## M & M Candy: I Want Green

### Susan CornwellLeon County Schools

#### Description

This lesson is a fun way to compare mathematical expectations and experimental probability, and then explain any difference in the two numbers. Students use colored candy pieces (such as M & M's) for their data collection, comparisons, and explanations.

#### Objectives

The student compares and explains the results of an experiment with the mathematically expected outcomes.

#### Materials

-Enough unopened packages of colored sets of candy, such as M&M’s, for each pair of students to share one package. The brand you choose should have sets of single colors. The worksheet is designed for green, red, yellow, brown, blue, and orange.
-Two paper towels or napkins for each pair
-Worksheet (attachment)
-Calculator (optional)
-Set of either colored socks, cubes, or marbles of four or more colors for demonstration
-Overhead transparency and marker (optional). You could use the dry-erase board or a piece of large chart paper.

#### Preparations

1. Attain and gather materials for the lesson. (Refer to the materials section for necessary materials.) Decide how and when you will distribute the materials to be most effective with your limited class time.
2. Make copies of the worksheet (see associated file) for students to record their work.
3. Study entire lesson prior to teaching it.

#### Procedures

Note: Students should have prior experience in computing experimental probability and mathematical expectations (theoretical probability).

PRIOR TO ACTIVITY -- GAIN ATTENTION
1. Ask students if they have a favorite color of M&M’s (or whatever brand of candy you use) and let them share their own favorite color.

2. Tell students that your favorite is green and you wonder what the probability of getting a green M&M is when you pick one from the bag without looking.

3. Tell students we plan to investigate this experimental probability and the mathematical expectation.

DEMONSTRATION
4. Use colored marbles, socks, or blocks to review with students how to compute an experimental probability. Demonstrate by drawing one of these items at a time from a concealed container. Make sure you return items to the bag before each drawing. Do this ten times and record these results. Compute the experimental probability for each color. For instance, if you draw three red items-- express the three as the numerator. The denominator is ten (total number of items drawn). Be sure students understand how to compute an experimental probability.

5. Compute the mathematical expectation (theoretical probability) as a demonstration for the class using the same set of items. Take all items out of the container. Count and record the number of red items. Count and record the total number of items. The number of red items is the numerator; the total number of items from the container is the denominator. This is the mathematical expectation for the red items. Be sure students understand how to make this representation. Discuss the difference in the experimental probability and the mathematical expectation for each color and why this may have occurred. Students should be aware that the more times the experimental probability data is collected, the closer we would expect this number to come to the mathematical expectation.

BEGIN ACTIVITY
6. Arrange students into groups of two students and distribute materials for the activity. Tell students not to eat any candy yet! Each group receives one worksheet (see associated file), one package of candy, two paper towels or napkins, calculator (optional).

(Teacher monitors and circulates among groups.)

7. Instruct students to draw, without looking, one piece of candy from their packages at a time and record the color with a tic mark beside the appropriate color. Students return the piece of candy to the bag, shake the bag, and repeat the process until they have drawn twenty times.

8. Tell the students to record the experimental probability based on their drawings: numerator is the number of times a particular color has been drawn; denominator is the total number of draws (20).

9. Next, have the students pour out all of the candy pieces from the packet, count the number of each color, and record the result. Students should express a ratio of the number of that particular color for the numerator; the total number of pieces of candy from the bag as the denominator.

10. Tell the two students in each team to compare and discuss the difference between the mathematical expectation and the experimental probability for the green candy and write an explanation for the differences.

11. Instruct the students to prepare to share their results with the class.

WRAP UP
12. Let students share their comparisons plus their explanations for attaining different numbers for the mathematical expectation and the experimental probability for the green candy. Record their data on an overhead projector or dry-erase board to elicit class discussion about the differences in their numbers.

13. Collect students’ worksheets for formative assessment, providing effective feedback, and return the worksheets to the students the following day.

#### Assessments

Use completed worksheets (see associated file) to formatively assess the student’s ability to compare experimental probability with the mathematical expectation and explain the results.

#### Extensions

Students could repeat the experiment using number cubes. Under teacher direction the class could determine the mathematical expectation for each of the six numbers on the cube to appear when rolled. Next students could roll the cube for a designated number of times and record how many times each number appeared face up. Next the students could record their results for experimental probability. Finally, discuss the students' findings and compare their experimental probabilities with the earlier determined mathematical expectation. Students could discuss that a few rolls of the cube may not provide the mathematical expectation.

ESE or ESOL students: Provide every two students with a penny. Determine the mathematical probability of heads landing face up after being flipped into the air (1/2). Instruct students to flip their penny ten times and record the number of times that heads appears face up. Express this erperiement as an experimental probability - numerator: # of times the penny landed heads up/ denominator: 10. Compare the mathematical probability with the experimental probability. If time permits, repeat the activity flipping the penny 50 or 100 times.

Another activity would be to sort out the hearts from a deck of cards and compute the mathematical probability of selecting an Ace, if the cards were face down, and one was chosen and then returned (1/13). Let the students place the cards face down and draw a designated number of times and determine the experimental probability of drawing an Ace from the set of hearts. Compare the two numbers. You could also determine the mathematical probability for drawing an Ace or Jack (2/13). Next draw and compute the experimental probability. Lastly compare the mathematical probability with the experimental probability. Students could repeat the activity with more draws. The expectation is that the more times the experiment is done, the closer the experimental probability would be to the mathematical probability.

#### Attached Files

Students compare experimental and theoretical mathemtaical probability.     File Extension: pdf