All about math department projects and events.

US Mathematics Students of the Trimester – Fall 2016

I am excited to announce a new award, the Upper School Mathematics Students of the Trimester!


Each math faculty member was free to choose whichever student of theirs they thought best exemplified what they are looking for in a model mathematics student. The official description of the award is as follows:

“Awarded to students who exemplify the math department’s core values of competence, confidence, and perseverance while helping their peers realize the relevance and importance of an exceptional mathematical education both for its beauty and for its practical application.”

The following students have been recognized as the Upper School Mathematics Students of the Trimester for Fall 2016.

Please join me in congratulating these outstanding mathematics students!

Cao,-AndyAndy Cao – Andy’s in depth thought, attention to detail and willingness to tackle the most difficult problems has made him an excellent Geometry student. He has shown a genuine interest in all of the material. His approach to class has helped push his classmates to excel along side of him.
Cavanaugh,-JuliaJulia Cavanaugh – I chose Julia as my student of the trimester for her work inside and outside of the classroom. Outside of the classroom she is a dedicated student who does her very best every class to prepare herself for our upcoming lesson. During class she is always focused, eager to participate, and extremely willing to help her peers out with their own work. She is a is truly a model student!
Fieldman,-ArvinArvin Fieldman – Many students work hard to find success. Arvin finds success and then some. He is not satiated with even strong understanding of material. He must dominate each topic. His work ethic and abilities have shown through in his careful yet creative group work as well as his ridiculously high and consistent quiz and test scores. Bravo Arvin.
Grabowski,-KyleKyle Grabowski – Kyle had a remarkable first trimester at Williston in geometry. His work is neat, thorough and incredibly accurate. He has a strong natural aptitude for mathematics and he thrives on the most demanding questions given both in class and on tests. On more than one occasion this fall, Kyle was the only student in two sections of geometry to get a challenge question correct. His inductive reasoning skills are impressive! 
Kim,-SimonSimon Kim – Simon has great energy in class everyday. He is always prepared and willing to help those around him. He is quick to answer questions and come up to the board. Simon is an all around great student who raises the bar everyday.
Marion,-SarahSarah Marion – Sarah is one of the most enthusiastic, most energetic students I have ever met. She is always the first student to ask a question; she stays after class to go over any concept she had difficulty with; and she is always prepared for class with her books and homework out in front of her almost as soon as she gets in the room. She has had a ton of success in Topics, mostly because of her diligent, conscientious approach to the material. It has been such a pleasure working with her this year!
Mizobuchi,-SomaSoma Mizobuchi – Soma’s work shows that he has set high standards for his academic performance. His Engineering Notebook is extremely complete and serves as a fine reference for his Engineering Labs and his code development. In writing programs, his code, and its logic, is always well thought-out and his program documentation is outstanding. Congratulations to Soma for a very impressive trimester!
Monaco,-GabbyGabby Monaco – Gabby has taken on a great challenge this year moving into AP Statistics from TPS. She is one of the most persistent students and works very hard to understand difficult concepts. She is not discouraged by setbacks, but resolves to work harder on the next challenge. I am hopeful that, with this continued level of effort, she will make great progress this year and continue to model strong study habits and the kind of determination that leads to success and innovation in the field of statistics.
Record,-GabbyGabby Record – Gabby did an outstanding job this fall in Calculus. She was always prepared and engaged in class. Her positive attitude along with interesting questions brought great energy to the class. I’m looking forward to her continued great work.
Spiegel,-AndrewAndrew Spiegel – Andrew is a talented programmer who accepted the challenge of applying is already significant background and skill to a new context, game design. Rather than executing a safe or simple final project, Andrew pushed himself to attempt a highly challenging game that relies on random room generation. The result was an amazingly polished and enjoyable project that I hope he will continue to build upon!
Thibault,-XavierXavier Thibault – Xavier has been an extremely strong AP Calculus student. He has very good insights into the material which he is always willing to share with the class. He learns material very quickly and is able to apply his knowledge well. He helps keep our class moving forward as he is always willing to participate in our discussions.
Zawacki,-MollyMolly Zawacki – Molly has been a phenomenal addition to our Discrete math course. She has shown leadership in the class, whether it is stepping up to present a hard solution or explaining her thinking to her peers. I can count on Molly to go out on a “mathematical limb” and offer new ideas, even if she is not sure they are correct. Mathematically, she has developed into a beautiful proof writing; creating clear and easy to read explanations for her work. All in all, she has really embraced the core goals of the course; to explore, conjecture and prove.

Past students of the trimester can be found right here: Fall 2013, Winter 2014, Spring 2014, Fall 2014, Winter 2015, Spring 2015, Fall 2015, Winter 2016, Spring 2016

Steiner Point Construction

This just in from Mrs. Hill:

Topics in Discrete Mathematics students got to review some lessons from geometry when they learned how to construct the Steiner Point of a triangle. The Steiner Point in a triangle is the point from which three branches lead out to the triangle’s vertices at perfect 120° angles. (See diagram below.) This Steiner Point has important modern day applications for creating the shortest (and therefore cheapest) possible fiber optic network between any three locations. If a triangle’s angle measures are all less than 120°, then the Steiner Point can be found inside the triangle using a geometric construction first developed by Italian mathematician Evangelista Torricelli in the early 1600’s. Torricelli (most famous for his work in physics but also an accomplished mathematician) was a protégé of Galileo, and in 1641 succeeded Galileo as the court mathematician to Grand Duke Ferdinand II of Tuscany.

Figure 2
Figure 2

Torricelli’s method for finding the Steiner Point in a triangle requires only the use of the traditional geometric construction tools – straightedge and compass. Using properties of equilateral triangles and inscribed angles from elementary Euclidean Geometry, the basics of the construction are shown below in Figure 1 (a,b,c). The students in Topics further learned how the Steiner Point was used in 1989 to connect Hawaii, Japan, and Guam via the Third Trans-Pacific Cable (TPC-3). Also below, Figure 2 shows a stamp issued by the Japanese Postal Service to commemorate the completion of TCP-3. As one can see from the picture, the three undersea cables meet at 120° angles under the western Pacific Ocean.

Figures 1a, 1b, and 1c
Figures 1a, 1b, and 1c

Student constructions:

Dispatch from Calculus

CalculusThis just in from Mrs. McCullagh:

Teaching Calculus to seniors and a few juniors, I feel an obligation to help move them toward independence and self-sufficiency in their learning. I want them to learn to support themselves as learners and know how to reach out for assistance. I use two primary methods to this end.

First, I provide full solutions for all homework. Students are expected to use these solutions to check their work as they complete each problem to be sure they not only have the correct answer, but more importantly, that they have supported their work appropriately. The other way they can use these solutions is as a hint on how to start a problem if they just need a little help. No one should come to class with a blank homework saying they did not know how to do the work. With this, they know when they need help and are expected to ask for it.
CalculusMy second strategy for student increased independence is to have all students at the boards at the same time to do problems together. By being visible, at the boards, they can and should look to others around them to confirm they are headed in the right direction. Each student becomes a source of information for everyone else. Students who might not take the lead sitting at their desks are now asked for help by their peers. Again, no one is left unable to start a problem. Help is all around.

Math Classroom Success Stories (16-17, T1)

At a recent math department meeting each teacher was asked to write about something that went particularly well in the first trimester. Here’s what they wrote!

Ms. Evelti: I had a student who came into my Video Game class reluctantly, unsure if she would be interested in the work. She ended up really excelling in the class both in the technical and creative aspects of the work. She brought humor and visual interest to the stories behind her games while challenging herself to include difficult interactive elements in her projects that extended and deepened her understanding of the topics we covered in class.

Mr. Seamon: As we moved into a different system for graphing (polar coordinates), I worried about the transition. It’s a reorientation of how to look at the basic space we’ve been working in and it’s been a challenge in the past to communicate the new “up” and “down”. This year I tried bringing in a scene from a science fiction classic (Ender’s Game) and it went over quite well, even though most of the students hadn’t read the book! Having a concrete picture of our new space for differentiation and integration has translated into a deeper understanding on the part of the students which has been expressed through impressive board work and high quiz scores.

Ms. Schneider: One of my favorite things to do in class is play a review game. Although I made up the game myself, it is similar to jeopardy where the students pick questions of different difficulty within a topic. The students are split up into teams, and if one group answers a question incorrectly other groups have an opportunity to steal the question. I absolutely love this game because the students work so well together in their groups and are extremely invested in each problem. They have smiles on their faces the entire time as well as they work meticulously to complete the problem within the time frame. The pure exhilaration of getting a question correct or having the opportunity to steal a question brings such a positive energy to the classroom. Every test that we have my students get excited because they know that means we get to play the “review game” the lesson beforehand.

Mrs. Conroy: It has been a treat to return to the Geometry classroom. The biggest change in this class over the past three years has been the use of technology. Now that each student in the class has their own surface loaded with the geometer’s sketchpad software, the variety of classroom activities available to the class are remarkable. Each day feels different. We are discovering geometry through investigations, constructions and traditional class framework notes. My ability to project figures from a variety of sources has led to a much more efficient classroom. Students can see examples in one note as well as on the board and we are able to spend so much more classroom time doing problems. This has not gone unnoticed by my students. They enter class wondering what will we be doing today. Some things never change. Students love to find the missing angles but proofs remain a challenge!

Mrs. Hill: My Topics class can be a bit of a raucous group. The students are all seniors who, for the most part, have not all had great success in mathematics. In this course, however, we are focusing on political and societal applications of mathematics, and the “math” kind of sneaks in under the radar. A young woman in that class has struggled in past math courses at the school, but has had tremendous results in this one due to her intense work ethic and willingness to participate. She talks about how she really understands the relevance of this course and can appreciate how math is used in the “real world.” It is so wonderful to see a person who, before now, has not seen a use for mathematics discovering how it can be relevant to her life.

Mrs. Whipple: During a recent lesson on proving congruent triangles, students in my geometry honors class where given a new type of problem using overlapping triangles. They were put into groups and sent to the white boards to work together to come up with the most efficient ways to prove that certain triangles were congruent. Afterwards, we talked about all the strategies that each group used in tackling the problem and which worked best. After sharing all their ideas and observations, they were given another extremely hard proof to work on together. Not only did they use the strategies that we talked about but the majority of the groups commented on how “this problem was much easier”, when it was actually much more challenging.

Ms. Briedis: In a recent class we were beginning a lesson on composite trig functions. The lesson started with absolute value functions and the students were amazed by how the absolute value of a trig function changed the way the graph looked. We began playing with trig functions such as f(x)=(x^2+1)sin(2pix), and they thought the graph was the incredible. The amazement on their faces was exactly what teachers thrive on. We began playing with different functions on, and each student began creating their own functions and then would share them with the class. We would then work on what the two functions would be that the overall function oscillated between. It was a really fun lesson that the students connected with. They were engaged and excited about the different functions they were creating and seeing from others. It was an overall thrilling time to see them so inspired about graphing.

Mrs. Baldwin: Our class has been investigating random phenomena through use of examples and simulations. The students are doing a great job figuring out what makes a process truly random as opposed to arbitrary or haphazard. We have been noticing that the word “random” is used often in a casual sense in everyday language and have begun to recognize cases where the word is used inappropriately. Students did a great job with a recent project in which they found a probability estimate through a little research and conducted a simulation in which they used a random number generator (or table) to conduct repeated trials. One example involved estimating the number of attempts needed to catch a toy in the claw machine when there is an 8% chance of grabbing the toy on any single attempt. The student discovered, through 20+ repeated trials of this simulation that it took about 12 attempts on average. This corresponded with the estimate published on the website. We will next investigate the theory behind these random phenomena and connect the underlying principles to our observations. It has been great working with these students who bring enthusiasm and a lot of creativity to class.

Mr. Matthias: Each year when the class starts Engineering & Robotics, they aren’t quite sure what they will be facing. There is some concern as we begin with a survey of Engineering and the Engineering process. Then, as we start ROBOTC programming, the class begins to feel more comfortable and confident about the material. We practice our programming with robots in “Engineering Labs” designed to give students practical experience with programming the movement of their robot to achieve certain goals. The Engineering Labs soon become one of the favorite activities of the class and students regularly ask if we are doing one in the day’s class. As a teacher, I am so thrilled that the class looks forward to this engaging hands-on learning activity.

Mrs. McCullagh: Looking back at trimester 1, I am particularly pleased with how the students adjusted to the abstract nature of Calculus. In this course they are asked to use the skills they have built in Algebra, Geometry, and Pre-Calculus. To that we add the concepts of Calculus. While challenging, the students did really well in working with limits and longer problems than they had seen in the past. We spent a block of classes exploring the definition of the derivative. The students have a very good intuitive understanding of what we mean by derivative being the instantaneous rate of change.

A call for stairwell art!

Let's make this space awesome!
Let’s make this space awesome!

Ms. Smith and the math department want student ideas for what to put up on the walls of the math department stairwell. Send your submissions to Ms. Smith ( by Friday, 12/16. You can also drop off physical submission in the box in the math office, located in Schoolhouse 21.

Math + Art = Awesome

Here’s Ms. Smith’s presentation:

Hello, I am Ms. Smith and I am one of the teachers in the math department. But today, I am not here to talk about math. I’m here to talk about math and art, like the mathematical murals projected behind me.

You may think that math is restricted to the realm of numbers and equations. While it is certainly true that numbers and equations form the building blocks of mathematics, they also give rise to things that look a lot like art.

Those equations give rise to parabolas, ellipses, circles, shapes that are found throughout the artistic world. Infinite repetition and self-similarity give rise to fractals, like the dragon curve. Computer programs can even give rise to art. They can generate, random, yet strangely structured images.

Math can give rise to art. And art can give rise to math.

So here’s where you come in. The brick stairwell to the math classrooms is empty. We want to fill it with math and art. We want your designs for the space, whether it’s math, art, or something in between. Until winter break, the Math Department will be collecting designs and ideas for the stairwell. You can submit your designs to the box in the math office or to

Thank you and happy sketching!

Williston Math Teacher to Present at Math Conference

JMMWilliston math teacher Mia Smith will be traveling to Atlanta this January to present a paper at the Joint Mathematics Meetings, the largest mathematics meeting in the world.

Smith authored the paper “Colorful Graph Associahedra” with Professor Satyan Devadoss while at Williams. From the abstract:

Mia Smith
Mia Smith

“Given a graph G, there exists a simple convex polytope called the graph associahedron whose face poset is based on the connected subgraphs of G. Motivated by ideas in algebraic topology and computational geometry, we define the colorful graph associahedron based on an assignment of a color parameter. We show it to be a simple abstract polytope, provide its construction based on the classical permutohedron and prove various combinatorial and topological properties.”

Congratulations, Ms. Smith!

What we’re reading – October 2016

We’re constantly working to improve our craft here in the Williston Mathematics Department. Here are some of the items we’ve been reading and watching lately:

Article: Study Shows Computer Science Gap Begins Early

Article: A Simple Solution to the STEM Crisis: Do We Have the Will to Lead the Way?

Article: How the Geometry of Movies Can Change the Way We Think

Video: Sal Khan speaks at TED about mastery-based learning

Video: A tiny origami robot can deliver medicine once it’s inside a person’s body

Origami in Japan!

John Montroll's Grasshopper
John Montroll’s Grasshopper

Origami is one of the primary reasons I have devoted my life to mathematics. When I was 5, my parents gave me my first Origami books and I was immediately hooked. I can remember waking up early every morning to fold all sorts of models. The geometry, logic,  necessary dexterity, and focus were all mesmerizing.

Origami has been a constant in just about all of my math courses. We have constructed specific angles in Geometry,  studied complex surface area and volume questions with the help of modular Origami in Multivariable Calculus, and worked on edge connectivity problems in Algebra 2. There are seemingly infinite numbers of ways to work Origami into just about any math lesson! You can read about a recent class module right here: Star Project Part 1, Part 2, Part 3, and Part 4.

Star folding!
Star folding!

Not only can Origami be applied almost anywhere in math, it’s incredibly fun. We’re only a few weeks into class and I’ve already had several students approach me to ask about when we’ll be folding. I can’t wait to get started with this year’s students!

Last fall, as part of my quest to learn as much as I can about Origami, I sat down with one of the world leaders of Math and Origami, Dr. Thomas Hull, Associate Professor at Western New England University and author of Project Origami: Activities for Exploring Mathematics. Dr. Hull was, as expected, provided an absolute wealth of information! We spoke about how to best design Origami lessons, how to choose topics, as well as his recent sabbatical trip to Japan. His input was invaluable in helping me design my Williston professional development application to travel to Japan and study Origami!

screen-shot-2016-09-28-at-11-04-02-amWilliston accepted my proposal and  I was able to travel to Japan in August to attend the Origami Tanteidan in Tokyo. Needless to say, I had an incredible time!

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Math Resource Center – Fall 2016

This just in from Mrs. Hill:

Hello everyone,

My name is Mrs. Hill, and this year I am coordinating the Mathematics Resource Center. We now have a whole gang of wonderfully helpful, articulate, and supportive math tutors who are ready and willing to help you in the MRC! In fact, we have so many tutors this year that we will be able to offer expanded hours, so that hopefully everyone can make use of this facility. If you have a bunch of questions before a test, or if you just want to feel like you have some support while you get your math assignment done, these are the perfect people to ask.

Also, this year, I will be spending time in the Math Resource Center helping out as well, In fact if you look at the schedule, you’ll see that I will be at the MRC a number of times over the two week cycle. So if you are a little nervous about asking another student for help, you can come find me instead! 

The Math Resource Center is located at the end of the hall on the second floor of the Schoolhouse (room 28), so come by soon,

Mrs. Hill


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