We began our day by playing "Liar's Bingo," a game that utilized sequences of six different numbers that were chosen in order to follow a very mysterious rule. The process can be described as follows:
Each group of students sat together with some of these six number sequences, attempting to find any obvious patterns between them. The numbers were either contained in a red box, or a black box, suggesting some relevant relationship between color and the number contained. Since no student found any predictable pattern, we were able to safely have volunteers come to the front of the room to Eliana where the student needed read the sequence of colors on their strip, but also lie about precisely one of the colors in their sequence. from this information, Eliana was able to deduce the exact value of the number that was lied about.
An example strip from "Liar's Bingo"
After successfully guessing many of the numbers, students then attempted to guess what the "trick" was behind Eliana's odd ability to read their minds! One student in the back row suggested that the colors were actually being assigned values such as "0" for red, while another student suggested that perhaps the sequence of numbers could be subdivided into groups of three. Both of these suggestions lead to an entire class that had the answer on the tip of their tongue! However, the morning logic warm up concluded and students shuffled out of the room and into their other classes speaking to each other about what the trick might be.
Hint: Consider the fact that every number in liar's bingo had either one or two digits. Now take a look at two distinct groupings of three numbers whose color seems to represent a particular value. Keep place value in mind! Hmm...
Here is a link where students can find their own "Liar's Bingo" cards:
In geometry, Group A started with a hexagon template and were assigned to create their own design within the confines of the template. Here, the expectation was that students would use geometric principles in order to make shapes within the hexagon using only a compass and straight edge. Thankfully they had also been working on this technique in art class and were able to create many interesting designs.
Group B on the other hand used all of the tools they had gathered during the week in order to construct a "Pi Sandwich." Here, the goal was to understand how ancient civilizations were able to calculate Pi to an arbitrary number of decimal places! The stock answer for many middle schoolers in terms of how one might obtain the value of Pi is by dividing the circumference of a circle by its diameter. However, Group B should now understand that you can't actually measure an irrational circumference to any degree of accuracy using this method. Instead, the group worked together with Frances in order to show that one can "sandwich" the circumference of a circle in between two hexagons (one that circumscribes the circle, and the other is inscribed.) This means that students were able to find lower and upper bounds for the exact value of Pi. Unfortunately, the Hexagon method seems to be insufficient as the bounds for Pi are still quite large with this method.
The first step of deriving Pi
With this understanding, students then worked in smaller groups to implement a similar method with a dodecagon (A shape whose perimeter is closer to the circumference of a circle.) While some groups didn't come to an exact numerical estimation of Pi with the dodecagon, many of them were able to see that one could use this method for an arbitrarily large "n-gon" and get the value of pi to as many decimal places as they had the patience for.
In graph theory, students were given an assortment of problems concerned with "tree graphs" and "bipartite graphs." Unbeknownst to many students, these are actually very applicable forms of mathematical organization. A bipartite graph is a graph that can be partitioned into two different categories in which each "vertex" from one category is related to a respective vertex in the other. A natural formulation of a bipartite graph is to think of a soccer game. Imagine one category that is the team to which each player belongs, and another category that consists of the players themselves. Draw an edge between a player and their team if and only if the player is in that team, and you will have a bipartite graph!
Here is another natural example of a bipartite graph:
Complete Bipartite graph
While trees and bipartite graphs are valuable tools to have, the techniques that students had to develop in dealing with these nuanced concepts might actually prove to be equally important. The act of actually proving some properties of trees and bipartite graphs (for example, that every tree is bipartite) is at once a difficult question and also a necessary one for any serious mathematician. For example, students learned about "if and only if" statements, the basic premises of "induction," and the use of definitions to form a "proof by contradiction." Many students were alarmingly talented at finding counterexamples and using their own intuitions to decide on whether or not an idea was correct. The students worked in groups and were not only kind to each other but were able to actively articulate their thoughts so that everyone was able to understand how they reached their conclusion. In many ways, the last day of graph theory only continued to prove that children are far more capable than many adults would like to give them credit for! Congratulations to everybody who worked on some very good questions today.
Some additional problems to consider:
1) Explain why or why not every Eulerian bipartite graph has an even number of edges.
2) A Spanning Tree is a tree graph that connects every vertex of a connected graph G. Explain why every connected graph must contain a spanning tree.
3) Think of some real-life examples of a tree graph!
In mathematical art, all of this week's work resulted in the completion of each student's personalized three dimensional shape! Students on their final day began to seem like very independent artists, setting up their own drafting space and working steadily to complete their projects for this week. For students who were unable to finish, you all know the process now! See if you can finish your shapes at home and think about what materials you might need.
Some of the artwork done by students
Some additional projects to consider include:
1) A small exercise in pythagorean color theory is the idea of how you can color the faces of each individual shape without any two adjacent faces containing the same color. The challenge here is to consider the minimum number of colors necessary to accomplish this goal.
2) A very important intersection between graph theory and the polyhedra we worked on this week is how each polyhedron can be drawn in a two dimensional graph! We can look at every corner of the polyhedron as a vertex, and every side of the shape as an edge that connects two vertices. Here is the graph for some familiar shapes, namely the hexahedron and tetrahedron:
As you guys can probable imagine, this can get very complicated very fast! Try doing the graphs for some of the other polyhedra we worked with this week. More importantly though, try to take your own shape and draw a graph of it.
**Challenge problem: Use some of the basic polyhedra that you know and draw their graphs. See if you can relate the number of faces, edges, and vertices in each graph.
In computer science, the students also completed their work with graphics and animation. As is apparent to anyone who saw their work, many of the students worked very intensively in order to create works of sophistication. More impressive is that students were able to learn an entirely new language and create something with it. Hopefully all of the projects have been shared with parents, and the students will use all of the language (as well as the downloadable software) they have learned about this week and continue to work on their craft.
One of the many computer science projects
All in all, it was a very successful day, and showed how valuable even a week can be in the development of any academic pursuit. I hope that all of the students continue to work independently but also strive to stay in touch with their like-minded peers. I also hope to see many of you back next year for another week of exploration and growth!
Good luck to everyone,
P.S If anyone has questions about any of these problems (or just want to talk about any of the math they're doing during the academic year!) you guys can all reach me at firstname.lastname@example.org.
have fun with these interesting questions and all of the other great questions you can get your hands on this year.
A final goodbye, till next year!