<p>Students solve problems by calculating probability based on the area represented by each of the possible outcome. Tree diagrams and other probability organizers are reviewed and discussed. The concepts of independent and dependent events are discussed. Activities use only independent compound events. Students will:&nbsp;<br>- use tree diagrams to list outcomes of compound events.&nbsp;<br>- calculate probabilities with tree diagrams.&nbsp;<br>- make and apply area models to calculate probability.&nbsp;<br>- understand the distinction between area models and organized charts of outcomes.&nbsp;<br>- define the following vocabulary terms: area model, experimental vs. theoretical probability, independent events, organized lists and charts of outcomes, sample space, simple event, and compound events.&nbsp;<br>&nbsp;</p>

<p>- What makes a tool and/or strategy suitable for a certain task?&nbsp;<br>- How are mathematical properties of objects or processes measured, calculated, and/or interpreted?&nbsp;<br>- How may data be arranged and represented to reveal the relationship between quantities?&nbsp;<br>- How can probability and data analysis be used to make predictions?&nbsp;<br>&nbsp;</p>

<p>- Compound Event: An event made up of two or more simple events.&nbsp;<br>- Dependent Events: Two events in which the outcome of one event affects the outcome of the other event.&nbsp;<br>- Independent Events: Two events in which the outcome of one event does not affect the outcome of the other event.</p>

<p>- student copies of vocabulary journal pages, copied back-to-back (M-7-2-1_Vocabulary Journal Page)&nbsp;<br>- copies of Matching Game (M-7-2-2_Matching Game) or grid paper with a large grid&nbsp;<br>- coins to flip with partners&nbsp;<br>- copies (one per group) of vocabulary graphic organizer, paper, or transparencies, (M-7-2-2_Four-Square Vocabulary Organizer)&nbsp;<br>- chart paper and markers for group use&nbsp;<br>- student copies of the supplemental area models worksheet (M-7-2-2_Lucky Winner and KEY)</p>

<p>- Observe students in small groups and throughout class discussions to see if they are gaining sufficient comprehension.&nbsp;<br>- To check if students comprehend the concepts, ask small groups to report on the new games they created for the Matching Game activity.&nbsp;<br>- Assign the optional homework assignment (M-7-2-2_Lucky Winner and KEY) to students who would benefit from more practice, and use the results to assess student development.&nbsp;<br>- A partner activity can be utilized as a warm-up for the following lesson. This would provide further insight into the students' understanding of the lesson materials.&nbsp;<br>- Check students' definitions in their vocabulary journals to ensure they have not written incorrect information.<br>&nbsp;</p>

<p>Active Engagement, Modeling, Formative Assessment&nbsp;<br>W: The lesson focuses on applying area models to determine outcomes and theoretical probabilities.&nbsp;<br>H: Engage students in the lesson by discussing the fairness of the game or the fairness of Calvin's seat-choosing approach.&nbsp;<br>E: Encourage student participation in creating sample-space organizers by asking questions and thinking aloud as they fill them in. To learn more, have students talk you through solving the Calvin problem.&nbsp;<br>R: Encourage students to construct area models representing all possible outcomes by presenting game scenarios in small groups. After the small groups have completed their tasks, work as a large group to determine the problem's sample space.&nbsp;<br>E: Assess student comprehension through questioning and observation during small-group work. Additionally, vocabulary journals can be checked.&nbsp;<br>T: To customize the lesson, refer to the Extension section for suggestions.&nbsp;<br>O: This lesson aims to teach students how to solve probability questions in multiple ways using one-stage and compound area models, as well as get students to understand the differentiate between organizational charts and area models. No dependent events are modeled, but students are introduced to the definitions and differences between independent and dependent events as a foundation for future lessons.&nbsp;</p>

<p>Place this lesson's vocabulary words on the board and invite students to define them in their vocabulary journals as the lesson develops. Keep a supply of vocabulary diary pages (M-7-2-1_Vocabulary Journal Page) available for students to use as needed. Write "Words you should know by the end of the lesson" above them. When students arrive for class, draw a huge spinner on the board or project it into the overhead screen.</p><figure class="image"><img style="aspect-ratio:154/143;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_10.png" width="154" height="143"></figure><p><strong>"You're going to play with a partner to see who gets to</strong> (do whatever fun for them as an incentive or reward). <strong>When the spinner lands on the shaded portion, the person on the right receives a point, while the person on the left gets a point when the spinner lands on the unshaded portion."</strong> (Instead of left and right, use boys and girls, earliest birthday, etc.) <strong>"When you get to 10 points, you win."</strong>&nbsp;<br><br>Students should respond instantly with, "That's not fair!"&nbsp;<br><br><strong>"We can all agree that that would be an unfair game. The shaded part of the spinner is significantly smaller than the unshaded part."</strong> This is an appropriate opportunity to talk about chance (experimental) versus expected (theoretical) outcomes with students. Expected (theoretical) value is a useful addition for challenging students at or going beyond the standard. Add expected value when possible. An extended activity for finding expected value is available at the end of the lesson.&nbsp;<br><br><strong>"Some probability scenarios are not as clear. Today we will look at tree diagrams, organized lists, and charts. We'll learn how to use area models to estimate outcomes and theoretical probabilities. These methods are used to study theoretical probability and provide predictions for a wide range of real-world applications, including games, sports, weather, attendance patterns, and many more."</strong><br><br>Begin the class with a story: <strong>"Calvin and his brother both like to ride in the front seat of the car. To avoid arguing about it every day, Calvin devised a game to determine who sits in the front seat. The boys individually flip a coin. If the coins match, Calvin wins and gets to sit up front. If the coins do not match, Calvin's brother will win. Do you think Calvin is being fair or trying to trick his brother?"</strong> Students will likely have two major opinions regarding this:&nbsp;<br><br><i>The game is unfair because there are 3 possible outcomes: H-H, T-T when Calvin wins, or one H and one T where his brother wins. Calvin has more chances.&nbsp;</i><br><i>The game is fair since it has 4 possible outcomes: H-H, T-T, T-H, H-T. So each boy has the same probability of winning.</i>&nbsp;<br>This is an excellent opportunity to test the idea of whether or not T-H is the same as H-T. Allow students to play 20 games with partners. They can keep track of the results using notebook paper. Allow them to record the "one of each" situation whichever they like at this point. It will become clear later that these are indeed unique outcomes. As you walk around the class, take note of students who are trying to figure out whether they flipped H-T or T-H. Because the coins are identical, you may notice some groups attempting to keep them "separate" as they toss, or tossing them one at a time and recoding the results as first and second because they recognize that the order probably makes a difference. Some students may mark their coins in some way for the same purpose.<br><br><strong>"How many people think the game is fair?"</strong> Ask individual students why they think this way. <strong>"How many thought it was unfair before they started but now think otherwise?"</strong> Again, ask for their explanation. <strong>"How many think the game is unfair?"</strong> Someone will undoubtedly think so if one person scored a significantly larger number of points.&nbsp;<br><br>Encourage students to discuss their results by writing the number of games won by Calvin and his brother on the board or overhead. Make sure to point out that when the data from the entire class is combined, it appears considerably closer to the theoretical probability. If possible, combine data from previous classes to observe the change in experimental probability. This is a good opportunity to discuss <i>chance</i> (experimental) versus <i>expected</i> (theoretical) <i>outcome</i>. Even though you would expect to get heads half the time with a single coin, you can flip it multiple times in a row and get heads every time. This is a fantastic extension topic for students who are at or going beyond the standard, and it is a great review of the previous investigation on the law of large numbers.&nbsp;<br><br><strong>"This situation, like many others, can be analyzed by making an organized list. The problem is that we haven't decided whether 'getting one of each' means one outcome or two different outcomes. Let's refer to them as coin 1 and coin 2."</strong> Mark the results H-H, T-H, H-T, T-T on the board. <strong>"This is known as the </strong><i><strong>sample space</strong></i><strong>, or a list of all possible outcomes. Because the scenario is straightforward in terms of probability, creating a list is simple. Who wants to come up and draw a tree diagram for this situation?"</strong>&nbsp;<br><br>Have a student make the tree diagram.</p><figure class="image"><img style="aspect-ratio:437/246;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_11.png" width="437" height="246"></figure><p>Discuss the situations in which tree diagrams are simple or difficult to make. Ask for examples. If it doesn't come up, try finding the probability of receiving a specific sum while rolling two number cubes and then begin drawing the tree diagram. It can soon become messy, so don't carry it all the way through. <strong>"Some of you have seen this situation organized this way before.</strong>" Begin creating the typical 6 x 6 chart for the outcomes. Write the numbers 1–6 down the left side and across the top. Ask for a volunteer to complete the table (with the sums).</p><figure class="image"><img style="aspect-ratio:507/267;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_12.png" width="507" height="267"></figure><p><strong>"This is known as an organizational chart or </strong><i><strong>sample-space</strong></i><strong> </strong><i><strong>organizer</strong></i><strong>. Typically, the specific outcomes are within the grid. When trying to compute probabilities, we circle all of the positive outcomes that make up the event. Remember that an </strong><i><strong>event</strong></i><strong> is the outcome or set of outcomes that is favorable. If I asked for the probability of getting a 5, you would circle all of the 5s."</strong><br><br>Write "P (getting a 4) =" on the board and ask what the probability is. When a student answers "3 out of 36," ask, <strong>"How did you get that?"</strong> Students should identify that the grid contains 36 sections, 3 of which have a sum of 4. This is also a good opportunity to review that probability can be expressed as a fraction in simplest form, a decimal, or a percent. <strong>"You are really using this sample-space organizer as an area model, which is what we are going to be learning about today."&nbsp;</strong><br><br><strong>"Let's return to Calvin and his brother's situation. I'm going to make a 10 × 10 grid. How many sections are there in the grid? Who would like to come up and divide the grid so that one side represents the probability of getting heads and one side tails?"</strong> Students will likely see that they just need a two-by-two grid. <strong>"Now, I'll write Heads and Tails next to the table</strong> (on the left or top) by each of the sections to keep them straight. I'll also write Coin #1 on this side <strong>so we can see that the horizontal (or vertical) line dividing the section represents the two possible results for the first coin."</strong> Now, write "Coin #2" on the other side of the square (top or left, whichever was not labeled previously). Label the two remaining columns (or rows) as Heads and Tails, and you should end up with something similar to the following, except that the inside will be blank.</p><figure class="image"><img style="aspect-ratio:173/139;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_13.png" width="173" height="139"></figure><p><strong>"Who can come up and label the outcomes inside the grid where they belong?"</strong> If no student volunteers at first, mark one of the sections with H-H. <strong>"Who wins in this situation?"</strong> Add the words "Calvin wins." Students will easily grasp the remainder. When the grid is completed, ask, <strong>"How many possible outcomes are there in total? Is the game fair or unfair? What are the probabilities for each section?"</strong> Write them on the grid.<br><br><strong>"This type of area model is commonly used to model probability situations involving more than one </strong><i><strong>dependent</strong></i><strong> or </strong><i><strong>independent event</strong></i><strong>. The two-coin toss activity is </strong><i><strong>independent</strong></i><strong> since the outcome of one coin does not affect the other. However, if I set up the game so that after getting heads on the first coin flip, I have to quit, but if I get tails on the first flip, I can flip again, these are </strong><i><strong>dependent events</strong></i><strong>. We are just going to look at independent events today, but area models can be used for both.</strong><br><br><strong>"You might have noticed that I used a 10 x 10 grid to create the area model. The grid is divided into 100 equal sections, making it simple to 'count' the probability of each section and express it as a percentage. Because the results of flipping a coin are \(50\over 50\) or \(1 \over 2\) and \(1 \over 2\), it is easy to see the dividing lines. However, making an area model does not need the use of a grid. The round spinner at the beginning of the lesson also served as an area model. Because it was clearly divided into \(1 \over 4\) and \(3 \over 4\), the probability was easy to calculate."</strong>&nbsp;<br><br>Make another area model by using the spinner and the flip of a coin. Don't use a grid. Simply begin with a rough square (see below). Explain how a square or rectangle is useful for a <i>compound</i> event since it has sides. A circle or other shape is typically only used for <i>simple</i> or one-stage events.</p><figure class="image"><img style="aspect-ratio:298/375;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_14.png" width="298" height="375"></figure><p><strong>"Let's say we're creating a new game and want to determine if it's fair. In our game, we have a spinner with three equal sections (A, B, and C) and a bag containing one clear marble and two blue marbles. The game consists of spinning the spinner and then drawing one marble from the bag."</strong> Distribute copies or display a transparency of the Matching Game (M-7-2-2_Matching Game.doc).<br><br><strong>"How many people think that getting a 'match' (for example the letter B and a blue marble [Bb] or getting the letter C and a clear marble [Cc]) is likely?"</strong> Many students may think it is possible because more than half of the letters on the spinner are Cs or Bs. <strong>"How many think it is equally likely to get a match as to not get a match?&nbsp;</strong><br><br><strong>"You will work in groups to design area models for the new game. After you've completed, consider whether the game is fair and be prepared to support your decision. If you think the game is unfair, design a new set of outcomes (with the same spinner and marbles). If you think it is already fair, see if you can create another game that is almost fair but seems close enough to trick someone into picking the less likely outcome."</strong> If groups are using the handout, the instructions are repeated on the sheet. If groups are using regular grid paper, put instructions on the board or overhead for students who struggle with multiple verbal instructions. As the groups begin to work, walk around the room to ensure that the area models are properly set up and that everyone is involved.&nbsp;<br><br>Allow the groups to present their reports. The script below should summarize the use of the area model and its difference from an organizational grid (as shown in the drawings.)</p><figure class="image"><img style="aspect-ratio:607/221;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_15.png" width="607" height="221"></figure><p>Some groups may have set up the area model more as a sample-space organizer than a real area model. Only show the finished chart seen above on the left. <strong>"Many groups answered the question with a chart similar to this one. This model separates the spinner/marble problem into nine equal sections because there are nine distinct, equally likely outcomes. That makes this a sample-space organizer rather than an area model."</strong> Encourage any discussion regarding how the fact that the separate, similar outcomes from the chart would still have the same probabilities as the area model.<br><br>Start a new square on the board. <strong>"There are just two possible results when choosing a marble: clear or blue. However, they are not equally likely. "What are the fractional probabilities for the two outcomes?"</strong> (<i>P(c) = \(1 \over 3\) and P(b) = \(2 \over 3\)</i>)<br><br><strong>"How can you divide the grid into \(1 \over 3\) and \(2 \over 3\)?"</strong> Continue to complete the chart with student participation. The final product of the class discussion should look like the grid on the right above.&nbsp;<br><br>Explain to students that the sample is a list of all possible outcomes for the problem they're working on. For this problem (spinning having A, B, and C with marbles clear, blue, and blue), students could list the outcomes using capital letters for the spinner sections and the first letter of the marble color to indicate all the potential ways one of each could be combined. For example: Ac means that you got an A on the spinner and drew a clear marble.<br>Have students help create the sample space shown below. Discuss the importance of having an organized way of creating it so that outcomes do not get missed.<br><br>Ac, Ab, Ab, Bc, Bb, Bb, Cc, Cb, Cb<br><br>Also, talk about how the sample space can assist predict an outcome. Describe how, in real life, predictions are not always correct because chance is still involved.&nbsp;<br><br>Before creating the final area model for the whole class, verify group work to ensure that the total of the probabilities of individual outcomes is 100%. Correct any misunderstandings in the reports. On the day after the lesson, provide a partner warm-up task to measure students' grasp of the differences between organized lists, charts, tree diagrams, and area models.<br><br>A possible question would be <strong>"Give an example of a compound probability situation and make an organized list, tree diagram, organizational chart, and area model for the situation."&nbsp;</strong><br><br>Another option is to ask, <strong>"When is it most convenient to use a tree diagram? When is it more convenient to use an area model? Provide examples of both."</strong>&nbsp;<br><br>A third activity could be a vocabulary presentation, in which each group is assigned one word and a graphic organizer (M-7-2-2_Four-Square Vocabulary Organizer) to complete and present to the class. Students could use a paper copy or transparency of the graphic, or draw an enlarged version on chart/poster paper for their presentation. During presentations, students in the class could complete their own definitions in their vocabulary journals or on handouts of journal pages (M-7-2-1_Vocabulary Journal Page or M-7-2-2_Four-Square Vocabulary Organizer), which could then be collected.&nbsp;<br><br><strong>Extension:&nbsp;</strong><br><br><strong>Routine:</strong> Emphasize the proper use of probability vocabulary in both student assignments and classroom discussions about probability problems. Throughout the lesson, the vocabulary journals should have included the following words: <i>area model, independent event, organized list, chart of outcomes, sample space, </i>and<i> simple and compound events</i>. Review these terms and ensure that students have accurately recorded them in their vocabulary journals. Another option is to have students share and compare their vocabulary entries in small groups for 15 minutes at the end of each lesson. Make extra copies of the journal sheets available for students who need them.<br>If students are struggling with the concept of area models or constructing them, send home the supplemental area models worksheet (M-7-2-2_Lucky Winner and KEY).&nbsp;<br><br>As a warm-up activity after the lesson's last day, have these students work with partners who understand probability to go over the additional worksheet. They could also make an organized list or tree diagram to accompany the area models. If students are having trouble grasping the concept of multistage probability situations, give them more simple compound situations to practice in groups (spinners, picking blocks, flipping coins, etc.).<br><br><strong>Expansion:</strong> Ideas for extension work for students meeting or going beyond the standard include:<br>Find the expected value for each player on the Who Will Be the Lucky Winner? worksheet (M-7-2-2_Lucky Winner and KEY).<br>Make an organized list, chart, tree diagram, and area model for a compound probability involving three events (for example, spin a spinner, roll a number cube, and flip a coin). Discuss the advantages and disadvantages of each strategy for estimating outcomes and probabilities.<br>Create a <i>conditional probability</i> situation and teach students how to use tree diagrams or area models to demonstrate conditional probabilities.<br>Given an area model, instruct students to create situations that could be represented by the model.</p>