Transcript Ch. 2

Section 2.1
Sets and Whole Numbers
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK
How do you think the
idea of numbers
developed?
How could a child who doesn’t know how to count
verify that 2 sets have the same number of
objects? That one set has more than another
set?
Sets and Whole Numbers - Section 2.1
A set is a collection of objects
or ideas that can be listed or described
A set is usually listed with a capital
letter
A set can be represented using braces
{}
A = {a, e, i, o, u}
C = {Blue, Red, Yellow}
A set can also be represented using a
circle
A=
a u
e o
i
C=
Blue
Red
Yellow
Each object in the set is called an element of the set
C=
Blue
Red
Yellow
Blue is an element of set C
Blue  C
Orange is not an element of set C
Orange  C

Definition of a One-to-One Correspondence
Sets A and B have a one-to-one
correspondence if and only if each element
of A is paired with exactly one element of B
and each element of B is paired with exactly
one element of A.
Set A
Set B
1
a
2
b
3
c
The order of the elements does not matter
Definition of Equivalent Sets
Sets A and B are equivalent sets if and only if
there is a one-to-one correspondence between
A and B
Set A
Set B
one
two
three
Dog
Cat
Frog
A~B
Finite Set
A set with a limited number of elements
Example: A = {Dog, Cat, Fish, Frog}
Infinite Set
A set with an unlimited number of elements
Example: N = {1, 2, 3, 4, 5, . . . }
Section 2.2
Addition and Subtraction of Whole Numbers
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK
Using Models to
Provide an Intuitive Understanding of Addition
Joining two groups of discrete
objects
3 books + 4 books = 7 books
Using Models to
Provide an Intuitive Understanding of Addition
Number Line Model - joining two continuous lengths
5+4=9
Properties of Addition of Whole Numbers
Closure Property
For whole numbers a and b, a + b is a unique whole
number
Identity Property
There exist a unique whole number, 0, such
that 0 + a = a + 0 = a for every whole
number a. Zero is the additive identity
element.
Commutative Property
For whole numbers a and b, a + b = b + a
Associative Property
For whole numbers a, b, and c, (a + b) + c = a + (b
+ c)
Modeling Subtraction
Taking away a subset of a set.
Suppose that you have 12 Pokemon cards and give away 7. How
many Pokemon cards will you have left?

o
Separating a set of discrete objects into two disjoint sets.
A student had 12 letters. 7 of them had stamps. How many letters
did not have stamps?

o

o

o

Comparing two sets of discrete objects.
Suppose that you have 12 candies and someone else has 7
candies. How many more candies do you have than the other
person?
Missing Addend (inverse of addition)
Suppose that you have 7 stamps and you need to mail 12 letters.
How many more stamps are needed?
Geometrically by using two rays on the number line
Definition of Subtraction of Whole Numbers
In the subtraction of the whole numbers a and b, a – b = c if
and only if c is a unique whole number such that c + b = a. In
the equation, a – b = c, a is the minuend, b is the subtrahend,
and c is the difference.
Restating the definition substituting whole numbers:
In the subtraction of the whole numbers 10 and 7, 10 – 7 = 3 if
and only if 3 is a unique whole number such that 3 + 7 = 10. In the
equation, 10 – 7 = 3, 10 is the minuend, 7 is the subtrahend, and
3 is the difference.
Comparing Addition and Subtraction
Properties of Whole Numbers
Which of the properties of addition hold for
subtraction?
1. Closure
2. Identity
3. Commutative
4. Associative
Properties of Addition of Whole Numbers
Closure Property
For whole numbers a and b, a + b is a unique whole
number
Identity Property
There exist a unique whole number, 0, such
that 0 + a = a + 0 = a for every whole
number a. Zero is the additive identity
element.
Commutative Property
For whole numbers a and b, a + b = b + a
Associative Property
For whole numbers a, b, and c, (a + b) + c = a + (b
+ c)
Section 2.3
Multiplication and Division of Whole Numbers
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK
How are addition, subtraction,
multiplication, and division
connected?
• Subtraction is the inverse of addition.
• Division is the inverse of multiplication.
• Multiplication is repeated addition.
• Division is repeated subtraction.
•
“Amanda Bean’s Amazing Dream”
Using Models and Sets to Define Multiplication
Multiplication - joining equivalent sets
3 sets with 2 objects in each set
3 x 2 = 6 or 2 + 2 + 2 = 6
Repeated Addition
Multiplication using a rectangular array
3 rows
2 in each row
3x2=6
Using Models and Sets to Define Multiplication
Multiplication using the Area of a Rectangle
width
length
Area model of a polygon
Can be a continuous region
Definition of Cartesian Product
The Cartesian product of two sets A and B, A X B
(read “A cross B”) is the set of all ordered pairs (x,
y) such that x is an element of A and y is an
element of B.
Example:
A = { 1, 2, 3 } and B = { a, b },
A x B = { (1, a), (1, b), (2, a), (2, b), (3, a), (3, b) }
Note that sets A and B can be equal
Problem Solving: Color Combinations for Invitations
Suppose that you are using construction paper to make
invitations for a club function. The construction paper
comes in blue, green, red, and yellow, and you have
gold, silver, or black ink. How many different color
combinations of paper and ink do you have to choose
from?
Use a tree diagram or an array of ordered pairs to match
each color of paper with each color of ink.
Blue
Green
Red
Yellow
Gold
Silver
Black
(B, G)
(B, S)
(B, Bk)
(GR, G)
(GR, S)
(GR, Bk)
(R, G)
(R, S)
(R, Bk)
(Y, G)
(Y, S)
(Y, Bk)
4 x 3 = 12 combinations
Using Models and Sets to Define Multiplication
Multiplication by joining
segments of equal length on a
number line
Number of
segments
being joined
4 x 3 = 12
Length of
one
segment
Properties of Multiplication of Whole Numbers
Closure property
For whole numbers a and b, a x b is a unique whole number
Identity property
There exists a unique whole number, 1, such that 1 x a = a x 1 =
a for every whole number a. Thus 1 is the multiplicative identity
element.
Commutative property
For whole numbers a and b, a x b = b x a
Associative property
For whole numbers a, b, and c, (a x b) x c = a x (b x c)
Zero property
For each whole number a, a x 0 = 0 x a = 0
Distributive property of multiplication over addition
For whole numbers a, b, and c, a x (b + c) = (a x b) + (a x c)
Suppose you do not know the fact 9 X 12.
A. How can you use other known facts to
figure out the answer?
B. Find as many different ways as possible
and explain why your way works.
Models of Division
• Think of a division problem you might
give to a fourth grader.
Modeling Division (continued)
How many in each group (subset)?
There is a total of 12 cookies. You want to give
cookies to 3 people. How many cookies can each
person get?
This is the Sharing
interpretation of division.
Models of Division
How many groups (subsets)?
You have a total of 12 cookies, and want to put 3
cookies in each bag. How many bags can you fill?
This is the Repeated Subtraction or
Measurement interpretation of Division.
Division as the Inverse of Multiplication
Factor
Factor
Product
9 x 8 = 72
Product Factor Factor
72÷8 = 9
So the answer to the division equation, 9, is
one of the factors in the related multiplication
equation.
This relationship suggest the following
definition:
Definition of Division
• In the division of whole numbers a
and b (b≠0): a ÷ b = c if and only if c is a
unique whole number such that c x b
= a. In the equation, a ÷ b = c, a is the
dividend, b is the divisor, and c is the
quotient.
Division as Finding the Missing Factor
When asked to find the quotient 36 ÷ 3 =？
You can turn it into a multiplication problem:
？x 3 = 36
Think of 36 as the product and 3 as one of the factors
Then ask,
What factor multiplied by 3 gives the product 36
？
Does the Closure, Identity, Commutative,
Associative, Zero, and Distributive Properties
hold for Division as they do for Multiplication?
Division does not have the same
properties as multiplication
Division by 0
a. Is 0 divided by a number defined?
(i.e. 0/4)
b. Is a number divided by 0 defined?
(i.e. 5/0)
Explain your reasoning.
Why Division by Zero is Undefined
When you look at division as finding the
missing factor it helps to give understanding
why zero cannot be used as a divisor.
3 ÷ 0 = ？No number multiplied by 0 gives 3.
There is no solution!
0 ÷ 0 = ？Any number multiplied by 0 gives 0.
There are infinite solutions!
Thus, in both cases 0 cannot be used as a divisor.
However, 0 ÷ 3 = ？ has the answer 0. 3 x 0 = 0
Section 2.4
Numeration
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK
A symbol is different from what it represents
The word symbol for cat is different than the actual
cat
Numeration Systems
Just as the written symbol 2 is not itself a
number.
The written symbol,
2, that represents a
number is called a
numeral.
Here is another familiar
numeral (or name) for
the number two
Definition of Numeration System
An accepted collection of properties and
symbols that enables people to systematically
write numerals to represent numbers. (p. 106,
text)
Egyptian Numeration System
Babylonian Numeration System
Roman Numeration System
Mayan Numeration System
Hindu-Arabic Numeration System
Hindu-Arabic Numeration System
• Developed by Indian and Arabic cultures
• It is our most familiar example of a numeration system
• Group by tens: base ten system
•10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Place value - Yes! The value of the digit is
determined by its position in a numeral
•Uses a zero in its numeration system
Definition of Place Value
In a numeration system with place value, the
position of a symbol in a numeral determines that
symbol’s value in that particular numeral. For
example, in the Hindu-Arabic numeral 220, the
first 2 represents two hundred and the second 2
represents twenty.
Models of Base-Ten Place Value
Base-Ten Blocks - proportional model for place value
Thousands cube, Hundreds square, Tens stick,
Ones cube
or
block, flat, long, unit
text, p. 110
2,345
Expanded Notation:
This is a way of writing numbers to show place
value, by multiplying each digit in the numeral
by its matching place value.
Example (using base
10):
1324 = (1×103) + (3×102) + (2×101) + (4×100)
or
1324 = (1×1000) + (3×100) + (2×10) + (4×1)
Expressing Numerals with Different
Bases:
Show why the quantity of tiles shown can be
expressed as (a) 27 in base ten and (b)102 in
base five, written 102five
we can group
(a) form groups of 10
these tiles into two
27
groups of ten with
7 tiles left over
(b) form groups of 5
102five
we can group these tiles
into groups of 5 and
have enough of these
groups of 5 to make one
larger group of 5 fives,
with 2 tiles left over.
No group of 5 is left over, so we need to
use a 0 in that position in the numeral:
Expressing Numerals with Different
Bases:
Find the base-ten representation for 1324five
1324five = (1×53) + (3×52) + (2×51) + (4×50)
= 1(125) + 3(25) + 2(5) + 4(1)
= 125 + 75 + 10 + 4
= 214ten
Find the base-ten representation for 344six
Find the base-ten representation for 110011two
Expressing Numerals with Different
Bases:
Find the representation of the number 256 in base six
256
- 216
40
-36
4
1(63) + 1(62) + 0(61) + 4(60)
1(216) + 1(36) + 0(6) + 4(1) = 1104six
0
6
=1
1
6 =6
62 = 36
3
6 = 216
64 = 1296
Roman Numeration System
Developed between 500 B.C.E and 100 C.E.
•Group partially by
fives
•Would need to add new symbols
•Position indicates
when to add or subtract
•No use of zero
Write the Hindu-Arabic
numerals for the numbers
represented by the Roman
Numerals:
ⅭⅯⅩⅭⅼⅩ
900 + 90 + 9 = 999
ⅼ (one)
Ⅴ (five)
Ⅹ (ten)
Ⅼ (fifty)
Ⅽ (one hundred)
Ⅾ (five hundred)
Ⅿ(one thousand)
Egyptian Numeration System
Developed: 3400 B.C.E
reed
Group by tens
New symbols would
be needed as
system grows
No place value
heel bone
One
Ten
coiled rope
One Hundred
lotus flower
One Thousand
bent finger
Ten Thousand
burbot fish
One Hundred Thousand
No use of zero
kneeling figure
or
astonished man
One Million
Babylonian Numeration System
Developed between 3000 and 2000 B.C.E
There are two symbols in the Babylonian Numeration System
Base 60
Place value
one
ten
Zero came later
Write the Hindu-Arabic numerals for the numbers represented
by the following numerals from the Babylonian system:
1
42(60 )
+
0
34(60 )
= 2520 + 34 = 2,554
Mayan Numeration System
Developed between 300 C.E and 900 C.E
•Base - mostly by 20
•Number of symbols: 3
•Place value - vertical
•Use of Zero
Symbols
=0
=1
=5
Write the Hindu-Arabic numerals for the numbers
represented by the following numerals from the Mayan
system:
8(20 ×18) = 2880
6(201) = 120
0(200) = 0
2880 + 120 + 0 = 3000
Summary of Numeration System Characteristics
System Grouping Symbols
Place
Value
Use of
Zero
No
No
Egyptian
By tens
Infinitely
many
possibly
needed
Babylonia
n
By sixties
Two
Yes
Not at first
Roman
Partially
by fives
Infinitely
many
possibly
needed
Position
indicates when
to add or
subtract
No
Mayan
Mostly
by twenties
Three
Yes,
Vertically
Yes
By tens
Ten
Yes
Yes
Hindu-
The End
Chapter 2