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1. Lesson Plan
Title:
Rational and Irrational Numbers – Are You for Real?
2. Subjects:
Mathematics
3. Grade:
Rising 8th
4. CSS:
7NS1.4
Differentiate between rational and
irrational numbers.
7NS1.5
Know that every rational number is either
a terminating or repeating decimal and be able to
convert terminating decimals into reduced fractions.
5. Keywords:
Real Number, Natural Number, Whole Number,
Integer, Rational Number, Irrational Number
6. Required
Materials:
Students:
Each student will need this set of shapes made from
blank paper:
 1 large, half-page sized rectangle
 2 smaller rectangles that fit inside the large
one
 1 large oval that fits inside a smaller rectangle
 1 medium-sized oval
 1 small oval
Additionally:
 Glue
 Rational or Irrational Worksheet
 Half sheets of paper for quick check
 Math Notebook
 Pencil
 Eraser
7. Lesson Goal: The student will identify the subsets of real numbers
and ways to express them. They will identify rational
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numbers as fractions or decimals, and convert
terminating decimals into reduced fractions.
8. Time:
45 minutes
9. Vocabulary:
Real numbers: The set of all numbers except
imaginary and complex numbers.
Natural numbers: The set of positive numbers,
beginning with 1, normally used for counting.
Whole numbers: The set of natural numbers plus 0.
Integers: The set of all positive and negative numbers
and zero.
Rational numbers: The set of real numbers that are
ratios of integers, including positive and negative
fractions, decimals, improper fractions and integers.
Irrational numbers: Ratios other than fractions,
terminating decimals or repeating decimals. Irrational
numbers include non-terminating decimals and
undefined ratios.
10. ELL
Differentiation
:
See the teacher guidebook for guidelines on ELL
Differentiation.
11. Lesson
Background
for Teacher:
Formulas: Computation will include converting
terminating decimals into reduced fractions.
Data: Rational numbers are real numbers that express
ratios. Fractions and decimals that either terminate or
repeat are rational numbers. Examples of rational
numbers are ½, 2.5, and 1.333…
Irrational numbers are decimals that do not terminate
or repeat. Examples of irrational numbers are √6 and
π. These are non-repeating and non-terminating
decimals.
Relevancy: This lesson helps students understand the
subsets of real numbers and ways to express them.
Students will construct a graphic organizer to illustrate
the sets and subsets of number systems used in
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algebra.
Previous Learning: Students should be able to
recognize decimals and fractions.
12. Math
Review for
Teacher:
Real numbers may be divided into 2 categories:
rational and irrational numbers.
Rational numbers are numbers that can be expressed
as a ratio of two integers, otherwise called a fraction.
Simple, compound, proper, and improper fractions, as
well as mixed numbers are all rational numbers.
Decimals are also rational numbers. Since a fraction
is another way to express division, (numerator divided
by the denominator), this division can be carried out
resulting in a decimal fraction. Only terminating
decimals, such as .5, and repeating decimals, such as
2.41414… are rational.
Integers are rational numbers, because all integers can
be expressed as quotients of themselves divided by 1.
Irrational numbers are numbers that cannot be
expressed as ratios. Numbers that are not fractions,
are non-terminating decimals, or are non-repeating
decimals are described as irrational. Pi is a wellknown irrational number because it continues to
divide infinitely.
Subsets of rational numbers are integers, which are
positive, negative, and 0; whole numbers, which are
all positive numbers and 0; and natural numbers,
which are positive counting numbers greater than or
equal to 1.
.
Sample Problem:
What number system(s) do the following numbers
belong to? Tell why.
(-9)
Answers to Sample Problem:
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(-9) real, rational, integer
real, rational
5.25
real, irrational
0
5.25 real, rational
11
0 real, rational, integer, whole,
natural
Task Analysis (TA):
11 real, rational, integer, whole,
Number systems may be thought of as umbrellas for
each sub-category. They may be listed in outline form natural
with each subsystem included in the system above it.
A perfect square is rational
because it results in an integer.
I. Real Numbers
II. Imaginary Numbers
The root of a non-perfect square is
A. Rational Numbers
B. Irrational Numbers
irrational because it does not
1. Integers
terminate or repeat.
a. Whole Numbers
1) Natural Numbers
Parallel Problem:
Think of a number that fits each number subsystem.
Is there a number that fits all subsystems? Why?
13. Linked w/
Lesson Plan:
N/A
14. Classroom
Organization:
See the teacher guidebook for guidelines on
Classroom Organization
15.
Anticipatory
Set: A model of
what the
Teacher may
say!
Students do the following “Think-Ink-Pair-Share”
activity:
We use a lot of kinds of numbers in math. Think of all
different kinds of numbers you can and write them
down.
[Allow students 1 to 2 minutes to write.]
Relevance:
Students work with various types of rational numbers
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and learn to convert between terminating decimals and
fractions.
The purpose of this lesson is to construct a graphic
organizer to illustrate the sets and subsets of number
systems used in algebra, then target rational numbers.
They will recognize rational numbers as fractions and
decimals, and convert between decimals and fractions.
16. Math
Notebook
Setup:
See the teacher guidebook for guidelines on Math
Notebook Setup
17. Script for
Teacher: A
model of what
the Teacher
may say!
Let’s share some of the numbers you thought of. Get
together in your groups of four and show each other
your lists. See if others have numbers that are like
yours in some way. Your task is to make a list that
names different kinds of numbers.
Students work for a few minutes, then share as a
whole class and put the list with examples on the
board.
Students come up with several types of fractions,
decimals, negative numbers, positive numbers,
exponents, square roots, etc. Help them organize the
lists into the categories of Real, Natural, Whole,
Integers, Rational Numbers, and Irrational Numbers.
The conversation will go something like this. If
students suggest categories in a different order, go
with their suggestions.
I see we have a lot of positive numbers that are larger
than 0. Is there a name for this kind of number?
[Natural or counting numbers.]
Put any numbers from your list in a box on your
paper. Label it “Natural”.
We can add 0 to the list of positive numbers and make
a new category. Does anyone know what this
category is called?
[Whole numbers.]
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If you have any whole numbers, label them. Do any of
your numbers have two labels? Why?
[The only difference between whole and natural
numbers is the inclusion of 0 in the whole numbers.]
Now I want to include negative numbers with 0 and
the positive numbers. I need a new category name.
[Integers.]
Label any of your own numbers that fit the “integers”
category.
You have done a lot of work with fractions and
decimals in the last couple of days. What number
systems do these belong to?
[Rational numbers and irrational numbers.]
What are some examples of irrational numbers? Find
any you had on your list.
[Irrational numbers cannot be simplified.]
Now label your rational numbers.
What do we know about real numbers?
If a real number can have this characteristic, lift your
pencil. If not, put your pencil down.
Are real numbers positive?
[Yes.]
Negative?
[Yes.]
Can real numbers be whole numbers?
[Yes.]
Negative numbers?
[Yes.]
Can real numbers be parts of numbers?
[Yes.]
Can real numbers be rational?
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[Yes.]
Fractions?
[Yes.]
Decimals?
[Yes.]
So, everything on our list is a subset of the real
numbers. Draw a huge circle around the list on the
board and label it “Real”.
Today we’re going to organize the real numbers into
various number systems to see where rational
numbers fit in. Then, we will convert between two
types of rational numbers.
Distribute the Real Number System Worksheet to
students. Students will need to follow along with the
presentation to complete their graphic organizer.
What number system includes all the types of numbers
in our list?
[Real numbers.]
There are types of numbers that are not real. These
are called imaginary or complex numbers, but we will
not use any imaginary numbers this year. So, real
numbers are all the numbers we work with in math.
First students should glue the title, “Real Number
System”, to the large rectangle on their worksheet.
Is there a number system that includes almost all the
rest of the numbers? Tell your partner which one you
think it is and why.
[Rational numbers includes all the rest of the number
systems except for irrational numbers.]
What number system is left?
[Irrational.]
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Let’s put it into the graphic organizer, as well. Use
the rectangles and be sure to label each one.
Next students will glue the two smaller rectangles to
the inside of the real number system and label them as
rational and irrational (make sure they put the labels at
the top of each rectangle).
Now, please write one number that fits into each
category in each corner, for a total of 4.
[Rational numbers could include:
fractions like 1/2
terminating decimals like 0.25
integers like -1
repeating decimals like 3.33…
Irrational numbers could include:
Pi (3.1416)
non-repeating, non-terminating decimals like
5.62301…
square root of a non-square number like √5]
We have removed the fractions and decimals from the
number systems. What systems are left? You have 3
sizes of ovals left. The largest oval should be used for
the system that includes the rest. What is it?
Students should glue the largest oval into the rational
rectangle and label it, “Integers”.
Ask students to write integers in the far left and far
right of the oval.
Integers include positive and negative whole numbers
and 0. What system encompasses two subsystems?
Students will glue the middle oval inside the larger
one and label it “Whole Numbers”.
Ask students to write whole numbers to the far right
and far left of the oval.
Whole numbers include 0 and positive numbers only.
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What subsystem is included in all the rest?
Students will glue the last oval inside the “Whole
Numbers” oval and label it “Natural Numbers”.
Ask students to write one example of a natural number
inside the oval.
Natural numbers are counting numbers: 1, 2, 3, etc.
Look at the graphic organizer you just designed. How
will you remember the systems of numbers?
Allow students to work together to come up with
mnemonic devices or other pictures or graphics. Have
students share their ideas with the class.
Ideas might include:
Natural numbers are what you count with: they
just seem natural.
Whole has an “o” so we include zero with the
positive numbers.
Integers include positive and negative whole
numbers as well as 0.
Is it possible for a number to belong to more than one
system?
[Yes.]
With your partner, tell all the systems each of these
numbers belongs to.
What subsystems does the number -7 belong to?
[Real, Rational, Integers.]
What subsystems does the number 5.45 belong to?
[Real, Rational, Integers.]
What subsystems does the number 6 belong to?
[Real, Rational, Integers, Whole, Natural]
Now, we will practice identifying all subsets that a
number belongs to. Use your graphic organizer.
Begin by determining whether the number is rational
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or irrational, and then look at any other possibilities.
Have students refer to their graphic organizer to help
in this process.
Independent Practice:
Ask students to complete the problems on their Real
Number System worksheet (attached). Students will
complete this individually. They will circle all the
subsets that each number belongs to.
After students have had time to finish, have them
check their answers in groups of four. Present any
questions or disagreements to the whole class for
resolution.
Answers to Independent Practice
Worksheet
1) 3.45 real, rational, integer
2) -5 real, rational, integer,
whole, natural
3) 10 real, rational, integer,
whole, natural
4)
real, rational
We did not have very many irrational numbers in our
worksheet. How can we devise a test to see if a
number is irrational or not? We know a few things
that irrational numbers are not: irrational numbers
are not rational, and irrational numbers cannot be
simplified.
5)
real, irrational
6)
real, rational
7)
4 real, rational, integer
Let’s look at some numbers to see if they are rational
or not.
8) 0 real, rational, integer,
whole
½ We already know that fractions are rational
because fractions are ratios. The definition of
rational number means it is a fraction.
9)
real, irrational
10) 2.35 real, rational
Can you turn these numbers into fractions? Work
with your partner to convert these into fractions.
0.5
0.89
1.333…
4.456295…
List these numbers on the board. When they have
finished, have students share their fractions and write
them beside the decimal.
TASK ANALYSIS (TA):
1. Convert any mixed
numbers to improper
fractions.
2. Multiply the numerators
(top numbers) together
and the denominators
(bottom numbers)
together.
3. Simplify to get the fraction
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0.5 =
One decimal place means tenths. This is
in its lowest terms.
rational. It makes a fraction. Can we simplify it?
[Yes, one-half.]
0.89 =
Two decimal places means hundredths.
This is rational. It makes a fraction. Can we simplify
it?
[No.]
1.33… = 1 A repeating decimal can stand for a
fraction because we know that it will be the same, no
matter how long we divide it. Divide 1. 33 into 100.
What do you get?
[1.33333…]
Task Analysis:
Test the decimals to see if they
are rational.
All rational numbers can be
converted to fractions if they:
Are terminating or repeating.
Can convert to fractions.
Can be simplified.
This repeating decimal can be simplified to .
This is rational. It stands for a fraction.
4.456295… = 4 What do we do with a decimal that
doesn’t end, but doesn’t repeat? We don’t even know
the rest of the digits in the decimal place.
[It is irrational. It does not make a fraction.]
So, a decimal that goes on forever in no order is an
irrational number.
What did we do to test the decimals to see if they were
rational?
[Terminating or repeating
Convert to fractions
Simplify]
So, we have certain conditions for rational numbers.
All rational numbers can be converted to fractions.
Try converting another decimal to a fraction.
0.2
What do we use for the denominator?
[Place value.]
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What decimal place is occupied by the 2?
[Tenths.]
So 0.2 is two tenths. Go ahead and write that as a
fraction.
[ ]
Individual Practice: Students work these problems
individually using the steps in the Task Analysis (TA).
After the teacher writes them on the board, students
need to write every problem in their own notebook.
Which of these are rational numbers? Prove it by
converting them to simplified fractions.
a. 0.999…
b. 3.14159…
c. 5.5
d. 12
e.
Whole group discussion: Have students come share
all the reasons why each number is rational or
irrational.
18. Learning
Reflection for
Student:
Can a number belong to the integer subset and the
irrational subset? Why?
19. Plan for at
home Ind.
Practice:
Homework card: Tell a family member about
irrational numbers and give them 3 examples.
20. Assessment
Based on Goal:
Teacher will observe students as they work through
the Individual Practice problems. Check for
understanding of rational and irrational numbers.
21. Additional
Notes:
Multiplication and division are the basis for working
with fractions. Students who still have difficulty with
basic facts need to be given extra outside
opportunities to learn them.
Answer Key for Individual Practice
a. 0.999 is rational, a
repeating decimal, it
simplifies to .
b. 3.14159 is irrational,
does not repeat or
terminate.
c.
5.5 is rational, a
terminating decimal, it
simplifies to .
d. 12 is rational, an integer,
can be expressed as
e.
is irrational, it does not
divide or terminate. Roots
of non-perfect squares are
irrational.
Plan 5 – Rational and Irrational Numbers – Are You for Real?
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Name ____________________________________________
Date _______________________________
Independent Practice
The Real Number System
Circle all of the subsets to which each given number belongs.
1) 3.45
real
rational
irrational
integer
whole
natural
2) (-5)
real
rational
irrational
integer
whole
natural
3) 10
real
rational
irrational
integer
whole
natural
4) √121
real
rational
irrational
integer
whole
natural
5) π
real
rational
irrational
integer
whole
natural
6) 2/3
real
rational
irrational
integer
whole
natural
7) (-4)
real
rational
irrational
integer
whole
natural
8) 0
real
rational
irrational
integer
whole
natural
9) √40
real
rational
irrational
integer
whole
natural
10) 2.35…
real
rational
irrational
integer
whole
natural
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