## Activities Archive

### A Problem Fit for a Princess: Apollonian Gaskets in History

The examination of an MTC logo takes us on a journey starting over 2,000 years ago in ancient Greece, passing through seventeenth century Bohemia, moving through twentieth century fractals, and ultimately forming the focus of this problem-solving session. Explore the Circles of Apollonius, the question posed by Princess Elisabeth of Bohemia to René Descartes, and Apollonian Gaskets.

This session is also suitable for a high school math circle or classroom.

### Acting Out Mathematics

In teams, participants will create body movements related to geometry facts and will use their body to create a convincing argument as to why the statement is true. Please bring your fun-meter, your creativity, your body, and open physical space (for moving) to this session.

### Animated Activities

In this session, we introduce participants to the programming language, Scratch, by giving them a rich task. As they explore possible approaches to achieving the goals of the task, they naturally uncover the three fundamental structures of programming (conditionals, loops, and sequential statements). We explore multiple approaches to the task, highlighting the creativity involved in programming, and even allow participants to declare their own variables to simplify coding work.

### Art Meets Math: Escher’s Tilings

Dutch artist M. C. Escher is well known for his amazing prints of interlocking lizards, fish transforming into birds, and angels and devils intertwined, just to name a few. His intricate tilings offer a beautiful and engaging way to explore ideas related to geometric transformations and symmetries.

### Balance Beans

If you start with some beans on a seesaw and you’re given certain additional beans to place on the seesaw, can you do it so the seesaw balances?

In this activity, students start by trying to solve various challenges involving different arrangements of beans on the seesaw and then design their own challenges. Next, they try to predict which arrangements will make the seesaw balance and which ones won’t (and why!).

### Bicycle Math

You are brought to a crime scene. You are told that a thief just made off with a bag full of diamonds, escaping on a bicycle. You come across a pair of bicycle tracks in the snow, no doubt made by the fleeing thief. But which way did the thief go? Just by looking at the shapes of the tracks, can you determine which way the thieving cyclist went: left to right or right to left?

### Big Numbers

Using the standard arithmetic operations (addition, subtraction, multiplication, division and exponentiation), what is the largest number you can make using three copies of the digit “9”? While this would be a pretty large number, with a little cleverness you can do far better. This session explores VERY large numbers!

### Bubbling Cauldrons

Place our numbers into the cauldrons in ascending order – you can choose which cauldron each one goes in. However, if two numbers in one cauldron add up to a third number in that same cauldron, they bubble up and cause an explosion! This means that all the numbers, leave the cauldrons, and you must start all over again.

Our goal is to find the largest number we can place in our cauldrons without them exploding… do you think you’re up for this daunting task?

### Can Voting Ever Really Be “Fair”?

What is “fair” when voting? In this session, the participants apply and analyze several established methods for determining the “voice” of the majority. They will discover these methods through an inquiry-based experience in a deep problem, and join an ongoing discussion that has gone on for hundreds of years about what is considered “fair voting.”

This session is also suitable for a high school student math circle or classroom.

### Can you “Spot it!”

Spot It! is a matching game for kids but the mathematics behind the game is both deep and accessible. This session is an open ended investigation of the game where participants discover the hidden structures, patterns, and formulas that make the game special.

It is particularly interesting to see how non-Euclidean geometry is at the heart of a real game. This in an opportunity to see how different types of mathematics can be used to model/design parts of the real world.

### Catalan Numbers

Suppose you have n pairs of parentheses and you would like to form valid groupings of them, how many groupings are there for each value of n? How many “mountain ranges” can you form with n upstrokes and n downstrokes that all stay above the original line? What about counting the number of ways to triangulate a regular polygon with n + 2 sides? We begin with a set of problems that will be shown to be completely equivalent. The solution to each problem is the same sequence of numbers called the Catalan numbers.

### Codes, Ciphers, and Secrets

Cryptography is the study and practice of secure communication, making it an interesting and relevant application. In addition to number theory concepts like modular arithmetic, this session uses locks, codes, and ciphers to explore functions (definition, inverse functions, linear functions).

### Coins in Twoland

In Twoland, the only money is coins with value 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, and so on. How many ways can change be given, following Twoland’s strict rules? In this whimsical session, participants will practice precision and organizational skills.

This session is also suitable for student math circles and the classroom.

### Coloring

Consider a 9 × 9 chessboard, and you wish to cover as much of it as possible using figures shaped like the one to the right, where each of the four squares is the same size as the squares on a chessboard. The pieces can be rotated or flipped over. What is the maximum number of non-overlapping pieces that you can fit? Here are nine problems exploring coloring, or covering, grids and other shapes.

This collection of problems can be used in a single problem solving session, or as individual teaser questions.

### Conway’s Rational Tangles

What do adding positive and negative fractions have to do with tying knots? In this entertaining lesson, students will use ropes to explore and identify mathematical operations that untangle knots and lead to new thinking. Simple operations of twists and rotations circle back to practicing the addition of positive and negative fractions.

### Count Me In

Counting problems can involve the counting of combinations, permutations, factorials, pathways using Pascal’s Triangle, partitions and complements, and many more. In this lesson many different types of counting problems (some easy and some harder) were given to groups of teachers to solve in whatever way they preferred.

### Counterexamples

What can you do with a paperclip? What can you do with a grid? Answers to these questions lead participants to explore their own conjectures. This professional development session will help everyone turn their classroom into a Thinking Classroom, where students use Conjectures and Counterexamples to power genuine mathematical experiences.

This session is suitable for all ages, math circles and classrooms.

### Cup Stacking

Begin with a row of cups and end with all of the cups in a single stack.

Rules:

1. Count the number of cups in a stack. That stack must jump that number of

spaces. For example, 1 cup can only move 1 space; 2 cups have to move 2

spaces; 3 cups have to move 3 spaces…

2. A cup or stack of cups cannot move into an empty space. They have to land

on another cup or stack of cups.

### Daydreams in Music: Patterns in Musical Scales

There is no shortage of examples of mathematicians and scientists who are also musicians. Perhaps it is the abundance of patterns and structure prevalent in music that underpin these common interests. This exploration of musical scales is an engaging session, accessible to elementary and secondary teachers with and without a musical background.

### Euler Characteristic Exploration

The purpose of this activity is to introduce students to elementary concepts in

graph theory in a hands-on, accessible manner. Students will create their own graphs, count

certain quantities related to their graphs, make conjectures regarding these quantities, and

learn how to explain why their conjecture is true. In the process they will discover the Euler

characteristic, a powerful tool for understanding planar graphs.

### Factor Game

A teacher challenges students to a game. The rules are explained as the game progresses. The player with the highest total wins! Students then play against each other. Afterwards, while analyzing the game, prime, composite, perfect, deficient, and abundant numbers are discovered and defined. Students again play the game using the strategies they determined.

In a professional development video, teachers focus on the topic of number systems and number theory using a game setting to investigate the properties of prime, composite, abundant, deficient, and perfect numbers. This video can be used in conjunction with The Factor Game lesson plan.

### Flipping Pancakes

The Pancake Problem, first posed in 1975, is a sorting problem with connections to computer science and DNA rearrangements, which leads to discussions of algorithms, sequences, and the usefulness of approximations and bounds.

The original problem was first posed by mathematician Jacob Goodman under the pen name “Harry Dweighter” (read it quickly) in 1975, and it has delighted math enthusiasts (including undergraduate Bill Gates) ever since!

### Fold & Cut

What shapes can result from the following fold-and-cut process?

Take a piece of paper.

Fold it flat.

Make one complete straight cut.

Unfold the pieces.

Are all shapes possible?

### Folding Perfect Thirds

Imagine you’re packing for a trip, and you’re planning on bringing your favorite tie. It’s too long to fit in your suitcase, even after folding it in half. You would fold it into fourths, but you don’t want all of those creases ruining your tie. You’ve decided folding it into thirds will be the perfect length to fit in your suitcase without noticeable creases on your tie. However, you don’t have a ruler or any means of making sure your tie is folded into perfect thirds. Is there anything you can do about this?

### Function Diagrams

What if we graphed with parallel axes instead of perpendicular ones? This intriguing transformation exchanges points and lines; replacing points being on the same line, with lines passing through the same point. By changing the way we usually graph functions, new patterns and deeper understandings can emerge.

### Game of Criss-Cross

The purpose for having students play the game of Criss-Cross is to motivate them to explain the underlying mathematical reason governing who wins or loses. This exploration should lead the students to form, test, and ultimately prove conjectures about how to win at Criss-Cross. The game illustrates a beautiful application of the Euler characteristic and gives them practice at elementary counting techniques as well.

### Gerrymandering

Merriam Webster defines gerrymandering as “the practice of dividing or arranging a territorial unit into election districts in a way that gives one political party an unfair advantage in elections.” This activity tries to make sense of that definition using a few examples.

### Getting Started with PROBLEM SOLVING: A Trio of Friendly Problems

The three problems presented for this extended lesson have both individual and cluster appeal. Since each of these problems can be visualized or acted out quite readily, the problems can be accessible at some level for virtually every middle school student. By offering all three of the problems to your students over a period of time, there is the opportunity to draw them into the problem-solving experience at a deeper level and allow students to practice and strengthen their ability to generalize and recognize underlying themes/parallels within various contexts. The Handshake Problem is a natural opener for the beginning of...

### Good Things Come in Unexpected Packages

“When my son opened his Christmas cracker, he discovered ‘The Mystery Calculator,’ a little two-player card game. Of course, I was intrigued and offered to exchange my fake mustache for his game. Little did I know where this gem of a gift would take me.” Exploring the math behind the Mystery Calculator leads to a low-setup, high-fun activity!

This session is also suitable for student circles or the classroom.

### Grid Power

“When I grew up in the Soviet Union, all we used for math was grid paper. Grid paper leads to discovery.” This is how Tatiana Shubin, San Jose State University, begins her lesson demonstrating the myriad of wonderful math questions arising from a simple sheet of grid paper. Attempting to count all squares of any size on a limited grid will require participants to persevere, organize their thinking and construct viable arguments.

### Hercules and the Hydra

After a late night reading about classical mythology (or watching “Clash of the Titans” yet again), you drift off to sleep and dream that you are face-to-face with a many-headed monster that is clearly not happy to see you, either. “Ahah,” you think, “I must be Hercules and that . . . thing must be the Hydra!” Following the Hydra’s rules, can you kill it?

This session is also suitable for student circles.

### How do you connect the dots?

Suppose you have a collection of dots on a page and want to find a way to connect them “automatically” in some meaningful way. How would you do it? This question has inspired mathematicians, computer scientists, statisticians, network engineers, and even artists and architects! We will learn about the exciting mathematics behind this problem primarily through drawing.

This session is suitable for the classroom or student circles.

### Humans, Zombies, & Other Problems Crossing the River

A town faces an epidemic of zombies! Luckily, the virus has just started to spread and the infected are able to stave off their hunger for human brains… for now. In fact, they’re willing to work alongside the remaining humans to help them get across a river to safety. Can you get all the humans and zombies across safely?

### Hungry for Change: Food Deserts in CT

In this lesson, we are tasked with determining a location for a new supermarket to address a possible “food desert” problem in Glastonbury. You will use these diagram to guide your analysis along with tools in Desmos and Geogebra. To qualify as “low access” in urban areas, at least 500 people or 33% of the population must live more than 1 mile from the nearest large grocery store. In rural areas, at least 500 people or 33% of the population must live more than 10 miles from the nearest large grocery store.

### Hyperbolic Footballs

We are all familiar with the basic Euclidean geometry of the plane, including the behavior of parallel lines and angles in triangles. This familiarity may lead us to think it is a law of nature that parallel lines are always the same distance apart and the sum of the angles of a triangle is 180°. The purpose of this activity is to disabuse us of this misconception. Included are materials and directions to build an attractive two-dimensional model of negatively curved (hyperbolic) space (called the hyperbolic football). We also describe an activity to use this model to explore the basic...

### I See Platonic solids In Your Future

What is the three-dimensional shape inside the classic Mattel Magic 8-Ball toy? This hands-on activity explores the intriguing fact that there are only five different types of Platonic solids, whereas there are infinitely many regular polygons.

This session is also suitable for student math circles or the classroom.

### I Walk the Line

Your regular commute begins at your house and ends at your office at the corner of 5th street and 6th avenue. You have been making this trip for years, but you are the restless (or adventurous) type, and you try to take a different route each day. At some point, you start to wonder how long it will take you to try all of the routes.

Oh, did I mention that you have to avoid the zombies?

### Intergenerational Wealth

Students will gather data on median household incomes from three towns across Connecticut (including their own town) and will calculate potential future wealth using exponential modeling. Students will compare outcomes of these models and discover factors that impact a household’s ability to accumulate and transfer wealth. Students will understand the complexity and interconnectivity of social issues. Note that this lesson is closer to a 1-week mini-unit, but that teachers may choose certain sections to focus on for a smaller 1-2 day exploration.

### Intersection Math

What is four times three? 12 you might say, but no longer! In a new type of math — intersection math— we will see that four times three is 18, two times two is 1, and that two times five is 10 (Hang on! That’s not new!) Let’s spend some fun time together remembering what it is like to figure things out for the first time, rekindle that joyous creative mathematical spark in each of us, and realise that we are each capable of ingenious and clever thinking. Let’s work out 1001 x 492 in intersection math together! The key...

### Introduction to Diophantine Equations

Diophantine equation – an equation whose roots are required to be integers. In this article we will only touch on a few tiny parts of the field of linear Diophantine equations. Some of the tools introduced, however, will be useful in many other parts of the subject. Suppose that dolls sell for $7 each, and toy train sets sell for $18. A store sells 25 total dolls and train sets, and the total amount received is $208. How many of each were sold? Easy, two equations and two unknowns. However what if the problem was changed to: Suppose a store...

### Introduction to Finite Systems: Z6 and Z7

You are no doubt familiar with solving linear and quadratic equations with real numbers. However, in much the same way as learning Latin, French, or Spanish gives the language learner a better appreciation of English, so the careful examination of solution techniques in finite number systems adds depth to the mathematics students’ understanding of equation solving. Precision is key when solving linear and quadratic equations in finite number systems where the only available digits are the remainders after dividing by 6 or 7. Discover why in Z7, adding 3 and 4 results in 0, and why the product of 3...

### KenKen

Kenken is a puzzle whose solution requires a combination of logic and simple arithmetic skills. The puzzles range in difficulty from very simple to incredibly difficult. Students who get hooked on the puzzle will be forced to drill their simple addition, subtraction, multiplication and division facts.

### Knights and Knaves: A journey to the land of logic

Logic provides the framework that allows us to agree on what is, or is not, a valid argument. It is possible to deepen understanding of the rules of logic by offering some well-chosen puzzles that highlight important ideas. This session’s puzzles all come from a lovely book entitled What is the Name of this Book: The riddle of Dracula and other logical puzzles, written by Raymond Smullyan, a mathematician, philosopher, magician, and author.

This session is also suitable for student math circles and the classroom.

### Liar’s Bingo

From recognizing a pattern to generating terms, to abstracting and making inferences, tasks based on patterns embody the “low-threshold, high-ceiling” trait of good problems. Liar’s Bingo is all about patterns. This session involves recognizing patterns and searching for underlying structure, number theory, numeration, and potentially binary arithmetic. Sometimes, as in the game of Liar’s Bingo, order seems to arise magically from something we first assume to be random or chaotic. In this case, we use the game of Liar’s Bingo to engage participants’ desire to find patterns, and supercharge that desire by demonstrating a magic trick that captivates attention by...

### License Plates and Divisibility

The divisibility rules are often “accepted without proof” by both teachers and students. The problem explored in this session involves a rich, novel way of looking at “amazing numbers,” to authentically develop notions around patterns of divisibility as a solution strategy. “This session offered opportunities for the group to use a wide range of strategies, depending on which approach a person chose in solving it. Some gravitated toward abstraction, which perhaps tilted their work toward MP7 and 8 in an algebraic sense. Others did more with modeling. All of us were involved in arguing, sense making, reasoning abstractly, and precision....

### Locked Out: A Breakout Box Session for Your Circle

Escape Rooms and “Bomb Disposal” activities are growing in popularity as a form of team building and entertainment. This session blends the two ideas to create a cooperative math activity where the challenge is to solve math problems whose solutions generate combinations to open a locked box. The math problems can be selected to fit any audience, and the activity appeals to problem solvers of all ages.

### Lockers: An Open-and-Shut Case

A classic Math Circle problem! At a large high school, there are 10000 lockers. The lockers are numbered, in order, 1, 2, 3, . . . , 10000, and to start, each locker is closed. There are also 10000 students, also numbered 1, 2, 3, . . . , 10000. The students walk the length of the corridor, opening and closing lockers according to a set of rules. How many lockers remain open? Which lockers? What if the rules were slightly different? Can you manipulate the rules to obtain specific outcomes? This collection of nine locker problems is suitable for...

### Magic, Latin, & Sudoku Squares

Squares and numbers, numbers and squares. There is something very satisfying about arranging numbers in a square formation, following specific rules, whether it is a Magic Square, Latin Square or Sudoku. This is probably why Sudoku puzzles are so popular. This session touches on some of the deep mathematics behind these special squares.

### Making Connections Between Forms of Quadratic Equations

Developed as part of the Math Circles of Inquiry project, the goal of this module is to help students in Algebra II become fluent in the various forms of a parabola equation based on the information that they are given. Students sometimes fail to understand that there are multiple ways to find an equation of a parabola. This module incorporates multiple activities both in the learning packet and using Desmos activities online to encourage students to discover and practice writing equations of parabolas in their various forms. At the end of this activity students will work more efficiently with equations...

### Match or No Match

In this session, participants will explore the Match-No Match game: two players each draw one chip out of a bag – if the color of the chips match Player 1 wins, if not Player 2 wins. Under what conditions is this a fair game? How do we know? How can we construct a fair game? What variations of this game are possible? Participants will explore these questions to determine how this game connects to other mathematical problems.

### Mathemagical Card Tricks

There are many card tricks based on simple mathematics as opposed to sleight of hand. In this session, participants will play with a number of such tricks, test them out and work on discovering the math underneath, with a goal to formalize the mathematics that makes the trick work.

### Mathematical Games

This session includes 15 games using manipulatives or paper and pencil. The goal is to decide which one of the two players has a winning strategy. To solve a game means to find a winning, or a non-losing, strategy for one of the players. An answer must include a detailed description of such strategy, and you have to explain what the winning player should do so that this player wins regardless of his opponent’s moves.

These games may be presented as a single circle session, or individually in a circle or classroom.

### Mathematical Magic for Muggles

Presented are several easy-to-perform feats that suggest supernatural powers such as telepathy, “seeing fingers,” predicting the future, photographic memory, etc. Each trick uses simple mathematical ideas that allow information to flow effortlessly and sneakily, among them simple, efficient “coding” parity and other invariants symmetry probability One can approach these activities in many ways. At first, you may want to figure out HOW to do a trick. Then, you want to know WHY it works. Finally, you should strive to understand REALLY WHY it works: is there a simple theme or principle behind your possibly complex explanation? Look for simple and...

### Measuring Up: “Perfect” Rulers

Is it possible to measure all possible integer lengths on a ruler without marking every integer on that ruler? This is an engaging and challenging problem for all. Beautiful mathematics can be revealed while delving deeper into this seemingly easy question.

### Mind Reading With Math

For the Math Mind Reading Trick, you’ll need a volunteer who’s willing to have their mind read. The person performing the trick holds out the four cards and askes their volunteer to pick a number (whole numbers only, no fractions allowed!) between 1 and 15 and keep it a secret. Next, the mind-reader asks the volunteer if their number is on the cards one-by-one. The volunteer answers the questions with yes or no answers, and with some magic and a little math, the mind-reader figures out their number!

### Modeling a Pandemic

How can mathematics help us understand the phenomenon of a pandemic? What factors impact the spread of a disease like Covid-19? Which of these factors can we influence? (society, in schools) How can mathematics help us act during a pandemic toward better outcomes?

### Mondrian Art Puzzles

You’re Mondrian’s mathematical boss. Instead of allowing Mondrian to randomly draw rectangles and colors -you lay out requirements: 1) Mondrian must cover an N by N canvas entirely with rectangles. 2) Every rectangle in the painting must have different dimensions. 3) Mondrian must use as few colors as possible, and rectangles with the same color cannot touch one another.

Under these rules, Mondrian must try to minimize his score. A painting’s score is the area of its largest rectangle minus the area of its smallest rectangle.

### Monty Hall Problem

A prize is hidden behind one of three doors. You choose the door where you think the prize is hidden. But before the door is opened, one of the other 2 remaining doors is opened to reveal no prize. You can choose to keep the door you chose earlier or switch to the other remaining door. What should you do?

### Mural Mathematics

Students will explore existing equity themed murals to identify themes, and the mathematics necessary to plan/create a city mural. Students will utilize Google maps tools to begin connecting Mathematical representations to precisely interpret circle properties and apply them to the mural design.

### One, Two, Three, Four: Building Numbers with Four Operations

What numbers can you make with 1, 2, 3, and 4, using the operations of addition, subtraction, and multiplication? Work on these problems builds arithmetic fluency and provides opportunities to identify patterns, develop and defend arguments, and create conjectures. This investigation also highlights how thin the boundary is between a fun warm-up activity for fifth graders and deep questions investigated by research mathematicians!

### Optimal Locations of Firehouses (Taxi-cab Metric)

This session asks participants to expand their notion of “distance,” using a nontraditional taxicab metric instead of the usual Pythagorean notion. Participants are guided to construct the equivalent of “circles” with this new metric and to look at the intersections of multiple such circles. In particular, two firehouses in Gridtown are a certain distance apart and at specific addresses. What firehouse should serve a given house with a specified address? What areas of town should each firehouse serve? What if there were three firehouses? The focus of the session is on a deeper understanding of the coordinate system and notions...

### Percents

Developed as part of the Math Circles of Inquiry project, this module has students grapple with different representations of percents in various contexts in order to solve real life problems. Students need fluency in percentages for real world applications such as shopping, eating at restaurants, commission based careers, etc. Understanding percent expressions in seventh grade is necessary to be able to create exponential functions in Algebra 1.

This module contains twelve activities to address the various fine points associated with percent standards.

### Pick’s Theorem

Austrian mathematician Georg Pick first stated this theorem in 1899. However it wasn’t brought to broad attention until 1969. In this exploration, participants will use rates of change to aid them in discovering Pick’s famous formula by finding a relationship between the area of the figure, the number of perimeter pegs, and the number of interior pegs.

This session is also suitable for student circles or the classroom.

### Piece of Cake; Delectable Fractions and Decimals

While exploring the relationship between fractions and decimals, participants will have the opportunity to practice operations with fractions, notice and explain patterns, review understandings of place value and number sense, and justify their reasoning.

You can get a taste of math research by repeating these two steps: Think about an interesting unsolved problem, and Do Something to try solving it. Now Think about what you notice, and Do Something to explore your results. Repeat.

### Pigeonhole Principle and Parity Problems

The pigeonhole principle states that if n pigeons are put into m cubbies, with n > m, then at least one cubby must contain more than one pigeon. Parity problems deal with odd and even integers. Here is a collection of problems that can be used in a single problem solving session, or as individual teaser questions.

Problems are suitable for a math circle or classroom.

### Place Value Problems

In this session, we’ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should understand profoundly. Several example puzzles are followed by a rich selection of over 30 additional problems to explore.

This collection of place value problems is suitable for student circles, teacher circles, or the classroom.

### Practical Probability: Casino Odds and Sucker Bets

Gambling casinos are there to make money, so in almost every instance, the games you can bet on will, in the long run, make money for the casino. However, to make people gamble, it is to the casino’s advantage to make the bets appear to be “fair bets,” or even advantageous to the gambler. Similarly, “sucker bets” are propositions that look advantageous to one person but are really biased in favor of the other. We’ll examine what is meant by a fair or biased bet and look in detail at some casino games and sucker bets. Various problems can be...

### Prejudiced Polygons

We adapt “Parable of the Polygons” (Vi Hart and Nicky Case), an online simulation on diversity and segregation, into an appropriate MTC session. The session is interactive, and offers multiple layers of content depending on the age and comfort level of students with conversations on social issues. These levels include: (1) an exercise in fractions, (2) an introduction to game theory, (3) an invitation for students to think about the benefits of diverse groups, and (4) a discussion of how individual biases, no matter how small, can lead to detrimental societal effects like segregation. Triangles and Squares live together in...

### Primes, Divisibility, and Modular Arithmetic

Number theory is all about adding and multiplying integers: pretty simple stuff, good for elementary school or for PhD mathematicians. Dr. Arnold Ross says of number theory, that the purpose is “to think deeply of simple things.” So let’s do that together.

This session includes multiple problem sets beginning with prime numbers, continuing to divisibility and its rules, and concluding with Modulo (Modular Arithmetic).

### Primes!

Mathematicians have long been fascinated by prime numbers and a great deal of number theory revolves around the study of primes. Develop a deeper understanding of these intriguing numbers by exploring the questions presented in this session.

### Probability

Some probability problems can be solved by drawing a picture; this approach is sometimes called geometric probability. Other approaches can include experimentation, looking at smaller cases, looking at extreme cases, recursion, or carefully listing possibilities.

This session includes ten problems that can be explored alone or in sets, providing material for several circle sessions or the classroom.

### Puppies & Kittens

A local animal shelter has a puppies and kittens available for adoption that you just happen to be itching to own! In this week’s “paw-some” activity, two players begin with a certain number of animals to choose from and take turns adopting animals. The player to adopt the last animal wins! We’ll be constructing a strategy for beating this game and exploring a bit of sequences.

### Puzzles, Bands, and Knots

This activity packed session starts with a fun Pythagorean Puzzle Proof. Then, Knot Theory is explored while experimenting with the Mobius Band, Knots and Links; Untangling Ropes and Rings, and acting out the Human Knot Experiment. These explorations are further connected to the coiling and knotting of DNA molecules.

These activities are suitable for the classroom or student circles.

### Queen’s Move

Students will explore a game between two players moving a chess Queen from place to place on a square grid. The Queen may move any number of spaces to the left, any number of spaces downward, and any number of spaces on the downward-left pointing diagonal. Each player takes turns using these moves. Whoever gets the Queen to the bottom-left square first wins!

### Quilt While You’re Ahead

Quilts are a familiar set of cultural artifacts for many people. Quilts also happen to be beautifully mathematical. “What sorts of symmetries can a quilt block possess?” Participants will design and examine quilt blocks, and develop a taxonomy of symmetry in order to compare the blocks according to the symmetries, both present and absent.

### Racial Profiling

To investigate situations in the real world, we sometimes create a mathematical model. A mathematical model is a simplified version of the real world that allows us to understand the real world a little better. Over time we can change this model so that it gets closer and closer to the real world. Today we are going to create a mathematical model that represents a police officer pulling over a car randomly to try and gain an understanding of a police officer conducting a traffic stop. Our essential question is “Do police officers disproportionately pull over Black, Hispanic, or minority...

### Rational Numbers

Developed as part of the Math Circles of Inquiry project, this module is an introductory activity for rational numbers, likely aligned with Grade 7. Students will be given five points on a number line and will be asked to estimate the values of each in a 3-part task and explain their reasoning. The activity is designed to have students then fluently add, subtract, multiply, and divide these rational numbers and justify the placement of their solutions on the number line.

### Recruiting Change for a Dollar

How many different ways are there to make change for a dollar? As mathematicians we often search for patterns in a problem. However, for this problem, there is no simple, predictable pattern to build to an answer, encouraging participants to reach outside their comfort zones and ponder alternative strategies in order to make progress.

This monetary problem is engaging, and classroom adaptable with multiple entry points.

### Semiregular Tilings

Can you find all possible semiregular tilings of the plane? A tiling of the plane covers the (infinite) plane, without gaps or overlaps, using congruent copies of one or more shapes. A semiregular tiling is a tiling of the plane with certain constraints: two or more regular polygons are used, polygons meet edge-to-edge, and the pattern of polygons around every vertex is the same. Questions about polygonal tilings of the plane can utilize a classical area of mathematics to highlight and connect middle and high school mathematics content standards, mathematical practices, and the nuanced nature of mathematical justification. This session...

### Shifting Gears: Approximations in Cycling

After studying James Tanton’s MTC session about bicycle tracks, avid bicyclist Michael Nakamaye started questioning the mathematics behind how a bike works.

How do gears work? How many teeth are there usually on the different gears? Why? How is a bike like a ratio machine?

### Simplex Locks

A simplex lock is a type of combination door lock that involves pushing-in buttons. Given the set of rules for using a 5-button simplex lock, how many different combinations are there?

### Skyscrapers

Skyscrapers come in so many different sizes! Sometimes you can’t see small skyscrapers if tall ones are in front of them. Using clues about how many skyscrapers you can see from each side you look at them, can you figure out the layout of the entire city?

### Solving Linear Equations: An M&M Mystery

Developed as part of the Math Circles of Inquiry project, this session is aimed at grades 7 or 8, but may be useful for high school algebra. It consists of worksheets and series of videos meant to get students to develop an understanding of solving linear equations, using the real world example of distributing M&Ms into jars.

### Squaring the Square

Each puzzle is a rectangle made up completely of smaller squares. These squares have numbers inside that represents the length of their sides. Just knowing a few of the squares side lengths, can you figure out all the size of all the squares in the puzzle?

### Supreme Court Handshakes

Developed as part of the Math Circles of Inquiry project, this session is a good introduction to the 8th grade or Algebra Math curriculum using inquiry based instruction. Students are asked to use their problem solving skills in order to determine the relationship between the number of Supreme Court justices and handshakes that occur when each pair shakes hands exactly once. Students will begin exploring with simpler numbers and work up to creating an algebraic expression to represent the function. This lesson allows for multiple representations by using a table, list, circle diagram, matrix and manipulatives.

### System of Inequalities: Math Dance

You want this year’s dance to be LIT! The dance committee has a goal of fundraising $3,500 through ticket sales. How many tickets do they need to sell?

Developed as part of the Math Circles of Inquiry project, this module presents an engaging problem which will allow students to investigate how to graph and solve a system of inequalities.

### Systems of Linear Equations

Developed as part of the Math Circles of Inquiry project, this short module explores a graphical solution to a system of equations. Students answer questions about lemonade sales and physically stand on the coordinates of a giant grid in order to see that plotting two equations on the same set of axes can give useful information. They will also gain experience in linear equation formats other than slope-intercept form and explore what the intersection points of the lines in a system of equations means.

### The 1-to-100 problem

This exercise can be used for middle school students and older. The original problem seems almost impossibly difficult, but there are obviously many ways to approach it by considering simpler problems. As they investigate it, students are actually drilling their multiplication and addition facts. In addition, depending on how detailed a study you want to make, the students will be forced to do a lot of simple algebra to obtain the results they need. They may also learn something about commutativity, associativity, and symmetric functions.

### The Dollar Game

A group of people, some that just met, have a dilemma. Some people owe money and some have money. Problem is that only people that know each other, connected by nodes, can give or lend a dollar. But they must give each person they know a dollar, even if that puts themselves in debt!! Find ways to give money in such a way so that everyone in the group has money or owes 0 dollars.

### The Futurama Theorem

In the television show Futurama, Professor Farnsworth and Amy decide to try out their newly finished “Mind-Switcher” invention on themselves. When they try to switch back, they discover a key flaw in the machine’s design: it will not allow the same pair of bodies to be used in the machine more than once. Is there a way to restore their minds back to their original bodies?

The Futurama theorem is a real-life mathematical theorem invented by Futurama writer Ken Keeler (who holds a PhD in applied mathematics), purely for use in the Season 6 episode “The Prisoner of Benda”.

### The Game of SET

SET is a fun game that can be enjoyed by kids as young as 6 and is challenging even for adults. It is rich in counting problems and is great for getting people to pose problems. It is also an example of a finite geometry and interesting to explore how well one’s geometric intuition works.

### The Jug Band

“Using just that 5 pint jug and that there 12 pint jug, measure me 1 pint of water!” Is this possible with just the two jugs? What about a 7 pint jug and 17 pint jug? Or p pint and q pint jugs?

### The Mad Veterinarian on Mathematical Safari

A Mad Veterinarian has created three animal transmogrifying machines…

While grappling with the posed questions, players will explore a set of problems, figuring out how and if the machines can complete a given transformation. Connections can be made to invariants, abstract algebra, graph theory, and Leavitt path algebra.

### The Roommate Game: An Exploration of Stable Matchings

College students need to be matched with a roommate. They each make a list of who they prefer to room with. Given the preference lists for each individual, can we find a matching that is stable? That is, would any pair ask to change rooms because they would rather room together than with their current roommates? Explorations lead to new questions or new avenues to investigate using various mathematical methods including, but not limited to, combinatorics, graph theory, or matrices.

### Tic-Tac-Toe 2.0

The game of Tic-Tac-Toe has roots going back centuries. Grid-style game boards have been found in Ancient Egypt, during the Roman Empire, and in our current age on restaurant placemats. Multiple avenues of exploration are possible with this simple children’s game. A related game called “Gobblet Gobblers” takes Tic-Tac-Toe to a whole new level!

### Tiling With Pentagons

A pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. The plane cannot be tiled with regular pentagons. However, are there any convex pentagons that can tile the plane? This session explores various pentagons and their tiling abilities. From 1918 to 1985, fourteen irregular pentagons that would tile the plane were discovered. On August 14, 2015, Casey Mann, Jennifer McLoud, and David Von Derau of the University of Washington Bothell announced that they discovered a fifteenth pentagon. Interestingly, in 2017, Michaël Rao proved that there were no more than these...

### Touching Infinity: Hyperbinary Numbers and Fraction Trees

Are there more fractions than counting numbers? Surprisingly, an investigation into binary notation can help us answer this question! This session explores the binary number system. Participants will investigate Hyperbinary numbers, create a Fraction Tree, and discover connections between them.

### Trigonometric Ratios in Right Triangles

Developed as part of the Math Circles of Inquiry project, this five to six day activity is designed to help students understand trigonometric ratios, by building on their understanding of similar triangles and ratios of corresponding sides. The purpose of this module is for students to spend time and energy developing the reasons the sine, cosine and tangent ratios are effective tools for solving right triangles, by analyzing patterns that emerge when the trig table is compiled from class generated data, and to understand the numbers stored in their calculator before they start using it to problem solve. An optional...

### Visualization in Algebra

Many topics in mathematics can be made much clearer when symmetric aspects are made clear or when nice alternative visualizations are possible. When this occurs, it helps both the student and the teacher. We will examine visualization and symmetry in a very general way by means of a set of problems.

There is a large amount of potential classroom material here, and almost any small part of it could be used for an entire class session.

### Winning the Lottery, An Expected Value Mystery

In 2005, while researching the expected value for lottery tickets in various states, a group of MIT students won millions of dollars in the Massachusetts $2 Cash Winfall drawing. Do you want to know how they did it? This teacher led activity starts with a lottery, explores expected value, and finally ties into finite projective geometries.

### Wolves and Sheep

The rules are simple: you want to place the sheep on the board so that the wolves can’t eat them. A wolf can eat a sheep if it has a direct path to it – or is in same row, column, or diagonal as that sheep. Can you place all your wolves and sheep on an nxn grid so all the sheep are safe?