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Math Labs with Activity – Sum of the First n Terms of an AP

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Math Labs with Activity – Sum of the First n Terms of an AP

OBJECTIVE

To verify that the sum of the first n terms of an arithmetic progression where a is the first term and d is the common difference is given by
Math Labs with Activity – Sum of the First n Terms of an AP 1

Materials Required

  1. A sheet of white paper
  2. A geometry box
  3. A tube of glue
  4. A long, colored paper tape of uniform width (say, 1 unit)
  5. A pencil

Theory
If a is the first term, d the common difference and l the nth term of an AP then
l=a + (n-1)d. … (i)
Now, the sum of n terms of an AP is given by
Math Labs with Activity – Sum of the First n Terms of an AP 2
[using equation (i)].

Procedure
Step 1: We shall verify the above formula for a general AP having the first term a and the common difference d for n = 10.
Step 2: Draw horizontal lines on the sheet of paper with a distance of 1 unit between two consecutive lines.
Step 3: Cut 10 small rectangular strips from the coloured paper tape, each of the same length (say, a units).
Step 4: Cut 45 other small rectangular strips from the paper tape, each of the same length (say, d units).
Step 5: Paste both types of strips on the white paper along the horizontal lines so as to obtain rectangles of lengths a,a + d,a + 2d,…,a+9d arranged sequentially, as shown in Figure 3.1.
Step 6: Extend the line DE to C by a units to construct the rectangle ABCD (as shown in Figure 3.1).
Step 7: Cut the portion of the rectangle ABCD which is covered with the coloured paper tape. We find that this portion completely covers the remaining portion of the rectangle ABCD.
Math Labs with Activity – Sum of the First n Terms of an AP 3

Observations and Calculations

  1. The length of the rectangle ABCD = (a + 9d) + a = 2a + 9d and the breadth of the rectangle ABCD =10×1=10 units.
    ∴  the area of the rectangle ABCD = 10(2a + 9d) units² … (ii)
  2. Area of the portion of the rectangle ABCD covered with coloured strips of paper tape = sum of the areas of the 10 rectangles
    = (a x 1) + [(a + d) x 1] + [(a + 2d) x 1]+…+ [(a + 9d) x 1]
    = a+(a + d)+(a + 2d) +…+ (a + 9d). … (iii)
  3. Area of the portion of the rectangle ABCD covered by the coloured strips = ½ (area of the rectangle ABCD).
    a + (a + d)+(a + 2d) +…+(a + 9d) = 10/2 (2a + 9d) [using equations (ii) and (iii)]
    i.e., a + (a + d) + (a + 2d) +…+ [a + (n -1 )d] = n/2 [2a+(n -1 )d] for n = 10.

Result
It is verified for n = 10 that the sum of the first n terms of an AP is given by
Math Labs with Activity – Sum of the First n Terms of an AP 4
Remarks:
The students shall apply the above method of verification for various values of n, taking different values of a and d as well.

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Math Labs with ActivityMath LabsScience Practical SkillsScience Labs

Math Labs with Activity – Sequence of Numbers is an Arithmetic Progression (AP)

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Math Labs with Activity – Sequence of Numbers is an Arithmetic Progression (AP)

OBJECTIVE

To verify using a graphical method that a given sequence of numbers is an arithmetic progression (AP)

Materials Required

    1. Two sheets of graph paper
    2. A ruler
    3. A pencil
    4. A long, colored paper tape of uniform width (say, 1 unit)
    5. A tube of glue

Theory
A sequence of numbers in which every term except the first term is obtained by adding a fixed number to its preceding term is called an arithmetic progression (AP).

Procedure
Step 1: Mark both the sheets of graph paper with squares (each side = 1 unit).
Step 2: Mark x- and y-axes on each sheet of graph paper.
Step 3: We shall first test if the sequence 2,5,8,11,14,… is an AP.
Step 4: Cut the paper tape in rectangular strips of lengths 2 units, 5 units, 8 units, 11 units, 14 units,….
Step 5: Using the x-axis as the base, paste the strips on graph paper I sequentially (as shown in Figure 2.1). Record your observations in the first observation table.
Math Labs with Activity – Sequence of Numbers is an Arithmetic Progression (AP) 1
Step 6: We shall now test if the sequence 2,6,9,13,16,… is an AP.
Step 7: Cut the paper tape in rectangular strips of lengths 2 units, 6 units, 9 units, 13 units, 16 units,….
Math Labs with Activity – Sequence of Numbers is an Arithmetic Progression (AP) 2
Step 8: Using the x-axis as the base, paste the strips on graph paper II sequentially (as shown in Figure 2.2). Record your observations in the second observation table.

Observations
I. For Figure 2.1
Math Labs with Activity – Sequence of Numbers is an Arithmetic Progression (AP) 3
Math Labs with Activity – Sequence of Numbers is an Arithmetic Progression (AP) 4

Conclusions

  1. The sequence 2, 5, 8, 11, 14, … forms a uniform ladder having equal steps (as shown in Figure 2.1) and has a common difference d = 3 units (see the first observation table). Hence, this sequence is an AP.
  2. The sequence 2,6,9,13,16,… forms a ladder but of unequal steps (as shown in Figure 2.2) and the common difference (d) does not exist (see the second observation table). Hence, this sequence is not an AP.

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Math Labs with Activity – Solve the System of Linear Equations

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Math Labs with Activity – Solve the System of Linear Equations

OBJECTIVE

To use the graphical method to obtain the conditions of consistency and hence to solve a given system of linear equations in two variables

Materials Required

  1. Three sheets of graph paper
  2. A ruler
  3. A pencil

Theory
The lines corresponding to each of the equations given in a system of linear equations are drawn on a graph paper. Now,

  1. if the two lines intersect at a point then the system is consistent and has a unique solution.
  2. if the two lines are coincident then the system is consistent and has infinitely many solutions.
  3. if the two lines are parallel to each other then the system is inconsistent and has no solution.

Procedure
We shall consider a pair of linear equations in two variables of the type
a1x +b1y = c1
a2x +b2y = c2
Step 1: Let the first system of linear equations be
x + 2y = 3 … (i)
4x + 3y = 2 … (ii)
Step 2: From equation (i), we have
y= ½(3 – x).
Find the values of y for two different values of x as shown below.

x13
y10

Similarly, from equation (ii), we have
y=1/3( 2 – 4x).
Then

x-12
y2-2

Step 3: Draw a line representing the equation x+2y = 3 on graph paper I by plotting the points (1,1) and (3,0), and joining them.
Similarly, draw a line representing the equation 4x + 3y = 2 by plotting the points (-1, 2) and (2, -2), and joining them.
Math Labs with Activity - Solve the System of Linear Equations 1
Step 4: Record your observations in the first observation table.
Step 5: Consider a second system of linear equations:
x – 2y = 3 … (iii)
-2x + 4y = -6 … (iv)
Step 6: From equation (iii), we get

x31
y0-l

From equation (iv), we get

x-3-1
y-3-2

Draw lines on graph paper II using these points and record your observations in the second observation table.
Math Labs with Activity - Solve the System of Linear Equations 2
Step 7: Consider a third system of linear equations:
2x – 3y = 5 …(v)
-4x + 6y = 3 … (vi)
Step 8: From equation (v), we get

x14
y-11

From equation (vi), we get

x03
y½5/2

Draw lines on graph paper III using these points and record your observations in the third observation table.
Math Labs with Activity - Solve the System of Linear Equations 3
Observations
I. For the first system of equations
Math Labs with Activity - Solve the System of Linear Equations 4
II. For the second system of equations
Math Labs with Activity - Solve the System of Linear Equations 5
III. For the third system of equations
Math Labs with Activity - Solve the System of Linear Equations 6
Conclusions

  1. The first system of equations is represented by intersecting lines, which shows that the system is consistent and has a unique solution, i.e., x = -1, y = 2 (see the first observation table).
  2. The second system of equations is represented by coincident lines, which shows that the system is consistent and has infinitely many solutions (see the second observation table).
  3. The third system of equations is represented by parallel lines, which shows that the system is inconsistent and has no solution (see the third observation table).

Remarks: The teacher must provide the students with additional problems for practice of each of the three types of systems of equations.

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What is meant by the Net of a Solid?

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What is meant by the Net of a Solid?

Nets

A net is a sort of skeleton – outline in 2-D, which, when folded results in a 3-D shape.
OR
A net of a 3-D figure is the shape that can be cut out of a flat piece of paper or cardboard and folded to make the 3D-shape.
OR
A solid figure may be changed into a plane figure by cutting some of the edges of the original figure and opening it out flat as a fold-out. Such a plane figure is called a net of the solid.

Nets for Building Different 3-D Shapes

Net of a Cube
The following figure is a net of a cube which is formed by six squares.
meant-net-solid-1Net of a Cuboid
Given below is the net of a cuboid. This contains two pairs of congruent rectangle.
meant-net-solid-2

Net of a Cone
The given below is the net of a cone. This is obtained by cutting a segment of a circle.
meant-net-solid-3 Net of a Cylinder
Given below is the net of a cylinder. It is a rectangle with two congruent circle whose circumference is the length of the rectangle.
meant-net-solid-4Net of a Triangular Prism
Given below is the net of a triangular prism. The net consists of three rectangles and two equilateral triangles.
meant-net-solid-5 Net of a Triangular Pyramid
This net consists of four triangles in which three are identical isosceles triangles and one is an equilateral triangle.
meant-net-solid-6 Net of a Tetrahedron
To construct this, make an equilateral triangle of any measure. Then join the midpoints of these sides to form the inner dotted triangle.
meant-net-solid-7 Net of a Square Pyramid
This net consists of a square with an isosceles triangle on each side of the square.
meant-net-solid-8

10 Lines on Peacock for Students and Children in English

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10 Lines on Peacock

10 Lines on Peacock: Peacock, additionally called peafowl, any of the three types of radiant feathered birds of the pheasant family, Phasianidae (request Galliformes). So, the male is a peacock, and the female is a peahen; both are peafowl. The two most-unmistakable types of peafowl are the blue, or Indian, peacock (Pavocristatus), of India and Sri Lanka, and the green, or Javanese, peacock (P. muticus), found from Myanmar (Burma) to Java. The Congo peacock (Afropavocongensis), which occupies the forested inside of the Democratic Republic of the Congo, was found in 1936 after a hunt that started in 1913 with the finding of a solitary feather.

You can read more 10 Lines about articles, events, people, sports, technology many more.

Set 1 – 10 Lines on Peacock for Kids

Set 1 is helpful for students of Classes 1, 2, 3, 4 and 5.

  1. Peacock is one of the most beautiful creatures of the earth, which is feathered.
  2. Peacock is a blue-colored bird, and its feathers are blue, green, and golden.
  3. In India, Sri Lanka, Indonesia, and Africa, peacocks are found.
  4. Peacocks look beautiful in their colorful feathers.
  5. Peacocks look lovely when it dances in the rain.
  6. Peacocks cannot fly much high.
  7. Peacock has a huge tail.
  8. In night peacocks go on the tree for protecting themselves.
  9. Peacock is the male one and peahen is the female.
  10. The life of a peacock is generally up to 10 to 25 years.

10 Lines on Peacock for Kids

Set 2 – 10 Lines on Peacock for School Students

Set 2 is helpful for students of Classes 6, 7 and 8.

  1. Peacock is a magnificent bird that is found in not many nations like India, Myanmar, Sri Lanka, African mainland.
  2. In India, in Jammu-Kashmir, Assam, Mizoram, and parts of the eastern Indian landmass, peacocks are found.
  3. The tail of the peacock is beautiful; mainly when it moves in the downpour.
  4. In the year 1963, peacock said to be the national bird.
  5. In mythology “Karthikeya” is referred to as a peacock. Ruler Krishna wore peacock’s plume on his head.
  6. Peacock is sacred in the Hindu religion.
  7. The feather of the peacock is likewise utilized in some plan and enrichment, hoops; ornaments made with peacock feathers are additionally very mainstream.
  8. Peacocks are said to be timid they evade individuals and attempt to stow away in shrubs and in different spots from the individuals who are looking to it.
  9. There are a few types of a peacock which have white feathers.
  10. Peacocks are exceptionally alert, on feeling any threat they begin yelling to caution different individuals from their peacock family.

Set 3 – 10 Lines on Peacock for Higher Class Students

Set 3 is helpful for students of Classes 9, 10, 11, 12 and Competitive Exams.

  1. Peacock is an adorable feathered creature. It has variation feathers that make it distinctive from other birds. Peacocks have a crown or peak on their head.
  2. They are called kings all things considered. They look delightful because of their vivid feather.
  3. Peacocks can live in dry and hot deserts and even can get by in frigid climates. In territories where there is a lot of freshwaters found, there in woodlands and bushes. Peacocks sleep on the lower parts of trees.
  4. Peacocks are exceptionally shy and timid.
  5. Peahen is the female one. Peacocks can not fly yet they can run extremely quickly since they have solid legs. They ordinarily make sharp and abrasive voices to caution different birds when they see any danger.
  6. On cloudy days they deliver a ton of views since they love downpours. In the rainy season when it downpours Peacocks open their feathers and move joyfully.
  7. They live up to 20 to 25 years. Their female peahen is smaller in size.
  8. As we realize that Peacocks are likewise found in fields and gardens. They, as a rule, eat grains. They are the companion of ranchers and foe of the snake and insects.
  9. Peacocks can not fly higher because their large tail is more massive than their body.
  10. They are likewise found in African landmass, Sri Lanka and Myanmar.

10 Lines on Peacock for Higher Class Students

FAQ’s on 10 Lines On Peacock

Question 1.
Can peacocks able to fly?

Answer:
No, the peacock cannot fly because of their large tail that stops them from flying.

Question 2.
Do snakes scare from peacocks?

Answer:
Peacocks do not like snakes, so they generally are tough on them, which leads the snakes to run away.

Question 3.
Who are called peacocks?

Answer:
The male peafowl is known as a peacock, and the female peafowl is the peahen.

Question 4.
Describe the nature of the peacocks.

Answer:
Peacocks are generally aggressive, territorial types of birds which usually said to be the king of all birds. They are also very protective of the creatures they are friendly with.