1. Section 2.1 Sets and Whole Numbers
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

2. 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?

3. 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
C =
Blue
Red
Yellow
A = o i e u a

4. Each object in the set is called an element of the set
Blue
Red
Yellow
Blue is an element of set C
Orange is not an element of set C

5. 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 1 2 3
Set B c b a
The order of the elements does not matter

6. 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
one
two
three
Set B
Frog
Cat
Dog A ~ B

7. A set with a limited number of elements
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, }

8. Addition and Subtraction of Whole Numbers
Section 2.2
Addition and Subtraction of Whole Numbers
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

9. Provide an Intuitive Understanding of Addition
Using Models to
Provide an Intuitive Understanding of Addition
Joining two groups of discrete objects
3 books + 4 books = 7 books

10. Provide an Intuitive Understanding of Addition
Using Models to
Provide an Intuitive Understanding of Addition
Number Line Model - joining two continuous lengths
5 + 4 = 9

11. 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)

12. 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?
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?
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

13. 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 = 10. In the equation, 10 – 7 = 3, 10 is the minuend, 7 is the subtrahend, and 3 is the difference.

14. Comparing Addition and Subtraction Properties of Whole Numbers
Which of the properties of addition hold for subtraction?
Closure
Identity
Commutative
Associative

15. 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)

16. Multiplication and Division of Whole Numbers
Section 2.3
Multiplication and Division of Whole Numbers
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

17. 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”

18. Using Models and Sets to Define Multiplication
Multiplication - joining equivalent sets
3 sets with 2 objects in each set
3 x 2 = 6 or = 6
Repeated Addition
Multiplication using a rectangular array
3 rows
2 in each row
3 x 2 = 6

19. 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

20. 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

21. 4 x 3 = 12 combinations Gold Silver Black Blue Green Red Yellow
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.
Gold
Silver
Black
Blue
(B, G)
(B, S)
(B, Bk)
Green
(GR, G)
(GR, S)
(GR, Bk)
Red
(R, G)
(R, S)
(R, Bk)
Yellow
(Y, G)
(Y, S)
(Y, Bk)
4 x 3 = 12 combinations

22. Using Models and Sets to Define Multiplication
Multiplication by joining segments of equal length on a number line
4 x 3 = 12
Number of segments being joined
Length of one segment

23. 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)

24. Suppose you do not know the fact 9 X 12.
How can you use other known facts to figure out the answer?
Find as many different ways as possible and explain why your way works.

25. Models of Division
Think of a division problem you might give to a fourth grader.

26. 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.

27. How many groups (subsets)?
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.

28. Division as the Inverse of Multiplication
Factor
Product
9 x 8 = 72 ÷ 72
8 = 9
Product
Factor
So the answer to the division equation, 9, is one of the factors in the related multiplication equation.
This relationship suggest the following definition:

29. 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.

30. Division as Finding the Missing Factor
When asked to find the quotient 36 ÷ 3 =？
？x 3 = 36
You can turn it into a multiplication problem:
Think of 36 as the product and 3 as one of the factors
Then ask,
What factor multiplied by 3 gives the product 36？

31. 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 does not have the same properties as multiplication

32. 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.

33. 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

34. Section 2.4
Numeration
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

35. A symbol is different from what it represents
The word symbol for cat is different than the actual cat

36. Just as the written symbol 2 is not itself a number.
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

37. 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

38. 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

39. 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.

40. 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

41. 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)

42. we can group these tiles into two groups of ten with 7 tiles left over
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 these tiles into two groups of ten with 7 tiles left over
(a) form groups of 10 27
(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: 102five

43. Find the base-ten representation for 1324five
Expressing Numerals with Different Bases:
Find the base-ten representation for 1324five
= 1(125) + 3(25) + 2(5) + 4(1)
1324five = (1×53) + (3×52) + (2×51) + (4×50) =
= 214ten
Find the base-ten representation for 344six
Find the base-ten representation for two

44. Find the representation of the number 256 in base six
Expressing Numerals with Different Bases:
Find the representation of the number 256 in base six
64 = 1296
63 = 216
62 = 36
60 = 1
61 = 6
256
- 216 40
-36 4
1(63) + 1(62) + 0(61) + 4(60)
1(216) + 1(36) + 0(6) + 4(1) = 1104six

45. Roman Numeration System
Developed between 500 B.C.E and 100 C.E.
Group partially by fives
Would need to add new symbols Ⅼ Ⅽ Ⅾ Ⅿ Ⅹ ⅼ Ⅴ
(one)
(five)
(ten)
(fifty)
(one hundred)
(five hundred)
(one thousand)
Position indicates when to add or subtract
No use of zero
Write the Hindu-Arabic numerals for the numbers represented by the Roman Numerals: Ⅽ Ⅿ Ⅹ ⅼ
= 999

46. Egyptian Numeration System
Developed: 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 figure or
astonished man
Group by tens
New symbols would be needed as system grows
No place value
No use of zero

47. Babylonian Numeration System
Developed between 3000 and 2000 B.C.E
There are two symbols in the Babylonian Numeration System
ten
Base 60
one
Zero came later
Place value
Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the Babylonian system:
42(601) + 34(600) = = 2,554

48. Mayan Numeration System
Developed between 300 C.E and 900 C.E
Symbols
= 1
= 5
= 0
Base - mostly by 20
Number of symbols: 3
Place value - vertical
Use of Zero
Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the Mayan system:
8(20 ×18) = 2880
6(201) = 120
= 3000
0(200) = 0

49. Summary of Numeration System Characteristics
Grouping
Symbols
Place Value
Use of Zero
Egyptian
By tens
Infinitely many possibly needed No
Babylonian
By sixties
Two
Yes
Not at first
Roman
Partially
by fives
Position indicates when to add or subtract
Mayan
Mostly
by twenties
Three
Yes, Vertically
Hindu-Arabic
Ten

50. The End
Chapter 2