CH. 5 Integers
CH. 6 Fractions and Rational
Numbers
5.1 Representation of Integers
Natural numbers ? Whole numbers ?
Integers
l Integers are used to describe:
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1.
2.
3.
4.
5.
debits & credits
profits & losses
changes in prices
temperature
Golf under par (-18)
5.1 Representation of Integers
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Representation of
Integers
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Example 1
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Example 2
Red (-)
Black (+)
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5.1 Representation of Integers
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DEF The Integers page 284
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The positive integers are the natural numbers.
The negative integers are the numbers -1, -2, -3,
…
, where -r is defined by the equality
s + (-s) = (-s) + s = 0
The integer 0 is neither positive nor negative and
has the property 0 + n = n + 0 = n for every
integer n. The integers consist of the positive
integers, the negative integers, and zero.
5.1 Representation of Integers
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Mail-Time
Representation of
Integers
“
Postman Delivers”
Checks +
Brings addition
Bills Takeaway subtraction
Are we better Off
+ By how much?
Or worse off?
- by how much?
1.
2.
3.
4.
5.
Postman delivers 1
check of $300.
Postman delivers 1 bill
$25.
Postman brings 6 bills
for $3.
Postman takes away 1
bill for $3.
Postman takes away 3
bills for $4 each.
5.1 Representation of Integers
l THM. The Negative of the Negative of
an Integer page 289
For every integer n, -(-n) = n
l DEF Absolute Value of an Integer
Distance from zero, always positive
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5.2 Addition and Subtraction of
Integers
l Remember
a = n(A), b = n(B), and A ∩ B = ∅ , then
a + b was defined as n(A ∪ B).
l Ex. 1
l 9 + (-3) = 5
5.2 Addition and Subtraction of
Integers
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THM: Adding
Integers pg. 296
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Let m and n be
positive
integers so that
-m and -n are
negative. Then
the following
are true:
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(-m) + (-n) = -(m + n)
If m>n, then m + (-n) =
m–
n
If m<n, then m + (-n) =
-(n –
m)
n + (-n) = (-n) + n = 0
5.2 Addition and Subtraction of
Integers
THM: Properties of the Addition of Integers page
297
Let m, n, and r be integers. Then the following hold.
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Closure Property
Commutative Property
Associative Property
Additive identity
Additive inverse
m + n is an integer
m+n =n+m
m + (n + r) = (m + n) + r
0+m=m+0=m
(-m) + m = m + (-m) =0
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5.2 Addition and Subtraction of
Integers
l Draw a number-line diagram to illustrate
each of these calculations.
(-3) + 7
(-5) + (-2)
l 3 + (-6)
l 5 + (-5)
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5.2 Addition and Subtraction of
Integers
THM: The Law of Trichotomy page 331
If a and b are any two integers, then
precisely one of these three possibilities
must hold:
a<b
or
a = b or
a>b
Example:
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a.
15 ? 101
c. 3 ? -(-3)
b.
-8 ? -9
d. 0 ? -8
5.2 Addition and Subtraction of
Integers
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Subtraction using colored counters page 302
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(-3) –
7 = -10
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You have (-3) until you remove 7 black, then
you have (-10)
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5.2 Addition and Subtraction of
Integers
l5–
(-4) = 9
or 5 = (-4) + 9
5.2 Addition and Subtraction of
Integers
l THM: Closure Property for the
Subtraction of Integers page 306
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The set of integers is closed under
subtraction.
5.3 Multiplication and Division of
Integers
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Multiplication is repeated ADDITION
THM: THE RULE OF SIGNS page 317
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Let m and n be positive integers so that –
m and –
n are negative integers. Then the
following are true:
m ⋅−
( n) =−(mn)
(−m) ⋅n =−(mn)
(−m) ⋅−
( n) = mn
a ⋅0 =0 ⋅a = 0 for any integer a
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5.3 Multiplication and Division of
Integers
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THM: Multiplication Properties of Integers
page 319
Closure Property
Commutative
Property
5.3 Multiplication
and Division of Integers
Associative Property
Multiplicative Property of One
Multiplicative Property of Zero
GIVE AN EXAMPLE OF EACH OF THE ABOVE
PROPERTIES.
5.3 Multiplication and Division of
Integers
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Example: Describe a mail-time,
situation that illustrates -44.
a.
b.
c.
At mail-time, you are delivered a check for
$44.
At mail-time, you are delivered a bill for
$44.
At mail-time, you are delivred a check for
$44 and a bill for $44.
5.3 Multiplication and Division of
Integers
2.
Example: The letter carrier takes away
4 checks for $16 each.
a.
b.
c.
d.
4 +16
4 x 16
(-4) x 16
4 x (-16)
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5.3 Multiplication and Division of
Integers
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THM: Rule of Signs
for Division of
Integers page 322
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Let m and n be
positive integers so
that –
m and –
n are
negative integers
and suppose that n
divides m. Then
the following are
true:
m ÷(−n) =−(m ÷n)
( −m) ÷n =−(m ÷n)
( −m) ÷( −n) = m ÷n
6.1 The Basic Concepts of Fractions
and Rational Numbers
A. Pies to represent & check for rational
number comprehension. (activity)
1.
Size whole parts
2.
What is 1
2
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3.
2
?
3
Leads in to fraction multiplication.
How many
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of
1
2
in ?
8
4
leads in to fractions division.
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6.1 The Basic Concepts of Fractions
and Rational Numbers
4.
What is 2 + 1 , 1 + 1 ?
4 4 4 3
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5.
Leads into like and unlike fractions in addition.
What is
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1 1
− ?
3 4
Leads in to like and unlike fractions in
subtraction.
6.1 The Basic Concepts of Fractions
and Rational Numbers
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DEF. Fractions page 347
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First introduce in measurement problems
to express a quantity less than a whole
unit.
The Number Line Model page 349
Game “
Fraction Wars”
6.1 The Basic Concepts of Fractions
and Rational Numbes
C. Fraction Strips
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6.1 The Basic Concepts of Fractions
and Rational Numbers
l PROPERTY The Fundamental Law of
Fractions
l Let a be a fraction. Then
b
a an
=
b bn
, for any integer n ? 0.
1 1g3 3
=
=
2 2g3 6
l Example
6.1 The Basic Concepts of Fractions
and Rational Numbers
l THM The Cross-Product Property of
Equivalent Fractions
a
c
and
The fractions
are equivalent if
b
d
and only if, ad = bc.
Example
a)
−23 2231
=
47 −4559
6.1 The Basic Concepts of Fractions
and Rational Numbers
l DEF. Fractions in Simplest Form
l A fraction
a
is in simplest form if a and b
b
have no common divisor larger than 1
and b is positive.
l Method 1: Divide successively by
common factors.
EX.
240
360
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6.1 The Basic Concepts of Fractions
and Rational Numbers
l DEF. Fractions in Simplest Form
(cont’
d)
l Method 2: Divide a and b by GCD (a, b)
EX.
28
40
6.1 The Basic Concepts of Fractions
and Rational Numbers
l Method 3:
Divide by the common factors in the
prime factorization of a and b.
−45
l EX.
135
l Method 4: Use a fraction calculator.
EX.
117
−468
6.1 The Basic Concepts of Fractions
and Rational Numbers
l Common Denominators
l EX.
3 5 2
+ +
4 8 3
l EX.
15 12
+
34 51
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6.1 The Basic Concepts of Fractions
and Rational Numbers
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DEF. Rational Numbers (end or repeat)
A rational number is a number that can be
represented by a fraction a/b, where a and b are
integers, b ? 0. Two rational numbers are equal if,
and only if, they can be represented by equivalent
fractions.
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DEF. Order Relation on the Rational Numbers
Let two rational numbers be represented by the
a
c
and , with b and d positive.
fractions
d
a b
c,
Then b is less than
written a < c ,
d
b d
if, and only if ad < bc.
6.1 The Basic Concepts of Fractions
and Rational Numbers
Comparing
Rational Numbers
Example:
1. 3
2
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2.
4
5
15
29
6
11
3.
4.
2106
7047
−10
13
234
783
22
-29
6.2 The Arithmetic of Rational
Numbers
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DEF: Addition of Rational Numbers page 365
(LIKE FRACTIONS)
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a
c
Let two rational
and numbers be represented by
b
b
fractions
with a common denominator.
Then their sum of the rational numbers is given
by
a c a +b
+ =
b b
b
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6.2 The Arithmetic of Rational
Numbers
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Draw
1 2
+
4 4
6.2 The Arithmetic of Rational
Numbers
l Addition of UNLIKE FRACTIONS
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Draw
2 1
+
3 2
6.2 The Arithmetic of Rational
Numbers
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Example
1.
3 5  2
 + +
4 6  3
2.
3 −7
+
8 24
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6.2 The Arithmetic of Rational
Numbers
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Draw 1
3
8
what is it as an improper
fraction?
355
133
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Give a mixed number for
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Give a mixed number for −15
4
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Compute
3
2
2 +4
4
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6.2 The Arithmetic of Rational
Numbers
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Which of the following are Proper Fractions?
1 8 2 4 15
, , , ,
5 7 2 7 17
6.2 The Arithmetic of Rational
Numbers
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DEF: Subtraction of
Rational Numbers
page 371
a
c
and be rational numbers.
b
d
a c e
Then − = if, and only if,
b d f
a c e
= + .
b d f
Let
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6.2 The Arithmetic of Rational
Numbers
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Example
1.
4 2
−
5 3
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2.
1
3
8 −4
3
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6.2 The Arithmetic of Rational
Numbers
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DEF: Multiplication
of Rational Numbers
page 373
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Let a/b and c/d be
rational numbers.
Then their product
is given by:
a b ab
⋅ =
c d cd
6.2 The Arithmetic of Rational
Numbers
Examples (decimal answers are not
acceptable.)
1.
2.
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5 2
⋅
8 3
1 1
3 ⋅5
7 4
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6.2 The Arithmetic of Rational
Numbers
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THM: The Invert and Multiply Algorithm for
Division of Fractions and Rational Numbers.
Page 378 a
c a d
c
b
÷ = ⋅ , where ≠0.
d b c
d
DEF: Reciprocal of a Rational Number
-The reciprocal of a nonzero rational number
c d
is .
d c
6.2 The Arithmetic of Rational
Numbers
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Example
1.
2.
3 1
÷
4 8
1
1
4 ÷2
6
3
FRACTION WARS
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Purpose: Reinforce estimation and comparison of
fractions in a game format.
Play 10 rounds of each goal with your partner.
GOAL 1: Form a fraction by placing the card with the
smaller number in the numerator. Player with the
smaller fraction is the winner.
GOAL 2: Form a fraction by placing the card with the
larger number in the numerator. Player with the larger
fraction is the winner.
GOAL 3: Place the first card in the numerator and the
second in the denominator. Player with the fraction
whose value is closest to 2 is the winner.
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6.3 The Rational Number System
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Density page 391
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There are infinitely many rational numbers
between two rational numbers.
Find at least two fractions between:
5
3
and
6
7
12
10
and
14
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CONCEPT MAPS
l Complete the following in your
groups:
l Chapter 5.
5 Whole-Number Arithmetic
Concept Map
l Chapter 6.
6 Fraction Concept Map
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