Test Activities Page

Audience
  • 1st - 2nd (12)
  • 3rd - 5th (30)
  • 6th - 8th (83)
  • 9th - 12th (85)
  • College Level (72)
  • For Teachers (81)
Topics
  • Algebra / Arithmetic (33)
  • Combinatorics (31)
  • Geometry (36)
  • Mathematical Games (26)
  • Mathematical Modeling (15)
  • Number Theory (25)
  • Parity / Invariants (2)
  • Probability and Statistics (3)
  • Problem Solving / General (39)
  • Social Justice Mathematics (5)
Supporting Materials
  • Facilitator Guides (88)
  • Featured in MCircular (25)
  • Handouts (38)
  • Lesson Plan (10)
  • Photos & Videos (28)
  • References (36)
  • Virtual Tools (17)
Session Styles
  • Integrates Technology (17)
  • Kinesthetic Element (11)
  • Manipulatives (33)
  • Multiple Representations (33)
  • Problem Posing (45)
  • Problem Sets (51)
  • Try a Smaller Problem (40)
  • Work Backwards (21)
Mathematical Practices
  • MP1 - Make sense of problems and persevere in solving them. (83)
  • MP2 - Reason abstractly and quantitatively. (53)
  • MP3 - Construct viable arguments and critique others' reasoning. (58)
  • MP4 - Model with mathematics. (58)
  • MP5 - Use appropriate tools strategically. (40)
  • MP6 - Attend to precision. (40)
  • MP7 - Look for and make use of structure. (72)
  • MP8 - Look for and express regularity in repeated reasoning. (61)

Activities Archive

Tic-Tac-Toe 2.0

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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...

Hyperbolic Footballs

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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...

Flipping Pancakes

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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...

Liar’s Bingo

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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...

Winning the Lottery, An Expected Value Mystery

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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...

Percents

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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...

Prejudiced Polygons

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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:...

Locked Out: A Breakout Box Session for Your Circle

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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...

A Problem Fit for a Princess: Apollonian Gaskets in History

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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...

Puzzles, Bands, and Knots

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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...

Lockers: An Open-and-Shut Case

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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....

Probability

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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...

Coins in Twoland

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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...

License Plates and Divisibility

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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...

Simplex Locks

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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?

Primes!

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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.

Hercules and the Hydra

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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 ....

Recruiting Change for a Dollar

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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...

The Mad Veterinarian on Mathematical Safari

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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.

Factor Game

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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...

Function Diagrams

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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.

Catalan Numbers

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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...

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

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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...

Can Voting Ever Really Be “Fair”?

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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...

Place Value Problems

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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...

Intersection Math

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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...

Animated Activities

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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,...

Bubbling Cauldrons

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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...

Counterexamples

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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....

Daydreams in Music: Patterns in Musical Scales

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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...

Coloring

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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....

Optimal Locations of Firehouses (Taxi-cab Metric)

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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...

Primes, Divisibility, and Modular Arithmetic

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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...

Count Me In

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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.

The Jug Band

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“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 Futurama Theorem

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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...

Humans, Zombies, & Other Problems Crossing the River

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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...

Gerrymandering

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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.

Mathematical Magic for Muggles

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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...

Grid Power

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“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...

Piece of Cake; Delectable Fractions and Decimals

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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...

Intergenerational Wealth

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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...

Big Numbers

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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!