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APPRECIATE ~ GRADE 7 / SEC 1 MATH
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PRIME NUMBERS 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? A.) 25.
Step by step finding Prime numbers . - Cross out the number 1 - Circle the number 2 and cross out all the other multiples of 1 . - Circle the number 3 and cross out all the other multiples of 3 . - Circle the number 5 and cross out all the other multiples of 5 . - Circle the number 7 and cross out all the other multiples of 7 . - Continue the process unit all unit all the numbers are either circled or
crossed.
HIGHEST COMMON FACTOR(common)How to find the highest common factor?
Find the highest common multiple of 15 and 75 ?
15,75 5,25 1,5
HCF = 3x5=15
35
Express 252 in PRIME FACTORS
252
2 x 126
2 x
2 X 63
22 3 X 21
3 3 x 7
From the above factor tree, We have 252 = 2x2x3x3x7
INDEX NOTATION Index notation is using the power of a certain number.e.g.) 252= (first, prime factorise the numbers)= 2x2x3x3x7= 22 x 32 x 7
12= 2x2x3 can be written as 12= 2 x 32
LOWEST COMMON MULTIPLE(max) Lowest common multiple of 65 , 175 , 135
65,175,135
Please note that we have to arrive to the answers to all be one at the last ladder
13, 35,271,35,271,1,27
1,1,1
5133527
LCM(LOWEST COMMON MULTIPLE)= 5x13x35x27 =61425
FINDING CUBE ROOT AND SQUARE ROOTS
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
2x2x3=4x3=12
Cube root
Do the same only at step to , instead of 2 brackets , it becomes 3 brackets .
IntegersPositive 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 .
DIVISION OF INTEGERS
3x6 3-2= (18 3)-2=6-2=4
6 2x4 + (-3)= 3 x 4 +(-3)= 12 + (-3)= 9
*ALWAYS DO FROM LEFT TO RIGHT
ALWAYS DO THE “ POWERS “ FIRST
(-4)2 (-8) + 3 x (-2)3
= 16/(-8) + 3 x (-8)= -2 + 3 x (-8 )= -2 + (-24)=-26
RATIONAL NUMBERS
a/b - b cannot be “0”
e.g : mixed numbers improper fraction
- 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.
**Irrational CANNOT BE EXPRESSED AS A FRACTION
ALGEBRA
• 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 (40-20) .
• BUT IF I AM x YEARS OLD , YOU ARE (x-20)years old
• OF BOTH , POSITIVE AND NEGATIVE INTEGERS . THE (-)MINUS SIGN IS ACTUALLY THE “NEGATIVE” SIGN .
ALGEBRA
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)
SUBTRACTION IN ALGEBRA
1) (+)3a-2b+2a-3b= 3a+2a – 2b – 3b
= 5a-5b
2)[3a+3b(a-bc)] FROM 3a-3b=(2a-3b) – (3a2 -3b2c)=2a-3a2-3b-3b2c=-1a3-3b3c
Step 1 : rearrange
Step 2 : evaluate
TERMS , VARIABLE , COEFFICIENTWhen x = 4 , When y = 6 When z = 10 ,
x+y= 4+6 = 10 z-(x+y) = 10-10 = 0 5x = 5x4 = 20
THE VALUE OF x IS CALLED A VARIABLE 5 IS ATTACHED TO x , SO 5 is the coefficient OF x.
E.G) 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 ALGEBRAIC EXPRESSIONS
RECALL : addition / subtraction of integers
e.g.) sum of 4 and 2 = 4+2 = 6 Subtract 2 from 5 = 5-2 = 3
Exponents often are used in the formula for area and volume. In fact, the wordsquared comes from the formula for the area of a square.
s
s
Area of a square: A = s2
The word cubed comes from the formula for the volume of a cube.
ss
s
Volume of Cube: V = s3
SQUARE ROOTS AND CUBE ROOTS
FACTORISATION
4p2 + 2pq= 2p(2p+q)
Common factor
1) Factorisation is the process of finding a term or an algebraic expression.
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 = 2y(x+3)+3(x+3)
=( 2y+3)(x+3)Same
FACTORISATIONFactorisation means taking out the common factors . Factorisation is NOT expansion . Factorisation vs expansion => opposite
OPERATION OPPOSITE
ADDITION SUBTRACTION
SQUARE SQUARE ROOT
FACTORISTION EXPANSION
CUBE CUBE ROOT
DIVISION MULTIPLICATION
EXPANSIONExpansion – final answer should not have fractions . (Using the “rainbow” method )
e.g) 3(2+x) = 6+x e.g) -3(2h-2k)+4(k-3h) = -6 -6k +4k – 12h = -6h-12h+6k+4k = -18h+10k
STEP 1 : Remove the bracket by doing EXPANSION .
STEP 2 : Rearrange to put the “like” terms together
NOTE : 2 SETS OF BRACKETS , 2 EXPANSIONS
ALGEBRA
Square root is the opposite of squareE.G.) p(square) is opposite of p
-DETAILS MUST BE STATED CLEARLY- Times (x) must be written in “bracket format” such as 3x4=
3(4)
2P= 2 x PP2= P x PP3= p x p x p3P= 3 X P
What algebraic expression can be used to find the perimeter of the triangle below?
a b
c
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
more than
less than
product
addition
subtraction
multiplication
of multiplication
quotient divisionWrite an algebraic expression for eachverbal expression:
a) Eight more than a number n. 8 + ntranslates to
b) Seven less the product of 4 and a number x. 4x-7 translates to
c) One third of the size of the area a. translates to or a3
1
3
a
Variables and Expressions
Find the perimeter of the triangle.If a is 8 , b is 15 and c is 17
a b
c
Perimeter = a + b + c Write the expression.
= 8 + 15 + 17 Substitute values.
= 40 Simplify.
= 8
= 17
= 15
SUBSTITUTION
FINDING THE UNKNOWN
e.g.) 3x – 2 = 43x= 4+23x=6x = 6/3X=2
(+)11-2k=17 -2=17-11 -2k=(-2) 17-11=6 6/-2 = -3(k)K=-3
2h +1.3=2.82h=2.8-1.32h=21.5-2=0.5h= 0.5
*If “ –” , do “+” If “x” do “/”
FINDING THE UNKNOWN II
3.14 => recurring number
FURTHER EXAMPLES ON EQUATIONS7 + 2x = 6x-52x=6x-5+72x=6x-122x-6x=-12-4x=-12X= -12/-4= +3
.
6hx + 12ky +9kx +8hy =6hx + 9kx + 12ky + 8hy = 3x (2h+3k) + 4y (3k+2h)=(3x+4y) (2h+3k)
*REARRANGETHE ONE WITH THE MOST COMMON FACTOR
ESTIMATION
1003 x 78 ~ 1000 x 80 = 80,000
1003 x 85~ 1000 x 90 = 90,000
~
~
*LESS THAN 5- ROUND DOWN / ignore (“0”)
*5 OR MORE – ROUND UP
1300 + 6~ 1000+10= 1010~
AREA AND PERIMETER
AREA)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 height)Trapezium = ½ x (a+b) x height (a&b 2 parallel lines)
AREA AND PERIMETER
PERIMETER)Triangle = plus (+) all outer sidesRectangle = plus(+) all outer sidesSquare = plus (+) all outer sides Circle= (circumference) π x diameter (πD)Parallelogram = Plus(+) all outer sidesTrapezium = plus(+) all outer sides
FORMULAS FOR MEASURING VOLUME
CUBE = Length x Length x Length CUBOID = Length x Breadth x Height PRISM = Base area x Height = 1/2 x Length x Breadth x HeightPARALLELOGRAM = Base x Height
CONE = 1/3 x x radius2 x heightSPHERE= 4/3 X x radius3
NUMBER SEQUENCENUMBER SEQUENCE PATTERN2,4,6,8,10,12 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
FINDING SEQUENCES
1st layer 1 = 12nd lay+0er 1+2= 33rd layer 1+2+3=64th layer 1+2+3+4= 10
30th layer ?1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30 = 465
FINDING SEQUENCESStep 1 : Find the pattern Step 2 : See how the pattern flows Step 3 : Continue the pattern
SOLVING INEQUALITIES
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
SOLVING INEQUALITIES12 < x + 5If we subtract 5 from both sides, we get:12 - 5 < x + 5 - 5 7 < xBut put an "x" on the left hand side ... so let us flip sides (and the inequality sign):x > 7Do you see how the inequality sign still "points at" the smaller value (7) ?
ANS: x > 7
VOLUME Volume of cuboid Length x breadth x heightVolume of cube Length x Length x Length Volume of pyramid 1/3 x Base x Height Volume of Cylinder Base x Height Volume of cone1/3 x Base x Height Volume of sphere 4/3 x π x r3
UNIT CONVERSION
Units : mm,cm,m,km,ha (perimeter) : mm2,cm2,m2, km2,ha2 (area)
10mm= 1cm1mm=0.1cm100cm= 1m1cm=0.01 m1000mm= 1m 1mm= 0.001 m 1 ha = 10000 m2
1kg=1000g (1k-1000 , g – grams)
VOLUME AND TOTAL SURFACE AREA
1. CUBE Volume : length3 Area : 6xlength2
2. CUBOID Volume : length x breadth x height Area: 2(lb + bh + hl ) 3. PRISM Area : Base area x height Volume: (Perimeter of base x h ) + 2base area 4. Cylinder Volume : πr2h Area: 2πr2 + 2πrh
UNIT CONVERSION
185mm= 185 x 0.1 cm = 18.5 cm21cm = 21 x 10mm = 210 mm21cm = 21 x 0.01m = 0.21 cm1 hectare = ?x?
CONVERSION
1m = 100cm (x100)1cm = 0.01m (/100)1m=0.001km(/1000)1000m =1km (x1000)1hour=60mins 1minute=60 seconds
RATIO (REPEATED IDENTITY)If a:b = 3:5 and a:c = ½ : 3/5 , find the ration of a:b:c.
a:c½:3/55:6
a:b3:5
LCM of 3 and 5 =15 a:b:c = 15:25:18