Being a part of Emory Math Circle is one of my favorite things about being a grad student. Emory Math Circle is a free math-enrichment program for middle and high school students, that takes place on weekends at Emory. Lessons last around 75 minutes and are geared toward developing logical thinking, creativity, and skepticism in our students. I’ve been an instructor for the math circle since 2017 and have been its director since 2018.
It can be difficult to describe Emory Math Circle, so let me show an example lesson instead. One of my favorite lessons is Unfair Thrones. It’s designed for middle school students with some problem-solving experience.
First, the Story: An empress has two children who will inherit her empire. To prevent future war, she wants to build two thrones for them that are as even as possible (having one throne larger than the other sparks jealousy, and then war). So call down two students to sit in chairs in front of the blackboard. Put a fraction bar above each chair, and have students call out numbers from 1 to 4 to fill in the fractions. So you might get 2/3 over one chair and 1/4 over the other chair. You can’t reuse a number, and if you get an “improper fraction” (say, 3/2) declare, “This throne is upside down!” and flip the fraction. Now, what is the difference between the two fractions? In this example, it’s 5/12. Can we make the thrones more “fair” by shrinking the difference?
This is the main goal of the activity: to find the minimum difference between two proper fractions that use the numbers 1 through 4 exactly once. I love this activity because finding the solution (for this first question, at least) isn’t so hard, but justifying why it’s optimal is harder. Many students will say 1/3 and 2/4 are the closest you can get, but why? Force students to challenge their own answer! Can they enumerate all the possible combinations fractions? This would be one way to prove their answer.
Just as the students are all solving this first questions–and the whole time they are working together, practicing explaining their thinking–the empress has another baby! Joyful news! But now we need another throne. This time, we will have three thrones, three fractions, using the integers from 1 to 6. We want to minimize the maximum difference between each pair of thrones. For example, if we write 1/2, 3/6, and 4/5, then 1/2 and 3/6 have no difference–they’re equal!–but 1/2 and 4/5 differ by 3/10. This is the maximum difference, and we want to minimize it. Mini-max problems are likely new to every student in the room, and will be a struggle to them at first.
Of course, we can continue adding new children as long as the lesson last. Some extension questions we might ask are:
- As the number of thrones increases, can we find a set of fractions that is more and more “fair,” or does the difference between thrones sometimes increase when a new child is born? (I.e., are the solutions monotonic decreasing?)
- If the empress has n children, how many possible sets of fractions are there? Do we need to check all of these to find the solution, or are some set-ups never optimal?
As a warm-up to this activity, I give students this worksheet as they come in the door. It reminds them how to compare fractions, and how to find their sums and differences. The students should have these skills already, but it’s still good to spend some time discussing the warm-up activity. You may be surprised how many different correct ways students can compare the sizes of two fractions.
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