Eight Grade Math

 In the previous post we have discussed about Math for 8th graders and In today's session we are going to discuss about Eight Grade Math.


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In Eight Grade Math, we study different topics related to numbers. We come across the chapter of Rational numbers. In rational numbers, we study about how rational numbers are formed, their standard form, the positive and the negative rational numbers and how to express the series of rational numbers on a number line.

Further we also learn in Eight Grade Math, about how to compare two rational numbers and how to perform the mathematical operations on rational expressions. We study different properties of rational numbers too.

Here are some of the properties of rational numbers:

1. Closure Property: By closure property we mean that if we have two rational numbers, then sum, difference, product and quotient of two rational numbers is also a rational number.

2. While talking about the Commutative property of rational numbers, we say that it holds true for addition and multiplication of rational numbers but not for the subtraction and the division operation, so if the two rational numbers are p1/ q1 and p2/q2, then we get :

(p1 / q1 + p2 / q2) = (p2 / q2 + p1 / q1)

(p1 / q1 * p2 / q2) = (p2 / q2 *  p1 / q1)

(p1/q1 - p2 / q2) < > (p2 / q2  - p1 / q1)

(p1/q1 divided by p2 / q2) < > (p2 / q2 divided by p1 / q1).

To know about the Lateral Area of a Cylinder, we can learn the formula from online math tutor help, which is available for free. We can also download the cbse syllabus for class 11, from Internet, which is available on website of cbse. 

Math for 8th graders

Here we are going to discuss about the topic rational number. We will study about the rational numbers in detail in 8th Grade Math.

We define the set of rational numbers as the numbers which we can write in the form of p / q, where p and q are any integers, such that q <> 0. Further exploring the definition of rational numbers we say that any number which can be either written in the form of terminating decimal number or a repeating decimal number, then we say that the given fraction number is a rational number. All the natural numbers, whole numbers, positive or negative integers and the fraction numbers are the part of the family of rational numbers.

Let us look at the properties of the rational numbers. They are as follows :

1. Closure Property: If we have a pair of rational numbers, then we say that the sum, product, difference and the quotient of the given two rational numbers is also a rational number. Thus we say it mathematically as follows : If p1/ q1, p2 /q2 are any two rational numbers, then we have

P1/ q1 + p2 / q2,

P1/ q1 - p2 / q2 ,

P1/ q1 * p2 / q2 .

P1/ q1 divided by p2 / q2 are also the rational numbers.

2. Associative property of rational numbers also holds true for the addition and multiplication, but not for the operation of division and subtraction. So we have

( P1/ q1 + p2 / q2 ) + p3 / q3 = P1/ q1 + (p2 / q2 + p3 / q3) ,

( P1/ q1 * p2 / q2 ) * p3 / q3 = P1/ q1 * (p2 / q2 * p3 / q3) .

Also we have the commutative property, Additive identity, Multiplicative identity, Additive and multiplicative inverse property which holds true for the rational numbers.

Questions based on Osmosis and Diffusion can be seen in the cbse latest sample papers.  

How to factoring quadratic equations

The factorization of a quadratic equation (mid - term splitting) leads to real roots of the equation. The quadratic equations when solved using the formula, it is the case where real and imaginary roots both are possible to occur. That is by using the direct formula also we are able to factorize the quadratic equation, but imaginary factors are possible. The direct formula for a quadratic equation wx² + vx + z is given as follows: x = [- v ± Sq root(v² – 4wz)] / 2w

The ±sign is put before the root sign because according to the property of root there are two possible roots when a root is solved. Thus we get either two real or two imaginary roots for our quadratic equation. In case of imaginary roots we can also say that the two roots are complex conjugate of each other. Let's us suppose an example of a quadratic equation to understand the factoring quadratic equations:

Suppose our quadratic equation is given as follows: 5x² + 8x = -6. First we have to arrange the given equation such that it is present in the standard form of a quadratic equation and then start solving it. In our equation we can - not factorize by using mid - term splitting and so we need to use the direct formula to solve for x as follows:

Substituting the values of w, v and z in the formula we get, 

x = [- 8 ± sq root(8² – 4 * 5 * (6))] / 2 * 5 ,

Thus we get x = -8/10 ±56i/10 .

According to the concept of Round to the Nearest Tenth, the values of the numbers are rounded to their nearest tenth number. For instance, 304 will be written as 300 and 256 is written as 260. These concepts are very important and have been explained in the icse sample papers 2013 in detail. (know more about How to factoring quadratic equations, here)

right triangles

In the previous post we have discussed about adjacent angles and In today's session we are going to discuss about right triangles. A right triangle in the mathematics can be defined as that triangle in which we have 1 angle equal to the right angle which means that 1 angle is equal to the 90 degree. The right triangles are also sometimes called as the right angled triangles. It should be known that the relationship among the different sides and the angles of any right angled triangle forms the basis for the trigonometry.

Now let us talk about the terminology in the right triangles. In any right angled triangle the side which is opposite to the right angle or we can say the 90 degree angle is known as the hypotenuse whereas the sides which are adjacent to the 90 degree angle are known as the legs of the right triangle. (know more about right triangles, here)

Also when the lengths of the 3 sides of the right angled triangle are the integers then that right triangle is known as the Pythagorean triangle whereas the lengths of the 3 sides are collectively called as the Pythagorean triple.

We all know that in any type of the triangle the area is calculated by multiplying 1 / 2 with the length of the base of that triangle and then multiplying that result with the height corresponding to that base of the triangle. But the case of a right angled triangle is very easy. In any right angled triangle, when 1 leg is considered as the base of the triangle then the other leg gives the height of the triangle. Thus the area in any right angled triangle can be calculated by just multiplying 1 / 2 with the product of the lengths of the 2 legs of the triangle.

In order to get more help in understanding the topics: right triangles, Equation for Force and icse board papers 2013, you can visit our next article.

adjacent angles

We say that the angle is formed when we have two rays going in the different directions and they have a same starting point called the vertex. The measure between the two rays is done in the terms of degrees. So we say that the angles are measured in degrees.

 Let us now look at the adjacent angles. The pair of angles formed such that :

a)    The two angles have a common vertex.

b)    The two angles have one  common arm and two uncommon arms.

c)    The common arm exist in between the uncommon arms.

If all the  above three conditions are satisfied , then we say that the pair of angles is called the adjacent angles.

In case the pair of adjacent angles are supplementary, then we conclude that the two uncommon arms of the adjacent angles form a straight line. Thus if the sum of the two adjacent angles  is supplementary, then we say that the two uncommon arms is in fact the straight line.

The reverse of the above statement is also true, which says that if the two adjacent angles are formed on the straight line, then the pair of angles are supplementary.

 We also call these pair of angles as the linear pair of angles.

So by the term linear pair, we simply mean the pair of the adjacent angles which are formed on the straight line and so the sum of this adjacent pair of angles is automatically 180 degrees.

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Equation of Circle

In mathematical geometry, a round shape in which all the points on the boundary are at same distance from the center is known as a circle. The general Equation of Circle is given by: S 2 + T 2 = r2, where ‘r’ is the radius of a circle (Here S is along to the horizontal axis and ‘T’ is along the vertical axis). All of the points on the boundary of a circle are at fixed distance from the center is known as the radius of a circle. The general equation for circle is given by: (know more about Circle, here)

⇒ (x – s)2 + (y – t)2 = r2;

Now put the value of s, t, and r is 4, 5, 6 respectively then we get:

⇒ (x – 4)2 + (y – 5)2 = (6)2; on further solving the equation of a circle we get:

⇒ x2 + 16 – 8x + y2 + 25 – 10y = 36; So the equation of a circle is:

⇒ x2 + y2 – 8x – 10y + 41 = 36; we can also write it as:

⇒ x2 + y2 – 8x – 10y – 5 = 0; This is the required equation of a circle.

Here we can also write the general form of a circle using the constant value in place of numbers. So the equation of circle using constant is given by:

⇒ x2 + y2 + Sx + Ty + U = 0, here we will also see the equation of a unit circle. We know that ‘1’ is the radius of unit circle, if the radius of a unit circle is more or less then ‘1’ then the circle is not unit circle. The general equation of a unit circle is given by:

⇒ x2 + y2 = 1; let’s discuss how to graphing linear equations. In mathematics there are many methods through we can easily plot the graph of linear equation. To study more about the linear equations go through the online tutor of tamilnadu board of higher secondary education and In the next session we will discuss about adjacent angles. 

isosceles triangle

By the term triangle we mean that it is a closed polygon which if formed by joining 3 line segments. There are different types of triangles based on the lengths of the line segments.

They are : Equilateral Triangle,  isosceles triangle, and scalene triangles. By the term Isosceles triangle, we mean that the triangle which has a pair of the two sides equal. In such a triangle, if we have pair of the  sides equal, we say that the pair of the angles which are formed by the  pair of sides is also equal.  So we come to the conclusion that in the isosceles triangle we have two sides equal and two angles equal. (If you want to get more information about isosceles triangle, Refer this)

If we drop a perpendicular from the angle which is unequal in the isosceles triangle, we observe that this perpendicular is the perpendicular bisector to the line segment which is opposite to this angle. Thus we also called this line as the median of the isosceles triangle.

We further observe that this median, which is also the perpendicular bisector is the line of symmetry of the given isosceles triangle. Any isosceles triangle has only one line of symmetry and this line of symmetry divides the  given isosceles triangle into two equal halves. We also call it the mirror half of the triangle.  On the other hand an equilateral triangle has 3 lines of symmetry and a scalene triangle has no line of symmetry.

If we need to learn about  What are Perpendicular Lines, we can visit the online math tutorial and take the help of the modules based on the above topic in the tutorial. We also have cbse syllabus 2013 online which can be downloaded to understand the pattern of the question paper in the upcoming examination. It guides the students to  know about the important questions  for the exams and In the next session we will discuss about Equation of Circle.