Prime numbers : Numbers that can only be divided by 1 and itself .
Composite numbers : Not prime numbers .
Prime numbers from 1-20 = 2,3,5,7,11,13,17,19
* 1 is not a prime number – because it can only be divided by itself .
Q.) How many prime numbers are there from 1-100?
Stepby stepfindingPrimenumbers .
- Cross outthe number1
- Circlethe number2andcross out allthe othermultiples of 1.
- Circlethe number3andcross out allthe othermultiples of 3.
- Circlethe number5andcross out allthe othermultiples of 5.
- Circlethe number7andcross out allthe othermultiples of 7.
- Continuetheprocess unit allunitall thenumbers areeither circled or crossed.
HIGHEST COMMON FACTOR(common)
How to find the highest common factor?
Find the highest common
multiple of 15 and 75 ?
Express 252 in PRIME FACTORS
2 x 126
2 x 2 X 63
2 3 X 21
3 3 x 7
From the above
We have 252 =
Index notation is using the power of a certain number.
e.g.) 252= (first, prime
factorise the numbers)
= 22 x 32 x 7
12= 2x2x3 can be
written as 12= 2 x 3
LOWEST COMMON MULTIPLE(max)
Lowest common multiple of 65 , 175 , 135
Please note that we have to
arrive to the answers to all be one
at the last ladder
LCM(LOWEST COMMON MULTIPLE)
FINDING CUBE ROOT AND SQUARE
1) FIRST PRIME FACTORISE THE NUMBER
EG.) square root of 144
= 24 x 32
2) Arrange them into 2 brackets
Square root ( 2x2x3) (2x2x3)
3) Solve what is in 1 bracket
Do the same only at
step to , instead of 2
brackets , it becomes
3 brackets .
Positive and Negative integers
. In the number line , the more left you go , the larger
the number gets(smaller value) .
. Zero is an integer by itself- not positive or negative.
*note that there is no such thing as +0 or -0 .
- BODMAS rule stated that everything should be from
left to right UNLESS there is a bracket .
ADDITION OF INTERGERS
Owe someone 4 dollars
and another 2 dollars.
DIVISION OF INTEGERS
= (18 3)-2
6 2x4 + (-3)
= 3 x 4 +(-3)
= 12 + (-3)
*ALWAYS DO FROM LEFT TO
ALWAYS DO THE “ POWERS “ FIRST
(-4)2 (-8) + 3 x (-2)3
= 16/(-8) + 3 x (-8)
= -2 + 3 x (-8 )
= -2 + (-24)
a/b - b cannot be “0”
e.g : mixed numbers
- Using the cancellation method ….
Such as: - 21/17 X 19 /7 = - ?
THE INTEGERS IN RATIONAL NUMBERS CAN BE BOTH POSITIVE AND NEGATIVE.
THE CHANGING OF SIGNS MUST BE INCLUDED!!!!!
REMEMBER WHEN DIVIDING A FRACTION OR FRACTIONS , SAME-CHANGE-INVERT !!
ALSO REMEMBER THAT EVEN IF THERE ARE 3 OR MORE FRACTIONS ONLY ONE DOESN’T
CHANGE – DURING DIVISION OF FRACTIONS ONLY !!
DENOMINATORS MUST BE THE SAME.
• Actually writing numbers in the form of letters
• IF YOU ARE 40 YEARS OLD , I AM 20 YEARS
YOUNGER THAN YOU , MY AGE WILL BE (40-
• BUT IF I AM x YEARS OLD , YOU ARE (x-
• OF BOTH , POSITIVE AND NEGATIVE
INTEGERS . THE (-)MINUS SIGN IS ACTUALLY
THE “NEGATIVE” SIGN .
Only like terms can combine into a single term
( BY ADDITION OR SUBTRACTION ONLY )
Like terms :
1) ab , 2 ab ( yes)
2) x , 2x2 (no)
3) 3p,7p (yes)
4) xy , 2x2y (no)
DIVISION IN ALGEBRA
27a / 3a
= 9 ( cancel the “a”)
TERMS , VARIABLE , COEFFICIENT
When x = 4 ,
When y = 6
When z = 10 ,
z-(x+y) = 10-10
5x = 5x4
THE VALUE OF x IS CALLED A VARIABLE
5 IS ATTACHED TO x , SO 5 is the coefficient OF x.
10a _ a is a variable and 10 is the coefficient OF a
6ab - ab is a variable and 6 is the coefficient of ab
2B - 2 of B’s
B2– 1b 2so , B is the variable and 1 is the coefficient of b2
ADDITION AND SUBTRACTION OF
RECALL : addition / subtraction of integers
e.g.) sum of 4 and 2 = 4+2
Subtract 2 from 5 = 5-2
Exponents often are used in the formula for area and volume. In fact, the word
squared comes from the formula for the area of a square.
Area of a square: A = s2
The word cubed comes from the formula for the volume of a cube.
Volume of Cube: V = s3
SQUARE ROOTS AND CUBE ROOTS
4p2 + 2pq
1) Factorisation is the process
of finding a term or an
2) The common factors of
several algebraic terms are
numbers or terms that are the
factors of all algebraic terms
3) An algebraic expression with
2 or more terms can be
factorised by taking out all the
common factors of the
expressions from the brackets.
2xy + 6y + 3x +9
Factorisation means taking out the common
Factorisation is NOT expansion .
Factorisation vs expansion => opposite
SQUARE SQUARE ROOT
CUBE CUBE ROOT
Expansion – final answer should not have fractions . (Using the
“rainbow” method )
= -6 -6k +4k – 12h
STEP 1 : Remove the
bracket by doing
STEP 2 : Rearrange to
put the “like” terms
NOTE : 2 SETS OF
BRACKETS , 2
Square root is the opposite of square
E.G.) p(square) is opposite of p
-DETAILS MUST BE STATED CLEARLY
- Times (x) must be written in “bracket format” such as
2P= 2 x P
P2= P x P
P3= p x p x p
3P= 3 X P
What algebraic expression can be used to find the perimeter
of the triangle below?
Perimeter = a + b + c
In this algebraic expression, the letters a, b, and c are called ________.variables
In algebra, variables are symbols used to represent unspecified numbers or values.
NOTE: Any letter may be used as a variable.
Variables and Expressions
It is often necessary to translate verbal
expressions into algebraic expressions.
English word(s) Math Translation
Write an algebraic expression for each
a) Eight more than a number n. 8 + ntranslates to
b) Seven less the product of 4 and a number x. 4x-7translates to
c) One third of the size of the area a. translates to ora
Variables and Expressions
Find the perimeter of the triangle.
If a is 8 , b is 15 and c is 17
Perimeter = a + b + c Write the expression.
= 8 + 15 + 17 Substitute values.
= 40 Simplify.
FINDING THE UNKNOWN
e.g.) 3x – 2 = 4
x = 6/3
6/-2 = -3(k)
*If “ –” , do “+”
If “x” do “/”
FINDING THE UNKNOWN II
3.14 => recurring number
FURTHER EXAMPLES ON EQUATIONS
7 + 2x = 6x-5
6hx + 12ky +9kx +8hy
=6hx + 9kx + 12ky + 8hy
= 3x (2h+3k) + 4y (3k+2h)
THE ONE WITH THE
1003 x 78
~ 1000 x 80
1003 x 85
~ 1000 x 90
*LESS THAN 5- ROUND
DOWN / ignore (“0”)
*5 OR MORE – ROUND
1300 + 6
AREA AND PERIMETER
Triangle = ½ x base x height
Rectangle = Length x Breadth
Square = Length x Length
Circle= π x radius x radius (πr2)
Parallelogram = Base x height (perpendicular
Trapezium = ½ x (a+b) x height (a&b 2 parallel
AREA AND PERIMETER
Triangle = plus (+) all outer sides
Rectangle = plus(+) all outer sides
Square = plus (+) all outer sides
Circle= (circumference) π x diameter (πD)
Parallelogram = Plus(+) all outer sides
Trapezium = plus(+) all outer sides
FORMULAS FOR MEASURING
CUBE = Length x Length x Length
CUBOID = Length x Breadth x Height
PRISM = Base area x Height
= 1/2 x Length x Breadth x Height
PARALLELOGRAM = Base x Height
CONE = 1/3 x x radius2 x height
SPHERE= 4/3 X x radius3
NUMBER SEQUENCE PATTERN
2,4,6,8,10,12 2 times table
1,3,5,7,9,11 Odd numbers/add 2
1,2,4,8,16,32 Power of 2
2,5,8,11,14,17,20 Add 3
0,10,20,30,40,50,60… Add 10 / 10 times table
1,3,6,10,15 Add 1 to the top
1,1,2,3,5,8,13,21 Add the 1st 2 numbers
to get the 3rd number
Step 1 : Find the pattern
Step 2 : See how the pattern flows
Step 3 : Continue the pattern
Symbol Words Example
> greater than x + 3 > 2
< less than 7x < 28
≥ greater than or equal to 5 ≥ x - 1
≤ less than or equal to 2y + 1 ≤ 7
12 < x + 5
If we subtract 5 from both sides, we get:
12 - 5 < x + 5 - 5
7 < x
But put an "x" on the left hand side ...
so let us flip sides (and the inequality sign):
x > 7
Do you see how the inequality sign still "points at" the
smaller value (7) ?
ANS: x > 7
Volume of cuboid
Length x breadth x height
Volume of cube
Length x Length x Length
Volume of pyramid
1/3 x Base x Height
Volume of Cylinder
Base x Height
Volume of cone
1/3 x Base x Height
Volume of sphere
4/3 x π x r3
Units : mm,cm,m,km,ha (perimeter)
: mm2,cm2,m2, km2,ha2 (area)
1mm= 0.001 m
1 ha = 10000 m2
1kg=1000g (1k-1000 , g – grams)
VOLUME AND TOTAL SURFACE AREA
Volume : length3
Area : 6xlength2
Volume : length x breadth x height
Area: 2(lb + bh + hl )
Area : Base area x height
Volume: (Perimeter of base x h ) + 2base area
Volume : πr2h
Area: 2πr2 + 2πrh
185mm= 185 x 0.1 cm = 18.5 cm
21cm = 21 x 10mm = 210 mm
21cm = 21 x 0.01m = 0.21 cm
1 hectare = ?x?