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

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