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## Math Sessions

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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 rich mathematical task.

A decade ago, 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.

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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.

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.

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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.

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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.

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.

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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.

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.

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.