# Evaluating Statements About Probability

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1 CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Evaluating Statements About Probability Mathematics Assessment Resource Service University of Nottingham & UC Berkeley For more details, visit: MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at - all other rights reserved

2 Evaluating Statements About Probability MATHEMATICAL GOALS This lesson unit addresses common misconceptions relating to the probability of simple and compound events. The lesson will help you assess how well students understand concepts of: Equally likely events. Randomness. Sample size. COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics: 7.SP: Investigate chance processes and develop, use, and evaluate probability models. This lesson also relates to the following Standards for Mathematical Practice in the Common Core State Standards for Mathematics, with a particular emphasis on Practice 3:. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 6. Attend to precision. 7. Look for and make use of structure. INTRODUCTION This lesson unit is structured in the following way: Before the lesson, students work individually on an assessment task that is designed to reveal their current understanding and difficulties. You then review their solutions and create questions for students to answer in order to improve their work. After a whole-class introduction, pairs of students work together to justify or refute mathematical statements. They do this using their own examples and counterexamples. Students then explain their reasoning to another group of students. In a whole-class discussion students review the main mathematical concepts of the lesson. In a follow-up lesson, students again work alone on a task similar to the assessment task. MATERIALS REQUIRED Each individual student will need a copy of the assessment tasks Are They Correct? and Are They Correct? (revisited), a mini-whiteboard, a pen, and an eraser. Each pair of students will need a copy of the sheet True, False or Unsure?, (cut up into cards), a large sheet of paper for making a poster, and a glue stick. There is a projector resource to support whole-class discussions. TIME NEEDED 5 minutes before the lesson, a -hour lesson, and 20 minutes in a follow-up lesson. Timings are approximate and will depend on the needs of the class. Teacher guide Evaluating Statements About Probability T-

4 If you do not have time to do this, you could select a few questions that will be of help to the majority of students and write these questions on the board when you return the work to the students at the start of the follow-up lesson. Common issues: Q. Assumes that both outcomes are equally likely Q2. Assumes that likelihood is based on the pattern of recent events For example: The student argues that if there are already four boys in the family, the next child is likely to be a girl. Q3. Relies on own experiences For example: The student states they have never thrown four sixes in a row. Q3. Assumes that some numbers on a number cube are harder to roll than others Suggested questions and prompts: What factors affect whether it will rain tomorrow? What does a probability of 0.5 mean? What is the probability that the baby will be a girl? What is the probability that the baby will be a boy? Does the fact that there are already four boys in the family affect the gender of the next child? Is it more difficult to throw a six than a two? Is it more difficult to throw a six, then another six or a two, then a three? What is the probability of rolling a six on a number cube? What is the probability of rolling a two on a number cube? Teacher guide Evaluating Statements About Probability T-3

9 SOLUTIONS Assessment Task: Are They Correct?. This statement is incorrect. It highlights the misconception that all events are equally likely. There are many factors (e.g. the season) that will influence the chances of it raining tomorrow. 2. Assuming that the sex of a baby is a random, independent event equivalent to tossing a coin, the statement is incorrect. It highlights the misconception that later random events can compensate for earlier ones. The assumption is important: there are many beliefs and anecdotes about what determines the gender of a baby, but tossing a coin turns out to be a reasonably good model. 3. This statement is incorrect. This highlights the misconception that special events are less likely than more representative events. Assessment Task: Are They Correct? (revisited). This statement is incorrect. It is not known whether the four sections on the spinner are in equal proportion. The probability of getting the red section would only be 0.25 if this were the case. 2. This statement is true. There are more students than days of the week. 3. This statement is incorrect. There are many factors that could affect whether or not the school bus is on time. There is also a chance that the bus is early. Collaborative Activity: True, False or Unsure? A. If you roll a six-sided number cube and it lands on a six more than any other number, then the number cube must be biased. False. This statement addresses the misconception that probabilities give the proportion of outcomes that will occur. With more information (How many times was the cube rolled? How many more sixes were thrown?) more advanced mathematics could be used to calculate the probability that the number cube was biased, but you could never be 00% certain. B. When randomly selecting four letters from the alphabet, you are more likely to come up with D, T, M, J than W, X, Y, Z. False. This highlights the misconception that special events are less likely than more representative events. Students often assume that selecting the unusual letters W, X, Y and X is less likely. C. If you toss a fair coin five times and get five heads in a row, the next time you toss the coin it is more likely to show a tail than a head. False. This highlights the misconception that later random events compensate for earlier ones. The statement implies that the coin has some sort of memory. People often use the phrase the law of averages in this way. D. There are three outcomes in a soccer match: win, lose, or draw. The probability of winning is therefore. 3 False. This highlights the misconception that all outcomes are equally likely, without considering that some are much more likely than others. The probabilities are dependent on the rules of the game and which teams are playing. See, for example, Teacher guide Evaluating Statements About Probability T-8

11 Are They Correct?. Emma claims: Is she correct? Explain your answer fully: 2. Susan claims: If a family has already got four boys, then the next baby is more likely to be a girl than a boy. Is she correct? Explain your answer fully: 3. Tanya claims: If you roll a fair number cube four times, you are more likely to get 2, 3,, 6 than 6, 6, 6, 6. Is she correct? Fully explain your answer: Student materials Evaluating Statements About Probability S- 205 MARS, Shell Center, University of Nottingham

12 Card Set: True, False Or Unsure? A. If you roll a six-sided number cube and it lands on a six more than any other number, then the number cube must be biased. B. When randomly selecting four letters from the alphabet, you are more likely to come up with D, T, M, J than W, X, Y, Z. C. If you toss a fair coin five times and get five heads in a row, the next time you toss the coin it is more likely to show a tail than a head. D. There are three outcomes in a soccer match: win, lose, or draw. The probability of winning is therefore 3. E. When two coins are tossed there are three possible outcomes: two heads, one head, or no heads. The probability of two heads is therefore 3. F. Scoring a total of three with two number cubes is twice as likely as scoring a total of two. G. In a true or false? quiz with ten questions, you are certain to get five correct if you just guess. H. The probability of getting exactly two heads in four coin tosses is 2. Student materials Evaluating Statements About Probability S MARS, Shell Center, University of Nottingham

13 Are They Correct? (revisited). Andrew claims: A spinner has 4 sections - red, yellow, green, and blue. The probability of getting the red section is Is he correct? Explain your answer fully: 2. Stephen claims: In a group of ten students the probability of two students being born on the same day of the week is. Is he correct? Explain your answer fully: 3. Thomas claims: The school bus will either be on time or late. The probability that it will be on time is therefore 0.5. Is he correct? Fully explain your answer: Student materials Evaluating Statements About Probability S MARS, Shell Center, University of Nottingham

14 Two Bags of Jellybeans I have two bags A and B. Both contain red and yellow jellybeans. There are more red jellybeans in bag A than in bag B. If I choose one jellybean from each bag I am more likely to choose a red one from bag A than from bag B. Projector Resources Evaluating Statements About Probability P-

15 True, False or Unsure? Take turns to select a card and decide whether it is a true or false statement. Convince your partner of your decision. It is important that you both understand the reasons for the decision. If you don t agree with your partner, explain why. You are both responsible for each other s learning. If you are both happy with the decision, glue the card onto the paper. Next to the card, write reasons to support your decision. Put to one side any cards you are unsure about. Projector Resources Evaluating Statements About Probability P-2

16 Mathematics Assessment Project Classroom Challenges These materials were designed and developed by the Shell Center Team at the Center for Research in Mathematical Education University of Nottingham, England: Malcolm Swan, Nichola Clarke, Clare Dawson, Sheila Evans, Colin Foster, and Marie Joubert with Hugh Burkhardt, Rita Crust, Andy Noyes, and Daniel Pead We are grateful to the many teachers and students, in the UK and the US, who took part in the classroom trials that played a critical role in developing these materials The classroom observation teams in the US were led by David Foster, Mary Bouck, and Diane Schaefer This project was conceived and directed for The Mathematics Assessment Resource Service (MARS) by Alan Schoenfeld at the University of California, Berkeley, and Hugh Burkhardt, Daniel Pead, and Malcolm Swan at the University of Nottingham Thanks also to Mat Crosier, Anne Floyde, Michael Galan, Judith Mills, Nick Orchard, and Alvaro Villanueva who contributed to the design and production of these materials This development would not have been possible without the support of Bill & Melinda Gates Foundation We are particularly grateful to Carina Wong, Melissa Chabran, and Jamie McKee The full collection of Mathematics Assessment Project materials is available from MARS, Shell Center, University of Nottingham This material may be reproduced and distributed, without modification, for non-commercial purposes, under the Creative Commons License detailed at All other rights reserved. Please contact if this license does not meet your needs.

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