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Sets and Whole Numbers
2.1 Sets and Operations on Sets
2.2 Sets, Counting, and the Whole
Numbers
2.3 Addition and Subtraction of Whole
Numbers
2.4 Multiplication and Division of Whole
Numbers
2.1
Sets and Operations on Sets
Slide 2-3
THREE WAYS TO DEFINE A SET
1.
Word Description:
The set of even numbers between 2 and
10 inclusive.
2.
Listing in Braces:
{2, 4, 6, 8, 10}
3.
Set Builder Notation:
{x | x = 2n, 1 ≤ n ≤ 5, n N}
Slide 2-5
Example 2.1 Describing Sets
Each set that follows is taken from the universe N of
the natural numbers and is described either in
words, by listing the set in braces, or with set-builder
notation. Provide the two remaining types of
description for each set.
a. The set of natural numbers greater than 12 and
less than 17.
{13, 14, 15, 16}; listing
n n  N and 12 < n < 17; set builder
Slide 2-4
Example 2.1 continued
b. {3, 6, 9, 12,…}
The set of all natural numbers that are
multiples of 3; word description.
x x  3n and n  N ; set builder
c. The set of the first 10 odd natural numbers.
{1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
x x  2n  1 and n  1, 2,...,10; set builder
Slide 2-5
VENN DIAGRAMS
Sets can be represented pictorially by Venn
diagrams, named for the English logician
John Venn.
• The universal set U is represented by a
rectangle.
• Any set A within the universe is
represented by a closed loop lying within
the rectangle.
U
A
Slide 2-6
DEFINITION: INTERSECTION OF SETS
The intersection of two sets A and B,
written A  B, is the set of elements
common to both A and B. That is,
A  B  {x | x  A and x  B}
Slide 2-7
DEFINITION: DISJOINT SETS
Two sets C and D are disjoint if C
and D have no elements in common.
That is, C and D are disjoint means
that C  D  .
Slide 2-8
DEFINITION: UNION OF SETS
The union of sets A and B , written
A  B , is the set of all elements that
are in A or B. That is,
A  B  {x | x  A or x  B}
Slide 2-9
2.2
Sets, Counting, and Whole
Numbers
Slide 2-10
TYPES OF NUMBERS



Nominal numbers
 A number can be an identification, or nominal
number, such as a ticket number.
Ordinal numbers
 Ordinal numbers communicate location in an
ordered sequence, such as first, second, third.
Cardinal numbers
 A cardinal number communicates the basic
notion of “how many,” such as four may tell us
how many tickets we have for an event.
Slide 2-11
Example 2.8 Showing the Order of
Whole Numbers
Use (a) sets, (b) tiles, (c) rods, and (d) the number
line to show that 4 < 7.
a.
Slide 2-12
Example 2.8 continued
Use (a) sets, (b) tiles, (c) rods, and (d) the number
line to show that 4 < 7.
b.
c. With rods, the order of whole numbers is
interpreted by comparing the
lengths of the rods.
Slide 2-13
Example 2.8 continued
Use (a) sets, (b) tiles, (c) rods, and (d) the number
line to show that 4 < 7.
d. On the number line, 4 < 7 because 4 is to the left
of 7.
Slide 2-14
2.3
Addition and Subtraction of
Whole Numbers
Slide 2-15
TWO CONCEPTUAL MODELS FOR
ADDITION OF WHOLE NUMBERS
• Number-Line (Measurement) Model
Slide 2-16
THE MEASUREMENT MODEL OF
WHOLE NUMBER ADDITION
On the number line, whole numbers
are geometrically interpreted as
distances.
3 +5
Slide 2-17
PROPERTIES OF
WHOLE NUMBER ADDITION
• CLOSURE PROPERTY
If a and b are any two whole numbers,
then a + b is a unique whole number.
• COMMUTATIVE PROPERTY
If a and b are any two whole numbers,
then a + b = b + a.
Slide 2-18
PROPERTIES OF
WHOLE NUMBER ADDITION
• ASSOCIATIVE PROPERTY
If a, b and c are any three whole
numbers, then
a  (b  c )  ( a  b)  c.
• ADDITIVE-IDENTITY PROPERTY
OF ZERO
If a is any whole number, then
a + 0 = 0 + a = a.
Slide 2-19
Example 2.11 Using Properties of
Whole-Number Addition
Which property justifies each of the following
statements?
(i)
8+3=3+8
(ii) (7 + 5) + 8 = 7 + (5 + 8)
(iii) A million plus a quintillion is not infinite.
(i) Commutative Property
(ii) Associative Property
(iii) The sum is a whole number by the closure
property and is therefore a finite value.
Slide 2-20
Example 2.12 Illustrating Properties
What property of whole-number addition are shown
on the following number lines?
a.
The commutative property: 4 + 2 = 2 + 4.
Slide 2-21
Example 2.12 continued
What property of whole-number addition are shown
on the following number lines?
b.
The associative property: (3 + 2) + 6 = 3 + (2 + 6).
Slide 2-22
Example 2.12 continued
What property of whole-number addition are shown
on the following number lines?
c.
The additive-identity property: 5 + 0 = 5.
Slide 2-23
SUBTRACTION OF WHOLE NUMBERS
Let a and b be whole numbers. The difference of a
and b, written a – b, is the unique whole number c
such that a = b + c.
That is, a – b = c if, there is a whole number c, such
that a = b + c.
Slide 2-24
TERMINOLOGY:
SUBTRACTION OF WHOLE NUMBERS
Slide 2-25
FOUR CONCEPTUAL MODELS FOR
SUBTRACTION OF WHOLE NUMBERS
• Take-Away Model
• Missing Addend Model
• Comparison Model
• Number-Line (Measurement) Model
Slide 2-26
TAKE-AWAY MODEL
Joel has 10 cookies. He gives 3 of
the cookies to his little sister. How
many cookies does Joel have left?
To answer, the student uses
subtraction.
10 – 3
Slide 2-27
COMPARISON MODEL
Roger has read 9 books this week.
His friend Jake has read 6 books this
week. How many more books has
Roger read than Jake?
To answer, the student uses
subtraction.
9–6
Slide 2-28
MISSING-ADDEND MODEL
Sarah has saved $15. She needs $22
to buy a blouse she noticed last week.
How much more money does she
need to buy the blouse?
To answer, the student uses
subtraction.
$22 – $15
(Note: $15 + ? = $22.)
Slide 2-29
NUMBER-LINE MODEL
Illustrate 8 – 5 on the number line.
Slide 2-30
2.4
Multiplication and Division of
Whole Numbers
Slide 2-31
FOUR CONCEPTUAL MODELS FOR
MULTIPLICATION OF WHOLE NUMBERS
• Array Model
• Rectangular Area Model
• Skip Count Model
• Multiplication Tree Model
Slide 2-32
ARRAY MODEL
Leah planted 3 rows of tomato plants
with 4 plants in each row. How many
tomato plants does she have planted?
3
ROWS
T
T
T
T
T
T
T
T
T
T
T
T
3 x 4 array
3  4  12
4 PLANTS PER ROW
Slide 2-33
RECTANGULAR AREA MODEL
Mr. Hu bought an 8 ft by 6 ft rug for
his house. How many square feet are
covered by this rug?
8 ft x 6 ft
= 48 sq ft
6
8  6  48
8
Slide 2-34
SKIP COUNT MODEL
“Skip” by the number 2 six times:
2, 4, 6, 8, 10, 12
6 ×2 = 12
“Skip” by the number 3 five times:
3, 6, 9, 12, 15
5 ×3 = 15
Slide 2-35
MULTIPLICATION TREE MODEL
Timmy has 3 tops (blue, red and
green) and 2 shorts (jean and khaki)
that can be mixed and matched. How
many outfits does he have?
CHOOSE
TOP
CHOOSE
SHORT
SIX
OUTFITS
Slide 2-36
THREE CONCEPTUAL MODELS FOR
DIVISION OF WHOLE NUMBERS
• Repeated-Subtraction Model
• Partition Model
• Missing-Factor Model
Slide 2-37
REPEATED-SUBTRACTION MODEL
Ms. Rislov has
28 students in
her class whom
she wishes to
divide into
cooperative
learning groups
of 4 students
per group.
28  4  7
Slide 2-38
PARTITION MODEL
28  4  7
Slide 2-39
Example 2.16 Computing Quotients with
Manipulatives
Suppose you have 78 number tiles. Describe how to
illustrate 78 ÷13 with the tiles, using each of the three
basic conceptual models for division.
a. Repeated subtraction. Remove groups of 13 tiles
each. Since 6 groups are formed 78 ÷ 13 = 6.
b. Partition. Partition the tiles into 13 equal-sized parts.
Since each part contains exactly 6 tiles, 78 ÷13 = 6.
c. Missing factor. Use the 78 tiles to form a rectangle
with 13 rows. Since it turns out there are 6 columns n
the rectangle, 78 ÷13 = 6.
Slide 2-40
DEFINITION:
THE POWER OPERATION
Let a and m be whole numbers where
m ≠ 0. Then a to the mth power,
written am, is defined by
a  a,
1
if m  1,
and
a  aa
m
a,
if m  1.
m times
Slide 2-41
Example 2.18 Working with Exponents
Compute expressing your answers in the form of a single
exponential expression am.
a. 74 • 72 = (7 • 7 • 7 • 7) • (7 • 7) = 7•7•7•7•7•7 = 76
b. 32 • 52 • 42 = (3 • 3) • (5 • 5) • (4 • 4) = (3 • 5 • 4)2 = 602
c. (32)5 = (3)2 • (3)2 • (3)2 • (3)2 • (3)2
= (3 • 3) • (3 • 3) • (3 • 3) • (3 • 3) • (3 • 3)
= 3•3•3•3•3•3•3•3•3•3
= 310
Slide 2-42
MULTIPLICATION RULES OF
EXPONENTIALS
Let a, b, m, and n be whole numbers
where m ≠ 0 and n ≠ 0.
a a  a
m
n
mn
a  b  ( a  b)
m
a 
m n
m
a
m
mn
Slide 2-43
ZERO AS AN EXPONENT
Let a be any whole number, a ≠ 0.
Then a0 = 1.
Slide 2-44
DIVISION RULES OF EXPONENTIALS
Let a, b, m, and n be whole numbers
where m  n  0, b  0 and a  b is defined.
m
b
m n
b
n
b
m
m
a  a

 bm   b 
   
Slide 2-45
Example 2.19 Working with Exponents
Rewrite these expression in exponential form am.
a. 512 • 58 = 512 + 8 = 520
b. 32 • 35 • 38 = 32 + 5 + 8 = 315
c. (35)2/34 = 35•2-4 = 36
Slide 2-46