<p>Students will gain experience with experimental probability through hands-on experiments and data collection. Students will comprehend and apply basic probability concepts. They will grasp and apply suitable vocabulary when describing elementary probabilities and experimental results. Students will:&nbsp;<br>- use frequency tables to organize data.&nbsp;<br>- understand and determine experimental probabilities.&nbsp;<br>- probability can be represented in a variety of forms (fraction, decimal, percentage).&nbsp;<br>- reason using probabilities associated with experiments.&nbsp;<br>- learn the vocabulary terms necessary to communicate the concepts of probability.&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>- sticky note or small piece of paper for each student (one colored dot on each)&nbsp;<br>- copies of the Vocabulary Journal pages, two per student (M-7-1-1_Vocabulary Journal)&nbsp;<br>- student copies of the Frequency Tables sheet (M-7-1-2_Frequency Tables)&nbsp;<br>- Equally Likely or Not? transparency (M-7-1-2_Equally Likely Transparency Master)&nbsp;<br>- eight to ten sets of Action Sorting Cards and a bag or envelope for each set (M-7-1-2_Action Sorting Cards and KEY)&nbsp;<br>- eight to ten Sorting Mats (M-7-1-2_Sorting Mat)&nbsp;<br>- exit ticket for each student (M-7-1-2_Lesson 2 Exit Ticket and KEY)&nbsp;<br>- chart paper and markers for group use&nbsp;<br>- student copies of the Super Seconds activity sheet (M-7-1-2_Super Seconds)&nbsp;<br>- two or more four-sided number cubes (if not available, use two sets of index cards numbered 1 to 4, or make a paper spinner numbered 1 to 4 and spin it twice)&nbsp;<br>- student copies of the Let’s Roll Pyramids sheet (M-7-1-2_Pyramid Sums)&nbsp;<br>- four to six Paint It Spinners pages (M-7-1-2_Paint It Spinners)&nbsp;<br>- four to six Paint It sheets (M-7-1-2_Paint It Record Sheet)</p>

<p>- Observe students' performance during the Action. Sorting activity to assess students' understanding of the probability of events&nbsp;<br>- Observations on the Super Seconds The activity and presentations will help to assess student comprehension.&nbsp;<br>The Lesson 2 Exit Ticket (M-7-1-2_Lesson 2 Exit Ticket and KEY) can be used to assess students' level of comprehension.<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, Formative Assessment&nbsp;<br>W: Conduct a random draw activity with the class to review experimental probability. Record the results and make predictions based on them.&nbsp;<br>H: Have students calculate experimental probabilities based on random draw outcomes. Then, expose the actual contents and the theoretical results are calculated and compared with the experimental results. Formats for organizing data from theoretical outcomes are presented.&nbsp;<br>E: Have students gain experience by organizing outcomes from a spinner activity in order to find the sample space. Allow each group to prepare their findings and present them to the class.&nbsp;<br>R: To review the lesson concepts, clarify any misconceptions each group has from the spinner activity, and allow them to modify their presentations.&nbsp;<br>E: Allow students to question each other in pairs and provide feedback on theoretical probability. This is a self-evaluation.&nbsp;<br>T: Review theoretical probability vocabulary. Have students create sample spaces for everyday situations. Use the Extension section to personalize the lesson to the students' requirements. The Routine section offers ideas for incorporating lesson concepts throughout the school year. The Small Group section offers additional learning and practice opportunities for students who could benefit from it. The Expansion section provides suggestions for students who are willing to go above and beyond the requirements of the standard.&nbsp;<br>O: This lesson introduces students to the key differences between experimental and theoretical probability. Students are also expected to grasp that experiments with a large number of trials produce more consistent results, which are similar to the outcomes obtained using theoretical probability.&nbsp;</p>

<p>The purpose of this lesson is to develop the concept that probability, or chance, refers to events that are uncertain but that have a pattern of regularity when trials are repeated several times. Students conduct experiments and record the frequency of each outcome for later use in calculating experimental probabilities. They make decisions based on experimental evidence about the predictability of the outcomes. When the experiments are repeated only a few times, results vary greatly. Students calculate experimental probabilities for both a small number of trials and large number of trials, and they begin to consider how the number of trials affects the usefulness of the probabilities.&nbsp;<br><br>As students enter the classroom, give them a sticky note or a little piece of paper with a large dot on it (red, yellow, or blue), or use colored dot stickers. Display the terms <i>experimental probability</i> and <i>equally likely</i> on a board, overhead, or chart paper, along with a spinner like the one seen below.</p><figure class="image"><img style="aspect-ratio:210/217;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_5.png" width="210" height="217"></figure><p><strong>"Think about what would happen if I spin the spinner once. Stand up if you think your dot color is the one I am most likely to spin."</strong> Students who are standing should hold their paper such that the dot can be seen. (mostly red dots should be standing).&nbsp;<br><br>Ask a few students to explain their thinking. Guide students to the conclusion that, while there are equal numbers of blue and yellow spaces (one each), there are two more red spaces. Discuss if the outcomes are equally likely or not.<br><br><strong>"Another method to predict what will happen is to spin the spinner multiple times and note how frequently each color appears. Based on the results, we can predict what will happen on future spins. This is known as experimental probability since we are actually performing the activity or experimenting to get our outcomes."&nbsp;</strong><br><br><strong>"In today's lesson, we will learn about experimental probability. We will go over different situations and practice calculating the experimental probabilities for the outcomes. We will also see if different actions have equally likely outcomes or not."</strong><br><br>Before beginning the spinner activity, review with students how a frequency table is used.</p><figure class="image"><img style="aspect-ratio:321/452;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_6.png" width="321" height="452"></figure><p>Explain that the first column is for the "target event," which is what they're searching for (spinning, rolling, etc). The second column is where they place their tally marks, and the third column shows the total number of tally marks in each row.&nbsp;<br><br><strong>Spinner Activity&nbsp;</strong><br><br><strong>"I'm going to spin the spinner 12 times and have you record the results. You will use a frequency table to record the results."</strong> Distribute student copies of the Frequency Tables (M-7-1-2_Frequency Tables). Instruct students to write each color in the left column and make a tally mark in the middle column for the color that appears each time they spin. The last column displays the total for the number tallies.<br><br><strong>"Before I spin, I want you to predict how many of each color you think I will spin." </strong>Allow numerous students to give their predictions. Spin 12 times, or have students spin. Use a paper clip or ruler with a hole near the end as a spinner pointer. Whatever you use, insert the point of a pencil or pen through the opening to hold it in the center of the spinner. Spin with one finger on the opposing hand. Demonstrate for students for later activities.&nbsp;<br><br><strong>"Did the results turn out as you expected?"</strong> Take the time to discuss how results can e uneven, especially with a limited number of trials.<br><br><strong>"I am going to spin 12 more times and have you record the results."</strong> After students have recorded the next 12 outcomes, explain how the experimental probability is calculated as a fraction and how it is represented (number of favorable outcomes / total outcomes). After writing the following sentences on the board, say the words aloud and ask students to repeat them. Remind them that a fraction is actually a ratio.<br><br>P(red) = _____ &nbsp; &nbsp; &nbsp;&nbsp;<strong>“The probability of spinning red is _____.”</strong><br><br>P(blue) = _____ &nbsp; &nbsp;&nbsp;<strong>“The probability of spinning blue is _____.”</strong><br><br>P(yellow) = _____ &nbsp;<strong>“The probability of spinning yellow is _____.”</strong><br><br><strong>"Because we had a total of 24 trials, the denominator for all of our probabilities was 24. The numerator will represent the number of times we got the color evaluated in our 24 spins, also known as the 'favorable outcome.' If possible, we'd like to simplify each ratio. Each ratio can also be expressed as a percentage or decimal."</strong> Display each of the three representations for the spins you recorded.&nbsp;<br><br><strong>"What outcomes could be expected with two spins?"</strong> Let students make suggestions. They should create double colors, such as (blue, blue), (red, yellow), and (yellow, blue). They don't have to mention every possibilities at this time. Encourage students to use one of the large 12-section frequency tables at the bottom of the sheet. As you do double spins on the spinner, ask them to keep track of the results using tallies (or have students take turns doing the spins). Do approximately 25 double spins. Discuss what students would conclude based on these 25 trials. You can do more spins if you have time. Calculate the probability of getting each of the different combinations that appeared. Ask students about their observations of the results. They may note that the matching colors (blue, blue), (red, red), (yellow, yellow) appear more frequently. They may also ask if outcomes like (red, blue) are equivalent to (blue, red). Ask if they think there are any combinations that did not appear during your limited number of trials (the answers will vary).&nbsp;<br><br><strong>Action Sorting Activity&nbsp;</strong><br><br><strong>"Who will volunteer to explain in their own words what equally likely events are?"</strong> Select one or more volunteers. Use the Equally Likely or Not? transparency (M-7-1-2_Equally Likely Transparency Master) as a discussion tool in class.<br><br>1. A baby is born ... boy or girl?<i> Equally likely.</i>&nbsp;<br>2. When shooting a basket, do you make it or miss? <i>Not equally likely unless you have an established shooting record of 50%.</i>&nbsp;<br>3. Roll a six-sided number cube ... prime or composite? <i>Not equally likely because there are three primes and only two composite numbers.</i>&nbsp;<br>4. Flip a fair coin ... heads or tails? <i>Equally likely.</i>&nbsp;<br>5. If you take the bus to school, you get a seat near the front or a seat near the back? <i>Not equally likely; it depends on your preferences and how full the bus is when you get on.</i>&nbsp;<br>Clarify any misconceptions that arise. Encourage students to explain their thinking on the examples that seem the most confusing.<br><br>Divide students into groups of three or four. Give each group a set of Action Sorting Cards and a Sorting Mat (M-7-1-2_Action Sorting Cards and KEY and M-7-1-2_Sorting Mat). Instruct students to discuss each action before putting the card in the "Equally Likely," "Not Equally Likely," or "We Disagree" part of their mat. Give them about five minutes to sort through the cards. As you move around the room, note which cards are in each group's Disagree category. If there are numerous cards in the "We Disagree" section, discuss one commonly disagreed-upon with the entire class. Allow groups a few more minutes to talk about any other cards they disagreed on. Allow them to get input from another group if they are still unable to reach a consensus.<br><br>Leave students in small groups to finish the following activity.&nbsp;<br><br><strong>Super Seconds Activity</strong><br><br>Each group will receive a piece of poster board or chart paper, as well as markers. They will also require at least one copy of the Super Seconds Activity Sheet (M-7-1-2_Super Seconds). Students must roll the number cubes 50 times and record the sum each time. Based on these results, they must calculate the experimental probability of receiving each sum and respond to the additional questions on the activity sheet. Explain to students that they must create a poster with the results of their experiment and the answers to be presented to the class. Each student will be responsible for presenting a piece of the information. For groups of four, the presentation parts could be divided as follows:<br><br>1. experimental data results&nbsp;<br>2. how data was used to calculate probabilities<br>3. questions 1 and 2 with explanations&nbsp;<br>4. question 3 with explanation.&nbsp;</p><p>Allow about 25 minutes to work before the presentations start (5 minutes to roll and record sums, 5 to 10 minutes to discuss the results and answers, and 10 to 15 minutes to create a poster and decide who will present which information). Each group presentation should last roughly 4-5 minutes.<br><br>As each group presents, ask questions to help them correct any misconceptions. Encourage the group to make any required changes or corrections to their presentation or poster afterward. Individual students comprehension levels can also be assessed through questioning.<br><br>After the groups have finished presenting, combine the data from all of the groups' rolls. Calculate the experimental probabilities for each sum using the class totals. Discuss how these probabilities differ from the group results, if at all, and the benefits and drawbacks of gathering more data to calculate the probabilities (which will lead to a later discussion of the law of large numbers).&nbsp;<br><br>Each student should be given an Exit Ticket (M-7-1-2_Lesson 2 Exit Ticket and KEY) around 5 to 10 minutes before the end of class. The responses provided by students can be used to assess their level of understanding and assign extra appropriate exercises, as mentioned in the instructional strategies section.<br><br><strong>Extension:</strong><br><br>Use these suggestions to personalize the 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 terms in their vocabulary journals: <i>equally likely, experimental probability, frequency table, not equally likely, outcomes,</i> and <i>trials</i>. Keep a supply of vocabulary journal pages on hand so that students can add them as needed. Bring up instances of chance and prediction from throughout the school year. Ask students to provide examples from throughout the year.<br><br><strong>Small Group: Pyramid Sums Activity&nbsp;</strong><br>This activity is intended for students who might benefit from an opportunity for additional learning and understanding of the concept or calculation of experimental probabilities. Students should roll two four-sided number cubes 25 times and note the sums. The sums will range between 2 and 8. Using the results and the Let's Roll Pyramids sheet (M-7-1-2_Pyramid Sums), students should calculate the experimental probabilities as fractions and percentages. Discuss the significance of each probability's numerator and denominator, as well as how percentages might be useful. Ask students to describe how the probabilities can be used to predict future rolls. Clarify any misconceptions.<br><br><strong>Expansion: Paint It Activity&nbsp;</strong><br>Use this activity with students who have showed a strong comprehension and proficiency in working with experimental probabilities. In the Paint It activity, students will spin two colored spinners. They will spin each spinner once and record the two colors they land on, as well as the color of paint that would result from mixing their colors. Consider the color combinations (red + red = red, yellow + yellow = yellow, blue + blue = blue, red + yellow = orange, blue + yellow = green, red + blue = purple). Students will need copies of Paint It Spinners and Paint It Record Sheet.doc to record data and calculate experimental probabilities (M-7-1-2_Paint It Spinners and M-7-1-2_Paint It Record Sheet).<br><br>Up to six players can participate. Each student chooses a color from red, yellow, blue, green, orange, or purple (or draws a slip of paper from a container, each with one of these colors). Students will play the game for 10-15 minutes. Players take turns spinning, moving from player to player in a clockwise direction. The player whose paint color combination matches the combination spun earns one point (i.e., if yellow and red are spun, the player with orange gets a point.) If there is no player for a specific color combination, no points will be awarded for that spin; however, the spin should still be recorded. The game continues till the time is up. The player who scores the most points wins. Have students respond to the questions on the game sheet.&nbsp;<br><br><strong>Technology Connection:</strong> If computers are available, direct students to <a href="http://www.saintannsny.org/depart/math/misterg.htm">http://www.saintannsny.org/depart/math/misterg.htm</a>. Allow them time to play and experiment. The website rolls two number cubes and creates a bar graph from the results. Students will create a journal page detailing their experiments, observations from various small and large numbers of rolls, their outcomes, and the resulting graphs. Extend the project by allowing students to physically roll two or three number cubes 20 to 30 times in the classroom, as well as 200 to 300 times, and then create a table and graph for each situation similar to those seen on the website.</p>