G OVT. OF T AMILNADU D EPARTMENT OF S CHOOL E DUCATION B RIDGE C OURSE C LASS VIII- M ATHS

M ATHEMATICIANS

P YTHAGORAS 569 B.C. – 475 B.C. Greece First pure mathematician 5 beliefs Secret society Pythagorean theorem

A RISTOTLE 384 B.C. – 322 B.C. Greece Philosopher Studied mathematics in relation to science

E UCLID 325 B.C. – 265 B.C. Greece Wrote The Elements Geometry today

A L -K HWARIZMI 780 A.D.-850 A.D. Baghdad (in Iraq) 1 st book on Algebra Algebra Natural Number Equation

Alileanardo-da-Vinci 1452 A.D A.D Italy Painter Architect Mechanic

Galileo-Galili 1564 A.D. – 1642 A.D. Italy Teacher Mathematician

ALGEBRA Ty pes of polynomials: Polynomials are named according to the number of terms present in them. Monomial: Binomial: Trinomial They are special names given to a polynomial.

Classify the following into monomials, binomials and trinomials: (A) x + 1 (B) 3m ² (C) 2x ²- x-4

F IND THE VALUE OF THE VARIABLES 31.5 = a y = 57 x = x - 32 = y = 119 A LGEBRA

A REA Area measures the surface of something. 1 metre 1 square metre 1m 2 1 cm 1cm 2 1 mm 1mm 2

A REA OF A RECTANGULAR LAWN 15 square metres The area is 5 metres long 3 metres wide Area of a rectangle= Length x Width = 15 m 2

Page 21 cm 30 cm = 30 x 21 Area of the page = 630 Field 65 m 32 m = 65 x 32 Area of field = 2080 cm 2 Area of rectangle = Length x Width m2m2 = 300 x 210 Area of the page = mm mm 210 mm

U NITS OF A REA 1 cm 1 square centimetre 1 cm 2 = 100 mm 2 1 cm 2 = 10 2 mm 2 1 cm = 10 mm

1 m 2 = 100 x 100 cm 2 1 m = 100 cm 1m 1 square metre 1 m 2 = cm 2 U NITS OF A REA 1 m 2 = 1000 x 1000 mm 2 1 m 2 = mm 2 1 m = 1000 mm 1 m 2 = cm 2 = mm cm 1000 mm

P ERIMETER Perimeter of a shape is the total length of its sides. Perimeter of a rectangle length width length width = length + width + length + width P = l + w + l + w P = 2l + 2w P = 2(l + w)

5.2 m 4.5 m 3 m3 m 3 m3 m 1.5 m 2.2 m Example L-shaped room Perimeter = A B = 19.4 m Area of A= 4.5 x 3 = 13.5 m 2 Area of B= 3 x 2.2 = 6.6 m 2 Total area = = 20.1 m 2

V OLUME is the amount of space occupied by any 3- dimensional object. 1cm Volume = base area x height = 1cm 2 x 1cm = 1cm 2

Side 2 Bottom Back Top Side 1 Front Side 2 Bottom Back Top Side 1 Front Length (L) Breadth (B) Height (H) Cuboid

T HE NET L L L L B H H H H L B BB BB H H

H B L H B H L L L H B L H B H L L Total surface Area = L x H + B x H + L x H + B x H + L x B + L x B = 2 LxB + 2BxH + 2LxH = 2 ( LB + BH + LH ) Total surface Area

C UBE Volume = Base area x height = L x L x L = L 3 L L L Total surface area = 2LxL + 2LxL + 2LxL = 6L 2

2(LxB + BxH + LxH) LxBxH Cuboid 6L 2 L3L3 Cube Sample netTotal surface area VolumeFigureName

V OLUME OF A C YLINDER

W HAT IS THIS ? It has 2 equal shapes at the base, but it is not a prism as it has rounded sides It is a Cylinder

E XAMPLE V = Base area x Height =  r 2 X h

The Circle

O A CIRCLE Eg. ball,bangle,lemon.coin

O A CENTRE : O RADIUS : OA B CENTRE RADIUS

. O In a plane,each point of the circle is at equal distance from a fixed point.The fixed point is called the centre of the circle. CENTRE = O

O The distance from centre to any point on the circle is called radius of the circle. A RADIUS = OA Radius

. O D I A M E T E R A Line segment passing through the centre of the circle and whose end points lie on the circle is called the diameter of the circle. A B DIAMETER = AB

. O The length of the circle or the distance around it is called circumference of the circle.

Relation between radius and diameter. O A B RADIUS = DIAMETER 2 OB = AB 2

E G. IF IF DIAMETER OF THE CIRCLE IS 10 CM THEN FIND ITS RADIUS ?. O Diameter = 10cm Sol. Radius =diameter / 2 radius=10 cm/ 2 radius =5 cm AB

LINES AND ANGLES

PARALLEL LINES Def: line that do not intersect. Illustration: Notation: l | | m A B | | CD l m A B C D

PERPENDICULAR LINES Def: Lines that intersect to form a right angle. Illustration: Notation: m  n Key Fact: 4 right angles are formed. m n

Def: a line that intersects two lines at different points Illustration: T RANSVERSAL t

V ERTICAL A NGLES Two angles that are opposite angles t  1   4  2   3  5   8  6   7

V ERTICAL A NGLES Find the measures of the missing angles 125  ? ? 55  t 125 

S UPPLEMENTARY A NGLES / L INEAR P AIR Two angles that form a line (sum=180  ) t  5+  6=180  6+  8=180  8+  7=180  7+  5=180  1+  2=180  2+  4=180  4+  3=180  3+  1=180

S UPPLEMENTARY A NGLES / L INEAR P AIR Find the measures of the missing angles ? 72  ? t 108 

A LTERNATE I NTERIOR A NGLES Two angles that lie between parallel lines on opposite sides of the transversal t 33   6 44  

A LTERNATE I NTERIOR A NGLES Find the measures of the missing angles 82  ? t 98  82 

A LTERNATE E XTERIOR A NGLES Two angles that lie outside parallel lines on opposite sides of the transversal t 22   7 11  

A LTERNATE E XTERIOR A NGLES Find the measures of the missing angles 120  ? t 60  120 

C ONSECUTIVE I NTERIOR A NGLES Two angles that lie between parallel lines on the same sides of the transversal t 33 +  5 = 180 44 +  6 =

C ONSECUTIVE I NTERIOR A NGLES Find the measures of the missing angles ? t 135  45 

GRAPH

G RAPH THE FOLLOWING LINES Y = -4 Y = 2 X = 5 X = -5 X = 0 Y = 0

A NSWERS Y X x = 5 x = -5

A NSWERS Y X y = -4 y = 2

M EAN

D EFINITION Mean Mean – the average of a group of numbers. 2, 5, 2, 1, 5 Mean = 3

M EAN IS FOUND BY EVENING OUT THE NUMBERS 2, 5, 2, 1, 5

Copy right © 2000 by Moni ca Yusk aitis M EAN IS FOUND BY EVENING OUT THE NUMBERS 2, 5, 2, 1, 5

M EAN IS FOUND BY EVENING OUT THE NUMBERS 2, 5, 2, 1, 5 mean = 3

H OW TO F IND THE M EAN OF A G ROUP OF N UMBERS Step 1 – Add all the numbers. 8, 10, 12, 18, 22, = 96 Step 2 – Divide the sum by the number of addends = 96/6 = 16

H OW TO F IND THE M EAN OF A G ROUP OF N UMBERS The mean or average of these numbers is 16. 8, 10, 12, 18, 22, 26

W HAT IS THE MEAN OF THESE NUMBERS ? 7, 10, 16 11

W HAT IS THE MEAN OF THESE NUMBERS ? 2, 9, 14, 27 13

W HAT IS THE MEAN OF THESE NUMBERS ? 1, 2, 7, 11, 19 8

W HAT IS THE MEAN OF THESE NUMBERS ? 26, 33, 41, 52 38

D EFINITION Median Median – the middle number in a set of ordered numbers. 1, 3, 7, 10, 13 Median = 7

H OW TO F IND THE M EDIAN IN A G ROUP OF N UMBERS Step 3 – If there are two middle numbers, find the mean of these two numbers = 46 2)2) 23 median

D EFINITION Mode Mode – the number that appears most frequently in a set of numbers. 1, 3, 7, 10, 13 Mode = 1

Find the mode for the following 7, 4, 5, 1, 7, 3, 4, 6, 7

Find the mode of the following frequency table:- If the data are arranged in the form of a frequency table the class corresponding to the maximum frequency is called the model class. The value of the variate of the model class is the mode. x F