WHAT IS ALGEBRA? It’s the part of mathematic in which letters and other general symbols are used to present numbers and quantities in formulae and equations

1. Algebraic Properties
Prepared by
Yousef Elshrek

2. • WHAT IS ALGEBRA?
• It’s the part of mathematic in which letters and other
general symbols are used to present numbers and
quantities in formulae and equations.

3. • WHY DO MOST SCHOOLS START TEACHING
ALGEBRAAROUND 8TH , 9TH OR 10TH GRADE ?
• Actually the reason students start learning algebra at this
time has to do with the development of their minds.
• It’s the perfect time to learn algebra.
• In early years children focus on concrete things, stuff
they take in through their five senses things like such as:
• trucks , candy, flowers, stars , dinosaurs
• In teen years , kids start to focus on abstract ideas.
• That’s why during this time people find themselves
asking the big question for example:
• Why am I here?
• Why am I on this planet?
• What’s the purpose of school?
• Why is the universe here?
• It’s also why teen-agers tune in to music lyrics-
because they’re searching for meaning.

4. • Teen-agers naturally start to develop abstract thinking and learning algebra
gives them another big push that path.
• So the answer is this :-
•You learn algebra during early adolescence because
that’s when your mind is most ready to grow in this way.

5. PROPERTIES

6. • QUESTION
• Why do you need to learn the properties when so many of them
seems obvious?
• ANSWER:
• Even though many of the properties – for example, the reflexive
and symmetric properties- seems so obvious that even most two-
year-olds
• Would understand them.
• So you need to memorize these properties so you can:
• Perform algebraic operations with confidence , and
• Prove algebraic principles

7. • BASIC NUMBER PROPERTIES
• There are four basic properties of numbers:
1. commutative
2. Associative
3. distributive
4. and identity.
• You should be familiar with each of these.
• It is especially important to understand these properties once
you reach advanced math such as algebra and calculus.

8. • COMMUTATIVE PROPERTY
• The commutative property tells you that:
• a. Addition.
• When two numbers are added, the sum is the same regardless of the order in
which the numbers are added.
• a + b = b + a
• 3 + 5 = 8 or 5 + 3 = 8.
• 8 = 8.
• Multiplication.
• When two numbers are multiplied together, the product is the same
regardless of the order in which the numbers are multiplied.
• a x b = b x a
• 3 x 5 = 15 or 5 x 3 = 15

9. • ASSOCIATIVE PROPERTY
• The associative property tells you that:
• a. Addition.
• When three or more numbers are added, the sum is the same
regardless of the way in which the numbers are grouped.
• A + b = b +a
• 6 + (4 + 3) = 13 or (6 + 4) + 3 = 13 b.
• Multiplication. When three or more numbers are multiplied, the
product is the same regardless of the way in which the numbers
are grouped.
• (a) x (b) =(b x (a)
• 6 x (4 x 3) = 72 or (6 x 4) x 3 = 72

10. COMMUTATIVE PROPERTY ASSOCIATIVE PROPERTY
not true for subtraction not true for subtraction
7 – 3 = 3 – 7 7 – (5 – 3) = (7 –5) – 3
4 ≠ – 4 7 – 2 ≠ 2 – 3
5 ≠ – 1
not true for division not true for division
10 ÷2 = 2÷10 8 ÷ ( 4÷2)= (8÷ 4) ÷ 2
5 ≠ 1/5 8 ÷ 2 = 2 ÷ 2
4 ≠ 1

11. • DISTRIBUTIVE PROPERTY
• The sum of two numbers times a third number is equal to the sum of
each addend times the third number.
• a(b + c) = a(b) x a(c)
• 5 x (7 + 2) = 45 or 5 x 7 + 5 x 2 = 45
• And
• a(b - c) = a(b) - a(c)
• 3(a +5 ) = 3a + 3x 5
3a = 15.

12. • IDENTITY PROPERTY
• a. Addition.
• The sum of any number and zero is that number.
a + 0 = a
12 + 0 = 12
• b. Multiplication, The product of any number and one is that
number.
a x 1
18 x 1 = 18
• Knowing these properties of numbers will improve your
understanding and mastery of math

13. • The Reflexive property means everything is always congruent (equal) to
itself….
• Say you have a rectangle ….with the four points A B C and D… and you have
line AB as the top line of the rectangle and line DC as the bottom line and AD
as the left side line followed by BC as the right side line then divide the
rectangle with a line making two equal triangles…..with line AC as one side of
both triangles….

14. • It is a given that line AB is congruent to line DC
• It is a given that line AD is congruent to line BC
• The Reflexive Property is that line AC is congruent to line CA
• We can prove that the triangle ABC is congruent to triangle CDA and
are so by using the Side-Side-Side Theorem by showing that side AC is
congruent to itself.
• The reflexive property tells you this simple truth :
• a =a
• Or , any quantity is equal to itself.
• 3=3 ms=ms
• ⅚=⅚ x3 = x3
• 0.9 = o.9 ac2 x = ac2 x

15. • WHAT DOES THE SYMMETRIC PROPERTY SAY
• The symmetric property says this:
• If a = b, then b = a.
• Or : if a first quantity is equal to a second quantity , then second quantity is
equal to the first quantity.
• Examples:-
• If 8/4 , 4/2 = 2.
• If 8/10 , 8/10 = 0.8.
• If 0/4 , 0 = 0/4.
• If x3 = x.x.x then x.x.x= x3.
• If a+b = 20 , then 20 =a+b

16. • WHAT DOES THE TRANSITIVE PROPERTY SAY?
• The transitive says this:
• If a = b , and b = c , then a = c.
• Or: If the first quantity is equal to a second quantity, and the second
is equal to a third quantity , then the first quantity is equal to the
third quantity.
• Examples:
• If 2+2 = 4 and 4 = 16/4
• If 2x3 = 12/2
• If x + b = c, and a = m + t , then x + b = m +b.

19. • IDENTIFY PROPERTIES
• Example
• Name the property shown by each statement.
1. (7 + 3x) + 2x = 7 + (3x + 2x) 2) 0 x5a = 0
Associative Property Zero Property of Multiplication
• Example
• Use properties to simplify expressions
3. 4+ (x + 13)
4 + (x +13) = 4+ (13 + x) Commutative Property
= 4 + 13 + x
Add 4 + 13
= 17 + x
4. 6(x+7)
6(x+7) = 6x + 6(7) Distributive property
= 6x + 42 Multiply

20. • APPLY PROPERTIES TO PROBLEM SOLVING
5. MUSEUMS Three friends are going to the science museum.
• The cost of admission is $x each. It will cost an additional $4 to view a movie on the 3-D
screen. Write and simplify an expression that represents the total cost for the three friends.
• Answer
• The cost of admission plus the movie can be represented by (x + 4).
• Multiply this cost by the number of friends, 3(x + 4)
• . 3(x+ 4) = 3(x) 3(4) Distributive Property
• = 3x 12 Multiply.
• So, the total cost for the three friends is $3x + $12.
6. MUSEUMS Refer to Example 5. A fourth friend will meet the group of friends at the
museum but will not go to the movie. Write and simplify an expression that represents the
total cost for the four friends.
• The cost for the fourth friend is $x. Add this to $3x $12
• 3x +12+ x = 3x + x + 12 Commutative Property
• = 4x + 12 Add
So , the total cost for four friends is $ 4 + $12