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Section 2.1
Sets and Whole Numbers
Mathematics for Elementary School Teachers - 4th EditionO’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 2.1A set is a collection of objectsor ideas that can be listed or
described
A = {a, e, i, o, u} C = {Blue, Red, Yellow}
A set is usually listed with a capital letterA set can be represented using braces { }
A set can also be represented using a circle
A = oi
eua C =
BlueRed
Yellow
Each object in the set is called an element of the set
C = BlueRed
YellowBlue is an element of set C
Blue C
Orange is not an element of set C
Orange C
Definition of a One-to-One CorrespondenceSets 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
1
2
3
Set B
c
b
a
The order of the elements does not matter
Definition of Equivalent SetsSets A and B are equivalent sets if and
only if there is a one-to-one correspondence between A and B
Set A
onetwo
three
Set B
FrogCat
Dog
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 EditionO’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 AdditionNumber Line Model - joining two continuous
lengths
5 + 4 = 9
Properties of Addition of Whole Numbers
Closure PropertyFor whole numbers a and b, a + b is a unique
whole number
Identity PropertyThere 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 PropertyFor whole numbers a and b, a + b = b + a
Associative PropertyFor whole numbers a, b, and c, (a + b) + c =
a + (b + c)
Modeling Subtraction Taking away a subset of a set.o Suppose that you have 12 Pokemon cards and give away 7. How many Pokemon cards will you have left?
Separating a set of discrete objects into two disjoint sets.o A student had 12 letters. 7 of them had stamps. How many letters did not have stamps?
Comparing two sets of discrete objects.o 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)o 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 NumbersIn 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.
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.
Restating the definition substituting whole numbers:
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 PropertyFor whole numbers a and b, a + b is a unique
whole number
Identity PropertyThere 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 PropertyFor whole numbers a and b, a + b = b + a
Associative PropertyFor 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 EditionO’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”
Multiplication - joining equivalent sets
3 sets with 2 objects in each set3 x 2 = 6 or 2 + 2 + 2 = 6
Repeated Addition
Multiplication using a rectangular array
3 rows2 in each row
3 x 2 = 6
Using Models and Sets to Define Multiplication
Multiplication using the Area of a Rectangle
width
lengthArea model of a polygon
Can be a continuous region
Using Models and Sets to Define Multiplication
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
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.
Problem Solving: Color Combinations for Invitations
GoldGold SilverSilver BlackBlackBlueBlue (B, G) (B, S) (B, Bk)
GreenGreen (GR, G) (GR, S) (GR, Bk)
RedRed (R, G) (R, S) (R, Bk)
YellowYellow (Y, G) (Y, S) (Y, Bk)
4 x 3 = 12 combinations
Multiplication by joining segments of equal length on
a number line
4 x 3 = 12
Length of one
segment
Number of segments
being joined
Using Models and Sets to Define Multiplication
Properties of Multiplication of Whole Numbers
Closure propertyFor whole numbers a and b, a x b is a unique whole number
Identity propertyThere 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 propertyFor whole numbers a and b, a x b = b x a
Associative propertyFor whole numbers a, b, and c, (a x b) x c = a x (b x c)
Zero propertyFor each whole number a, a x 0 = 0 x a = 0
Distributive property of multiplication over additionFor 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)
This is the Sharing interpretation of division.
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?
Models of Division
This is the Repeated Subtraction or Measurement interpretation of
Division.
You have a total of 12 cookies, and want to put 3 cookies in each bag. How many bags can you fill?
How many groups (subsets)?
Division as the Inverse of Multiplication
Factor Factor Product
9 x 8 = 72
÷72 8 = 9
Product Factor Factor
This relationship suggest the following definition:
So the answer to the division equation, 9, is one of the factors in the related multiplication equation.
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
Think of 36 as the product and 3 as one of the factors
What factor multiplied by 3 gives the product 36 ?
When asked to find the quotient 36 ÷ 3 = ?
You can turn it into a multiplication problem: ? x 3 = 36
Then ask,
Division does not have the same properties as multiplication
Does the Closure, Identity, Commutative, Associative, Zero, and Distributive
Properties hold for Division as they do for 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.
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
Why Division by Zero is Undefined
Section 2.4
Numeration
Mathematics for Elementary School Teachers - 4th EditionO’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK
The word symbol for cat is different than the actual cat
A symbol is different from what it represents
Here is another familiar numeral (or
name) for the number two
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.
Definition of Numeration SystemAn accepted collection of properties and
symbols that enables people to systematically write numerals to represent numbers. (p. 106, text)
Hindu-Arabic Numeration System
Egyptian Numeration System
Babylonian Numeration System
Roman Numeration System
Mayan 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 ValueIn 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
orblock, flat, long, unit
text, p. 110
2,345
Expanded Notation:
1324 = (1×1000) + (3×100) + (2×10) + (4×1)
1324 = (1×103) + (3×102) + (2×101) + (4×100)
Example (using base 10):
or
This is a way of writing numbers to show place value, by multiplying each digit in the numeral by its matching place value.
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
(a) form groups of 10we can group these tiles into two groups of ten with 7 tiles
left over(b) form groups of 5 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.
27
No group of 5 is left over, so we need to use a 0 in that position in
the numeral: 102five
102five
Find the base-ten representation for 1324five
Find the base-ten representation for 344six
Find the base-ten representation for 110011two
= 1(125) + 3(25) + 2(5) + 4(1)
1324five = (1×53) + (3×52) + (2×51) + (4×50)
= 125 + 75 + 10 + 4= 214ten
Expressing Numerals with Different Bases:
Find the representation of the number 256 in base six
64 = 129663 = 21662 = 36
60 = 161 = 6
256- 216
40-36
4
1(216) + 1(36) + 0(6) + 4(1) = 1104six
1(63) + 1(62) + 0(61) + 4(60)
Expressing Numerals with Different Bases:
Roman Numeration SystemDeveloped between 500 B.C.E and 100 C.E.
ⅬⅭⅮⅯ
Ⅹ
ⅼⅤ
(one)
(five)
(ten)
(fifty)
(one hundred)
(five hundred)(one thousand)
•Group partially by fives•Would need to add new symbols
•Position indicates when to add or
subtract•No use of zero
Ⅽ Ⅿ Ⅹ Ⅽ ⅼ Ⅹ
900 + 90 + 9 = 999
Write the Hindu-Arabic numerals for the numbers represented by the Roman
Numerals:
Egyptian Numeration SystemDeveloped: 3400 B.C.E
One
Ten
One Hundred
One Thousand
Ten Thousand
One Hundred Thousand
One Million
reed
heel bone
coiled rope
lotus flower
bent finger
burbot fish
kneeling figureor
astonished man
Group by tens
New symbols would be needed as system grows
No place value
No use of zero
Babylonian Numeration SystemDeveloped between 3000 and 2000 B.C.E
There are two symbols in the Babylonian Numeration System
Base 60Place value one ten
42(601) + 34(600) = 2520 + 34 = 2,554
Zero came later
Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the
Babylonian system:
Mayan Numeration SystemDeveloped between 300 C.E and 900 C.E
•Base - mostly by 20•Number of symbols: 3•Place value - vertical•Use of Zero
Symbols
= 1= 5
= 0
Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the
Mayan system:
0(200) = 0
6(201) = 120
8(20 ×18) = 2880
2880 + 120 + 0 = 3000
Summary of Numeration System Characteristics
SystemSystem GroupinGroupingg
SymbolsSymbols Place Place ValueValue
Use of Use of ZeroZero
EgyptiaEgyptiann
By tens
Infinitely many
possibly needed
No No
BabyloniBabylonianan
By sixties Two Yes Not at first
RomanRoman Partiallyby fives
Infinitely many
possibly needed
Position indicates
when to add or subtract
No
MayanMayan Mostlyby twenties
ThreeYes,
VerticallyYes
Hindu-Hindu-ArabicArabic
By tens Ten Yes Yes
The EndChapter 2