Addition Preliminaries

Physical Interpretation

Addition Via Counting

To illustrate this, take for instance two bags of marbles. Say one bag has 13 marbles and another 15. Ask your child to put the two bags together, and count how many there are. Now take two bags, one with 13 buttons and another with 15 buttons. Ask your child again to count how many there are. This shows the total number of objects does not depend on their type or kind. Redo this addition experience with varying numbers.

Addition via Counting may lead to the additon table

+	1	2	3	4	5	6	7	8	9
1	2	3	4	5	6	7	8	9	10
2	3	4	5	6	7	8	9	10	11
3	4	5	6	7	8	9	10	11	12
4	5	6	7	8	9	10	11	12	13
5	6	7	8	9	10	11	12	13	14
6	7	8	9	10	11	12	13	14	15
7	8	9	10	11	12	13	14	15	16
8	9	10	11	12	13	14	15	16	17
9	10	11	12	13	14	15	16	17	18

If you do or review it with a student, observe how adding n+1 to number is one more than adding n to the number. That provides a mechanism for filling in the table by rows (horizontally from right to left) and by columns (vertically from top to bottom).

Students should know the sums of all pairs of digits from 1 to 9, as well as how to add 0 and 10 to a single digit number. Filling in the addition table is a good exercise.

Addition with Decimals - How to Justify

Besides column methods for addition shown below, there are column methods for subtraction, multiplication and even long division. All methods, some more directly than others, take advantage of decimal place value.

Single to Triple Digit Examples - with carries

Provide the following experiences (and similar ones).

By counting 3 ones added to 4 ones is 7 ones. On paper this can be written in the shorthand form

                3
              + 4
            ----------
                7
            ----------

By counting 8 ones added to 7 ones yields 15 ones. On paper this can be written in the shorthand form

                8
              + 7
            ----------
               15
            ----------

By counting, 12 ones added to 25 ones is 37 ones or three tens and 7 ones. On paper this can be written in the shorthand form

   12         (read backwards: 2 ones plus 1 ten )
 + 25         (read backwards: 5 ones plus 2 tens)
 ------     added together gives
   37                           7 ones plus 3 tens
 ------

(Carrying Illustrated). Now lets try 144 plus 85 without counting. On paper this can be written in shorthand form and longhand form as follows.

 244      ( 4 ones plus 4 tens plus 2 hundreds )
  85      ( 5 ones plus 8 tens)
 -----     added together gives
           9 ones plus 12 tens plus 2 hundreds
 -----

But the 12 tens is the same as 2 tens plus 1 hundred. Thus replacement suggests the result

 

                ____________________________________\_
                                                    / 

                                                          

  244          (read  2 ones plus 4 tens plus 2 hundreds ) |
   85          (read  5 ones plus 8 tens)                  |
----------     added together gives                        |
  329           9 ones plus 12 tens plus 3   hundred       |
---------    or 9 ones plus  2 tens plus 2+1 hundreds      |
  1          or 9 ones plus  2 tens plus 3   hundreds     \|/


               /____________________________________________
               \  

shorthand   +      Longhand Representations, respectively, 
                of the addition

The one in the last row denotes a carry. The on the left and the words on the right represent the same number. Here the shorthand expressions requires less work to write, but its justification requires a knowledge of the longhand form. In the following examples, we write the shorthand form, then do the calculation with the help of the longhand representation and lastly return to the shorthand form. With practice, the long representation and explanation of the shorthand will be understood, and it need not be written down. The calculation is then done and represented via its shorthand representation.

Examples:

Addition without Conversions (Carries)

Steps

add units
add tens
add hundreds

Conclusion: 243 + 452 = 695.

Addition with Conversions (Carries)

Steps

Add units: 9 and 6 give 15 = 5 + 10 or a 5 in the unit column with one ten to carry over into the ten's column.
Add tens: 5 + 2 + the 1 from the carry give 8.
Add hundreds: 4 + 8 hundreds give 10 + 2 hundreds or 2 in the hundred column with a one thousand to carry over to into the thousands column.
Add thousands. the carry of 1 + 2 + 4 = 7. Done
Conclusion: 2459 + 4826 = 7285.

Number Theory

A. Start of Number Theory

Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions

B. Number Theory
Continued

Decimal Place Value
Place Value Reinforcement
Addition Method
Comparison Method
Subtraction Methods
Multiplication Methods
Division Methods
Long Division Continued
Remainder Arithmetic I
Primes & Composites
Primes Factorization Theorem
Primes & Composites
Prime Factorization Examples
Counting Whole No. Factors
Prime Factorization Aids
Square Roots & Primes
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Infinite Decimals Expansion Arithmetic
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II

Related Site Folders

Euclidean-Geometry/Complex No.s
Complex Numbers More 2

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