Session Style: Problem Posing

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Mathematical Practices
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  • MP1 - Make sense of problems and persevere in solving them. (82)
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  • MP8 - Look for and express regularity in repeated reasoning. (61)

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...

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...

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.

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.

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.

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...