Rational and Irrational Numbers 1 - Montgomery County Public Schools

CONCEPT DEVELOPMENT

Mathematics Assessment Project

CLASSROOM CHALLENGES

A Formative Assessment Lesson

Rational and

Irrational

Numbers 1

Mathematics Assessment Resource Service

University of Nottingham & UC Berkeley

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For more details, visit:

? 2012 MARS, Shell Center, University of Nottingham

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detailed at - all other rights reserved

Rational and Irrational Numbers 1

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:

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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:

3.

Construct viable arguments and critique the reasoning of others.

INTRODUCTION

The lesson unit is structured in the following way:

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Before the lesson, students attempt the assessment task individually. You then review students¡¯

work and formulate questions that will help them improve their solutions.

The lesson is introduced in a whole-class discussion. Students then work collaboratively in pairs

or threes to make a poster on which they classify numbers as rational and irrational. They work

with another group to compare and check solutions. Throughout their work students justify and

explain their decisions to peers.

In a whole-class discussion, students revisit some representations of numbers that could be either

rational or irrational and compare their classification decisions.

Finally, students work individually to show their learning using a second assessment task.

MATERIALS REQUIRED

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Each individual student will need a mini-whiteboard, an eraser, a pen, and a copy of the

assessment task Is it Rational?

Choose how to end the lesson. Either provide a fresh copy of the assessment task, Is it Rational?

for students to review and improve their work, or provide a copy of the assessment task,

Classifying Rational and Irrational Numbers.

For each small group of students provide a copy of the task sheet Poster Headings, a copy of the

task sheet Rational and Irrational Numbers, 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, 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 are also some projector resources to help with whole-class discussion.

TIME NEEDED

15 minutes before the lesson for the assessment task, a 1-hour lesson, and 10 minutes in a follow-up

lesson (or for homework). All timings are approximate, depending on the needs of your students.

Teacher guide

Rational and Irrational Numbers 1

T-1

BEFORE THE LESSON

Assessment task: Is it Rational? (15 minutes)

Have the 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 follow-up lesson.

Rational and Irrational Numbers 1

Student Materials

Alpha Version June 2011

Is it Rational?

Remember that a bar over digits indicate 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.

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5

!

5

7

0.575

!

Give each student a copy of Is it Rational? Introduce the

task briefly, and help the students understand what they

are being asked to do.

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Spend 15 minutes answering these questions.

5

5+ 7

10

2

5.75....

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

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Show all your work on the sheet, and make sure you

explain your answers really clearly.

(5+ 5)(5" 5)

(7 + 5)(5" 5)

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I have some calculators if you wish to use one.

!

? 2011 MARS University of Nottingham UK

S-1

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.

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.

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 the next page. These have

been drawn from common difficulties observed in trials of this unit.

We suggest that you write your own lists of questions, based on your own students¡¯ work, using the

ideas below. You may choose to write questions on each student¡¯s work. If you do not have time to

do this, select a few questions that will be of help to the majority of students. These can be written on

the board at the end of the lesson.

Teacher guide

Rational and Irrational Numbers 1

T-2

Common issues:

Suggested questions and prompts:

Student does not recognize rational numbers

from simple representations

? 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?

For example: The student does not recognize

integers as rational numbers.

Or: The student does not recognize terminating

decimals as rational.

Student 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.

Student does not recognize irrational

numbers from simple representations

For example: The student does not recognize 5

is irrational.

Student assumes that all fractions are rational

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For example: The student claims

10

2

is rational.

Student does not simplify expressions

involving radicals

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For example: The student assumes

? Use a calculator to find

1

9

, 29 , 39 ¡­ as a

decimal.

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? What fraction is 0.8 ?

! ! is 1 ?

? What kind of!decimal

3

!

? Write the first few square numbers. Only

these perfect square integers have whole

! roots. So which numbers can

number square

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?

(5+ 5)(5" 5) is irrational because there is an

irrational number in each bracket.

!

Student does not recognize that some

representations are ambiguous.

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

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5.75 ), and that represent irrational numbers

(non-terminating non-repeating decimals).

? 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

the decimal expansion of ¦Ð.

!

Teacher guide

Rational and Irrational Numbers 1

T-3

Common issues:

Suggested questions and prompts:

Student 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

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? How do you write

What about

4

9

1

3

as a decimal?

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? Does every rational number have a

! decimal expansion?

terminating

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0.57 cannot be written as a fraction.

Student does not know how to convert

repeating decimals to fraction form

For example: The student makes an error when

converting between representations (Q2b.)

Student 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.¡±

Student does not understand that repeating

non-terminating decimals are rational, and

non-repeating 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.

Student explanations are poor

For example: The student provides little or no

reasoning.

Teacher guide

? 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:

7

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? x = 0.7 , 10x = 7.7 , 9x = 7 , x = 9 .

? 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

What about

1

3

as a decimal?

4

?

9

? Does every rational numbers have a

! decimal expansion?

terminating

? Does

! every irrational number have a

terminating decimal expansion?

? Which non-terminating decimals can be

written as fractions?

? 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?

Rational and Irrational Numbers 1

T-4

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