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REAL RATIONAL
NUMBERS
(as opposed to fake numbers?)
and Properties
Part 1 (introduction)
STANDARD: AF 1.3 Apply algebraic
order of operations and the commutative,
associative, and distributive properties to
evaluate expressions: and justify each step in
the process.
Student Objective:
• Students will apply order of operations to
solve problems with rational numbers and
apply their properties, by performing the
correct operations, using math facts skills,
writing reflective summaries, and scoring
80% proficiency
Vocabulary
Set
Set Notation
Natural
numbers
Whole
Numbers
Integers
A collection of objects.
{ }
Counting numbers {1,2,3, …}
Natural numbers and 0.
{0,1,2,3, …}
Positive and negative natural
numbers and zero {… -2, -1, 0, 1, 2, 3, …}
A real number that can be expressed
Rational
as a ratio of integers (fraction)
Number
Any real number that is not rational.
Irrational
 2 , 
Number
Real Numbers All numbers associated with
the number line.
Essential Questions:
• How do you know if a number is a
rational number?
• What are the properties used to
evaluate rational numbers?
Two Kinds of Real Numbers
• Rational Numbers
• Irrational Numbers
Rational Numbers
• A rational number is
a real number that
can be written as a
ratio of two
integers.
• A rational number
written in decimal
form is terminating
or repeating.
EXAMPLES OF
RATIONAL NUMBERS
•16
•1/2
•3.56
•-8
•1.3333…
•-3/4
To write a fraction as a decimal, divide
the numerator by the denominator. You
can use long division.
numerator
denominator
denominator numerator
Additional Example 3A: Writing Fractions as
Decimals
Write the fraction as a decimal.
11
9
The fraction
1 .2
9 11 .0
–9
20
–1 8
2
The pattern repeats.
Writing Math
A repeating decimal can be written
with a bar over the digits_that
repeat. So 1.2222… = 1.2.
11
is equivalent to the decimal 1.2.
9
Additional Example 3B: Writing Fractions as
Decimals
Write the fraction as a decimal.
7
20
0.3 5 This is a terminating decimal.
20 7.0 0
–0
70
–6 0
1 00
–1 0 0
0 The remainder is 0.
The fraction
7
is equivalent to the decimal 0.35.
20
Irrational Numbers
• An irrational
• Square roots of
number is a
non-perfect
number that
“squares”
cannot be written
as a ratio of two
17
integers.
• Irrational numbers
written as
• Pi- īī
decimals are nonterminating and
non-repeating.
Irrational numbers can be written only as
decimals that do not terminate or repeat. They
cannot be written as the quotient of two
integers. If a whole number is not a perfect
square, then its square root is an irrational
number. For example, 2 is not a perfect square,
so 2 is irrational.
Caution!
A repeating decimal may not appear to
repeat on a calculator, because
calculators show a finite number of digits.
A fraction with a denominator of 0 is
undefined because you cannot divide by zero.
So it is not a number at all. So it is not a real
number, it is not rational or irrational.
Real Numbers
Rational numbers
Integers
Whole
numbers
Irrational numbers
Rational Numbers
Natural Numbers - Natural counting numbers.
1, 2, 3, 4 …
Whole Numbers - Natural counting numbers and zero.
0, 1, 2, 3 …
Integers - Whole numbers and their opposites.
… -3, -2, -1, 0, 1, 2, 3 …
Rational Numbers - Integers, fractions, and decimals.
Ex:
-0.76, -6/13, 0.08, 2/3
Rational Numbers on a Number Line
Integers
Whole Numbers
Natural Numbers
|
– 4
|
|
|
|
|
|
|
|
– 3
– 2
– 1
0
1
2
3
4
Negative
numbers
Positive
numbers
Zero is neither
negative nor positive
Making Connections
Biologists classify animals based on
shared characteristics. The horned
lizard is an animal, a reptile, a lizard,
and a gecko. Rational Numbers are
classified this way as well!
Animal
Reptile
Lizard
Gecko
Venn Diagram: Naturals, Wholes, Integers, Rational
Real Numbers
Rationals
6.7
5

9
0.8
Integers
11
Wholes
Naturals
1, 2, 3...
5
0
3
2
7
Make a Venn Diagram that displays the following
sets of numbers:
Reals, Rationals, Irrationals, Integers,
Wholes, and Naturals.
Reals
Rationals
2
3
-3
-2.65
Integers
-19
Wholes
1
0
6
Naturals
1, 2, 3...
4
Irrationals

2
Reminder
• Real numbers are
all the positive,
negative, fraction,
and decimal
numbers you
have heard of.
• They are also
called Rational
Numbers.
• IRRATIONAL
NUMBERS are
usually decimals
that do not
terminate or repeat.
They go on forever.
• Examples: π
2
3
Additional Example 2: Determining the
Classification of All Numbers
State if each number is rational, irrational,
or not a real number.
A.
21
irrational
B.
0
3
rational
0
=0
3
Additional Example 2: Determining the
Classification of All Numbers
State if each number is rational, irrational,
or not a real number.
C. 4
0
not a real number
Check It Out! Example 2
State if each number is rational, irrational,
or not a real number.
A.
23
23 is a whole number that
is not a perfect square.
irrational
B.
9
0
undefined, so not a real number
Rational Numbers on a Number Line
To graph a set of numbers means to draw, or plot, the
points named by those numbers on a number line.
The number that corresponds to a point on a number
line is called the coordinate of that point.
Identify Coordinates on a Number Line
Name the coordinates of the points graphed on the
number line.
The dots indicate each point on the graph.
Answer: The coordinates are {–9, –7, –6, –3}.
Identify Coordinates on a Number Line
Name the coordinates of the points graphed on the
number line.
The bold arrow on the graph indicates that the graph
continues infinitely in that direction.
Answer: The coordinates are {11, 12, 13, 14, …}.
Identify Coordinates on a Number Line
Name the coordinates of the points graphed on
each number line.
a.
Answer: {6, 9, 11, 12}
b.
Answer: {–0.5, 0, 0.5, 1, 1.5, …}
Graph Numbers on a Number Line
Graph
Answer:
.
Graph Numbers on a Number Line
Graph {–1.5, 0, 1.5, …}.
Answer:
Graph Numbers on a Number Line
Graph {integers less than –6 or greater than or equal
to 1}.
Answer:
Graph Numbers on a Number Line
Graph each set of numbers.
a. {–5, 2, 3, 5}
Answer:
b.
Answer:
Graph Numbers on a Number Line
c. {integers less than or equal to –2 or greater than 4}
Answer:
Properties
A property is something that is true for all
situations.
Four Properties
1. Distributive
2. Commutative
3. Associative
4. Identity properties of one and
zero
Distributive Property
A(B + C) = AB + BC
4(3 + 5) = 4x3 + 4x5
Commutative Property
of addition and multiplication
Order doesn’t matter
Ax B= B xA
A+B = B +A
Associative Property of
multiplication and Addition
Associative Property  (a · b) · c = a · (b · c)
Example: (6 · 4) · 3 = 6 · (4 · 3)
Associative Property  (a + b) + c = a + (b + c)
Example: (6 + 4) + 3 = 6 + (4 + 3)
Identity Properties
If you add 0 to any number, the number stays
the same.
A + 0 = A or 5 + 0 = 5
If you multiply any number times 1, the
number stays the same.
A x 1 = A or 5 x 1 = 5
Example 1: Identifying Properties of Addition and
Multiplication
Name the property that is illustrated in each
equation.
A. (–4)  9 = 9  (–4)
(–4)  9 = 9  (–4)
The order of the numbers changed.
Commutative Property of Multiplication
B.
The factors are grouped
differently.
Associative Property of Addition
Example 2: Using the Commutative and
Associate Properties
Simplify each expression. Justify each step.
29 + 37 + 1
29 + 37 + 1 = 29 + 1 + 37
Commutative Property
of Addition
= (29 + 1) + 37
Associative Property of
Addition
= 30 + 37
Add.
= 67
Exit Slip!
Name the property that is illustrated in each equation.
1. (–3 + 1) + 2 = –3 + (1 + 2)
2. 6

y

7=6
●
7
●
y
Associative Property of Add.
Commutative Property of Multiplication
Simplify the expression. Justify each step.
3.
22
Write each product using the Distributive Property.
Then simplify
4. 4(98) 392
5. 7(32) 224