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Chapter
3
Numeration Systems
and Whole Number
Operations
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.
3-2 Addition and Subtraction of
Whole Numbers
• Number relationships including comparing and
ordering.
• The meaning of addition and subtraction by
studying various models and in turn learn addition
and subtraction facts.
• Properties of addition and subtraction and how to
use them to develop computational strategies.
• The inverse relationship between addition and
subtraction.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide
2
Addition of Whole Numbers
Counting On
Counting on is an addition strategy where
addition is performed by counting on from one of the
numbers, for example, 5 + 3 can be computed
by starting at 5 and counting 6, 7, 8.
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Slide
3
Addition of Whole Numbers
Set Model
Suppose Jane has 4 blocks in
one pile and 3 in another. If she
combines the two groups, how
many objects are there in the
combined group?
Note that the sets must be
disjoint (have no elements in
common) or an incorrect
conclusion can be drawn.
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Slide
4
Definition
Addition of Whole Numbers
Let A and B be two disjoint finite sets.
If n(A) = a and n(B) = b, then a + b = n(A U B).
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Slide
5
Number-Line Model
1. Josh has 4 feet of red ribbon and 3 feet of white
ribbon. How many feet of ribbon does he have
altogether?
2. One day, Gail drank 4 ounces of orange juice in
the morning and 3 ounces at lunchtime. If she
drank no other orange juice that day, how many
ounces of orange juice did she drink for the
entire day?
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Slide
6
Number-Line Model
Students need to understand that the sum
represented by any two directed arrows can be
found by placing the endpoint of the first directed
arrow at 0 and then joining to it the directed arrow
for the second number with no gaps or overlaps.
The sum of the numbers can then be read.
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Slide
7
Definition
Less Than: For any whole numbers a and b, a is
less than b, written a < b, if, and only
if, there exists a natural number k
such that a + k = b.
a ≤ b means a < b or a = b.
a > b is the same as b < a.
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Slide
8
Whole Number Addition Properties
Closure Property of
Addition of Whole Numbers
If a and b are whole numbers, then a + b is a whole
number.
The closure property implies that the sum of two
whole numbers exists and that the sum is a unique
whole number.
For example, 5 + 2 is a unique whole number, 7.
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Slide
9
Whole Number Addition Properties
Commutative Property of
Addition of Whole Numbers
If a and b are any whole numbers, then
a + b = b + a.
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Slide 10
Whole Number Addition Properties
Associative Property of
Addition of Whole Numbers
If a, b, and c are any whole numbers, then
(a + b) + c = a + (b + c).
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Slide 11
Whole Number Addition Properties
Identity Property of
Addition of Whole Numbers
There is a unique whole number, 0, the
additive identity, such that for any whole
number a, a + 0 = a = 0 + a.
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Slide 12
Example
Which properties are illustrated in each of the
following?
a. 5 + 7 = 7 + 5
Commutative property of addition
b. 1001 + 733 is a unique whole number.
Closure property of addition
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Slide 13
Example (cont)
Which properties are illustrated in each of the
following?
c. (3 + 5) + 7 = (5 + 3) + 7
Commutative property of addition
d. (8 + 5) + 2 = 2 + (8 + 5) = (2 + 8) + 5
Commutative and associative properties
of addition
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Slide 14
Mastering Basic Addition Facts
Counting on: Start with the greater addend then
count on the smaller addend. For example: 4 + 2,
start with 4, then count on another two, 5, 6.
Doubles: After students master doubles (such as
3 + 3), doubles + 1 and doubles plus 2 can be
learned easily. For example, if a student knows
6 + 6 = 12, then 6 + 7 is (6 + 6) + 1 = 12 + 1 = 13.
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Slide 15
Mastering Basic Addition Facts
Making 10: Regroup to form a group of 10 and a
leftover. For example: 8 + 5 can be added as
follows:
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Slide 16
Mastering Basic Addition Facts
Counting back: Usually used when one number is
1 or 2 less than 10. For example, because 9 is 1
less than 10, then 9 + 7 is 1 less than 10 + 7 or 16.
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Slide 17
Subtraction of Whole Numbers
Subtraction of whole numbers can be modeled in
several different ways:
 Take-Away Model – views subtraction as a
second set of objects being taken away from the
original set
 Missing Addend Model – an algebraic-type of
reasoning is used where students compute a
difference by determining the value of an
“unknown” addend.
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Slide 18
Subtraction of Whole Numbers
 Comparison Model – students determine “how
many more” of one quantity exists than another.
 Number-Line Model – subtraction is represented
by moving left on the number line a given number
of units.
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Slide 19
Subtraction of Whole Numbers
Take-Away Model
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Slide 20
Subtraction of Whole Numbers
Missing-Addend Model
8−3=
This can be thought of as the number of blocks that
must be added to 3 in order to get 8.
The number 8 – 3 is the missing addend in the
equation
3+
=8
5
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Slide 21
Subtraction of Whole Numbers
Missing-Addend Model
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Slide 22
Definition
Subtraction of Whole Numbers:
For any whole numbers a and b, such that
a ≥ b, a − b is the unique whole number c
such that b + c = a.
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Slide 23
Subtraction of Whole Numbers
Comparison Model
Juan has 8 blocks and Susan has 3 blocks. How
many more blocks does Juan have than Susan?
8−3=5
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Slide 24
Subtraction of Whole Numbers
Number-Line (Measurement) Model
5−3=2
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Slide 25
Properties of Subtraction
It can be shown that if a < b, then a − b is not
meaningful in the set of whole numbers. Therefore,
subtraction is not closed on the set of whole
numbers.
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Slide 26
Introductory Algebra Using WholeNumber Addition and Subtraction
Sentences such as 9 + 5 = ☼ and 12 − ◊ = 4 can
be true or false depending on the values of ☼ and
◊.
For example, if ☼ = 10, then 9 + 5 = ☼ is false.
If ◊ = 8, then 12 − ◊ = 4 is true.
If the value that is used makes the equation true, it
is a solution to the equation.
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Slide 27