Sketching and modeling

Hello students,Previously we have discussed about descriptive statistics examples and in this blog we are going to discuss the Sketching and modeling which comes under school of secondary education andhra pradesh. In the mathematics we draw the graphs, diagrams, figures and shapes and by doing this we make the sketch of the digram and then model that diagram. The sketch is comes under the theory of categories of mathematics. Its a D category sketch with the combination of set of limits cones and a set of co limit cones where model is the C category called factor and the equation for that M : D → C. In the year 1968, sketches came into existence by the great mathematician Charles Ehresmann.
Sketching is the process of making sketches and modeling gives the whole idea or descriptions about those phases which are sketched.
We can sketches the polar curve and after then we also model them with the several types of modeling techniques. We can sketch the straight lines, quadric graphs, cubic graphs etc.
It is not easy that every problem of calculus will be solved in algebraically form, that's why we use the polar forms that are in the form of (r, θ). There are the some steps that are helpful to sketch the polar curves.
The mathematics sketching is the new approach by which we can create and explore the dynamic illustrations. Every sketched shape can be modeled easily. For the statistical equation we use the regression, semi parametric models and many more.
We can make any sketches for any type of observations, just like if we want to make the sketch of two bikes moving down a road and one of them with fix velocity and another one with fix acceleration. Then we can draw the road and two bikes and label the bikes with their names that are related with each other on the basis of mathematical expressions.
In the next session we will discuss about adding unlike fractions calculator and You can visit our website for getting information about chemistry tutor. 

Rectangular coordinate system

In mathematics,Previously we have discussed about column addition worksheets and In today's session we are going to discuss about Rectangular coordinate system which comes under andhra pradesh education, Graph can be considered as an important tool that provides the visual representation of the data. Basically graphs are used for representing the relation between two or more variables like x and y variables. In mathematics, normally two dimensional graphs are used for representing relationship between two or more variables. In the concept of graph, rectangular coordinate system is popularly known as Cartesian coordinate system or x-y coordinate system. According to rectangular coordinate system definition, it is a system in which coordinate point having a distance form a set of perpendicular lines. A rectangular coordinate system consist four quadrants, dual perpendicular lines that is a horizontal line and a vertical lines and they also has the origin point.
In graph horizontal lines are called as x – axis and vertical lines of the graph are called as y axis of the graph. When the both x – axis and y axis of the graph crosses each other than that point are known as origin point. In mathematics, rectangular coordinate system is used for representing all the values which are generated by the graphical equations. To represent the value of an equation we need to create a table of values which are generated by the graphical equations.
Rectangular coordinate system is split into four quadrants. In this system all the values are associated with the ordered pair. In ordered pair x coordinate considered as first value and y coordinate considered as second value. In rectangular coordinate system, the 1st quadrant contains the positive value of x and y axis. In the same aspect 2^nd quad contains the negative x and positive y. In the same aspect the quad 3^rd and 4^th is working as inverse of quad 1^st and quad 2^nd.
Example: Plot the given ordered pairs on the graph?
                           X (2, 3), Y (-2, - 3)
Solution: In the 1^st ordered pair both value of x axis and y axis contains the positive value that why this pair plotted on the 1^st quad. In the same aspect 2^nd ordered pair contains the both negative value of x and y axis then this pair lies in 3^rd quad.
In the next session we will discuss about Sketching and modeling and You can visit our website for getting information about algebra tutor.

Pythagorean theorem

Previously we have discussed about calculate the volume of a sphere where the radius is 9 meters. and In today's session we are going to discuss about Pythagorean theorem which comes under school of secondary education andhra pradesh, Pythagorean Theorem was given by the ‘Pythagoras’ a Greek mathematician. Pythagorean Theorem can be defined as the square of a hypotenuse is equal to the sum of the square of the base and square of the perpendicular that is the opposite side of the hypotenuse in a right angled triangle. If we define it in the form of expression then it will be denoted as,
(Hypotenuse) ² = (base) ² + (perpendicular) ².
For Example: A Right angled triangle named as XYZ and if ‘Z’ is a hypotenuse of right angled triangle and ‘Y’ is base and ‘X’ denotes the perpendicular then Pythagorean theorem can be expressed as (Z) ² = ( X) ²+ (Y) ² .
We can explain it by taking an example as;
If the base = 12 inch (Y = 12 inch) and perpendicular = 5 inch (X = 3 inch) then find the length of hypotenuse ‘Z’ in meters by using the Pythagorean Theorem?
Solution: Pythagorean Theorem proof
(Hypotenuse) ² = (base) ² + (perpendicular) ²,
Then (Z) ² = (X) ² + (Y) ²,
(Z) ² = (12) ² + (5)²,
Z = √ (12) ² + (5)²,
Z = √ (144) + (25),
Z = √ 169,
Z = 13 inch.
So according to the calculation, the length of the hypotenuse in a right angled triangle XYZ is 13 inch.
Pythagorean Theorem is used in the case when two sides of the right triangle are given and for calculating the third side of the tight angled triangle, we are use the Pythagorean Theorem.
In the next session we are going to discuss Grade VII, Rectangular coordinate system and You can visit our website for getting information about algebra help online.

Basic constructions

Hello students, Previously we have discussed about who invented calculus and in this blog we are going to discuss basic constructions which comes under andhra pradesh education board. In basic construction generally we include the construction of similar triangles, tangents to circles and the basic constructions in geometry which include: -
-Construction of various angles such as 30°, 45°, 60°, 90^o.
-Bisecting an angle.
-Constructing a line parallel to a given line through a given point.
-Construction of perpendicular bisector of a line.
-Construction of closed figures like squares, rectangles, quadrilaterals etc.
-Construction of in circle, circum circle and ex circle of a given triangle.
All of the above construction uses compass and ruler.
If we talk about more simple geometry then basic construction include: -
-Constructing of line (basic construction in geometry).
-Constructing of triangle.
-Construction of circle.
Let’s take one construction from the above list to see the basic construction in geometry.
Construction of triangle: - Triangle is a three sided close figure. Triangle can be constructed using the ruler, compass and protractor.
Some steps to construct the triangle are: -
Step 1: - A triangle has three side then XY = a units YZ = b units and ZA = c.
Step 2: - Draw the line XY = a units using ruler.
Step 3: - Take a compass and measure c units with the help of ruler.
Step 4: - With X as center, cut an arc above the line segment XY with the help of compass.
Step 5: - Again take a compass and measure b units.
Step 6: - With Y as center cut an arc above the line segment XY.
Step 7: - Both the arcs meet at a point that is Z.
Step 8: - Join XZ and YZ. Now, XYZ is the required triangle.

In the next session we are going to discuss Grade VII, Pythagorean theorem and You can visit our website for getting biology help.

Congruence

Previously we have discussed about tangent line approximation and In today's session we are going to discuss about Congruence which is a part of school boards in india , In geometry if we are given two figures and the question is asked that are the two geometrical figures congruent, for this we need to recall the congruence definition. According to Congruence, two or more given geometrical figures are congruent if they have all the sides of the equal measurement and the angles formed by all the line segments are exactly equal. We say the two figures are congruent, if the two figures are placed one over another, they overlap each other.
Let us first take two circles, the two circles are said to be congruent, if they are drawn with the same radius.
 In case of the square, we say that the two squares are congruent, if they are formed with the help of the same line segment. As we know that all the angles of squares are equal to 90 degrees each, so we say that the two squares are congruent if they have the length of each side of the same measure.
Now we take the two rectangles, the two rectangles are said to be congruent, if the length of one rectangle is equal to the length of another rectangle and the breadth of one rectangle is equal to the breadth of another rectangle. All the rectangles have all the angles equal to 90 degrees. If we place one rectangle over another, the two rectangles are found to be exactly of the same measure and they overlap each other.
 In case of the triangles, we know that the triangles are formed by joining 3 line segments. Here we say that the two triangles are congruent, if the three sides of the lines are congruent   and the corresponding sides of the triangle are also equal, then the triangles are said to be congruent.
 In the next session we are going to discuss Basic constructions and You can visit our website for getting information about chemistry answers.

Reflections/translations on a coordinate plane

Hello students in mathematics we study the concept of the transformation. There are many types of Transformation schemes that are used in various fields. Some transformations are: -
-Rotation
-Reflection
-Translation
The transformation can be stated as the word that means changing in shape of the object, in this a shape of object can be change, and position of object can be change.
-Reflections on a coordinate plane mean reflection on the xy axis or xy coordinate or xy plane. The x coordinate represents the horizontal position and y coordinate represents the vertical position.
Reflection of any object means reverse image of that particular object or flipping of the object. We see our image in the mirror is the example of Reflection and reflection on a coordinate plane means a particular object that are on xy plane will reflect on the -x-y plane.
-Translations on a coordinate plane mean shifting of one shape from xy plane to another plane. In it only changes in position not in shape. In the translation transformation the object or figure is shifted only one position to another and their shape remain unchanged. In the translation we can move the shape into any direction like in upward, downward, on the right side and left and wherever we want. (know more about cbse board, here)

The reflexion and translation can be more understand by graphically. The shape may be triangle, rectangle and anything for both the transformation. Reflected and translated figures are represented by the ' symbol like we have original figure name is ABCD then the transformed figure will be represented as the ABCD'. To solve both we have to provide the numeric data so that we can draw the any shape on a coordinate plane for transforming.


In the next session we will discuss about Congruence. 

Tessellations

Hello friends today we will discuss about tessellation which you need to study in grade VII. Generally in mathematics you didn't heard this term but it has a relation with mathematics. A tessellation is a pattern made by repeating shapes. Making Tessellations require a creativity of an art with capability of solving puzzles. In this world there are many natural tessellations also present.
In modern world very few people are aware of the term Tessellations. The connection between math and art is very strong and frequent but very few people are aware of that. Tessellation also covers a regular space without overlapping and without leaving any space. For example a chessboard is a Tessellation, made of squares with no gap in between and without overlapping. The pattern of the brick on a wall is a Tessellation made by rectangle.
Whenever we deal in mathematics we aware of terms like algebra, calculus, trigonometry ...etc. if we talk about the term tessellation we are not aware of this term. But as we know that there is a very strong relation between art and mathematics and that is only because of the term like tessellation, so we can say that Tessellations has a relationship with mathematics. The tessellation definition is that shape of repeated things. (know more about cbse latest sample papers, here) The repeated things can be anything, if we take an example of a chessboard then it is a pattern made by squares as squares are repeated regularly in the chessboard but the thing we need to notice is that there is no gap between the square, so if we are drawing any tessellation then the thing we need notice that there is no overlapping between the figure and there is no gap between the figure. In the next session we will discuss about Reflections/translations on a coordinate plane. 

Complementary/supplementary angles

In this unit we are going to learn about Complementary angles, supplementary angles. We say that the angle is formed, if the two rays goes in the different directions and have the same starting point. This starting point is called the vertex of the angle.  Now we will see what complementary angles are and what are supplementary angles? Moreover we will study that what is the complement of the given angle and what is the supplement of the given angle. (know more about cbse class 12 board papers, here)
We mean, by the term Complementary angles, we mean that the   sum of two angles is equal to 90 degree. If the measure of one angle is given say x and we need to find the Complement of the given angle then we say that the complement of the given angle is 90 – x.  Thus if the two angles are complementary, it means that their sum is 90 degrees and so it forms a right angle. The two adjacent angles if joined together if form a right angle, and then we say the two angles are complementary.
We mean, By the term supplementary angles, we mean that the   sum of two angles is equal to 180 degree. If the measure of one angle is given say x and we need to find the supplement of the given angle then we say that the supplement of the given angle is 180 – x. Thus if the two angles are supplementary, it means that their sum is 180 degrees and so it forms a straight angle.  We also observe that if the two angles are supplementary, then their sum is 180 degrees and so we can say that the two angles form a linear pair. In the next session we will discuss about Tessellations. 

Properties of 2-d and 3-d figures

Hello students. In this session we are going to discuss ob the topic of Properties of 2 d and Properties of 3 d figures. But before starting it you should be familiar with the terms of 2D and 3D. 2D is the shapes that can be drawn on the plane paper and 3d shapes can not be and to be more precise 2d shapes is 2d dimension and 3d shapes is 3 dimensions. We can not handle the 2d shapes in our hand because they are the flat shapes but we can handle the 3d shapes.
The most important difference between them is that a 3d shapes have three axises such as x, y and z whereas 2d shapes have just two axises that is x and y.   (know more about cbse sample papers, here)
Let’s talk about the properties or dimension of 2 d and 3 d figures.
2d figures has two properties or dimensions such as length and width and 3d figures has three properties or dimensions such as length, width and depth.
Circle, triangle, square, rectangle and any polygons follow the 2d properties, because they are 2d shapes.
Sphere, prism, cuboids, cube, cylinder, pyramid and cone follow the 3d properties because they are 3d shapes.
Although every shape of 2d and 3d have different properties for example: -
Some 2d Shapes                                  Properties
Square                                                 4 sides, closed figure, 2 sets of parallel line
Triangle                                                3 sides, closed figure
Rhombus                                              2 sets of parallel line, polygon, 4 sides of equal measure

Some 3d Shapes                                  Properties
Cube                                                    6 congruent faces, 12 vertices,
Cylinder                                               2 circle faces
Rectangular prism                                 6 faces, 8 vertices
Although we have so many 2d and 3d figures and shapes, but here it is not possible to discuss all of them.
I hope that make sense.
In the next session we will discuss about Complementary/supplementary angles. 

Regular/irregular geometric shapes

In the geometrical mathematics, we all are very well aware that it is a collection of several kinds of shapes and figures. These figures and shapes are very helpful in solving various kinds of problem that are related to our daily routine life. In geometry, these shapes and figures are categorized into different category according to their properties like quadrilateral, circle, and polygons and so on. Here we are going to discussing about the geometrical shape 'polygon'. The concept of polygon helps the school's student to understand the geometrical concept very well. (know more about cbse latest sample papers , here)
Polygon is a geometrical shape which comes from the Greek word. Polygon word formed by the combination of two words, 'poly' which mean is “many” and 'gon' which mean is “angle”. A polygon is a shape that is drawn on the 2-D plane with help of straight lines or sides. In the simple mean we can say that polygon is the two dimensional shapes. These shapes are formed by straight lines and the shape is closed. It means that in this shape all the lines are connect to each other. In the polygon, according to the property of geometrical shape we can categorize the polygon into the following manner:
A) simplex or complex
B) Concave or convex
C)
Regular geometric shapes or irregular geometric shapes
Here we are going to discussing about the regular and irregular shape.
Regular geometric shapes are those shapes in which all the existing angles are equal and all the sides are equal. In the same aspect we can say that those figures whose existing angles and shapes are not equal in measure then these shapes are consider as a
Irregular geometric shapes. In the case of regular, equilateral triangle is the one of the best example because this triangle is made up of three straight sides, it is closed in shape and all the angles and sides are equal to each other. In the next session we are going to discuss Grade VII, Geometric concepts and In the next session we will discuss about Properties of 2-d and 3-d figures. 

Geometric concepts

As we all are very well aware that geometry is the one of the oldest branch of mathematics. It plays an important role in mathematics to solve various kinds of problem. The term “geometry” is comes from the Greek word “Geometron” which is formed by the combination of two word “Geo” and “Metron”. The word “Geo” refers to Earth and the word 'Metron' refers to measurement. The need of the  Geometric concepts was felt in several years ago when person realizes to measure their land when they want to sold or buy their land. The basic application of geometrical construction were made centuries before the mathematical principles on which construction were based and recorded.
In the basic concept of geometry we generally studied about the points, lines, planes, closed flat shapes and many others. These are the most basic concept of geometry that provides the helps in describing, designing and constructing any type of visible objects. In todays life, the concept of geometry used in various field like architects, engineers field, planning for constructing the buildings, bridges roads and many other geometrical architecture cal things. In the below we show you the some of the basic
Concepts of geometry:
A) Point: In geometry a point can be defined as a mark of point which has no length, no breath and no height. It occupies the position and location but no magnitude. This point helps in creating various other type of geometrical shape.
B) Line: The line can be defined as a distance between the two points. A line has no width. Usually a line refers as a straight line. (know more about icse board, here)
C) Line – segment: This line segment different form the line. In line segment, the part of a line with two definite end points is defined but in line these definite points are not define. These line segments are very helpful to form various geometrical shapes like square, rectangle and so on.
As like above there are other geometrical concept are defining in mathematics. Like interesting lines, parallel lines, perpendicular lines and so on. In the next session we are going to discuss Grade VII, Tessellations.

Properties of lines

In this session we are going to learn about Properties of lines To study about the  properties of lines and angles we first talk about line as the group of endless points which are collinear and extends endlessly in both the directions. The lines do not have any fixed length and has arrows at both the ends, which indicates that the line is extending in both sides.
Here are some of the properties of the lines and angles:
If we have a pair of lines which intersect each other at only one point, then we say that the lines are intersecting at a point. If the two lines are intersecting then the pair of opposite angles so formed are called  vertical opposite angles. We must remember that the pair of  vertical opposite angles are  equal. Another pair of  lines we talk about are  parallel lines, the pair of lines are called parallel, when we find that the pair of lines  do not meet at all , or it is said intersecting lines  at infinite which we cannot  see at all.  Thus we say that the lines are parallel then the  perpendicular distance between  the two lines at all the points are equal. (know more about cbse sample papers, here)

Next type of lines are called coinciding  lines, when we have two lines on the plane, such that its all points are coinciding so that when the lines are plotted, the two lines overlap each other. Two lines are called perpendicular to each other, when we find that the pair of lines form an angle of 90 degrees  between each other, then the  lines are called perpendicular to each other. Remember that the line segment is the part of the line which has a fixed length and  so it has two end points. In the next session we will discuss about Geometric concepts. 

inequalities

Inequality is a collection of operators which is used to represent the inequality of algebraic equations. An inequality is a statement of algebraic expression to calculate the value of unknown variables. In general aspect we can say that inequality is used to calculate the algebraic expression that is not same in both sides of equal sign. The term inequality can be applied to any type of statement by using the various types of symbol like ‘>’ (greater then), ‘<’ (less then), ‘<=’ (less then equal to), ‘>=’ (greater then equal to) and so on. The concept of inequality helps the students of Grade VII to understand basic concepts of mathematics.
Here we show you the fundamental properties of inequalities to understand the concept of inequalities:
a)      x, y and z are the real numbers if x ≤ y then x + z ≤ y + z.
b)      x, y and z are the positive real numbers if x ≤ y then xz ≤ yz.
A solution of an inequality is a number which is substituted for the variable makes the inequality a true statement. In the mathematics there are various properties defined for inequality to solve equations. In the next session we are going to discuss Multistep problems.
a) Transitive property: if a > b and b > c then a > c.
b) Addition property: if a > b then a + c > b + c.
c) Multiplication property: if a > b then ab > ac.
d) Subtraction property: if a > b then a – c > b – c.
The above given properties of inequality helps the students to Graphing inequalities into the graph. Inequalities can be performed by solving the inequalities into the algebraic expressions. There are some rules given below:
a)      Adding and subtracting the same number on both sides.
b)      After performing the above rule interchange the sides and changing the orientation of the given inequality symbols.
c)      If needed, then perform the multiplication and division of same positive or negative number on both sides of equal sign then changing the orientation of the inequality symbol.

In the next session we are going to discuss Multistep problems. 

Simplifying numerical expressions

This unit is for Grade VII. We will learn how to solve the expressions. We know that the expressions are the numerals joined together with the help of different operators. When more than one operator appears in the expression and some fixed laws are not designed for which operation is to be performed first, we will get variety of outcomes for all the persons who operate the calculations. So we will set the pattern by which hierarchies of the operators are decided and we all get the symmetry of the output for any expression whichever is solved.  Simplifying numerical expressions is the planned and the systematic methods of calculating the mathematical expressions. The order of solving the equations is as follows: BODMAS, where we have,
B – Bracket or braces,
O- Of operation,
D- Division,
M- Multiplication,
A-Addition,
S- Subtraction,
It means that when we solve the mathematical expressions, we will first open the braces and so whichever expression appears in the braces will be solved first. Next step of solution includes the calculating an “of” operator.  After solving the operator of, we will take up all the division calculations in the expression. After completing division, comes multiplication of the terms which are joined by multiplication “*” operator. After multiplying, we will solve the operation of Addition of the terms in the expression and in the end is left the operator of subtraction, if any exist in the main given expression. Thus proceeding in the systematical way and following the steps of the calculation as explained above all the expressions will follow a set pattern and hence the results we get will be universally same, who so ever does it and whenever they are solved. (know more about cbse text books, here)

Thus we say that the above method of solving the equations will help us to get symmetry of outputs
In the next session we will discuss about inequalities. 

Linear non-linear functions

Linear functions can be defined as the functions that are denoted by the straight line or their equation is presented in the form of y = m x + c .Sometimes Linear functions are also describe as the equation that has variable with no power means highest power of the variable is one .This topic helps to understand basic concepts of grade VII.
When we talk about the non linear function then it is denoted as the functions whose graphs are not a straight line. These non linear functions are also known by different names as quadratic equation or cubic equations etc. means when the power of the variable in the given equation is greater than one then these types of equations are known as non linear equation. As if we define non linear equations as p a² + q a + r = b is an example of quadratic equation that is defined within the non linear equation or a cubic equation that is also the non linear equation is defined as,
b = p a³ + q a² + r a + s but we should keep in mind that value of ‘a’ will never be zero.
If we want to know that which equation is linear or which one is non linear then, check the exponent value on the variable ‘x’, if value of exponent is not greater than one then the expressions is defined as the linear function otherwise expression is non linear function . (know more about cbse question papers, here)
If the expression in the form of f (a) = p a + q then these types of functions are surely linear functions.
The other way to find the linear or non linear function is vertical line set.
In the next session we are going to discuss Simplifying numerical expressions. 

Operations on monomials

This Unit is for  Grade VII, here we are going to learn about  monomials operations. A monomial is an algebraic expression which has only one term.  Now we will learn about Operations on monomials. All mathematical operators namely addition, subtraction, multiplication and division can be performed on monomials. Before we discuss the methods of operation we must know the terms like terms and unlike terms. The terms with same variables are called like terms as 2xy and 5yx are like terms, on another hand 3xz and 3x are unlike terms.ely, without considering if they are like terms or not
When we perform addition and subtraction of the two monomials, then we add the terms which are like and the unlike terms are simply written in original form  with their operators. If the two monomials are multiplied or divided, we perform the operation of multiplication with constants and the variables separately
 Addition and subtraction of monomials is possible with like terms only, if the terms are unlike, then  only the terms are represented as follows:
Add 3xy and 4xz will give 3xy + 4xz Ans
On another hand if we have  to add 4x , 3y  and 8x, we get
=4x + 3y  + 8x
= 4x + 8x + 3y = 12x + 3y Ans.
Now we take the problem of subtraction: Find the difference between 6x and 2x will give:
= 6x – 2x = 4x Ans
If the problem is as follows  Subtract 4x from 6y, we get
6y – 4x  Ans as the two monomial terms are not like. (know more about cbse board papers, here)

 In case of multiplication, we can multiply the terms without considering if they are like or unlike. In multiplication, the numerals of the two monomials are multiplied and the powers of the like variables are added up. Similarly when we divide one monomial with another, in that too we need not to have like terms. We will simply divide the numeral with the numeral and the powers of the same variables are subtracted. Let us make it more clear with the following examples:
 Multiply 4x²y and 3y, we can write the above expression as = 4x²y * 3y =4 * 3 * x²y * y
= 12 x²y².
Divide  12x²y and 3y, we can write the above expression as = 12x²y / 3y = (12/3)* x² * y^1-1.
 = 4 x² * y⁰ , but we know that any number raise to the power 0 is 1, we get  4 x^{2    Ans}


In The Next Session We Are Going To Discuss Linear non-linear functions. 

Geometry

In this unit we are going to study Geometry of Grade VII. In earlier classes we have studied about lines and angles.
If the two lines are drawn in such a way that they do not meet at any point are called parallel lines. Also we conclude that the perpendicular distance between the two lines is equal at all the points. Now if ‘l’ and ‘m’ are any two lines such that l || m. Here if a line ‘n’ is drawn such that it intersect both the lines at  a point, then line ‘n’ is called the transversal at parallel lines ‘l’ and ‘m’.  Now we observe the following situations to occur if two parallel lines are intersected by a transversal:
a)      The pair of corresponding angles is equal.
b)       The pair of interior alternate angles is equal.
c)      The pair of exterior alternate angles is also equal.
d)      The pair of angles on the same sides of the transversal is supplementary. It means that the sum of angles on the same side of the transversal is 180 degrees. (know more about cbse books, here)
 Thus we conclude that if any two lines are given and we need to check that they are parallel or not, it means that we need to check any one of the above given conditions are satisfied, then the lines are parallel. So we will try to check that which of the above condition we can show are equal. More over we come across the problems in which we are given that the pair of lines is parallel and there exist a transversal passing through both the lines. Further any one angle among all the angles is given and we need to find the value of all the other angles so formed. We can find all the angles by applying the property of corresponding angles are equal and the interior alternate angles are equal, when the lines are parallel.
In the next session we will discuss about Operations on monomials. 

Multistep problems

In this unit we are going to learn how to solve multistep problems. This unit is designed for Grade VII.  When we have certain equations, which are having one variable, we proceed in the way that in every step, we move towards separating all constant values from the variables. This can be done in single steps, when the equations are small. But in the bigger equations which include multi operators existing in the equation, can be solved step by step. In the initial step, we will first shift all the variables to one side of the equation. While shifting we must remember that the positive term changes to a negative term and the negative term changes to a positive term. Here we can do the same process in another way, if the term which is negative on the right side of the equation has to be shifted to the right side of the equation, and then we simply add positive value of the same term on the both side of the equation. Thus the positive and the negative term of the right side of the equation will be cancelled, on another hand the positive of the same value will be added to another side of the equation.

We can proceed in the same method for subtraction too. Thus any negative value from one side of the equation is to be removed, for this we will add the same value on the both sides of the equation. This becomes possible as the addition and the subtraction are inverse of each other.

 Now we look at multiplication and division sign which appears in the equation. To separate the variable, we need to see that the constant value with the variable consists of the multiplication operator or the division operator. If the operator is of multiplication, divide both sides of the equation by the same number and if the operator is of division, multiply both sides of the equation by that number. It will give the resultant value of the variable.

In the next session we are going to discuss Geometry. 

 

Arithmetic sequences

Arithmetic sequences can be defined as a sequence of a number. It means that sequence of number specify the difference between the consecutive number that is constant. Suppose there is an arithmetic sequence 2, 5, 8, 11, 13, 16…….with the common difference of 3. In mathematics, sometime arithmetic sequence is known as arithmetic progressions. The calculation of arithmetic sequence is very easy to understand. This topic helps to understand basic concepts of grade VII. If the initial number of an arithmetic sequence is ‘x₁’ and the difference of the successive members is ‘df’, then the ‘x_n’ term of the sequence is given by:
                         x_n = x₁ + (n – 1) df,
And in the mathematical terms, it can be defined as,
                         x_n = x_m + (n – m) df.
A finite portion of an arithmetic sequence is called as a finite sequence and sometime it is known as arithmetic progression. The total of different arithmetic progression is known as arithmetic series. The number added or subtracted at each stage of an arithmetic sequence is known as the common difference “df”. In simple language, an arithmetic sequence can be defined as a set of number that follows a particular pattern. The term and pattern in the number sequence depends on the behavior of the common difference means on ’df’. Shown below are some of the instances, (know more about ICSE Board Syllabus, here)
a) If the common difference is positive in the sequence then the terms of the sequence grow towards the positive infinity.
b) If the common difference is negative in the sequence then the terms of the sequence grow towards the negative infinity.
The total of arithmetic sequence can be defined as an arithmetic series.
Sequence of ‘n’ numbers = x₁ + (x₁ + df) + (x₁ + 2 * df) + ……… + (x₁ + (n – 1) df),

In the next session we will discuss about Multistep problems. 

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.