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    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
  1. 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.
  2. 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.
  3. 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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