About the author
Mr. M . S . M . Dahlan
M r . M . S . M . D a h l a n is a veteran teacher i n m a t h e m a t i c s
who
has a l m o s t
35-year
experience
in
teaching
m a t h e m a t i c s i n g o v e r n m e n t schools. A n d also he has
taught i n an a t o l l s c hool i n t h e R e p u b l i c o f M a l d i v e s
f o r t w o years.
M r . D a h l a n has served as a Project O f f i c e r attached t o
the M a t a r a D i s t r i c t E d u c a t i o n a l D e v e l o p m e n t Project
( M E D P ) f u n d e d b y S I D A , f r o m 1993 t o 1995.
H e has served i n the Subject Specialist Panel o f the
Text B o o k s E v a l u a t i o n C o m m i t t e e o f the E d u c a t i o n a l
Publication Department, Isurupaya, Battaramulla. In
a d d i t i o n t o that he w a s a resource p e r s o n attached t o
the B i l i n g u a l E d u c a t i o n P r o g r a m m e o f the N a t i o n a l
Institute o f M a h a r a g a m a .
H e r e t i r e d f r o m g o v e r n m e n t service i n 2008 after a
c o m m e n d a b l e service at St. T h o m a s ' C o l l e g e M a t a r a
for over
fifteen
years a n d presently attached t o t h e
Academic Staff o f Ceylinco-Sussex College, Matara.
I S B N 978-955-590-096-6
NATION \ l
1
SGIENCI
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H
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47/5, M a i t l a n d Place
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NATION\i
National Science Foundation
s
Science Books Series - 11
Some Elementary Hints in
Mathematics
M. S. M. Dahlan
NATIONAL
SCIENCE
FOUNUAHON
Published by
National Science Foundation
47/5, Maitland Place
Colombo-07.
© National Science Foundation 2000
No responsibility is assumed by the National Science Foundation
of Sri Lanka for statements and opinions expressed by the author
in this book
ISBN 978-955-590-096-6
Publication Coordination: Upuli Rathnayake
Cover page design: Dimuthu Weerasuriya
Printing Division
National Science Foundation
Science Books Series - 1 1
Some Elementary Hints
Mathematics
CONTENTS
R e p r e s e n t a t i o n o f N u m b e r s i n a Place- V a l u e T a b l e
1
R e p r e s e n t a t i o n o f D e c i m a l N u m b e r s i n a Place - V a l u e T a b l e
1
Estimation o f Decimal Numbers
2
Integers-Addition & Subtraction
2
Some Properties o f M u l t i p l i c a t i o n
5
H i n d u Lattice Multiplication
6
Division
7
Classification o f Triangles
8
Classification o f Quadrilaterals
9
A p p l i c a t i o n o f B O D M A S Sequence
10
Powers & Exponents
11
Variables & Expressions
12
Substitution
13
Equations
15
Prime Factorization
18
Greatest C o m m o n F a c t o r ( G C F )
19
Least C o m m o n M u l t i p l e ( L C M )
20
Perimeter & A r e a
21
Use o f V e n n Diagrams to Solve Problems
23
M a i n R e p r e s e n t a t i o n a l V a l u e s i n Statistics
25
Representation o f Data in Frequency Tables
26
Quartiles & Inter-quartile Range
27
Construction o f Bar Graphs
28
Metric System
29
S o m e C o m m o n t e r m i n o l o g y used i n M a t h e m a t i c s
Representation of Numbers in a Place-Value Table
H o w to write a number in Standard Form and Expanded Form
Ones (Units) Zone
Thousands Zone
H
T
0
H
T
0
2
3
8
9
6
1
How lo write a larger number
Millions Zone
Thousands Zone
H
Ones Zone
T
O
H
T
O
H
T
O
6
4
9
3
5
7
3
4
W o r d N a m e : Sixty-four million nine hundred and thirty-five
thousand seven hundred and thirty four.
Standard Form: 6 4 935 7 3 4
Expanded Form: 6 0 0 0 0 0 0 0 + 4 0 0 0 0 0 0 + 9 0 0 0 0 0 + 3 0 0 0 0
+5 000+700+30+4
Representation of Decimal Numbers
Thousands
Hundreds
2
1
-
Tens
Ones
-
Tenths
Hundredths
Thousandths
,9
2
-
1
6
7
2
S o m e Elementary Hints in Mathematics
Standard Form: 2 192.167
Expanded Form: 2 0 0 0 + 100 + 9 0 + 2 + 0.1 + 0.06 + 0.007
Estimation of Decimal Numbers
Estimate: 3.78 + 5.97round to the nearest whole number
3^78 + 5|97
4
+
6=10
Estimate: 6 . 2 4 9 - 0 . 8 1 9
6.249-0.819
1
6
1
i - l = 5
Integers
A l l the integers can be written as a set o f { . . . - 4 , - 3 , - 2 , - 1 , 0 , 1 , 2 ,
3,4...}
The
called ellipses, means that the set continues without
end, following the same pattern.
The positive integers are often written without + sign. So + 2
and 2 are the same. O n the number line, 0 , is considered the
starting point with the positive numbers to the right and
negative numbers to the left. Zero is neither negative nor"
positive.
—i—i—r—r—i—i—i—i—i—i—i—i—i
-6
-5 -4 -3
-2 -1. 0
t
t
I
r> negative <j
I
1
t
2
3
4
5
••'
L__r>positive <q
•
6
t
|
Some Elementary Hints in Mathematics
3
T h e arrows show that the numbers continue without end. -4 is
read "negative 4 " and + 2 is read "positive 2". T o graph an
integer, you locate the number and draw as a dot at that point
on the line. Letters are sometimes used to name points on a
number line. The integers that correspond to the letter'is called
the ordinates o f that particular point.
Example: Graph points A , B, C and D on a number line. Their
corresponding ordinates
are 8, - 3 , -7 and 3 respectively.
C
<—i
o
_8 -7
IB
1
1
-6- -5
-4
1
G
-J3
A
i
t
i
i
i
D
a
i
i
i
i
• fc.
-2
-1
0
1
2
3
4
5
6
7
8
A d d i n g ; integers w i t h same s i g n :
T h e s u m o f t w o p o s i t i v e integers is p o s i t i v e
T h e s u m o f t w o negative integers is negative
Example:
i) 3 +2 = 5
i i ) (-6) + (-2) = (-8)
Representing the above t w o examples o n a n u m b e r l i n e
4
Some Elementary Hints in Mathematics
Subtracting integers
The opposite of an integer is called its additive inverse
Example: I f a number is 6, its additive inverse is (-6).
I f a number is -7 its additive inverse is +7.
Property o f integers and additive inverse:
The sum o f an integer and its additive inverse is 0.
Arithmetic
Algebra
6 + (-6) = 0
X + (-X) = 0
Some Pattern of Properties
7 x 2 = 14
1 x 3.5 = 3.5
1 x 0.004 = 0.004
7 x 2 0 = 140
10 x 3.5 = 35
10 x 0.004 = 0.04
7 x 2 0 0 = 1400
100 x 3.5 = 350
100 x 0.004 « 0.4
7 x 2 0 0 0 = 14000
1000 x 3.5 = 3 5 0 0
1000 x 0.004 = 4
Some Properties of Addition
Commutative property of addition: The order of Addends does not
change the sum
3 + 4= 4+3
Associative Property of addition: The ways the Addends are grouped
does not change the sum.
(2+ l)+5 = 2 + (l +5)
Some Elementary Hints in Mathematics
Identity Property: The sum of any number and zero equals the
number.
5 + 0 = 5 and 0 + 5 = 5 '
Example:
Solve
12+ (48+8)
1 2 + ( 4 8 + 8)
= 1 2 + ( 8 + 48)
Commutative Prop. = ( 1 2 + 8 ) + 48
Associative Prop.
= 2 0 + 48
= 68
Some Properties of Multiplication
Commutative Property of Multiplication
The order of the factors does not change the Product
6x8=8x6
Associative Property of Multiplication
The way the factors are grouped does not change the product
5x(8x7) = (5x8)x7
Zero Property of Multiplication
The product of any factor and zero equals zero. 49 x 0 = 0
5
Some Elementary Hints in Mathematics
6
Hindu L a t t i c e Multiplication
th
I n the 1 2 century, mathematicians in India used a lattice
method to multiply.
Trnv is the lattice method for 5 x 876 = 4 3 8 0
Step 1. Place the factors outside the grid
8
7
6
5
Step 2. Find the products o f 5 x 8, 5 x 7, 5 x 6
8
7
6
Step 3. W r i t e the sum o f the diagonals. Start from right.
8
7
4
3
6
8
Therefore the answer is 4 3 8 0
Solve: 3 4 2 x 2 5
3
731 x 36
4
7
3
1
Some Elementary Hints in Mathematics
7
Division
47 divided by 3 = 47/3
Quotient
15
Divisor
47-*
Dividend
2 _
17
JL5_
2
+
Remainder
T h e answer o f 47/3 is 15 R 2
R = remainder
Therefore the dividend = quotient x divisor + remainder
Squaring of L a r g e r l u m b e r s when the Unit Place values
are 5
Example 1:
25x25
<
Multiplier
Multiplicand
Step 1 :
M u l t i p l y the unit place number o f the multiplicand .by
the unit place number o f the multiplier. That is 5 x 5 = 25
Step 2:
M u l t i p l y the unit place number o f the multiplicand b y
the next consecutive number that occurs after the tenth place
number o f the multiplicand. That is 2 x 3 = 6.
Therefore the answer is 25 x 25 = 625
Example 2
65 x 65
5 X 5 = 25
6 x 7 = 42
Therefore the answer is 4225
8
Some Elementary Hints in Mathematics
Example 3
125 x 125
5 x 5 = 25
12x13=156
Therefore the answer is 15 625
Classification o f Triangles
W h a t is a triangle? A triangle is a 3-sided closed plane figure
bounded by straight lines.
Classification of triangles by the lengths of the sides
Equilateral triangle
Isosceles triangle
Scalene
triangle
A l l 3 sides the same
2 sides the same length
N o sides the
length
same length
Identification of triangles by the kind of angles
Right-angled triangle
1 right angle
acute- angled triangle
3 acute angles
obtuse-angled
1 obtuse angle
Some Elementary Hints in Mathematics
9
Classification of Quadrilaterals
Quadrilaterals: a four sided closed plane figure bounded
by straight lines.
rectangle
square
J
•L
n
r
4 right angles
4 equal sides
in length
parallelogran.
D
4 right angles
opposite side equal
in length
a
rhombus
opposite side equal
in length
opposite sides parallel
trapezium (trapezoid)
kite
A l l sides equal i n
only one pair o f
2 pairs o f equal
length
parallel sides
adjacent sides
opposite sides parallel
10
Some Elementary Hints in Mathematics
Application of BODMAS sequence in mathematical
calculation
To
accurately
solve mathematical
problems
with
multiple
operations a specific sequence is being followed. This process
is called B O D M A S sequence.
B
Bracket
O -
Of
D
Division
M
Multiplication
A
Addition
S
Subtraction
Example:
solve 3 x 60/4 + 1/3 o f (25 - 10) - 5
Step 1: do the operations given within the brackets
Step 1
r
3 x 60/4 + 1/3 o f | ( 2 5 - 1 0 ) | - 5
Step 2: do the operation containing ' o f 3 x 60/4 -r| 1/3 o f 15| - 5
t
Step 3: do the division
3 x! 60/4
Step 2
+ 5-5
-*—
Step 3
Step 4: do the multiplication 3 x
15 + 5 - 5
Step 4
11
S o m e Elementary Hints in Mathematics
Step 5: do the addition
145 + 5 - 5
Step 5
-4—
Step 6: do the subtractions
50-5
Step 6
Therefore the answer is 45
Powers and Exponents
Generation
family tree
number
Person
P
1
Parents
FM
FM FM
FM FM FM FM
1x2 = 2
Grand parents
Great-grand parents
G reat- great- grand
2x2 = 4
2 x 2 x 2 = 8
FM FM FM FM
FM FM FM FM
Parents
Study the pattern in the number column o f the table. W h a t is
the next value would be? W h e n two or more numbers are
multiplied, these numbers are called factors o f the product.
W h e n the same facfor is repeated, you may use an exponent to
simplify the notation. '
:
16 = 2 x 2 x 2 x 2
>• 2
4
four is the exponent
4
A n expression like 2 is called a p o w e r and is read; 2 to the
power 4 ' or ' 2 to the fourth power'. The 2 in this expression is
the base.
Powers are often used to write a product in a shorter form.
Example:
3
3 x 4 x 4 x 3 x 4 x 3 x 3 = 4 x3
4
12
Some Elementary Hints in Mathematics
Variables a n d E x p r e s s i o n s
In mathematics w e also use substitution. Let us consider the
numerical expression
5 + 6.it has a value o f 1 l.how ever the expression x + 4 does
not have a value until a value for x is given. B u t when you
substitute the value x = 10 or put 10 i n place o f x i n the
v
expression, it becomes 10 + 4, which is equal to 14.
In
order
to evaluate,
or
find
the value
o f a numerical
expression, w e need to follow an order of operations. That is
you need to know which operation to do first when there is
more than one operation i n the expression.
Order o f operations:
1.
D o all operations within grouping symbols first;
start w i t h innermost grouping symbols.
2.
D o all powers before other operations.
3.
N e x t , do all multiplications ;ind division in order
from left to right
4.
Then do all addition and subtraction.
Example:
2
evaluate (4 + 8) / 3 x 5 + ( 2 + 7)
Y o u may have to follow the order o f operations to do this step.
D o operations in the parentheses first.
2
(4 x 8)/3 x 5 + ( 2 + 7) = 12/3 x 5 + 11
= 4 x 5 + 11
= 20+11
= 31
Some Elementary Hints in Mathematics
13
Algebra is a language o f symbols. In algebra, w e use letters,
called
variables
to represent
unknown
quantities.
I n the
expression x + 4, x is a variable.
Expressions
that
contain
variables
are called
algebraic
expressions.
In order to evaluate algebraic expressions, you must k n o w how
to read expressions. !
means
3 xa
ab
3a
means
a xb
5x2y
means
5 x2 xy
a[b(cd)]
means
a x (b x c x d)
t/3b
means
t / (3 x b)
Substitution
Algebraic Expressions: Expressions involving algebraic
symbols.
2x + 3 <
5x + 2 y + 1 • *
an expression with one variable.
an expression with two variables.
Solving expressions with one variable
Example: find the value o f 2x + 1, when x = 3
Substitute the numerical values 3 for the algebraic symbol x or
variable.
2x + 1 := 2 x 3 + 1
= 6+1
=
7
14
Some Elementary Hints in Mathematics
Solving expressions in two variables
Example:
find the value of 3 a - 2b, when a = 5 and b = 3
Let us substitute the values of a and b
Then3a-2b = 3 x 5 - 2 x 3
= 15-6
= 9
Substitution related to equations
Example:
in the equation y = 6m - 2 find the value of y
when m = '/z.Let us substitute the value m = 1/2 in the equation
Then y = 6m - 2
= 6x1/2-2
= 3-2
= 1
Example:
Evaluate x + y - 4 i f x = 5 and y = 3
First replace each variable in the expression with its value.
Then use the order of operations.
x + y- 4 = 5+ 3- 4
= 8-4
= 4
Example:
Evaluate 2a + 3b if a = 4 and b = 12
2a + 3b = 2(4) + 3(12)
= 8 + 36
= 44
2
Example:
Evaluate b / 3a if b = 6 and a = 3
The bar, which means division, is also a grouping symbol.
Evaluate the expressions in the numerator and denominator
separately before dividing.
S o m e Elementary Hints in Mathematics
2
b /3a
2
=
6
=
36/9
=
15
/ 3 x 3
4
Equations
A situation that can be represented b y a m a t h e m a t i c a l sentence
w i t h a n equal sign is called an e q u a t i o n .
Equation
50- 30= y
T h e v a l u e that m a k e s sentence (statement) t r u e is c a l l e d t h e solution o f
that e q u a t i o n . I n t h e a b o v e e q u a t i o n 5 0 - 3 0 = y , 2 0 i s t h e s o l u t i o n o f
that e q u a t i o n . T h e process o f f i n d i n g t h e s o l u t i o n i s c a l l e d solving the
1
equation.
S o m e e q u a t i o n s d o n o t h a v e the v a r i a b l e a l o n e o n o n e side o f the e q u a l
s i g n . F o r e x a m p l e , let us l o o k at t h e e q u a t i o n 4 5 + a = 6 4
T r y 19 f o r v a l u e o f a
T h e n y o u get 4 5 + 1 9 = 6 4
64 = 64
true
Addition Property o f Equality:
I f y o u add the same n u m b e r t o each side o f an equation, then the
sides r e m a i n equal.
Arithmetic
15 = 5
5 + 2 = 5 + 2
7 = 7
Algebra
a = b
a +c= b + c
16
Some Elementary Hints in Mathematics
Subtraction Property o f Equality:
I f you subtract the same number from each side o f an equation,
then the sides remain equal.
Arithmetic
Algebra
5 = 5
a= b
5-2=5-2
3 = 3
a - c= b - c
Examples:
1. solve y - 34 = 15
y - 3 4 + 3 4 = 15 + 34
y
2.
=
49
solve p + 17 = 4 8
p+17-17
p
=48-17
=31
Solving statement problems
Kanthi is thinking o f a number. She divides her number by 4
and subtract 6.The result is 3.What number Kanthi is thinking
of?
Let us think
• divided by 4 — •
subtract 6 — • result is 3.
K.anthis number as y
N o w let us write the equation:
y/ 4 - 6 = 3
y / 4 - 6 + 6 = 3+6
y/4 = 9 '
4 x y / 4 = 9x4
y = 36
Some Elementary Hints in Mathematics
17
Writing Expressions and Equations
Verbal Phrase
Algebraic Expression
8 more than a number
n+ 8
A number decreased by 10
m - 10
The sum o f twice a number and four
2y + 4
A number divided into 5 groups
m / 5
Verbal statements may be translated into equations. The equations
can often be used to solve a problem.
Verbal Statement
Algebraic Equation
Three is five more than a number
3= a+ 5
Four times a number is one hundred
4n = 100
Example:
Ravi has Rs.5 more than the twice the amount Kapila has. Ravi
has Rs. 15. W r i t e an equation to represent this problem.
Let y = the amount Kapila has
T w i c e the amount Kapila has
Rs.5 more than that—
•
>
Ravi's amount, Rs.15, equal this
2y
2y + 5
> 2y + 5 =15
H o w to find the value o f y
2 y + 5 = 15
2y+ 5 - 5 = 15-5
2y / 2 = 10 / 2
y=
5
<
equation
subtract 5 from each side
divide each side by 2
18
Some Elementary Hints in Mathematics
Prime Factorization
In mathematics, w e use factoring to separate a number
into
smaller parts. T h e basic elements o f a number are its factors.
W h e n a whole number greater than 1 'uas exactly two factors, 1
and itself, it is called a prime number. For example, 5 is a prime
number since it has two factors, I and 5.
A n y whole number, except 0 and 1 , that is not prime can be
written as a product o f prime numbers. W h e n a whole number
greater than
composite
1 has more than two factors, it is called a
number. For example 6 is a composite number
since it has four factors, 1 , 2 , 3 and 6.
The numbers 0 and 1 are neither prime nor composite. Notice
that 0 has an endless number o f factors and that I has only one
factor, itself.
T o find the prime factors o f any composite numbers begin by
expressing number as a product o f two factors. Then continue to
factorize until all the factors are prime. W h e n a number is
expressed as a product o f factors
that
are all prime. T h e
expression is called the prime factorization o f the number.
Example:
84 = 2 x 2 x 3 x 7
90 = 2 x 3 x 3 x 5
The diagrams below each show a different way to find the prime
factorization o f 24. These diagrams are called factor trees.
24
24
2
x
3
x
2
x
2
3
x
2
x
2
x
2
Some Elementary Hints in Mathematics
19
24
Every number has a unique set o f prime factors. Notice that the
bottom row o f "branches" in each factor tree is the same except
for the order in which the factors are written.
Greatest Common Factor
The greatest o f the factors common to two numbers is called
the greatest c o m m o n factor ( G C F ) o f the numbers. Y o u can
use prime factorization to find the G C F . Consider the prime
factorization o f 84 and 9 0 below.
84
2
2
x
^
2
90
x ^
x
3
42
2 ^
x
7
2
x
3
x ^
x
45
3 x
T h e integers 84 and 9 0 have 2 and 3 as common factors. T h e
product o f these prime factors, 2 x 3 or 6 is the G C F o f 84 and
90.
Example: Use prime factorization to find the G C F o f 84,126
and 2 1 0 .
W r i t e each number as a product o f prime factors.
5
Some Elementary Hints in Mathematics
20
84 = 2 x 2 x 3 x 7
126 = 2 x 3 x 3 x 7
2 1 0 = 2 x 3 x 5 x 7 the common factors 2, 3, and 7
Thus, the G C F is 2 x 3 x 7 or 4 2 .
L e a s t C o m m o n Multiple ( L C M )
The least o f the common multiple o f two or more numbers is
called the least c o m m o n multiple ( L C M )
Example:
Find the L C M o f 2 , 4 and 6
Method 1:
Multiples o f 2 = 0, 2, 4, 6, 8, 10, 12
Multiples o f 4 = 0, 4, 8, 12, 16, 2 0 ,
Multiples o f 6 = 0, 6, 12, 18, 2 < 3 0 ,
C o m m o n multiples are 12, 2 4 , 36, 4 8 ,
T h e r e f o r e L C M is 12
Method 2: Prime factorization can be used to find the L C M o f a
set o f numbers.
A common multiple contains all the factors o f each number in
the set. The L C M contains each factor the greatest number o f
times it appears in the set.
2
=
2
x
1
21 = 3 x 7
4
=
2
x
2
25 = 5 x 5
6
=
2
X
9 = 3x3
1
2
x
2 x 3 = 12
The greatest power o f 3 is 3
2
The greatest power o f 5 is 5
2
The greatest power o f 7 is 7
l
Some Elementary Hints in Mathematics
2
21
2
Therefore the L C M o f 21,25,9 is 3 x 5 x 7 ' = 1 575
Method 3:
find the L C M o f 6, 8, and 12
2 - 6 , 8 , 12
2
3,4,6
3
G o on dividing the set o f numbers
3,2,3
2
by prime factors
1,2,1
Then you get
2x2x2x3
Therefore the L C M is 24
P e r i m e t e r and A r e a
The perimeter o f a geometric plane figure is the sum o f the
measures o f all sides. T o find the perimeter o f the rectangular
piece o f glass you can add up the length o f all the sides or you
can use an equation. The equation for the perimeter o f any
rectangle is P = 2a + i2b where ' a ' represents the length and ' b '
represents the width.
Example:
11 cm
P = 2a + 2b
4cm
= .
2 ( 1 1 ) + 2(4)
=
22 + 8
=
30cm
A square is a special rectangle in'which the lengths o f all sides
are equal. The values for a and b in the perimeter equation are
the same number. For this reason, the perimeter equation for a
square is often written as P = 4s, w h e r e ' s ' is the length o f a
side.
22
Some Elementary Hints in Mathematics
Example:
5 cm
P = 4s
5 cm
5 cm
-4(5)
= 20cm
5cm
Squares and rectangles are special types o f parallelograms. Each
pair o f opposite sides o f a parallelogram are parallel and have the
s a m e length.
T o find the perimeter o f a parallelogram, you add the length o f
the sides.
Example:
7cm
P = 2a + 2b
= 2 ( 7 ) + 2(6)
= 14+12
= 26cm.
7cm
T o find the perimeter o f a triangle, you add the length o f all 3
sides.
Example:
9cm
23
Some Elementary Hints in Mathematics
Area
The area of a geometric plane figure is the measure of the surface
enclosed by the figure. The area of any rectangle can be found by
multiplying the length and width.
Rectangle
square
parallelogram
b
a
A= ab
x
A=x
b
2
A' = bh
Trapezium
b
a
A = l/2(a + b).h
Use of Venn Diagrams to Solve Problems.
Example: There are 27 students in a class. 14 of them are
members of the Tennis Club, 7 are members of the Basketball
Club. 3 of them are members of the both the Tennis Club and
Basketball Club. How many students are not in either Club.
To solve the above problem you can use a Venn Diagram to
show the above information as follows.
24
Some Elementary Hints in Mathematics
A d d to find the number o f students in either Clubs or in both
Clubs.
4 + 3 + 11 = 18
subtract to find the total number o f students who are not in either
Club.
27-18 = 9
N o w try this problem
N u m b e r o f students passed in Maths, Science ,and English is
represented in the above V e n n Diagram. Find the following data
by using the above diagram.
25
Some Elementary Hints in Mathematics
1.
2.
N u m b e r o f students passed in Maths?
• N u m b e r o f students passed in all 3 subjects?
,
3.
Numbers o f students passed in Maths and English? *.
4.
N u m b e r o f students passed in Science and Maths only?
M a i n R e p r e s e n t a t i o n a l Values in Statistics
•
Example: Grade nine students were asked how many times they
have eaten ice-cream lastweek.
the results were 3, 0, 1, 1 , 5, 2, 0, 1, 8
W r i t e the numbers in order from the least to the gratest.
0, 0 , 1 , 1 ,
1,2,3,5,8
the R a n g e is the difference between the greatest and the least
number.
8-0 = 8
the Mode is the number that occurs most often.
There fore the mode is 1
The
Median
is the middle number,if they are arranged in
ascending (increasing) or discending (decreasing) order.since the
data
has
a nine
numbers
the
middle
number
is
the
fifth
one.therefore the median also one here.
0, 0 , 1 , 1 , ( 1 ) 2 , 3 , 5 , 8
i f there are two middle numbers in a group o f data.to find the
median o f
0 , 1 , 2 , 3,|5, 6,|7, 7, 8, 9 add the two middle numbers and divide
the number b y 2.
26
Some Elementary Hints in Mathematics
Hence 5 + 6 = 1 1
is 5.5
11 / 2 = 5.5
therefore the median
M e a n : the sum o f data divided by the number o f data. W h e n w e
consider the above example the mean =
( 0 + 1 + 2 + 3 + 5 + 6 + 7 + 7 + 8 + 9 ) divided by 10 = 4 8 / 10 Hence the
answer is 4.8
R e p r e s e n t a t i o n o f D a t a in a F r e q u e n c y T a b l e
These are often used in statistics. Statistics is a branch o f
mathamatics that deals with collecting, organizing and analyzing
data.
One type o f table used in statistics is a frequency table.
A frequency rable tells how many times each piece o f data
occurs in a set o f information.
The tabic below shows the result o f a survey done by a sports
club regarding some games
Game
Soccer
Hockey
Cricket
Tennis
Tally
fHJ fW fW fW /
rw rw rw rw a
fW fHJ fW ///
rw rw rw ////
Frequency
21
22
18
19
W h e n statisticians study the data and make conclusions from the
numbers they are observing, w e call it data analysis. Some times
when there is a wide range o f data, statisticians w i l l group the
data into intervals.
Some Elementary Hints in Mathematics
27
In a large set o f data, such as exam results, it is helpful to
separate the data into four equal parts called quartiles.
Interquartile Range:
T h e inter quartile range is the range o f the middle half o f data.
Example:
Step 1. Find the median o f the data since the median separate the
data into halves.
3, 3, 5, 6, 6, 7, 10, 10, 10, 12, 12, 14, 16, 17, 20
Median
Step 2. Find the median o f the upper half. This number is called
U p p e r Q u a r t i l e , indicated by U Q .
Step 3. Find the median o f the lower half. This number is called
the L o w e r Q u a r t i l e , indicated by LQ.
Therefore the interquartile Range = U Q - L Q = 1 4 - 6 = 8
According to the above data the interquartile Range is 8.
28
Some Elementary Hints in Mathematics
Construction of B a r Graphs
In a bar graph the data are represented in the form o f bars.
• No. of Student
English
French
German
Russian
Horizontal Axis represents the languages chosen
Vertical Axis represents the number o f students
Use the above graph to answer the following questions.
1.
2.
W h a t language is mostly spoken by the six graders?
H o w many students have chosen French?
3.
H o w many students have chosen Russian?
4.
H o w many students are in grade six?
5.
What is the difference between the number o f students
who have chosen English and the number o f students who
have chosen German?
Some Elementary Hints in Mathematics
2 9
The Metric System
The metric system was created by a French scientist in
the late 1 8 century, as a standard Measurement. The United
State is the only large nation of the world that does not
commonly use this system, the Metric system is a decimal
system, that means it is based on 10.
th
The standard unit of length in the Metric system is the
meter(m).the standard of capacity is the liter ( L ) and the
standard unit of mass is the gram (g).
Prefix
kilohectodecadecicentimilli-
Symbol
k
h
,da
d
c
m
Meaning
Example
1000
100
10
0.1
0.01
0.001
1km = 1000m
lhL = 1 0 0 L
l d a g = lOg
ldg = 0 . 1 g
1cm = 0.01m
lmL = 0 . 0 0 1 L
The diagram bellow shows how you change one unit to another by multiplying or b y dividing.
xlO
xlO
i
xlO
11
kilo ( k m )
hecto- ( h m )
t
11
ii
deka- (dam)
meter ( m )
It
divided by
10
xlO
It
divided by
10
xlO
ii
10
11
deci- (dm)
Jt
divided by
xlO
IL
divided by
10
i
centi (cm)
milli (mm)
It
divided by
10
I
divided by
10
Some Elementary Hints in Mathematics
Relationship
between
31
units of measurement in the
metric system
W h e n you multiply b y 10 ,100 and 1000 you can determine the
answer b y moving the decimal point right as many places as
there are zeros in the multiplier.
1 0 0 x 6.54 = 6 5 4 . = 654
1 0 0 0 x 6 . 5 4 = 6 5 4 0 . = 6540
W h e n you divide b y 10,100, and 1000 you move the decimal
point left as many places as you have zeros in the divisor.
78/1
=
78
78/10
=
7 £ / =• 7.8
7 8 / 100 =
0.78 = 0.78
7 8 / 1000=
0.078 = 0.078
Common Termminology used in Mathematics
Base
Absolute
Adjacent
6)cJO
Basis
Addition
d r s q j iSS*®
Basic
Adjustment
Sdi®3di©
Bearing
OC.25}®
Addi'ive Bond
Bisector
A d d i t i v e Inverse
Bisection
C3®0®l5^®C3
0®C®(5^2»C3
Binary
<S®C3
Algebraic
9c5c3
Bilateral
Algebraic Expression
Algebraic Function
86c3 @C1C3
Bar Graph
Allied Angles
Binomial
<f9oe.
Alternate Angle
Capacity
Altitude
Circle
QoSSM©
Qaeftscs
A n g l e o f Depression
Circumference
oSSca
Corresponding A n g l e
Angle o f Elevation
A n t i - c l o c k w i s e (Counter Clockwise)
Oo®aQt)C&
Congruent
Approximate
Curve
Axioms
Curvature
Axis
Chord
Axis of Symmetry
Component
A x i s o f Rotation
Composite
Arithmetic
Co-ordinate
A r i t h m e t i c Progression
Concave
A r m s o f an A n g l e
Arc
Convex
6>3C3C3
cotee
Complex
Equal
Complementary
Equality
Coefficient
Equilibrium
Constant
Equivalent
Conclusion
Equilateral
»®0S)
o®q>Ss«5>3cD
Concept
Equation
Consequent
Element
Consecutive
Estimation
j9®3S5C3
Construction
Exterior
3a£d
Direct Numbers
esc|ta
C 3 o S s a
esSSzsd-eSca
Factor
Division
Factorization
csoOs) © o S ®
Dividend
Fraction
0)30X9
Divisor
Formula
Denominator
eodca
F l o w Chart
Direction
QOO®
Function
Geometry
Distribution
Distributive Law
Geometrical
Domain
£te©
Geometrical Progression
Decimal
C_<B®
Geoboard
Diameter
Graph
Graphical
C5oSS3CK3
Fixed
Digit
Discount
Frequency
ge&5»o*s>
(§©-e6:fate)d ©cs^Sca
Height
Lamina
epdcodca
Hemisphere
Linear
©bales
Hexagon
Least Common Multiple
2^6)3 ©C33£ Q2€S3Z3Ddc3
Histogram
6dQ ©tJS)C3
Logarithm
Homogeneous
C3®daSc3
Locus
Horizontal
Sdtf
Lattice
C3C5C3
Hypothesis
Linear
©baWt&sd
Index
Line
©babe)
Mean
®OS2S5BC3
Median
®033C3*C)C3
Intersection
Mode
®3C0C3
Inverse
Maximum
CC30®
Inequality
Minimum
cfO®
Irrational Numbers
Monomial
Integers
Infinity
-
Isosceles T r i a n g l e
es®c*So3e~
Improper Fraction
8 ® ® ED3CD
^©zs^-eSca
Matrix
Interior A n g l e
S s S d ©ExJcSca
Magnitude
Intercept
Interest
Multiple
Numerator
©C33f3c3
Numerical
Intermediate
Negative
Inter - Quartile Range
Notation
Irregular
Nomenclature
Grid
Gradient
C2-eS3!3X5c5
ep^ig®-eSc3
C3oS)33as5®23
2S13®Z5>d«SC3
Octagon
Opposite A n g l e
Proportional
g&SjS) © z s o W
C3®3^C3DSS)
Property
Ordered Pair
Proper Fraction
C33®a2S3G CD3CD
Origin
Principle
DeQfr®c3
Obtuse A n g l e
®203 ©JSJ€SC3
Obtuse A n g l e d Triangle
®s>3 © o d ^ S a
Octahedron
«frf3 E5K3C3
Quality
G2-C&C3
Odd Numbers
® e j © £ i C3oS333
Qualitative
(^•«S3Sf®S)
Parallel
C3®02rfCDd
Quantity
g®3«Sc3
Parallelogram
C3®3lrfet>CbQC3
Quantitative
g®o-eS3s5®S)
Percentage
gSoamcs
Quadrilateral
©Qdga
Perfect
o8gt$-©S
Quadrant o f C i r c l e
SacJcO 03C.23C3
Pentagon
OoOsgca
Quadratic Equation
Perimeter
oSSSca
Perpendicular
Plane
^©S)i«sc3
Positive
Quartile
Quotient
Pictogram
SS@C3
Radius
Cfdc3
Probability
c3®to3Sa»©
Random
£f!Og/C3C3®eD38
Product
Process
Rate
3§C33©gc3
Ratio
Place V a l u e
Range
Profit
Real
Progression
Reciprocal
Proportion
C5®3QO30SO
Cf2gc33EDC3
odefodcs
Rectangle
Simultaneous
cs®eoo®
Rectangular
eaa^eaxJ-eSagaaxKJ
Simplification
gdSa®
Rectilinear
eso*c3 ©bolca
Statement
gzswcsxa
Recurring
Statistics
C3o£)B3CX3
Reflex Angle
Set
3>Cmca
Regular
oSS
Straight L i n e
Relation
Surface
oaaftSca
Relationship
e3®S>2rfQO»0
Square
o®©^dgca
Relative
C33©C3*»I»
Substitution
c?30<f(B S S ®
Successive
cfzgcaacD
Representation
Representative
Value
fidtaca
ejeDca
Right Angle
Symmetry
ea®®fic3
Systematic
g®Od
gco-eSs)
Right Angled Triangle
e a a ^ S Q J S ® @©a>J-eSca
Tally
Rotation
gS-eSca
Tetrahedron
(325-eS
Rhombus
©dD®Sesc3
Theory
ScJQaafaxa
Rhomboid
®Cb®3acDC3
Theorem
g©®caca
Rounding Off Numbers
C5oS)B3 O x O Q ®
Term
oca
Remainder
©C&BC3
Triangle
^©sxf-esca
Scale
08®3<SC3
Triangular
^©csji-^SzB
Scalene Triangle
5 » ® ooc;
Transversal
fiBcari
Sector
©afcrfgss S ) « £ Q c a
Transformation
Sequence
cj^®ca
Translation
Caf033d<SC3
Similar Triangle
C3®d|8 @©S)3*-GSC3
Tessellation
©OcaeoocS-eSca
Similarity
ca®di8o»£)
fijerai-efica
®B5bQ
Unit
Value
q'Goca
U n i t Fraction
Variable
Se)e»c3
Universal
Variation
S&csrc
Unitary
Vertex
Seleses
Unequal
Uniform
Vertical
dzS>3S5)3d
Bdd
Vertically Opposite A n g l e g S g z s ©sDieSca
Unknown
Vector
©©qSzsjcs
Unlike
Volume
oS®3©
SOLIDS
Cube
usages
Cuboid
Cone
Cylinder
632512336X3
Prism
Pyramid
Tetrahedron
§C*®C3
Sphere
Complex
8d@£Jc3
©codecs
About the author
Mr. M . S . M . Dahlan
M r . M . S . M . D a h l a n is a veteran teacher i n m a t h e m a t i c s
who
has a l m o s t
35-year
experience
in
teaching
m a t h e m a t i c s i n g o v e r n m e n t schools. A n d also he has
taught i n an a t o l l s c hool i n t h e R e p u b l i c o f M a l d i v e s
f o r t w o years.
M r . D a h l a n has served as a Project O f f i c e r attached t o
the M a t a r a D i s t r i c t E d u c a t i o n a l D e v e l o p m e n t Project
( M E D P ) f u n d e d b y S I D A , f r o m 1993 t o 1995.
H e has served i n the Subject Specialist Panel o f the
Text B o o k s E v a l u a t i o n C o m m i t t e e o f the E d u c a t i o n a l
Publication Department, Isurupaya, Battaramulla. In
a d d i t i o n t o that he w a s a resource p e r s o n attached t o
the B i l i n g u a l E d u c a t i o n P r o g r a m m e o f the N a t i o n a l
Institute o f M a h a r a g a m a .
H e r e t i r e d f r o m g o v e r n m e n t service i n 2008 after a
c o m m e n d a b l e service at St. T h o m a s ' C o l l e g e M a t a r a
for over
fifteen
years a n d presently attached t o t h e
Academic Staff o f Ceylinco-Sussex College, Matara.
I S B N 978-955-590-096-6
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