Educating Risa

Jim and Daegan did some simple activities last night about probability. If you’ve got a pair of dice, paper and pencil, and some kind of counter (we used cuisinaire rods, but anything from pennies to beans to bingo chips will do), you can do this too. I found it interesting how Daegan would form hypotheses (often incorrect ones) and through hands-on experience, correct his reasoning.

Jim began with one die, and asked Daegan what numbers could be rolled with one die. “1, 2, 3, 4, 5, 6.” said Daegs. “Right.” Jim theh wrote 1 through 6 on a paper, and asked Daegan how many ways you could roll a 1, or a 2, or a 3, etc. using only one die. Daegan thought this was all too easy: “Just one!" That’s simple!”

Jim moved on to two dice. “So what numbers can I roll with two dice? What’s the smallest number?” “Two!” shouted Daegan. “And what’s the biggest number?” “Ten!” Daegan shouted again. Jim turned the dice to 5 and 5, and said again, “Are you sure that’s the biggest roll?” “Oh, no…(pause)…I mean twelve!” Daegan corrected himself. Jim then wrote the numbers 2 through 12 on the page, and asked Daegan “How many ways can I roll a two?” “One! So it’s ones under each number!” Daegan hypothesized incorrectly. To help Daegan see why this was wrong, Jim set the dice up to go through all the possible rolls of two dice in a systematic way, starting with the first dice as 1, and the second as 1 (i.e., 1-1), then the first dice as 1 and the second as 2 (i.e. 1-2), then 1-3,…to 1-6. He had Daegan add the totals of these rolls and put a tally mark (tick) under the appropriate number. So, for 1-1, Jim put a tally mark under the 2, for 1-2, under the 3, etc. Jim then moved on to 2-1, 2-2, 2-3, etc. all the way through 6-6. This is what you end up with (above the dotted line—the tally marks underneath were the first attempt at replicating these theoretical results with actual rolls of the dice):

Jim then represented this data on a bar graph (which is a concept Daegan is working on elsewhere in his math):

You can see that 7 is the most common role, though 6 and 8 aren’t far behind, and 2 and 12 are the least common rolls. Jim added 1 and 13 to the graph to emphasize the symmetry and show that there are no ways to roll those numbers using two dice.

Jim then created a bar graph for Daegan to fill in with cuisinaire rods (counters) as Jim rolled. This was a great exercise, as it showed Daegan that the general shape of the graph from real rolls was as expected—more rolls in the middle numbers (6, 7, 8 ) than the ends (2, 12). But it also showed that the real world does not match the theoretical perfectly. Jim emphasized that with just a few rolls you may well get a very odd-looking graph, but the more rolls you do, the more likely it is that the graph will resemble the one we came up with originally, the probability graph.

And indeed, by the end, we had more sevens rolled than any other number, and in general more of the middle numbers, though again, the graph did not match the probability graph perfectly.

Have fun playing math!