Fun with Probability! The Probable Pen in the Cereal Box Teaching Suggestions

The following document describes one way of presenting the Probable Pen in the Cereal Box in a classroom situation. Because of the nature of this lesson, students in grades K-9 can participate. Simply stop the lesson at the appropriate point for your classroom. From the global perspective, only the data gathering step and minimal analysis is required to include your results with the rest of the participating classrooms. If do not want to have your students perform the analysis, you can do it yourself in about 10 minutes.

The lesson is divided into stages. They are:

1. Preparation
2. Introduction
3. Performing the Experiment
4. Consolidating the data for your classroom
5. Optional Activities

Preparation

There are two simple steps to prepare for this lesson. First, you will need to print out and duplicate the student worksheet and classroom worksheet. These can be downloaded from the Worksheet Download page. While you're at it, print the submission worksheet so you can see what you will be submitting.

Second, you will need a six-sided die for each of your students. You may also want to obtain a large coffee can or cereal box (possibly decorated with the SnowFlakes logo) and six colored pens or markers (blue, green, yellow, orange, red and purple) for the initial demonstration. While the students will use a die to model buying boxes of cereal with colored pens, they may find it helpful to link this to a "live" demonstration.

Introduction (10-20 minutes)

To introduce this lesson, you can:

1. Talk about buying cereal boxes with prizes in them.
2. Simulate purchasing a box of cereal and randomly choosing a pen from the coffee can or cereal box.
3. Stress that each box can have any color pen, and you can't tell by looking at the box what color pen is in it (randomness).
4. Discuss the probability of getting any specific colored pen (1 in 6) when you purchase a box of cereal.
5. Discuss predictions and how they vary from actual values.
6. Talk about the number of boxes you would expect (expected value) to buy before you obtained one pen of each color (this is one of the excercises the students will perform individually).
7. Explain that since we don't have enough cans and pens to go around, we will be using a six-sided die as a substitute. Each face of the die cooresponds to a different colored pen: 1=Blue, 2=Green, 3=Yellow, 4=Orange, 5=Red, 6=Purple. You may want to ask the question, "What does rolling the die once mean?" Answer: "Buying one box of cereal and finding out what color of pen is inside."

Performing the Experiment (10 minutes)

1. Each student predicts the number of pens he or she will get for each color, and the total number of pens, recording the value on the student worksheet, row #1. (How many times will you roll a one, a two, etc.. and how many total times will you roll the die?)
2. Each student rolls their die repeatedly, placing a tally mark in the appropriate column of row #2. Important: They must stop after they have placed a tally mark in the last empty box. If they continue you will get invalid results.
3. Each student counts their tallys and records the total for each pen in row #3. They must also calculate (or count) the total number of rolls (or pens) and record it in the last box of row #3.
4. Each student records his or her own prediction of the number of total rolls (by color and total) for the entire class. Students will hopefully use their own results, possibly the results of their neighbors and the number of students in the classroom to predict.

Consolidating the data for your classroom (15 minutes)

1. On the classroom worksheet, collect the data from each student. This is best done orally in a group setting so the students will hear about the experiences of their classmates. If you have more than 20 students, use multiple copies of the classroom worksheet. (Group option: If your students like working in groups of 4-5, you may want to have each group complete a classroom worksheet on their own, consolidating the totals on a classroom summary sheet - see Optional Activities below).
2. Calculate the total row (pens and grand total) on the bottom of the classroom worksheet. (Next to last row).
3. Have students copy these values back to their student worksheet and compare to their predictions.

Optional Activities (presented in no particular order - please pick and choose!)

• Answer the questions at the bottom of the student worksheet (either individually or in a group setting).
• Have class calculate a class average. How many students were within 1 or 2 rolls of the average? How about more than 5 rolls from the average? More than 10? Discuss standard deviation. Have the class calculate the standard deviation.
• Have class calculate the expected value (sum of the recipricals of probability for each color): 6/1 + 6/2 + 6/3 + 6/4 + 6/5 + 6/6 = 14.7. Compare that value to their results.
• Create a histogram of the student totals. Does this histogram resemble any known curves (such as a normal distribution?) Why or why not?
• Calculate and compare the ratios of colored pens to total pens (should be near 1/6 for each). Why is there a difference or why is it the same? Are your dice "fair?"
• Submit your results and compare them with the averages and graphs prepared by the project coordinator. Results will be calculated daily, with a cut-off of 5:00 PM Central Standard Time (GMT -6:00), and results posted by midnite. If you want your results included, wait one day. Otherwise, you can use the results to date immediately. See the Results page for more information.

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