Math bolg on grade VIII

Friends, today i am going to teach you one of the interesting topic of mathematics, formulas for measurements . In  grade  VII  we  study  about  some  geometrical  figures  like  -  line, Triangle, circle  etc ,  but  in  grade VIII of maharashtra board we  study,  what  kind  of  formulas  are  we  use  to  measure  these  geometrical  elements and  learn  formulas for measurements which is  a very important part of online tutoring . Here  measurement means finding  distance , area  and  volume  of  geometric figures  and  so  we  divide  measurement  of  geometrical  figures  in  three  units –

• measuring   distance 
• measuring   area  of   geometrical  figures
• measuring  volume of  geometrical  figures

So,  first  of  all  we  discuss  formula  of  measurement of  distance  :
 1 )   if  there is  two  points P( x₁ , y₁ )  and   Q(x_{2 ,}y₂)  in  a  plane ,   then  formula  to   measuring distance  between  these  two  points   PQ  is  –
    sqrt ((x₂ – x₁ )²  +  ( y₂ –  y₁ )² )
    here   sqrt  means  square  root
2)  if  there  is  two  points  P( x₁ , y₁ , z₁  )  and   Q(x_{2 ,}y₂ , z₂ )  in  a  space ,  then formula  to  measuring  distance  between  these  two  points  PQ  is –
    sqrt ((x₂ – x₁ )²  +  ( y₂ –  y₁ )²  + (z₂   -  z₁ )² )
    here   sqrt  means  square  root and to know more about measurements click here,
Then  we  further  proceed  to   discuss   measuring  area  of   geometrical  figures  :

1.  Area  of   triangle     =    1    *   b  *  h
                                          2

                       here   b  refers    base  length   of  triangle   and   h  refers  to  height  length  of  triangle
2.   Area  of   square       =     a²
               here  a  refers  to  one  side  of  square
3.    Area  of  rectangle   =      a* b
                here  a  refers  to  length  of  rectangle and  b  refers  to  width  of  rectangle
4.    Area   of   parallelogram   =   b  * h
                here  b  refers  to  one  side  length  and  h  refers  to  width  between  two  sides
5.    Area  of   circle   =   pi * r²
                here  pi = 3.14  and   r  refers  to  radius  of  circle
6.    Area  of  eclipse  =  pi * r₁  * r₂
                here  pi = 3.14  and  r₁  and  r₂ are  two side  radius  of  eclipse
7.     Area  of   sphere  =  4 * pi * r²
                here  pi = 3.14  and  r  refers  to  radius  of  sphere
These  are  standard  area  formulas  which  use  to  measure  area  of   geometric  figures  and    we  further  proceed  to  discuss  formula  of  measuring  volume  of  geometrical  figures –

1. Volume  of  cube  =  a³ 

         here  a  refers  to  each  side  of  cube
2.  Volume  of  Cuboid  = a * b * c
         here  a  shows  length , b  shows width  and  c  shows  height
3.  Volume  of  cylinder  =  pi *  r² * h
         here  pi = 3.14  , r  refers  to  radius  of  cylinder  and  h  refers  to  height  of  cylinder
4.  Volume  of  pyramid   =   1    *  b *  h                            
                                              3
         here  b  refers  to  width  of  pyramid  and  h  refers  to  height  of  pyramid
5.  Volume  of   sphere    =    4    *   pi  *  r² *  h
                                              3
         here  pi = 3.14  ,  r   refers  to  radius  of  sphere  and   h  refers  to  height  of  sphere


These  are   standard  formulas for measuring  volume  of   geometrical  figures and  You can also refer Grade IX blog for further reading on Basic constructions.Read more maths topics of different grades such as Range in Grade VIII  in the next session here. 

Learn Geometry and Transformation on a coordinate plane

Hello Friends, in today's class we all are going to discuss about one of the most interesting and a bit complex topic of mathematics, geometry. In VIII grade gujarat board students  are  familiar  with algebraic expressions  of  mathematics , but  geometry  is  new  subject  for  them  . First  of  all   one  question  arises  in  student’s  mind  what is  geometry ? Basically  when  we  study  about  shape like Triangle, size  and figure  of  space   than  this  study  is  called  as  a  geometry (Learn more about natural numbers here),

like when two points  meets  it  makes  line  and  four  lines  make  rectangle  , square  etc  .  For  understanding  the properties  of  geometry  what  grade VIII  student  supposed  to  do,  they  have  to  study  about  coordinate  system  and   Transformations on a coordinate plane . coordinate  system  is  basically  a  system   which tell   position  of   points  or  position  of  geometry element . We  can  understand coordinate  system  by  coordinate  plane  .  So,  first  of all  we  have  to  understand coordinate plane . coordinate  plane  is  basically   two  intersecting  lines (combination  of  horizontal  and  vertical  lines)  where  horizontal line  is  known  as  a  x  coordinate  and  vertical  line  is  known  as  a  y coordinate , for  example  we  have  a  line  on  a  plane   whose  one  point  coordinate  is  (0,1)  means  x  direction  coordinate  is  0  and  y  direction coordinate  is  1 .  But  when  we  want  to study  about  3D  coordinate  of  space,  we  have  to  include  z coordinate  on  plane  which  shown  outward  direction  form plane  and  in  3d ,  coordinate  plane  work  as  a  ( x, y, z )  coordinates .
                                       Now , when  we  move  some  point  with  same  distance  in  same  direction  on   plane ,  then  this  process  is  called  as  a  Transformations on a coordinate plane . We  can  understand  Transformations on a coordinate plane  with  an  example  like  when  we  move  table  from  one  place  to  another in  room ,  then  we  have   to   move  each  bottom legs  of  table    with  same  distance in  same  direction .  In  mathematical  point  of  view , when  we  move  one  object P (x , y)   with      a  units  in  horizontal  direction ( moves  right )  and  b  units  in  vertical   direction ( moves  upward)   on   coordinate  plane ,  then   after  this  Transformations on a  coordinate plane  process   coordinate  of  object  P( x +a , y +b) . Here  one  think  is  noted  that  in  horizontal  direction  moves  right  shows  positive  movement  and  moves  left  shows  negative  movement  of  object   and  in  vertical  direction   upward  movement  shows  positive  movement   and  downward  movement  shows  negative  movement  of  object .  So, we  take  some  example  to  understand  Transformations on a coordinate plane –
Let  one  point  A (2 , 3)  is  moves  right  with  2 unit  and  moves  up  with  5  unit  then  after  transformation  coordinate  of   A ( 2 + 2 ,  3 + 5 )  or   A ( 4 , 8 ) .
Similarly when  A( 2, 3 )  is  moves  left with  1 unit  and  move  up  with  3  unit  then  after   transformation   coordinate  of  A ( 2 – 1 ,  3 + 3)   or   A ( 1 ,  6)
So, I  hope  u  understand  about  geometry  portion  of Eighth Grade syllabus and Read more maths topics of different grades such as Basic constructions
in the next session here.

Factors and Number Sequences in Grade VIII

We have studied a lot about numbers till now. We have come across different types of numbers in various grades of different education board, which includes Natural Numbers (used for counting), whole numbers (used for measurements), Even and Odd Numbers. Now we will proceed and learn more about Math problems associated with it.
Let us first talk about Factors of a given number. Any number which exactly divides any number is called the factor of any number. Let us take 16.
 16 is exactly divisible by 1, 2, 4, 8, and 16 . So 1,2,4,8, and 16 itself are the factors of 16.
Also we know that 1 is the factor of all the numbers as any number multiplied by 1 gives the number itself.
Multiple of a number: A number is multiple of all its factors, which means if we multiply any factors we get the number. In the above mentioned example we find that 16 comes in the table of 1, 2, 4 ,8 and 16 , so 16 is the multiple of all its factors. To find the multiples of any number we simply write its table as
 multiples of 5 are 5, 10, 15, 20, 25, ------- and so on.
Primes : Any number is called a prime number if it has only 1 and itself as two of its factors. It means that a number is prime, if it is divisible by 1 and itself.
Composite Numbers: The number which has factors other than 1 and itself are called Composite Numbers. The smallest composite number is 4 as the factors of 4 are 1, 2 and 4 itself.
 Co- primes: If we have two numbers, not necessary primes, such that they do not have any common factor except 1 are called co- prime numbers. Ex 4 , 5.  Here common factors of 4 & 5 is only 1. So they are co- prime numbers.
Twin Prime Numbers: A pair of  numbers which have two consecutive odd numbers are called twin primes. Ex: 3, 5   and  11, 13 are consecutive odd as well as consecutive prime numbers.Know more about factors here,
We come across such number sequences in our day to day mathematical work.

We should remember the following facts related to numbers:
a. 1 (one) is a number which is neither prime, nor composite.
b. 2 is the only even number which is prime. All other prime numbers are odd numbers.
c. 2 is the smallest Prime number.
 Prime factorization: To find all the factors of any number which are all primes they are called prime factorization. Suppose we need to find the factors of 36.
 36 can be written as 36 = 2 * 18, but 18 is not a prime,
 further we write 18 as 2 * 9, but 9 is again not prime,
 again we write 9 as 3 * 3, which are primes,
so we can write 36 = 2 * 2 * 3 * 3, where we get all the numbers as prime factors.

We observe that factors can be written in different order, but the product remains the same. This is called unique property of factorization.  Factorization also helps in finding the L.C.M. and H.C.F. of any given numbers. Also various numerical sequences are formed using the factors.

So children, this is a brief article about prime numbers, factors, multiples, and number sequences and If you want to know about Variables in VIII Grade and also about Congruence in Grade VII then there are many websites available on the internet where you can get detailed knowledge about all the topics.

Probability in Grade VIII

When we talk about the possibility of occurrence of any event, it is called probability and its an important part of cbse syllabus.
Any occurrence of the experiment is called an Event. Events can be dependent events or independent events.
Children if we talk about a independent event, in any experiment if the event occurs independently then it is called independent event. For example, if we throw a dice, the possibility is of  getting any numbers 1, 2, 3, 4, 5 and 6. But we find the possibilities are independent that is it does not depend on anything else, Two events are called to occur independently, if they to not depend on one another. Suppose 2 coins are thrown, we find that on both the coins we have the possibility of getting either a head or a tail. The possible outcomes are ( HH, HT, TH, TT ) but all these 4 outcomes are not dependent on one another, so they are  called Independent events.

Now let's talk about dependent events. When we conduct any experiment such that the observation of 2nd depends upon the observation of first experiment, then the event is called dependent event. You will study this type of math questions in higher grades.
Conditional Probability of any event is the probability condition when we have two events A and B associated with the same random experiment. Then probability of occurrence of  of event A under the condition that event b had already taken place is called conditional event.
In Grade VIII we will learn about basic experiments of probability.To know more about it refer this,
Let us take an experiment of throwing a coin. The possible outcomes for this event will be Head and Tail. Now a question arises that

•     what is the probability of getting a head in this event?

        We observe that the event has one probability out of two for getting a head, so
        P (Getting head ) = 1/2

•  what is the probability of getting a tail in this event?

      We observe that the event has one probability out of two for getting a tail, so

      P (Getting tail ) = 1/2

   Similarly if we study the pack of cards, we know that there are 52 cards in a pack. out of which 13 are ♣ ,13 are ♥, 13 are ♠ , 13 are ♦
   So probability of getting a ♣, if one card is drawn = 13/52  = 1/4
        probability of getting a ♥, if one card is drawn = 13/52  = 1/4

       probability of getting a ♠, if one card is drawn = 13/52  = 1/4

       probability of getting a ♦ , if one card is drawn = 13/52  = 1/4

     

      Further we elaborate and see, what is the probability of getting a red card, if a card out of a pack of 52 cards is drawn?
             P ( getting a red card ) = 26/ 52
                                               = 1/2
    what is the probability of getting a black card, if a card out of a pack of 52 cards is drawn?

             P( getting a black card ) = 26/ 52

                                               = 1/2

   In this way we solve various other problems related to it. To get information about Percent and Rates in IX Grade  and Mean in Grade VIII you can refer Internet.

Pythagoras Theorem in Grade VIII

Hello friends, in today's session we are going to learn about Pythagorean Theorem. This theorem is very useful and gives a very simple relationship between the three sides of a right angle triangle.

The theorem states that :

In any Right Angle Triangle, the area of the square whose Side is the Hypotenuse is equal to the sum of the areas of the square whose sides are the two legs. In algebraic form the theorem can be written as: a ² + b ² = c ², where a and b are the small sides and c is the hypotenuse of the right triangle.

In a right triangle if we know any two sides then we can easily find the other one with the Pythagoras equation.

If we generalize this theorem then we will get the law of cosines, which makes possible the computation of the third side of the triangle.

The converse of the theorem is also true:

For any three positive numbers a,b, and c such that a ² +b ² = c ², there exists a triangle with sides a,b and c, and every such triangle has a right angle between the sides of lengths a and b.

The Pythagoras equation can also be used to check what type of triangle we have, whether it is acute, obtuse or right angled. We can check this by using the following results:-

If a ² + b ² = c ², then the triangle is said to be right angled.

If a ² + b ² > c ², then the triangle is said to be acute angled.

If a ² + b ² < c ², then the triangle is said to be obtuse angled.

Pythagorean triplet is a set of three positive integers a, b and c, they are written in the form ( a, b, c). These triplets satisfy the Pythagoras equation a ² + b ² = c ².

For example - (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37) are some among those which come below 100.

So let's solve some examples based on Pythagoras theorem.

1. if two sides of the triangles are 3 and 4, then find the length of the hypotenuse?

By using Pythagoras theorem. a ² + b ² = c ². in here a = 3 and b = 4, so the value of c will be c ² = 9 + 16.

c ² = 25

c = 5.

2. if one side of the triangle is 9 and the hypotenuse is 41, then find the other side.

In this problem a = 9 and c= 41, then the third side will be b.

b ² = c ² - b ^2.

b ² = 1681-81

b ² = 1600

b = 40.

3. check whether the triangle with given sides is a right triangle or not. a = 11, b = 60 and c = 61.

We can check whether it is a right triangle or not by using the Pythagoras theorem.

61² = 11² + 60^2.

3721 = 121 + 3600.

3721 = 3721,

so these three sides satisfy the Pythagoras theorem, so the triangle formed will be right angled.

Number System in Grade VIII

Hello friends, in today's session we are going to discuss about the Number System. The Number System is defined as a set of numbers arranged together in a such a manner so that we can perform different operations like addition, multiplication etc.

The number system is classified as follows:

1. Natural Numbers
2. Integers
3. Rational Numbers
4. Polynomials
5. Real Numbers
6. Complex numbers.

Let's just take them one by one and understand what are these:

Natural numbers is a set of all positive whole number greater than zero.

The set of natural numbers is denoted by N. There are various laws which the set of natural numbers follow:

Commutative axioms: a + b = b + a; a · b =b · a.

Associative axioms: a + (b + c) = (a + b) +c ; a · (b · c) = (a · b) · c.

Distributive axioms: a · (b +c ) = a · b +a · c; (b · c) n = b n · c n.

Identity axioms: a + 0 = a ; a · 1 = a ; a ⁰ = 1.

Now we come to Integers which are defined as for every natural number “a” there exists a Integer “-a” and the set of these numbers is represented by Z.

To perform arithmetic operations on integers we can look at generalizations given below to make our work simple.

−a + −b = −(a + b)

a + 0 = a

a − b = a + −b

−a · b = a · −b = −(a · b)

−a · −b = a · b

√4 can mean either 2 or −2.

The rational number system is a set of numbers used to represent the fractions. It allows the division to be done by all numbers except zero. It is written in the form a / b. where b cannot be equal to zero.

Polynomials are usually not called as numbers but their properties are very much similar to numbers.

Algebraic numbers is the system which includes all the rational numbers and is included in the set of real numbers.

The real numbers are those numbers which really exists. The set of real numbers is usually represented by R. Real numbers can also be defined as those ones which can be written in the decimal form. All the numbers we have studied till date are called as the real numbers, but now what are imaginary numbers. When the root of a  negative number is taken then the result is known as an imaginary number.

For Example – the number 24 is real because it can be written as 24.0. ½ is also a real number because it can be written as 0.5. even the number 1/3 is also real because it can be written as 0.3333.... whereas complex numbers or imaginary numbers are those numbers which are written in the form a + i b. where i is the imaginary unit i.e. a number whose square is minus one.

Tessellations in Grade VIII

Hii guys, so today we are going to study about a very new and different topic of Maths which is Tessellations. How many of you know what a Tessellations is? I believe not many but still a few might know. So let's start without wasting any time.

Tessellation is the process in which we create a two dimensional plane by repeatedly joining the geometric shapes without any overlaps and gaps. In simple language a tessellation is something in which we have some shapes and we join them together to form a regular sheet without any overlaps and gaps. We can see tessellations throughout our history, from ancient architecture to modern art.

For Example the Jigsaw puzzle we used to play when we were little. Its the best example we can get for our topic. It corresponds with the everyday term tiling which refers to applications of tessellations, often made of glazed clay. It also exist in the nature like the Honeycomb which also has a Tessellation structure.

For Example -

a tessellation of triangles

a tessellation of squares

a tessellation of hexagons

The above given figures are some examples of Tesselations, for more understanding.

Now let's study the different topics that comes under it.

When discussing about the tilings a form of tessellation, we see that they are multicolored, so we need to specify whether the colors are the part of tiling or just the part of an illustration.

Let's come to the theorem in our topic, it is known as the four color theorem.

The theorem states that every tessellation in every Euclidean plane, with a set of four available colors, each tiled when colored in one color such that no tiles of equal color meet at a curve of positive length. Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. To produce a coloring which does, as many as seven colors may be needed.

Any arbitrary Quadrilateral when taken can also form a tessellation with 2-fold rotational centre at the mid point of all the four sides of the Quadrilateral. In a similar manner, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational center. We can easily divide this by one diagonal, and take one half of the triangle as fundamental domain. Such a triangle formed will have the same area as of the quadrilateral and can be constructed from it by cutting and pasting.

Now we come over to the types of tessellations, we classify then in two types – the regular tessellation and irregular tessellation.

A regular tessellation is one which is highly symmetric and is made of congruent regular polygon. There are only three regular tessellations which exist. They are of Equilateral triangles, squares and regular hexagons. We even have a more accurate one which is edge-to-edge tessellation. In this type of tessellation the side of one polygon is fully shared with the sides of another polygon. There is no room for partial sharing of sides.

The most common example of the apperiodic pattern is the Penrose tilings which is formed using two different types of polygons and adds beauty to the walls. We also have the self dual tessellation and the example for it is the Honeycomb. Another example for it is shown in the figure below.

 

Tessellations is not only used in architecture but also to design the computer models. In the computer graphics, tessellation technique is used manage the datasets of polygons and divide them into polygon structure. This is generally used for rendering. The data is generally tessellated into triangles also known as triangulation.

The CAD application in our computers is generally represented in analytical 3D curves and surfaces.

The mesh of a surface is usually generated per individual faces and edges so that original limit vertices are included into mesh. The following three basic fundamentals for surface mesh generator are defined to ensure the approximation of the original surface that suits the need for the further processing

• The maximum allowed distance between the planar approximation polygon and the surface. This parameter ensures that mesh is similar enough to the original analytical surface (or the polyline is similar to the original curve).
• The maximum allowed size of the approximation polygon (for triangulations it can be maximum allowed length of triangle sides). This parameter ensures enough detail for further analysis.
• The maximum allowed angle between two adjacent approximation polygons (on the same face). This parameter ensures that even very small humps or hollows that can have significant effect to analysis will not disappear in mesh.

Some geodesic domes are designed by tessellating the sphere with triangles that are as close to equilateral triangles as possible.

So as we have read above that a honeycomb is a natural tessellation, there are some more tessellations in our nature like when the lava flows, it often displays columnar jointing, due to the contraction forces that are present, it causes the lava to break into cracks when it is cooled. These cracks often gets broken in the hexagonal columns. The tessellation is even present in the flower petals, leaves, tree bark, or a fruit. It is not easily visible but when we look through the leaf etc. with the help of a microscope then we can easily see the formation.

It seems like what we have learned till now seems to be theoretical, so let's now have something mathematical in it.

Now the topic that comes under it is, Number of Sides of a Polygon versus the number of Sides at a Vertex.

If we assume an infinite long tiling, then let a be the number of sides of a polygon and b be the average number of sides which meet at the vertex. Then ( a – 2 ) ( b – 2 ) = 4. For example, we have the combinations (3, 6), (31/3, 5), (33/4, 42/7), (4, 4), (6, 3) for the tilings.

For a tiling which repeats itself, one can take the averages over the repeating part. In the general case the averages are taken as the limits for a region expanding to the whole plane. In cases like an infinite row of tiles, or tiles getting smaller and smaller outwardly, the outside is not negligible and should also be counted as a tile while taking the limit. In extreme cases the limits may not exist, or depend on how the region is expanded to infinity.

For a finite tessellation and a polyhedra we have

( a – 2 ) ( b – 2 ) = 4 (1 – X/f ) ( 1 – X/v ).

In this expression f represents the number of faces of a polygon and v represents the number of vertex and X is the euler characterstic.

The formula follows observing that the number of sides of a face, summed over all faces, gives twice the total number of sides in the entire tessellation, which can be expressed in terms of the number of faces and the number of vertices. Similarly the number of sides at a vertex, summed over all vertices, also gives twice the total number of sides. From the two results the formula readily follows.

In most cases the number of sides of a face is the same as the number of vertices of a face, and the number of sides meeting at a vertex is the same as the number of faces meeting at a vertex. However, in a case like two square faces touching at a corner, the number of sides of the outer face is 8, so if the number of vertices is counted the common corner has to be counted twice. Similarly the number of sides meeting at that corner is 4, so if the number of faces at that corner is counted the face meeting the corner twice has to be counted twice.

A tile with a hole, filled with one or more tiles is not permissible, because the network of all sides inside and outside is disconnected. However it is allowed with a cut so that the tile with the hole touches itself. For counting the number of sides of this tile, the cut should be counted twice.

The another topic that we are going to study is the Tessellation of other spaces. As well as tessellating the 2-dimensional Euclidean plane, it is also possible to tessellate other n-dimensional spaces by filling them with n-dimensional polytopes. Tessellations of other spaces are often referred to as honeycombs.

So this is all we have for the topic tessellations and I hope that you would have understood what a tessellation is. This topic we have learned is totally a theoretical one, but it do have some problems to do, and I expect that you will be able to understand how to solve them by using the only formula which we have learned so far in this topic.