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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.

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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.

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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 a whole new level!

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.

Imagine that all the numbers from 1 to 100 inclusive are written on the blackboard. At every stage, you are allowed to erase two numbers that appear on the board (let’s call the numbers you erased x and y) and in place of the two erased numbers, write the number x+y +xy. Repeat this operation until only a single number remains. What are the possible values for that remaining number?

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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. Every time the Supreme Court justices get together, everyone shakes hands with each other. How many total handshakes will take place at one gathering?

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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.

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?

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?

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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.