# Summer Immersion 2015

This full-day immersion was based around a mathematical ice-breaker, two in-depth, exploratory problem-solving sessions, and a collaborative class-planning session.

# September session, 2015

Session Leader: Hanna Bennett (University of Michigan)
Title: Semi-Regular Tilings

In the August workshop we explored tiling the plane with copies of a single polygon. In this September session we'll revisit that work and expand on it by allowing ourselves to work with congruent copies of TWO different polygons. In particular, we'll see what tilings we can get using two regular polygons. We will answer the following questions: How many different such tilings are there? How do we know these work, and that there are no others?

An important tool in answering these questions will be symmetries of the plane. We'll tie this work to the Common Core State Standards concerning transformations and discuss what parts of our mathematical exploration can be used in your classrooms.

If time allows we will also work with "Escher-esque" hexagonal tilings, a mathematical-artistic activity you can bring into classrooms at various grade levels.

Resources
• The warmup, Pizza Party Challenge, can be found at www.collaborativemathematics.org
• See a summary of the tiling task on the NRICH website. Note: they have tile templates here for download, but the size is different than the ones we used at the MTC session.
• PDFs of the different tiles to be cut out.
• When Hanna first solved this problem a few years ago, she wrote out a pretty complete solution for herself. See her complete solution here. Note that parts of her justification may take a different approach than we took during the Circle.

# October session, 2015

Session Leader: Scott Schneider (University of Michigan)
Title: What color is your hat?

Imagine that you are in a group of people wearing hats of various colors, where you can see the colors of all the hats but your own. If you are allowed to strategize with the group beforehand but otherwise cannot communicate in any way after putting on the hats, how might you determine the color of your own hat? Can you make a guess that is more likely to be correct than if you had guessed at random? This is the premise of several intriguing puzzles with surprising solutions that we will consider-- and act out!--during this session.

Resources

# November session, 2015

Title: Lattice squares and a whole lot more

In this session we start by considering lattice squares. That is, squares whose vertices are on integer points of the coordinate plane. This leads us to a number theoretic question: What numbers can be expressed as the sum of two perfect squares?

Resources

# January session, 2016

Session Leader: Hy Bass (University of Michigan)
Title: Relations Between Different Math Problems: What are they like? What are they good for?

Sometimes you (or your students) practice new mathematical knowledge/skills by solving many variations of the same kind of problem, wherein what varies is not the math, but the story context. We are going to play with some problems that have more complex and subtle kinds of mathematical relations, and explore the range of such possible mathematical connections.

This is helpful for cultivating what I like to call "connected mathematical thinking." Moreover it has the advantage that, when you recognize how two problems are essentially the "same" mathematically, then you only have to solve one of them.

Resources

# February session, 2016

Session Leader: Amanda Sereveny (Riverbend Community Math Center)
Title: Mathematical Origami

What shapes can be folded from paper? During this session, we will explore questions relating to this one while folding geometric models. During our journey, we will consider questions relating to angles, patterns of mountain and valley folds, facts about triangles and parallel lines, and origami folding axioms that can be used for geometric constructions.

Resources

# March session, 2016

Session Leader: Yunus Zeytuncu (University of Michigan, Dearborn)
Title: Frobenius Problem (or What Football Scores are Possible?)

If the Michigan Wolverines football team finished a game with 31 points, can we confidently tell how many touchdowns they scored? What happens if the NCAA eliminates the point(s)-after-touchdown, can a team still score 28 points at the end of a game? We will find answers to these questions in the next WCMTC.

Resources

# April session, 2016

Session Leader: Nina White (University of Michigan)
Title: Mathematical Games to Challenge and Engage

We'll explore a variety of mathematical games that both support computational skills and engage and challenge our mathematical reasoning. You'll get to enjoy these activities yourself and the presenter will include resources and ideas for bringing them into you classrooms.

Resources

# May session, 2016

Session Leader: Gabriel Frieden (University of Michigan)
Title: Oiling up your math (and drawing) skills with Euler

Leonhard Euler (pronounced "Oy-ler") was a famous 18th century mathematician. In this session, we will explore some of his discoveries related to the mathematical concept of a "graph" (note: this has nothing to do with x-y coordinates!). Along the way, we will cross bridges, draw without lifting our pencil, and even learn about soccer balls.

Resources