Spotlight on Classes: Analytic Number Theory

Each week, the students are able to list their preferences for which classes they would like to take. They take one course for four hours a day, and another for two hours a day. Each week, new classes are offered and the students have the opportunity to immerse themselves in different advanced mathematical topics.

While each of BEAM’s classes explores incredibly interesting and advanced ideas, we’ve put together a spotlight on just one of the six courses offered this week: Analytic Number Theory with Cory. 

The class has just seven students, with a teacher and a teaching assistant—the kids get to really get to know one another and the staff members! In just three days, the class has gone from not knowing what a function is to being able to find the limit as n approaches infinity of f(n) = (-1)n/n: zero. The students are basically doing calculus without using the word “calculus” itself! 

Sitting in on the class, what is most impressive to our staff, even more than the sheer speed and intensity of the class, is how engaged each student is. Cory will give short lectures, but for a majority of the class, the students work in small groups in order to work on difficult problem sets. Mostly, they are teaching themselves the topics! Today in class, we were beyond impressed to see how each of the students was able to find patterns and draw conclusions from sequences of numbers.

When Anaís and Cynthia talked about the class, they said that they couldn’t believe how much they were learning in such a short period of time!

Jack agreed, saying: “I’ve learned more math in the past three days than I did in the last two months.” 

 Cynthia, Jack, Anaís, and Alvin work together on the board

Cynthia, Jack, Anaís, and Alvin work together on the board

Week two classes are now officially over at BEAM 7 Union College. We can't wait to see how much learning takes place in week three!

Weekend Trips at BEAM 7

BEAM works a little differently in that our academic week is Tuesday-Saturday and our weekend is Sunday and Monday. We do this so that when we go on fun trips, they’re less crowded and kids can really get the most out of their day off!

This past Sunday was our first field trip. We went to Grafton Lakes State Park, where we hiked and hung out at the beach. Most of the camp went for a fun, short hike around the lake and spent several hours at the lake, where they swam, built “sand volcanoes,” played sports, and just hung out with one another. Four of our students—Anais, Brandon, Cynthia, and Emmanuela—went on our hardcore hike. They hiked about 8 miles and climbed a fire tower! Here’s a photo of them with the amazing view they got at the top. 

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Here are a few photos of all of our students hanging out around the lake:

We came back incredibly exhausted, but excited for the next day: Zoom Flume Water Park!

At Zoom Flume, we spent the day riding crazy slides and rides and enjoying ourselves in the wave pool. Since we were at the park at the same time as BEAM 7’s Bard group, we got to know some of the other students. We don't have many photos from the water park, because we didn't want to get our phones wet!

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The weekend was pretty jam-packed, but we’re already back and starting a whole new week of classes! 

Our First Week at Bard!

What does it feels like to be a middle school student taking challenging math courses? Ask BEAM students as they have the answers to most of your questions whether mathematical or not. 

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For the most part, BEAM students have been describing their first week of Math classes as both challenging and fun. As their regular schedule starts at 7:45 AM with a walk to the dining hall, our young adolescents are always reminded of the busy schedule ahead of them.

 

However, to remain focused in their classes and maximize their performance in some of their favorite activities, our students are learning healthy habits such as correct meal portions, staying hydrated, and good communication skills. 

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Life at Bard can be described as anything but ordinary. Through daily interactions with their peers, our students are learning how to respect each other's ideas beyond classrooms.

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No matter the situation that these thinkers are placed in, their eagerness to ask questions and help each other surpass their current mental and physical state. Their hard work and dedication has indeed marked a successful ending of their first week at Bard College.

Welcome BEAM Bard 2018!


As always, this time of the year is highly anticipated by not only prospective students but also by instructors. As the BEAM staff made their way to Bryant Park on this July 7th, excitement seized every face. We could not wait to meet or be reunited with the young intellects who have decided to explore their curiosity for Math by attending BEAM 2018. 

 BEAM students getting to know staff and each other. 

BEAM students getting to know staff and each other. 

While parents signed forms and students found their road buddies, the BEAM students embarked on the new journey to Bard College. Upon their safe arrival at their new home, the faces that once seemed too nervous to separate from families members now experienced a form of joy that stemmed from the new friendships made on the road. 

 BEAM students on swing outside Student Center.

BEAM students on swing outside Student Center.

Before the grand tour of the school's campus and their classrooms, students were first brought to the new home now known as Keen. As bags were being rolled in dorms and students met their remaining staff, this experience became more realistic to most. After the kids had settled in, they were given a tour of their beautiful campus and they enjoyed the community's swing.

After that, we headed to what seems to be one of our most liked places on campus--- the dining room. After a few countless bowls of vegetables, fruits, and desserts, we headed back to the dorms for the preparation of next day's evening activities. 

As the sun settled down and eyes became wearier, BEAM students headed to bed with the intentions of resting for their full first day.

Here are all 40 of our students upon arrival to campus! We're sure this is going to be an amazing summer. 

A Typical Weekday at BEAM 7

 Bobae!

Bobae!

Today, we’re showing you what a typical day at BEAM looks like! With things happening from 8AM to 10PM, we’re pretty busy all day long. This guest post is written by Bobae, a counselor at BEAM, who just finished her first year at Harvard. 

8-9am: We begin with breakfast—what better way to get energized for the jam-packed day? The students can leave for breakfast with the counselors between 8:00 and 8:30. We head out to the cafeteria, where the kids get buffet-style options.

9-11am: The kids put their cell phones away until 7pm. (Don’t worry, if there’s an emergency, we will always make an exception!) From 9-11, the kids are in math class with a teacher and a teaching assistant. The classes are designed to let the students figure things out for themselves, and, frequently, the teaching assistants are not told the material ahead of time—so that they learn alongside the students! We like a collaborative and interactive learning environment.

Number Theory

11am-12:15pm: The students and counselors participate in fun activities, ranging from sports to arts and crafts to any other fun activities the kids can come up with. There are eight activities offered every day, but students only have time to participate in two! They can request whichever activities they prefer, and are assigned to their choices each morning.

Learning Latin dance
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12:15-1pm: Lunch. It’s a well-deserved and refreshing break from all of the learning!

1-3pm: We’re back in class. We all work together on cool math problems and ideas.

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3:05-4:05pm: We have a second activity. The kids take another well-deserved brain break, followed by a ten-minute snack break at 4:05.

4:15-6:15: We’re in our last two-hour chunk of classes: the home stretch! Even though this may seem like a lot of class, the kids still stay incredibly engaged and enthusiastic through the entire day.

6:15-7pm: We eat our last meal: dinner. The main part of the day is finished!

7-8:20pm: After dinner, the kids get free time to socialize, play sports, do laundry, shower, or take naps. One of the most popular evening free time activities is just relaxing in the dorm lounge:

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8:25-9:25: We have our last academic activity of the day: modules. Modules are a self-paced, self-motivated way of learning: We study topics that are more relevant to school math classes. Our past students say that modules were incredibly helpful in future math classes.

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9:30-10:00: Hall meeting! Each floor meets with its two counselors to discuss the day: how it went, and any announcements for the next day. As a counselor, this is definitely one of my favorite parts of the day; I love getting to see the kids at the end of their day as they reflect on how camp is going.

10:30: Room check. Students must be in their rooms and ready for bed.

10:45: Lights out! We want to make sure the campers get a good night’s sleep before the next day begins.

And now…time to do it all over again!
 

Icebreakers at BEAM 7 Union!

Day 1: Today was a day full of fun icebreaker activities and rule explanations. For our icebreaker activities, we played a round of human bingo, a team-building game on birthdays (that involved absolutely no speaking at all!), a camp-wide rock-paper-scissors tournament, and a name game where we would all assign adjectives to our names and memorize each other’s names and adjectives. We managed to find descriptors for students across the alphabet, from Adorable Angie to Zebra Zeina and everything in between: Lovely Luixander, Jazzified Jasmine, Magic Mekhi, and Bubbalicious Brandon!

Check out some photos we took during icebreakers!
 

Move-In Day

Day 0: Welcome to BEAM! Our seventh graders moved into Union College’s Richmond dormitory last Sunday, July 8. The day was jam-packed with fun activities as the kids unpacked and got to know each other.

We first met up with them at Bryant Park. The counselors and staff members took small groups to lunch. We began to get to know the kids as they ate and socialized with each other and us.

 Staff at Bryant Park. 

Staff at Bryant Park. 

After lunch, we boarded the bus to drive up to Schenectady. The ride was long, but many of us talked the whole way back. You could feel the excitement on board!

We arrived on campus around 4:00. We unpacked and toured the dorms and the campus itself. After a nice dinner in the cafeteria, the counselors and kids met to go over a few rules before the kids got ready for bed before their 10:45 lights-out time.

Here are all 40 of our students upon arrival to campus!  We're sure this is going to be an amazing summer. 

What do 6th graders love about BEAM? Open Math Time!

With BEAM 6 Los Angeles' first two weeks in the book, we thought we'd take a minute to talk about one of the students' favorite parts of the day: Open Math Time! Every day at BEAM 6 includes four hours of academics: two hours of class and two hours of Open Math Time (OMT). During OMT, students choose from a menu of options, or propose their own work. They can do basically anything mathematical! Students can work individually or in groups; as long as the time is productive, any arrangement is fair game! OMT exists for a number of reasons:

 Zavier (BEAM '11), right, supports students in a BEAM 6 LA classroom.  

Zavier (BEAM '11), right, supports students in a BEAM 6 LA classroom.  

  •  BEAM classes don't assign homework, because it just doesn't seem right for a summer program! But, it's important to practice new ideas to really learn them. So, students use some of their OMT for short, required Problem Sets (PSets) from each class. 
  • BEAM believes in flexibility and choice. We want students to build the best possible summer for themselves!
  • Middle schoolers are learning a lot about themselves. Students may never have been given the flexibility to choose their groupings or tasks. This is a huge opportunity for growth. 
  • Strong mathematicians become stronger by spending more time on math. By sharing with students the joy of self-paced, self-directed learning, we give students the tools they need to challenge themselves to learn independently. 

Ever since our first summer of BEAM 6, students have raved about OMT. It becomes many students' favorite part of the day. And our BEAM 6 LA 2018 students are no different!  Here are some reviews from our early program survey:

As you can see from the check boxes. students have a suggested menu of options for how to spend their OMT. Most students chose the following options:

PSets From Class

Every member of the faculty gives students about 15 minutes worth of extra work at the end of each class in the form of problem sets. Problem sets (PSets) are designed to help students practice and solidify new skills, extend understanding, and explore curiosity. For example, a student in a KenKen class might solve a more challenging puzzle, while a student learning to program in Python might add a new command to the program they're writing. Students working on PSets can head to their teacher's classroom to chat about the work and ask for support. 

Want a flavor of a PSet? Here are some sample problems from two of our classes:

Exponents: The Super-Powers of Numbers 

Find a number with five or more digits that is 7 in Z11 (in other words, it has a remainder of 7 when divided by 11).

Voting: How to Run a Country

A group of 4 friends want to do the same activity during activity time.  Their options are to do any of the following: soccer (S), drama (D), frisbee (F), board games (B), or painting (P) and they decide to use Borda Count to decide the activity they should do.  Consider the table of preferences below:

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a) 5 points are assigned to a 1st place vote, 4 points to a 2nd place vote etc.  What activity is the Borda count winner in this instance? (Make sure to give the Borda scores to justify your answer).

b) Is it possible for only one person to changing their ranking (in any way) so that painting is the only Borda count winner? If so, give the person and their new ranking and the new Borda counts of each alternative.  If it is not possible, thoroughly explain why it is not possible.

100 Problem Challenge

 Students work on a problem during OMT with their counselor Paula, right under the 100 problem board. 

Students work on a problem during OMT with their counselor Paula, right under the 100 problem board. 

This is exactly what it sounds like: a hundred problems selected by Dan Zaharopol (BEAM Founder and the Executive Diretor of the Art of Problem Solving Initiative, Inc.) to challenge the students mathematically and bring them just to the edge of mathematical proof. There’s a big board with 100 spaces for the 100 problems, and we write in the names of students who’ve completed problems. If all 100 are solved before the end of week 5, the entire program wins a prize!

Try out a sample problem for yourself:

Problem  1

A friend tells me that she has three children, and that if you multiply their three ages you get 72.  “I don’t have enough information to figure out how old they are!” I reply.

“All right,” she says, “if you add their ages, you get the street number from my address.”

I know where she lives, so I do some calculations.  “I still don’t know how old they are!” I reply.

“Well,” she replies, “the oldest is really good at chess.”

Now I know how old they are.  How old are her children, and what is her street address?

 The state of the 100 problem challenge at the end of week 1. 

The state of the 100 problem challenge at the end of week 1. 

Art of Problem Solving/Alcumus

 Students, hard at work!

Students, hard at work!

Another major goal of BEAM is that our students (new to enrichment math) have access to the same resources and opportunities that other middle schoolers all over the country already have! So, we connect them all with accounts for the Art of Problem Solving (AoPS), and encourage students to start with Alcumus. Alcumus is a free, self-paced tool that works students through math skills, starting at pre-algebra. It's fun and it's something students can use for years to come!

Becoming familiar with AoPS and Alcumus know will have major benefits to BEAM 6 alums during 7th grade, as AoPS offers a free, online pre-algebra course to any BEAM 6 alum who completes the Alcumus pre-work to demonstrate to BEAM that they plan to take the course seriously. And as our students age, we sign individual students up for AoPS classes whenever they need an enrichment boost, and are looking to challenge themselves (the combinatorics and number theory classes are a particular hit!). 

We're so glad students are enjoying OMT and the rest of BEAM 6. It's clear that BEAM 6 Los Angeles is off to an incredible start!  Want to read more? Check out what our own Executive Director, Dan Zaharopol, had to say on his blog:

Wow! Gabriel can cube!!

About 50 weeks ago, Gabriel (center, below) showed up at BEAM 7 Union College and met Yi Ling (right, below). Yi Ling loves to cube (solve a Rubik's cube) so she taught Gabriel (and others) on the first night.Last summer, Yi Ling actually performed in the Talent Show: solving two cubes as fast as she could

Gabriel learns to cube

The picture above shows Gabriel and his first cubing accomplishment: solving one face of the cube. All summer, Gabriel practiced in spare moments, including almost every night during evening free time. He went back home and kept it up. Eventually, he got good enough that he signed up for the New York City Spring 2018 Cubing Competition through the World Cubing Association. His average cubing time was 18.91 seconds, and his single best time was 14.25 seconds!

Gabriel even posted a video of that record time. Check it out:

Amazing, Gabriel. We just think it's so cool that you picked this skill and did the work to become an expert!

A Day in the Life of BEAM 6

It's Friday which means we are nearly 20% of the way finished with BEAM 6 Los Angeles 2018. Where did the time go??

So, what is a week like at BEAM 6? Each day features:

  • Breakfast
  • Morning class
  • Open Math Time
  • Activities
  • Lunch
  • Afternoon class
  • Open Math Time
  • Activities

Let's dig deep on each of those blocks! 

The day begins with a balanced breakfast, served a la carte in the cafeteria. 

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Academics

Then, it's off to class!  Students at BEAM take four classes, one each in the following tracks: Logical Reasoning, Math Fundamentals, Math Team Strategies, and Applied Math. At the beginning of the summer, students selected which version of each class they wanted to take. Their options were:

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  • Logical Reasoning:
    • KenKen Puzzles and More
    • Ultimate Brain Puzzles!
    • Elementary, My Dear!
    • Liars, Truthtellers, and More
  • Math Fundamentals:
    • Fractions and Food
    • Exponents: the Super-Powers of Numbers
    • Patterns, Lines, and Number Rules
  • Applied Math:
    • Introduction to Cryptography
    • Voting: How to Run a Country
    • Computer Programming
  • Math Team Strategies:
    • Using Patterns to Solve Problems
    • Counting Without Counting and Fantastic (Number) Beasts
    • Words, Meet Numbers: An Algebra Story

    Don't those sound fun? How did students choose?? Each instructor gave a course description and a sample problem! Here's one set to consider!

    Elementary, My Dear!

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    Course Description:

    Ever wonder how detectives like Sherlock Holmes solve complicated mysteries? It's all in the details. In this class we'll explore the different techniques used to solve complicated riddles and how to apply them in tough mathematical problems.

    Sample problem:

    Mr. Red, Mr. Blue, and Mr. White meet at a restaurant for lunch. Under their coats they are wearing either a red, blue, or white shirt. Mr. Blue says, “Hey, did you notice we are all wearing different colored shirts from our names?” The man wearing the white shirt says, “Wow, Mr. Blue, that’s right.” Can you tell who is wearing what color shirt?

    Open Math Time

    One of the big goals of BEAM is that students spend time doing math they enjoy! So, during Open Math Time, students get a menu of options they can pursue. We encourage students to keep going back to this menu of options during 7th grade, whenever they're looking for a challenge!  During Open Math Time, students might:

    • Reinforce learning from their classes with Problem Sets  
    • Try out the weekly challenge problem
    • Work on the 100 Problem Challenge (more on that later!)
    • Explore the Art of Problem Solving, probably by checking out Alcumus

    Students can work independently, or in groups!  It's really up to what makes each student the most productive. 

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    Activities

    Morning activities last for the whole week! They're a time to learn new skills or spend a whole week on a passion. This week's options are:

    • Board Games: Old classics and new! 
    • Decorate Your Binder 
    • Watch the World Cup: Watch live and old soccer games!  
    • Learn to Solve A Rubik’s Cube
    • Rooftop FUN: Steal The Bacon and Sharky Sharky 
    • Learn to Play Settlers: A trading and building board game set in the mythical world of Catan!
    • Learn Dominion: A strategy card game where each time you play it’s different! 

    Afternoon

    Then, it's on to lunch (buffet catering from local restaurants), the next class, the next block of Open Math Time, and afternoon activities, which change every day. 

    All in all, it's a busy, exciting day, and we'll have much more to share over the upcoming weeks!

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