Session Style: Manipulatives

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  • MP1 - Make sense of problems and persevere in solving them. (82)
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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.

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.

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.

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.

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...

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.



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.

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.