<p>This lesson builds on the use of game strategy to teach and practice using organizational charts and area models to solve probability problems. Students investigate the theoretical outcomes of an action, select a strategy, then play the game several times to test their strategy. They will:&nbsp;<br>- determine all possible outcomes from actions with different probabilities.&nbsp;<br>- to arrange and evaluate data sets, make a organized list, chart, or diagram.&nbsp;</p>

<p>- How may data be arranged and represented to reveal the relationship between quantities?<br>- How can probability and data analysis be used to make predictions?<br>&nbsp;</p>

<p>- Conditional Probability: The conditional probability that event <i>A</i> will occur, given that event <i>B</i> has occurred, is written P(<i>A</i>|<i>B</i>) (read as “<i>A</i> given <i>B</i>”).<br>- Histogram: A bar graph in which the labels for the bars are numerical intervals.</p>

<p>- six-sided number cubes (enough for one set per two students)&nbsp;<br>- copies of Set the Animals Free Labsheet 1 and Labsheet 2 (M-7-2-3_Set the Animals Free Labsheet 1 and M-7-2-3_Set the Animals Free Labsheet 2)&nbsp;<br>- copies of Set the Animals Free Record Sheet (M-7-2-3_Set the Animals Free Record Sheet)&nbsp;<br>- copies of Set the Animals Free Sample-Space Organizer (M-7-2-3_Set the Animals Free Sample-Space Organizer)&nbsp;<br>- counters or game markers for the game (plain or animal-related)&nbsp;<br>- grid paper&nbsp;<br>- chart paper&nbsp;<br>- colored pencils or markers&nbsp;<br>- copies of Exit Ticket (M-7-2-3_Exit Ticket and KEY)&nbsp;<br>- student copies of the Histogram &amp; Bar Graph Comparison template (M-7-2-3_Histogram &amp; Bar Graph Comparison)</p>

<p>- Throughout the game, monitor student responses and ask questions about strategies and theories. This will assist determine whether students are on track with lesson goals.&nbsp;<br>- Use the exit ticket (M-7-2-3_Exit Ticket and KEY) to assess students' comprehension of the concepts.<br>&nbsp;</p>

<p>Active Engagement, Modeling, Formative Assessment&nbsp;<br>W: The lesson focuses on examining frequency and fairness in probabilities using larger sample sets, such as a pair of number cubes.&nbsp;<br>H: Hand out Set the Animals Free labsheets and a recording sheet to engage students in the lesson. Discuss where to place the animals and what is most fair.&nbsp;<br>E: Have students play the Set the Animals Free game three times. Allow them to move their animals between games.&nbsp;<br>R: Introduce the sample-space organizer and discuss Set the Animals Free game strategy and outcomes. Following completion of the organizer, students should develop a summary chart and a histogram.&nbsp;<br>E: Monitor students' work while they complete the organizer and histogram. Allow them to take the exit ticket examination to further evaluate their progress.&nbsp;<br>T: Use feedback during monitoring to determine whether the lesson should be adjusted. Make the necessary changes to the lesson plan based on the suggestions provided in the Extension section.&nbsp;<br>O: The lecture aims to improve students' grasp of theoretical probability and help students learn several strategies for recording and evaluate experimental outcomes. After collecting and organizing data, students should be able to formulate hypotheses or conjectures. Students can also test theories using computer simulation and analysis rather than performing the experiments.&nbsp;</p>

<p><strong>"What are all the possible outcomes from rolling two number cubes and adding them?"</strong> Allow students a minute to write them down. Some students may claim 36 because it is the number of all the possible rolls. If this happens, ask:&nbsp;<br><br><strong>"What is the smallest sum you can get?"</strong> (<i>1 + 1 = 2</i>)&nbsp;<br><strong>"What is the largest sum you can get?"</strong> (<i>6 + 6 = 12</i>)&nbsp;<br><strong>"Would there be the same number of outcomes if you were to find the difference?"</strong> (<i>No.</i>)&nbsp;<br><strong>"What is the smallest number you could get for a difference?"</strong> (<i>0</i>)&nbsp;<br><strong>"How many ways could the smallest difference occur?"</strong> (<i>6</i>)&nbsp;<br><strong>"What is the largest difference?"</strong> (<i>5</i>)<br>Do not recommend that they create a table or chart before playing the game. Students may incorrectly assume that zero is the most likely outcome, or make another inaccurate assumption. Playing the game and observing the outcomes (and attempting to come up with the best strategy to win) will make the experience more meaningful.&nbsp;<br><br><strong>"Today we are going to play a game called 'Set the Animals Free.'"</strong> Have a student hand out one. Set the Animals Free Labsheet 1 (M-7-2-3_Set the Animals Free Labsheet 1) for each student, one Each group will receive the Animals Free Record Sheet (M-7-2-3_Set the Animals Free Record Sheet), as well as six game counters for each student.<br><br><strong>"We'll start with the smaller version, which requires subtracting the number cubes. You will play with a partner or in groups of no more than three people. You will each receive six animal markers, a labsheet with six cages, and one recording sheet for the group. To play, roll two number cubes and subtract the smaller number from the larger. If there is an animal in the cage marked with that number on your lab sheet, you set the animal (the marked) free. The first person to set all of the animals free wins."</strong> Check your understanding here. <strong>"Be sure to record all of your rolls on your tally chart."</strong><br><br><strong>"If you all put one animal in each cage, do you think this would be a fair game?"</strong> Students must agree that it is. Have a brief discussion about what "fair" means, which means that everyone has the same chance of receiving all possible outcomes. At some point during the lesson, remind out that many games rely not only on chance, but also on strategy and, in some cases, skill (as in sports).&nbsp;<br><br><strong>"Predict where your animals will land the most often.&nbsp;</strong><br><br><strong>"Do you think it would be more fun if you could choose where to place your animals? In other words, certain cages can have more than one animal. Will this impact the game's fairness?</strong><br><br><strong>"Would the game still be determined by chance or strategy alone?</strong><br><br><strong>"Write down the number of animals in each person's cage on the Set the Animals Free Labsheet 1. This will be used to examine strategy later. Roll one number cube to determine who goes first; then, take turns rolling the number cubes. If you roll and there is no animal in the cage, you will forfeit your turn. Please play only one game and then stop."</strong> Allow students approximately 5 minutes to play one game.&nbsp;<br><br>After one game, ask if the games were close or not. <strong>"Did anyone, without saying what they are, come up with any strategies or changes they would make in the next game?</strong><br><br><strong>"I'd like you to play two more games now. Place your markers in the cages as you choose, but don't look at the other player's board until you start."</strong>&nbsp;<br><br>While students are playing, walk around the room and watch where they placed their markers, as well as ask them what strategies they are considering. Remind them that the data collected from only three games may not accurately reflect the theoretical probabilities from the sample space.<br><br>Distribute the Set the Animals Free Sample Space Organizer (M-7-2-3_Set the Animals Free Sample Space Organizer).<br><br><strong>"Fill in the Sample-Space Organizer for subtraction</strong> (top organizer only) <strong>and make a chart of your own on the back side to summarize the outcomes."</strong> The ability to create the chart depends on the previous lesson (Lesson 2). As a result, if students have not completed that lesson, you may need to demonstrate them the structure and assist them with filling in the organizer. When students have completed both parts, create a chart on the board showing the number of ways to get the differences from zero to five.&nbsp;<br><br>Students should record the following results in their Sample-Space Organizer:</p><figure class="image"><img style="aspect-ratio:504/259;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_16.png" width="504" height="259"></figure><p>Assist students if necessary in making a summary chart on the back of their work, similar to the one below.</p><figure class="image"><img style="aspect-ratio:216/495;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_17.png" width="216" height="495"></figure><p><strong>"Which cage number would be the best for animals? Do you think you should put all of your animals in that one cage? Why or not?"</strong> Typical strategies include putting all of the animals in cage number 1 because that is the most likely to occur. Other students will put what appears to be a normal distribution in multiple cages, with the most in cage 1 and gradually fewer in cages 2, 3, 0, and so on. Many people will choose to place none in cages 4 and 5.&nbsp;<br><br>After students have shared some strategies, have them play three additional games, with each participant using one of the two (or three) most popular/agreed-upon strategies and sticking to it for each game. Some strategies include:<br><br>all in cage #1<br>three each in cages #1 and #2<br>two in each of cages #1 and #2, and one in each of cages #0 and #3<br>one in each cage ((if students continue to believe that spreading them out would be a better strategy to win)<br>Remind students to keep track of the outcome of each roll throughout the exercise in their tally chart. Combine the outcomes of two or three strategies from all of the groups. <strong>"Did any strategy appear to be better than the others?"</strong><br><br>(The histogram section might be assigned as homework, with the group histogram completed the next day.) <strong>"Now, on your labsheet, you will create a frequency histogram of your results; use your tally chart."</strong> Remind students on the differences between a bar graph and a histogram. Consider that histograms usually display intervals of data, and the bars representing the intervals will touch rather than have space between them.<br><br>Note: There is a small group activity at the end of the lesson that allows you to review and practice drawing a histogram. If a large number of students need review, use the small-group histogram activity before beginning this lesson. Use it after the lesson if only a few students appear to be struggling after instruction and guided practice during the lesson.&nbsp;<br><br>Once students have completed their histograms, collect tally information from them on the board. Select students to complete a larger version of the frequency histogram. <strong>"What does the frequency histogram tell you about the probability of receiving each difference? Do they all seem equally likely?"</strong><br><br>This is a good opportunity to go back to the Sample-Space Organizer and look at the organized list of all the outcomes of subtracting two number cubes. Remind students that the whole-class histogram represents <i>experimental </i>probability, and the outcomes will not always match what is predicted using <i>theoretical </i>probability.&nbsp;<br><br><strong>"Now we will play a bigger version of the game using the sum of the two number cubes." </strong>Distribute copies of Set the Animals Free Labsheet 2 (M-7-2-3_Set the Animals Free Labsheet 2).&nbsp;<br><br><strong>"Fill in your Sample-Space Organizer with all of the possible outcomes for the sum of two number cubes." (It appears below the sample-space organizer for the subtraction of two number cubes.) Create a summary chart on the back of the sheet, just like you did for the subtraction part.</strong><br><br><strong>"Which number of cages would be best for your animals?"&nbsp;</strong><br><br><strong>"Place your markers on your game sheet without looking at the other players' sheets. Play three games with your own strategy, which is likely to differ from your partner's. Stick to your strategy for each of the three games."</strong>&nbsp;<br><br>If possible, use the number-cube roller program at <a href="http://roll-dice-online.com">http://roll-dice-online.com</a> to investigate the trend of the outcomes as the number of trials increases. (The program can roll a maximum of 5000.) As in previous classes, investigate the outcomes as the number of trials is gradually raised (for example, 10, 20, 50, 100, 500, etc.) as well as the overall percentages of outcomes when a large number of trials are repeated.<br><br>As students play the game, ask each team to write up its findings on a transparency and present them to the class. Ask team members to describe why they chose the strategies they did, as well as the strategy the winner used. Remind them that they will be selected at random to present, so everyone should be ready to explain the group's findings to the class (Random Reporter method). Use the exit ticket (M-7-2-3_Exit Ticket and KEY) to assess students' comprehension of instructional objectives.&nbsp;<br><br>Some students may leave the activity at the basic level, noticing that the number 5 does not appear frequently whereas the number 1 does. Listing the outcomes in an organized chart helps to demonstrate why this is the case. All students can speculate on which technique they believe is the greatest, but the supporting evidence can differ substantially in sophistication.&nbsp;<br><br><strong>Extension:&nbsp;</strong><br><br><strong>Routine:</strong> Continue to highlight and model the use of vocabulary words, as well as providing chances for students to discuss mathematical ideas with their partners and the class. Emphasize the proper use of probability vocabulary in both student work and classroom discussions about probability scenarios. Throughout the class, the vocabulary journals should have had the following words: <i>probability, fair game, frequency histogram, outcome, tally chart, sample-space organizer, </i>and<i> tree diagram</i>.<br>The unit's major goal is to teach students how to generate organized lists of the outcomes of compound, independent actions and how to translate them into probability, so students may want to practice this in other contexts (flipping coins, spinning spinners, etc.). Allow students to create their own probability game (see end-of-unit performance evaluation) and play it with a partner. Is the game fair? Help them in creating an area model or organized list of all possible outcomes. Students should be asked to describe whether the results seem to be close to the expected outcomes and provide evidence to support their opinion.&nbsp;<br><br><strong>Technology Connection:</strong> If student computers are available or you can project images from a computer, use the following link as an additional activity:<br><a href="http://www.shodor.org/interactive/lessons/HistogramsBarGraph/">http://www.shodor.org/interactive/lessons/HistogramsBarGraph/</a>&nbsp;<br><br>This website provides an interactive bar graph and histogram lesson that may be used individually or in class. The similarities and differences between bar graphs and histograms are highlighted. Students can make bar graphs and histograms with data from the online lesson, or you can use your own class data.&nbsp;<br><br><strong>Expansion 1:</strong> Hand out the Histogram &amp; Bar Graph activity template to students (M-7-2-3_Histogram &amp; Bar Graph Comparison). Begin by asking students to discuss what they already understand about bar graphs. Have a student write this on chart paper. Do the same with the histogram. Identify any missing details and clear up any misconceptions. Summarize each and ask students to write a full definition for each term on their templates in the Definition spaces.<br><br><strong>Expansion 2:</strong> Gather month-of-birth information 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 similar to the one below:</p><figure class="image"><img style="aspect-ratio:244/855;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_18.png" width="244" height="855"></figure><p>After gathering the data, have students examine the number of people for each month and also for each season (Winter: December-February, Spring: March-May, Summer: June-August, Fall: September-November). Discuss the best way to label the scale on the vertical axis of each graph. Students should complete the bar graph and histograms on their template sheets and compare them to their group members' graphs.<br><br><strong>Expansion 3:</strong> Students who are at or going beyond the standard may be assigned the following problem: Jeremiah designed a game in which each player flips a coin and rolls a number cube. If both coins are heads, the player receives twice the number on the cube. If both coins are tails, the player loses all points for that turn. Any other coin toss results in the player receiving only the number on the cube. If a player quits before getting two tails, he or she keeps all of the points from that turn. The first person to reach 50 points wins.</p><p>Create an organized list, chart, or tree diagram that shows all possible outcomes for Jeremiah's game. Play the game multiple times. More than two people can play, but each player should have an individual strategy (examples: quit after two rolls, quit as soon as your points double, quit when you have 10 points, etc.) <strong>"What strategy do you think a player should use before stopping the rolls and flips during his/her turn?"</strong>&nbsp;<br><br>Consider using computer-generated number cubes and coin tosses to speed up students' data collection and reinforce their theories. There are numerous opportunities for students to investigate the use of technology. Students could be encouraged to build their own game as an end-of-unit performance assessment.</p>