• Monitor student responses during the game and ask questions regarding strategies and theories. This will help determine if students are on track with the lesson goals.
• Use the exit ticket (M-7-2-3_Exit Ticket and KEY.doc) to assess student understanding of the concepts.

“What are all the possible outcomes from rolling two number cubes and adding them?” Allow students a minute to write them down. Some students may say 36 because that is the number of all the possible different rolls. If this happens, ask:

• “What is the smallest sum you can get?” (1 + 1 = 2)
• “What is the largest sum you can get?” (6 + 6 = 12)
• “Would there be the same number of outcomes if you were to find the difference?” (No.)
• “What is the smallest number you could get for a difference?” (0)
• “How many ways could the smallest difference occur?” (6)
• “What is the largest difference?” (5)

Do not suggest making a table or chart before they play the game. Students may incorrectly assume that zero is the most likely or some other incorrect assumption. Playing the game and observing the outcomes (and trying to come up with the best strategy to win) will give the experience more meaning.

“Today we are going to play a game called ‘Set the Animals Free.’” Have a student hand out one Set the Animals Free Labsheet 1 (M-7-2-3_Set the Animals Free Labsheet 1.doc) to each student, one Set the Animals Free Record Sheet (M-7-2-3_Set the Animals Free Record Sheet.doc) to each group, and six game counters to each student.

“We will start with the smaller version, which involves subtracting the number cubes. You will play with a partner or in a group of three, at the most. You will each have six animal markers, a labsheet with six cages, and one recording sheet for the group. To play the game, you roll two number cubes and subtract the smaller number from the larger. If there is an animal in that number cage on your lab sheet, you set the animal (the marker) free. The first person to set all the animals free wins.” Check for understanding here. “Be sure to record all of your rolls on your tally chart.”

“If you all put one animal in each cage, do you think this would be a fair game?” Students should agree that it is. Have a short discussion about “fair” meaning everyone has the same chances to get all possible outcomes. At some point in the lesson, differentiate that many games rely not only on chance but also on strategy and often skill (as in sports).

“Make a prediction about where your animals will land most often.

“Do you think it would be more fun if you could decide where to put your animals? In other words, some cages can have more than one animal. Would that change the fairness of the game?

“Would the game still be affected by chance or strategy alone?

“Record the number of animals in each person’s cage on the Set the Animals Free Labsheet 1. You will use this to analyze strategy later. Roll one number cube to see who goes first; then take turns rolling the number cubes. If you roll and there is no animal in that cage, you forfeit your turn. Please play only one game and stop.” Give students about 5 minutes to play one game.

After one game, ask if the games were close or not. “Did anyone, without saying what they are, think of any strategies or things you would change in the next game?

“I want you to play two more games now. Place your markers in the cages in any way you would like, but do not look at the other player’s board until you start.”

While students are playing, walk around the room and observe where students placed their markers and ask individuals what they are thinking about strategies. Remind them that the data gathered from playing only three games will not necessarily mirror the theoretical probabilities from the sample space.

Hand out the Set the Animals Free Sample-Space Organizer (M-7-2-3_Set the Animals Free Sample-Space Organizer.doc).

“Fill in the Sample-Space Organizer for subtraction (top organizer only) and make a chart of your own on the back side to summarize the outcomes.” Being able to make the chart depends on the previous lesson (Lesson 2). Therefore, if students have not done that lesson, you may need to show them the format and help them fill in the organizer. When students have had time to do both parts, make a chart on the board showing the number of ways to get the differences from zero to five.

Students should record the results below in their Sample-Space Organizer:

Assist students, if necessary, to make a summary chart on the back of their paper similar to the one below.

“Which cage number would be the best to put animals in? Do you think you should put all your animals in that one cage? Why or why not?” Typical strategies are to put all the animals in cage number 1 because that number is the most likely to occur. Other students will put what looks like a normal distribution in more than one cage, with the most in cage 1, and progressively fewer in cages 2, 3, 0, etc. Many will opt to put none in cages 4 and 5.

After students have shared some strategies, have them play three more games, with each player using one of the two (or three) most popular/agreed-upon strategies and sticking to it for each game. Some strategies are:

• all in cage #1
• three each in cages #1 and #2
• two in each of cages #1 and #2, and one in each of cages #0 and #3
• one in each cage (if students still insist that spreading them out would be a better strategy to win)

Remind students to keep track of the outcome of each roll for the entire activity in their tally chart. Pool the results of the two or three strategies from all the groups. “Did any strategy appear to be better than the others?”

(The histogram portion could be assigned as homework with the group histogram compiled the following day.) “Now, on your labsheet, you will create a frequency histogram of your results; use your tally chart.” Remind students of some of the differences between a bar graph and histogram. Include the fact that histograms usually display intervals of data, and the bars representing the intervals will touch rather than have space between them.

Note: There is a small group activity at the end of the lesson which offers an opportunity to review and practice drawing a histogram. Use the small-group histogram activity prior to this lesson if many students need the review. Use it after the lesson if only a few students appear to be having difficulty after instruction and guided practice during the lesson.

Once students are finished with their histograms, collect tally information from students on the board. Select students to fill in a large version of the frequency histogram. “What does the frequency histogram tell you about the probability of getting each difference? Do they all seem equally likely?”

This would be a good time to refer back to the organized list of all the outcomes of subtracting two number cubes on the Sample-Space Organizer. Remind students that the whole-class histogram is experimental probability and the outcomes will not exactly match what is expected using the theoretical probability.

“Now we will play a bigger version of the game using the sum of the two number cubes.” Hand out copies of Set the Animals Free Labsheet 2 (M-7-2-3_Set the Animals Free Labsheet 2.doc).

“Fill in your Sample-Space Organizer for all the possible outcomes for the sum of two number cubes. (It is below the sample-space organizer for the subtraction of two number cubes.) Also make a summary chart on the back of the sheet just as you did for the subtraction section.

“Which number cages would be best to put your animals in?

“Place your markers on your game sheet without looking at the other player’s sheet(s). Play three games using your own strategy, which is likely to be different from your partner’s. Stick to your strategy for each of the three games.”

If possible, use the number-cube roller program at http://roll-dice-online.com to explore the trend of the outcomes as the number of trials is increased. (The program can roll 5000 maximum.) As in previous lessons, explore the outcomes as the number of trials is gradually increased (for example, 10, 20, 50, 100, 500, etc.) and the overall percentages of outcomes as a large number of trials is repeated.

As students play the game, ask each team to write up its findings on a transparency to present to the class. Ask team members to explain which strategies each player chose, why, and which strategy the winner used. Remind them that they will be called on randomly to present, so everyone should be prepared to explain the group’s findings to the class (Random Reporter method). Use the exit ticket (M-7-2-3_Exit Ticket and KEY.doc) to assess student understanding of lesson objectives.

Some students may leave the activity at a basic level, realizing that the number 5 does not come up very often and the number 1 does. Listing the outcomes in an organized chart helps reinforce why this is the case. All students can conjecture regarding a strategy they think is the best, but the supporting evidence can vary greatly in sophistication.

Extension:

• Routine: Continue to emphasize and model the use of vocabulary words and provide opportunities to communicate mathematical ideas to partners and to the class. Emphasize the proper use of probability vocabulary in both student work and classroom discussions involving probability situations. Throughout the lesson the following words should have been added to the vocabulary journals: probability, fair game, frequency histogram, outcome, tally chart, sample-space organizer, and tree diagram.

Learning how to create organized lists of outcomes of compound, independent actions, and understanding how to translate them into probability is the main objective of the unit, so practicing this in other situations (flipping coins, spinning spinners, etc.) could be students’ next step. Allow students to make up their own probability game (see the end-of-unit performance assessment) and play the game with a partner. Does the game seem fair? Help them create an area model or organized list of all the possible outcomes. Students should be asked to explain whether the results seem close to the expected outcomes and support their response.

• Technology Connection: If student computers are available or you can project images from a computer, use the following link as a supplemental activity:

http://www.shodor.org/interactive/lessons/HistogramsBarGraph/

This site offers an interactive bar graph and histogram lesson for use by an individual or the class. The similarities and differences between bar graphs and histograms are emphasized. Students can create bar graphs and histograms using data in the online lesson or you can use your own class data.

• Expansion 1: Provide the Histogram & Bar Graph activity template to students (M-7-2-3_Histogram & Bar Graph Comparison.doc). Start by asking students to share what they already know about bar graphs. Have a student record this on chart paper. Do the same for the histogram. Identify any missing details and clarify misconceptions. Summarize each and have students write a complete definition for each term on their templates in the Definition spaces.
• Expansion 2: Collect month-of-birth data from the group (either just students’ birth months if the group is the whole class or, if the group is small, students’ birth months and those of their immediate family members). Have a student record the birth months in a tally chart like the one below:

After collecting the data, have students analyze the number of people for each individual month, and also for each season (Winter: December–February, Spring: March–May, Summer: June–August, Fall: September–November). Discuss the appropriate choice for labeling the scale on the vertical axis of each graph. Ask students to complete the bar graph and histograms on their template sheets and compare them with their group members’ graphs.

• Expansion 3: Students who are at or going beyond the standard may be given the following problem: Jeremiah created a game where each player flips a coin and rolls a number cube. If the coins are both heads, the player gets two times the number on the cube. If the coins are both tails, the player loses all the points for that turn. Any other coin toss means the player just gets the number on the cube. If a player quits any time before getting two tails, s/he keeps all the points for that turn. The first person to reach 50 points wins.

Create an organized list, chart, or tree diagram to show all possible outcomes for Jeremiah’s game. Play the game several times. More than two people can play; however, each player should choose a strategy that is different (examples: quit after two rolls, quit as soon as your points double, quit when you have 10 points, etc.) “What strategy do you think a player should use before stopping the rolls and flips during his/her turn?”

Consider using computer-generated number cubes and coin tosses to speed students’ data gathering and to reinforce their theories. There are many opportunities for students to explore the use of technology. The end-of-unit performance assessment (designing their own game) could be suggested to students at this point.