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
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:
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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.
!
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.
!
!
!
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
!
For example: The student claims
10
2
is rational.
Student does not simplify expressions
involving radicals
!
For example: The student assumes
? Use a calculator to find
1
9
, 29 , 39 ¡ as a
decimal.
_
? 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
!
!
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
!
!
? 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
!
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
!
? 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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