Symbols Representing Numbers

Numeration Systems

The meaning of the words numeral, numeration, base, additive, and positional are learned in the numeral system. Some of the numeral systems are the Egyptian, Roman, Babylonian, Mayan, and Hindu-Arabic. All these systems are represented by different symbols. The Web-site wichita.edu contains a lot of information about the numeral systems and their symbols along with the history of each one of them.
Printed or written symbols represent numerals, and an organized collection of numerals is called a numeration system. Base is the number of objects used in the grouping process. The Mayan and Hindu-Arabic numeration systems are positional numeration systems. In a positional numeration system, the position of each digit indicates a power of the base. The Egyptian and Roman are additive numeration systems which means that each symbol is represented as many times as needed.

Egyptian Numerals

                       1,000,000            100,000              10,000            1000              100              10                 1

"The Egyptian is an additive numeration system." (Mathematics for Elementary Teachers ).

Reading and Writing Numbers

Whole numbers from 1 to 20 have single word names. The names for the numbers 21 to 99 with the exceptions of 30, 40, 50, etc., are hyphenated names. Numbers with more than three digits are read by naming the periods.

Example:

                154,         209,          100

            Million   Thousand   Hundred

One hundred fifty four million, two hundred-nine thousand, one hundred.

Rounding Numbers

There are some rules for rounding numbers.

1.   Locate the digit with the place value to be rounded, and check the digit to the right.
2.  If the digit to the right is 5 or grater, then each digit to the right is replaced by 0 and the digit with the given value is increased by 1. 
3.  If the digit to the right is 4 or less, each digit to the right of the digit with the given place value is replace by 0.

Example:

Rounding 375,296,588 to the million, ten thousand, and thousand place values.

Million place

               375,000,000

Ten thousand place

               375,300.000

Thousand place

               375,297,000

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



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.                                                   

Models and Properties of Products

Multiplication of Whole Numbers

For any whole numbers r and s, the product or r and s is the sum with s occurring r times. This is written as

                                    r x s = s + s + s + . . . + s

                                                     r times

If r ¹ 0 and s ¹ 0, r and s are called factors.

Models for Multiplication

Repeated Addition

Three groups of 6 to illustrate 6 + 6 + 6.

Rectangular Array

Shows the figures pushed together to form a 3 x 6 rectangle.

Tree Diagram
A tree diagram is useful for certain types of multiplication. Here is a learning about making tree diagrams video: Tree Diagram Video

Partial Products
When a two-digit number is multiplied by a two-digit number, there are four partial products.
Example:
                                17
                             x 13
           Partial          21   (3 x 7)
         products         30   (3 x 10)                             
                                70   (7 x 10)
                              100   (10 x 10)
                              221

                                          Number Properties

Closure Property of Multiplication

For any whole numbers a and b, a x b is a unique whole number  (the product of any two whole numbers is also a whole number).

               

                      (whole #) (whole #) = whole #

                      (even #) (even #) = even #

                      (odd #) (odd #) = odd #

Identity Property for Multiplication

For any whole number b, 1 x b = b x 1 = b, and 1 is a unique identity for multiplication (when number 1 is multiplied by another number, it leaves the identity of the number unchanged).
Example:

              1 x 7 = 7                   14 x 1 = 14                  1 x 0 = 0

Commutative Property for Multiplication

For any whole numbers a and b, a x b = b x a (any product of two numbers may be interchanged (commuted) without affecting the product.

Example:

                347 x 26 = 26 x 347

Associative Property for Multiplication

For any whole numbers a, b, and c, a x (b x c) = (a x b) x c (in any product of  three numbers, the middle number may be associated with and multiplied by either of the two end numbers.

Example:

                   2 x (3 x 10) = (2 x 3) x 10 = 60

Distributive Property for Multiplication over Addition

For any whole numbers a, b, and c, a x (b + c) = a x b + a + c.

Example:
                 7 x 8 = 7 x (7 + 1) = 49 + 7 = 56
                                Distributive property

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

Mental Calculations and Estimation of Products

Mental Calculations

Computative numbers and substitution/distributive property (numbers that are easy to calculate).

Compatible Numbers
Rearrangement of numbers.
                         1.         5 x 346 x 2 = 5 x 2 x 346 = 10 x 346 = 3460
              
                         2.         2 x 25 x 5 x 2 = 2 x 2 x 25 x 5 = 100 x 5 = 500
                                          
Substitutions

Distributive property is useful for mental calculations of products.

Example:

               14 x 102 = 14 x (100 + 2) = 1400 + 28 = 1428

                                            Distributive property

Equal Products

This method is based on the fact that the product of two numbers is unchanged when one of the numbers is divided by a given number and the other number is multiplied by the same number.

Example:

              12 x 52 =

                          12/2 x (52 x 2 ) = 

                                                   6 x 104 = 624 

Estimation of Products

Rounding

Products can be estimated by rounding one or both numbers. 
  
                         1.      71 x 58 » 70 x 60 = 4200

                               

                         2.      205 x 29 »  200 x 30 = 6000

                                          or » 210 x 30 = 6300

Compatible Numbers

Estimation of products combined with mental calculations become a useful tool when using compatible numbers. For example, to estimate 4 x 237 x 26, we might replace 26 by 25 and use a different ordering of the numbers.

                4 x 237 x 26 » 4 x 25 x 237 = 100 x 237 = 23,700

Front-End Estimation
Is related to rounding. keep the leading digit and everything else becomes a zero. For example, to estimate 62 x 83, keep the leading digits 60 x 80 so the estimated product is 4800.

Example:
               62 x 83 » 60 x 80 = 4800
Kids Olr  is a good website to collect information about math.             
Most of the information on this blog was taken from the text book: Mathematics for Elementary Teachers and from other sources.

Division and Exponents

Division

Multiplication and division are inverse operations. We arrive at division facts by knowing multiplication facts. There are three terms to describe division: dividend, divisor, and quotient.

    

                                         12   ¸   4   =   3

                                     dividend        divisor     quotient

Division of Whole Numbers

For any whole number r and s, with  s ¹ 0, the quotient of r divided by s, written    r ¸ s, is the whole number k, if it exists, such that r = s x k.

Example:

               18 ¸ 3 = 6 because 3 x 6 = 18

Please visit the website Itcoline.net if you want any extra information about this subject.

True and False Statements:

         A.  5 ¸ 0 = 0  false since  0 x 0 ¹ 5

         B.  0 ¸ 5 = 0   true since   5 x 0 = 0

         C.  0 ¸ 0 = 0   false since  0 x ? = 0   0 ¸  0  =  any number   many answers (not a unique solution).



Concepts of Division

There are two concepts of division: The sharing (partitive) concept and the measurement (subtractive) concept.

Sharing/partitive

Illustrating the sharing/partitive concept of division. Suppose you have 30 colored chips, which you want to divide among 6 friends. How many colored chips would each friend receive?
          30 ¸  6 = 5

                       1    2    3    4    5    6

                        I     I     I     I     I     I                      
                        I     I     I     I     I     I          

                        I     I     I     I     I     I         five each      
                        I     I     I     I     I     I

                        I     I     I     I     I     I

Measurement/subtractive

Illustrating the measurement/subtractive concept of division. Suppose you have 30 colored chips and want to give 6 colored chips to as many friends as possible. How many friends would receive colored chips?

            30 ¸ 6 = 5

                              1   I I I I I I

                              2   I I I I I I

                              3   I I I I I I          five friends

                              4   I I I I I I

                              5   I I I I I I

                                 
 Division Theorem

For any whole numbers a and b, with divisor b ¹  0, there are whole numbers q (quotient) and r (remainder) such that
                                                divisor

                                         a  =   bq   +   r remainder
                                     dividend      quotient

and 0 £ r < b.

Example:

               5 x 2 + 2 = 12    «    12 ¸ 5 = 2 r 2

The remainder r is always less than the divisor b. If r = 0, then the quotient a ¸ b is the whole number q.



Mental Calculations and Estimation

Equal Quotients

Multiply or divide both the dividend and divisor by the same number.

Examples:

           1.    400 ¸ 16 = (400 ¸ 4) (16 ¸ 4) = 100 ¸ 4 = 25

           2.    77 ¸ 21 = (77 ¸ 7) (21 ¸ 7) = 11 ¸ 3 = 3 r 2


Compatible Numbers

Using compatible numbers is useful for mental estimating quotients (round to easy numbers).

Example:

              92 ¸ 9 » 90 ¸ 9 = 10

Rounding
Rounding is one method of estimating a quotient.

             145 ¸ 23 » 150 ¸ 25 = 6

Front-End Estimation
This technique can be used to obtain an estimated quotient of two numbers by using the leading digit of each number. The following example use both the compatible numbers and front-end estimation.

             783 ¸ 244  » 700 ¸ 200 = 7  ¸ 2 = 3 ½

Exponents

Exponentiation
The operation of raising numbers to a power is called exponentiation. A number in the form bⁿ is said to be in exponential form;  bⁿ  is called the nth power of  b. For any number b and any whole number n, with b and n not both zero, 

                                          bⁿ  = b x b x b x b x b x . . . x b     

                                                         b occurs n times

 Where  b  is  called  the  base  and  n  is  called  the  exponent.  in case  n = 0  or  n = 1,  b⁰ = 1 and  b¹ = b.

                                                                                                               

Laws of Exponents

For any number a and all whole numbers m and n, except for the case where the base and exponents are both zero,

              aⁿ x a^m  = a^{n + m}

                 aⁿ ¸ a^m  = a^{n - m}         for a ¹ 0

For more information about law of exponents go to the Oak Road Systems website.

Evaluating Exponential Expressions

Multiplication and division can be performed easy with numbers that are written as powers of the same base. To multiply, we add the exponents; to divide, we subtract the exponents.

Examples:

            1.    3²² x 3⁸ = 3³⁰

            2.    7³ x 7⁸ = 7¹¹

               3.    4⁶ ¸ 4² = 4⁴

               4.    10³²¸  10¹⁵ = 10¹⁷

For repeated multiplication we use exponents:

                                           3⁵ = 3 x 3 x 3 x 3 x 3

                                           6⁴ = 6 x 6 x 6 x 6

Order of Operations

Parentheses (or grouping symbols) first;  then numbers raised to a power (exponents) are evaluated; next products and quotients (multiplication/division) are computed from left to right; finally sums and differences are calculated from left to right.

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