Sums and Differences

Models for Addition Algorithms

 An algorithm is a step-by step procedure for computing. Algorithms for addition involve two separate procedures: ( 1 ) adding digits and ( 2 ) regrouping, or "carrying," so that the sum is written in positional numeration. the two numbers being added are called addends and the answer is called a sum.

Example:

            17 + 32 = 49

            addends   sum

Addition Algorithms

Left-to-Right

To compute 792 and 747 first add 7 and 7. Then add 9 and 4, because of regrouping 4 is scratched out and replaced by 5, then add 2 and 7.

                               792                792              792

                             +747              +747            +747  

                             14                   143             1439

                                                      5                 5

Partial Sums

In this method, the digits for each place value are added, and the partial sums are recorded before there is any regrouping.

Examples:

                     1.      476                         2.       4 hundreds  +   7 tens   + 6

                            +447                                   4 hundreds  +   4 tens   + 7

                                13                                   8 hundreds  + 11 tens   + 13

                              11            Regrouping      9 hundreds  +   2 tens   +   3

                              8                     

                              923                                = 923

Right-to-Left

(aka the traditional algorithm)

Example:

                             738

                           +295

                           1033

Number Properties

Closure Property for Addition

For every pair of numbers in a given set, if an operation is performed, and the result is also a number in the set, the set is said to be closed for the operation. If the operation does not produce an element of the given set, then the set is not closed for the operation.

Identity Property for Addition

The number zero is called identity for addition because when is added to another number, there is no change.

Examples:

                   1.    0 +0 = 0          2.     17 + 0 = 17              3.    0 + 5 = 5

For any whole number a,  0 + a = a + 0 = a

Associative Property for Addition

In any sum of three numbers, the middle number may be added to ( associated with ) either of the two end numbers.

Example:

                  147 + ( 20 + 6 ) = ( 147 + 20 ) + 6

For any whole number a, b, and c,   a + ( b + c ) = ( a + b ) + c

Commutative Property for Addition

When two numbers are added, the numbers may be interchanged ( commuted ) without affecting the sum.

Example:

                  257 + 498 = 498 + 257

Here is a video that might be helpful with number properties.

"Number Properties Video"

                                     Models for Subtraction Algorithms
Subtractions Concepts and Algorithms

Subtraction is usually explained as the taking-away of a subject of objects from a given set. Subtraction and addition are inverse operations. There are three concepts of subtraction: the take-away concept, the comparison concept, and the missing addend concept.

Take-Away Concept of Subtraction

                                                

                                         X X X X X   -   XXX   =   XX

                                     Take-away concept showing  5 - 3 = 2

Comparison Concept of Subtraction

Compare one set to another to determine the difference.Compare and see how many more the set of 12 has than the set of 8.

                                                                             4

                                          X X X X X X X X X X X X

                                          X X X X X X X X

                                     Comparison concept showing  12 - 8 = 4

Missing Addend Concept of Subtraction

How many more are needed?

                                               

                                       12 - 7 = ____             12 - ____ = 5

Algebra Den
This web-site offers detailed information about addition, subtraction, multiplication, and division.
Most of the information on this blog was taken from the text book: Mathematics for Elementary Teachers and from other sources.