Measurement in Grade VII

Hello friends, today I will make you familiar with few interesting math topics that you study in Grade VII. Today, we will go through formulas for measurements, graphing data to demonstrate relationships and Arithmetic sequences. Before, this I will make you familiar with the starting topic of Grade VII that you study in Algebra like:   Problems involving positive/negative powers and solving problems in percent; proportional relationships. After the brief introduction of what you study today let’s move on the topic. The very first topic that we will study today is Formulas for measurement. In this we study the different formula that are used for calculating different things like we learn the formula that are used to calculate the area of square, circle, and etc. not only you learn formulas but you also learn how to calculate them using the formula.

Let’s start with the circle and see its different formula used to calculate different things. We use several terms when we deal with circles like circumference, diameter, radius and many other. Circumference is the distance around the circle and the diameter is defined as the distance across the circle. The radius is generally defined as the half of the diameter or we can define it as the distance from the center to a point on the circle.

The different formula of measurement related to circle is circumference, radius and diameter. The radius is defined as ‘r’ and the diameter is defined as ‘d’ and d = 2r.

Circumference of circle (c)= pie(d) ,

                                         = pie(2r)

 And the area of circle is given as(A) = pie(r²) the value of pie is equivalent to 3.14.

For example r = 2cm, then calculate the d, c and A of the circle.

As you know d =  2r, so,

D = 2 X 2 cm,

D = 4cm,

C = pie(d) so,

C = pie X4,

C= 3.14 X4,

C = 12.56 cm. and in the same way Area(A) = pie(r²),

So, A = 3.14 X 2X2,

A = 12.56 cm².

 

In this grade kids learn formula for the rectangle also. The volume of solid rectangle is calculate with help of formula given as:

Volume = Length X Width X height

V = lwh

And the surface formula is give as: 2lw + 2lh + 2wh.

Let's see an example that how to calculate the volume of rectangle. Suppose you have length = 12, breadth = 2, and height = 5, and we can easily calculate the volume by putting the values in the formula, like

V = lbh

= 12 cm X 2 cm X 5 cm,

= 120 cm³.

If we have measurements in different units then first change all of them in same unit and then only apply the formula on the given terms. If we don’t take all the terms in the same form then you will not get the right or correct answer. In order to get the correct answer analyze the problem properly and then implement it in the formula.

Next formula for measurement we will study abiut the Trapezoid, its Area is given as: (b1 + b2/2) X h.

Perimeter of Trapezoid is given as: area +b1 + b2 +c

or,

P = a + b1 + b2 +c.

 Students of class VII also learn formula for calculating different measurement like Cylinder and Cones. The Volume of Cylinder is given as: pie(r²) X h.

The surface formula of Cylinder is: S 2(pie) r X  h or

S = 2(pie)rh + 2(pie) r²

The cone volume is calculated with the following formula:

Volume = 1/3 pie r² X h,

V = 1/3pier²h .

The Surface is calculated with help of S= (pie)r² + (pie)rs

S = pie r² + pie rs .

Let’s see an example of cylinders and see how to determine its surface. Suppose we have r = 5cm and h 8cm, then we can easily determine the surface of the cylinder using the formula:

The surface formula of Cylinder is: S=  2(pie) r X  h

Put all the given values in formula and doing so we get,

S = 2 X 3.12 X 5 X 8,

S= 251.2 cm.

In this way you can easily calculate the surface of the cylinder. Using different formula for measurement you can determine different things like volume, surface, area, circumference and etc.

Now, move to the next topic we will study today it is called as Arithmetic sequence. In mathematics, this sequence is defined as a sequence of numbers that has a constant difference between every two consecutive terms. In  simple words we can define the same as a sequence of numbers in which each term except the first term is the result of adding the same number, which is called as common difference, to the preceding terms.

Let’s see an example of the topic that my dear students can easily understand the topic properly and easily and examples are the best way to clear your doubts and concepts. The following examples are problems involving arithmetic sequences

Problem Number One:

 

The sequence 5, 11, 17, 23,….. is an arithmetic sequence in which the common difference between each term is of 6. And if 6 is added to each term of the sequence it gives the next number of the sequence.  Arithmetic series is defined as the indicated sum of the terms of an arithmetic sequence. The sequence 5, 11, 17, 23, 29, 35 is an arithmetic sequence. 5 + 11 + 17 + 23 + 29 + 35 is the corresponding arithmetic series.

 

In arithmetic sequence to calculate the nth term of the series we have a formula, which is given below:

An = A + (n - 1) d

Where, An = is the nth term, in the case of our problem it is the 40th term

A = the first term of the sequence , in our problem it is 2.

n = number of terms, in our problem it is 40.

d = the interval of the terms, or the difference of the next term.

Let’s see an example and implement the above formula in order to calculate the nth term.

Question: We have first three terms of an arithmetic sequence as 2, 6 and 10, then find the 40th term?

Answer: using the above formula, An = A + (n - 1) d

We can determine that what is given to us and we first we calculate the value of d.

To get d; d = 6 - 2 = 4.

A = 2,

n = 40,

and d = 4

Now, substitute the different values in the formula and we will get the final answer:

An = 2 + (40 - 1 ) 4

An = 2 + (39) 4

An = 2 + 156

An = 158.

The 40th term of the arithmetic sequence is 158.

Now, see one more problem of the Arithmetic sequence, suppose we have first term of an arithmetic sequence is -3 and the eighth term is 11, find d and write the first 10 terms of the sequence.

Solution: in the above problem we have,

A = -3,

n = 8 

A = 11

On substituting these values in the formula we will get,

11 = -3 + (8 - 1) d

11 = -3 + 7d

14 = 7d

d = 2

The first ten terms are -3, -1, 1, 3, 5, 7, 9, 11, 13, 15

 

After this move to other topic of Arithmetic sequence, it is called as sum of arithmetic sequence. Sum of the first ‘n’ terms of an arithmetic sequence with first term A and nth term An is;

Sn = n/2 (A + An) or this formula maybe rewritten as

Sn = n (A+An)/2

In the simple terms we can define the same as: "the number of terms multiplied by the mean value or average of the first and last terms."

If we have any arithmetic sequence with the first term A and common difference d, then the sum of the first n terms is given as:

Sn = n/2 2a + (n - 1 ) d

Now, see one more example to understand this concept.

Question: Determine the sum of all the odd integers from 1 to 1111, inclusive.

Solution :  in order to calculate the solution of the above problems we can simply use the following method,

Since the odd integers 1, 3, 5, etc, taken in order from the arithmetic sequence with d = 2, we can first find n from the formula for the nth term;

1111 = 1 + (n - 1) 2

1111 = 2n -1

1112 = 2n

n = 556

S = 556/2 ( 1 + 1111)

= 278 ( 1112)

= 309, 136.

In this way we get the solution of the above problem. Hopefully you understand both the topics that I taught you today. In the next section we will put light on some other interesting topics.