Algebra For Kids

The word "algebra" derives from the Arabic al-jabr meaning "the reunion of broken parts." Algebraic concepts were used as early as 1650 B.C. in Egypt and Ancient Babylon, and were brought into Europe by the Arabs. And now, thousands of years later, judging by the mail in my inbox, it is a subject that many students need help with. Learn more at the following sites.

Algebra 1: Graphing Linear Equations

There is a lot of interactive material buried within this Java-enhanced site. Across the top of your window, you'll easily find the graph-your-own-functions plotter and the algebraic calculator. But down the left-hand side are lots of goodies hidden from sight until you move your mouse over the menu buttons. For example, under "Assess" you'll find quizzes, tests and a not-to-missed project graphing voter turnout in the most recent presidential election. Click on "Help Desk" for the bulletin board, algebra chat room, and complete algebraic glossary. Actual algebra lessons can be found under "Topics".

Algebra - Fun with Calendars

Want to amaze your friends and family? Start with a monthly calendar and a volunteer from the audience. Ask your volunteer to pick 4 days that form a square such as March 17, 18, 24 and 25. Your volunteer should tell you only the sum of the four days (for example 84), and you'll be able to tell her which four days she picked. How is it done? With algebra, of course!

Girls to the Fourth Power: Algebra Program

What a scream! Meet Gwynn and Alison, two sisters doing the mall "California style" each trying to outspend the other. How much do they have to spend? Well, Gwen got a $50 present from her uncle, and managed to borrow $20 from Dad. Alison squeezed a birthday advance out of Mom. Then Dad went on a business trip and bought Gwynn a really nice present, but only bought Alison a crummy teddy bear. How are they ever going to figure it all out? It's Al-Jabra to the rescue!

Article Source: http://EzineArticles.com/?expert=Barbara_Feldman

some skills in division

to divide any number by 50 = multiply it by 2 then divide by 100 

ex : 150/50= 150 * 2 = 300 then 300 /100 = 3 

to divide any number by 500 = multiply it by 2 then divide by 1000

ex: 1500/500= 1500*2 = 30000 then 30000/1000 = 3

to divide any number by 5 = multiply it by 2 then divide by 10

ex :15/5 = 15*2 = 30 then 30/10 = 3 

to divide any number by 25 = multiply it by 4 then divide by 100 

to divide any number by 250 = multiply it by 4 then divide by 1000

to divide any number by 75 =  divide it by 3 then multiply it by 4 then divide by 100 

ex : 150 / 75 = 150/3=50 then 50 *4 = 200 then 200 / 100 = 2

fraction problem

Find the rational number which equal 3/5 and sum of its numerator and dominator equal 24

solution :

we know that the fraction value not change if we multiply its numerator and dominator by any similar number for instance 3 / 5 = 3 * 2 / 5 * 2 =6 / 10.

now let the rational number = x/y 

then x/y = 3/5 so by using the blue property above x/y = 3n / 5n

 ( n is const) so x = 3n and y = 5n 

we know that numerator and dominator equal 24 so x + y = 24 so 3n + 5n = 24 so 8n = 24 so n = 3

then x = 3*3 =9 and y = 3 *5 = 15 then the number is 9/15

Adding Fractions with Different Denominators

there are many ways to adding Fractions with Different Denominators.all of them are about making Denominators similar

to adding Fractions with Different Denominators :

• Find the Least Common Denominator (LCD) of the fractions
• Rename the fractions to have the LCD
• Add the numerators of the fractions
• Simplify the Fraction if possible

example 5/6 + 3/2

determine the greatest common factor of 2 , 6 which is 6

divide the GCF by denominators and multiply by numerator

6/6 = 1 , 1 * 5(the numerator ) = 5 then 5/6 = 5/6 (Least Common Denominator)

6 / 2 =3, 3* 3 (the numerator ) = 9 then 3/2 = 9/6 (Least Common Denominator)

5 / 6 + 3/2 = 5/6 + 9/6 = 14/6 = 7/3

other solution 

we can solve this by other easy way ( right  if we adding two fractions only)

find the common Denominator by multiply two dominator by each other 

2 * 6 = 12

multiply the dominator of the first fraction by numerator of the second fraction and multiply the dominator of the second fraction by numerator of first fraction 

6 * 3 = 18 and 2 * 5 = 10 then 18/12 + 10/12 = 28/12 = 7/3

easy way to solve equations

equation is very important concept in our live.because we use it in every time.some of children find it hard to understand ,so it is an easy way to understand it . balance scale is very useful tool to understand it.

in the picture above 12g + 50g = 62g so 

 Xg  + 50g = 62g  then X = 12g . 

example :

2g + Xg = 10g so if we put 2g + Xg in side and 10g in the other side in the balance scale then to make balance X should be 8g

  

cartesian product

  it is said that (a,b) is ordered pairs ,a is called first coordinate and b is called second coordinate. 

let (a,b) and (c,d) are ordered pairs then

 (a,b) =  (c,d) if and only if a = c and b = d 

 (a,b) ≠  (c,d) if and only if a ≠ c and b ≠ d 

cartesian product :
let A and B are two sets then cartesian product for A and B is the set of all ordered pairs (a,b) so that a ∈ A and b ∈ B ie A x B = {(a,b) :a ∈ A and b ∈ B }. If number of elements of A = n and number of elements of B = m then number of elements of A x B = n x m 

example : A ={ 1,2,3 } , B = {x , y } find A x B and B x A

numbers of elements of A = 3 and numbers of elements of B = 2 then numbers of elements of A x B and

 B x A = 3 x 2 = 6 elements 

 A x B = { (1,x),(1,y),(2,x),(2,y),(3,x),(3,y) }

 B x A  = { (x,1),(x,2),(x,3),(y,1),(y,2),(y,3)}

basic property :

1 - A x  ∅ = ∅
to prove this :
 let ( x ,y) ∈ A x  ∅ so x ∈ A and y ∈ ∅ this is contradiction so A x  ∅ = ∅

2- A x B ≠ B x A
to prove this let x ∈ A x B then y ∈ B because B ≠ ∅  then (y ,x) ∈ B x A  then A x B ≠ B x A

3- A x B = ∅ then A = ∅ or B = ∅

4- A x (B ∪ C) = (AxB) ∪ (AxC)

to add 1 to n number

n(n+ 1)

this formula helps us to adding 1 to n number easily

for instance 1 + 2 + 3 + 4 + 5+ 6  +7 + 8 + ..... + 25 = 325

by use formula

n = 25

25 * 26 / 2 = 325

this formula is can be work with it in the big sequence start from 1

i think now it is very easy 

easy way to add numbers 1 to 100

I will ask you what is the mean for values from 1 to 100 ?this is very easy question .

the mean = Sum of the values / how many numbers then  

but now the problem is to get the Sum of the values from 1 to 100 ,to do this by easy way write half of 100 twice then read the new number. in this example 100 / 2 = 50 then the Sum of the values from 1 to 100 = 5050 

so the mean = 5050 / 100 = 50.5

example : 1 + 2 + 3 + 4 + ...... + 1000 = 500500 because 1000/2 = 500 then the sum = 500500

you can do this with 10000 , 100000 , .........................

zero makes the numbers like mercury

We know that the decimal system consists of 0 ,1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 ,9. also we know that from x digits we can get X ! different numbers for instance from 4703 we can get 4! = 4  x 3 x 2  x 1 = 24 number

this numbers are :   

0347 ، 0374 ، 0437 ، 0473 ، 0734 ، 0743 

3047 ، 3074 ، 3407 ، 3470 ، 3704 ، 3740 

4037 ، 4073 ، 4703 ، 4730 ، 3407 ، 4370 

7034 ، 7043 ، 3704 ، 7340 ، 7403 ، 7430

Also we know that we can get from  x digits x^x numbers (different and similar) for instance 4703 we can get 4^4 = 265 number

now from 4703 i can make :

• Smallest number consists of four digits = 0000
• Smallest number consists of four different digits =   0347
• biggest number consists of four digits = 7777
• biggest number consists of four different digits = 7430
• biggest odd number consists of four different digits = 7403
• biggest even number consists of four different digits = 7304
• smallest even number consists of four digits = 0000
• smallest number consists of four digits not equal zero = 0003
• smallest number consists of four digits bigger than 1000 = 3407
• biggest number consists of four digits smaller than 1000 = 0743  

now you can make more by use zero then there is nothing wrong when I say zero makes the numbers like mercury

MR Hesham

   

Various forms of the rational number

Relation between the integer numbers set and rational numbers set

We know that I = {.........................,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, ............................ }

                     Q=  {X :X= p/q and p,q ∈ I and q ≠ zero }

then we can say that every integer number is also rational number his denominator = 1 then we can

  I = {.........................,-5/1,-4/1,-3/1,-2/1,-1/1, 0/1, 1/1, 2/1, 3/1, 4/1, 5/1, ............................ } 

but not every rational number is also integer number but in one case if the numerator is divisible by the denominator for instance 12 /4 is rational number and also integer number because 12 /4 = 3 and also 15/5=3 , 20/4= 5, 90/3 = 30 and 21/7 = 3 

                            -----------------------------------------------------------

Various forms of the rational number :

the value of any rational number is not changed if we multiply or divide denominator and numerator by one number not equal zero 

ex : find two numbers equal 3/2

decimal form for rational number

to write the rational number in the decimal form we should make the denominator Multiples of 10 by use Previous property

ex : write 5/2 in the decimal form.

ex2: write 6/8 and 1/6 in decimal form.

Percentage form for rational number

to write the rational number in the percentage form we should make the denominator 100 only

ex : write 5/20 , 5/16 and 5 (12/125) in Percentage form

Simplest form for rational number 

 to write the rational number in the simplest form we should divide denominator and numerator by highest common factor (hcf)

write 8/12 in the simplest form

              






SOLVED PROBLEMS

find X to make 75/X and 15/(X+1)

75/X is natural number if X = { 1, 3, 5, 15, 25, 75} because 75 is divisible by { 1, 3, 5, 15, 25, 75}only

15/ (X+1) is integer number if X = {0 ,2 ,4 ,14}because 15 is divisible by {1,2,5,15}only

operations on sets (set theory) and 4 solved problems

let X = { 1 , 2 , 3 ,4 } and Y = { 3 , 4 , 5 , 6 , 7 , 8 } then find  X ∪ Y , X ∩ Y , X - Y , Y - X 

1 - (X ∪ Y ) union

∪ this symbol is called union means all elements in X and all elements in Y and  no need to repeat the elements

then  X ∪ Y  = { all elements in X and all elements in Y  don't repeat the elements  } ={1, 2, 3 ,4 ,5,6, 7 ,8 }

2- (X ∩ Y) intersection

 ∩ this symbol is called intersection means the common elements between X & Y 

then   X ∩ Y = {  the common elements between X & Y}= { 3 , 4 }
If the intersection of two sets A and B is empty, that is they have no elements in common, then they are said to be disjoint, denoted: A ∩ B = ∅. For example the sets {1, 2} and {3, 4} are disjoint, written
{1, 2} ∩ {3, 4} = ∅.

3 -  (X - Y )difference

 - means elements in X but  not in Y (notes that X is first )

then X - Y = {elements in X but  not in Y }= {1 , 2}

4-( Y - X ) difference

- elements in Y but  not in X (note that Y is first)

then  Y - X  = {elements in Y but  not in X}={5, 6 ,7 ,8}

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let A & B are to sets then A is a subset of  B (A ⊂ B) if all elements of A are found in B

ex : A = { 5 , 8 , 9 } &  B = { 4 , 6 ,8 , 9 , 10 , 11 }

then A⊂B because all elements of A (5, 8 ,9 ) are found in B
if not all elements of A not found in B then  A is not a subset of B (A B)

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Other basic concepts 

• universal set (U) : is a set which contains all objects such as if A = { 5 , 8 , 9 } then U = N (natural number set)
• Complement of set A(A^c)   : it is the set of all number in (U) not in (A)
• Symmetric difference of sets A and B is set of elements in A not in B and elements in B not in A For instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4}
• Power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } .

some solved problems :

1) zero ∈ { X - 3  ,   4 } therefore X = ....................................

because zero ∈ { X - 3  ,   4 } then X should equal 4 or X - 3 but zero ≠ 4 then zero should 

equal X - 3 

then X - 3 = zero then X = 3 

2) { 3 , X + 1} ⊂ {4 , 3 } therefore X = ...................................

BECAUSE 3 = 3 THEN  X + 1 = 4  then X = 3

3) { 2,1}∪ { 3 , X - 1 } = {1,2,3,4}THEN X = ...................................

1 = 1 , 2=2 and 3 =3 then X  -1 = 4 then X = 5 

4)  {2X , 8 }∩ {5,12}= {12}then X = ....................................

the common element is 12 then 12 should be in the two sets then12 = 2X because 8 ≠ 12 then 

X = 6 

number zero secrets

lot of people know that divide by zero has no meaning  but they did not know why? so here in this article i will explain basic concepts about zero and i will try to show why divide by zero has no meaning.

we know 12 / 3 mean that we want to divide 12 to 3 parts then every part equal 4 so too 12/2 = 6 finally 12 / 1 = 12 it is look like we want to put 12 pens in one Packet then we will put all pens in the packet but can we put 12  pens in 0 packets ? it isn't logical because there is no packets to put pens in it so this is the reason for why  divide by zero has no meaning .

is 0/0 = 1 or not ?

lets consider this 3X = 15 then 3 x X = 3 x 5 then X = 5 yes this is right accordingly, 0 x 5 = 0 x 7 but 7 ≠ 5 this mean 0/0  ≠ 1 because if 0/0 = 1 then it should be 5 =7  this is error.Also we know 3 x = 12 equal  x = 12/3, 5 x = 25 equal X = 25/5 and 0 X= 0  equal x = 0/0 but we know multiply 0 by any number = 0 this means that every number can equal 0/0 so 0/0 has meaning but is undefined 

ex : 0 x X = 0 then X = 0/0 here x may equal 1 ,2 ,3 ,4 , any other number because if 0x 2= 0 then 2 = 0/0 and 3 x 0 =0 than 3= 0/0 and so on i cannot define x  = 2 or 4 or 5 or ................

now we know why 0/0 undefined , how ca we define it?

we use limits 

let  f(x) = ( x²- 4 ) / (x - 2) is undefined at x = 2 but lim ( x²- 4 ) / (x - 2) = 2 +2 = 4 then we define f(x) = 4 at x = 2

   

sets of numbers

The first concept  is the set

The set is Gathering distinct and well defined things, these things called the element of the set. In the set there is no need to repeat the elements for example A = {1, 2, 3, 4, 5, 6} this equal A = {1, 1, 2, 2, 3, 5, 6, 4}. We should note not all Gathering of things called set by mathematical meaning for example we can't call the important numbers, Beautiful girls and intelligent students are sets because we can't judge whether thing is belong or not belong to the set.

The second concept is belonging

It is said that a  A If a is element of set A and it is said that a is not belongs to A Or a A If a is not element of the set A 

   Ex: A = {1, 2, 3, 4, 5, 6} then 1A and 4A but 7A  because 7 not element of A

The set of numbers:

1) The real number set (R)

This set has four subsets :

•    The natural number set

(N) = { 0,1,2,3,4,5,6,7,8,………….. } note that in some references N={ 1,2,3,4,5,6,7,8,………….. }.

• The integer numbers set

 (I) =  { …………,-5,-4,-3,-2,-1,0,1,2,3,4,5,……………………}.

• The rational numbers set 

(Q) = {p/q, q≠0, p, qI} such as 5/6, 5/9, 10/9 and 12/7

• The irrational numbers set (Q*) 

is the set of numbers cannot be written p/q and p/qI such as .

2) The complex numbers set (c)

x is called a complex number if x = a+ ib and i^2 = 1 and a ic called the real part of x and b is called the imaginary part of x 

ex : x = 5 + 9i 

real part = 5 and imaginary part = 9

x = 5 - 3i 

real part = 5 and imaginary part = -3 

wish to be useful
mr Hesham

why x ^zero = 1

Not many students know the cause of it but it is very simple

 (10000^ zero) = 1 because 10000^zero = 10000^(1-1)= 10000^1 /10000^1 = 1

very easy 

operations on Fractions: addition &subtractuon

first if the denominators are similar then we just add the numerator

ex : 3/5  + 6/5 

here the denominators are similar then total is (3+6)/5=9/5

ex 2: 12/7 + 5/7=  17/7

ex3:   6/5   - 7/5=   -1/5

second if the denominators are not similar then we should make Unification denominators before make the operation

ex :  3/2 + 5/3 = 

( (6/2)× 3  + (6/3) ×  2  )  /(2 ×3)= ( 4 + 9 ) / 6 = 13 / 6

ex: 5/6 - 7/8 = 

((40/6)× 5 - (40/8) × 7) / ( 6×8)=  ( 48 - 35 ) / 40 =13 / 40

very easy 

enjoy

mr Hesham

Fractions

A Fraction consists of two numbers separated by a line. 

the top number called numerator  and the Bottom called denominator

 ex : 3 / 10 

numerator = 3 and denominator = 10

to get the meaning of fractions and its importance ,Suppose that we want to Divide three oranges on ten person then we say 3 / 10  the numerator is 3 and the denominator is 10 

ex :  divide 5 apples on 3 persons 

     5 is numerator and 3 is denominator  or  3/5

easy way to understand Addition and subtraction

 When we explain the addition and subtraction to a child, in some cases have difficulty in understandingThis is an easy way I use it with the kids give me great resultsI  hope to be helpful to you

We consider the  positive number as profit and consider the negative number as loss

ex : 4 = we gained  4  and  - 4 = we loss 4 

ex : 4 + 4 

here the two number are positive then we gained 4 and 4 then we gained 8 

ex :  -4  +  6  

here   4 is negative (loss) but 6 is positive (profit) then i gained 6 but loss 4 then i gained 2 = +2

  

ex :  9 - 5

here i have 9 as profit but i lost 5 then i gained 4 = +4 

enjoy

mr Hesham

What is Algebra

Algebra is one of the three wings of mathematics, the other two being arithmetic and geometry. While most of the common calculations done are though arithmetic and geometry, algebra plays a crucial role in exemplifying the abstract and unknown entity.
Algebra is derived from two words “al-gbr” which is Arabic in origin. Al-Khwarizmi born and brought up in Baghdad the capital of Iraq was a great mathematician and a renowned astronomer of his time around 789 A.D., is known in contemporary mathematics world as the father of algebra. However Babylonian legend promulgates us that the word algebra was in some form present in form of algorithms, linear and quadratic equations were frequently used to calculate the production of grains, position of stars and other forms of estimates. ‘Al-Jabr wa-al-Muqabilah’ was perhaps the first book to be written on algebra in the later stages algebra got due attention from the famous English scientist Sir Isaac Newton in his book Arithmetica Universalis.
Two most famous forms of algebra commonly used are
1) General form of quadratic equation
Ax2 + bx +c=0
2) A,b,c are variables and x is a constant
General form of linear equation
Ax+by+c=0
The basic terms used in algebra in the contemporary education system are x,y,z and a,b,c. Algebra uses a number of different sets of equations such as linear equations and quadratic equations.
Algebra uses same standard sets of operational addition multiplication division and subtractions as in arithmetic. However the integer’s multiplications are different and can be comprehended with an example,
Let “a” be a number which has to be multiplied by a negative integer “-b” the outcome will be a negative “-ab”
Similarly a negative integer “-a” when multiplied by a negative integer “-b” the result being a positive “ab”.
The key being a negative and a negative will give a positive result, a positive and a negative will give a negative result and a positive integer when multiplied by a positive one will give a positive outcome.
While dealing with algebraic abstracts all the signs used in arithmetic and geometrical mathematic can be used but for multiplications {[( )]} instead of ‘X’ are used / is for division + for addition and – for sub traction. The entities within the brackets must be solved and opened in the order of hierarchy {[(.
Algebra is extensively used in deriving theorems such as in integration and calculus where dx/dy is one of the popular ones in order to calculate the propagation of numbers.
The algebraic equations find their way not only in the field of mathematics but also in other subjects as well like physics biology and chemistry. In fact all the major equations are based on algebraic formulae and assumptions. For example
E=mc2 the famous Einstein equation is also a product of algebraic assessment.
Algebra is a calculated way to derive equations for different purposes which can calculate the exact outcome from a tiny entity to a maximum infinite entity. In many ways algebra exemplifies a holistic approach to a calculation unlike its other two counterparts which are limited to a lengthy singular derivation process. Algebra cuts short the tedious derivative process of calculation to single formulae where you can just fill up the blanks and derive the actual outcome without going into a much tedious process of derivations.

Skills in multiplying 2

To multiplying any number of two-digit with number 11

put the total of the two digit between the two digits

ex :  34×11=374 

3 + 4 = 7 then output is 374

if the total of the two digit is greater than 9 then we add one to number left and put the single digit of the total between the two number 

ex : 67 × 11 = 737

6 + 7 = 13 then we add one to 6 to become 7 then we put single digit of total between the two digits to become 737

Skills in multiplying

 1-  To square No component of nines only Quickly and without multiplying :

We write from the left a number of Nines one less than the number of Nines in the number and then write 8 and then write the number of zeros equal to the Nines, which prescribe the number and then write one

ex1 :  999 × 999 square of the number quickly without multiplying we write only 99 then 8 to become the number 998 and add two zeros become number 99 800 and finally add 1 and becomes the final output: 998 001

ex2 :to square ( 9999 × 9999)  we write only 999 then 8 to become 99980001