Classifying Rational and Irrational Numbers

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: ? 2015 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

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 homework a day or more before the formative assessment lesson. This will give you an opportunity to assess the work and identify students who have misconceptions or need other forms of help. You should then be able to target your help more effectively in the subsequent lesson.

Is it Rational?

Remember that a bar over digits indicates a recurring decimal number, e.g. 0.256 = 0.2565656...

1. For each of the numbers below, decide whether it is rational or irrational.

Explain your reasoning in detail.

!

5

5 7

Give each student a copy of Is it Rational?

0.575

I'd like you to work alone for this part of the lesson.

Spend 15 minutes answering these questions. Show all your work on the sheet and make sure you explain your answers really clearly.

I have some calculators if you wish to use one.

It is important that, as far as possible, students answer the questions without assistance. Help students to understand that they should not worry too much if they cannot understand or do everything because, in the next lesson, they will work on a related task that should help them make progress.

5

5+ 7

10 2

5.75....

(5+ 5)(5 ! 5)

(7 + 5)(5 ! 5)

Student materials

Rational and Irrational Numbers 1 ? 2014 MARS, Shell Center, University of Nottingham

2. Arlo, Hao, Eiji, Korbin, and Hank were discussing 0.57.

This is the script of their conversation.

Student

Statement !

Arlo:

0.57 is an irrational number.

S-1 Agree or disagree?

Assessing students' responses

Collect students' responses to the task. Make some notes on what their work reveals about their current levels of understanding and any difficulties they encounter. The purpose of this is to forewarn you of the issues that will arise during the lesson, so that you may prepare carefully.

We suggest that you do not score students' work. The research shows that this is counterproductive, as it encourages students to compare scores and distracts their attention from how they may improve their mathematics.

! Hao:

Eiji:

Korbin:

Hank:

No, Arlo, it is rational, because 0.57 can be written as a fraction.

!

Maybe Hao's correct, you know.

Because

0.57 =

57 .

100

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?

!

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. !

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 simple representations

For example: The student does not recognize integers as rational numbers.

Or: The student does not recognize terminating decimals as rational numbers.

? 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?

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 5 is irrational.

?

Use a calculator to find

1 9

,

2 9

,

3 9

...

as

a

decimal.

_

? What fraction is 0.8?

?

What

kind

ofdecimal

is

1 3

?

? Write thefirst few square numbers. Only

these perfect square integers have whole number square roots. So which numbers can

you find that have irrational square roots?

Assumes that all fractions are rational

For example: The student claims

10 2

is rational.

? Are all fractions rational? ? Show me a fraction that represents a

rational/irrational number?

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.

? What happens if you remove the parentheses? ? Are all expressions that involve a radical

irrational?

Explanations are poor

? Suppose you were to explain this to someone

For example: The student provides little or no reasoning.

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

Common issues:

Suggested questions and prompts:

Does not recognize that some representations are ambiguous

? The dots tell you that the digits would continue forever, but not how. Write a

For example: The student writes that 5.75... is

rational or that it is irrational, not seeing that

5.75... is a truncated decimal that could continue

in ways that represent rational numbers (such as

5.75), and that represent irrational numbers

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 .

(non-terminating non-repeating decimals).

Does not recognize that repeating decimals are rational

For example: The student agrees with Arlo that

?

How do you write

1 3

as a decimal?

What about

4 9

?

0.57 is an irrational number.

? Does every rational number have a terminating decimal expansion?

Or: The student disagrees with Hao, claiming

0.57 cannot be written as a fraction.

Does not know how to convert repeating

? What is the difference between 0.57 and 0.57 ?

decimals to fraction form

1

? How do you write 2 as a decimal?

For example: The student makes an error when

? How would you write 0.5 as a fraction?

converting between representations (Q2b.)

? Explain each stage of these calculations:

?

x = 0.7 ,

10x = 7.7 ,

9x = 7,

x=

7 9.

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.

? Remember that a bar indicates that a decimal number is repeating. Write the first ten digits

of these numbers: 0.45 , 0.345 . Could you figure out the 100th digit in either number?

?

How do you write

1 3

as a decimal?

What about

4 9

?

? 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?

Teacher guide

Classifying Rational and Irrational Numbers

T-4

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