLOludk3XQGZmcw09Mq--BkTa3nhFhJhLGtqePNzdD9S923z4Qt5FWfS_4aAuVlEALw_wcB}{Miro} gallery.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\includegraphics[scale=.05]{figures/inftycube_ex.jpg}
\caption{Origami Infty Cube}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.05]{figures/InftyCubeGroup.jpg}
\caption{The group of the inifinity cube is isomorphic to cyclic group $\mathbb{Z}/\mathbb{Z}6$}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.05]{figures/Z6table2}
\caption{Students will consolidate by determining what group their crafted infinity cube models and folding/writing out the corresponding Cayley table for  $\mathbb{Z}/6\mathbb{Z}$}.
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\subsection{The Rubik's Cube Group}
\begin{figure}
\centering
\includegraphics[scale=.25]{figures/cubelayout.jpg}
\caption{We will model the Rubik's Cube in our session.}.
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.10]{figures/spinner.jpg}
\caption{The Rubik's Square Spinner group is isomorphic to $\mathbb{Z}/4\mathbb{Z} \rtimes \mathbb{Z}/4\mathbb{Z}$.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Prep}
\begin{itemize}
\item Print out/bring \href{https://lang.just-coloring-pages.com/educational-and-culture-coloring-pages/rubiks-cube-coloring-pages/rubiks-cube-coloring-pages-24/}{Cube Coloring Page}, markers,  tape, brain
\end{itemize}

\begin{abstract}
We introduce the Rubik's Cube as a group and define its valid configurations via a 4-tuple design demonstrated by coloring a paper cube. 
\begin{itemize}
\item How do we define the Rubik's Cube as a group?
\item How do permutations relate to $S_n$?
\item How can we describe our cube moves?  
\item What data do we need to determine the configuration of a cube?  
\item How do we determine the \emph{valid} configurations of the cube? 
\item How can we we illustrate this data?
\end{itemize}


\end{abstract}

Keywords/vocab: Permutation Group, Group action, Alternating Group, conjugacy class,  Normal subgroup, Direct sum
\subsubsection{Consolidate/Assess: Build a \href{https://www.youtube.com/watch?v=IPE49GfXdVw}{Rubik's Square Spinner}} 
\begin{itemize}
\item Materials: 1 sheet origami paper
\item Consider: \emph{How can we mathematically model this (origami and algebraic) structure?}
\item Post pics of completed cubes and spinners on TB Miro
\item Post pics of completed cubes and spinners on TB Miro gallery.
\end{itemize}


\section{Culminating Projects}
 \subsection{Reflection:} 
 Consider these resources:
\begin{itemize}
\item \href{https://artofproblemsolving.com/blog/articles/knowing-vs-understanding-rubiks-cube-difference}{Knowing Versus Understanding: How the Rubik’s Cube Taught Me the Difference}
\item \href{https://coffeeandjunk.com/knowing-something/}{Feynman's Thoughts} 
\end{itemize}
and reflect on your learning and \emph{how you can apply it} via google form.
\subsection{Prompt and Resources}
Creativity and innovative
thinking are essential life
skills to be developed!
\begin{itemize}
\item Design a toy, game, or play experience applying or teaching the algebraic concepts you have learned throughout this course
\begin{itemize}
\item Resources:
\begin{itemize}
\item \href{https://www.youtube.com/watch?v=QvuQH4_05LI}{Tips to be a better Problem Solver by 3b1b}
\item \href{https://web.stanford.edu/~mshanks/MichaelShanks/files/509554.pdf}{Stanford d.school: Design Thinking}
\item\href{https://www.gse.harvard.edu/ideas/news/19/05/designing-dream-toy}{HGSE: Designing the Dream Toy}
\item \href{https://www.unicef.org/southafrica/partnerships-children/lego-foundation-learning-through-play}{UNICEF $\times$ LEGO: Learning through Play }




\end{itemize}
\item Inspiration
\begin{itemize}
\item \href{https://mathforlove.com/}{mathforlove curriculum}
\item \href{https://www.youtube.com/playlist?list=PL5jDTd07plNBkAhTmX7ZJjKJNEf-W579j}{modular origami}; \href{https://www.youtube.com/playlist?list=PLAC715B71FABCF06C}{origami diagramming}; \href{https://www.karakuriworkshop.com/projects-8}{Karakuri Gallery}
\item \href{https://www.imaginary.org/physical-exhibits}{IMAGINARY Exhibits}
\item \href{https://www.media.mit.edu/groups/lifelong-kindergarten/projects/}{MIT Lifelong Kindergarten Projects}

\end{itemize}
\item Playtesting Session/Student Showcase
\item Final \href{https://padlet.com/account/setup}{Padlet} Gallery
\end{itemize}
\end{itemize}
\begin{figure}
\centering
\includegraphics[scale=.75]{figures/rubiksrubric.png}
\caption{Sample rubric used to "judge" culminating projects}
\end{figure}

\section{Documentation}
\begin{itemize}
\item "Galleries" of origami and culminating projects (instagram?)
\item Record live sessions (youtube)
\item Create slideshow presentation for Prof. (mandatory - \textbf{Due 5/01/25})
\item Prepare documentation of experience for college interdisciplinary studies showcase in May 2025 (optional)
\end{itemize}
\begin{figure}
\centering
\includegraphics[scale=.75]{figures/presdirections.png}
\caption{Guidelines for college slideshow presentation - will be introducing TB as an organization}
\end{figure}
\cite{alg}
\cite{TL}
\cite{cube}

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