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.
The following link provides lessons to try after playing Prime Climb with your students: https://primeclimbgame.com/teach/
These lessons especially target Common Core Math Standards 3.OA.C.7, 3.OA.D.8, 3.OA.D.9, 4.OA.A.3, and especially 4.OA.B.4 & 6.NS.4, as well as Math Practices 1 and 7.
In this Math Teacher’s Circle session, run by Dr. Angela Antonou of the University of St. Francis and Dr. Brittany Stephenson of Lewis University, we explored the world of continued fractions. This activity put complex fractions in their place. We began with a visual exploration of continued fractions through a jigsaw puzzle activity. Participants were provided with one large rectangle with integer dimensions and several squares of various sizes (also with integer dimensions). The first task was to try to fill the rectangle using the least number of squares.
After group discovery and discussion about the least number of squares needed to fill the rectangle, participants kept track of how many of each size of the squares they used to fill the rectangle. We observed that we were able to create a fraction using the number of squares of each size used to fill the rectangle and then successively taking the sum and the reciprocal in a particular way. For example, if we used 5 of the largest squares, 4 of the next largest, and 2 of the smallest, we would obtain the continued fraction 5 + 1/(4+1/2).
After completing the jigsaw activity process, participants discovered that writing a continued fraction in this way based on the number of squares needed to fill the original rectangle was equivalent to writing a fraction for the length divided by the width of the original rectangle. In particular, each group began with a 16 by 13 rectangle and found that we could fill the rectangle with one square of size 13 by 13, four squares of size 3 by 3, and three squares of size 1 by 1. This created the continued fraction 1 + 1/(4+1/3)), which equals 16/13 (the size of the original rectangle).
After observing this relationship between continued fractions and the original rectangle, we continued to play around with continued fractions. We observed how continued fractions relate to Fibonacci numbers and can be used to create the Fibonacci spiral!
Another cool application of continued fractions that we explored is how certain irrational numbers, such as square roots, can be written in terms of continued fractions with a periodic pattern that repeats forever. We can also use continued fraction forms to compare two fractions to determine which is larger. Throughout this session, we had many interesting discussions about the uses of continued fractions. For more information, here is a helpful website we used as resource that details many more applications of continued fractions: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#section6.4
Join us for our next Math Teachers’ Circle Meeting at Saint Xavier University!
RSVP Here: http://bit.ly/MTCOct2019
We had a great meeting on May 6 at Lewis University – led by Kristen Schreck from Saint Xavier University. We discussed the Sierpinski Triangle and Carpet Fractals. Some people even won some beautiful constructions done on a 3-D printer. Here are some images from the fun and creative event.
A teacher professional development workshop coming up in Chicago on 23 March. You are listed as part of the leadership of the Southwest Chicago Math Circle; we have seen a strong affinity between these programs and the Math Teachers’ Circles in both philosophy and content.
This program is open to both in-service mathematics teachers and pre-service student-teachers with classroom experience. Let me explain that there is funding for hotel rooms for participants who live beyond easy commuting distance.
This outreach comes from the Teacher Leadership Program of the Park City Mathematics Institute, a program of the Institute for Advanced Study in Princeton NJ. PCMI is a 3 week summer program which includes rich mathematics content and also applied pedagogy, “reflecting on practice.” We are in our fourth year of weekend outreaches, spreading the PCMI teacher program beyond the small number who can attend the summer session.
If you have questions please contact Matt Roseberg at email@example.com
We are canceling our January Meeting due to weather… stay safe and stay warm and we hope to see you at our February 25th meeting at Trinity Christian College. RSVP now: http://bit.ly/MTCFeb2019
We had 3 great sessions this fall!
Our Septemeber session was run by Dr. Angela Antonou and Dr. Rita Patel.
During this Math Teachers’ Circle event, we explored new visual representations of functions and discover properties of numbers and functions using these new representations.
Our October Meeting was run by a guest leader from our Sister Circle, Brian Seguin.
During this session, we explored patterns and a Number Machine which takes in two numbers and produces a new number.
Our November Meeting was run by Dr. Amanda Harsy.
The session titled, “Voting Fairly,” was well timed with election day. During this event, we explored various voting systems. We applied and analyzed several established methods for determining the “voice” of the majority.
Participants also created and analyzed their own voting methods!
We look forward to more events this spring! Remember to RSVP for our January event!