Lesson 27
Mathematics Assessment Project
Formative Assessment Lesson Materials
Rational and Irrational Numbers 1
MARS Shell Center
University of Nottingham & UC Berkeley
Alpha Version
If you encounter errors or other issues in this version, please send details to the MAP team
c/o map.feedback@mathshell.org.
© 2011 MARS University of Nottingham
Rational and Irrational Numbers 1
Teacher Guide
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Rational and Irrational Numbers 1
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Mathematical goals
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Alpha Version August 2011
This lesson unit is intended to help you assess how well students are able to identify rational and irrational
numbers. In particular, this unit aims to help you identify and assist students who have difficulties in:
•
•
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Classifying rational and irrational numbers.
Moving between different representations of rational/irrational numbers.
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Common Core State Standards
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This lesson involves mathematical content in the standards from across the grades, with emphasis on:
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N-RN: Use properties of rational and irrational numbers.
This lesson involves a range of mathematical practices, with emphasis on:
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3.
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Introduction
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The lesson unit is structured in the following way:
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Construct viable arguments and critique the reasoning of others.
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Materials required
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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.
During the lesson, students work collaboratively in pairs or threes, making a poster on which they
classify representations of numbers as rational and irrational. They then work with another group to
check their solutions. Throughout their work students justify and explain their decisions to peers. In
whole-class discussion, students compare their decisions about classification of some ambiguous
representations.
Finally, students work individually again to improve their solutions to the assessment task.
•
•
Each individual student will need a mini-whiteboard, an eraser and a pen, and two copies of the
assessment task Is it Rational?
For each small group of students provide a copy of the task sheet Poster Headings (cut up into
headings), a copy of the task sheet Rational and Irrational Numbers (cut up into cards), a large (C or
D) sized sheet of poster paper, a sheet of scrap paper, and a glue stick.
Have calculators and several copies of the Hint Sheet (cut up) available in case students wish to use
them.
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
Fifteen minutes before the lesson for the assessment task, a one-hour lesson, and ten minutes in a follow up
lesson (or for homework). All timings are approximate, depending on the needs of your students.
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© 2011 MARS University of Nottingham
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Rational and Irrational Numbers 1
Teacher Guide
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Before the lesson
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Assessment task: Is it Rational? (15 minutes)
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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 will then be able to target your help
more effectively in the follow-up lesson.
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Assessing students’ responses
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Collect students’ responses to the task. Make some notes
on what their work reveals about their current levels of
understanding and difficulties. The purpose of this is to
forewarn you of the issues that will arise during the
lesson, so that you may prepare carefully.
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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
!
!
!
!
It is important that students answer the questions without
assistance, as far as possible.
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Student Materials
Is it Rational?
Spend fifteen minutes working individually,
answering these questions.
Show all your work on the sheet.
Make sure you explain your answers really
clearly.
I have some calculators if you wish to use one.
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Rational and Irrational Numbers 1
Give each student a copy of Is it Rational? Introduce the
task briefly, and help the class to understand what they are
being asked to do.
Students should not worry too much if they cannot
understand or do everything, because in the next lesson
they will engage in a similar task, which should help
them. Explain to students that by the end of the next
lesson, they should expect to answer questions such as
these confidently. This is their goal.
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Alpha Version August 2011
5
5+ 7
10
2
5.75....
!
!
!
!
(5+ 5)(5" 5)
(7 + 5)(5" 5)
© 2011 MARS University of Nottingham UK
S-1
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 further 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.
© 2011 MARS University of Nottingham
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Rational and Irrational Numbers 1
Teacher Guide
Alpha Version August 2011
Common issues:
Suggested questions and prompts:
Student does not recognize rational numbers
from simple representations
• A rational number can be written as a fraction.
Is it possible to write 5 as a fraction? 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 decimal as rational
For example: The student may think 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 2
,
9 9
, 39 … as a decimal
_
• What fraction is 0.8 ?
• What kind of decimal is 13 ?
!!!
!
• Write the first few square numbers. Only these
! integers have whole number
perfect square
square roots. So which numbers can you find
that have irrational square roots?
Student assumes that all fractions are rational
!
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For example: The students writes that
is
2
rational.
• Are all fractions rational?
• Show me a fraction that represents a
rational/irrational number?
Student does not recognize that some
!
representations are ambiguous.
• 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 π.
For example: The student writes that 5.75... is
rational.
Or: The student writes that 5.75... is irrational. The
student does not recognize that 5.75... is a truncated
!
decimal could continue in multiple ways, some of
which represent rational numbers (such as 5.75 ),
! are irrational numbers (because
and others of which
! non-repeating decimals).
they are non-terminating
1
3
!
Student does not realize that repeating
decimals
are rational
• How do you write
For example: The student agrees with Arlo that 0.57
is an irrational number.
• Does every rational number have a terminating
decimal!expansion?
Or: The student disagrees with Hao, and says that
0.57 cannot be written as a fraction. !
What about
4
9
as a decimal?
?
!
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!
© 2011 MARS University of Nottingham
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Rational and Irrational Numbers 1
Teacher Guide
Common issues:
Student does not know how to convert repeating
decimals to fraction form
For example: The student makes an error when
converting between representations (Q2b.)
Alpha Version August 2011
Suggested questions and prompts:
• What is the difference between 0.57 and 0.57 ?
1
as a decimal?
2
• How would you write 0.5 as a fraction?
• How do you write
• Explain each stage of these calculations in
detail:
x = 0.7 , 10x = 7.7 , 9x = 7 , x =
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 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
Student explanations are poor
For example: The student provides little or no
reasoning.
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.
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• 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
?
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• Does every rational numbers have a terminating
decimal!expansion?
• 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?
Or: Student assumes that the number is irrational
because there is an irrational number in the sum, e.g.
(5+ 5)(5" 5).
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!
© 2011 MARS University of Nottingham
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Rational and Irrational Numbers 1
Teacher Guide
Alpha Version August 2011
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Suggested lesson outline
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Introduction to Classifying Rational and Irrational Numbers (10 minutes)
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Give each student a mini-whiteboard, a pen and an eraser. Use these to maximize participation in the
introductory discussion.
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Explain the structure of the lesson to students.
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Recall your work on irrational and rational numbers [last lesson].
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You’ll have a chance to review that work later today.
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Today’s lesson is to help you improve your solutions.
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Explain to students that this lesson they will make a poster classifying rational and irrational numbers. Display
the projectable resource Classifying Rational and Irrational Numbers on the board.
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I’m going to give you some headings, and a large sheet of paper.
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You’re going to use the headings to make this classification poster.
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Then you’re going to get some cards with numbers on them. You have to decide whether the number is
rational or irrational, and where it fits on your poster.
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You will classify these examples of rational and irrational numbers.
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Check students’ understanding of the terminology used for decimal numbers:
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On your whiteboard, write a number with a terminating decimal.
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Can you show a number with a non-terminating decimal on your whiteboard?
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Show me a number with a repeating decimal.
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Show me the first six digits of a non-repeating decimal.
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Write 0.123 on a large post-it note. Model the classification activity using the number 0.123 .
0.123 . Remind me what the little bar over the digits means. [It is a repeating decimal that begins
0.123123123... ; the digits continue in a repeating pattern; it does not terminate.]
In which row of the table does 0.123 go? Why? [Row 2, because the decimal does not terminate but
does repeat.]
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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.]
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Show me on your whiteboard: is
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0.123
rational or irrational? [Rational.]
Students may offer different opinions on the rationality of 0.123 . If there is dispute, accept students’ answers for
either classification at this stage in the lesson. Make it clear to students that the issue is unresolved, and will be
discussed again later in the lesson.
Explain to students how you expect them to work together, collaboratively. Display the slide Instructions for
Working Together.
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Take turns to choose a number card.
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When it is your turn to choose, decide if there is a unique place for your number card on the poster.
Give reasons for your decision.
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If the others in your group agree with you, they should repeat your reasons in their own words.
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If they disagree, they should explain why. Then figure out the answer together.
© 2011 MARS University of Nottingham
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Rational and Irrational Numbers 1
Teacher Guide
Alpha Version August 2011
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When you have reached an agreement, write your reasoning on the number card.
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If the card belongs on the poster, put it in place. If not, put it to one side.
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All students in your group should be able to give reasons for every placement.
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Don’t glue things in place or draw in the lines on the poster yet – you may change your mind later.
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Remember. There are some cards that could go in more than one category. Write your reasoning on
the card. Keep these cards separate. Don’t place them on the poster if they can go in more than one
place.
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Collaborative small-group work (20 minutes)
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Organize students into groups of two or three.
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For each group provide a cut up copy of the sheet Poster Headings, a set of cards cut from the sheets Rational
and Irrational Numbers, a sheet of paper for making a poster, and a sheet of scrap paper. Do not distribute the
glue sticks yet.
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Students take turns to place cards and collaborate on justifying these placements. Once they have agreed on a
placement, the justification is written on the card and either placed on the poster or to one side.
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During small group work, you have two tasks, to find out about students’ work, and to support their thinking.
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Find out about students’ work
Listen carefully to students’ conversations. Note especially difficulties that emerge for more than one group. Do
students assume that all fractions represent rational numbers? Do students manipulate the expressions involving
radicals to show that the number represented is rational/irrational? Or do they evaluate the expressions on a
calculator? Figure out which students can convert a non-terminating repeating decimal to a fraction. Listen for
the kind of reasoning students give in support of their classifications. Do they use definitions? Do they reason
using analogies with other examples? You can use what you hear to choose students to call on in the plenary,
and in particular, identify two or three cards as a focus for that discussion.
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Support student thinking
If you hear students providing incorrect classifications or justifications, try not to resolve the issues for them
directly. Instead, ask questions that help them to identify their errors and to redirect their thinking. You may
want to use some of the questions and prompts from the Common Issues table. If students are relying on a
calculator to place the cards, encourage them to explain the answers displayed on it.
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If a student struggles to get started, encourage them to ask a specific question about the task. Articulating the
problem in this way can sometimes offer a direction to pursue that was previously overlooked. However, if the
student needs their question answered, ask another member of the group for a response.
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There is a Hint Sheet for four of the numbers.
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You might refer students from one group to those in another who have dealt with an issue well. If several groups
of students are finding the same issue difficult, you might write a suitable question on the board, or organize a
brief whole-class discussion focusing on that aspect of mathematics.
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Are all fractions rational? Show me a fraction that is rational/that is irrational.
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What does a calculator display 0.7777777778/0.1457878342 tell you about the number?
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Prompt students to write justifications next to the cards. If you hear one student providing a justification, prompt
the other member(s) of the group to either challenge or rephrase what they heard.
If any groups of students finish early, ask them to use the blank card to try to make up an example for a cell with
no card.
A couple of minutes before the end of the activity, ask students in each group to copy numbers from the cards
they have chosen to keep off the poster onto a sheet of scrap paper.
© 2011 MARS University of Nottingham
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Rational and Irrational Numbers 1
Teacher Guide
Alpha Version August 2011
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Comparing solutions (10 minutes)
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Ask one student from each group to swap with a student in another group, taking with them the sheet of scrap
paper on which they have written their group’s non-classified numbers.
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In the new groups, students compare the numbers they have not classified, to see if there are any differences.
Ask students to share their reasons for the numbers they have not classified.
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Gluing posters (5 minutes)
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Distribute glue sticks, and ask students to return to their original small groups. Ask students to discuss with their
group any changes they might want to make. Once students are satisfied with their answers, they can glue the
cards in place. Remind students to leave off the poster any numbers that they think can go in more than one
place.
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Whilst students work on this, you have time to choose numbers from the task that your students have
experienced difficulties with, to use as a focus for the whole class discussion. Write these numbers in marker pen
on large post-it notes.
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Whole-class discussion (15 minutes)
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The intention is that this discussion focuses on the justification of one or two examples that students found
difficult, rather than checking students all have all solutions correct.
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Check that each student has a mini-whiteboard, a pen and an eraser. Display the slide Classifying Rational and
Irrational Numbers Poster. Write the numbers you have chosen as a focus on large post-it notes. You will then
be able to move these numbers from one position to another on the projected classification table until students
have reached agreement about proper placement.
Using one of the numbers you have written on a sticky note, ask students to tell you on their whiteboards where
to place it on the classification table. Place your sticky note on the part of the table indicated by one student, and
ask that student to justify the placement of a card. Now ask another student to comment on their justification.
Call on students who gave different answers on their whiteboards.
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Bradley, where did you place this card? What was your reasoning?
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Do you agree, Lia? You do? Please explain Bradley’s answer in your own words.
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Kanya, can you tell us why you think differently?
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Now write another number that is a rational non-terminating repeating decimal.
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Move the cards from one cell to another until students reach agreement on the placement of the card. If there is
no disagreement, provoke some by placing a card in the wrong places and asking students for an explanation.
The discussion below is indicative. You may feel that other issues should be the focus of discussion for your
students.
Sample discussion
You might begin by checking the placement of the non-terminating repeating decimal 0.123 .
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Write on your mini-whiteboard. Is
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Is
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Alicia, where did you place 0.123 ? Why might you think it goes there?
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0.123
0.123
terminating or non-terminating?
rational or irrational?
As part of this discussion, you might ask students to show you on mini-whiteboards how to change from the
repeating non-terminating decimal representation to a fraction. If necessary, ask students to cooperate in
modeling use of the standard algorithm for changing a repeating decimal to a fraction.
Focus on the decimal representations 0.123 , 0.123 , 0.123... , 0.123 , and 0.123 (rounded correct to three decimal
places). Ask students to use their mini-whiteboards to involve them actively.
© 2011 MARS University of Nottingham
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Rational and Irrational Numbers 1
Teacher Guide
Alpha Version August 2011
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What is the difference between 0.123 and 0.123 ?
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Is 0.123 rational or irrational? How do you know?
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What about 0.123 / 0.123 ? How do you know?
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What is the difference between 0.123 and 0.123... ? What difference do those little dots [the ellipsis]
make?
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On your mini-whiteboard, show me a way that 0.123... could continue that shows it’s a rational
number.
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Show me a way that 0.123... could continue that would make it an irrational number.
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Push students to identify explicitly the difference between decimals, in which there is a repeating pattern of
digits, and non-terminating decimals in which there is a pattern, but no repeats.
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[Either refer to one of the student’s whiteboards, or write 0.123456789101112131415... on the board.]
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What would the next six digits be, Jenny? How do you know?
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Tell me about the pattern in the digits of this decimal number.
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Is this a repeating decimal?
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Do the same digits repeat in the same order again and again?
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Consider the other ambiguous representations:
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What about if the number is rounded to 0.123 ? Is this a rational or an irrational number?
Show me a number that could be rounded to 0.123 that is a rational number.
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Show me a number that could be rounded to 0.123 that is an irrational number.
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Did you find any other representations that might be rational, or might be irrational, depending on
how the number continued? [The calculator display looks like the first digits of pi, but could be a
rational number.]
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Explain how the calculator display number might continue.
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Show me a rational version of the calculator display number.
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If there is time, consider 0.9 and how, as this can be written as 1, the distinction between terminating and nonterminating decimal representations becomes blurred.
Finally, discuss the empty cells on the classification table (irrational terminating decimal, irrational nonterminating repeating
decimal, rational non-terminating non-repeating decimal).
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Which numbers go in this cell?
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Why is it not possible to place numbers in this cell? [No terminating decimals/non-terminating
repeating decimals are irrational numbers, and no non-terminating non-repeating decimals are
rational numbers.]
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Next Lesson: Improving individual solutions to the assessment task (10 minutes)
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Distribute students’ solutions to the assessment task Is it Rational? along with a fresh copy of the task sheet.
Invite students to revise their work on the assessment task.
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I would like you to read through your original solutions, and think about what you have learned this
lesson.
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Produce a new solution, using what you have learned.
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If you have not added questions to individual pieces of work, then write your list of questions on the board.
Students should select from this list only the questions they think are appropriate to their own work.
© 2011 MARS University of Nottingham
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Rational and Irrational Numbers 1
Teacher Guide
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Solutions
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Assessment task: Is it Rational?
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1.
5
Alpha June 2011
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 0.
5
7
!
!
5
7
0.575
is rational, since it can be written as the ratio of two integers
a, b, with b " 0.
!
! !
0.575 is rational.
!
1
7
5
575
!
Using decimal place value, 0.5 = , 0.07 =
, 0.005 =
, so
.
0.575 =
2
100
1000
1000
!
5 is irrational.
5
!
35 "5
,
.
7 "1
Students might argue
! that if!a number s!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
! 5.
!
5+ 7 is
5+ 7
irrational.
is!irrational from (d). The sum of a rational and irrational number is always irrational.
!
10
2
!
10
is irrational.
2
Students may think it is rational because it is represented as a fraction.
!
!
5.75.... From this trunctated decimal representation, it is not possible to decide whether
the number represented by 5.75… is rational or irrational.
5.75....
!
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. 5.757 and 5.7576777879710711...
!
The product of these two irrational factors is rational.
(5+ 5)(5" 5)
Use a!calculator !
will allow the student to establish that the product is rational, but not
why. (5 + 5)(5 " 5) = 25 " 5 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) = 35 " 7 5 + 5 5 " 5 = 30 " 2 5 . Since
!
!
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5 is irrational, 2 5 is irrational, and
30 " 2 5 is irrational. Use of a calculator will not help to show the irrationality of
30 " 2 5 .
!
© 2011 MARS !
University of Nottingham
!
!
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Rational and Irrational Numbers 1
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Teacher Guide
Alpha June 2011
2.
Agree or
disagree?
Student
Statement
Arlo:
0.57 is an irrational number.
Hao:
No, Arlo, it is rational, because 0.57
can be written as a fraction.
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.
Agree
0.57 is rational since it can be written as a
fraction. Disagreeing with Hao, the
student may believe that 0.57 = 57 , as
100
Eiji does, which is a common error.
Maybe Hao’s correct, you know.
Eiji:
Because
0.57 =
Agree
57
100 .
Eiji is correct that Hao’s answer is correct;
0.57 is rational. He is incorrect to say that
0.57 =
57
, the common error identified
100
immediately above.
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 = 0.57575757... 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.
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2.
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!
!
b. 0.57 is rational.
57
Let x = 0.57 . Then 100x = 57.57 . So 99x = 57 . Thus x = .
99
!
© 2011 MARS University of Nottingham
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!
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Rational and Irrational Numbers 1
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Teacher Guide
Alpha June 2011
Collaborative small-group work
Classifying Rational and Irrational Numbers
Rational numbers
Irrational numbers
7
=0.875 and is a terminating decimal.
8
This cell is empty: there are no
terminating decimal representations of
irrational numbers.
7
is a fraction and so rational.
8
!
0.123 =
!
!
Terminating decimals
!
123
and so rational.
1000
It is a terminating decimal in its given form.
2
(8 + 2 )(8 " 2 ) = 64 " 8 2 + 8 2 " 2 = 64 " 2 = 62.
62 is an integer, and so rational, with a terminating decimal
representation.
!
8
2
=
2 2
2
=2
2 is an integer, and so rational, with a terminating decimal
representation.
!
2 " 8 = 2 "2 2 = 2"2 = 4
4 is an integer, and so rational, with a terminating decimal
representation.
!
255
256
© 2011 MARS University of Nottingham
11
Rational and Irrational Numbers 1
Teacher Guide
Rational numbers
Irrational numbers
2
= 0.6
3
This cell is empty: there are no non-terminating
repeating decimal representations of irrational
numbers.
Students might already know this decimal
conversion, or perform division to calculate it.
!
Alpha June 2011
22
= 3.142897
7
Students might use division to calculate this
commonly used approximation of π. Using a
calculator, they would be unlikely to find all the
repeating digits, and so would have no reason
to suppose it is a repeating decimal.
!
!
1000x = 123.123
1000x " x = 123
999x = 123
123
x = 0.123 =
999
Non-­‐terminating repeating decimal x = 0.123
0.123 = 0.1 " 0.023
x = 0.023
10x = 0.23
1000x = 23.23
1000x " 10x = 23.23 " 0.23
990x = 23
23
x =
990
1
23
0.123 =
+
10
990
99
23
=
+
990
990
122
61
=
=
990
495
257
258
!
© 2011 MARS University of Nottingham
12
Rational and Irrational Numbers 1
Teacher Guide
Alpha June 2011
Irrational numbers
This cell is empty: no rational number has a
non-terminating non-repeating decimal
representation.
" is irrational, with a non-terminating,
non-repeating decimal representation. We
expect students will have been told of the
irrationality of " , and do not expect this
claim to be justified.
Rational numbers
Non-­‐terminating non-­‐repeating decimal !
3
4
!
is !
irrational since 3 is not a perfect
square, and only integers that are perfect
squares have rational roots. An irrational
divided by a rational is irrational.
8 is irrational since 8 is not a perfect
square, and only integers that are perfect
squares have rational roots.
!
!
!
2 + 8 is the sum of two irrational
numbers. It is irrational, since
2 + 8 = 2 + 2 2 = 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
4 2 "2 8 .
259
260
There is not enough information to assign the following!number cards to a unique category:
261
0.123 ....
262
This is a non-terminating decimal, but the ellipsis does not determine whether it is repeating or non-repeating.
263
It could turn out to be a rational number, such as 0.1234 .
264
265
It could turn out to be an irrational number, such as 0.12301001000100001… (a patterned tail of zeros and ones,
with one extra zero between ones each time.)
266
0.123 (rounded correct to three decimal places) and the calculator display: 3.14159265
267
270
The first number 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.
271
0.9 is a separate case.
272
273
It is clearly a rational number, but its categorization as a non-terminating repeating decimal rather than as a
terminating decimal is dubious.
274
Students might argue that 0.1 = 1 , 0.2 = 2 , … , 0.9 = 9 = 1 .
275
Alternatively, they could show that if x = 0.9 , 10x = 9.9 so 9x = 9 and x = 9 = 1 .
276
A similar argument could be made for any of the numbers with terminating decimal expansions.
268
269
9
9
9
9
© 2011 MARS University of Nottingham
13
Rational and Irrational Numbers 1
Student Materials
Alpha Version Aug 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
!
!
!
!
5
5+ 7
10
2
5.75....
!
!
!
!
(5+ 5)(5" 5)
(7 + 5)(5" 5)
© 2011 MARS University of Nottingham UK
S-1
Rational and Irrational Numbers 1
Student Materials
Alpha Version Aug 2011
2. Arlo, Hao, Eiji, Korbin, and Hank were discussing 0.57 .
This is the script of their conversation.
Agree or disagree?
!
Student
Statement
Arlo:
0.57 is an irrational number.
!
Hao:
No, Arlo, it is rational, because 0.57 can be written
as a fraction.
!
Eiji:
Korbin:
Hank:
Maybe Hao’s correct, you know. Because
57
.
0.57 =
100
!
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?
!
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.
!
© 2011 MARS University of Nottingham UK
S-2
Rational numbers Non-­‐terminating repeating decimal
Student Materials
© 2011 MARS University of Nottingham UK
Irrational numbers
Terminating decimal Non-­‐terminating non-­‐repeating decimal Classifying Rational and Irrational Numbers Rational and Irrational Numbers 1
Alpha Version Aug 2011
Poster Headings
S-3
Rational and Irrational Numbers 1
Student Materials
Alpha Version Aug 2011
Rational and Irrational numbers (1)
7
8
_
0.123...
0.9
!
!
2
3
22
7
(8 + 2)(8 " 2)
!
!
!
"
8
8
2
!
!
!
© 2011 MARS University of Nottingham UK
S-4
Rational and Irrational Numbers 1
Student Materials
Alpha Version Aug 2011
Rational and Irrational numbers (2)
2 "
!
8
2 +
8
3
4
!
!
0.123
0.123
Calculator display:
rounded correct to
3 decimal places
!
!
0.123
0.123
Try ....
!
© 2011 MARS University of Nottingham UK
S-5
Rational and Irrational Numbers 1
Student Materials
Alpha Version Aug 2011
Hint Sheet
0.123
x = 0.123
What is 1000x?
What do you have to add to x to get the same number as 1000x?
!
!
3
4
Are all fractions rational?
Is 0.5π rational?
Can you write 0.5π as a fraction?
!
Calculator display:
Does this display represent π?
How do you know for sure?
0.9
1
?
9
2
What is !
?
9
What is
© 2011 MARS University of Nottingham UK
S-6
Classifying Rational and Irrational Numbers
Rational Numbers
Irrational Numbers
Terminating
decimal
Non-terminating
repeating decimal
Non-terminating
non-repeating
decimal
Alpha 1
© 2011 MARS, University of Nottingham
Projector resources:
1
Instructions for Working Together
1.
Take it in turns to choose a number card.
2.
When it is your turn:
–  Decide whether there is a unique cell for your number card on the
poster.
–  Give reasons for your decision.
3.
If the others in your group agree with you, they should repeat your reasons
in their own words.
4.
If they disagree, they should explain why. Then figure out the answer
together.
5.
When you have reached an agreement:
–  Write your reasoning on the number card.
–  If the card belongs on the poster, put it in place.
–  If not, put it to one side.
Alpha 1
© 2011 MARS, University of Nottingham
Projector resources:
2