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.

Mental Calculations and Estimation of Sums and Differences

Mental Calculations

Mental calculations help us to solve or to find answers fast. In order to accomplish mental calculations we need to combine several abilities: the skills to understand and use algorithms, place values, and number properties. There are various procedures to perform mental computations.

Compatible Numbers

Find pair of numbers whose sum or difference is easy to compute.

Example:  

              17 - 12 + 43 = 17 + 43 = 60 - 12

Substitutions

A number is broken down into a convenient sum or difference of numbers. Here are some ways to compute the sum 127 + 38.

                              127 + (3 + 35) = (127 + 3) + 35 = 130 + 35 = 165

                              127 + (30 + 8) = (127 + 30) + 8 = 157 + 8 = 165

                              (125 + 2) + 38 = 125 + (2 + 38) = 125 + 40 = 165

Equal Differences

The difference between two numbers is unchanged when both numbers are increased or decreased by the same amount.

Example:  11 - 7 = 4

                                    11 + 9 = 20

                                      7 + 9 = 16   

                                                         20 - 16 = 4

Add-Up Method

Add up from the smaller to the larger number. 

         53 - 17

From 17 to 20 is 3, and from 20 to 53 is 33. S the difference is 3 + 33 = 36.

Estimations of Sums and Differences

The three most common techniques for estimating are rounding, using compatible numbers, and front-end estimation.

Rounding
An approximate sum or difference. The type of problem will determine to what place value the numbers will be rounded.
       1.    624 - 289 - 132 »
                                      600 - 300 - 100 = 200

       2.    4723 + 419 + 1040 »
                                         5000 + 400 + 1000 = 6400

       3.    812 - 245 »
                            800 - 200 = 600

Compatible Numbers
Sometimes a computation can be simplified by replacing one or more numbers by approximations in order to obtain compatible numbers. For example, to approximate 342 + 250, we might replace 342 by 350.

                                            342 + 250 » 350 + 250 = 600

Front-End Estimation
Is similar to left-to-right addition, but involves only the leading digit of each number.
Example:
                 433 + 684 + 288 » 400 + 600 + 200 = 1200

Basic Math Skills

Most of the information on this blog was taken from the text book: Mathematics for Elementary Teachers.