Selecting objects blindly from a bag or container is one of the classic models of probability used in the elementary school classroom. I find that it is a nice situation to begin our exploration and discussion of the big ideas in probability: experimental probability, theoretical probability, and the Law of Large Numbers.
I'll begin this lesson with a hands on demonstration using actual colored blocks that I place in brown bag. during the course of the discussion I'll encourage my students to begin to speak using correct terminology, and try to elicit a formal definition of probability from my them as well:
In this discussion of theoretical probabilities, I'll make sure to add and subtract blocks from the bag in order to get my PST to talk about certainty [ P(event) = 1], impossibility [P(event) = 0], and for them to see that in any defined experiment the total sum of all possible outcomes has to equal 1.
After we examine the theoretical
probabilities for some situations, we will then actually make
some predictions and draw a block from the bag. In examining the
discrepancy what we thought should happen and what did happen
will allow us to transition into a discussion of experimental
probabilities. This discussion should lead us to the question:
This essential question is one that we will examine using a choosing marbles applet. This applet is a part of the project designed by The Shodor Education Foundation "Project Interactive" . Click on this link to open the "Marbles" applet.
This applet is well designed and has options to control: the ammounts and colors of marbles in the bag, the number of marbles drawn, the number of trials to be performd by one click of the trial button, whether to draw with or without replacement, and whether or not order matters when drawing. The applet also keeps track of the total number of trials performed, the experimental frequencies, the experimental probabilites (as percentages), and the theoretical probabilites (as percentages). These features really make it easty for PST to focus on the relationships between the percentages, and allows them to perform and analyze a much higher number of trials than they could do by hand.
Activities
For my use with my methods class, I try to begin by keeping the experiments simple enough so the PST can explore our essential question on their own. This means that I'll begin by having them change the defaults to read: "choose one marble at a time","order does not matter", and "replace marbles". I'll ask them at the beginning to set the number of trials to a small number like 5.
1. For their first exploration, I'll suggest that they choose two colors and have one of each in the bag. I want them to think about the theoretical probability of that situation - that is, there should be an equal number of each color drawn. I want them to perfrom trials in sets of 5 and continue to add repititions, observing each time what the relationship between the two percentages is. I'll have them describe the changes in the numbers over time, and to stop when their numbers have come to a point of consistensy. We'll discuss how many trials it takes for each pair of students to reach thier consitency point, and collect that data on the board. From this we can start to make a hypothesis about this relationship.
2. In order to test our hypothesis, I'll have the students change the number and color of marbles in their bag to an arrangement of their own choosing. I'll then ask them to perform trials until they feel that their numbers have become consistent again. we will again examine their arrangements and number of trials to see if we can confirm our hypothesis.
From these explorations, I want to be able to talk about the Law of Large Numbers, We will examine it's mathematical definition and talk about how we might discuss this concept with children in the guise of examining probabilities.
3. I'll give the PST who are quick to discover the relationship between the exprimental and theoretical probabilities, an opportunity to do some exploration with some of the other options of the applet. They can do some explorations without using replacement and see how their numbers compare with other experiments, as well as looking at what happens when we draw more than one marble at a time.
This applet can also be extremely ueful to examine some more sophisticated concepts in probability including dependant events. I tend to use area models to examine the theoretical probabilities involved with dependant events with my PST in the methods class, so I probably would not use this applet in that situation. I would also be able to employ more of the features of this applet if I were to teach a content course for elementary mathematics.