DICKINSON STATE UNIVERSITY
DICKINSON ND
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ADDITION OF WHOLE NUMBERS
Introduction
The four fundamental operations of arithmetic; addition, subtraction, multiplication
and division form the foundation for solving real life problems. For example, the
following problem utilizes three of the four basic operations:
Four fifth graders decided to buy a large pizza costing $8.95. They pooled their
resources. Jill has $3.20, Betty has $3.00, Amy $3.25 and Mary $2.80. They
tipped the waitress $2.00. They then bought four ice cream cones at 25 cents each.
How much money is left?
In this page we will examine only the addition of two whole numbers focusing on
its basic properties. Addition is the first operation the child encounters. Hence,
an understanding of this operation is very crucial toward understanding the other
three operations. This understanding involves not only the question when
to add, but also the question how to add.
A correct answer of the question when to add can only follow from a carefull
analysis of the problem like the the one that is shown above. We believe that the
only effective way to learn such analysis must involve solving an adequate number of
word problems of increasing complexity.
Because of the nature of our enterprise we will discuss in more details only the
second question, that is how to add. The following are some of the common
models used to arrive at the answer.
The Set Model
The basic meaning of addition is that of combining two distinct smaller sets of
certain objects to form a larger set of these objects, and then to determine
how many of them is in this larger set. For example, Jimmy has four marbles and his
sister gave him five more. How many marbles does he have now?
The important advantage of the set model is that it can be illustrated with
simple and explicit diagrams easy for children to understand. Such a diagram for
the example with marbles is shown in Fig. 1.
Fig. 1
To find how many marbles Jimmy has altogether, we combine the two sets to form
a larger set. We can then use counting or "counting on" to find the answer. First
the child uses counting to find the answer. He or she starts at one and continue
counting until all the marbles are counted. Later he or she uses "counted on" in
which the child starts counting on with the second set as follows: the child says
five (the first set has four elements), six, seven, eight, nine. And this is the
answer.
In the next stage of development the child uses pictures to represent the actual
objects to find the answer. He or she may draw the picture of marbles in the two
sets and then combine them creating a diagram similar to Fig. 1.
Our Single Digit Addition Applet is designed to visualize
the process of addition. The child can create there the two sets of squares of the
different colors in the separate two rows. An example of it related to the marbles
problem is shown in the upper part of Fig. 2. To add these sets, the squares from one
row should be moved to the other row. The transfer can be continued as long as there
are enough squares in the row loosing its squares or the accepting row is filled.
Notice that only the squares on the right ends of the rows can be moved. Finally
the single set which may extend on both rows is created.
Fig. 2
For the initial sets with more than ten elements you may use the Double Digit
Addition Applet which is a little bit more sophisticated. There the elements
of sets are in the packages of tens or even hundreds if possible.
Both applets support the decimal system of numbers for very obvious reason.
The final stage is for the child to understand the formal definition of addition.
We can use set notation to do this. We let n(A) = a, the number of
objects in the first set represented by the letter A and let n(B) =
b, the number of objects in the second set represented by the letter B.
Then the number of objects in the two sets is n(A U B) or a + b.
Thus, for any two whole numbers a and b
a + b = c.
The numbers a and b are called addends or summands and c is
called the sum. The symbol U used above means the sum of sets. As long as these
sets are disjoint (do not have any common elements) the sum of two sets is made of
elements of the first and the second set.
Children should be given ample opportunities using concrete objects, pictures
and diagrams and a wide variety of activities to get a firm understanding of
the meaning of addition.
The Whole Number Line Model
Another model commonly used to find the sum of two numbers is the whole number
line model. A whole number line is constructed by a solid horizontal line with
an arrow pointing in the right direction indicating that the whole numbers are
unending. This is illustrated below in the upper part of Fig. 3.
Fig. 3
To find the answer to the marbles problem, we draw an arrow four units long
starting at zero (red) and another arrow five units long (blue) from where the
first arrow ends. We now have two arrows end to end. Where the second arrow ends
will correspond to a number which gives the answer. In our problem this corresponds
to the nine.
This model, however, is more abstract. Its generalization to the real number line
model is very useful in the more advanced math.
The Formal Properties of Addition for Whole Numbers
- Commutative Property
For any whole numbers a and b , a + b = b + a
This means that if a child knows that 6 + 4 = 10, then he or she also
knows that 4 + 6 = 10. Notice that the commutative property is so very natural
not only for children but also for adults that if you try to teach them about it,
they may not understand what you are talking about. When teaching advanced math on the
university level it is very often difficult to convince the students that certain
operations on vectors or more sophisticated objects do not have this property.
If we assume that the addition table shown in Fig. 6 contains the 100 basic facts
of whole number addition to be mastered by novices, the comutative property reduces
learning to only 55 basic facts.
- The Identity Property
For any whole number a, there is a number 0 called the identity element
such that a + 0 = 0 + a = a
This means that if zero is added to any whole number, the number remains the same.
Again, for nonmatematicians this property is very natural. It boils down to the
simple observation which in the every day language says that if you add "nothing"
to the given whole number, this number stays unchanged.
- Associative Property
For whole numbers a, b, and c (a + b ) + c = a + (b + c).
This property is also very natural and should be used with children without any
formal introduction. It simply means that it does not matter if you first add
a to b and the result of it to c or you first add b
to c and the result of it to a. The formal introduction of the
property on this level will most probably cause a lot of confusion. Otherwise
it helps children to find sums such as 9 + 8 as follows:
9 + 8 = 9 + (1 + 7) = (9 + 1) + 7 = 10 + 7 = 17,
or 8 + 7 as:
8 + 7 = 8 + (2 + 5) = (8 + 2) + 5 = 10 + 5 = 15.
Thus, if a child forgets any basic fact, he or she can use implicitly the
associative property to arrive at the answer as shown in the above examples.
All the properties of whole number addition are very natural for people. They
should be used implicitly to help the novices to understand and master addition
and later subtraction operations. Their formal formulation and formal use should
be posponed untill the students have matured enough in math to understand some
abstract ideas, otherwise we will create a lot of confusion and unnecessary
problems. Further implicit and informal demonstrations of these properties with
help of the Single Digit Addition Applet can be found below.
Implicit Demonstrations of Addition Properties
With the help of the Single Digit Addition Applet (or any other suitable device) we
can demonstrate the commutative property without any formal definition or even
naming it. Adding 6 + 4 and 4 + 6 we obtain the two results shown in
Fig. 4. These results are obviously identical because the color arrangements do
not have any influence on the final number of elements in the upper row of the
applet.
Fig. 4
In Fig. 5 the two stages of the problem 9 + 8 are shown. They can be
interpreted in the terms of the associative propert as
9 + 8 = 9 + (1 + 7) = (9 + 1) + 7 = 10 + 7 = 17
Fig. 5
The Addition Table
The principle of the addition table is simple. If you want
to add two numbers, localize one of them in the left most column (white
background with red numbers) and move along the row that starts there.
When you reach the square (green bcakground with the black numbers) which is
immediatly below your second number localized in the top row (white background
with red numbers), you have found the answer. This is really the count on
method. Adding 5 + 4 you start at the red five in the most left
column and count six, seven, eight, nine, moving after each count to the next
green square. When your count is nine, you are straight below the red four in
the top row. This means, you have added 4 to your first number 5.
The addition table is shaped by the addition properties. At the intersection of
the fifth row and the fourth column there is nine. This is because 5 + 4 = 9.
But 4 + 5 due to the commutative property of addition is also 9.
Consequently at the intersection of the fourth row and the fifth column we
have 9 too. It works the same way for the other pairs of numbers.
Fig. 6
Take a diagonal of the addition table that starts at a certain number in
the left most column and ends at the same number in the top row. All numbers
along this diagonal are the same, and equal to the starting number. This is the
result of the associative property. For example, starting at 4 in the
left most column and using this property we can write
4 = 3 + 1 = (2 + 1) + 1 = 2 + (1 + 1) = 2 + 2 =
(1 + 1) + 2 = 1 + (1 + 2) = 1 + 3 = 4.
Notice, after each application of the associative property we are moving one
step up along this diagonal. Thus, if the child knows that 2 + 2 = 4
and has the ability to apply implicitly the associative property, he or she
can find how much is 3 + 1. Acquiring such ability demands, of course,
some prctice. But it will pay off hansomely because these kind of operations
are used very often in math. Such practice does not even demand use of any
parenthesis. Then, what we have done more formally above, can be replaced by
the following simplified form
4 = 3 + 1 = 2 + 1 + 1 = 2 + 2 = 1 + 1 + 2 = 1 + 3 = 4
that is more practical for a mental exercise. Later more advanced applications
like
8 + 7 = 8 + 2 + 5 = 10 + 5 = 15
should be practiced. Generally, if the sum of two single digits a and b
is greater than ten, take the smaller number and replace it with two numbers such
that when one of them is added to a or b it gives ten. Now, add
the other number to ten and you have the final result. This method also exploits
the fact that in the decimal system of numbers, the addition of any single digit
number to ten is very easy. Our Single Digit Addition Applet also uses this fact.
Note in the addition table that all numbers on the main diagonal (the one that
runs from the upper left hand corner to the lower left corner) are
all doubles. They were created by the additions 1 + 1, 2 + 2, ... .
The results of such additions can be quickly learned by counting by twos. For
example adding 4 + 4, one counts in the four steps: two, four, six,
eight. And this is the result. It works like taking the one from the first
4 and another one from the other 4 and adding them together.
Next, you do it three more times acumulating all units taken from both fours.
Moving down by a single square along any other diagonal which is parallel to
the main diagonal, increases the number in the square by two. Simultaneously
each of the addends increases by one. Thus, if child already knows that
5 + 6 = 11 he or she should be able to find out quickly that 6 + 7 =
13. The same result can be found, of course, using implictly the associative
property:
6 + 7 = 3 + 3 + 7 = 3 + 10 = 13.
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