<p>This lesson covers the methods for determining experimental probabilities as well as calculating theoretical probabilities. Students will learn how to distinguish between equally and unequally likely events, as well as their associated probabilities. Students will:<br>- compare experimental and theoretical probability.<br>- use experimental probability to make predictions and conjectures.<br>- understand the difference between equally and unequally likely events.<br>- write a concept for equally likely outcomes and provide examples of basic experiments with equally likely outcomes.<br>&nbsp;</p>

<p>- How are mathematical properties of objects or processes measured, calculated, and/or interpreted?&nbsp;<br>- How can probability and data analysis be used to make predictions?&nbsp;<br>&nbsp;</p>

<p>- Complementary Event: The opposite of an event. That is, the set of all outcomes of an experiment that are not included in an event.&nbsp;<br>- Equally Likely: Two or more possible outcomes of a given situation that have the same probability. If you flip a coin, the two outcomes—the coin lands heads up and the coin lands tails up—are equally likely to occur.&nbsp;<br>- Outcome: One of the possible events in a probability situation.&nbsp;<br>- Probability: A number from 0 to 1 that indicates how likely something is to happen.</p>

<p>- two to three sheets per student of vocabulary journal pages, copied back-to-back (M-7-2-1_Vocabulary Journal Page)<br>- buckets (not clear) or paper bags to draw objects from<br>- Stations Walk transparency (M-7-2-1_Stations Walk)<br>- chart paper with one event from the Stations Walk transparency on each sheet (M-7-2-1_Stations Walk)<br>- colored objects (cubes, counters, etc.) or Color Cards (M-7-2-1_Color Cards)<br>- Color Mystery Record Sheet paper copies and transparencies (M-7-2-1_Color Mystery Record Sheet)<br>- Color Mystery Graph, paper copies and transparencies (M-7-2-1_Color Mystery Graph)<br>- red, blue, black, and yellow markers<br>- transparency markers for groups (red, blue, black, and yellow)<br>- calculators<br>- copies of Partner Quick Quiz (M-7-2-1_Partner Quick Quiz and KEY)</p>

<p>- Examine student vocabulary journals for new entries and check the definitions for accuracy.<br>- Observe student performance throughout class discussions and small-group activities to determine which students require further assistance and which concepts require additional work.<br>- Use the Partner Quick Quiz (M-7-2-1_Partner Quick Quiz and KEY) to examine students' comprehension of probability concepts.<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, Formative Assessment&nbsp;<br>W: The lesson focuses on probability, including various kinds of probability and the likelihood of events.&nbsp;<br>H: Use the Stations Walk activity to encourage students to consider and discuss the likelihood of various events.&nbsp;<br>E: Use the Color Mystery activity to distinguish between experimental probability and theoretical probability. Discuss equally and unequally likely events involving the colored blocks from the activity. Introduce the concept of complimentary events and discuss how they are equally likely to occur.&nbsp;<br>R: Use the Carnival Ducky game or a similar scenario to help students rethink and clarify conceptual ideas such as probability, equal likely, and unequal likely.&nbsp;<br>E: The Partner Quick Quiz can be used to assess students' understanding.&nbsp;<br>T: Tactile activities such as randomly selecting blocks from a bag, collecting and comparing data, and displaying data in a bar graph engage students in learning concepts from many perspectives. The lesson can be altered as needed using the suggestions in the Extension section.&nbsp;<br>O: The lesson begins with a brief overview of the difference between experimental and theoretical probability, followed by a hands-on experiment in which all students contribute to the conclusions. Trends are identified and predictions are produced using the students' own data. As a group, students discover that as the data set gets larger, the experimental probability gets closer to the theoretical probability, as predicted by the law of large numbers.&nbsp;</p>

<p>Give students two or three vocabulary journal pages (M-7-2-1_Vocabulary Journal Page) to keep in their folders or binders. Discuss how important it is to comprehend and use the appropriate vocabulary terms while communicating mathematical ideas. Explain to students that they will keep a vocabulary journal throughout the unit. When students encounter a new vocabulary word, they can record it in their journals. At the end of each lesson, students should review and update their journals with any new terms they learned.&nbsp;<br><br><strong>"In today's lesson, we'll look at why two or more events are equally likely or not. We will examine probability situations and determine the probabilities. We will compare two types of probability and how they are related. In the next lesson, we'll practice strategies for understanding more complex probability situations."</strong><br><br>Prepare ahead of class by putting one pair of events from the Stations Walk transparency (M-7-2-1_Stations Walk) on each sheet of chart paper and posting the sheets around the classroom. Display the Stations Walk transparency when students arrive.&nbsp;<br><br>Consider a question about yourself that students would be familiar with. You could include favorite things, subjects you teach, marital status, age, and so on. For example:&nbsp;<br><br>If students know a teacher lives a few blocks from school, they can ask, <strong>"Is it equally likely or unequally likely that I will walk or drive home from school today?"&nbsp;</strong><br><strong>"Is it equally likely or unequally likely that I like doing math problems or I don't?"</strong><br>Encourage students to share and discuss the reasoning and relevant facts they used to make their conclusion. Before beginning the Stations Walk Activity, have a brief discussion about what equally likely and unequally likely events are. Modify the definitions as needed.&nbsp;<br><br>Divide students into eight groups to begin the Stations Walk (one group each station).&nbsp;<br><br><strong>"I'll lead you to numerous stations for this exercise. Please only bring one marker per group. At the first station, your group will read the events given, write at the top of the work space whether they believe the events are equally likely or unequally likely, and explain why.</strong><br><br><strong>"Remember to provide space for other groups to respond below your comments. At my signal, you will rotate clockwise to a new station and record your response and reasoning below the previous group's response."</strong> Have students rotate through four stations prior to the class discussion.&nbsp;<br><br><strong>"When you finish with the last station, I will assign someone from your group to read the events and someone to summarize all of the responses. You will have approximately one minute to work at each station."</strong><br><br>As groups summarize the responses on their posters, ask questions and involve the class in discussions to get them thinking critically about each situation and what could have helped them decide which were equally likely. Some of the situations may appear obvious (such as problems 2, 5, and 6). Encourage students who disagree to discuss their argument and provide specific examples. This type of discussion can be an effective way to teach the concepts of equally likely and unequally likely. To finish, ask students to write the definition of <i>equally likely</i> on the poster. Students should stay with their groups for the next activity.&nbsp;<br><br>In the Color Mystery activity, students will gather information and make predictions about what is inside a container. They will also be asked to change the contents of the container to make specific outcomes equally likely.<br><br>Prepare one bucket or paper bag (the size of a lunch bag) for each group. (They shouldn't be transparent containers.) Inside each container, place 24 colored blocks, counters, folded color cards (M-7-2-1_Color Cards), or other uniformly shaped colored objects. Each 24-piece set should have 12 blue, 8 red, and 4 yellow objects. Prepare paper copies and transparency copies of the Color Mystery Record Sheet (M-7-2-1_Color Mystery Record Sheet) and Color Mystery Graph (M-7-2-1_Color Mystery Graph) for each student. Do not give any information about colors or amounts to the students; only the sort of object should be known.<br><br><strong>"Can anyone explain the difference between experimental and theoretical probability?"</strong> Discuss briefly.<br><br><strong>"The first task for your groups is to collect experimental data. Without looking at what is in your container, draw one</strong> (name the object you placed in the containers) <strong>and write down the color on the record sheet in the Data Set 1. Replace the</strong> (object) <strong>and repeat until you've picked and replaced 20 times. Make a note of the color on your record sheet after each time you select. Stop after 20 tries and try to assess whether each of the colors you drew is equally likely."</strong> Do this activity for approximately 3 to 4 minutes.&nbsp;<br><br><strong>"What colors do you think are in the container?&nbsp;</strong><br><br><strong>"Do you think there is equal number of each color in the container?</strong><br><br><strong>"Do you think each color has an equal chance of being drawn next? How does the relative number of each color influence your prediction of the outcome?&nbsp;</strong><br><br><strong>"What do you think the probability of selecting _____ next is? Why?"</strong>&nbsp;<br><br>The complete contents of the container should not be revealed at this time.&nbsp;<br><br>Ask the groups to present their findings. Ask the class <strong>"Do all of your colors appear to be equally likely. Why, or why not?"&nbsp;</strong><br><br><strong>"All of your containers have the same contents. Why do you think the results of the groups are so different?"</strong> Students should understand that the results of experiments will vary. If they do not, recommend doing additional trials.<br><br>Ask, <strong>"Based on all of the groups' results, who wants to make a prediction about the ratio of colors in our containers?"</strong> Make a few predictions on the board and refer to them later when the contents are revealed.&nbsp;<br><br><strong>"Would having more data help us identify the color ratio and whether they are equally likely? Why, or why not?&nbsp;</strong><br><br><strong>"Each group will collect more data. Draw 20 more</strong> (objects) <strong>to complete the second data set.&nbsp;</strong><br><br><strong>"Add the total number of each color to the table on your record sheet.</strong><br><br><strong>"Calculate the fractions and percentages from the two sets of combined data. Finally, only answer questions 1–3."</strong> For students who struggle with multi-step instructions, write the instructions on the board or circle the parts on the overhead.&nbsp;<br><br>When students have completed questions 1–3, reveal that the total number of blocks in the container is 24. Have students respond and discuss question 4.<br><br>Before students respond to question 5, reveal the exact quantity of each color of block (12 blue, 8 red, 4 yellow). In the discussion after question 5, mention that the percentages they discovered for their 40 trials are the experimental probabilities. Additionally, emphasize the corresponding forms of probability (fraction, decimal, and percentage). Discuss the percentages for the actual numbers of colored blocks (the theoretical probabilities). Compare the predictions made earlier to the actual ratios shown.&nbsp;<br><br>The concept of equally likely should be reviewed in terms of the fraction of different colored blocks in the container.&nbsp;<br><br><strong>"How do the fractions (or percentages) of colors relate to the likelihood of choosing a specific color?"&nbsp;</strong><br><br>Give students 1–2 minutes to calculate the theoretical probabilities stated below. Check that students grasp the form P(event) and the question being asked. Make sure to explain that "P(not yellow)" is the <i>complement</i> of an event and what it signifies.<br><br>1. P(red)<br>2. P(blue)<br>3. P(red or yellow)<br>4. P(not yellow)<br>5. P(not blue or yellow)<br><strong>"Now that you know the exact number of each color, it is clear that the three colors are not equally likely to be drawn. Take a minute to consider how we could add or remove blocks to make red and blue blocks equally likely to be drawn, without changing the total number of blocks and without the probability of any color being \(1 \over 2\)"</strong> (Several correct responses are possible. For example, adding and subtracting enough blocks to end with a total of 1 blue, 1 red, 22 yellow, or 2 blue, 2 red, and 20 yellow, etc.)&nbsp;<br><br>The second part of this activity compares experimental and theoretical probabilities and shows how the law of large numbers applies to probability problems.<br><br><strong>"We're going to start the second part of the activity. You will be combining the data that each of our groups collected. You'll need the Color Mystery Graph sheet</strong> (M-7-2-1_Color Mystery Graph). <strong>I'm going to ask each group for the findings of the experiments, and we'll all note it on the table, totaling it as we go. This is a great opportunity to practice mental arithmetic, so be prepared to be called on for each new total and do not use a calculator."</strong> This should take approximately 5 minutes.<br><br><strong>"Next, we'll calculate the percentage of each color that was drawn. We will begin with the first 40 trials, starting with blue. We discovered that the number of blue items drawn was XX, with a total of 40 trials. So we divide XX by 40 to calculate the percentage of blue items drawn. We'll do the same thing with red. We know there were YY red items drawn in the first 40 trials, therefore we divide YY by 40 to calculate the percentage of red items drawn in the first 40 trials. You will do the same with yellow.</strong><br><br><strong>"The first space in the 80 column represents the total number of blue items drawn, ZZ, across 80 trials. So we'll take ZZ and divide by 80 to calculate the percentage of blue items drawn in 80 trials. Complete the table during working hours.&nbsp;</strong><br><br><strong>"Before we begin this step, I would like you to spend 3 minutes discussing with your group what you think our class results will be. Will they differ from your results? What specific information or data will influence your prediction?"</strong> Ask some groups to share their responses.&nbsp;<br><br>After a few minutes of work, interrupt and show students how to graph the first column or two.<br><br><strong>"After you have finished the table, go ahead and graph the percentages from the table."</strong> Allow students around 10-15 minutes to complete the table and graph. This could potentially be assigned as homework if time does not allow for completion during class.&nbsp;<br><br>When the students have completed, ask them to share their observations. The discussion should lead to the conclusion that the inclusion of extra data sets brought the experimental probability closer to the theoretical probability. Explain how this forms the foundation of the law of large numbers. Use specific instances, such as the difference between the results of 10 fair coin flips and 10,000 flips of the the same coin.<br><br>This is a great opportunity to use a computer-generated simulator, such as the one found at <a href="http://www.shodor.org/interactivate/activities/Marbles">http://www.shodor.org/interactivate/activities/Marbles</a>. The simulator draws marbles from a bag. Adjust the numbers of the various colors to match the experiment numbers. Start with 10 trials, then 50, 100, 1000, and so on. Students should pay special attention to the differences between experimental and theoretical probabilities. When you achieve a very large number of trials, you will notice that the experimental probability still fluctuates above and below the theoretical probability. This could also be an extension activity, as described below. Also demonstrate that every time a fixed number of trials is performed, the probability changes, just as the class groups had different results.<br><br>Introduce a new scenario to students. Use the one below, or make your own.&nbsp;<br><br>The school carnival included a game with 120 rubber duckies swimming in a pool. Each is labeled at the bottom with a phrase stating what prize, if any, a player earns by selecting that duck on his or her turn. After each person's turn, the ducks will be returned to the pool, ensuring that there are always 120 ducks in the pool when a player makes his or her selection. Ducks are marked as follows:&nbsp;<br><br>40 are marked "no prize."&nbsp;<br>20 are marked "stuffed animal."&nbsp;<br>12 are marked "small toy."&nbsp;<br>8 are marked "$10."&nbsp;<br>The rest are marked "candy bar."<br>Ask students to record responses to the following or similar questions about a situation of their choice. Assume 80 participants played the game (picked one duck to win a prize).&nbsp;<br><br><strong>"Are there any outcomes which are equally likely to be selected?"</strong> (<i>No prize or candy bar because each has 40.</i>)&nbsp;<br><strong>"Which prize do you predict will be won the most often?"</strong> (<i>Candy bar since it contains the most markings other than no prize.</i>)&nbsp;<br><strong>"What is the probability of that prize being selected?"</strong> (<i>There are 40 out of the total of 120 ducks, so \(40 \over 120\) or \(1 \over 3\).</i>)<br><strong>"How many times out of 80 do you predict that it will be selected?"</strong> (<i>Since the probability is \(1 \over 3\), 80 ÷ 3 is about 26 or 27 times</i>)<br><strong>"Which prize do you think will be selected least often?"</strong> (<i>$10 because there are only 8 ducks marked for this</i>)<br><strong>"What is the probability of this prize being selected?"</strong> (<i>There are 8 out of 120 ducks in total, so \(8 \over 120\) or \(1 \over 15\).</i>)<br><strong>"How many times out of 80 do you predict it will be selected?"</strong> (<i>\(1 \over 15\) of 80 times, so 80 ÷ 15 is between 5 and 6</i>)<br><strong>"Find the probabilities for the remaining prizes."</strong><br>P(<i>No prize</i>) = <i>\(1 \over 3\)</i>; P(<i>stuffed animal</i>) = <i>\(20 \over 120\)</i> or <i>\(1 \over 6\)</i>; P(<i>small toy</i>) = <i>\(12 \over 120\)</i> or <i>\(1 \over 10\)</i><br><br>The challenge question for any group that finishes early is: <strong>"How many of each prize should they have ready if they expect 100 people to play the game?"</strong><br><br><i>Stuffed animal: \(1 \over 6\) = 16.666 or about 17%, so <u>17</u> (out of 100 tries)</i><br><br><i>Small toy: \(1 \over 10\) = 0.10 or 10%, so <u>10</u> (out of every 100 tries)</i><br><br><i>$10: \(1 \over 15\) = 0.0666 or about 7%, so <u>7</u> (out of every 100 tries)</i><br><br><i>Candy: \(1 \over 3\) = 0.333 or about 33%, so <u>33</u> (out of every 100 tries)</i><br><br><i>No prize for 33 people.</i><br><br><i>Check: the total is 100</i><br><br>Challenge question for students at or going beyond the standard: <strong>"How many of each prize do you think they should have ready if they expect 180 people to play the game?"</strong><br><br><i>Stuffed animal: 30</i><br><br><i>Small toy: 18</i><br><br><i>$10: 12</i><br><br><i>Candy: 60</i><br><br><i>No prize for 60 people.</i><br><br><i>Check: the total is 180</i><br><br>Allow approximately 15 to 18 minutes to finish the activity. Circulate throughout the classroom, asking students guided questions and addressing any misconceptions. Inform students that they will be divided into two or three groups at random to present their answers.<br><br>Throughout the Stations Walk and Part 1 of the Color Mystery activity, students should be given regular opportunities to discuss, ask questions, and correct their work. Part 2 of the Color Mystery activity allows students to rethink and clarify concepts such as probability, equal likely, and the law of large numbers.&nbsp;<br><br>To assess students' comprehension of the concepts covered in this lesson, have them work for 5 to 10 minutes on the Partner Quick Quiz (M-7-2-1_Partner Quick Quiz and KEY).<br><br><strong>Extension:</strong><br><br><strong>Routine:</strong> Students share ideas by working on problem-solving strategies with partners and groups. The emphasis is on communicating probability concepts with correct vocabulary. Students will make a vocabulary journal. Throughout the lesson, students will record definitions and examples of each word in their journals. This lesson should include the following terms: <i>outcome, complementary event, experimental probability, theoretical probability, equally likely, unequally likely, </i>and<i> law of large numbers</i>.&nbsp;<br><br><strong>Small Groups:</strong> Students who struggled with the Partner Quick Quiz or who still require extra learning can work on a simple problem to gain a better understanding of the concepts in this lesson.&nbsp;<br>For example, flip a coin once. If you get heads, ask students, <strong>"Does this mean I'll always get heads because I got heads 100% of the time (1 out of 1)?"</strong><br><br><strong>"What happens if I also flip heads on my second toss? Can I be confident that it will always be heads as it is still 100%?&nbsp;</strong><br><br><strong>"On the third toss, I got tails. So does that mean \(2 \over 3\) of the time I can expect heads and \(1 \over 3\) of the time expect tails? What else should I consider?"</strong> (Students might say, "You need to flip it a lot more times before you decide the probabilities.")&nbsp;<br><br><strong>"I'm going to toss a coin ten times. Predict how many heads and tails you think I'll get. Note it down on a piece of paper."</strong> Toss the coin and compare the results with the students' predictions. Ask why they made their predictions in this way. Use this opportunity to explain that, over a large number of trials, the theoretical probability of heads and tails is 50% for each.&nbsp;<br><br>Have students flip coins and keep a record. After every five (or ten) trials, have them calculate the probabilities for heads and tails. Discuss how their probabilities get closer to 50% each time as they undertake additional trials. Instead of flipping coins, instruct students to use a computer-based random-number generator (<a href="http://www.random.org/integers/">http://www.random.org/integers/</a>) or coin flipper <span style="background-color:rgb(255,255,255);color:rgb(8,42,61);">(</span><a href="http://www.random.org/coins/">http://www.random.org/coins/</a><span style="background-color:rgb(255,255,255);color:rgb(8,42,61);">)</span>.<br><br><strong>Expansion 1:</strong> For students that performed well on the Partner Quick Quiz and shown proficiency throughout the lesson's block activity, apply the following expansion. Continue to work with the container of blocks from the lesson. <strong>"Using the addition of blocks to your container (without removing any blocks), figure out how to make drawing red and blue blocks equally likely, while maintaining the present probability of selecting yellow. Examine your answers and explain the relationship between the new and original numbers of blocks of each color."</strong><br>(Possible solution: <i>Add 3 blue, 7 red, and 2 yellow to get the overall totals 15 blue, 15 red, and 6 yellow (blue \(15 \over 36\) = \(5 \over 12\), red \(15 \over 36\) = \(5 \over 12\), yellow \(6 \over 36\) = \(1 \over 6\)</i>). Other solutions involve any number of block additions that result in multiples of the totals in the first solution.)<br><br><strong>Expansion 2:</strong> To challenge students who demonstrated proficiency and advanced thinking during the class, use the Carnival Ducky Pool scenario from the lesson, but add the following information:<br><strong>"The carnival committee expects about 220 students to play the game and each will pay $1.00 to play. The game's rewards cost the following amounts:</strong><br><br><strong>Candy = $0.25</strong><br><strong>Small Toy = $0.65</strong><br><strong>Stuffed animal = $1.10</strong><br><strong>$10 bill = $10”</strong><br><br>Have students respond to the following questions:&nbsp;<br><br><strong>"How much money will the Ducky game earn from ticket sales?"</strong> (220 x 1.00 = $220)&nbsp;<br><strong>"How many of each prize should the committee expect to give away?"</strong><br><i>No prize: about 73</i><br><br><i>Candy: about 73</i><br><br><i>Small Toy: 22</i><br><br><i>Stuffed animal: about 37</i><br><br><i>$10: 15</i><br><br><i>Check: total is 220</i><br><br><strong>"How much will they spend on these prizes? Show your work."</strong><br><i>Candy: about 73 × 0.25 = $18.25</i><br><br><i>Small Toy: 22 × 0.65 = $14.30</i><br><br><i>Stuffed animal: about 37 × 1.10 = $40.70</i><br><br><i>$10: 15 = 15 × 10 = $150</i><br><br><strong>"Will the committee earn enough from the 220 tickets they plan to sell to pay for the prizes you predict they will need to give away?"</strong> (<i>No, because the total amount earned is $220 and the total cost of prizes is $223.25.</i>)<br><br><strong>Technology Connection:</strong> <a href="http://www.shodor.org/interactivate/activities/Marbles/">http://www.shodor.org/interactivate/activities/Marbles/</a>&nbsp;<br>This website contains a colored-marble simulation activity that provides a good overview of the law of large numbers. Use the classroom computer to display the simulation, or have students work on PCs to create their own simulation. As an extension, students can investigate what happens when the color ratios changed, more or fewer colors are used, and the number of marbles drawn at once is changed.<br><br>Another website link is <a href="http://nces.ed.gov/nceskids/createagraph/default.aspx">http://nces.ed.gov/nceskids/createagraph/default.aspx</a>, where students can generate the graph from the Color Mystery activity.</p>