# Classifying Rational and Irrational Numbers

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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: 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 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

4 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: Does not recognize rational numbers from simple representations For example: The student does not recognize integers as rational numbers. Or: The student does not recognize terminating decimals as rational numbers. Does not recognize non-terminating repeating decimals as rational For example: The student states that a nonterminating repeating decimal cannot be written as a fraction. Does not recognize irrational numbers from simple representations For example: The student does not recognize is irrational. Assumes that all fractions are rational For example: The student claims 10 is rational. 2 Does not simplify expressions involving radicals For example: The student assumes (5+ 5)(5 5) is irrational because there is an irrational number in each parenthesis. Explanations are poor For example: The student provides little or no reasoning. 5 Suggested questions and prompts: A rational number can be written as a fraction of whole numbers. Is it possible to write 5 as a fraction using whole numbers? What about 0.575? Are all fractions less than one? Use a calculator to find decimal. _ 0.8? What fraction is What kind of decimal is 1, 9 1? 3 2, 9 3 as a 9 Write the first few square numbers. Only these perfect square integers have whole number square roots. So which numbers can you find that have irrational square roots? Are all fractions rational? Show me a fraction that represents a rational/irrational number? What happens if you remove the parentheses? Are all expressions that involve a radical irrational? Suppose you were to explain this to someone unfamiliar with this type of work. How could you make this math clear, to help the student to understand? Teacher guide Classifying Rational and Irrational Numbers T-3

5 Common issues: Does not recognize that some representations are ambiguous For example: The student writes that is rational or that it is irrational, not seeing that is a truncated decimal that could continue in ways that represent rational numbers (such as 5.75), and that represent irrational numbers (non-terminating non-repeating decimals). Does not recognize that repeating decimals are rational For example: The student agrees with Arlo that 0.57 is an irrational number. Or: The student disagrees with Hao, claiming 0.57 cannot be written as a fraction. Does not know how to convert repeating decimals to fraction form For example: The student makes an error when converting between representations (Q2b.) Does not interpret repeating decimal notation correctly For example: The student disagrees with Korbin, who said that the bar over the decimal digits means the decimal would go on forever if you tried to write it out. Does not understand that repeating nonterminating decimals are rational and nonrepeating non-terminating decimals are irrational For example: The student agrees with Hank, that because 0.57 is non-terminating, it is irrational and does not distinguish non-repeating from repeating non-terminating decimals. Suggested questions and prompts: The dots tell you that the digits would continue forever, but not how. Write a number that could continue but does repeat. And another And another Now think about what kind of number this would be if subsequent digits were the same as the decimal expansion of π. How do you write What about? as a decimal? Does every rational number have a terminating decimal expansion? What is the difference between 0.57 and 0.57? 1 How do you write 2 as a decimal? How would you write 0.5 as a fraction? Explain each stage of these calculations: x = 0.7, 10x = 7.7, 9x = 7, x = 7 9. Remember that a bar indicates that a decimal number is repeating. Write the first ten digits of these numbers: 0.45, Could you figure out the 100th digit in either number? How do you write What about ? Does every rational number have a terminating decimal expansion? Does every irrational number have a terminating decimal expansion? Which non-terminating decimals can be written as fractions? as a decimal? Teacher guide Classifying Rational and Irrational Numbers T-4

6 SUGGESTED LESSON OUTLINE Introduction (10 minutes) Give each student a mini-whiteboard, pen, and eraser. Use these to maximize participation in the introductory discussion. Explain the structure of the lesson to students: Recall your work on irrational and rational numbers [last lesson]. You ll have a chance to review that work in the lesson that follows today. Today s lesson is to help you improve your solutions. Display Slide P-1 of the projector resource: Classifying Rational and Irrational Numbers Terminating decimal Rational Numbers Irrational Numbers Non-terminating repeating decimal Non-terminating non-repeating decimal Projector Resources Explain to students that this lesson they will make a poster classifying rational and irrational numbers. I m going to give you some headings and a large sheet of paper. You re going to use the headings to make this classification poster. You will be given some cards with numbers on them and you have to classify them as rational or irrational, deciding where each number fits on your poster. Check students understanding of the terminology used for decimal numbers: On your whiteboard, write a number with a terminating decimal. Can you show a number with a non-terminating decimal on your whiteboard? Show me a number with a repeating decimal. Rational and Irrational Numbers 1 Show me the first six digits of a non-repeating decimal. Write on a large sticky note. Model the classification activity using the number Remind me what the little bar over the digits means. [It is a repeating decimal that begins ; the digits continue in a repeating pattern; it does not terminate.] In which row of the table does go? Why? [Row 2, because the decimal does not terminate but does repeat.] Ok. So this number is a non-terminating repeating decimal because the bar shows it has endless repeats of the same three digits. [Write this on the card.] Show me on your whiteboard: is rational or irrational? [Rational.] P-1 Teacher guide Classifying Rational and Irrational Numbers T-5

11 SOLUTIONS Assessment task: Is it Rational? 1. Number Possible student reasoning 5 5 is rational. It can be written as a fraction, the ratio of two integers a, b, with b 0, e.g. Students sometimes discount whole number rationals, especially , is rational, since it can be written as the ratio of two integers 7 a, b, with b is rational. Using decimal place value, 5 is irrational = Students might argue that if a number is not a perfect square then its root is irrational. Or they might argue that 4 < < 9, and there is no integer between 2 and 3. This doesn t establish that there is no fractional value for though. At this stage of their math learning, students are unlikely to produce a standard proof by contradiction of the irrationality of is irrational because the sum of a rational and irrational number is always irrational is irrational. Students may think it is rational because it is represented as a fraction From this truncated decimal representation, it is not possible to decide whether the number represented by is rational or irrational. This is a difficulty that students often encounter when reading calculator displays. The dots here indicate that the decimal is non-terminating, but the number might have a repeating (rational) or non-repeating (irrational) tail. Students could give _ examples to show this, e.g and (5+ 5)(5 5) The product of these two irrational factors is rational. Using a calculator will suggest that the product is rational, but not why. (5+ 5)(5 5) = = 20. Removing the parentheses and simplifying does provide a reason why the expression is rational. (7 + 5)(5 5) The product of these two irrational factors is irrational. (7 + 5)(5 5) = = Since 5 is irrational, 2 5is irrational and is irrational. Use of a calculator will not help to show the irrationality of Teacher guide Classifying Rational and Irrational Numbers T-10

12 2. Student Statement Agree or disagree? Arlo: 0.57 is an irrational number. Disagree Agreeing with Arlo, that 0.57 is an irrational number, may indicate a standard misconception. Many students believe that a number with an infinitely long decimal representation must be irrational. Hao: No, Arlo, it is rational, because 0.57 can be written as a fraction. Agree Hao is correct = Eiji: Maybe Hao s correct, you know. Because 0.57 = Agree with statement, but disagree with reason. Eiji is correct in that Hao s answer is correct, but he is incorrect to say that 0.57 = 57. This is a common error. 100 Korbin: Hang on. The decimal for 0.57 would go on forever if you tried to write it out. That s what the bar thing means, right? Agree 0.57 = does go on forever. Not recognizing this is a misunderstanding of standard notation. Hank: And because it goes on forever, that proves 0.57 has got to be irrational. Disagree Thinking that an infinitely long decimal tail means a number can t be rational is a common student error. 2. b is rational. Let x = Then 100x = So 99x = 57. Thus x = = Teacher guide Classifying Rational and Irrational Numbers T-11

13 Collaborative small-group work Rational numbers Irrational numbers Terminating decimal 7 8 = and is a terminating decimal. This cell is empty: there are no terminating decimal representations of irrational numbers = 123 and so rational and terminating (8 + 2)(8 2) = = 64 2 = rational, with a terminating decimal representation. 8 = = 2 2 is rational, with a terminating decimal representation. 2 8 = = 2 2 = 4 4 is rational, with a terminating decimal representation. Nonterminating repeating decimal 2 3 = 0.6 Students might already know this decimal conversion, or perform division to calculate it = Students might use division to calculate this commonly used approximation of π. Using a calculator, they may not find all the repeating digits and so would have no reason to suppose it is a repeating decimal. x = x = x x = x = 123 x = = x = x = x = x 10x = 122 x = = This cell is empty: there are no non-terminating repeating decimal representations of irrational numbers. Teacher guide Classifying Rational and Irrational Numbers T-12

14 Nonterminating nonrepeating decimal Rational numbers This cell is empty: no rational number has a non-terminating non-repeating decimal representation. Irrational numbers π is irrational, with a non-terminating, non-repeating decimal representation. We expect students will have been told of the irrationality of π and do not expect this claim to be justified. 3 is irrational since 3 is not a perfect square and only 4 integers that are perfect squares have rational roots. An irrational divided by a rational is irrational. 8 is irrational since 8 is not a square number and only integers that are square numbers have rational roots is the sum of two irrational numbers. It is irrational, since = = 3 2 and the product of a non-zero rational number and an irrational number is irrational. Some sums of irrational numbers are rational, such as There is insufficient information to assign the following number cards to a unique category: This is a non-terminating decimal, but the ellipsis does not determine whether it is repeating or non-repeating. It could turn out to be a rational number, such as It could turn out to be an irrational number, such as (a patterned tail of zeros and ones, with one extra zero between ones each time.) (rounded correct to three decimal places) and the calculator display: could be a rounded version of either a rational or an irrational number, because it might be a rounded terminating decimal, rounded non-terminating repeating decimal (both rational), or a non-terminating non-repeating decimal (irrational). This is also the case with the calculator display, which shows the first 9 digits of pi. 0.9 is a separate case. It is clearly a rational number, but its categorization as a non-terminating repeating decimal, rather than as a terminating decimal is dubious. Students might argue that 0.1= 1 9, 0.2 = 2 9,, 0.9 = 9 9 =1. Alternatively, they could show that if x = 0.9, 10x = 9.9 so 9x = 9 and x = 9 9 =1. A similar argument could be made for any of the numbers with terminating decimal expansions. Teacher guide Classifying Rational and Irrational Numbers T-13

15 Assessment task: Classifying Rational and Irrational Numbers 1. Number Possible student reasoning is a rational number. A student might establish this by showing that is 21 can be written as a fraction of integers e.g. or by stating that it is a 100 terminating decimal, and therefore rational is the ratio of two integers a, b, with b 0 and so is rational. Or 12 alternatively, 3 12 = 0.25, which is a terminating decimal and so is rational Students may argue that 12 is irrational because 12 = 3 4 = 2 3 and the product of a non-zero rational and an irrational number is irrational = = 3. The quotient of an irrational and a non-zero integer is 2 irrational The ellipsis does not determine how the decimal digits will continue, only that they do. The number could be irrational (the digits continue and do not repeat) or rational (the digits continue and repeat). Check whether students confuse non-repeating with non-patterned : a number in which the pattern of the digits is e.g is irrational because although patterned, the non-terminating tail will never repeat itself. ( 12 4)(4 + 12) ( 12 4)(4 + 12) = = 4. This is a rational number (rounded to 2 d.p.) There is no way of telling whether this is rational or irrational. For example: is irrational (providing the pattern continues) but is rational. Teacher guide Classifying Rational and Irrational Numbers T-14

16 2. In this question, the student s explanations are more important than the truth value assigned to the statement. These explanations may indicate the student still makes common errors or has alternative conceptions. Student Statement Otis 3 is a rational number because 8 it can be written as a fraction. Lulu 3 is irrational because 3 is 8 irrational. Agree or disagree? Disagree Agree Otis is incorrect. Students who agree with Otis may not understand that some fractions are irrational! That error comes from a partial definition of rational. A fuller definition of rational number is that it is a number that can be written as a ratio of integers. Lulu s statement is correct, but her justification is incomplete. 3 is irrational because 3 is irrational 8 and the quotient of an irrational number and non-zero integer is irrational. Leon is rational because you can write it as the fraction Agree with statement, but disagree with reason. Leon is correct that is rational, but his claim that it is the fraction is incorrect. Students may produce the correct rational representation of the recurring decimal: = 286. Alternatively, students may 999 argue that 286 = Joan is an irrational number because that decimal will carry on forever. Disagree Joan is incorrect to claim that is irrational. A repeating non-terminating decimal is a rational number. Ray (rounded to three decimal places) might be rational or irrational. Agree Ray is correct. If were rounded from e.g then it would be a rational number. If it were rounded from then, since this pattern does not repeat, the number would be irrational. Arita is rational - the little dots show the digits carry on in the same pattern forever. Disagree Arita is incorrect. The ellipsis shows only that the digits do continue, not how could represent either a rational or an irrational number Teacher guide Classifying Rational and Irrational Numbers T-15

17 Is it Rational? Remember that a bar over digits indicates a recurring decimal number, e.g = For each of the numbers below, decide whether it is rational or irrational. Explain your reasoning in detail (5+ 5)(5-5) (7+ 5)(5-5) Student materials Classifying Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

18 2. Arlo, Hao, Eiji, Korbin, and Hank were discussing This is the script of their conversation. Student Statement Agree or disagree? Arlo: 0.57 is an irrational number. Hao: No, Arlo, it is rational, because 0.57 can be written as a fraction. Eiji: Maybe Hao s correct, you know. Because 0.57 = Korbin: Hang on. The decimal for 0.57 would go on forever if you tried to write it. That s what the bar thing means, right? Hank: And because it goes on forever, that proves 0.57 has got to be irrational. 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. If you think it is not rational, explain how you know. Student materials Classifying Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

19 Poster Headings Classifying Rational and Irrational Numbers Non-terminating non-repeating decimal Terminating decimal Non-terminating repeating decimal Rational numbers Irrational numbers Student materials Classifying Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

20 Rational and Irrational Numbers (1) 7 8 _ (8 + 2)(8-2) p Student materials Classifying Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

21 Rational and Irrational Numbers (2) rounded correct to 3 decimal places Calculator display: Student materials Classifying Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

22 Hint Sheet Figure out the changes from one line to the next: x = x = x = 43 x = How can this help you figure out as a fraction? 3 4 What kind of number is 3? What is your definition of a rational number? Are all fractions rational numbers? Calculator display: 0.9 Write 1 9 as a decimal. Write 2 9 as a decimal. Does this calculator display show? Why might you say yes? Why might you say no? What fraction would you consider next? Continue the pattern. Student materials Classifying Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

23 Classifying Rational and Irrational Numbers 1. For each of the numbers below, decide whether it is rational or irrational. Explain your answers. Number Reasoning ( 12-4)(4 + 12) (rounded to 2 d.p.) Student materials Classifying Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

24 2. Some students were classifying numbers as rational and irrational. Decide whether you agree or disagree with each statement. Correct any errors. Explain your answers clearly. Student Statement Agree or disagree? Otis 3 8 is a rational number because it can be written as a fraction. Lulu 3 8 is irrational because 3 is irrational. Leon is rational because you can write it as the fraction Joan is an irrational number because that decimal will carry on forever. Ray (rounded to three decimal places) might be rational or irrational. Arita is rational - the little dots show the digits carry on in the same pattern forever. Student materials Classifying Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

25 Classifying Rational and Irrational Numbers Terminating decimal Rational Numbers Irrational Numbers Non-terminating repeating decimal Non-terminating non-repeating decimal Projector Resources Classifying Rational and Irrational Numbers P-1

26 Instructions for Placing Number Cards Take turns to choose a number card. When it is your turn: Decide where your number card fits on the poster. Does it fit in just one place, or in more than one place? Give reasons for your decisions. When it is your partner s turn: If you agree with your partner s reasoning, explain it in your own words. If you disagree with your partner s decision, explain why. Then together, figure out where to put the card. When you have reached an agreement: Write reasons for your decision on the number card. If the number card fits in just one place on the poster, place it on the poster. If not, put it to one side. Projector Resources Classifying Rational and Irrational Numbers P-2

27 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.

### Translating Between Repeating Decimals and Fractions

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Translating Between Repeating Decimals and Fractions Mathematics Assessment Resource Service University

### Representing the Laws of Arithmetic

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing the Laws of Arithmetic Mathematics Assessment Resource Service University of Nottingham

### Translating between Fractions, Decimals and Percents

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Translating between Fractions, Decimals and Percents Mathematics Assessment Resource Service University

### Evaluating Statements about Rational and Irrational Numbers

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Evaluating Statements about Rational and Irrational Numbers Mathematics Assessment Resource Service

### Representing Variability with Mean, Median, Mode, and Range

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing Variability with Mean, Median, Mode, and Range Mathematics Assessment Resource Service

### Representing Data with Frequency Graphs

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing Data with Graphs Mathematics Assessment Resource Service University of Nottingham & UC

### Classifying Equations of Parallel and Perpendicular Lines

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Classifying Equations of Parallel and Perpendicular Lines Mathematics Assessment Resource Service University

### Representing Data 1: Using Frequency Graphs

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing Data 1: Using Graphs Mathematics Assessment Resource Service University of Nottingham

### Solving Linear Equations in Two Variables

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Solving Linear Equations in Two Variables Mathematics Assessment Resource Service University of Nottingham

### Interpreting Distance-Time Graphs

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Interpreting Distance- Graphs Mathematics Assessment Resource Service University of Nottingham & UC

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Evaluating Statements About Probability Mathematics Assessment Resource Service University of Nottingham

### Evaluating Statements About Length and Area

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Evaluating Statements About Length and Area Mathematics Assessment Resource Service University of Nottingham

### Representing Polynomials

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing Polnomials Mathematics Assessment Resource Service Universit of Nottingham & UC Berkele

### Classifying Solutions to Systems of Equations

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Classifing Solutions to Sstems of Equations Mathematics Assessment Resource Service Universit of Nottingham

### Representing Data Using Frequency Graphs

Lesson 25 Mathematics Assessment Project Formative Assessment Lesson Materials Representing Data Using Graphs MARS Shell Center University of Nottingham & UC Berkeley Alpha Version If you encounter errors

### Decomposing Numbers (Operations and Algebraic Thinking)

Decomposing Numbers (Operations and Algebraic Thinking) Kindergarten Formative Assessment Lesson Designed and revised by Kentucky Department of Education Mathematics Specialists Field-tested by Kentucky

### Place Value (What is is the Value of of the the Place?)

Place Value (What is is the Value of of the the Place?) Second Grade Formative Assessment Lesson Lesson Designed and revised by Kentucky Department of Education Mathematics Specialists Field-tested by

### Lesson Plan. N.RN.3: Use properties of rational and irrational numbers.

N.RN.3: Use properties of rational irrational numbers. N.RN.3: Use Properties of Rational Irrational Numbers Use properties of rational irrational numbers. 3. Explain why the sum or product of two rational

### Time needed. Before the lesson Assessment task:

Formative Assessment Lesson Materials Alpha Version Beads Under the Cloud Mathematical goals This lesson unit is intended to help you assess how well students are able to identify patterns (both linear

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### Mathematical goals. Starting points. Materials required. Time needed

Level N of challenge: B N Mathematical goals Starting points Materials required Time needed Ordering fractions and decimals To help learners to: interpret decimals and fractions using scales and areas;

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number

### Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %

Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the

Beads Under the Cloud Intermediate/Middle School Grades Problem Solving Mathematics Formative Assessment Lesson Designed and revised by Kentucky Department of Education Mathematics Specialists Field-tested

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### 2 Mathematics Curriculum

New York State Common Core 2 Mathematics Curriculum GRADE GRADE 2 MODULE 3 Topic G: Use Place Value Understanding to Find 1, 10, and 100 More or Less than a Number 2.NBT.2, 2.NBT.8, 2.OA.1 Focus Standard:

### TRU Math Conversation Guide

Release Version Alpha TRU Math Conversation Guide Module A: Contextual Algebraic Tasks This TRU Math Conversation Guide, Module A: Contextual Algebraic Tasks is a product of The Algebra Teaching Study

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### Irrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers.

Irrational Numbers A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Definition: Rational Number A rational number is a number that

### Private talk, public conversation Mike Askew King s College London

Private talk, public conversation Mike Askew King s College London Introduction There is much emphasis on the importance of talk in mathematics education. In this article I explore what sort of talk we

### Lesson on Repeating and Terminating Decimals. Dana T. Johnson 6/03 College of William and Mary dtjohn@wm.edu

Lesson on Repeating and Terminating Decimals Dana T. Johnson 6/03 College of William and Mary dtjohn@wm.edu Background: This lesson would be embedded in a unit on the real number system. The set of real

### Handouts for teachers

ASKING QUESTIONS THAT ENCOURAGE INQUIRY- BASED LEARNING How do we ask questions to develop scientific thinking and reasoning? Handouts for teachers Contents 1. Thinking about why we ask questions... 1

Ohio Standards Connection: Government Benchmark A Identify the responsibilities of the branches of the U.S. government and explain why they are necessary. Indicator 2 Explain the structure of local governments

### 0.75 75% ! 3 40% 0.65 65% Percent Cards. This problem gives you the chance to: relate fractions, decimals and percents

Percent Cards This problem gives you the chance to: relate fractions, decimals and percents Mrs. Lopez makes sets of cards for her math class. All the cards in a set have the same value. Set A 3 4 0.75

### Unit 6 Number and Operations in Base Ten: Decimals

Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,

Grades 4-6 Mathematics Formative Assessment Lesson Designed and revised by Kentucky Department of Education Mathematics Specialists Field- tested by Kentucky Mathematics Leadership Network Teachers If

### Chapter 4 -- Decimals

Chapter 4 -- Decimals \$34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789

### Mathematical goals. Starting points. Materials required. Time needed

Level S6 of challenge: B/C S6 Interpreting frequency graphs, cumulative cumulative frequency frequency graphs, graphs, box and box whisker and plots whisker plots Mathematical goals Starting points Materials

### 1. The Fly In The Ointment

Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 5: Dividing Decimal Fractions, Part 2. The Fly In The Ointment The meaning of, say, ƒ 2 doesn't depend on whether we represent

### Decimal Fractions. Grades 6 and 7. Teacher Document. We acknowledge the valuable comments of Hanlie Murray and Sarie Smit

Decimal Fractions Grades 6 and 7 Teacher Document Malati staff involved in developing these materials: Therine van Niekerk Amanda le Roux Karen Newstead Bingo Lukhele We acknowledge the valuable comments

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### OA3-10 Patterns in Addition Tables

OA3-10 Patterns in Addition Tables Pages 60 63 Standards: 3.OA.D.9 Goals: Students will identify and describe various patterns in addition tables. Prior Knowledge Required: Can add two numbers within 20

### Sample Fraction Addition and Subtraction Concepts Activities 1 3

Sample Fraction Addition and Subtraction Concepts Activities 1 3 College- and Career-Ready Standard Addressed: Build fractions from unit fractions by applying and extending previous understandings of operations

### Math Circle Beginners Group October 18, 2015

Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### Materials: Student-paper, pencil, circle manipulatives, white boards, markers Teacher- paper, pencil, circle manipulatives, worksheet

Topic: Creating Mixed Numbers and Improper Fractions SOL: 5.7- The student will add and subtract with fractions and mixed numbers, with and without regrouping, and express answers in simplest form. Problems

### Fractions as Numbers INTENSIVE INTERVENTION. National Center on. at American Institutes for Research

National Center on INTENSIVE INTERVENTION at American Institutes for Research Fractions as Numbers 000 Thomas Jefferson Street, NW Washington, DC 0007 E-mail: NCII@air.org While permission to reprint this

### Decimal Notations for Fractions Number and Operations Fractions /4.NF

Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.

### Planning Guide. Grade 6 Factors and Multiples. Number Specific Outcome 3

Mathematics Planning Guide Grade 6 Factors and Multiples Number Specific Outcome 3 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg6/html/pg6_factorsmultiples/index.html

### Tom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.

Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find

### Section 1.5 Exponents, Square Roots, and the Order of Operations

Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares.

### Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### Using Proportions to Solve Percent Problems I

RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving

### Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern.

INTEGERS Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern. Like all number sets, integers were invented to describe

### Using games to support. Win-Win Math Games. by Marilyn Burns

4 Win-Win Math Games by Marilyn Burns photos: bob adler Games can motivate students, capture their interest, and are a great way to get in that paperand-pencil practice. Using games to support students

### 47 Numerator Denominator

JH WEEKLIES ISSUE #22 2012-2013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational

### Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain

### Objective To introduce the concept of square roots and the use of the square-root key on a calculator. Assessment Management

Unsquaring Numbers Objective To introduce the concept of square roots and the use of the square-root key on a calculator. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts

### Math Released Set 2015. Algebra 1 PBA Item #13 Two Real Numbers Defined M44105

Math Released Set 2015 Algebra 1 PBA Item #13 Two Real Numbers Defined M44105 Prompt Rubric Task is worth a total of 3 points. M44105 Rubric Score Description 3 Student response includes the following

### Calculator Practice: Computation with Fractions

Calculator Practice: Computation with Fractions Objectives To provide practice adding fractions with unlike denominators and using a calculator to solve fraction problems. www.everydaymathonline.com epresentations

### MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

### Guidance paper - The use of calculators in the teaching and learning of mathematics

Guidance paper - The use of calculators in the teaching and learning of mathematics Background and context In mathematics, the calculator can be an effective teaching and learning resource in the primary

### Subject: Math Grade Level: 5 Topic: The Metric System Time Allotment: 45 minutes Teaching Date: Day 1

Subject: Math Grade Level: 5 Topic: The Metric System Time Allotment: 45 minutes Teaching Date: Day 1 I. (A) Goal(s): For student to gain conceptual understanding of the metric system and how to convert

### Comparing Fractions Objective To provide practice ordering sets of fractions.

Comparing Fractions Objective To provide practice ordering sets of fractions. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management

### Mathematics Instructional Cycle Guide

Mathematics Instructional Cycle Guide Fractions on the number line 3NF2a Created by Kelly Palaia, 2014 Connecticut Dream Team teacher 1 CT CORE STANDARDS This Instructional Cycle Guide relates to the following

### Playing with Numbers

PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also

### Lesson Description. Texas Essential Knowledge and Skills (Target standards) Texas Essential Knowledge and Skills (Prerequisite standards)

Lesson Description This lesson gives students the opportunity to explore the different methods a consumer can pay for goods and services. Students first identify something they want to purchase. They then

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### Number Factors. Number Factors Number of factors 1 1 1 16 1, 2, 4, 8, 16 5 2 1, 2 2 17 1, 17 2 3 1, 3 2 18 1, 2, 3, 6, 9, 18 6 4 1, 2, 4 3 19 1, 19 2

Factors This problem gives you the chance to: work with factors of numbers up to 30 A factor of a number divides into the number exactly. This table shows all the factors of most of the numbers up to 30.

### Balanced Assessment Test Algebra 2008

Balanced Assessment Test Algebra 2008 Core Idea Task Score Representations Expressions This task asks students find algebraic expressions for area and perimeter of parallelograms and trapezoids. Successful

### 1. I have 4 sides. My opposite sides are equal. I have 4 right angles. Which shape am I?

Which Shape? This problem gives you the chance to: identify and describe shapes use clues to solve riddles Use shapes A, B, or C to solve the riddles. A B C 1. I have 4 sides. My opposite sides are equal.

### Multiplication and Division with Rational Numbers

Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up

### Unit 5 Length. Year 4. Five daily lessons. Autumn term Unit Objectives. Link Objectives

Unit 5 Length Five daily lessons Year 4 Autumn term Unit Objectives Year 4 Suggest suitable units and measuring equipment to Page 92 estimate or measure length. Use read and write standard metric units

### 3. Logical Reasoning in Mathematics

3. Logical Reasoning in Mathematics Many state standards emphasize the importance of reasoning. We agree disciplined mathematical reasoning is crucial to understanding and to properly using mathematics.

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### Current California Math Standards Balanced Equations

Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.

### Unit 11 Fractions and decimals

Unit 11 Fractions and decimals Five daily lessons Year 4 Spring term (Key objectives in bold) Unit Objectives Year 4 Use fraction notation. Recognise simple fractions that are Page several parts of a whole;

### Area and Perimeter: The Mysterious Connection TEACHER EDITION

Area and Perimeter: The Mysterious Connection TEACHER EDITION (TC-0) In these problems you will be working on understanding the relationship between area and perimeter. Pay special attention to any patterns

### BBC Learning English Talk about English Business Language To Go Part 1 - Interviews

BBC Learning English Business Language To Go Part 1 - Interviews This programme was first broadcast in 2001. This is not a word for word transcript of the programme This series is all about chunks of language

Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

### Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 1 Real Numbers

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 1 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

### GRADE 8 MATH: TALK AND TEXT PLANS

GRADE 8 MATH: TALK AND TEXT PLANS UNIT OVERVIEW This packet contains a curriculum-embedded Common Core standards aligned task and instructional supports. The task is embedded in a three week unit on systems

### Pre-Algebra Lecture 6

Pre-Algebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals

### Georgia Standards of Excellence Curriculum Frameworks. Mathematics. GSE Algebra II/Advanced Algebra Unit 1: Quadratics Revisited

Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Algebra II/Advanced Algebra Unit 1: Quadratics Revisited These materials are for nonprofit educational purposes only. Any other use

### Understand numbers, ways of representing numbers, relationships among numbers, and number systems

Equivalent Fractions and Comparing Fractions: Are You My Equal? Brief Overview: This four day lesson plan will explore the mathematical concept of identifying equivalent fractions and using this knowledge

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Rational Number Project

Rational Number Project Fraction Operations and Initial Decimal Ideas Lesson : Overview Students estimate sums and differences using mental images of the 0 x 0 grid. Students develop strategies for adding

### Mathematical goals. Starting points. Materials required. Time needed

Level N9 of challenge: B N9 Evaluating directed number statements statements Mathematical goals Starting points Materials required Time needed To enable learners to: make valid generalisations about the

### HIGH SCHOOL ALGEBRA: THE CYCLE SHOP

HIGH SCHOOL ALGEBRA: THE CYCLE SHOP UNIT OVERVIEW This packet contains a curriculum-embedded CCLS aligned task and instructional supports. The task is embedded in a 4-5 week unit on Reasoning with Equations

GRADE 8 English Language Arts Proofreading: Lesson 6 Read aloud to the students the material that is printed in boldface type inside the boxes. Information in regular type inside the boxes and all information

### Aim To help students prepare for the Academic Reading component of the IELTS exam.

IELTS Reading Test 1 Teacher s notes Written by Sam McCarter Aim To help students prepare for the Academic Reading component of the IELTS exam. Objectives To help students to: Practise doing an academic

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Ummmm! Definitely interested. She took the pen and pad out of my hand and constructed a third one for herself:

Sum of Cubes Jo was supposed to be studying for her grade 12 physics test, but her soul was wandering. Show me something fun, she said. Well I wasn t sure just what she had in mind, but it happened that