Chapter 3

Whole NumbersSection 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 AB=Ø, then a+b is the number of elements in AB. 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.

____ Find the sum of 723468+164501

(a) 887909.1476 (b) 887969 (c)

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.

164501

723468

b

Which of the following sets have the closure property for addition?

{7,8,9,10,11,…}

{7,8,9,10}

{2,4,8,16,…}

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?

Closed, Two numbers bigger than or equal to 7 will give a number bigger than or equal to 7

Not Closed, 7+8=15

Not Closed, 2+4=6

=

5 + 3 = 8

What addition problem is represented by the following?

4 + 3 = 7

4 3

7

Combine Measures (measures)

Number line - What addition problem is represented by the following?

0 1 2 3 4 5 6 7 8

2 + 5 = 7

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

part part

b

?

c

b + c = ?

whole

part part

b

a

?

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 ?

0 1 2 3 4 5 6 7 8

8 – 6 = 2

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

6 – 2 = 4

Measures

9 – 4 = 5