We had a great turnout for our Math Teachers’ Circle Immersion Workshop! Over the three days, we had 31 teachers and professors participate. We had a great time working on rich mathematics and spending time with other math educators! We hope to see you all back for our future meetings. The next meeting will be at St. Xavier University on September 15th from 6:00 pm to 8:30pm. Remember to RSVP!

In case you missed this great workshop, below are summaries of our 8 Sessions. We have also posted the resources for these sessions on the Resources page of this site. To see more pictures from our workshop, visit our Facebook Page!

**Session 1**: Problem Solving (Melissa Loe and Brenda Kroschel)

For this initial session, a series of interesting and challenging mathematical problems were presented. As participants became familiar with others in their collaborative groups, a series of useful problem solving strategies emerged, including the use of special or simpler cases, a restatement of the problem, extending and generalizing an initial solution, and implying a “wishful thinking” to assume a desired outcome before working out ways to achieve it by modifying aspects of the given problem.

**Session 2**: Liar’s Bingo (Melissa Loe and Amanda Snooks)

Introduced as “Liar’s Bingo,” the leaders began by challenging participants to report a series of black and red numbers but to lie for exactly one of the colors. This activity required participants to analyze a series of examples involving strips of numbers to produce an underlying pattern. Initially, multiple groups shared different examples that could be traded and combined to uncover the hidden pattern. Eventually, participants shared their discoveries, including clues and strategies that helped them along the way.

**Session 3**: Twenty-five Point Affine Geometry (Melissa Loe and Rita Patel)

This session focused on a finite geometry with a total of 25 points. Participants began with a few simple definitions and worked to form and prove a variety of conjectures related to this particular geometry. Having a finite (and relatively small) number of points and lines allowed for an in-depth analysis of this geometry.

**Session 4**: Fibonacci Fun (Amanda Snooks and Angela Antonou)

In this session, participants were encouraged to analyze movement within a tiling of regular hexagons and discovered surprising links to the Fibonacci sequence. We also learned about an imaginary language called ABEEBA, and applied path counting as a way of determining the number of words of a particular length in the language of ABEEBA.

**Session 5**: Coins in Two-Land (Brenda Kroschel and Kristen Schreck)

What if all coins had values that were multiples of 2 (1, 2, 4, 8, 16, 32, etc.)? In this activity, participants were first challenged to express positive integers as the sum of these coins (“two-ples”) being careful to use at most one of each coin – in order to obey the rules of Oneville. Later extensions switched the focus to Twoville, a town that allowed up to two coins for each numerical value. Participants worked collaboratively to discover many interesting patterns. Switching back to base-10 numerals, a similar analysis was completed.

**Session 6**: Exploding Dots (Brenda Kroschel and Amanda Harsy)

If you ever wondered about the meaning behind the common algorithms used for the four basic operations, then Exploding Dots is worth your time and effort. After an initial example linked to binary (base 2), the speakers challenged workshop participants to transfer the idea of exploding dots to standard arithmetic in base 10. Familiar but perhaps mysterious concepts such as “trading,” “borrowing,” “partial products,” and “long division” come to life in the process of using exploding dots to model the four basic operations. Classroom teachers covering operations with integers will especially appreciate the notion of an “antidot.”

**Session 7**: Pascal’s Triangle (Angela Antonou and Rita Patel)

Many teachers are familiar with Pascal’s Triangle. However, most have not spent significant time identifying a myriad of numerical patterns contained within this famous mathematical construction. Groups of participants worked collaboratively to find, justify, and extend multiple such patterns. Patterns included those linked to odd and even terms, the sum of rows, and other geometric shapes within the triangle.

**Session 8**: Classroom Scenarios (Kristen Schreck and Amanda Harsy)

Sometimes our students overgeneralize based upon a few examples. In one of three “classroom scenarios,” participants were challenged to respond to a student who claimed that “as the perimeter of a rectangle increases, the area also increases.” After some time to work collaboratively, different groups offered their responses to this student with the goal of helping the student discover why the statement was not true in general. Two other scenarios presented. One of these encouraged participants to find multiple ways to justify a general formula for the sum of the first n odd integers. Solutions included drawings, algebraic formulas, and pattern analysis. The other explored a creative method for finding a fraction between two other fractions.

Thanks again to AIM for giving us the funds to provide this workshop and our monthly Math Teacher Circle Sessions at no cost to the teachers! It was a great time of fellowship and mathematics! We hope to see you back later this year! If you are interested in joining our mailing list, join our circle!

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