• Assess student performance during the M&M activity to determine level of comprehension.

• Observe student dialogue and discussion during the Counting Deer activity using the suggested strategies.

• To assess student learning, use Lesson 3 Exit Ticket (M-7-1-3_Lesson 3 Exit Ticket and KEY.docx).

This lesson was designed to allow students to experience ways in which probabilities are useful for predicting what happens over a long time period. They use probability calculations and proportional reasoning to make predictions on how outcomes would occur over time. Students also use these predictions for decision making, and determine how these types of decisions are used in everyday life. Sample problems were chosen to create a wide variation in the distribution of outcomes in a small number of trials, yet when data was pooled, the larger sample space resulted in more consistent results (law of large numbers).

As students enter the classroom hand each student a small bag of M&M candies (fun-size bag). [Note: the candy does not have to be M&Ms. You could use anything (not necessarily candy) with multicolored pieces.] Tell students not to open the bag until instructions have been given. Ask students if they can guess what colors are in the bag. They will probably be able to guess them all (red, yellow, orange, green, blue, and brown). Ask them how they were able to predict the colors (because they have previous experience with M&Ms and with repeated/predictable outcomes).

“In today’s lesson we will be looking at how we can use probability to make predictions and decisions.”

“Based on your past experience eating M&Ms, can you predict the percentage of each color you will find in your bag? Specifically, do you think all colors will be in the same proportion in the bag?”

Hand out the M&M Color Guide (M-7-1-3_M&M Color Guide.docx). “Pour your M&Ms onto your Color Guide sheet. Separate them by color and record them in column 1, and record the total number of M&Ms at the bottom of the column. In column 2 you will write a ratio for each color compared to the total number of M&Ms. In columns 3 and 4 you will rewrite your ratio as a decimal and as a percentage.” Give students 5 to 10 minutes to complete these steps. Walk around the room to clarify directions. Once all students have correctly completed the steps, allow students to eat their candies.

“I’m going to give you two minutes to compare your results with three people around you.”

“Did you get the same percentages for each color?” (Most will say no.)

“Why do you think this is the case?” (They each have a very small sample compared to the large batch mixed at the factory, so there is a large variation possible.)

Hold up the large bag of M&Ms. “I have a bag of M&Ms too. Do you think I have the same number of each color as you do? Do you think I will have the same percentages as you?” (Students will likely say that yours will be very similar; actually the M&M company uses a specific percentage for each color.)

“You are going to use your ratios or percentages to predict the number of M&Ms of each color that are likely to be in my bag. You will do this by using proportional reasoning.” Show the following example and others if necessary:

If you found 20% red in your bag, the ratio of red to total could be represented as . If you know I have 440 total M&Ms and an unknown number of red (r), the ratio in my bag could be represented as . We can use the proportion below to make a mathematical prediction about the number of red candies that will be in the large bag.

We can use cross products or a scale factor to solve the proportion:

Count the total number of M&M candies in your large bag, and the amounts of each individual color. Reveal to students only the total number of candies to use in their predictions. Allow students to complete their predictions on the M&M Color Guide.

After students have completed their calculations, reveal the number of each color from the large bag for students to compare to their own predictions. To complete the M&M activity, use a transparency of the student color guide or draw one on the board. Combine all the class data for the small bags and recalculate the color predictions based on the combined data. Ask students to record three observations on the back of their copy of the color guide. Call on several students to share an observation they made. One observation you want to focus on is that the combined data should have given predictions much closer to the large bag’s numbers than the small bags did. When this observation is mentioned, discuss how the law of large numbers works. Bring it up yourself during the activity summary if a student does not.

Separate students into eight small groups to do the Counting Deer activity. Introduce this activity by explaining how the Department of Natural Resources and other agencies need to count fish and other animal populations for a variety of reasons. Ask students to imagine trying to count every fish in a specific lake, every duck in a marsh, or every deer in a wooded area. Ask them for suggestions on how they could accomplish this and be accurate enough that decisions could be made based on their count. Let students offer suggestions.

Introduce the capture-tag-recapture strategy: a sample of wildlife is captured and tagged, then released back into the natural setting. The total number marked is recorded and compared to the total number of animals in the area, which is still unknown (use a variable such as a or x) and recorded as a ratio. At another time, more of the same type of wildlife is captured from the same area. The number that are still marked from being captured previously is compared to the total number captured that day (marked and unmarked combined) and recorded as a ratio. By using these two ratios to form a proportion (as in the M&M color predictions), the approximate number of total animals living in that area can be calculated.

Each group will need a container of beans and a copy of the Counting Deer sheet (M-7-1-3_Counting Deer Lab Sheet.docx). Students begin by counting the total number of marked deer (brown beans) in their forest (container). Important note: They should not count all the deer until the end of the activity. Let students know a few groups will be selected to share their findings at the end of the activity. Provide approximately 15 minutes to complete the lab sheet.

While students work on completing their Counting Deer sheets, check with each group to clarify the process or answer questions. If you see a group using an incorrect or illogical strategy, ask some leading questions to guide them to adjust their thinking. If a group uses a unique strategy that is logical, ask them to explain it to you. Make a mental note of a few groups that have good strategies to share. When groups share their strategies with the class, encourage the remainder of the class to record strategies different from their own, or to adjust their own work if they believe they did not use an appropriate strategy. Remind students that more than one strategy usually works for mathematical problems (for example: solving a proportion with cross products, or solving it using scale factor).

Note: There is no answer key for the Counting Deer Lab activity because the answers are based on results from the lab sheets, which will vary by group.

At the conclusion of the lesson, give each student 5 to 10 minutes to complete the Lesson 3 Exit Ticket (M-7-1-3_Lesson 3 Exit Ticket and KEY.docx).

Extension:

Use these suggestions to tailor this lesson to meet the needs of your students during the unit and throughout the year.

• Routine: Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. During this lesson the following term should be entered in students’ vocabulary journals: law of large numbers. Keep a supply of vocabulary journal pages on hand so students can add pages as needed.

Put a warm-up problem on the board once or twice per week that contains either experimental data or a sample space, and ask students to use it to make a prediction or decision. Use these examples to help students consider how much of their lives or their parents’ lives are based on predictions that come from data and the calculation of probabilities (likelihood or percentage of chance). Students should discuss specific examples of this when they encounter them in the news, magazines, and other media sources.

• Expansion: The actual percentages used by the M&M company in an industrial-size mixed batch of candies are 24% blue, 20% orange, 16% green, 14% yellow, 13% red, and 13% brown, which could be used to calculate theoretical probabilities. Have students write a one-page paper or poster describing how many M&Ms would be needed to get so close to the actual percentages that they could be used for probability calculations. Create (using a simulation, such as a color spinner, if helpful) a larger experimental data set. Explain how the law of large numbers may apply and/or be important in a different real-life situation.
• Small Group: Match Me Up

Students who might benefit from additional practice may be put into small groups to play Match Me Up. Have students use index cards to create a matching game. Each new word from the unit should be written on a card. Direct students to write a definition, diagram, or example for each vocabulary term on a second card. Each vocabulary word card should have a card to match it. Have students mix up the cards thoroughly and place them face down in a rectangular matrix pattern on the table. On players’ turns, they must draw two cards, trying to find a match of a vocabulary word and its mate (definition, diagram, or example). If they do not make a match, they must replace the cards. If they find a match, they keep the cards and take another turn. The player with the most cards, after they have all been matched, wins.

• Technology Connection: Probability Games

If students have access to computers, have them go to the following Web site and select a game. http://www.betweenwaters.com/probab/probab.html

Suggest that they choose the “Coin Flip,” “Dice Roll,” or “Key Problem.” Have students read the directions at the top of the “explain” screen and play the game. They should set the results keeper to “session.” On a piece of paper, they should record the results for playing the game 20, 50, 100, and 500 times. Ask students to predict which are the most likely ways for a person to win and why.

If time permits, ask them to go back and read the lower section of the explanation and summarize it in their own words. The text explains which are the most likely ways to win at the game they selected and why.

• Take It Home: There are many real-world situations that rely on probability alone as a means by which to make decisions. Surveys such as the U.S. census are used to gather information about a large population when it is impossible to collect all of the data that exists. The data is used to make predictions as well as decisions affecting the population being studied. Genetics is another area in which probability plays a role in decision making. Traits of offspring such as eye color are determined by the characteristic traits of the parents. Often the likelihood of an offspring having a specific trait can be diagrammed using a square or rectangular grid. Students can analyze these grids to make statements of the likelihood of the occurrence of a particular trait in an offspring.

Ask students to find a probability example at home. They can either bring it to school to explain it to the class or write about it. The example should relate to one or more of the probability concepts discussed and practiced in class. Some examples include:

• a news story or advertisement that involves a simulation and description of experimental data
• a game where students can describe the probabilities at work and/or the types of predictions a player can use based on probability
• U.S. census data used by the government to make decisions
• political polling data used to make campaign decisions
• interest surveys such as the Nielson television ratings, used by networks and advertisers to make decisions
• a record of stock market movement and types of decisions based on predictable behavior

The activity could be extended to interviewing family members or neighbors about ways in which they use probability and prediction in their career or home life. Students could provide a verbal or written report summarizing the responses from the interview. Students should discuss the implications of this activity.