Fun with Probability! The Probable Pen in the Cereal Box Extended Curriculum

This document describes an extended curriculum for the Fun with Probability! project. It was developed for C&I 430, another course I'm taking at the good ol' University of Illinois, taught by my advisor and statistical mentor, Dr. Kenneth Travers. The course focuses on teaching mathematics and how that teaching relates to cognitive thinking, NCTM standards and curriculum development. Further, the course helps us to explore how technology can be used in the mathematics classroom to help enrich the learning experience.

This curriculum is designed for the seventh or eighth grade, and is intended to be taught over a five day period, and assumes a 50 minute classroom period. It is based fundamentally on the "short" curriculum prepared for the Fun with Probability! project. Refer to the project description to get a general feel for this project.

This lesson should be presented early in the probability/statistical section of the curriculum, and acts as an introduction to some of the larger concepts. Please refer to this discussion of the lesson's relationship to the NCTM standards. The lesson assumes the students have a basic working knowledge of Microsoft Excel, or a comparable spreadsheet.

Required Materials
Day 1 - Introductory lecture and class discussion
Day 2 - Team execution of the Fun with Probability! simulation
Day 3 - Lecture/Demonstration of using Microsoft Excel
Day 4 - Team execution of Excel portion of project
Day 5 - Group sharing, discussion and project extensions

The materials required for this lesson are:

• Demonstration materials (cereal box or coffee can, six colored pens)
• Six-sided dice for the simulation
• Computers equipped with Microsoft Excel (or some other spreadsheet). You will need one for each group of four students in the classroom. The computers are only need on days 2-4 of the project.

Day 1 - Introductory lecture and class discussion

This day is spent in lecture/discussion format. First, the instructor presents the cereal box problem to the students, preferably by demonstration. This should lead immediately to a discussion of probability, as in "What is the probability that I will get out a red pen?". The teacher can also discuss the differences between replacement and non-replacement, and the effect on the probability.

Next, the discussion should turn toward the thrust of the cerealbox problem: How many boxes do you have to buy to get one of every kind of pen? Obviously, the reasons the students choose values is actually more important than the values themselves. We are trying to the get students talking about predicting results, and that phrase should enter the conversation at some point.

Now it's time to actually try it out. Execute the experiment with the cerealbox and the pens and have the students count the number of draws. Record the result on the blackboard, and re-open the prediction discussion. Depending on the first trial, the discussion can go many ways. Repeat this process several times until the students start to figure out that it is somewhat "random."

At that point you can start discussing how the class might go about finding out the answer experimentally. This will lead to a discussion of models, and what some possibilities are for simulating the environment. One issue to remember is that cereal boxes are not infinite in availability. Does that have any affect on the model?

Leave the students with this simple homework assignment: Think about the ways you would like to simulate this problem. We will divide into groups tomorrow and try it out.

Day 2 - Team execution of the Fun with Probability! simulation

Have 4-person workgroups defined when class starts the next day. Each group is responsible for the following:

1. Decide what model you will use to simulate the problem.
2. Have your model approved by the teacher.
3. Create a data capture tool on paper. They are responsible for providing the total number of pens for each simulation as well as the total number of blue, red, etc... pens for all of their simulations. (e.g. 5 trials resulted in 12, 15, 8, 10 and 14 pens; there were 8 blue, 10 green, 6 red, etc...)
4. Execute your model 40 times (they may do this individually, in pairs or as a team)
5. Record your results on paper, then copy them to a spreadsheet (the spreadsheet is supplied and provides a consistent data-entry format).
6. Turn your spreadsheet in on the provided floppy disk.

Notes.
1. If the students don't have enough time to complete this stage, time can be taken from the Excel instructional lesson of Day 3.
2. The instructor will be required to compile the student results into a single spreadsheet, as well as generate some random data. An example spreadsheet is provided. The goal is to have about 500 simulations for the students to work on.

Day 3 - Lecture/Demonstration of using Microsoft Excel

The third day is back to the chalk-board (or actually the overhead display panel if you have one). You will be teaching the students about histograms, and how they are created using Microsoft Excel. You will also demonstrate the Chart Wizard and how it is used.

The histogram is a way of analyzing data, useful in this probability experiment. The format of the data, after combining all of the groups looks like this:

The resulting histogram should resemble...

Microsoft Excel supports this capability by using an iterative formula that evaluates a function over a range of cells. Combined with the IF() function, this provided the required histogram. The formula looks like this:

{SUM(IF(COLUMN()=$A$1:$J$100, 1, 0))}

The curly braces {} represent an iterative formula, basically applying the formula to every single cell in the $A$1:$J$100 range. The IF() function compares the value in each of those cells to the column number, and returns a one if it matches, zero if not. The SUM() function adds all of these together. If this formula is copied to column 40, for example, one will be added for each cell that contains the value 40.

The Chart Wizard lets the student select the range of values to graph (in our case the histogram) and then steps them through identifying which rows contains labels, etc...

This day is spent teaching the students these tasks so they can apply them on Day 4.

Day 4 - Team execution of Excel portion of project

Each team receives a computer and a spreadsheet with all of the data. The spreadsheet is actually a collection of sheets, each with identical format, just different data. There is one sheet for every group (containing their 40 values); a sheet containing all of the class data; and a sheet which includes the class data, plus 2000 computer-generated data sets. The groups' job will be to perform the following tasks on their own group's data, the class data and the class + computer-generated data:

1. Calculate the average number of cereal boxes purchased.
2. Calculate the standard deviation.
3. Create a histogram table.
4. Calculate the percentage of occurrences each histogram value represents.
5. Create a graph of the histogram.
6. Calculate total number of each colored pen. Graph that total.

The students will have seen demonstrations of the first four tasks during the previous day's lecture. The fifth task however, will be a complete surprise to them. They will have to find the data, put in formulas to calculate the totals, and use the Chart Wizard to graph them.

Day 5 - Group sharing, discussion and project extensions

The final day is an opportunity for the students to share their work with their classmates, and for the class to have a general wrap-up discussion about the lesson. The teams can see the different values arrived at by the various groups, and can discuss why they vary (or don't vary, depending on the situation). The last half of the period is spent discussing expected value, and how to calculate the "correct" answer. The students can think about why their answers are different from the expected value.

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