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Chapter 3
Whole Numbers
Section 3.2
Addition and Subtraction of Whole
Numbers
Addition of Whole Numbers
The concept of whole number addition can be described (or defined) in terms of
sets.
If a set A contains a elements and a set B contains b elements, and AB=Ø,
then a+b is the number of elements in AB. In the equation a+b=c, a and b
are called the addends and c is called the sum.
Because the sum of two whole numbers represents the number in a set the result
is always going to be a whole number. This property is called the Closure
property of addition of Whole Numbers.
An example of the closure property would be to answer this multiple choice
question.
b
____
Find the sum of 723468+164501
(a) 887909.1476
(b) 887969
(c)
723468
164501
We know the correct answer is (b) without doing any calculation since adding two
whole numbers will produce another whole number. What is the smallest set that
contains the number 5 and is closed under addition? (A set being closed means it
has the closure property)
{5,10,15,20,25,…} If you add two multiples of 5 together you get a multiple of 5.
Which of the following sets have the closure property for addition?
{7,8,9,10,11,…} Closed, Two numbers bigger than or equal to 7 will give a number bigger than or equal to 7
{7,8,9,10}
Not Closed, 7+8=15
{2,4,8,16,…}
Not Closed, 2+4=6
Visual Models for Addition
There are two categories for how we model addition, one is used when the
problem uses numbers that represent a count, the other when the numbers
represent a measure.
Combine Sets (count) What addition problem does this represent?






5
+


=

3
=





 

8
What addition problem is represented by the following?
4
3
4+3=7
7
Combine Measures (measures)
Number line - What addition problem is represented by the following?
2+5=7
0
1
2
3
4
5
6
7
8
Rods - What addition problem is represented by the following?
2+3=5
2
3
Subtraction of Whole Numbers
Subtraction of whole numbers is characterized in terms of addition.
For any whole numbers a and b with a  b, we say that a – b = c if and only if
a = b + c for a whole number c. The number c is called the difference of a
and b.
The concept of subtraction of whole numbers is illustrated with two ideas “the
whole” and “the part”. See the examples below.
whole
whole
?
a
b
c
b
?
part
part
part
part
b+c=?
a–b=?
Visual models for Subtraction
Like addition visual models for subtraction are based on either the numbers
being used as a count or a measure. I will mention three types of them here,
take away, comparison and missing part.
Take Away Sets
What is the subtraction problem illustrated here?
5–2=3
A better way to illustrate this although it is dynamic (hard to represent
on paper) is the following:
5–2=3
Take Away Measures
What subtraction problem is represented below ?
8–6=2
0
1
2
3
4
5
6
7
8
6–2=4
A more dynamic way to show this is:
6–2=4
Compare Sets
This is used to see how much bigger one set is than another. What is the
problem illustrated below?
9–4=5
Compare Measures
What problem is represented by the following?
7–3=4
Missing Parts
This can be used for both sets and measures. What problems are illustrated by
each of the following?
Sets
Measures
6–2=4
9–4=5