# Class Opener – Day 61 – Slicing the Cake

Today’s class started with a review of laws of exponents, including negative exponents. I like to send students up to boards often in class, and sometimes use a deck of cards to have students determine their fate. There are 3 large boards in my room, and those who drew Aces, twos and threes were each directed to a board – 3 at each board. Before I began to bark out problems, it’s time to claim our space –

Split the board into 3 parts so that each person has an equitable space.

This was no problem for most teams, as there is always someone willing to take control and draw big, vertical lines down the board. But you can easily determine the trios which had more than one “type A” personality, as erasing sabotage, arguing, and even boxing-out occurred to just split up the board….they’re freshmen….

After the first group of students had completed their problems, the fours, fives and sixes then went to the board. Remembering some lessons from a Contemporary Math class I had taught at Rowan University a few years back, this seemed like a great time to expose students to the divider-chooser method for fair division. And while we are splitting up a board, this would be ideal for splitting up a cake, land, or other assets.  Here are the instructions:

• In your team of 3, assign roles of player 1, player 2 and player 3.
• Player 1 – approach the board and divide it into 2 “fair” pieces, without help from the others.
• Player 2 – choose one of the 2 pieces to claim as your own. Player 1 now owns the other piece. Both players should stand in fron of their pieces.
• Players 1 and 2 – divide your area into 3 “fair” pieces.
• Player 3 – choose 1 slice from the areas of players 1 and 2 to claim as yours.
• Each player now has 2 “equitable” pieces.

Sometimes it’s fun to do 5 minutes of a math nugget they may never see again, but it’s worth the discussion it generates. It was interesting to see how some players chose to work horizontally, rather than vertically – and we even had a triangular arrangement (seen below). But these aren’t really practical for doing exponent problems, so we eventually went back to a traditional division.

North Carolina State provides a helpful file which summarizes a number of fair-division methods, including the Lone Chooser method for 3 people, and you can also easily search “Fair Division Methods” to find more interesting ways to divide assets.