During the first Math Teachers’ Circle of the 2022-2023 academic year (at College of DuPage), we explored quilts and quilt patterns. We examined all the symmetries possible in a 4×4 quilt. In addition, we began to explore how many possible quilts could have such symmetries (hint: it’s A LOT!). Some participants used statistics to create random quilts and the question came up: if we continued to use random pattern generation, what is the probability of getting a quilt with SOME type of symmetry (versus no symmetry)?
The Southwest Chicago Math Teachers’ Circle will be back this year to provide enriching, engaging professional development and collaboration. During the 2022-2022 academic year, we will be running four Math Teacher Circle Sessions. Each will be in person and all will be at College of DuPage on (or close to) the third Tuesday of the month 6-8 pm.
Description: In this Math Teachers’ Circle Session, we will explore mathematics of quilt patterns. Participants will create quilt blocks and discover their symmetries. Facilitated by inquiry-based methods, we will classify the types of symmetries a quilt block can have and investigate how many blocks exist in these categories. At the conclusion of the session, participants will apply what they learned to analyze actual quilts!
Mathemagical card tricks were the center of attention at the May 2022 Southwest MTC session. Participants were first wowed with card tricks performed by co-leaders Christina Jamroz (University of St. Francis) and Marie Meyer (Lewis University) before they were challenged to discover the mathematics behind the magic.
The first trick performed was Pick-A-Pair. In this trick, the magician correctly predicted the rank of 2 cards (ace through nine only) based on a unique number given by the victim from the 2 hidden cards. Participants figured out the magician was using an algebraic expression which encoded each rank as the digits place of the unique 2-digit number to perform this trick. Afterwards, each group of participants designed their own trick that relied on creative algebraic expressions.
The second trick performed was the Fitch-Chenney. In this trick, the magician and an accomplice work together to correctly predict 1 hidden card out of 5 randomly selected cards from a full deck. Participants relied on the Pigeonhole principle, permutations, modular arithmetic, and a total ordering of the deck in order to understand the communication between magician and accomplice for this trick.
The final trick performed was Pick a Card. In this trick, the magician correctly predicted a secret card from 27 cards by simply asking the victim to identify the column of their secret card 3 times and redistributing the 27 cards in columns between each identification. Participants figured out how the magician narrowed down the card with each iteration, and they used this strategy along with modular arithmetic to develop their own versions of this trick.
Students at the middle school, high school, and college level would enjoy this activity.
Please RSVP for the upcoming Math Teachers’ Circle session being held at College of DuPage in Glen Ellyn, IL this upcoming May 3, 2022! It will be held 6-8 pm and includes professional development hours as well as FREE DINNER!
Description: Many amazing card tricks are based on mathematics. In this Math Teachers’ Circle Session, participants will be introduced to several of these magic card tricks. Then, we will have the opportunity to try them ourselves and explore the mathematics involved. Through guided inquiry, we’ll discover the mathematics behind the card tricks and learn why they work.
On April 7, the Southwest Chicago MTC struggled with determining what could make a fair district map. In the meeting, everyone was given fake maps, a rectangle with small squares throughout. Each square contained a star or diamond, representing the political affiliations of the individuals in that “block.” Participants tried to draw “fair” maps consisting of five districts, each with nine blocks.
We learned that it is much easier (and maybe more fun?) to try to make an UNFAIR map. After debating over what measures should be used to determine whether a drawn map should be considered fair, we discussed some measures that are currently being used (or that can be used). This included the concepts of efficiency gap and compactness. For the efficiency gap, we try to reduce the difference between the proportion of “wasted votes” for each party. This relies on determining fairness based on political affiliation of the individuals, which can cause ethical concerns when drawing maps based on this criteria.
For compactness, we instead aim to construct districts based on individuals living “near” each other. One way to do this is to try to make the districts as circular or as square-like as possible. This introduced the idea of the Polsby-Popper Score for measuring compactness. We also introduced the Convex Hull (or rubberband) method of determining compactness. Below is an image one group made which had a PERFECT Convex Hull score.
We had a great time exploring the concepts of drawing district maps. This, of course, is a hot topic since it determines the power of political parties in the House. In fact, there are many legal challenges to the recently drawn maps (based on the 2020 Census). You can find these and more resources in the links below:
Please consider RSVPing for the upcoming Math Teachers’ Circle session being held at University of St. Francis in Joliet, IL this upcoming April 7, 2022! It will be held 6-8 pm and includes professional development hours as well as FREE DINNER!
Description: Redistricting is the process of redrawing district lines every 10 years to account for changing populations. A potential consequence of this process is racial and/or partisan gerrymandering. Racial gerrymandering, which is illegal, occurs when the lines are drawn to reduce the impact of minority voters. Partisan gerrymandering occurs when district lines are drawn to maximize a political party’s representation (whether or not it represents the voting body). In this session, we will attempt to draw “fair” district maps. In doing so, we’ll learn about the various measures we could use to quantify what it means it means to be fair in this context.
In the first Math Teacher’s Circle since COVID, Dr. Rita Patel of College of DuPage and Dr. Michael Smith of Lewis University led the participants through an activity titled Liar’s Bingo. Participants were given several slips of papers where they needed to determine the relationships among the numbers on each individual slip and relationships among the numbers among all slips they had. Each of the numbers on the slips were either black or red.
Thereafter, participants were asked to lie about one color on their slip of paper. Drs. Patel and Smith guessed the number that was lied about. Participants then tried to figure out how the leaders knew the numbers.
Throughout the session we had many thoughtful discussions about relationships among numbers. Students at the junior high and high school level would enjoy this activity.
We hope you are all healthy during these strange times! We have good news and are excited to announce that AIM is organizing a weekly Math Teachers’ Circle Network Online Series that is open to all MTC members across the country!
We will meet Tuesdays from 5 to 6 p.m. Pacific, beginning next Tuesday, May 19, when James Tanton will kick off the series with a session on “How to Fold Into Thirds”:
It is pretty easy to accurately fold a tie or a strip of paper or length of chord in half, or quarters, or even eighths. But how do you fold a strip into accurate thirds? Or how about sevenths? (Does this issue actually come up in everyday life?) Let’s play with some fun folding math and figure out how to become masters in folding lengths into most any fraction you like! Wild math to behold!
P.S. More details will be coming soon on a Nationwide Virtual MTC Workshop to take place this summer in July. Thanks to Chris Bolognese (Columbus MTC), Sloan Despeaux (Smoky Mountain MTC and NC Network of MTCs), Anne Ho (Tennessee Eastern & Appalachian MTC), and Lizi Metts (Middle Tennessee MTC) for organizing this workshop in conjunction with AIM.
In this month’s Math Teachers’ Circle session, run by Dr. Christina Jamroz of the University of St. Francis and Dr. Rita Patel of College of DuPage, we investigated Magic Squares. We began the session by examining a magic square of order five to infer some of its defining properties. Participants concluded that a magic square of order n must contain each of the numbers 1 through n^2 exactly once. Furthermore, every row, column, and main diagonal in the square must add to the same number, called the magic sum.
The session continued with groups working to construct magic squares of order two, three, and four. We observed that we could not construct a 2 x 2 magic square, but found eight magic squares of order three. We also realized that all of these magic squares were related by a rotation or reflection. During this discussion, we made conjectures related to patterns we noticed. One of these conjecture concerned the middle number of an odd order magic square. Is this middle number always going to be the median of the list of numbers used to fill the square? After considering more examples, participants found that this is not always true.
The construction of a magic square of order four led to a discussion about strategies we could use to create magic squares. Participants noted that knowing the magic sum would aid them while working on this problem. We observed that all of the magic squares of order three had the same magic sum. Does this generalize to all magic squares of a fixed order? This led to the discovery that the magic sum of a magic square of order four was always 34. Then, we deduced that the magic sum of a magic square of order n was n(n^2+1)/2.
At the conclusion of this session, we reflected on how we can use this topic in our classrooms. Furthermore, we realized that we have seen magic squares before, with participants mentioning Sudoku puzzles, art, and math history.
In this Math Teacher’s Circle session, run by Dr. Amanda Harsy of Lewis University, we explored the world of prime numbers by playing Prime Climb. First, we made observations about the game board, specifically reflecting upon the coloring of the board. We played a simplified round of the game, and considered the mathematical operations behind the game.
Through group discovery and discussion, we noted effective strategies for moving along the game board. For example, we found that a player can land on any number up to twenty using addition on their first turn. Furthermore, you can land on any number that can be written as a product of two numbers that is less than, or equal to ten.
Upon reflection, we realized the game of Prime Climb is relevant to our classrooms in many ways. First, the game can provide our students with practice finding factors and multiples. Prime Climb could be particularly helpful for struggling learners who need to practice their fact fluency. We also discussed that the color coordination of the dice and the game board spaces would be helpful for students who have difficulties with fact fluency. Colors can be a useful tool in helping students to make connections while studying difficult concepts, such as prime numbers. Furthermore, this game can help students discover that there is not a pattern of primes. Finally, we discussed the relevance of playing this game to explore the commutative property with various operations.