Overview. Essential Questions. Grade 7 Mathematics, Quarter 4, Unit 4.2 Probability of Compound Events. Number of instruction days: 8 10

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1 Probability of Compound Events Number of instruction days: 8 10 Overview Content to Be Learned Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Understand that the probability of a compound event is the fraction of outcomes in the sample space. Identify outcomes in a sample space for compound events using methods such as organized lists, tables, and tree diagrams. Design and use a simulation to generate frequencies for compound events. Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Use organized lists, tables, tree diagrams, and simulation to represent sample spaces and find probability. Use more than one strategy to check answers when appropriate. 4 Model with mathematics. Use area models, tree diagrams, organized lists, and simulations to visualize compound probabilities. 5 Use appropriate tools strategically. Create and use appropriate tools (e.g., spinners, number cubes, coins, colored cubes) to represent outcomes. Use technological tools to simulate compound probabilities to deepen understanding of concepts. Essential Questions What is meant by a compound event in the context of finding probability? Give an example. How can you use lists, tables, tree diagrams, or simulation to represent the outcomes of compound events? How can you use models/tools to simulate a particular probability event? What does the sample space tell you about the probability event? How do you use the sample space to calculate the probability of a compound event? Providence Public Schools D-87

2 Probability of Compound Events (8 10 days) Standards Common Core State Standards for Mathematical Content Statistics and Probability 7.SP Investigate chance processes and develop, use, and evaluate probability models. 7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes ), identify the outcomes in the sample space which compose the event. c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? Common Core State Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They D-88 Providence Public Schools

3 Probability of Compound Events (8 10 days) Grade 7 Mathematics, Quarter 4, Unit 4.2 are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In earlier grades, students used both categorical and measurement data to answer simple statistical questions, but they paid little attention to how the data were collected. In Grade 6, students developed an Providence Public Schools D-89

4 Probability of Compound Events (8 10 days) understanding of statistical variability. They summarized and described distributions. In Grades K 5, students organized, represented, and interpreted data for one or more categories in various ways. Current Learning Students gain experience in the use of diagrams, especially trees and tables, as the basis for organized counting of possible outcomes from a situation of chance. Students use proportional reasoning and percentages when they extrapolate from random samples and use probability. After the basics of probability are understood, students set up a model and use simulation (by hand or with technology) to collect data and estimate probabilities for a real situation that is sufficiently complex that the theoretical probabilities are not obvious. Future Learning In Algebra I, students will summarize, represent, and interpret data for one or more categorical data sets. In Geometry, students will understand independence and conditional probability and use these concepts to interpret data. In Algebra II, students will use probability to evaluate outcomes of decisions in more complex situations. Additional Findings Students intuitive understanding of independence is measured by their ability to recognize and justify when the occurrence of one event has no influence on the occurrence of another. A study revealed that some students harbored the pervasive misconception that the outcomes of a coin toss can be controlled. Similar misconceptions were evident in other studies of middle school students. Misconceptions of the kind illustrated above have been characterized more generally as representativeness a belief that a sample or sequence of outcomes should reflect the whole population. (Adding It Up, p. 293) According to Principles and Standards for Mathematics, in grades six through eight, all students should compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models. (p. 248) Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. D-90 Providence Public Schools

5 Probability of Compound Events (8 10 days) Grade 7 Mathematics, Quarter 4, Unit 4.2 The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Use organized lists, tables, tree diagrams, and simulations to represent sample spaces and find probability. Identify outcomes in a sample space for compound events. Design and use a simulation to generate frequencies for compound events. Use the sample space to calculate the probability of a compound event. Learning Objectives Students will be able to: Instruction Conduct experiments to test predictions involving compound events. Find probabilities of compound events. Determine theoretical probability and compare it to experimental probability of compound events. Use theoretical probabilities to make predictions of compound events. Use an area model to analyze the theoretical probabilities for compound outcomes. Simulate and analyze probability situations using an area model involving compound outcomes and distinguish between equally likely and non-equally likely outcomes. Use an area model to analyze the theoretical probabilities for compound outcomes. Demonstrate understanding of the concepts and skills in this unit. Resources Connected Mathematics 2, Pearson/Prentice Hall, 2008: What Do You Expect? Investigation 1: Evaluating Games of Chance, Student Book (pages 5-19) Investigation 2: Analyzing Situations Using an Area Model, Student Book (pages 20-37) Teacher s Guide Implementing and Teaching Guide Teaching Transparencies Assessment Resource Book Additional Practice and Skills Workbook Special Needs Handbook Parent Guide Prentice Hall Teacher Station Software Providence Public Schools D-91

6 Probability of Compound Events (8 10 days) Exam View Software (Students can enter web-codes) Teaching with Foldables (Dinah Zike; Glencoe McGraw Hill 2010) Available with the Algebra resources Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials Graphing calculators; paper clips or bobby pins for spinners, opaque containers (2 per pair), colored cubes or marbles (1 blue, 2 yellow, 1 green, and 2 red per pair), number cubes (2 per pair) coins, large sheets of paper and markers for recording student work (optional). Key Vocabulary compound event fair game Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: likely, outcome, experimental probability theoretical probability, tree diagram, and area model. Living word walls assist all students in developing content language. Word walls should be visible to all students, focus on the current unit s vocabulary, both new and reinforced, and have pictures, examples, and/or diagrams to accompany the definitions. Teachers should review the Mathematics of the Unit found on page 3 of all CMP2 teacher editions. For planning considerations read through the teacher edition for suggestions about scaffolding techniques, using additional examples, and differentiated instructional guidelines as suggested by the CMP2 resource. The focus of investigations 1 and 2 is the probability of compound events. The problems do not mention that the types of events are compound. The CMP2 resource uses the term two-stage outcomes or twostage events instead of the word compound. In the previous unit, students made predictions, conducted experiments and found the theoretical probabilities of simple events. In Investigation 1, students work with experimental and theoretical probabilities dealing with compound events. You may want to tell your students that a compound event consists of two or more simple events. For example, tossing a die is a simple event. Tossing two dice is a compound event. In Problem 1.1, be sure students understand how to score points in the game. Students need to be clear that they score a point(s) regardless of whose turn it is to spin the spinner. To help students to understand the context of Problem 1.2, play the game as a whole class, letting each student quickly have a turn to draw colored cubes or marbles. Have a student record the class data on the board. Note: This unit has a unit project in which students are asked to design a game that should make a profit for the school. The launch of Problem 1.2 would be a good time to foreshadow the upcoming unit project and provide context to the payoff embedded in the Red and Blue game. Problem 1.2 uses nonlinguistic representations to represent knowledge. Students create a pictographic representation by making a tree diagram to show the possible outcomes for the Red and Blue game. D-92 Providence Public Schools

7 Probability of Compound Events (8 10 days) Grade 7 Mathematics, Quarter 4, Unit 4.2 In the summary of Problem 1.3, have students identify similarities and differences to compare the theoretical and experimental probabilities for the Multiplication Game. Compile all of the class data and generate a graph that shows the relationship between the number of trials (x-axis) and the experimental probability of rolling an odd product expressed as a decimal or percent (y-axis). On the same graph, graph the theoretical probability of rolling an odd product, P(odd product) = 1. Students will see that as the 4 number of trials increases, the experimental probability begins to approach the theoretical probability of rolling an odd product, which is 1 = 0.25 or 25%. 4 Note: The List feature of a graphing calculator would be an effective and efficient tool for this teaching strategy. This visual will emphasize the Law of Large Numbers, which tells us that as we conduct more and more trials, the probabilities drawn from the experimental data should grow closer to the actual theoretical probabilities. In the last unit of study, students used organized lists and tree diagrams for finding probabilities. Area models are useful for finding probabilities in situations involving successive events, such as selecting a container to draw from and then drawing a cube. Unlike tree diagrams, an area model is particularly powerful in situations when the outcomes are not equally likely events. In Problem 2.1 part D, students are asked to make predictions of what might happen if the game was played 36 times. This is an opportunity to discuss and apply proportional reasoning understandings that were developed in earlier grade 7 units. Be sure that students see these connections. The area model used in this unit has a direct connection to earlier work in the 6th-grade unit Bits and Pieces II, where students used an area model to develop their algorithms for adding fractions. For example, in Problem 2.2, students determine the probability of ending in Cave A as the sum of the areas Be on the lookout for strategies that subdivide the area model into the smallest part that the areas all have in common (least common denominator) and then count up the number of parts that have been designated as Cave A and connect these actions to the algorithm for adding fractions. In Problem 2.3, consider giving groups large chart paper and requiring an area model for each possible arrangement to verify their solution. This nonlinguistic representation is used to elaborate on knowledge. Students create a pictographic representation (area model; see example below) to analyze statistical situations. Fidelity with this skill will support students as they continue to develop their knowledge of probability and statistics. Container 1 Container 2 green green blue blue Providence Public Schools D-93

8 Probability of Compound Events (8 10 days) Incorporate the Essential Questions as part of the daily lesson. Options include using them as a do now to activate prior knowledge of the previous day s lesson, using them as an exit ticket by having students respond to it and post it, or hand it in as they exit the classroom, or using them as other formative assessments. Essential questions may be included in the unit assessment. CMP2 has online resources that may be helpful in planning for all units of study. Visit and sign on to SuccessNet. You will find the Common Core Additional Investigations and Common Core Investigations Teacher s Guide under the worksheet tab. Notes D-94 Providence Public Schools