Task: True or False? Grade Level: 8 APS Mathematics Standard: Data Analysis, Statistics, and Probability Mr. Hall gave a quiz with ten true-false questions. When he corrected the quizzes, two students, who will remain anonymous, had identical answers. Unfortunately for them, all of those answers were incorrect. Does Mr. Hall have reason to be suspicious? Explain why or why not. Would he be more or less suspicious if all the answers had been correct? Use math language to explain your answer. Implementation Notes: The mathematics assessments are not designed to be reading tests, when implementing the task, teachers should read the tasks orally to their students, and make sure that all students are clear on what the task is asking and answering any questions that students may have. Teachers should not lead the student to a specific strategy or solution. Students may use manipulatives to work the problem to a solution. Students should not be limited by the amount of space at the bottom of the task to complete the problem. Allow students as much paper as they need to solve the task. Although recommended times are given for each assessment, the assessments are untimed and teachers should allow students as much time as needed to complete each task. Students must do individual work on the performance assessments. (No group work or collaborations are allowed.) Context From the Task Author: This task was given to students during a unit on probability. Previously the class had studied how to find the probability of independent events (which answers on a true-false test would be). We had recently learned how Pascal's Triangle could be used to find probabilities and many of the students used that knowledge when solving this task. What the task accomplishes Students enjoyed this task. They could easily relate to the situation. Beyond the problem of finding the probabilities, this task gave opportunity for great connections as the student could analyze why Mr. Hall should or should not be suspicious. What students will do Students will recognize that this is an independent event with an outcome of 1:2. Those students who understand the patterns and applications of Pascal s Triangle will quickly begin work on the 10 th row of the pattern to determine the probabilities for the 10-correct/incorrect answers. Students will realize that the probability of getting 10 correct answers is the same for getting 10 incorrect answers. Time Required: Most students completed the task within the 45-minute class period. Interdisciplinary Links: I wrote this task after discussing cheating on quizzes with a teammate. Perhaps students will consider this task the next time they are tempted to copy off a neighbor's quiz. Pre-Assessment Activity: This task should be given at the end of a unit on probability. Students will need to do the task Pascal s Triangle included with this task. Teachers should start with students identifying the patterns within the triangle and then do probability of independent events so students have experience applying the concepts. There are excellent resources available on probability concepts and activities. The I Hate Mathematics! Book Marilyn Burns APS/RDA/CHF: Performance-Based Mathematics Assessment 2001-02 Page 1
Math for Smarty Pants Marilyn Burns Probability Games and Other Activities Ivan Moscovich How Math Works by Carol Vorderman. A Note about Teacher Resources: The first place teacher s should look for pre-assessment activities is the curriculum materials they are currently using in the classroom. The materials that are suggested here are additional supplemental activities. Teaching Tips Students need to recognize when the outcome of one event, in this case the answers on a quiz, is independent of the outcome of a previous event. They must then be able to multiply the probabilities for each event together or be able to "read" Pascal's Triangle to find their solution. Explaining whether or not Mr. Hall should be suspicious brings a higher level of thought process into the problem. Some students were not able to get there, while others analyzed the situation carefully. Suggested Materials: Calculators, paper and pencil, some may need manipulatives to determine the 1:2 outcome of the True-False Quiz. Calculator Note: Students should use calculators to solve this problem, it would be too time consuming to have students compute their calculations by hand. The intent of the task is to determine if students can use formulas and interpret data. Task Modification for Special Needs Students: To modify the task, adjust the number of questions on the quiz to 5.If the student uses manipulatives, encourage them to use the manipulatives to help them write/represent their answers on the paper. Teachers may modify the task for gifted students adjusting the number of questions on the quiz to 20. If any of the task had to be modified for a student, remember to mark the student s computer score sheet as modified. Possible Solution Step 1: Initial Analysis The student determines that answering True/False questions are a 1:2 probability (correct/incorrect). The student determines that each answer is an independent event and can use Pascal s Triangle to determine the probability of choosing 10 incorrect answers out of 10 questions. Step 2: The student represents Pascal s Triangle. Row 10: 1 10 45 120 210 252 210 120 45 10 1 Step 3: Analysis of Pascal s Triangle (See the task Pascal s Triangle for an in-depth analysis.) The student adds the sum of row 10. The sum is 1024 The student compares the results of scoring 10 out of 10 answers correct, 9:10 correct, etc. to 0:10 correct. 1:1024 if 10:10 correct 10:1024 if 9:10 correct 45:1024 if 8:10 correct 120:1024 if 7:10 correct 210:1024 if 6:10 correct 252:1024 if 5:10 correct 210:1024 if 4:10 correct 20:1024 if 3:10 correct 45:1024 if 2:10 correct 10:1024 if 1:10 correct 1:1024 if 0:10 correct Step 4: Conclusion - The student determines that the probability of getting 0 correct on the quiz is 1:1024 and concludes that Mr. Hall should be suspicious of the students for cheating. APS/RDA/CHF: Performance-Based Mathematics Assessment 2001-02 Page 2
The probability of a student having all the answers wrong on a 10 question true-false quiz is 1 out of 1024 or (1/2) 10 or 1/1024. The probability of having all the answers correct (if you were randomly guessing) is also 1 out of 1024. But Mr. Hall should be more suspicious if all the answers are all wrong than if they are all right. Right answers could be the result of someone studying. Benchmark Descriptors: The benchmark descriptors and rubric are designed to help the teacher analyze student thinking and understanding at each of the four performance levels (Novice, Apprentice, Practitioner, and Expert). The descriptors are generalizations of what student work could look like. It is not possible to anticipate every answer a student can give, so in scoring student work the teacher must use these generalizations to come to their own conclusions as to where a student is performing on the assessment. It is recommended that teachers use the task specific rubric given for the assessment to identify the specific math skills that make up each section of the four performance levels for the task. Teachers should also review the benchmark papers provided to get a sense of the mathematics that students will use to solve the task. If the student does not attempt to solve the task or the work on the problem is completely unrelated to the task, the student s work on the task is considered Unscorable and should not be assigned a performance level of Novice, Apprentice, Practitioner, or Expert. General Rubric Novice The novice will not understand the concept of finding probabilities of independent events and will usually not even address any mathematics in the task. S/he is likely to answer the task by saying that Mr. Hall should check to see if the students were sitting next to each other. Apprentice The apprentice will understand the need to do something with the probability of right and wrong answers. S/he is likely to get bogged down in the process, and perhaps add the denominators of the fractions instead of multiplying. Practitioner The practitioner will successfully find the probability of getting all of the answers right or all of the answers wrong. S/he will use repeated multiplication of ½ rather than using powers of ½ to find the solution. The practitioner will not give an in-depth analysis of whether or not Mr. Hall should be suspicious but will base the reasoning on whether the students sat together. Expert The expert will clearly explain, using appropriate math language, the method used to find the probability of getting all the answers wrong or right. S/he will use either powers of ½ or Pascal s Triangle to find those correct probabilities. The expert will discuss in greater depth whether or not Mr. Hall should be suspicious. S/he may include a student-made version of Pascal s Triangle for a representation. APS/RDA/CHF: Performance-Based Mathematics Assessment 2001-02 Page 3
Mathematical Standards Alignment: All of the APS Performance-Based Mathematics Assessments being distributed district-wide in grades 2, 5, and 8 are aligned to the current Albuquerque Public Schools and NCTM Mathematics Standards. Each task lists all of the standards that align with that task. The performance tasks are math rich and align to many aspects of the standards. To focus the teacher on the key mathematics that is intended for the problem, Target Performance Standards have been indicated for each task. The dominant mathematics strand that the task covers is listed first. Every task covers Strand I: Global Mathematical Processes and is listed last. APS Mathematical Content and Performance Standards Target Performance Standards Grade 8 Mathematics Standards: 1. Interprets data and makes conclusions from data. Grade 7 Mathematics Standards: 2. Determines simple probability in experimental and theoretical situations. 3. Determines probability of dependent and independent events in experimental and theoretical situations. Task Proficiency: This task is designed to determine an 8 th grader s ability to interpret data and draw conclusions based on an independent probability experiment. This is a multi-step task and 8 th graders will need to show their understanding of probability throughout the task. The Practitioner can accurately read Pascal s Triangle and determine the likelihood of two students answering 10 True- False questions incorrectly. The Apprentice will not be able to accurately interpret the information from Pascal s Triangle and base their conclusion on inaccurate data. The Expert will be able to analyze Pascal s Triangle and provide a more detailed conclusion. The Expert may interpret the probabilities of each event in terms of percents. Strand IV - Data Analysis, Statistics, and Probability: The student identifies patterns and special features of data and events of chance through experiences with meaningful mathematical problems that focus on comparing, predicting, representing data, and making decisions to communicate mathematical understanding. Benchmark (6 8): The student designs a data question with two variables and collects, represents and analyzes the data. The student uses a variety of graphical representations to display data and understand measures of center and spread. The student makes conjectures and computes simple probability outcomes using a variety of tools. Performance Standards: Sixth Grade: Probability Develops and evaluates inferences, predictions, and arguments that are based on data. Seventh Grade: Probability Applies counting principles to determine sample space (e.g., tree diagrams, fundamental counting principle, combinations, and permutations). Determines simple probability in experimental and theoretical situations. Determines probability of dependent and independent events in experimental and theoretical situations. Explains and uses appropriate terminology to describe complementary and mutually exclusive events. Eighth Grade: Statistics Interprets data and makes conclusions from data. Strand II Number Sense and Operations: The student demonstrates number sense through experiences with meaningful mathematical problems that focus on number meaning, number relationships, place value concepts, relative effects of operations, and multiple representations to communicate sound mathematical thinking. APS/RDA/CHF: Performance-Based Mathematics Assessment 2001-02 Page 4
Benchmark (6 8): The student understands problems involving fractions, decimals, and percents and develops, analyzes, and explains a variety of algorithms and methods to solve problems. Performance Standards: Seventh Grade: Exponents and Square Roots Explains and models the value of exponents and square roots. Simplifies and evaluates (solves) numerical expressions involving exponents (e.g., 2³ = 2 x 2 x 2 = 8). Eighth Grade: Exponents Simplifies and evaluates, if solvable, algebraic expressions for all types of real numbers including exponents and common square roots. Examines, describes, and models exponential patterns that reflect growth and decay (e.g., Represent doubling 1 every day for 10 days in exponential form). Number Theory Develops and evaluates arguments involving real numbers, their patterns and operations. Strand V Patterns, Functions, and Algebraic Concepts: The student demonstrates an understanding of algebraic skills and concepts through experiences with meaningful mathematical problems that focus on discovering, describing, modeling, and generalizing patterns and functions, representing and analyzing relationships, and finding and supporting solutions. Benchmark (6 8): The student uses tables, graphs, and symbolic representations of patterns. The student understands and uses variables and linear equations in algebraic problem solving. Performance Standards: Sixth Grade: Patterns Predicts sequences and patterns involving varying rates of change (e.g., growth over time). Variables, Expressions, and Equations Solves one-step equations using the concept of balance when quantities are added, subtracted, or divided to both sides of an equation. Seventh Grade: Variable Expressions Identifies and uses variable expressions and formulas to solve a variety of real-life situations (e.g., Simple Interest: I =prt). Represents, describes, and analyzes numerical patterns and linear relationships using tables, graphs, words, and standard algebraic notation. Functional Relationships Develops and tests strategies for solving two-step equations. Translates hypotheses into formal methods of solving algebraic equations. Eighth Grade: Functions Develops exponential functions to represent real-life situations (e.g., compound interest problem). Represents, describes, and analyzes numerical patterns and relationships using tables, graphs, words, and standard algebraic notation. Linear Equations and Inequalities Identifies and models real-life situations using multiple representations. Simplifies algebraic expressions including rational expressions. Develops and tests strategies for solving multi-step equations. Solves equations for specified variables (e.g., solve for h if A = bh/2). Strand I Global Mathematical Processes: The student understands and uses mathematical processes. APS/RDA/CHF: Performance-Based Mathematics Assessment 2001-02 Page 5
Benchmark (K 12): The student uses problem solving, reasoning and proof, communications, connections, and representations as appropriate in all mathematical experiences. Performance Standards: Grades Kindergarten through twelve: Problem Solving and Reasoning Develops resourcefulness and perseverance in problem solving in mathematics and other disciplines. Recognizes when to use previously learned strategies to solve new problems. Develops and uses strategies (e.g., breaking complex problems into simpler parts) for solving given problems. Monitors, discusses, and reflects on the process of mathematical problem solving. Reasoning and Proof Makes and investigates mathematical conjectures and uses them successfully in developing and evaluating mathematical arguments and proofs. Uses the concept of counterexample to test the legitimacy of an argument. Develops a logical sequence of arguments leading to a valid conclusion or solution to a problem (e.g., statement/reasons, proof, informal proof, and algebraic steps). Communication Works in teams to share ideas, to develop and coordinate group approaches to problems, and to communicate findings. Communicates mathematical thinking coherently and clearly to others. Analyzes and evaluates mathematical thinking and strategies of others. Connections Relates applications to mathematical language in various modalities. Identifies and connects functions with real-world applications. Identifies how seemingly different mathematical situations may be essentially the same (e.g. the intersection of two lines is the same as the solution to a system of linear equations). Investigates and explains the mathematics required for various careers. Recognizes and applies mathematics in contexts outside the mathematics course. Representations Develops a repertoire of mathematical representation (e.g. pictures, written symbols, oral language, realworld situations, and manipulative models) that can be used purposefully and appropriately interchangeably. Selects, applies, and translates among mathematical representations to solve problems. Uses representations to model and interpret physical, social, and mathematical phenomena. Uses manipulatives, calculators, computers, and other tools as appropriate in order to strengthen mathematical thinking, understanding, and power to build upon foundational concepts. NCTM STANDARDS (Grades 6-8) In grades 6 8 all students should: NUMBER AND OPERATIONS Standard: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation. Use factors, multiples, prime factorization, and relatively prime numbers to solve problems. Standard: Compute fluently and make reasonable estimates. Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results. ALGEBRA Standard: Understand patterns, relations and functions. Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules. APS/RDA/CHF: Performance-Based Mathematics Assessment 2001-02 Page 6
Relate and compare different forms of representation for a relationship. Standard: Use mathematical models to represent and understand quantitative relationships. Model and solve contextualized problems using various representations, such as graphs, tables, and equations. DATA ANALYSIS AND PROBABILITY Standard: Understand and apply basic concepts of probability. Understand and use appropriate terminology to describe complementary and mutually exclusive events. Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations. Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models. PROBLEM SOLVING Standard: Build mathematical knowledge through problem solving. Standard: Solve problems that arise in mathematics and in other context. Standard: Apply and adapt a variety of appropriate strategies to solve problems. REASONING AND PROOF Standard: Recognize reasoning and proof as fundamental aspects of mathematics. Standard: Make and investigate mathematical conjectures. Standard: Develop and evaluate mathematical arguments and proofs. Standard: Select and use various types of reasoning and methods of proof. COMMUNICATIONS Standard: Organize and consolidate their mathematical thinking through communications. Standard: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. Standard: Analyze and evaluate the mathematical thinking and strategies of others. Standard: Use the language of mathematics to express mathematical ideas precisely. CONNECTIONS Standard: Recognize and use connections among mathematical ideas. Standard: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. Standard: Recognize and apply mathematics in contexts outside of mathematics. REPRESENTATIONS Standard: Create and use representations to organize, record, and communicate mathematical ideas. Standard: Select, apply, and translate among mathematical representations to solve problems. Standard: Use representations to model and interpret physical, social, and mathematical phenomena. APS/RDA/CHF: Performance-Based Mathematics Assessment 2001-02 Page 7
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