Lesson 3

The mathematical purpose of this activity is to represent sample spaces for compound events.

Arrange students in groups of 2. Tell students that standard number cubes will be referenced throughout this unit. A standard number cube is a cube with each side labeled with one of the numbers 1 through 6. Tell them that “1 on the first cube and 3 on the second cube is different from 3 on the first cube and 1 on the second cube.” If possible, roll two different (by color or size) number cubes to demonstrate. Monitor for students who use an organized list, a table, or a tree diagram to record the sample space.

The purpose of this discussion is to help students think about sample space and to have students share any methods that they used to find the sample space. Ask previously identified students to share their representation with the class. If students did not use an organized list, table, or tree diagram they will encounter the three representations later in this lesson.

Here are some questions for discussion.

• “What strategy did you use to make sure you found all of the outcomes?” (I used an organized list to make sure that I did not miss any of the outcomes.)
• “What strategy makes more sense for you to use?” (I thought that the organized list was easier to use. I started with a tree diagram and I noticed a pattern that made it easy for me to create an organized list.)
• “How did you make sure you found all of the outcomes without repeating any?” (I knew that there were 36 total outcomes so I just made sure I did not have any repeats in the 36 outcomes that I wrote down.)
• “What is the probability of rolling 2 on the first die and a 2 on the second die?” (\(\frac{1}{36}\))
• “What is the probability that the first die rolled is a 5?” (\(\frac{6}{36}\))

The mathematical purpose of this activity is to compare different methods that can be used to represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams.

Arrange students in groups of 2. Tell students that the activity involves comparing three different methods for finding the sample space of an experiment.

Some students may create a tree diagram but may not understand how to quantify the number of outcomes in the sample space. Prompt students to look at their tree diagram and to count each individual outcome. Show students that each branch at the end of the tree represents an outcome in the sample space.

The purpose of this discussion is for students to understand how each of the three representations can be used to represent the same sample space.

Here are some questions for discussion.

• “How many cells will there be in Tyler's method for each table?” (\(2 \boldcdot 3\) first, then \((2 \boldcdot 3) \boldcdot 4\))
• “How many paths are in Lin’s method?” (The first row has 2 groups of 1 thing each. Second row has 2 groups of 3 things each. Third row has 6 groups of 4 things each.)
• “Which method do you think you would use to solve this problem? Explain your reasoning.” (I would use the organized list. I just start with the sample space from the first spinner and then add to the list using the sample space from the second spinner, and so on. It is similar to a tree diagram, but I don’t actually create the tree diagram on paper.)

If time permits display each representation for all to see.

Organized list: ALW, ALX, ALY, ALZ, AMW, AMX, AMY, AMZ, ANW, ANX, ANY, ANZ, BLW, BLX, BLY, BLZ, BMW, BMX, BMY, BMZ, BNW, BNX, BNY, BNZ

Table:

Tree diagram:

Here are some questions for discussion.

• “What are the benefits of each representation?” (The organized list summarizes the same information that is in the table and the tree diagram and it is easy to see all of the information at once. The table organizes the information so that you can see where it comes from. The tree diagram helps you to see that you have all of the possibilities.)
• “Why wouldn’t you use one representation over another?” (I prefer to use the organized list over the tree diagram because it is hard for me to see each of the combinations in the tree diagram. However, sometimes I do use the tree diagram to help me create my organized list.)
• "How could you determine the new number of outcomes without writing out the sample space when we add the fourth spinner?" (I would double the number of outcomes because I know that there are two additional possibilities for each existing outcome.) 

The mathematical purpose of this activity is for students to record the sample space for experiments involving multiple events and to explore different methods for recording sample spaces. Identify students who use an organized list, table, or tree diagram. Monitor for students who:

• do not use an organized list, table, or tree diagram
• use the same type of representation for each of the experiments
• use more than one type of representation
• use different types of representations for a single experiment

Arrange students in groups of 2. Tell students:

• “All coins and number cubes mentioned in this unit are assumed to be fair unless otherwise noted. This means that there is a an equal chance for each of the outcomes to result.”
• “After completing each question, compare your solution and your solution method to those of your partner.”

Allow students a few minutes to think of solutions on their own, before sharing with their partner.

Select previously identified students to share in this order:

• do not use an organized list, table, or tree diagram
• use the same type of representation for each experiment
• use more than one type of representation
• use different types of representations for a single experiment

The purpose of this discussion is for students to analyze different methods for recording sample spaces and to determine probabilities from a sample space using different representations. Here are some questions for discussion.

• "Why did you choose to use the same representation each time?" (I remember using tree diagrams in middle school, so I just used them again and again.)
• "Why did you choose to use different representations?" (After using the tree diagram twice, I realized that I could make an organized list quickly without having to make the tree diagram.)
• "Why did you use more than one representation for a single experiment?" (I used a tree diagram, but then I organized my results in a table. I thought that the table would help me to see the information more easily.)
• "What are the advantages of using one representation over another?" (I thought that the tree diagram was the easiest way for me to make sure that I listed all of the possibilities. However, it is difficult to read the different possibilities in a tree diagram so using an organized list is an easier way to summarize the same information.)
• “For the experiment in question one, what is the probability that the outcome is heads? Explain your answer using the representation you used.” (The probability is \(\frac{1}{2}\). I made a tree diagram and for every roll of the number cubes, half of the outcomes had heads and half of them had tails.)
• “For the experiment in question two, what is the probability that the outcome is three heads and one tail? Explain your answer using the representation you used.” (The probability is \(\frac{4}{16}\). I created an organized list. After writing HHHH, I recorded all the possible outcomes that had exactly three heads. There were four of them.)
• “For the experiment in question three, what is the probability that the outcome is R1, R2, G1 or G2? Explain your answer using the representation you used.” (The probability is \(\frac{4}{20}\). I looked in my table and each of the 4 outcomes listed only appeared one time.)