All about math department projects and events.

2017 AP Calculus BC Free Response Questions & Answers!

Screen Shot 2016-05-13 at 11.36.37 AMA few days after each AP Calculus BC exam, the College Board releases the free response questions from the exam. They don’t release their very succinct answer keys for a few more weeks… so… I had my students make their own answer keys as well as screen recordings of their solutions!

All 2017 released free response questions and answer keys are online right here. Questions and answers for past years can be found right here.

Here’s the direct link to the 2017 AP Calculus BC free response questions.

Here are the answer keys and videos that my students created:

Many more Williston student screencasts can be found online right here.

AP Stats Bingo!

This just in from Mrs. Baldwin!

As we prepare for the AP statistics exam, we need to review several terms and concepts. One way we can do this is with BINGO. Students complete their grids with a list of terms, in an arbitrary (not random) order. Then they are given definitions and examples. They must match the definitions and examples with the core correct terms in order to win. It is very exciting!

Sidewalk math!

With awesome weather all around, we just had to work on math outside, in chalk, on sidewalks around campus!

From Mrs. Baldwin:

The Trig/Prob/Stats class learned about describing data with numerical summaries and graphical displays. We took our work outside to practice these skills. We studied a data set of travel times to work for 20 NYC residents. We found that the median travel time was 22.5 minutes, the minimum was 5 minutes and the maximum was 85 minutes. Students also learned to use a new measure of spread called the interquartile range. This measures the range of the middle half of observations. We found that the middle half of travel times for these New Yorkers vary from 15 minutes to 42.5 minutes. Working outside in the chalk brought a kinesthetic element to our learning that was fun and engaging.

From Mr. Seamon:

Math Classroom Success Stories (16-17, T2)

Here’s a glimpse into the world of the Williston math department from the second trimester of 16-17:

Mr. Matthias: The loved the level of engagement my Engineering & Robotics students showed during the last Trimester. Students asked many questions and demonstrated success with the last set of Challenges. I will certainly miss each one of them!

Ms. Baldwin: Three students wanted some extra help before their final assessment for the Winter term. We were not able to meet in person, but planned a time to meet using Skype for Business. We spent about an hour the night before the test going over problems and addressing their questions about all that we have been studying in probability. They were able to share their screens with me and with other members of the group and I did the same with them. Our time together was extremely productive and it was so convenient to meet in this way. We got a lot of good studying done and had a few laughs at the same time. It’s good to have one more way to connect with kids and support their learning.

Mrs. Whipple: One of my students, who works really hard, was discouraged that they were not getting the grades they would have liked on every test. We worked all trimester on their strategy and their confidence when approaching the material and by the final assessment they received a near perfect score!

Ms. Schneider: One of my favorite memories from class this past trimester was when one of my students became the teacher for part of the period. We often begin class by reviewing what we have learned in our previous lessons leading up to that day. This frequently includes discussing the homework assignment. At times the students get into small groups to review; however, on this day one student came to the front and lead the class throughout this activity. She walked the class through each problem, and kept every student engaged. Not only did her classmates gain valuable insight through her explanations and leading questions, but this student, who actually is considering a career in teaching herself, showed excellent leadership skills!

Mrs. Conroy: My Geometry assessment consisted of two parts, a group portion and an individual portion. The group portion of the assessment required students to stretch their problems solving abilities while doing geometry in a collaborative setting. As I described it to the students, “There is little you can do to prepare for this section. It will challenge you. Embrace the challenge.” The first question on the group portion was particularly challenging and involved proving triangles congruent after creating a diagram from specific instructions. Each group had the correct diagram but then the problem became interesting. Not a single group earning full credit on the problem but what I witnessed in the classroom during that question was the best math we had done this year. Students were questioning each other, everyone was participating and incredible thoughts were being debated. I was thrilled to sit back, listen and watch young minds at work. Well done my Geometry students. I am proud of your fighting spirit!

Mrs. Hill: I found a stats textbook that used a real trial from 1964 to illustrate the problems of assuming independence to calculate probabilities. A woman had been mugged in CA, and the prosecutor used the assumed probabilities of a man “driving a yellow car,” and being “over 6 feet tall,” and “having a beard,” etc to calculate that the odds of the defendant NOT committing the crime were less than one in a million. Unfortunately though, as the appeals court later determined, the prosecutor was wrongly assuming independence of events when, in fact, there was no way to be sure of that fact. It was a real life example of issues of conditional probability we had been discussing in class. Moreover, we also got the chance to discuss how, in modern times, DNA evidence is based heavily on probabilities. We were not all in agreement as to the legitimacy of that approach.

Mr. Seamon: The math team has been enjoying a very active and successful year! In additions to competing in the 6 rounds of the New England Math League, returning to the Harvard Math Competition, as well as participating in the AMC8/10/12 competitions, the team has also added in the Middle School NEML competition as well as heading to Yale for their spring HS competition. Not only is the team competing in more competitions than ever, the team is scoring as well as ever currently holding strong at 28th our of 140 teams in NEML, scoring in the top 1/3 of teams at the HMMT, and also qualifying a student for the American Invitational Mathematics Exam!

Mrs. King: I have a student who has been away at ski school during the entire second trimester and will return next Monday. Before she left her family and I had a discussion about what math class she should take, an Algebra 1 class at ski school or work with a tutor to complete our curriculum. Wanting to come back fully prepared for the third trimester she chose to work with a tutor and complete our notes, homework, quizzes and tests. I set up One Note Notebooks for both her and the tutor. After a little bit of a slow start she was off and running. The tutor and I communicated each week about what was due, what was coming up or any questions or concerns that we had. The tutor was wonderful and read all of the notes and assisted Arden after she did her assignments. Arden did a great job! It was great that she was willing to take on extra work so that she would be able to transition back into class next week. I can’t wait to have her back in class.

Ms. Smith: At the end of our unit on transformations of functions, my Pre-calculus students spent a class period designing a mathematical roller coaster. That is, using their knowledge of the parent functions and transformations, they created one continuous, piecewise-defined function that traced the vertical height of the roller coaster with respect to horizontal distance travelled. As students discovered, the trickiest part was ensuring that the functions linked up, that is, there were no unplanned gaps in the track. However, after a period of work there was a wide range of functions (or should I say roller coasters). Highlights included underground tunnels, death drops, and even a loop-the-loop made using logarithmic, exponential and even elliptic functions.

Mrs. McCullagh: We finished the winter trimester with a project in Calculus. The assignment was for each student, or student pair, to decide what they wanted to hang and from where and then find the minimum amount of wire needed to hang their object. They needed to decide how far apart their two attachments should be and how far down they wanted the object suspended. They needed to find, using calculus, the minimum amount of wire needed for their own scenario. It is a challenging calculus problem for students as they are learning how to solve maximizing/minimizing problems. Then they needed to present their findings with all calculations clearly shown and diagrams labeled with the minimum and extremes. They also needed to produce a model made to scale. The projects were outstanding! We had a target hung from a tree, donuts hung for a birthday party, a chair hung is a bedroom, a rubber ducky hung (just because), as well as a number of others. The students all reported that they learned a lot from the project. It is great to have their work on display!

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!

All about math department projects and events.