Rational and Irrational Numbers 1

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 Beta Version

For more details, visit: ? 2012 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

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:

? 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:

? 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

? 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

Is it Rational?

Alpha Version June 2011

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.

!

5

!

5

7

0.575

Give each student a copy of Is it Rational? Introduce the ! ! 5

task briefly, and help the students understand what they are being asked to do.

! 5+ 7

Spend 15 minutes answering these questions. I'd like you to work alone for this part of the lesson.

10

!

2

5.75....

!

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

! (5+ 5)(5" 5)

explain your answers really clearly. I have some calculators if you wish to use one.

(7 + 5)(5" 5) !

? 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

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

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

? 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?

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.

?

Use a calculator to find

1 9

,

2 9

,

3 9

...

as

a

decimal.

_

? What fraction is 0.8?

?

What

kind

of!de!ci!mal

is

1 3

?

? Write the!first few square numbers. Only these perfect square integers have whole number squa!re roots. So which numbers can

you find that have irrational square roots?

Student 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?

Student 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 bracket.

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

irrational?

Student 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 the decimal expansion of .

!

(non-terminating non-repeating decimals).

!

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.

?

How do you write

1 3

as a decimal?

What about

4 9

?

? Does every rational number have a termina!ting decimal expansion?

Or: The student disagrees with Hao, claiming

!

0.57 cannot be written as a fraction. !

Student does not know how to convert

? What is the difference between 0.57 and 0.57 ?

repeating decimals to fraction form !

?

How do you write

1 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 ,

1!0x = 7.7 ,

9x = 7 ,

x

=

7 9

.

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.

? 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 numbers have a termina!ting decimal expansion?

? D!oes every irrational number have a terminating decimal expansion?

? Which non-terminating decimals can be

written as fractions?

Student explanations are poor

For example: The student provides little or no reasoning.

? 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

Rational and Irrational Numbers 1

T-4

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