CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Classifying Rational and Irrational Numbers Mathematics Assessment Resource Service University of Nottingham & UC Berkeley For more details, visit: http://map.mathshell.org © 2015 MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at http://creativecommons.org/licenses/by-nc-nd/3.0/ - all other rights reserved Classifying Rational and Irrational Numbers MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students are able to distinguish between rational and irrational numbers. In particular, it aims to help you identify and assist students who have difficulties in: • Classifying numbers as rational or irrational. • Moving between different representations of rational and irrational numbers. COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics: N-RN: Use properties of rational and irrational numbers. This lesson also relates to the following Standards for Mathematical Practice in the Common Core State Standards for Mathematics, with a particular emphasis on Practices 3 and 6: 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. INTRODUCTION The lesson unit is structured in the following way: • Before the lesson, students attempt the assessment task individually. You then review students’ work and formulate questions that will help them improve their solutions. • After a whole-class introduction, students work collaboratively in pairs or threes classifying numbers as rational and irrational, justifying and explaining their decisions to each other. Once complete, they compare and check their work with another group before a whole-class discussion, where they revisit some representations of numbers that could be either rational or irrational and compare their classification decisions. • In a follow-up lesson, students work individually on a second assessment task. MATERIALS REQUIRED • Each individual student will need a mini-whiteboard, pen, and eraser, and a copy of Is it Rational? and Classifying Rational and Irrational Numbers. • Each small group of students will need the Poster Headings, a copy of Rational and Irrational Numbers (1) and (2), a large sheet of poster paper, scrap paper, and a glue stick. • Have calculators and several copies of the Hint Sheet available in case students wish to use them. • Either cut the resource sheets Poster Headings, Rational and Irrational Numbers (1) and (2), and Hint Sheet into cards before the lesson, or provide students with scissors to cut-up the cards themselves. • You will need some large sticky notes and a marker pen for use in whole-class discussions. • There is a projector resource to help with whole-class discussions. TIME NEEDED 15 minutes before the lesson for the assessment task, a 1-hour lesson, and 20 minutes in a follow-up lesson. All timings are approximate and will depend on the needs of your students. Teacher guide Classifying Rational and Irrational Numbers T-1 BEFORE THE LESSON Assessment task: Is it Rational? (15 minutes) Have students do this task in class or for Is it Rational? homework a day or more before the formative Remember that a bar over digits indicates a recurring decimal number, e.g. 0.2 56 = 0.2565656... assessment lesson. This will give you an 1. For each of the numbers below, decide whether it is rational or irrational. Explain your reasoning in detail. opportunity to assess the work and identify ! students who have misconceptions or need 5 other forms of help. You should then be able to target your help more effectively in the 5 7 subsequent lesson. Give each student a copy of Is it Rational? 0.575 I’d like you to work alone for this part of 5 the lesson. Spend 15 minutes answering these 5+ 7 questions. Show all your work on the sheet 10 and make sure you explain your answers really clearly. 2 I have some calculators if you wish to use 5.75.... one. (5 5)(5 5) It is important that, as far as possible, students + ! answer the questions without assistance. Help (7+ 5)(5! 5) students to understand that they should not Student materials Rational and Irrational Numbers 1 S-1 worry too much if they cannot understand or do © 2014 MARS, Shell Center, University of Nottingham everything because, in the next lesson, they will 2. Arlo, Hao, Eiji, Korbin, and Hank were discussing 0. 57. This is the script of their conversation. work on a related task that should help them Student Statement ! Agree or disagree? make progress. Arlo: 0. 57 is an irrational number. Assessing students’ responses ! Hao: No, Arlo, it is rational, because 0. 57 can be written as a Collect students’ responses to the task. Make fraction. some notes on what their work reveals about ! Eiji: Maybe Hao’s correct, you know. their current levels of understanding and any 57 Because 0.57 = . difficulties they encounter. The purpose of this 100 Korbin: Hang on. The decimal for 0. 57 would go on forever if is to forewarn you of the issues that will arise !you tried to write it. That’s what the bar thing means, right? during the lesson, so that you may prepare ! Hank: And because it goes on forever, that proves 0. 57 has carefully. got to be irrational. ! We suggest that you do not score students’ work. The research shows that this is a. In the right hand column, write whether you agree or disagree with each student’s statement. b. If you think 0. 57 is rational, say what fraction it is and explain why. counterproductive, as it encourages students to If you think it is not rational, explain how you know. compare scores and distracts their attention ! from how they may improve their mathematics. Instead, help students to make progress by asking questions that focus attention on aspects of their work. Some suggestions for these are given in the Common issues table on pages T-3 and T-4. These have been drawn from common difficulties observed in trials of this unit. Student materials Rational and Irrational Numbers 1 S-2 © 2014 MARS, Shell Center, University of Nottingham Teacher guide Classifying Rational and Irrational Numbers T-2 We suggest you make a list of your own questions, based on your students’ work. We recommend you either: • write one or two questions on each student’s work, or • give each student a printed version of your list of questions and highlight the questions for each individual student. 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 on the board when you return the work to the students in the follow-up lesson. Common issues: Suggested questions and prompts: Does not recognize rational numbers from • A rational number can be written as a fraction simple representations of whole numbers. Is it possible to write 5 as For example: The student does not recognize a fraction using whole numbers? What about integers as rational numbers. 0.575? • Are all fractions less than one? Or: The student does not recognize terminating decimals as rational numbers. Does not recognize non-terminating repeating • Use a calculator to find 1 , 2 , 3 … as a 9 9 9 decimals as rational decimal. For example: The student states that a non- _ • What fraction is 0. 8? terminating repeating decimal cannot be written • What kind of€ decimal € € is 1 ? as a fraction. 3 € Does not recognize irrational numbers from • Write the first few square numbers. Only simple representations these perfect square integers have whole number square€ roots. So which numbers can For example: The student does not recognize 5 you find that have irrational square roots? is irrational. Assumes that all fractions are rational • Are all fractions rational? € • Show me a fraction that represents a 10 For example: The student claims is rational. 2 rational/irrational number? Does not simplify expressions involving • What happens if you remove the parentheses? radicals • Are all expressions that involve a radical € For example: The student assumes irrational? ( 5+ 5)(5− 5) is irrational because there is an irrational number in each parenthesis. Explanations are poor • Suppose you were to explain this to someone € unfamiliar with this type of work. How could For example: The student provides little or no you make this math clear, to help the student reasoning. to understand? Teacher guide Classifying Rational and Irrational Numbers T-3 Common issues: Suggested questions and prompts: Does not recognize that some representations • The dots tell you that the digits would are ambiguous continue forever, but not how. Write a number that could continue but does repeat. For example: The student writes that 5.75... is And another… And another… rational or that it is irrational, not seeing that • Now think about what kind of number this 5.75... is a truncated decimal that could continue would be if subsequent digits were the same in ways that represent rational numbers (such as € as the decimal expansion of π. 5. 75), and that represent irrational numbers € (non-terminating non-repeating decimals). Does not recognize that repeating decimals • How do you write 1 as a decimal? 3 € are rational What about 4 ? For example: The student agrees with Arlo that 9 • Does every rational number have a 0. 57 is an irrational number. terminating€ decimal expansion? Or: The student disagrees with Hao, claiming € 0.