**October 8, 2015: Paper folding and Salt Mounds
**This month we first explored how a collection of lines can seemingly make a curve. We plotted a point on wax paper then successively made creases by folding points on a common edge of the paper to this original point. After enough folds, it appeared that we made a curve, which was conjectured to be a parabola. You can investigate this dynamically here.

We then took various planar shapes and poured salt on them to form salt mounds. These mounds formed 3-D solids with different salt ridges. We tried to conjecture where the peaks would occur, where the salt ridges occurred, and how symmetry influenced these ridges. You can investigate the ridge lines for a triangle here. Special thanks to Park City Mathematics Institute for this activity idea.

**September 8, 2015: Binary Representations
**This month’s focus was different applications of bases, specifically binary (or base 2). We started by looking at “Coins in Two-Land“, where currency comes in powers of 2: 1, 2, 4, 8, 16, 32, 64, etc. We posed questions about a system where we could only use each currency value at most once and found ways to represent small values like 5, 17, and 21 using these coins. We then developed three different algorithms to represent any value in terms of these coins. We also considered how this system would differ if we were allowed to use coins at most twice instead of at most once.

We also looked at Liar’s Bingo, which consists of strips of cards that each possess 6 2-digit numbers, some red some black. We asked participants to first try to “order” the cards, which was intentionally ambiguous since there were many ways to think of ordering (such as lexicographically, by total, by maximum value, etc.). We then did some “magic” by having a participant read the colors from left to right but intentionally lying on one color. For example, if the cards showed BRBBRB (black – red – black – black – red – black), then the participant might lie as follows: BRBBR**R**. Magically, the value where the lying took place was determined. Participants tried to figure out the math behind the “magic” and then explored other questions such as how once could engineer their own deck. Read more about this activity on page 18 from the MTC Newsletter from our Ohio Circle colleagues Bob Klein and Steve Phelps!

**April 14, 2015: A Cut Above the Rest**

This month’s focus was looking at dissections of shapes. We started by analyzing the prompt, “divide each shape below into two equal shapes”. These were from Matt Enlow’s Twitter page.

Groups explored what it meant to “divide”, what it meant to be “equal”. For example a cut could be curved, a single straight cut, or a polygonal cut along the grid lines. In addition, equal shapes could either refer to congruent shapes, those of equal area, or those of equal perimeters. We decided to explore those shapes of equal area. Three main questions were additionally posed by the group with the definitions stated above:

1)** How many** possible cuts divide the shape into two equal areas? Do all of these cuts go through the same point? [It was determined that not all cuts are coincident].

2) Is it possible to make a cut that divides the shape into two equal areas **and** perimeters? [Groups felt that this was possible but did not have an algorithm for it]

3) Is it possible to have a shape that when cut into two equal areas will always result in **more than two pieces**? [Groups looked at shapes like spirals but did not have a specific answer]

After this warmup was explored, we looked at a related problem called the “Brownie Problem”. While a tray of brownies is cooling, a thief removes some of the brownie. Divide the remaining portion into two equal areas with a single cut. Groups talked about the connections between this problem and the warmup and the difference between *constructing* a cut and proving that such a cut *exists*. We concluded the session with reviewing what mathematical practices were evident in these tasks and the balance between exploring specific cases and generalizing. Plus we ate some brownies for dessert!

**March 11, 2015: Derangements**

An *arrangement* is when *n* objects are placed in some order. There are *n*! or (*n*)(*n*-1)(*n*-2)…(2)(1) such arrangements. But how many ways are there to rearrange *n* objects to that no object returns to its correct position? For example, for the set {1, 2, 3}, only {3, 1, 2} and {2, 3, 1} are derangements.

To start off thinking about such counting problems, we considered a classic premise: 100 people go to board a plane with 100 seats. Everyone knows where they are to sit; however, the first person decides to sit in any of the 100 seats at random. Every other person that follows sits in their seat if available or sits at random otherwise. Before solving a particular problem, we generated questions about this premise. Our questions can be found on this Padlet wall. We quantified these questions as follows:

**Level 1: Already know the answer
**

**Level 2: Don’t know the answer but know an approach to solving**

**Level 3: Don’t know the answer and don’t know how to get started**

After discussing these questions and seeing themes of the types of questions asked, we explored the following: “What is the probability that the *last person* is in their correct seat”? Many groups started analyzing this problem in smaller cases, surprisingly finding the answer to be consistently 1/2 regardless of 2, 3, 4, or 5 people on the plane. Some used tools strategically such as a spreadsheet, to be precise in their organization. One participant even provided a recursive limit argument showing that the probability was 1/2 regardless of the number of people.

With the warmup in place, we explored the general problem of the probability of a derangement of a set of size *n*. To make sense of the problem, each person was given a unique hat. We then each selected a hat at random and determined if such a collective choosing was a derangement. Groups started by looking at smaller cases and looking for any patterns, such as in the table shown. While groups did not find an exact numerical pattern, they were confident that the limiting probability would be somewhere between 1/3 and 1/2.

It turns out that the limiting probability of a derangement as *n *goes to infinity is 1/*e*, where *e* is Euler’s Constant. This is approximately 36.8%. For more on derangements, check out this link.

**February 10, 2015: Midpoints on Quadrilaterals and Optimal Rulers
**

Draw any quadrilateral and connect consecutive midpoints. What do you notice and what do you wonder about the resulting shape? What if you do this process yet again? This was our warmup problem for the night, which led to many different observations and approaches to try to justify these observations. In particular, teachers wondered if the resulting shape was always a parallelogram. There were many different arguments provided as to why this must be so ranging from dynamic geometry, tessellations, coordinate geometry, and triangle similarity. There was a nice discussion about what specific mathematical tools are most useful in this investigation.

The main problem of the night was to make a ruler more optimal. Rather than having marks at every inch on a ruler, could some marks be erased and the ruler still be used to measure all lengths as before? Participants looked for patterns with smaller cases and some lower and upper bounds were determined to help narrow the search. Some groups were able to articulate some algorithms for trying to generate marks in the best locations.

**Pictures:**

**January 13, 2015: Trains and Number Partitions
**

The primes are very important as they are the multiplicative building block of natural numbers. That is, 28 can be written

*uniquely*as the product 2*2*7. However, what about writing 28 as a sum of natural numbers? Well, 28 = 1 + 27 (or 27 + 1) or more colorfully, 28 = 3 + 5 + 7 + 7 + 6, say. Essentially, how many ways can a natural number be expressed as a sum of other natural numbers? How does the order in which these addends are written affect the answer? To start understanding this question, we explored a warmup regarding Leo the Rabbit (slide below). Leo can hop upward either 1 or 2 steps each turn. How many ways can he ascend up a staircase of 10 steps?

**Pictures:**

**November 12, 2014: Graph Theoretic Games**

This month we explored graph theoretic games. When we think of a graph, we often think of a plot of a function or perhaps a geometric figure. However, a graph really is a network of points (or nodes) and segments (and edges). We played two specific games called Criss-Cross and Sprouts, both of which have players take turns drawing an edge. We explored the following questions:

1. Do both games end?

2. How can you predict who wins based on the starting configuration?

3. Is there a minimum or maximum number of moves giving the starting configuration?

4. How are the two games related?

**Handout: **Graph Theoretic Games

**Pictures:**