Congratulations BEAM NYC 9th Graders!

Each year, New York City’s rising freshman apply to high school, and last year was a year unlike any other.

The high school admissions process in New York City marks a pivotal step in students’ academic careers. Unlike in most other school districts, all New York City public school students must apply to high school. What school a student attends determines a lot about what opportunities they will have, including opportunities to study advanced math and science that will prepare them to study STEM in college. Consider, for example, that most high schools in New York City don’t offer precalculus, much less calculus. BEAM works with students and families to ensure that students can successfully navigate the application process and apply to high schools with strong course offerings and good support, which open the most doors to a successful STEM career.

The admissions process in 2021-2022 continued to be affected by the pandemic, with changes to admissions criteria and timelines creating another year of bureaucratic confusion in the New York City public school system. Many middle schools, especially under-resourced schools still grappling with the effects of virtual schooling, were unable to provide the individualized admissions support students and families needed.

BEAM worked with our 8th graders to ensure that students and families had access to the information and advice they needed, including updated deadlines and requirements. BEAM also offered admissions office hours, where students could seek additional virtual support on applications and essays.

With BEAM’s help, our 8th graders earned admission to great high schools in the spring, including Bard High School Early College and Beacon High School! Eighteen students were admitted to specialized high schools, including Stuyvesant High School and Bronx High School of Science.

Admissions results:

• 50% of BEAM 9th graders are attending schools that BEAM ranks as Tier 1*

• 72% of students are attending schools that BEAM considers Trusted+**

• 18 students were admitted to specialized high schools, including Brooklyn Technical High School, Stuyvesant High School, and Bronx High School of Science.

* Tier 1 high schools offer Advanced Placement calculus or its equivalent (like the opportunity to take a college-level math course), and more than 85% of graduates are prepared for college. BEAM counts only about 40 high schools citywide, or about 7% of New York City high schools, as Tier 1; all are highly selective for admissions.

** Trusted Schools have good course offerings and support. Trusted+ Schools have earned this status with BEAM, or higher.

Way to go BEAM 9th Graders! 

Here’s a complete list of high schools admissions for BEAM students:

• University Neighborhood High School

• Urban Assembly School for Criminal Justice

• Urban Assembly School for Applied Math and Science

• Pace High School

• High School of Economics and Finance (3)

• High School for Service & Learning at Erasmus

• Francis Lewis High School

• The High School of Fashion Industries

• Medgar Evers College Preparatory School

• East Side

• Benjamin Banneker Academy

• Academy for Careers in Television and Film

• University Heights Secondary School

• NYC iSchool

• NYC Museum School

• Millennium Brooklyn HS (2)

• Manhattan Center for Science and Mathematics

• Leon M. Goldstein High School for the Sciences

• High School for Dual Language and Asian Studies

• Bard High School Early College Queens (3)

• Beacon High School (3)

• Academy of Finance and Enterprise

• Academy of American Studies (2)

• Urban Assembly Media High School

• The College Academy

• The Boerum Hill School for International Studies

• New Visions Charter High School for Advanced Math

• Midwood High School

• KIPP NYC College Prep High School

• High CSchool for Health Professions and Human Services (2)

• Harlem Village Academy Leadership Charter School

• Frederick Douglass Academy

• East West School of International Studies

• DeWitt Clinton High School (2)

• Coney Island Prep

• Celia Cruz Bronx High School of Music

• A School Without Walls

• Academy for College Preparation and Career Exploration

• Immaculate Conception School

Specialized High School Admissions included:

• Stuyvesant High School (3)

• The Brooklyn Latin School (3)

• High School for Mathematics, Science and Engineering at City College (3)

• Brooklyn Technical High School (6)

• Bronx High School of Science

• Queens High School for the Sciences at York College

• Fiorello H. LaGuardia High School of Music and Art

We’re so proud of our 9th graders, and wish them all the best this academic year. We’re here for you!

Jane Street and Hudson River Trading Summer STEM Scholars

We are immensely proud to announce the recipients of the inaugural Jane Street and Hudson River Trading Summer STEM Stipends, awarded to BEAM students in partnership with Jane Street Capital and Hudson River Trading.

At BEAM, we are working to make sure our students get access to the same sorts of STEM enrichment opportunities that their affluent peers already have, especially academic summer programs. To that end, we help students find and apply to rigorous summer programs like Mathcamp, PROMYS, and more. This year, thanks to our corporate partners, we have been able to extend that support even further.

Designed for BEAM Pathway high school students, the Summer STEM Stipends are designed to address the fact that students who attend summer programs often cannot take summer jobs, which they and their families may be counting on. The stipends reduce the pressure of having to choose between getting paid for a summer job and participating in an intensive summer program. Attending such programs allows our students to explore diverse topics in STEM and prepares them for college and beyond.

+ Alexis M., LA

Canada/USA Mathcamp

What excites me about STEM in general is the complete vastness of it. Before, this used to annoy me as whenever I tried to learn something, there seemed to be a lot left about it to learn. This still holds true today, but instead of being annoyed, I'm pumped to try learning everything that there is to know about the subject despite the futility of the attempt. My interest in math has grown ever since I first became a part of BEAM, but to be honest, my interest in STEM did not truly start until this year. I was somewhat intimidated by the vastness of STEM and even math. If something was too vast of a subject, I'd simply learn the basics and move on. This was certainly not mollified during quarantine, a test to my motivation. However, during the past year, I have gotten the chance to meet and reacquaint myself with some amazing people, both through programs like BEAM and my school. They showed me that not only was there a huge lack in what I know, but that there was a reason for me to learn more about anything that I find even remotely interesting. To put it bluntly, I was made painfully aware just how much I don’t know about math. The reason why I am going to Mathcamp is based on this fact. I chose it in order to learn more about math in not only number theory, but in everything about it. I want to be able to be confident in what I know while being proud to show it, and I believe that Mathcamp will help me reach this goal.

+ Andrew M., NYC

BCC pre-college cybersecurity and NYU Center for Mathematical Talent

My name is Andrew and I have been interested in STEM, specifically math, for a long time. My Algebra 1 class in middle school is what first piqued my interest because I began to view math as a kind of puzzle or maze, which is what made math really fun to solve. My deep love for math led me to BEAM, where I applied for and got accepted into Summer Away at the end of my 7th-grade year. While I was excited to be learning new things (and going away from home for three weeks), to say I wasn't scared would be a lie. However, Summer Away ended up being one of the most memorable and enjoyable experiences that I have ever had.
After the summer, BEAM helped me throughout middle school and now in high school. I have attended Saturday classes, educational events, and am part of a community where I am surrounded by like-minded people and positive mentors. This summer I plan to attend two programs: a pre-college cybersecurity program at Bronx Community College and the NYU Center for Mathematical Talent Summer Math Program for Young Scholars. While I hope to pursue a career in computer science, I am very interested in the world of cybersecurity so these two programs will help me further my education, allowing me to branch out and see what I am truly passionate about.

+ Aria N., LA

GRACE Summer Program

My name is Aria, and I am 14 years old, currently a rising 4th former (10th grade) at The Groton School in Massachusetts. I grew up in downtown Los Angeles where, as an only child, I spent a lot of time with my mom and dad, often doing activities that combine fun and hands-on learning, such as creating abstract art with colorful paper cutouts, making 3D objects with origami, making pretend foods with homemade edible & colorful playdough, and cooking/baking. I was amazed to discover that many of the activities involved math and science, especially cooking and baking which requires real science, bringing together math and chemistry to create delicious food! I love that in math, there is always one clear answer, even though there can be many different ways of solving it. Being a part of BEAM since 2019 has given me access to an awesome environment where I have grown as a student of STEM. BEAM has also helped me learn how a team can effectively work together to solve problems.
I will be a part of the GRACE program this summer at The Groton School, which will allow me to go directly to AP chemistry and AP biology without having to take additional classes during the next school year. I will also have an opportunity to participate in weekday activities such as swimming, hiking, canoeing, theater, and art programs. On the weekends we’ll be able to go off campus for cultural and sightseeing adventures. It will be a great way to bond with my fellow students through not only shared academic interests, but also through the fun extracurricular activities.
I am very excited to continue learning and growing with BEAM in 2022.

+ Ashley V., LA

Pomona College Academy for Youth Success

Since an early age, I seemed to easily understand all the math lessons at school, or got the hang of it quickly even when harder stuff was thrown at me. Overall, I didn't hate math, but at the same time I didn't love math, because, well, it wasn't very exciting. Then, my 6th grade year (2019), I got the opportunity to apply for BEAM. Pretty quickly I realized that BEAM math isn’t like the normal school problems or lessons. These were challenging riddles, puzzles, patterns, anything you can imagine, and you bet I wanted to solve as many as I could. Math was actually exciting! Then, I did BEAM Summer Away during the pandemic online. It was just like the previous year, but harder, which you can guess made it even more fun. BEAM has changed the way I see math forever. Because of them, I can proudly say that I love exploring math and can't wait to explore it more in years to come.
This year I decided to apply for Pomona College Academy for Youth Success. There I will be able to live and take classes at Pomona College, allowing me to experience college life for four weeks. I chose this program because the classes that they offered were intriguing. For example, an elective I picked as my first choice is called "Trillionaire Grindset: Scams and Schemes." I'm also the oldest sibling in the family, so I have to not only be responsible and set a good example for myself, but also for my younger siblings. Being able to live and take classes in Pomona, will give me time to focus on myself fully. Overall, I am looking forward to the summer and hope to have fun!

+ Brandon C., NYC

Center for Talented Youth Engineering

My name is Brandon. I am currently going to Bard High School Early College as a freshman. I really love math/STEM. I am especially excited about learning new things and different ways of solving problems. My interest in math started in 5th grade with multiplication and division. I was pretty good at them and fast, too. But what really influenced my love for math was BEAM. At BEAM I learned interesting topics like logic and problem solving. The best part was that I was surrounded by people who liked what I liked and were obsessed about math as much as me.
This summer I am in a program called Center for Talented Youth, which I learned about through BEAM. Through CTY, I was assigned a personal advisor who will help me go to my dream college and be a great help, and I have access to a bunch of math/STEM classes. What I really hope to get from this program is to find what I am passionate about. I can try multiple classes and explore new topics. In the end I think I’ll find myself liking even more topics.

+ Cindy T. NYC

NY Bioforce

Hi, my name is Cindy! I started my STEM journey when I was in the 6th grade. At the time, I’d always done well in math and science class, but I wouldn’t say that I had a passion for them. During the summer of 6th grade, I attended BEAM Discovery where I learned advanced math, and while I did very much enjoy it, I didn't know if I wanted to continue. This changed in 7th grade when I attended BEAM Summer Away. Attending BEAM for the two summers allowed me to see the beauty in math and STEM especially when I realized that mistakes are a part of the process of learning. I was able to step out of my comfort zone, and the experiences that BEAM gave me spurred my curiosity and desire to learn more. After BEAM, I was determined to further explore my interests in STEM through programs and internships. In the past few years, I’ve participated in GOALS for Girls, HK Maker Lab at Columbia University, and the Navigators internship at the Intrepid Museum.
This summer, I am interning at NY Bioforce, a life science research program. I hope to gain valuable lab research skills during my time in the program and to form a lifelasting mentor-mentee relationship with my mentor.

+ Daniel M., NYC

Cooper Union STEM and NYU Center for Talented Youth

My math journey started when I was around 5 or 6 years old. I found I had a knack for chess while playing with my dad and older sister. Realizing I could beat my older sister in a game I had just learned how to play was an incredible feeling and really jump started my love for puzzles and logic games. From then on, I would always try to get my hands on any brain teasers or small puzzles, trying to take things apart or put them back together when I inevitably broke them. By around age 10 I was also learning origami, trying to fold different dinosaurs and model airplanes in my spare time. These two are what pushed me to learn more in math and try to figure out things for myself, not just wait around until I was taught more in school.
BEAM Discovery is what kicked my learning into high gear and set me up for learning at super fast paces. Before Discovery, I had been mostly homeschooled and never experienced the normal speed of school or math class. Instead of being able to learn at my own pace, I found myself having to pay attention for more than an hour and trying to teach myself a topic I did not fully understand once I got home. While this did not feel ideal at the time, BEAM allowed me to take leaps forward in my education and get ready for incredibly more challenging schools and summer camps.

+ Daniel T., LA

Summer Multicultural Engineering Training

Challenging myself in the field of mathematics has always been my passion, but I didn’t always have the opportunity. I am grateful to have been a part of BEAM starting when I was in 7th grade. Ever since then I have learned about math that I didn't even know existed before. While I have always “liked” math, this school year I became truly passionate about math. This year I decided to self-study for the AP Calculus BC exam while learning algebra 2 and pre-calculus. It was a bit of a challenge but I really enjoyed it. I finally got the challenge I was looking for. I learned so much new math that I have begun looking into careers in the field of math and science as a result. I came across the field of engineering and that's when my interest to learn more about engineering began. As a result I looked into engineering summer programs to learn more about this field. I am really excited to learn about engineering this summer and I am really thankful for having these opportunities.

+ Emmanuela B., NYC

NYU Center for Mathematical Talent

What excites me most about math is its endless possibilities. You can find math in arts, in literature and throughout history. The concepts of math dominate our whole world, especially because technology and the internet have become so widespread.
BEAM has given me the opportunity to take classes that let me think about math abstractly. I always loved math for its straightforwardness, but being a part of BEAM made me realize just how complicated math really is. For instance, the Counting to Infinity class I took this past year where we discussed what infinity is, and how we can reach it (if we even can). I am so used to getting definite answers in math that the class really made me think harder about numbers and equations that I didn’t know existed. All of this increased my interest in math because I became so eager to learn the unknown. I had to toss out all I thought I knew about math to think from scratch, to relearn what math was or what it wasn’t. This is what makes math most engaging; it's never the same, especially as you get to higher level math.
This summer I will be attending NYU Center for Mathematical Talent for three weeks. I chose to do this program because similar to BEAM, I know it will give me a wider experience across STEM fields and how math is being used to help society. At NYU I will have a chance to explore many STEM careers through their hands-on approach! This will definitely help me as a rising senior when deciding my major for college or picking interests for college. I am super excited to see how I will grow!

+ Helen G., NYC

Cooper Union Summer STEM

I have always had an interest in math/STEM. I first was involved with BEAM during the summer of 7th grade when I did BEAM Summer Away, where I learned college-level math for three weeks, plus did fun activities and met new people. One of the classes that I really enjoyed was Binary Numbers since they’re used for computers and depending on the series of numbers you write there is a different meaning. I have also taken BEAM Saturday classes where the things I've learned include finance, classic puzzles, and mass point geometry theorem. All of these classes have challenged me to think about things in a different way whether that be to find new solutions to a challenging problem or just make educated predictions to see if this week will be a good week for the stock market.
BEAM has also been very helpful in helping me find summer programs. My main interest in summer programs was anything related to engineering or business. Last year I had the opportunity to be at Cooper Union for the summer and learn about civil engineering. This year I will participate in Cooper Union again but this time I am taking the next-gen construction materials class. Being able to do a program is very important to me because I enjoy being busy and I'm also learning about topics that are going to benefit me in the future and that continue to increase my love for STEM. I am hoping to continue to participate in summer programs for the rest of my high school time and maybe get involved in other programs during the year that have to do with STEM.

+ Jacqueline O.-V., LA

Pomona College Academy for Youth Success

I've always been a fan of math but one class during BEAM Discovery is what made me love it. What I like about math is solving problems and ending up with a clear, universal answer. Some of the classes were so engaging that I willingly took notes for them even when they weren't required! This summer, I'll be attending PAYS, Pomona College Academy for Youth Success. I hope that by the end of this program, I make new friends and learn skills outside of just academics.

+ Lorraine A., NYC

School of Interactive Arts

Something that excites me about math is there is a right answer. But there are also many ways of solving questions, which is useful if you don't know how to do them at first. Something I do outside of BEAM that helps my interest in STEM is learning how to code. I have been learning JavaScript in BEAM's Saturday classes and teaching myself HTML. School of Interactive Arts is the summer program I'm joining because of the coding classes. I enjoy playing games and this summer program is going to help me learn how to create a game. I'm also interested in coding and want to learn more about it. After the program, I hope to make a game about animals and have more knowledge about coding.

+ Maxine H., LA

Brown Pre-College

I have always been encouraged to excel in STEM subjects by my family; however, my love for math in particular fluctuated throughout elementary and middle school. Sometimes it seemed like math was only about tedious exercises strictly working through the textbook; it wasn’t until 6th grade that my love for mathematics was solidified thanks to an amazing teacher who taught me how exciting math can be. This developed into a passion for math when I attended the BEAM summer camp and discovered all of these new aspects of math I had never explored before. Math became an exciting challenge of puzzles and games and cryptology and theories where I was immediately drawn into the challenge. This summer I have the wonderful opportunity to attend an online course at Brown Pre-College about Mathematical Game Theory. I would never have pursued this if not for BEAM’s encouragement and support. While I’m nervous about the level of work offered by a university course, I know that I am up for the challenge.

+ Maya P., NYC

Fordham STEP

Math excites me because it allows me to use my problem-solving skills to find answers to problems. Math helps me to think logically and to think about every step of the problem, in order to find the correct answer. STEM excites me because it combines some of my favorite things: science, engineering, and math. In addition, it allows me to use my hands to create and build new inventions, and to work with other people to solve a problem. At BEAM, I attended classes, including and a patterns and sequences class, that helped my interest in math grow because they allowed me to think logically and to use my critical-thinking skills to accurately solve difficult word problems. BEAM also introduced to me a course about astrobiology called, The Art of Inquiry. In The Art of Inquiry, I learned how space travel affects the human body and how space affects the world around us. This course helped my interest in STEM grow because it allowed me to think about how everything in the universe is connected. Also, my astrobiology group, in the Art of Inquiry, is one of the finalists in the WorldBuild AI Competition.
The program I am attending this summer is the Fordham STEP Program. I chose to attend this program because it introduces students to the math and science courses that they will be taking in September. I hope to learn about the introductions to geometry and biology and to learn more about the different career paths in STEM, to help me figure out which career path I should take in the future.

+ Rawin H., NYC

Canada/USA Mathcamp

My passion for mathematics all started at BEAM Summer Away, and since then it has only grown more and more. I have been to various other math programs such as MathPath, and PROMYS and had a lot of fun while I was at both of them. The thing that I find interesting about math is that it is the closest thing to an absolute truth we can find in our lives. It is also really challenging so it makes me feel great whenever I solve a particularly challenging problem.
The program I am attending this summer is Canada/USA Mathcamp and I really want to go to this one because it is supposed to be a more advanced version of MathPath and that was a place where I encountered so many new people and concepts for me to think about. I am hoping to replicate that experience this summer at Mathcamp.

+ Sean C., LA

All Star Code

Hello, I'm Sean, and I am an avid reader! I've loved BEAM for a long time now. Because of BEAM I now look at math differently. Once I was more interested in creative writing than STEM, but I now realize that STEM gives the same infinite possibilities as creative writing does. At first glance I didn't realize STEM could connect with many other careers/interests/paths. BEAM helped me see that.
The program I've chosen is called All Star Code. I chose this program because I can learn how to script code, which opens up so many new possibilities. What could I come up with once I know how to create art, or a game, or anything else within the realm of code? I'm excited for this!

+ Zain E. J., NYC

School of Interactive Arts

Something that excites me about Math/STEM is the challenges, logical thinking, and progression. There is always something new to learn! BEAM's summer program helped me understand math in a logical way like a puzzle. I have been doing math for many years at a Kumon Center. Both of these helped my interest in math grow, as I now tutor math to younger children from grades K-8.
I am attending a few programs this summer such as SIA Game Design Summer Camp, and Inspirit AI. I chose these programs because of the hands-on learning, and challenging activities. I hope to learn a lot about STEM and AI in these two programs.

A Special Summer Wrapped!

This summer was unlike any other, and we were so excited to welcome nearly 300 BEAM students to our in-person programs! While we are tremendously proud of the work we did in 2020 and 2021 to run summer programs online, our experience this summer reaffirmed that in-person programs create invaluable experiences and memories for our students.

Mathematical Wins

At BEAM Discovery LA, Executive Director of LA Programs Jacob Castaneda said students had a real “appetite to keep solving problems.” In addition to tackling a variety of problems in the classroom, students completed the 100 Problem Challenge this year (100 difficult, puzzle-like math problems that involve teamwork, pattern-finding, and strategy) to earn an ice cream field trip! 

Want to try one of the 100 Problem challenges? Check out this blog post from last year, showing Problem 64 of the 100 Problem challenge.

Students work on extra math challenges during Open Math Time.

How long do students spend working on one problem at BEAM? It can be days! When asked what the longest amount of time spent on a single math problem was, the median response at BSA La Verne was just over 2 and a half days. Ky’Yah Y, who spent 3 days on her longest problem at BSA La Verne, said: “It was thrilling and left me always wanting to know more.” At BD LA, Montserrat spent 4 days tackling his longest problem, but kept at it thanks to a little help from his friends. He wrote: “I struggled a lot with some problems taking me days but I had support from teachers and peers which helped me keep trying.”

At BEAM Discovery LA, students got an introduction to game theory in the best way – winning games by learning and applying mathematical strategies! Students loved playing a game alternating taking 1 or 3 chips from a pile, where the player who took the last chip won. New ideas were added to each round, to help students build a strategy that won every time. What if there were 2 chips? 5 chips? As many as 12 chips? Students uncovered a pattern to find out who would win if there were 100 chips!

Tanasia provides extra guidance in her class, Exponents: Superpower Numbers

This summer, Tanasia became BEAM's first-ever student to become a faculty member at one of our programs when she stepped in to teach a class on exponents at BEAM Discovery New York City (after the original instructor was unable to complete the program). In her class, “Exponents: Superpower Numbers,” students explored exponents and modular arithmetic – a system integral for computer science and cryptography. Since this is often BEAM students’ first encounter with modular arithmetic, the curriculum leans into the feeling of the unknown by introducing students to a far away planet called Planet Z11, where there are only 11 numbers from 0 to 10. Students get the opportunity to work in small groups to explore the properties of the planet, and learn how those rules may apply to other planets, like Z13. Students loved the activity so much they asked for more problems to do! They spent a whole week of Open Math Time (when students get to do math of their choosing) solving them, even pulling in their friends from other classes to teach them about Z Planets!

Interested in sitting in on a BEAM class? Watch Sam’s Cryptography Class here (13 minutes).

Building Community

One of our successes from online programs was an emoji “badge” system to reward students for specific math challenges and interactions with staff members (like telling a counselor their favorite joke). BEAM LA staffer Joesphine Ochoa brought this virtual success to life this summer in the form of stickers, to reward students for solving particularly challenging problems. There were several mathematical achievements students could earn stickers for, including completing a Problem of the Week (an extra set of five challenge problems completed during Open Math Time), solving problems from the 100 Problem Challenge, and working in a group of three or more to accomplish the challenges. Students also earned keychains with additional charms depending on what they had accomplished. Two students solved 20 problems each of the 100 Problem Challenge, and 7 students solved all 5 Problem of the Week sets!

BEAM Summer Away, La Verne

Quotable Quotes from Our Students:

“I would let them know how comfortable BEAM’s community is. For me, BEAM gave me so much room to grow. It felt like no matter where I felt I was on my math journey, BEAM pushed me further without comparing myself to others.”

— Sage T, BEAM Summer Away Union

“It is insanely fun and you learn a lot.”

— Andrew B, BEAM Summer Away La Verne

“BEAM teaches you many things and treats you like you’re at home.”

— Jeffren R, BEAM Discovery LA

“It’s worth it because aside from getting to meet new people, the staff are very cool and fun to be around. Also there’s a lot of challenging questions that I probably would have never solved before.”

— Dayana S, BEAM Discovery NYC

Jane Street College Scholars

We are immensely proud to announce the recipients of the inaugural Jane Street College Scholarships, awarded to BEAM students in partnership with Jane Street Capital.

Jane Street College Scholarships will allow BEAM students to attend great-fit colleges with strong STEM course offerings even when those schools do not meet 100% of financial need.

Christian C. — University at Albany (SUNY)

My journey into STEM started in middle school. I was the typical “gifted child” who didn’t really need to try too hard to get good grades. I was introduced to BEAM through my 7th grade math teacher.

My love for learning came about through BEAM, as the classes themselves and the way they were taught helped me learn about math in a way that kept me interested in learning more.

Throughout high school, I attended BEAM Saturday classes to continue learning. One Saturday class I remember fondly is Game Theory, since it taught me that even something like a game of counting up to 12 by up to 3 at a time can have complex mathematical strategy behind it.

The most exciting thing about math to me is learning more about it. There will always be something I don’t know or don’t fully understand and the process of learning that thing makes me feel like I’ve done something to improve myself.

I will be going to the University at Albany (part of the State University of New York system), where I plan to major in mathematics. This scholarship will allow me to continue my education without debt.

Ezra C. — Bard College

I first discovered BEAM as a 6th grader excited about math, typing "math programs for middle schoolers" into my search bar. I found BEAM, which felt like a perfect fit, and sent an email to them. A couple months later, I found myself in a great environment, full of supportive people, and super fun math. Since then, BEAM has been a great support system for me, fueling my interest both in math and in other subjects. Even in times of uncertainty, I've experienced growth because of my support from BEAM.

What most stood out to me about my experience was the world of math that opened up to me. For the first time, I explored subjects outside of what was normally taught. I developed a fondness for subjects like Number Theory and was excited to study math at a higher level.

Since BEAM, I have also explored things outside of math. I have taken part in internships related to science research in subjects like Genetics and Thermophysiology. At Bard College, where I will start this fall, I hope to delve further into these topics, while having a double major in math and economics.

Storm R. — Rochester Institute of Technology

BEAM has been an amazing help in my STEM journey. In 6th grade, I was recommended for BEAM by my teachers. I’ll admit, as a 6th grader, going to a math camp over the summer didn’t exactly have me jumping for joy. While I thought math was fairly simple once you got the formula down, it wasn’t my favorite subject. In the end, I would attend BEAM’s Discovery program for the summer. I still had my doubts, but it turned out to be lots of fun! It wasn’t as strict as I’d imagined, and there was so much wiggle room for experimentation and mistakes. At BEAM, you could explore to your heart’s content, rather than be stuck with one math class you either GET, or you don’t. Couple that with traditional extracurriculars sprinkled throughout the day and you have a recipe for an amazing and fun math camp! This is where I’d wager my love for STEM, er, stems from.

After Discovery, I took many Saturday classes with BEAM, and it fostered my skills in math. The classes were varied, providing the chance to experiment with the topics you wanted to learn. It was also a good social tool, giving me the chance to make many wonderful friends who were interested in similar topics. And, they had their own ideas, so discussing problems was endlessly enjoyable.

STEM itself is pretty great, and I’ve grown to love it more and more. Take the last part of STEM, the ‘M’, which stands for mathematics. Math is something that a lot of people love or hate. Whether or not they like it, it’s no secret that it’s absolutely vital in day-to-day life. The best part for me is having an especially hard problem that I have all the tools to solve, and finally coming to a solution. Especially when you can deliberate with others about finding an answer, there’s no feeling like it.

In the future, I plan to go into the ‘T’ part of STEM, technology. I want to become a game developer first and foremost. (If that doesn’t work out (which it will!) I’d like to become a biochemist.) Computers are absolutely amazing, and they have so much potential. I’ve always liked computers, especially with my love of video games. Thinking about the inner workings of consoles fascinates me. I also love looking into the design of games, seeing what went into making a certain level structure, and how things are animated. These are all things that help me appreciate the tech industry even more. All in all, I vow to do amazing things with every opportunity I’m given. I’ll become the ultimate computer scientist and amaze the world with what I can do!

Jane Street is a global trading firm and market maker with offices in New York, London, Hong Kong, and Amsterdam; trading a wide range of financial products. As a firm of collaborative problem solvers, Jane Street leverages quantitative analysis, technology, and expertise to provide consistent liquidity across market conditions.

Highlights from Summer

BEAM Summer Away at Union College, New York

BEAM Summer Away at Bard College, New York

BEAM Summer Away at University of La Verne, California

BEAM Discovery, Los Angeles

BEAM Discovery, New York City

...And Now for Some Math

To recap from the newsletter:

You are eating a special candy bar. You and a friend each take turns eating squares of blue, pink, and green candy starting from the left and moving right. You can bite off as many squares as you want, so long as they’re all the same color of candy. So, for example, your first move could be to bite off 1, 2, 3, or 4 squares. The winner is the person who eats the green square of candy (which is extra delicious!).

Start by playing around with it and learning how it works. Then, start asking yourself some questions. Can you figure out how to win the game every time you play? Do you want to go first or second to try to win? Can you figure out how to find a winning strategy even if we changed the board you’re playing on?

The Solution

If you start from the beginning, this problem is a mess: there are lots of possibilities to consider. In fact, in our weekend class “Entry to Abstract Math,” this problem was used specifically to introduce the idea of working backwards from the end. Once you have that insight, it becomes quite easy!

Of course, immediately before the final bite, this will be what remains of the candy bar:

You want your opponent to leave you with just that green square. How can you guarantee that? Well, if your opponent is faced with this board, then they have no choice but to leave you with that delicious final square!

Now, if your goal is to leave your opponent with this board, then you want to eat the square right before it, so we’ll put a little bite symbol on the squares you should bite.

To make sure you have the chance to bite that square, you must leave your opponent with just one square before it so they have no choice but to leave you with this configuration:

Which means you want to bite the square right before it:

This time, though, to make sure you can bite that marked pink bite, you need your opponent to leave you with all three pink squares. Otherwise, your opponent could bite to the pink bite and leave you in a losing position!

So how are you guaranteed to get all three pink squares? Simple, you must leave your opponent with just the single square before the pink region:

To guarantee that, you need to bite immediately to the left of that blue square:

If you want to be guaranteed that bite, your opponent must leave you with both pink squares. Otherwise, they could eat both! So you must leave them with just the one square before:

Which means you must bite the square immediately before it, which you can do on the first move:

Thus, the winning strategy is to go first, and bite up to each mark on your turn. If you go first, you will always be able to bite up to a mark (and your opponent will not!)

In addition to teaching about working backwards, this problem helps to introduce the ideas behind combinatorial game theory. In combinatorial game theory, not just can you analyze winning positions in games and develop winning strategies, but you can also develop a theory that lets you assign to each game a certain value that encodes how it works. These values (they’re not quite numbers) can be added together, and in fact, games can be added together as well in a compatible way!

In other words, this problem, which is already fun, is just the beginning!

—

Thanks to Mira Bernstein for creating and sharing this problem.

Entry Points Takes Off!

Entry Points, BEAM’s newest program, combines the same kind of high-quality math programming, and community and mathematical identity building for students as our local programs in New York City and Los Angeles. It is designed, also like our local programs, to help students discover the beauty of mathematics and develop real expertise in mathematical problem solving.

The key difference is that local teachers implement the Entry Points curriculum in their own classrooms, rather than BEAM faculty in BEAM classrooms. 

Students are using Beast Academy (Art of Problem Solving’s highly regarded online program) in their classrooms. One teacher told us with Beast Academy, which uses comics to teach math, students feel like they “get to do math” rather than that they “have to do math.”

While BEAM faculty led students using Beast Academy in the previous school year, this year local teachers implemented the program with the help of BEAM coaches. Coaches checked in with teachers monthly, learning about their experiences, providing support, and helping find solutions to challenges that arose.

One teacher commented: “[My coach] listens to the needs of my students. She is able to give me supporting activities that bring in more reluctant students.” Another commented, “My coach does a great job with efficiently utilizing our time together…She also follows up with resources and thoughts after we meet. I never feel burdened by our meetings, only empowered.”

In Entry Points, we’re also helping students and families create communities where they have fun doing math together.

One particular highlight of the program this year was Family Nights, hosted in English and Spanish, where families were invited to play games, meet Entry Points staff, and even have the chance to win a raffle! Families also had the opportunity to have conversations about math-problem solving skills. Staff asked, “What did you feel while playing these games?” One student reflected: “It felt fun, and the [brain teasers have] a lot in common with math.” Another family member said: “Tengo que ejercitar [mi mente] mas a menudo” (I have to exercise my mind more). We’ve been sending out monthly newsletters with activities, including puzzles for families to do together at home to continue practicing outside of the classroom.

Oodle, a Wordle variation, solved in teams at a Family Night!

Entry Points is designed to build on the success of BEAM programming in New York City and Los Angeles, while providing a pathway for BEAM to reach more students in diverse settings and to support teachers nationwide.

We’re looking for partner schools for the 2022-2023 school year! Please share this article with your networks! Learn more and connect with BEAM here.

...And Now for Some Math

In our Winter/Spring 2022 newsletter, we shared one of Dan’s Challenge Problems to BEAM Pathway students:

Crystal is filling in the blanks in the following addition problem:

_ _
+_ _
____
_ _ _

First, she chooses a random two-digit number for the first line. Then, she chooses a random two-digit number for the second line. Finally, she chooses a random three-digit number for the third line. What is the probability that the addition comes out correct?

Remember: Numbers don't start with 0.

Let’s start!

To figure out the probability, we need to figure out the total number of ways to fill in two two-digit numbers and one three-digit number, and the total number of ways to fill in the addition correctly. Then the probability is

(# of ways to fill in correctly)
(total # of ways to fill in)

The total number of ways to fill it in is not too hard. There are 90 two-digit numbers. You can figure that out either because you have all the options from 10 to 99, or you can think of it as 9 options for the first digit (because you can't use zero), times 10 options for the second digit: 9*10 = 90. There are 900 three-digit numbers (going from 100 to 999, or 9*10*10). So in total, you have 90 options for the first two-digit number, times 90 options for the second, times 900 options for the three-digit number: 90*90*900 = 7,290,000.

Now we need to find the total number of ways to fill it in correctly. Here are two ways to find this total.

Solution 1:

Suppose we fill in a first two-digit number. Then to find an addition that works, all we need to do is fill in a second two-digit number so that the total is a three-digit number. The only thing that could go wrong is the sum might be too small: it might be less than 100. So which ones do work? Well...

• If the first two-digit number is 10, then the second two-digit number can be 90-99 to get a three-digit number as the answer. That's 10 possibilities.

• If the first two-digit number is 11, then 89-99 works, giving 11 possibilities.

• If the first two-digit number is 12, then 88-99 works, giving 12 possibilities.

• And so on . . .

• Until, if the first two-digit number is 90, then every single other two digit number 10-99 works, giving 90 possibilities.

• It's the same for 91, 92, all the way up through 99: each one gives 90 possibilities because any two-digit number you add gives a three-digit number.

So there are 10 ways to make the addition work if your first two-digit number is 10. There are 11 ways to make it work if your first two-digit number is 11. The total number of ways to make it work is:

10 + 11 + 12 + ... + 89 + 10*90.

The 10*90 comes up because for all ten numbers 90-99, all 90 two-digit numbers make it work.

How do we add these numbers? Well, you can use the "rainbow method" you might have learned sometime, or you can remember the formula

1 + 2 + 3 + ... + n = n(n+1)/2.

So...

1 + 2 + 3 + ... + 89 = 89(90)/2 = 4005.

1 + 2 + 3 + ... + 9 = 9(10)/2 = 45.

And then we subtract these to get 10 + 11 + 12 + ... + 89 = 4005 - 45 = 3960.

Add that to 10*90 = 900 and we get 4860 ways to correctly fill in two two-digit numbers and a three-digit number.

The probability is then 4860/7,290,000 = 0.06667%. That translates to 1/1500, so about one out of every 1500 times, Crystal will randomly fill in the addition correctly!

Solution 2

There's another way to figure out the total number of ways to fill in the addition correctly. Consider options for the three-digit number:

• If it's 199 or bigger, it doesn't work, because the largest possible number we can get is 99 + 99 = 198.

• If it's 198, there is exactly one way to fill in the 2-digit numbers: 99 + 99.

• If it's 197, there are exactly two ways: 99 + 98 or 98 + 99.

• If it's 196, there are exactly three ways: 99 + 97, 98 + 98, or 97 + 99.

• If it's 195, there are exactly four ways: 99 + 96, 98 + 97, 97 + 98, or 96 + 99.

• And so on . . .

• Until with 109, there are 90 ways to do it: 99 + 10, 98 + 11, 97 + 12, ... , 10 + 99.

• For 108, there are 89 ways to do it: 99 + 11, 98 + 12, ..., 11 + 99

• And so on . . .

• Until 100, where there are 81 ways to do it: 90 + 10, 89 + 11, ... , 10 + 90

• Below 100, it is no longer a three-digit number.

So to find all the ways that work, the final step is to add up all of the ways we've found so far. That means we must add

1 + 2 + 3 + 4 + ... + 90 + 89 + 88 + ... + 81.

There are lots of ways to do this addition. We could use the "rainbow method" or follow this formula:

1 + 2 + 3 + ... + n = n(n+1)/2.

So we can add 1 + 2 + 3 + ... + 90 like this:

90(91)/2 = 4095.

We still need to add 89 + 88 + ... + 81. To do this, we can add 1 + 2 + ... + 89 = 89(90)/2, and subtract 1 + 2 + 3 + ... + 80 = 80(81)/2. That gives 4005 - 3240 = 765.

So in total, there are 4095 + 765 = 4860 ways to correctly fill in the addition. We got the same result as before, which again gives approximately a 1/1500 probability.

Happy Pi Day!

Happy 𝝅 Day! We’re celebrating with Reuleaux polygons and the remarkable property that ties them to 𝝅. We’ll also take a look at some of the many applications of these interesting shapes, touch on some big mathematical results, and even share a few open questions in the world of Reuleaux polygons!

Reuleaux Polygons and 𝝅

A Reuleaux triangle is a "curve of constant width." That just means that whenever you hug them with two parallel lines, they're always the same width apart:

Because of that, you could use them as wheels or in a whole bunch of other places where you might "normally" use a circle (or where circles won't do, like a drill that makes square holes). More on that below!

You can build a Reuleaux triangle by taking three intersecting circular discs, each with its center on the boundary of the other two, like this:

Alternatively, you can think of it as an equilateral triangle where each side of the triangle has been replaced with a circular arc whose center is on the opposite vertex.

But, it’s not only triangles! There are infinitely many Reuleaux polygons which we can build in a similar way. Start with any regular polygon with an odd number of sides, replace each edge with a circular arc whose center is on the opposite vertex, and voila — you have yourself another Reuleaux polygon!

So what about this connection to 𝝅? Well, it’s not too different from a circle’s. You probably know the formula for finding the circumference of a circle:

c = πd

Divide by d to get ^c/_d= π. This gives us a fundamental property of circles: when you divide the length around by the diameter, you always get π no matter the size of the circle.

For a Reuleaux triangle, we refer to the perimeter p instead of the circumference, and we talk about the width w instead of the diameter. (These are just different names for the same concepts.) To find the perimeter, notice that the angle at the vertex of an equilateral triangle is 60 degrees. Then the arc on the opposite side of the triangle is 1/6 of the way around a circle with a radius of w. Because the whole circumference would be 2wπ, that means that the length of that arc is:

^2wπ/₆ = (^π/₃)w

Add up all three arcs that make the Reuleax triangle, and you get πw for its perimeter.

Divide that by the width, and we find that:

^p/_w = ^πw/_w = π

This means that, amazingly, Reuleaux triangles have the same relationship with π that a circle has! In fact, the same calculation works if we start with ANY regular n-gon where n is odd, so really this relationship holds for ALL Reuleaux polygons. (Read on to discover an even bigger generalization of this result.)

Applications

The fact that Reuleaux polygons are curves of constant width makes them popular for use in various settings. Since they roll nicely, coins that are the shape of a Reuleaux polygon can still be used in vending machines! Some examples include the Canadian Loonie (a Reuleaux 11-gon) and the 2 pula coin from Botswana (a Reuleaux 7-gon).

Like with the commonly used circular manhole covers, the constant width of Reuleaux polygons means that with just a small lip underneath, they can’t fall into the holes that they are made to cover.

Photo by Owen Byrne

So why aren’t they commonly used as manhole covers? Well, if you need to roll them on their side, the center of gravity makes them harder to roll… But that’s exactly the property that makes them desirable for making pencils with a Reuleaux cross-section. These ones will be less likely to roll off your desk.

Similarly, a fire hydrant valve in the shape of a Reuleaux triangle can be helpful in preventing mischief, since you’ll need a special tool to get it open.

Spot the Reuleaux triangle bolt head?

Reuleaux triangles also have a really cool relationship to squares! They can form a rotor within a square, that is, they do a full rotation inside the square while maintaining contact with all four sides.

In the early 1900’s, the Watt’s Brothers Tool Works used this relationship to patent a drill bit that can drill square holes! You can view the original patent!

And if you’re wondering where these shapes get their name, it comes from Franz Reuleaux (1829-1905), the German mechanical engineer who studied and designed mechanisms that convert rotation around a fixed axis into a back-and-forth or up-and-down motion. One of the possible applications is a film advance mechanism, like the one seen here:

Bigger Math Results and Open Questions

Barbier’s Theorem, first published in 1860, showed that every curve of constant width has the same relationship to what we’re celebrating today, namely, ^p/_w= π.

Wondering about the area of a Reuleaux triangle? It’s known to be ¹/₂(π-√3)w². If you’d like to try deriving it yourself, you might consider breaking it into an inner equilateral triangle and the three curvilinear regions formed between this triangle and the circular arcs.

The Blaschke–Lebesgue Theorem proves that the Reuleaux triangle has the smallest area of all curves of a fixed constant width.

And if you’re looking for open questions…

The optimal packing density of Reuleaux triangles in the plane is conjectured but unproven!

In 3-dimensions, it’s unknown which surfaces of constant width have the smallest volume!

Or maybe you’d just like to see some surfaces of constant width in action — check out this cool video from Numberphile!

BEAM Summer Away Faculty Reflections

BEAM is currently hiring summer faculty for Summer 2022. If you are interested in working at one of our programs, please apply here.

BEAM’s summer staff members are what make our summer programs special. Knowing how much we get from our staff, we recently asked some of our long-time Summer Away faculty what they’ve gotten from working with us. The answers we got were resoundingly encouraging.

What we consistently heard was that our faculty learn from our students just as our students learn from them. Our staff learn how to be flexible, how to connect with students in a way that makes them more excited about math, and how to bring the spirit of mathematical exploration and discovery back to their year-round jobs. Here’s what our faculty have to say:

Tanya Finkelstein  

For Tanya, one of the most special things about BEAM students is how much they can figure out on their own. With only a definition, they can really run! Of course, letting students run requires curricular flexibility and a willingness to be confused that most teachers are taught not to have. It’s a learning experience:

“The most important way you’ll grow as a BSA faculty is learning not to have expectations, which so many teachers come in with. Teachers are taught to have expectations of themselves that they’ll always have the answers and know how to guide their students to the solution. The most important thing kids need from you is to guide them along their own path to the answer without an expectation of what that path will look like. Some of the best teachers I’ve had in my life were kids. A teacher should be ready to feel challenged. It’s important to be confused yourself and make sense of things in front of the students so they can see a model of exploration. This is how you grow over time with BEAM.” – Tanya Finkelstein

Siddhi Krishna

Siddhi has taught both Cryptography and Knot Theory every year she’s been with us, but her classes have never looked the same twice. Siddhi attributes this to the fact that the direction the class goes depends on what the students are buzzing about any one summer. Bringing the exploratory culture of BEAM classrooms to her year-round teaching has ultimately translated to better outcomes for her year-round students. In her own words:

“I’ve learned how to listen much more closely to what students are saying. I’ve also learned to ask better questions. When a student says something, you want to draw the best thing out of them without telling them the answers. What BEAM does well is create a culture where students aren’t expected to know the answers. Trying to inject that spirit into my teaching at the university level has been important. Making it clear I don’t expect you to know the answers, but I do expect you to ask questions and try to bridge that gap, has made a difference in what my students get out of my college classes.” – Siddhi Krishna

Andre Mathurin

For Andre, teaching with BEAM was a stark contrast to the formulaic, almost sterile culture of mathematics education that exists in most high schools. Being able to think about big concepts and problems without the overhead of midterms or standardized exams was not only personally motivating for Andre, it also helped him learn to better connect with students in his year-round position. 

“My first summer I had this idea that it would be super structured. I realized that you can have a game plan and that’s fine, but the need to be flexible and meet students where they are is important. That’s one of the big learnings I got from BEAM, how to leverage students by getting them to work together and make it about discovering math versus me being the one to point things out to them. One of the incredible things about BEAM students is that they really can do this. They leave the summer program being better thinkers and better communicators.” – Andre Mathurin

Cory Colbert

Cory has worked with BEAM every summer since 2014, and he’s taught topics ranging from Analytic Number Theory to Aviation Theory. Working with BEAM has not only given Cory the same tools for leading a discovery and exploration-based classroom as other staff, it’s given him a vehicle to reflect on how he thinks about equity in STEM more broadly:

“I used to not think very explicitly about equity, diversity and inclusion, especially in 2014. I think I had gotten so used to being the only Black person in a classroom (or one of the only), and one of only 2 Black graduate students in math at my school (out of over 100), that I think I grew numb to how much STEM struggles with diversity… Recently, with regards to teaching, my focus has been on combating bias and micro-aggressions in the classroom. For example, my women students are less likely to volunteer an answer than their male counterparts. So, I'll make extra efforts to encourage women to share their results with the class, which are usually right, by the way... My hope is that maybe BEAM faculty can learn how to be effective anti-racist teachers both at BEAM and elsewhere.” – Cory Colbert

BEAM is currently hiring summer faculty for Summer 2022. If you are interested in working at one of our programs, please apply here.