equations

This unit is for the students of Grade VII. In this unit we will learn about equations. We must remember that equations with variables, are inter related terms. We form the equations to express the mathematical statements in form of the expressions. The mathematical expressions joined with the numerical operators are called equations, when there exist two sides of the equation LHS and the RHS. Also by word equation we mean whatever placed on the left side of the equation is equal to the expression placed on the right side of the equation. By word variable, we mean the unknown value which may change every time. When we solve an equation with variable ‘x’ and another equation with same variable, then value of ‘x’ may change from equation to equation, which may satisfy the equation.
The basic purpose of solving any equation is to find the value of the value of the variable which is unknown in the given equation. When we say that we need to find the value of the unknown variable in the equation, we mean that the value we calculate must satisfy the equation. Thus we say that the  when we put the value of the unknown variable in the equation,  the value we get after solving both the sides of the equation must be equal.  (know more about cbse board books, here)
There are different ways to solve the equations and finding the values for the unknown variables in the equation. To find the value, we may adopt hit and trial method, where we put the values of the variables and then check it that particular value satisfies the given equation or not. The value that satisfies the given equation actually is the required solution to the given equation.
Another method to solve the equation is by shifting the variable to one side and all the constants to another side of the equation and  In the next session we will discuss about  Arithmetic sequences. 

Formulas for measurement

In the previous post we have discussed about Problems in measurements and In today's session we are going to discuss about Formulas for measurement.
Formulas for measurement
There are different shapes for which measurement formula is applied
Shape 1 Rectangle Measurement is done in the form of Area and Perimeter


Properties of rectangle:

In a  angle right angle triangle and lines are parallel to each other. Opposite sides are equal to each other .In the above figure L=Length and B=Breadth
Rectangle:
Area = Length X Breadth
A = L*B

Perimeter = 2 X Lengths + 2 X Breadth
P = 2L + 2B
P=2(L+B)
Shape 2:

In  the parallelogram opposite sides are equal a , b are the length of the two sides with the height as H where comes the 90 degree angle

Parallelogram Area:
Area = Base X Height
A = b*h


Shape 3
The triangle have three sides  a , b , c  with a height ‘h’ and b as base
Triangle
Area = 1/2 of the base X the height
a = 1/2 (B*H)
Perimeter = a + b + c
(add the length of the three sides)
 Learn Formulas for measurement with the help of examples

Examples: Find the area and perimeter of a rectangle with sides 4cm, 2cm?
Solution: Area =Length * breadth
Area = 4 * 2
Area =8 cm²
Perimeter=2(L+B)
P=2(4+2)
P=2*6
P=12 cm

Example: Find the area of the parallelogram where base =3 cm and height =8cm?
Solution: Area =Base * height
Area =3* 8
Area =24 cm²

 (know more about icse board, here)

Example :Find the area and perimeter of the triangle with Dimensions in centimetres  (3 , 4 , 6) and height as 3cm ?
Solution : Area = 1/2 of the base X the height
a = 1/2 (4*3)
Area =12/2
Area =6 cm²

Perimeter = a + b + c
P= 3+4+6
P=13 cm

Example: Find the length of a rectangle where area is 24cm² and breadth is 4 cm ?
Solution : Area =L*B
24=L*4
6cm =L

Example :Find the height of a triangle where base is 5 m and area is 35 m^{2?
Solution :}  Area =(b*h)/2
35= (5*h)/2
70=5*h
14m=h

Problems in measurements

Problems in measurements are very common problems that are mostly encountered while doing math calculations on a daily basis. This part of measurement can be broadly classified into three types of problems:
1.Distance : It  generally includes problems with mm, cm, km...
2. Time: It generally includes problems with seconds, minutes, hours...
3.Mass: It generally includes problems with Milli gram, gram, kilo gram...
This part is specially designed for Grade VII and now, we will move to word problems measurement:
Problems 1: If One liter is equal to 1000 cubic centimeter and we have a Milk container containing 34.6 liters of milk, how much does this in cubic centimeters?
Solution 1: 1 Lit = 1000 cc.
34.6Lit = 34.6 * 1000 = 34600 cc.
Problem 2 : If we increase the Radius of a circle by 200 %, how many times will the area be increased for that particular circle?
Solution 2 : Area of circle = ∏r>2. That is the area is directly proportional to the square of the value of radius, when we increase the value of the radius by 200 % that means that now the radius is 3r from earlier r and when we square this 3r it will give us 9r>2, making the value of circle increase by 9 times.
Problem 3: It takes 30 minutes to pile up the bundle of 100 files, there are 2000 files in the store room. How much time will it take? (know more about cbse sample papers, here)

  Solution 3: It takes 30 min to pile 100 files, which means to pile 1000 files it will take 300 min and similarly to pile 2000 files it will take 600 minutes, which is equal to 10 hours.
Thus we, hope that now things related to this topic are more clear with the pupils. In the next session we will discuss about Formulas for measurement.

Measurements in Mathematical World

Measurements in maths play a very crucial role in our day to day life. In earlier times, we used to calculate the distance and lengths by feet or palms, which was not a standard unit of measurement. To overcome this difference, standard scale of measurements was developed. It can be represented in form of a scale as follows:
Milli
Centi
Deci
Unit
Deca
Hecto
Kilo
The above scale shows that milli is the smallest unit and kilo represents the largest unit.  For length measurements meter is the standard unit and the above table can be written as:
Milli Meter
Centi Meter
Deci Meter
Meter
Deca Meter
Hacto Meter
Kilo Meter
In Grade VII, we will learn about converting one unit to another. When we do certain mathematical operations on measurements, the units must be same. So for this reason we need to convert one unit to another. Suppose we need to add 13 mm and 4 cm. In such cases as the units are not same, we need to convert either of the units. If we convert mm to cm, we need to divide it by 10 and if we convert cm to mm, we will multiply it by 10.
So 13 mm becomes 13/10 cm
= 1.3 cm
Now 1.3cm and 4 cm can be added easily.
= 1.3 + 4 cm
= ( 1.3 + 4.0 ) cm
= 5.3 cm Ans
we conclude that in the above table if we convert from upper to lower form the  digits are divided by 10 every time. Eg if mm is to be changed to cm, then divide by 10, mm to deci meter then divide by 100... and so on
Similarly if we move from bottom to top, then we need to multiply by 10 every time. eg: if km is to be converted to hectometer then the digits are multiplied by 10, km to deca meter, then multiply by 100 ... and so on.
In the next topic we are going to discuss Problems in measurements

Graphing solutions of inequalities

Today we will be studying Graphing Solutions Of Inequalities, Graph Solution Of Inequalities, Measurements-
To understand Graphing solutions of inequalities, Graph solution of inequalities, Measurements we must first understand the meaning of inequalities, inequalities can be defined as that one expression can be greater than or less than some other expression, the mathematical symbols used in inequalities are less than(<), greater than (>), less than equals to (≤) and greater than equals to (≥). The inequalities can transformed in many other ways. The direction of inequalities does not change if any number is added to both sides or if number is subtracted from both sides or both sides are multiplied and divided on both sides.
An inequality is of the form y > mx + b or y < mx + b, here x and y are the coordinates and 'm' is the slope and 'b' is the intercept.
Examples of inequalities-
y ≤ 2x + 3 or 2x – 3y < 6
As we have understood the meaning of inequalities, so now we will try to plot some of the inequalities on a graph. The examples are show below-

1)Let us take an example y ≤ 2x + 3 and plot it on the graph.
First we will plot the equal part of inequality that is y = 2x + 3, it is a straight line so after plotting
the graph will look like

                                                     This graph is for the equality part that is y = 2x + 3 but we also have to plot less than part of the inequality. To plot the less than part that is y < 2x + 3 we just shade the region.
So either we shade the region below the line on region above the, in this inequality we will shade the region below the line as 'y' is less than 2x + 3 .So after shading the region below the line graph will look like:                                                     

                                                            
                                   





So this is how we see Graph solution of inequalities.
In the next session we are going to discuss Measurements.

Relation between objects in space

SPACE:

• space is a three dimentional structure
• space is boundless and limitless.
• space contains object exist in space
• many events exist in space

objects:

• object is a physical body which is a collection of masses.

mass and weight of an object:

Mass is a measure of how much material is in an object, but weight is a measure of the gravitational force exerted on that material in a gravitational field; thus, mass and weight are proportional to each other, with the acceleration due to gravity as the proportionality constant. (know more about icse board papers 2013, here)
w=m*g
w is the weight
m is the mass.
g is the acceleration due to gravity.
g=9.8 m/sec>2

 Relation between objects in space
Relation between objects in space follows the newtons  gravitational law of gravitational
Newton's Universal Law of Gravitation states that any two objects exert a gravitational force of attraction on each other. The direction of the force is along the line joining the objects
Newton's law of gravitation resembles Coulomb's law of electrical forces, which is used to calculate the magnitude of electrical force between two charged bodies and this signifies the Object And Space Relationship.

.The magnitude of the force is proportional to the product of the gravitational masses of the objects, and inversely proportional to the square of the distance between them .
F(1,2)=(G*m1*m2)/r>2
m1 & m2 are the objects in space
m1 exerts  force F12 on m2
m2  exerts force F21 on m1
G is  the gravitational contant which is 6.67 x 10>- 11 N m>2/kg>2

• The inertial mass of an object determines the amount of force needed to produce a given acceleration of that object.
•  The gravitational mass determines the force of gravitational attraction between two bodies.

Gravitation is a natural phenomenon in which physical bodies attract with force propotional to the mass.
it is resposible for keeping all the object  in the universe in there orbit , otherwise all objects would have collide

In the next session we will discuss about Graphing solutions of inequalities. 

Congruence in Grade VII

Hello students today we are going to discuss Congruence. In mathematics you have to deal with many topics, congruence is one of them. If there are two or more objects or figures that have the same size and same shape, they are called congruent objects or figures. In other words the figures are called isometric, i.e. they have same size and same shape. In congruence we can reposition the objects in other shapes than they known as translation, rotation and reflection and combination of all called is transformation. Let’s take an example of triangle that will surely help you to understand this:-If      Two triangles have same corresponding size and same corresponding shape than they are congruent. As shown in figure we have two triangles ABC and DEF and in first triangle AC=7, CB=6 and in second triangle DF=7, FE=6. This implies both the triangles have same size and same shape so we can show their relationship in this way:-

∆ABC≅∆DEF

To determine congruence we have some comparisons, they are:-

-Angle side angle.

-Angle angle side.

-Side angle side.

-Side side side.

-Right angle hypotenuse side.

If any one of the comparisons exists in any figure than they are said to be congruent figures. Generally congruence may be defined in many fields like:-

-In relation.

-In modular arithmetic.

-In groups and subgroups.

-In number theory.

-In general relativity.

-In graphs.

And congruence is present in those shapes where equality is there; like in figures, objects, relations, groups and in many fields.

 

The above described congruence information will surely helpful for grade VII students. In the next topic we are going to discuss Relation between objects in space.