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October 03, 2014

Probability for 6th graders?

As an introduction to fractions the children were playing the pizza party game. They read the rules and the first game they played helps them get accustomed to the pieces (1/2, 1/3, 1/4, 1/6, 1/8). There are fraction sheets corresponding to each of the fractions as shown below.
The children throw a dice with these fractions on it, they pick up the fraction sheet that faces up. They then try to fill the sheet by waiting for their fraction to come up in their turn.

The game is of course terribly unfair for the children who get smaller fractions (and yet children do enjoy playing it), but I wondered if we could use this as an opportunity to see probability in action. I asked each child to write down the fraction (not person) who won the game in their group. Once the game was played a few times I made a tally of how frequently a fraction won.

As each student read our her/his findings it soon became clear that most of the time 1/2 was winning, once in a while 1/3 won and very infrequently 1/4 won. With these rules 1/6 and 1/8 did not win any games played. 

We had a conversation of whether the rules were fair to each fraction. Then we moved to the question of why it is not fair. It was nice that some of the children were able to think this through. The first child who got it said that its because 1/8 fraction needs 8 pieces and you need to get 1/8 8 times, vs 1/2 where you only need to get 1/2 2 times. Getting something two times is 'easier' than getting something 8 times. In a few minutes most children were also giving their explanations along the same lines.

I proposed a different set of rules for the next game to try to help the 1/6 and 1/8 (and get them comfortable with equivalent fractions). If you get any piece that lines up with the lines on your sheet you can take that many pieces i.e. on a 1/8 fraction sheet, if you roll 1/2 you can take 4 pieces, if you roll 1/4 you can take 2 pieces and if you roll 1/8 you can take one piece. But, if you roll 1/3 or 1/6 you need to pass. I asked if this set of rules would even the odds...the children were unsure so we went for a few rounds of the game with the new rules.

We had just enough time to come together and tally the results. 1/8 followed by 1/6 were the most common winners, 1/4 was next followed by 1/2. 1/3 was the least common fraction to finish first.

I asked them to think up game rules that would both be interesting and fair to the fractions. A couple of days later, they came up with some games. Some of the games  proposed were repeats (apparently arrived at independently). I asked the children to rate the games in two parameters, their interest in playing the game and if they thought the game was fair. 

Game 1: Like the first game, but you can take a piece of your fraction even if you get a nearby fraction e.g. 1/6 for 1/8. 
Q & A: What about 1/3 & 1/2? Yes you can take these pieces as well.
Interesting - 13, Fair - 1

Game 2: Have all pizza common, you place the fraction you get. The person who finishes a pizza first wins.
Interesting - 11, Fair - 8

Game 3: Like first game, but if you get a fraction you can't use, you give it to another person who has that card.
Interesting - 11, Fair - 11

Game 4: You just need to make a full pizza based on whatever your dice gives.
Interesting - 14, Fair - 10

Game 5: You can give and take pieces from others. 
Q & A: On what basis? Not clear, it needs to be figured out. (Calvin Ball!)
Interesting - 16, Fair - 6

5 children were not participating in the fair/not fair question as they were in doubt, but a large number of the remaining were able to guess that 1 was unfair; 2 was fair (to people); got tricked in 3 because it seemed that you were being nice (so it must be fair!); that 4 is fair. I am unclear about the rules of 5, clearly the children found the lack of clarity appealing :), but were less sure of its fairness!

Need to update this blog when we play the games again.

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