Probability. Overview. Number of instruction days: (1 day = 53 minutes) Algebra 2, Quarter 4, Unit 4.1

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1 Algebra 2, Quarter 4, Unit 4.1 Probability Overview Number of instruction days: (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Use sample spaces to describe probable outcomes of events using real-world examples. Identify two events as independent and relate this finding to conditional probability using real-world examples. Use a Probability Multiplication Rule to determine if two events are independent. Construct and interpret two-way frequency tables of data for two categorical variables and decide whether events are independent or dependent using real-world examples. Recognize and explain the concepts of conditional probability and independence in everyday language and situations. Calculate and interpret the conditional probability for real-world examples using the formula P(A and B)/P(B), where B is the given condition. Apply the Addition Rule to calculate probabilities and interpret the answer in context. 1 Make sense of problems and persevere in solving them. Use manipulatives such as spinners to understand the basis of a problem and subsequently solve the problem. 2 Reason abstractly and quantitatively. Understand how a random sample can be used to draw conclusions about a population. 4 Model with mathematics. Use real-world examples, such as dart boards, spinners, dice, and deck of cards, to explain probability and to solve problems. Collect data from a random sampling, such as the favorite subjects (from a selection of three or four subjects) of students at your school. 6 Attend to precision. Use precise language in describing conditional probabilities. Providence Public Schools D-97

2 Algebra 2, Quarter 4, Unit 4.1 Probability (11-13 days) Essential Questions How can you use conditional probability and multiplication to verify the independence of events? What are some applications where probability is used to analyze and make decisions? What are the similarities and differences between finding the probabilities of dependent events and finding those of independent events? How can you use a Venn diagram to describe sets and subsets of a sample space? What is the purpose of constructing a two-way frequency table of a real-world sample? Where is probability used to analyze and make decisions in the real world? D-98 Providence Public Schools

3 Probability (11-13 days) Algebra 2, Quarter 4, Unit 4.1 Standards Common Core State Standards for Mathematical Content Statistics and Probability Conditional Probability and the Rules of Probability S-CP Understand independence and conditional probability and use them to interpret data [Link to data from simulations or experiments] S-CP.1 S-CP.2 S-CP.3 S-CP.4 S-CP.5 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Use the rules of probability to compute probabilities of compound events in a uniform probability model. S-CP.6 S-CP.7 Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model. Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model. Providence Public Schools D-99

4 Algebra 2, Quarter 4, Unit 4.1 Probability (11-13 days) 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. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 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 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 D-100 Providence Public Schools

5 Probability (11-13 days) Algebra 2, Quarter 4, Unit 4.1 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. 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. Clarifying the Standards Prior Learning In seventh grade, students were introduced to probability. They used random sampling to draw inferences about a population. Students investigated chance processes and evaluated probability models. In eighth grade, students constructed and interpreted categorical data in a two-way table. They calculated the relative frequencies to describe possible association between two variables. In seventh grade, students investigated chance processes and evaluated probability models. They found probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Students understood that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of an event occurring. They collected data using models and simulations. Students developed a probability model and used it to find the probabilities of events. Probability was a supporting cluster in seventh grade. Current Learning Geometry students continue their earlier work with probability concepts, extending and formalizing the initial work in middle school. Students use simulations, concepts of counting, independence and dependence of events, and conditional probability to solve problems and make decisions. Application of probability is a critical area in geometry. Understanding independence and conditional probability using them to interpret data is classified as additional content by the PARCC Model Frameworks for Mathematics. Using the rules of probability to compute probabilities of compound events in a uniform probability model is also defined as additional content. Providence Public Schools D-101

6 Algebra 2, Quarter 4, Unit 4.1 Probability (11-13 days) Future Learning In Precalculus, students will continue working with probability and statistics. The work in this unit is foundational to the statistical inferences, which will also be studied in Precalculus. Careers in which probability is important include actuarial work in the insurance industry, medical applications, research, and meteorology. The lottery and gaming industries also rely on these concepts. Additional Findings Students struggle with understanding concepts related to probability. According to research in Adding It Up, When fifth, sixth, and seventh graders were asked to determine conditional probabilities, the performance of the sixth and seventh graders was dramatically lower when the task involved selection without replacement compared to the selection with replacement. (p. 292) To overcome potential misconceptions about probability, students benefit from real-world experimentation. According to A Research Companion to Principles and Standards for Mathematics, it is helpful to adopt a problem-solving approach to probability. Give students opportunities to investigate probability problems or chance situations on their own and to conduct their own stochastics projects. (p. 224) D-102 Providence Public Schools

7 Probability (11-13 days) Algebra 2, Quarter 4, Unit 4.1 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. 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. Define a sample space and events within the same space. Identify two events as independent. Explain properties of Independence and Conditional Probabilities in context. Define and calculate conditional probabilities. Construct and interpret two-way frequency tables of data for two categorical variables. Calculate probability from two-way frequency tables. Use probabilities from two-way frequency tables to evaluate independence of variables. Apply the concepts of independence and probability in real-world situations. Calculate conditional probabilities in context using the following definition: the conditional probability of A given B as the fraction of B s outcomes that also belong to A. Identify two events as mutually exclusive or not. Calculate probabilities of mutually exclusive or non-mutually exclusive events using the appropriate Addition Rule. Providence Public Schools D-103

8 Algebra 2, Quarter 4, Unit 4.1 Probability (11-13 days) Instruction Learning Objectives Students will be able to: List outcomes of a sample space and apply counting rules. Explore and use independent events in problem situations. Use conditional probability to determine if two events are independent. Construct and interpret two-way frequency tables of data. Compute and interpret probabilities of compound events in a uniform probability model. Find probabilities of events that are mutually exclusive and events that are not mutually exclusive. Review and demonstrate knowledge of important concepts and procedures related to probability. D-104 Providence Public Schools

9 Probability (11-13 days) Algebra 2, Quarter 4, Unit 4.1 Resources Geometry, Glencoe McGraw-Hill, 2010, Student/Teacher Editions (See the Supplemental Unit Materials Section of this binder) Section 13-1 (pp ) Section 13-5 (pp ) Section 13-6 (pp ) Glencoe McGraw-Hill Online Geometry CCSS Geometry Lab 15: Two-Way Frequency Tables (See the Supplemental Unit Materials Section of this binder) Geometry Chapter 13 Resource Masters (pp. 5 16, 31 42) Interactive Classroom CD (PowerPoint Presentations) Teacher Works CD-ROM TI-Nspire Teacher Software Exam View Assessment Suite Independent Events and Checking for Independence Problems (See the Supplementary Materials section of this binder.) 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 section below for specific recommendations. Materials Spinners, dice, coins, and/or other concrete models, TI-Nspire calculators, interactive white board such as SMART Board (optional), dynamic geometry software, overhead projector. Instructional Considerations Key Vocabulary combinations event independent events sample space set subset Providence Public Schools D-105

10 Algebra 2, Quarter 4, Unit 4.1 Probability (11-13 days) outcome compound event complement dependent Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: experimental probability, conditional probability, random, and theoretical probability. Living word walls will 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. For planning considerations read through the teacher edition for suggestions about scaffolding techniques, using additional examples, and differentiated instructional guidelines as suggested by the Glencoe resource. Materials for this unit are found in the Glencoe Geometry textbook and associated resources. Use nonlinguistic representations such as word walls, foldables, or graphic organizers to support the learning of vocabulary. Vocabulary support is essential to student success in Geometry. A foldable study organizer, such as the one illustrated on page 898, can also be used to assist students with note taking skills. Students can create and use a foldable to take notes, define terms, capture key concepts and ideas, and write illustrated examples. Carefully select items from the textbook sections or resource masters to be sure not to confuse students with other probability concepts. The focus in this unit is for students to develop understanding of probabilities of independent events and use them to interpret data, develop understanding of mutually exclusive events, application of the Addition rule for probability, and finding conditional probability. Additional CCSS supplementary district resources for this probability unit are available in the Supplemental Unit Materials Section of the binder. They are also accessible online on the Glencoe McGraw-Hill Math Geometry Student and Teacher textbooks or on the following website: To access the CCSS Supplements using the online textbooks, select the CCSS icon on the homepage of the online textbook and then choose the corresponding lesson or lab to access the supplementary Geometry Glencoe CCSS Lab 15: Two-Way Frequency Tables identified in the resource section. Experiences of a probabilistic nature are critical as students begin to grapple with the different representations in an effort to make good decisions and to be confident in their predictions. Reviewing experimental and theoretical probability allows you to choose a variety of hands-on experimental activities. Use nonlinguistic representations such as spinners, technology, coins, dice, and D-106 Providence Public Schools

11 Probability (11-13 days) Algebra 2, Quarter 4, Unit 4.1 other concrete models to allow students to visually confirm the theoretical probability of selecting a particular item and then compare that calculation to the experimental results of actually conducting the event for a specific number of trials. Spinners or other concrete models can be used as quick simulation tools to activate prior knowledge of theoretical and experimental probability. Facilitate a discussion regarding the independence of events. Key points of the discussion are identified in the Independent Events and Checking for Independence activities in the Supplemental Unit Materials section of this binder. Consider using a kinesthetic representation such as Take Your Corners to activate prior knowledge of mutually exclusive events. This nonlinguistic approach allows students to show their opinion in a nonthreatening way through physical movement. Classify one corner of the room as mutually exclusive and other corner as non-mutually exclusive. The students are then presented with a series of events which need to be classified as mutually exclusive or not. The students physically move to the corresponding corner. This activity can be modified as appropriate using all four corners of the room. To find the probability of event A or B, we must first determine whether the events are mutually exclusive or non-mutually exclusive. Then we can apply the appropriate Addition Rule: Addition Rule 1: When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event. P(A or B) = P(A) + P(B) Addition Rule 2: When two events, A and B, are non-mutually exclusive, there is some overlap between these events. The probability that A or B will occur is the sum of the probability of each event, minus the probability of the overlap. P(A or B) = P(A) + P(B) - P(A and B) The following examples may serve as additional resources for clarification of the topic or CCSS. An example of a Uniform Probability Model is drawing a card from a deck (with replacement) is 152. Randomly select a card. Given that the card is red, what is the probability that it is a queen? The concept of probability of mutually exclusive events can be confusing for students. In the Glencoe Geometry teacher edition on page 939 are Tips for New Teachers strategies for using nonlinguistic representations such as a Venn diagram to visually show how two mutually exclusive events do not have overlapping circles and how two events which are not mutually exclusive will have overlapping circles. On page 933 of the Geometry teacher edition, the Focus on Mathematical Content emphasizes that students should be aware that P(AlB) P(BlA). Providence Public Schools D-107

12 Algebra 2, Quarter 4, Unit 4.1 Probability (11-13 days) A Venn diagram similar to the one below can help students practice using their understanding of probability by identifying similarities and differences and using a Venn diagram to classify data. In this example, they review data on how many students in class have skate shoes, inline skates, and/or skateboards. Data similar to this can be used to reinforce sample space and conditional probability. Sample Space Skateboards 18% 3% 12% 15% Skate Shoes 17% 4% 10% Inline Skates 21% Consider asking students questions such as: Is having inline skates (S1) and skate shoess (S2) independent? A sample solution for this scenario is: P(S1) = 0.50 P(S1 І S2 ) = so the events are not independent. There are numerous websites available to help students understand probability. The website listed below clearly explains the Probability Rules including conditional probability and independence: science.psu.edu/stat200/node/11. Additionally, other textbooks, such as Stats: Modeling the World (Pearson, Addison Wesley) and For All Practical Purposes (Bedfordd Freeman), are excellent resources for supporting probability concepts. D-108 Providence Public Schools

13 Probability (11-13 days) Algebra 2, Quarter 4, Unit 4.1 The Glencoe Online Personal Tutors, can also be used to supplement classroom instruction and are located on the website As you assess students using the 5-minute check transparencies, a cues, questions, and advance organizers strategy is being used, since students are answering questions about content that is important. Some of the questions help students review prior knowledge, and these should be used at the beginning of a lesson; other questions, for use during and after the lesson, help students reinforce knowledge Notes Providence Public Schools D-109