<p>In this lesson, students use proportionality and a basic understanding of probability to make predictions and evaluate hypotheses about the outcomes of experiments and simulations. Students will:&nbsp;<br>- determine the situations in which prediction is useful.&nbsp;<br>- investigate how elementary probabilities, whether experimental or theoretical, can help predict future events connected to real-world issues.&nbsp;<br>- use experimental probability to develop predictions.&nbsp;<br>- understand that what happens in a predictable pattern over a long period of time can be used for decision-making purposes.&nbsp;<br>- learn that the greater the number of trials in a random experiment, the closer the experimental probability is to the theoretical probability (the law of large numbers).&nbsp;</p>

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

<p>- Compound Event: Two or more simple events.<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 landing heads-up and the coin landing tails-up—are equally likely to occur.<br>- Likely Event: The event that is most likely to happen. The probability of a likely event is generally between \(1 \over 2\) and 1.<br>- Outcome: One of the possible events in a probability situation.<br>- Probability: A number between 0 and 1 used to quantify likelihood for processes that have uncertain outcomes (such as tossing a coin, selecting a person at random from a group of people, tossing a ball at a target, or testing for a medical condition).<br>- Proportion: An equation showing that two ratios are equal.<br>- Random Sample: A sample in which every individual or element in the population has an equal chance of being selected. A random sample is representative of the entire population.<br>- Sample Space: The set of possible outcomes of an experiment; the domain of values of a random variable.<br>- Simple Event: One outcome or a collection of outcomes.</p>

<p>- student copies of Vocabulary Journal pages (M-7-1-1_Vocabulary Journal)&nbsp;<br>- small bags of plain M&amp;Ms (enough for each individual or one bag per small group of students)&nbsp;<br>- one large bag of M&amp;Ms (16 ounces or more)&nbsp;<br>- student copies of M&amp;M Color Guide (M-7-1-3_M&amp;M Color Guide)&nbsp;<br>- transparency of M&amp;M Color Guide, or draw it on the board or chart paper (M-7-1-3_M&amp;M Color Guide)&nbsp;<br>- eight bags or containers that seal, set up with a mixture of white and brown dry beans. (Marbles, dry pasta, or any other small markers of two different colors will also work). Each container should have a different number ranging from 100 to 300 items, with a much smaller number of brown beans than white.&nbsp;<br>- student copies of the Counting Deer sheet (M-7-1-3_Counting Deer Lab Sheet)&nbsp;<br>- transparency of the Counting Deer sheet (M-7-1-3_Counting Deer Lab Sheet)&nbsp;<br>- student copies of Lesson 3 Exit Ticket (M-7-1-3_Lesson 3 Exit Ticket and KEY)</p>

<p>- Assess student performance during the M&amp;M activity to determine their level of comprehension.&nbsp;<br>- Use the following ways to observe student interaction and discussion during the Counting Deer activity.&nbsp;<br>- To evaluate student learning, use the Lesson 3 Exit Ticket (M-7-1-3_Lesson 3 Exit Ticket and KEY).<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, Explicit Instruction&nbsp;<br>W: Create an activity that demonstrates how to use probability to create predictions and decisions.&nbsp;<br>H: Explain how identifying a ratio or proportion in a small sample can lead to a prediction about a large sample of the same thing. Engage students in the class by having them calculate percentages and extrapolate using M&amp;Ms candy.&nbsp;<br>E: Have students work in groups to complete a ratio assignment. They will conduct multiple experiments to calculate and predict a ratio involving larger numbers.&nbsp;<br>R: Allow groups to review their learning by asking questions and discussing strategies for the activity. Students will present their results to the entire class.&nbsp;<br>E: Have students self-evaluate using probability and ratios.&nbsp;<br>T: Use the Extension section to customize the lesson to match the needs of your students. The Routine section provides real-life situations throughout the year from which students can calculate and predict probabilities. Encourage them to play games and track expected outcomes to improve their probability abilities. The Small Group section can be used with students who would benefit from more practice with lesson concepts. The Expansion section is intended for use with students who are ready to move beyond the requirements of the standard.&nbsp;<br>O: This lesson teaches students how to use probability in many contexts and will provide them with experience with probabilities generated over a long period of time or with large numbers.&nbsp;</p>

<p>This lesson was aimed to provide students an understanding of how probabilities can be used to predict what will happen over time. They use probability calculations and proportional reasoning to predict how outcomes would occur over time. Students also use these predictions to make decisions and assess how these types of decisions are used in everyday life. Sample problems were chosen to provide a wide range of outcomes in a small number of trials; yet, when data was pooled, the larger sample space produced more consistent results (law of large numbers).&nbsp;<br><br>As students enter the classroom, give each one a little bag of M&amp;M candies (fun-size bag). [Note: the candy doesn't have to be M&amp;Ms. You could use anything (not necessarily candy) with multicolored pieces.] Tell students not to open the bag until they have received instructions. Ask students to guess what colors are in the bag. They'll probably be able to guess all of them (<i>red, yellow, orange, green, blue, and brown</i>). Ask them how they were able to predict the colors (<i>because of their previous experience with M&amp;Ms and with repeated/predictable outcomes</i>).&nbsp;<br><br><strong>"In today's lesson we will be looking at how we can use probability to make predictions and decisions."&nbsp;</strong><br><br><strong>"Based on your previous M&amp;M eating experience, can you predict the percentage of each color in your bag? Specially, do you think all colors will be in the same proportion in the bag?"</strong><br><br>Distribute the M&amp;M Color Guide (M-7-1-3_M&amp;M Color Guide). <strong>"Pour the M&amp;Ms onto your Color Guide sheet. Separate them by color and record them in column 1, then record the total number of M&amp;Ms at the bottom of the column. In column 2, write a ratio of each color compared to the total number of M&amp;Ms. In columns 3 and 4, write your ratio as a decimal and a percentage."</strong> Allow students 5–10 minutes to complete these steps. Walk around the room to explain the directions. Allow students to eat their candies after they have completed all of the steps correctly.&nbsp;<br><br><strong>"I'm going to give you two minutes to compare your results with three people around you."&nbsp;</strong><br><br><strong>"Did you get the same percentages for each color?"</strong> (Most will say <i>no</i>.)&nbsp;<br><br><strong>"Why do you think this is the case?"</strong> (<i>They each have a small sample compared to the large batch mixed at the factory, so there is a large variation possible.</i>)<br><br>Hold up the big bag of M&amp;Ms. <strong>"I've got a bag of M&amp;Ms too. Do you think I have the same number of each color as you do? Do you think I'll get the same percentages as you?"</strong> (Students will likely claim that <i>yours will be very similar</i>; actually the M&amp;M company uses a specific percentage for each color.)&nbsp;<br><br><strong>"You'll use your ratios or percentages to estimate how many M&amp;Ms of each color are likely to be in my bag. You'll do this by applying proportional reasoning."</strong> Show the following example, and others as needed:<br><br>If you discovered 20% red in your bag, the ratio of red to total may be expressed as \(20 \over 100\). If you know I have 440 total M&amp;Ms and an unknown number of red (r), the ratio in my bag can be represented as \(r \over 400\). We may use the following proportion to generate a mathematical estimate about how many red candies will be in the huge bag.<br><br>\(20 \over 100\) = \(r \over 400\)<br><br>To solve the proportion, we can use either cross products or a scale factor.</p><figure class="image"><img style="aspect-ratio:468/258;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_9.png" width="468" height="258"></figure><p>Count the total number of M&amp;M candies in your large bag, as well as the number of each color. Reveal to students <u>only the total number of candies</u> they can use in their predictions. Allow students to complete their predictions for the M&amp;M Color Guide.<br><br>After students have completed their calculations, expose the number of each color from the large bag so they can compare it to their own estimations. To finish the M&amp;M exercise, use the transparency of the student color guide or draw one on the board. Combine all of the class data for the small bags and recalculate the color predictions based on the combined data. Ask students to make three observations on the back of their copy of the color guide. Call on a few students to discuss an observation they made. One observation you want to focus is that the combined data should have produced predictions considerably closer to the large bag's numbers than the small bags did. When this observation is mentioned, explain how the law of large numbers works. If a student fails to mention it during the activity summary, bring it up yourself.&nbsp;<br><br>Divide students into eight small groups to complete the Counting Deer task. Introduce this task by describing why the Department of Natural Resources and other agencies must count fish and other animal populations for a variety of purposes. Ask students to imagine trying to count every fish in a particular lake, every duck in a marsh, or every deer in a wooded area. Ask them for recommendations on how they could accomplish this while remaining accurate enough to make decisions based on their count. Allow students to provide suggestions.<br><br>The capture-tag-recapture strategy involves capturing and tagging a sample of wildlife before releasing it back into its natural environment. The total number marked is recorded and compared to the total number of animals in the area, which is still unknown (use a variable such as <i>a</i> or <i>x</i>), and the ratio is calculated. At another time, more of the same type of wildlife is captured in the same location. The number that are still marked from being captured previously is compared to the total number captured that day (marked and unmarked combined) and recorded as a ratio. By combining these two ratios to make a proportion (as in M&amp;M color predictions), the approximate total number of animals living in that area can be calculated.</p><figure class="image"><img style="aspect-ratio:540/270;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_8.png" width="540" height="270"></figure><p>Each group will require a container of beans and a copy of the Counting Deer sheet (M-7-1-3_Counting Deer Lab Sheet). Students begin by counting the total number of <strong>marked</strong> deer (brown beans) in their forest (container). Important note: They should not count all of the deer until the end of the activity. Inform students that a few groups will be selected to present their findings at the end of the activity. Allow approximately 15 minutes to complete the lab sheet.<br><br>While students are working on their Counting Deer sheets, check in with each group to clarify the process or answer questions. If you notice a group utilizing an inaccurate or illogical technique, use some leading questions to help them adjust their thinking. Make a mental note of a few groups with useful strategies to share. When groups share their strategies with the class, encourage the rest of the class to record strategies that differ from their own, or to adjust their own work if they believe they did not utilize the appropriate strategy. Remind students that many strategies are frequently effective for mathematics problems. (for example: solving a proportion with cross products, or solving it using scale factor).&nbsp;<br><br>Note: There is no answer key for the Counting Deer Lab activity because the answers are based on the results of the lab sheets, which will vary by group.<br><br>At the end of the lesson, allow each student 5 to 10 minutes to complete the Lesson 3 Exit Ticket (M-7-1-3_Lesson 3 Exit Ticket and KEY).<br><br><strong>Extension:</strong><br><br>Use these suggestions to personalize this lesson to your students' needs throughout the unit and year.&nbsp;<br><br><strong>Routine:</strong> Discuss the significance of understanding and using the appropriate vocabulary words to convey mathematical ideas clearly. During this lesson, students should record the following term in their vocabulary journals: <i>law of large numbers</i>. Keep a supply of vocabulary journal pages on hand so that students can add them as needed.<br>Once or twice a week, post a warm-up problem on the board that includes either experimental data or a sample space, and allow students to utilize it to make a prediction or decision. Use these examples to help students understand how much of their lives or their parents' lives are based on predictions that come from data and the calculation of probabilities (likelihood or percentage of chance). Students should discuss specific examples of this as they come across them in the news, magazines, and other media sources.&nbsp;<br><br><strong>Expansion:</strong> The M&amp;M company's actual percentages in an industrial-sized mixed batch of candies are 24% blue, 20% orange, 16% green, 14% yellow, 13% red, and 13% brown, which can be used to determine theoretical probabilities. Students should create a one-page paper or poster stating how many M&amp;Ms would be needed to get so close to the real percentages that they could be used for probability calculations. Create a larger experimental data collection (with the help of a simulation, such as a color spinner). Explain how the law of large numbers might apply and/or be important in a particular real-life scenario.&nbsp;<br><br><strong>Small Group: Match Me Up</strong>&nbsp;<br>Students who could benefit from further practice may be divided into small groups to play Match Me Up. Students should utilize index cards to make a matching game. Each new word in the unit should be written on a card. Students should create a definition, diagram, or example for each vocabulary term on a second card. Each vocabulary word card should have a matching card. Students should properly mix up the cards before placing them face down in a rectangular matrix pattern on the table. On each turn, players must draw two cards and try to find a match between a vocabulary word and its mate (definition, diagram, or example). If the cards do not match, they must be replaced. If they find a match, they keep the cards and play another turn. The player with the most cards after they have all been matched wins.&nbsp;<br><br><strong>Technology Connection: Probability Games</strong><br>If students have access to computers, direct them to the following website and select a game.<br><a href="http://www.betweenwaters.com/probab/probab.html">http://www.betweenwaters.com/probab/probab.html</a>&nbsp;<br><br>Suggest they select "Coin Flip," "Dice Roll," or "Key Problem." Students should read the directions at the top of the "explain" screen and then play the game. They should change the result keeper to "session." On a piece of paper, they should write down the results of playing the game 20, 50, 100, and 500 times. Ask students to predict the most likely ways for a person to win and explain why.&nbsp;<br><br>If time allows, have them go back and read the lower section of the explanation and describe it in their own words. The text discusses the most likely approaches to win the game they choose and why.<br><br><strong>Take It Home:</strong> There are many real-world situations that rely on probability alone as a means by which to make decisions/ Surveys, such as the U.S. census, are used to collect information about a large population when all of the data that exists cannot be collected. The data is utilized to create predictions and decisions affecting the population being studied. Genetics is another area in which probability plays a role in decision making. The characteristics of the parents influence the features of their offspring, such as eye color. A square or rectangular grid is commonly used to depict the likelihood of an offspring having a specific feature. Students can use these grids to generate statements about the possibility of a specific characteristic in an offspring.&nbsp;<br>Instruct students to discover a probability example at home. They can either bring it to school to explain it to the class or write about it. The example should be relevant to one or more of the probability concepts covered and applied in class. Examples include:&nbsp;<br><br>a news story or advertisement that includes a simulation and description of experimental data&nbsp;<br>a game in which students can describe the probabilities at work and/or the kinds of predictions a player can make based on probability<br>U.S. census data is utilized by the government to make decisions.<br>political polling data are utilized to make campaign decisions<br>internet surveys such as the Nielson television ratings, used by networks and advertisers to make decisions<br>a record of stock market movements and types of decisions based on predictable behavior&nbsp;<br>The activity could be expanded to include interviews with family members or neighbors about how they use probability and prediction in their careers or daily life. Students could provide a verbal or written report summarizing their interview responses. Students should talk about the implications of this activity.</p>