Chapter 2: Sets and Whole-Number
Operations and Properties
2.1
Addition and
Subtraction of Whole Numbers
2.1.1. Using Models and Sets to Define Addition
2.1.1.1.
How are addition
and subtraction different?
2.1.1.2.
How are addition
and subtraction alike?
2.1.1.3.
2.1.1.3.1. Definition of the union of two sets: The union
of two sets A and B is the set containing every element belonging to set A or
set B and is written A È
B (read “the union of A and B”)
2.1.1.3.2. See examples on p. 68
2.1.1.4.
Your turn p. 68: Do the practice and the
reflect
2.1.1.5.
Intersection of
Two Sets
2.1.1.5.1. Definition of the intersection of two sets:
The intersection of two sets A and B is the set containing every element
belonging to both set A and set B and is written A Ç B (read “the intersection of A and B”)
2.1.1.5.2. Two sets are said to be disjoint or mutually
exclusive if their intersection is the empty set
2.1.1.5.3. See examples p. 69
2.1.1.6.
Your turn p. 69: Do the practice and the
reflect
2.1.1.7.
Addition of
Whole Numbers
2.1.1.7.1. Definition of the addition of whole numbers:
In the addition of whole numbers, if A and B are two disjoint sets, and n(A) = a and n(B) = b, then a + b = n(A È B). In the
equation a + b = c, a and b are addends,
and c is the sum
2.1.1.7.2. See example p. 70
2.1.1.8.
Your turn p. 71: Do the practice and the
reflect
2.1.2. Basic
Properties of Addition
2.1.2.1.
Closure
Property of Addition
2.1.2.1.1. Guarantees that the addition of two whole numbers
results in another unique whole number
2.1.2.1.2. Another way to say this is: The whole numbers are closed under addition
2.1.2.1.3. Closed means that the number obtained after an operation on
any numbers in the set yield a number in the set
2.1.2.1.4. Unique means that a + b gives one and only one result
2.1.2.1.5. For whole
numbers a and b, a + b is a unique whole number
2.1.2.2.
Identity
Property of Addition
2.1.2.2.1. a + 0 = 0 + a = a
2.1.2.2.2. Zero is called the additive identity element
2.1.2.2.3. There
exists a unique whole number, 0, such that 0 + a = a + 0 = a
for every whole number a. Zero is
the additive identity element
2.1.2.3.
Commutative
Property of Addition
2.1.2.3.1. a + b = b + a
2.1.2.3.2. Order is not important in addition
2.1.2.3.3. For whole
numbers a and b, a + b = b + a
2.1.2.4.
Associative
Property of Addition
2.1.2.4.1. (a + b) + c = a + (b + c)
2.1.2.4.2. The sum of three numbers is the same regardless of
which two are added together first
2.1.2.4.3. For whole
numbers a, b, and c, (a + b) + c = a +
(b + c)
2.1.2.5.
Definition
of Greater Than (>) and Less Than (<) for Whole Numbers
2.1.2.5.1. Given whole numbers a and b,
a is greater than b, symbolized as a > b, if and only if there is a whole
number k > 0 such that a = b + k.
Also, b is less than a (b < a), whenever a > b
2.1.3. Modeling
Subtraction
2.1.3.1.
Taking away
2.1.3.2.
Separating
2.1.3.3.
Comparing
2.1.3.4.
See examples p. 73-75
2.1.4. Using
Addition to Define Subtraction
2.1.4.1.
Subtraction
as the Inverse of Addition
2.1.4.1.1. Subtraction is the inverse operation of
addition
2.1.4.1.2. Definition of subtraction of whole numbers:
In the subtraction of 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
2.1.4.2.
Your turn p. 76: Do the practice and the
reflect
2.1.4.3.
Subtraction
as Finding the Missing Addend
2.1.4.3.1. Addition model: addend + addend = sum
2.1.4.3.2. Subtraction model: sum – addend = missing addend
2.1.4.3.3. Re-stated Subtraction model: addend + missing addend
= sum
2.1.4.3.4. Using this model: if you know addition facts, then
you know subtraction facts
2.1.5. Comparing
Subtraction to Addition
2.1.5.1.
See example p. 78
2.1.5.2.
Your turn p. 78: Do the practice and the
reflect
2.1.6. Addition
and Subtraction Facts and Fact Families
2.1.6.1.
Fact is a
digit plus a digit
2.1.6.1.1. How many addition facts are there?
2.1.6.1.2. If I know the additive identity (0 + a number is that
number), how many more facts do I need to learn?
2.1.6.1.3. If I know zero and my ones, how many facts are left?
2.1.6.1.4. If I know zero, ones, and doubles, how many facts are
left?
2.1.6.1.5. If I know the commutative property of addition and
doubles, how many facts do I need to learn?
2.1.6.1.6. If I know the commutative property, doubles, zero,
and ones, how many facts are left for me to learn?
2.1.6.1.7. Which list is more intimidating - - 100 facts or the
one we just figured out above? Remember
to share this with your future students!
It will empower them!! J
2.1.6.2.
Fact Families
2.1.6.2.1. Another way of organizing and learning addition facts
2.1.6.2.2. Fact families deal with sums and the facts that make
up those sums
0 Þ 0 + 0
1 Þ 1 + 0 0 + 1
2 Þ 2 + 0 1 + 1 0
+ 2
3 Þ 3 + 0 2 + 1 1
+ 2 0 + 3
4 Þ 4 + 0 3 + 1 2
+ 2 1 + 3 0 + 4
5 Þ 5 + 0 4 + 1 3
+ 2 2 + 3 1 + 4 0 + 5
6 Þ 6 + 0 5 + 1 4
+ 2 3 + 3 2 + 4 1 + 5 0 + 6
etc.