Probability Games

The understanding that students are expected to gain by playing probability games is that some events are more likely to occur than others. Probability games can help them compile their own data, search for patterns that help them predict probable outcomes and use the patterns they see to generate rules for making such predictions. They, of course, don't translate their rules or patterns into formal ratios usually associated with probability yet. Simple statements about the range of answers anticipated in a given situation are all that is expected.

We have seen the bell curve appear naturally while playing a probability game with 12 variables.

We have also played a probability game with three variables, in which they had to discover that the one variable was clearly more prominent. It was learned that this game actually had four variables, instead of three.

For this next probability game I followed Love 2 Learn 2day 's example,  and put 20 colored cubes into our "Probability Pouch" ( you could make this pouch or this pouch use any container that will hold your tokens and that the student can reach in without seeing the container's contents) and it was James' job to try to figure out how many of each color was in the pouch. I didn't tell him what or how many colors I'd used...only that 20 tokens were in the pouch. The rule was that he must shake the bag and to put the game markers back in the bag after each draw. After every 10th draw, I asked him to stop, look at the data he'd collected so far, and predict what he might get over the next ten draws.
Which colors would you draw least?
Which the most?
 Look at the data again. Is there any pattern to it?
How many of each did you think you'd get?
Was it possible that there was another color in the bag?
What do you think the bag contains? Why?
How many more of one color did you have than another color?
Do you want to change your prediction?

For the first 10 draws, he got 4 red and 6 blue. He predicted that the bag held 10 red, 8 blue and 2 of another color.
For his second 10 draws, he got 4 red, 5 blues and 1 yellow. He predicted that the bag held 11 red, 8 blue and 1 yellow.
For his third 10 draws, he got 7 red, 2 blue and 1 yellow. He predicted that the bag held 13 red, 6 blue and 1 yellow.
For his fourth 10 draws, he got 6 red, 3 blue and 1 yellow.
His final prediction was that the bag held was 12 red, 6 blue and 2 yellow, which was exactly what was in the bag.
He was delighted to find out that his predictions all along were not far from the actual amount, and that his final prediction was right on the mark.
What a lovely and fun way to teach probability and he is asking to play again!

Probability with Two Colors (a variation of coin flipping)

I won this lovely Probability Pouch from Love2Learn2day, so of course I had to have some fun with it. You could use any sort of pouch, or even a paper lunch bag or a cup. It did seem exciting to Quentin, however, to use this colorful bag with such a colorful name; "Probability Pouch." Using the name also reinforces the word "probability"each time we used talked about the pouch.

I wanted to start with a two chance probability, most often called coin flipping as coins are the most readily available two-sided math manipulative to use. There are two-color math manipulatives you can buy, as well, but I decided to use our game pieces from our Othello game (another great math game) since they have two different color sides.

I started by asking him what the possible combinations could be and he easily told me two blacks, two whites and one black and one white. We then made three columns of a piece of paper and labeled them WW, BB and BW.
He then began pulling them out of the Probability Pouch in pairs and then putting a check under the appropriate column.

These two-sided pieces effectively mislead students into thinking they have only three outcomes to consider. Some students might figure our that BW occurs more times because it is a composite of the two possibilities BW and WB but it is more likely at this age (1st grade) that they will not be able to provide an explanation of why this column has more checks.

I asked him to suppose why this was to give him an opportunity to think about it, to focus his attention on the possibility that since something seems to be happening, there may be a reason for it. This develops mathematical thinking.We will spent more time exploring probability, which may solve or only deepen the mystery for him.

source: 

• Mathematics; a Way of Thinking, Bob Baratta-Lorton

Watching the "Bell Curve" Develop Naturally

You can give your students experience with the bell curve with this simple, fun game. Sketch out 12 rows of about 8 circles and have 12 markers put on the first row of circles. We used gummy frogs, but you can use any math manipulative. Take turns rolling 2 dice. Your student might right away realize that you could never get the number 1. Early on in the game, the bell curve emerges, and continues through the game until the number 7 reaches the top of the chart. Throughout the game you can ask these questions to keep them thinking, "Which are likely to win the race the most times? Which are likely to come in second? Why is that? Which ones will come in last or next to last? Why? Is this a fair race? Why or why not?" With this game the youngest students get to practice counting skills and connecting a written number with the number of dots and a numeral and move the frog that corresponds. Students who are a little older get to practice following directions, cooperating with partners and the older students are interested in seeing the results of probability and statistics.

sources:

• Frog Pond Math GEMS guide