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NCERT Class 9 Maths Lab Manual - Form a Cube and Find the Formula for its Surface

Find Probability of Unit’s Digit of Telephone Numbers Experiment Class 9 Maths Practical NCERT

The experiment to determine Find Probability of Unit’s Digit of Telephone Numbers are part of the Class 9 Maths Lab Manual provides practical activities and experiments to help students understand mathematical concepts effectively. It encourages interactive learning by linking theoretical knowledge to real-life applications, making mathematics enjoyable and meaningful.

Maths Lab Manual Class 9 CBSE Find Probability of Unit’s Digit of Telephone Numbers Experiment

Determine Find Probability of Unit’s Digit of Telephone Numbers Class 9 Practical

Objective
To find experimental probability of unit’s digits of telephone numbers listed on a page selected at random of a telephone directory.

Materials Required

  1. Telephone directory
  2. Ruler
  3. Notebook
  4. Pen

Prerequisite Knowledge
Basic knowledge of probability.

Theory

  1. If E is an event that happen when an experiment is performed, then the experimental or empirical probability of the event E is given by
    P(E) = \(\frac { Number\quad oftrials\quad in\quad which\quad the\quad event\quad E\quad happened }{ Total\quad number\quad of\quad trials } =\frac { n(E) }{ n(S) }\)
    or
    Probability of an event E = P(E) = \(\frac { Number\quad of\quad trials\quad in\quad which\quad the\quad event\quad occurred }{ Total\quad number\quad of\quad trials }\)
  2. The probability of happening of an event always lies from 0 to 1, i.e. 0 ≤ P(E) ≤ 1.
    In percentage, it lies from 0% to 100%.
  3. If probability of an event say A is 1, i.e. P(A) = 1, then event A is called a certain event or sure event.
  4. If probability of an event say B is 0, i.e. P(B) = 0, then event B is called an impossible event.
  5. The sum of all the probabilities of all possible outcomes of an experiment is 1.

Procedure

  1. Taking a telephone directory, select any page at random.
  2. Suppose the count of total telephone numbers on the selected page is N.
  3. Unit place of any telephone number can be occupied by any one of the digits 0,1,2, ……. 9.
  4. Now, using tally marks, prepare a frequency distribution table for the digits at unit’s place.
  5. Now, using the table, write the frequency of each of the digits 0,1,2, …….. 9.
  6. By using the formula for experimental probability, find the probability of each digit.

Demonstration

1. Firstly, by using tally marks, prepare a frequency distribution table for the digits 0,1,2,…. 9

Digits 0 1 2 3 4 5 6 7 8 9
Tally marks frequency n0 n1 n2 n3 n4 n5 n6 n7 n8 n9

2. From the table, note down the frequency of each digit from 0 to 9.
3. We get that digits 0, 1,2,…, 9 are occurring n0, n1, n2, …, n9 times respectively.
4. Considering the occurrence of each digit as an event E, the probability of event E is
P(E) = \(\frac { Number\quad of\quad trials\quad in\quad which\quad event\quad occurred }{ Total\quad number\quad of\quad trials }\)
Hence, respective experimental probability of occurrence of 0, 1, 2, …, 9 is given by
P(0) = \(\frac { { n }_{ 0 } }{ N }\), P(1) = \(\frac { { n }_{ 1 } }{ N }\), ……., P(9) = \(\frac { { n }_{ 9 } }{ N }\)

Observations
Total telephone numbers on a page (N) = ………….
Number of times 0 occurring at unit’s place (n0) = …………..
Number of times 1 occurring at unit’s place (n1) = …………….
Number of times 2 occurring at unit’s place (n2) = …………..
Number of times 3 occurring at unit’s place (n3) = ………….
Number of times 4 occurring at unit’s place (n4) = ………….
……………..
……………….
Number of times 9 occurring at unit’s place (n9) = …………….
Hence, experimental probability of occurrence of 0 = P(0) = \(\frac { { n }_{ 0 } }{ N }\)
Now, experimental probability of occurrence of 1 = P(1) = \(\frac { { n }_{ 1 } }{ N }\)
P(2) = \(\frac { { n }_{ 2 } }{ N }\)
……….
……….
P(9) = \(\frac { { n }_{ 9 } }{ N }\)

Result
We have got the experimental probability of unit’s digits of telephone numbers listed on a page selected at random of a telephone directory.

Applications
The concept of experimental probability is useful in

  1. deciding premium tables by insurance companies. .
  2. stock market to forecast the performance of a company, by metreological department to forecast weather.

Viva-Voce

Question 1.
How will you define an event?
Answer:
An event for an experiment is the collection of some outcomes of the experiment.

Question 2.
How will you define the empirical probability P(E) of an event E?
Answer:
P(E) = \(\frac { Number\quad of\quad trials\quad in\quad which\quad E\quad has\quad happened }{ Total\quad number\quad of\quad trials }\)

Question 3.
What are the maximum and minimum values of the probability of an event?
Answer:
Maximum and minimum values of the probability of an event are 1 and 0 respectively.

Question 4.
What is the complement of an event E?
Answer:
1 – P(E)

Question 5.
What is the probability of a certain event?
Answer:
1

Question 6.
How many events can occur when a coin is tossed?
Answer:
Two events, i.e. head or tail.

Question 7.
How will you define a sure event?
Answer:
If probability of an event say A is 1, i.e. P(A) = 1, then event A is called a certain event or a sure event.

Question 8.
Is the sum of all the probabilities of all possible outcomes of an experiment 1?
Answer:
Yes, the sum of all the probabilities of all possible outcomes of an experiment is 1.

Suggested Activity
Find the experimental probability of getting a tail in tossing an unbiased coin 5,10,15,20,25,30 times.

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NCERT Class 9 Maths Lab Manual 

Obtain the Formula for the Surface Area of a Sphere Experiment Class 9 Maths Practical NCERT

The experiment to determine Obtain the Formula for the Surface Area of a Sphere are part of the Class 9 Maths Lab Manual provides practical activities and experiments to help students understand mathematical concepts effectively. It encourages interactive learning by linking theoretical knowledge to real-life applications, making mathematics enjoyable and meaningful.

Maths Lab Manual Class 9 CBSE Obtain the Formula for the Surface Area of a Sphere Experiment

Determine Obtain the Formula for the Surface Area of a Sphere Class 9 Practical

Objective
To obtain the formula for the surface area of a sphere.

Materials Required

  1. Cardboard/Wooden strips
  2. Thick sheet of paper
  3. Ruler
  4. String
  5. Measuring tape
  6. Adhesive
  7. A ball
  8. Scissors/Cutter
  9. A pin

Prerequisite Knowledge

  1. Basic knowledge about shape of a sphere.
  2. Basic concept of a circle and its area.

Theory

  1. A sphere is three dimensional figure (solid figure) which is made up of all points in the space, which lie at constant distance, from a fixed point called the centre of the sphere and the constant distance is called its radius, (see Fig. 31.1)
    A line segment through the centre of a sphere and with the end p’ ints on the sphere is called a diameter of the sphere, (see Fig. 31.1)
    NCERT Class 9 Maths Lab Manual - Obtain the Formula for the Surface Area of a Sphere 1
  2. For concept of circle refer to Activity 23.
    Area of a circle = πr²

Procedure

  1. Take a spherical ball and by placing it between two vertical boards or wooden strips, find its diameter (d). (see Fig. 31.2)
    NCERT Class 9 Maths Lab Manual - Obtain the Formula for the Surface Area of a Sphere 2
  2. Mark the topmost part of ball and fixing a pin. (see Fig. 31.3)
  3. Wrap the ball (spirally) with string completely, by taking support of pin such that no space is left uncovered, (see Fig. 31.3)
  4. Mark the starting and ending points on the string (for accuracy, it should be thinner) measure the length between these two marks and denoted by l.
    NCERT Class 9 Maths Lab Manual - Obtain the Formula for the Surface Area of a Sphere 3
  5. Now, unwind the string from the ball.
  6. Draw four circles on the thick sheet of paper with radius equal to the radius of the ball, i.e. r.
  7. Now, fill the circles one-by-one with string which wound around the ball, (see Fig. 31.4)
    NCERT Class 9 Maths Lab Manual - Obtain the Formula for the Surface Area of a Sphere 4

Demonstration
Let the length of string which covers a circle (radius r) is denoted by a.
The string which had completely covered the surface area of the ball, has been used completely to fill the region of four circles. This conclude that
Length of string needed to cover sphere of radius r = 4 x (Length of string needed to cover one circle)
i.e. Surface area of sphere = 4 x Area of circle of radius r
Surface area of sphere = 4πr²

Observation
Diameter (d) of the spherical ball = ………….. units
Radius (r) = ………….. units
Length of string / used to cover ball = …………. units
Length of string a used to cover one circle = ………….. units
Hence, l = 4 x ……………
Surface area of a sphere of radius, (r) = 4 x Area of a circle of radius = 4πr²

Result
We have verified the formula for the surface area of sphere experimentally.
Note:

  1. Measure the diameter of ball carefully.
  2. Wrap the ball completely so that no space is left uncovered.
  3. Thinner the string more is the accuracy.

Application
This activity is useful in finding the cost of painting, repairing, constructing spherical and hemispherical objects.

Viva-Voce

Question 1.
What is the formula for finding the surface area of a sphere having radius r units?
Answer:
4πr²

Question 2.
Is a circle an another name of a sphere?
Answer:
No

Question 3.
What do you mean by a hemisphere?
Answer:
The equal half part of sphere is called a hemisphere.

Question 4.
Can we take a disk as an example of a sphere?
Answer:
No

Question 5.
Is sphere a two dimensional figure?
Answer:
No, sphere is a three dimensional figure.

Question 6.
Are the total surface area and curved surface area of a sphere equal?
Answer:
Yes

Suggested Activity
By using this activity, calculate the surface area of sphere of radius 10.5 cm.

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NCERT Class 9 Maths Lab Manual 

Draw Histograms for Classes of Equal Widths and Varying Widths Experiment Class 9 Maths Practical NCERT

The experiment to determine Draw Histograms for Classes of Equal Widths and Varying Widths are part of the Class 9 Maths Lab Manual provides practical activities and experiments to help students understand mathematical concepts effectively. It encourages interactive learning by linking theoretical knowledge to real-life applications, making mathematics enjoyable and meaningful.

Maths Lab Manual Class 9 CBSE Draw Histograms for Classes of Equal Widths and Varying Widths Experiment

Determine Draw Histograms for Classes of Equal Widths and Varying Widths Class 9 Practical

Objective
To draw histograms for classes of equal widths and varying widths.

Materials Required

  1. Graph paper
  2. Geometry box
  3. Adhesive
  4. Sketch pens
  5. Scissors
  6. A piece of cardboard

Prerequisite Knowledge

  1. Knowledge about collection of data.
  2. Concept of histogram.

Theory

  1. Collection of Data
    The facts or figures which are numerical or otherwise; collected with a definite purpose (i.e. observation/information collected), are called data. Data is the plural form of the Latin word datum. Some basic terms related to data are given below:

    • The data collected by the investigator himself for a definite plan or purpose is known as
      primary data.
    • The data collected by someone and used by any other person is known as secondary data.
    • The data obtained in original form is called raw data or ungrouped data.
    • We may condense data into classes or groups such a presentation is known as grouped data.
  2. Histogram
    A histogram is the graphical representation of a grouped frequency distribution with continuous classes in the form of rectangles with class intervals as bases and the corresponding frequencies as heights. There being no gap between any two consecutive rectangles.
    Histogram is commonly used to show frequency distribution.

Procedure
1. Firstly, collect the data from daily life such as weights/heights of children in a class, then make a frequency distribution table.
Case I For classes of equal widths,

Class a-b b-c c -d d -e e-f
Frequency f1 f2 f3 f4 f5

Case II For classes of varying widths,
Here, d – f = 2(a-b)

Class a-b
(width x)		 b-c
(width x)		 c-d
(width x)

d-f
(width 2x)
Frequency f1 f2 f3 f4
Modified frequency f1 f2 f3 F’ = \(\frac { { f }_{ 4 } }{ 2 }\)

2. Take a graph paper having dimensions (20cm x 20cm) and paste it on a piece of cardboard sheet.
3. Now, draw two perpendicular axes X’OX and Y’OY on the graph paper, (see Fig. 32.1)
4. At equal distances, mark classes on X-axis and frequency Y-axis, (see Fig. 32.1)
NCERT Class 9 Maths Lab Manual - Draw Histograms for Classes of Equal Widths and Varying Widths 1
5. Draw rectangles of equal widths and heights f1, f2, f3, f4 and f5 on intervals (a-b), (b-c), (c-d), (d-e) and (e-f), respectively, (see Fig. 32.2)
NCERT Class 9 Maths Lab Manual - Draw Histograms for Classes of Equal Widths and Varying Widths 2
6. Draw rectangles of heights f1, f2, f3 and F’ on intervals (a-b), (b-c), (c-d) and (d-f) respectively, (see Fig. 32.3)
NCERT Class 9 Maths Lab Manual - Draw Histograms for Classes of Equal Widths and Varying Widths 3

Demonstration
On taking different numerical values of a, b, c, d, e and f, histogram of equal widths and varying widths can be drawn.

Observations
Case I For the classes of equal widths,

  1. The intervals are
    a-b = ………..
    b-c = …………
    c-d = …………..
    d-e = ……….
    e-f = ………..
  2. f1 = …………
    f2 = ………..
    f3 = ………….
    f4 = ………….
    f5 = …………..

Case II For the classes of varying widths,

  1. The intervals are
    a-b = ………..
    b-c = …………
    c-d = …………..
    d-f = ……….
  2. f1 = …………
    f2 = ………..
    f3 = ………….
    f4 = ………….
    F’ = \(\frac { { f }_{ 4 } }{ 2 }\) = …………..

Result
We have drawn histogram of equal widths and varying widths by using the collected data.

Application
Histograms can be used pictorially in representing large data in a concise form.

Viva-Voce

Question 1.
How will you define the statistics?
Answer:
Statistics deals with the collection, organisation, analysis and interpretation of data.

Question 2.
What do you mean by data?
Answer:
The facts or figures which are numerical or otherwise collected with a definite purpose are called data.

Question 3.
What do you understand by a primary data?
Answer:
When the information was collected by the investigator herself or himself with a definite objective in her or his mind, the data obtained is called primary data.

Question 4.
How will you define a secondary data?
Answer:
When the information was gathered from a source which already had the information stored, the data obtained is called secondary data.

Question 5.
In what form, data is presented graphically?
Answer:
Data can be presented graphically in the form of bar graph, histogram and frequency polygons.

Question 6.
How will you define a histogram?
Answer:
A histogram is the graphical representation of a grouped frequency distribution with continuous classes in the form of rectangles with class intervals as bases and the corresponding frequencies as heights.

Question 7.
What is the singular form of data?
Answer:
Datum

Question 8.
What do you understand by grouped data?
Answer:
Grouped data is a statistical term used in a data analysis. A raw data set can be organised by constructing a table showing the frequency distribution of the variable {whose values are given in the raw data). Such a frequency table is often reffered to as grouped data.

Suggested Activity
Draw histograms for number of students in each section of class IX of your school.

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NCERT Class 9 Maths Lab Manual 

Form a Cone from a Sector of a Circle Experiment Class 9 Maths Practical NCERT

The experiment to determine Form a Cone from a Sector of a Circle are part of the Class 9 Maths Lab Manual provides practical activities and experiments to help students understand mathematical concepts effectively. It encourages interactive learning by linking theoretical knowledge to real-life applications, making mathematics enjoyable and meaningful.

Maths Lab Manual Class 9 CBSE Form a Cone from a Sector of a Circle Experiment

Determine Form a Cone from a Sector of a Circle Class 9 Practical

Objective
To form a cone from a sector of a circle and to find the formula for its curved surface area.

Materials Required

  1. A piece of wooden hardboard
  2. Acrylic sheets
  3. White paper
  4. Adhesive tape
  5. Scissors
  6. Geometry box
  7. Marker

Prerequisite Knowledge

  1. Concept of a circle.
  2. Concept of sector of a circle.
  3. Concept of a cone.

Theory

  1. For concept of a circle refer to Activity 23.
  2. Sector of the Circle:
    The region between an arc and the two radii joining the centre to the end points of the arc is called a sector.
    Sectors are of two types – minor sector and major sector. Minor sector is the sector of circle, which is less than a semi-circle and major sector is the sector of circle, which is greater than a semi-circle, (see Fig. 28.1)
    Area of sector = \(\frac { Arc\quad length }{ Circumference\quad of\quad circle } \times { Area\quad of\quad the\quad circle }\)
    NCERT Class 9 Maths Lab Manual - Form a Cone from a Sector of a Circle 1
  3. Cone: A right circular cone is a solid generated by revolving a line segment which passes through a fixed point and which makes a constant angle with a fixed line.
    In other words, if a right angled triangle is revolved about one of the two sides forming a right angle, keeping the other sides fixed in position, then the solid so obtained by revolving the line segments is called a right circular cone.
    NCERT Class 9 Maths Lab Manual - Form a Cone from a Sector of a Circle 2
    In Fig. 28.2, a right angled ∆OPA on revolving about the segment OP, generates a right circular cone in which ABC is a circle.

Procedure

  1. Take a piece of wooden hardboard of suitable size and by using adhesive, paste a white paper on it.
  2. From acrylic sheet, cut out a circle of radius l. (see Fig. 28.3)
    NCERT Class 9 Maths Lab Manual - Form a Cone from a Sector of a Circle 3
  3. Now, cut out a sector having angle θ° from the circle, (see Fig. 28.4)
    NCERT Class 9 Maths Lab Manual - Form a Cone from a Sector of a Circle 4
  4. To form a cone, bring together both the radii of the sector and by using a adhesive tape, attach the ends and fix it on the hardboard. (see Fig. 28.5)
    NCERT Class 9 Maths Lab Manual - Form a Cone from a Sector of a Circle 5

Demonstration

  1. Radius of the base of cone = r
  2. Slant height of the cone = Radius of circle = l
  3. Circumference of the base of cone = Arc length of sector = 2πr
  4. Now, curved surface area of cone = Area of the sector
    = Area of sector = \(\frac { Arc\quad length }{ Circumference\quad of\quad circle } \times { Area\quad of\quad the\quad circle }\)
    = \(\frac { 2\pi r }{ 2\pi l } \times \pi { l }^{ 2 }\)
    = πrl

Observation
By actual measurement,
The slant height (l) of the cone = ………… and radius (r) = …………
∴ Arc length, (l) = ………….
Area of the sector = ………….
curved surface area of the cone = …………
Hence, curved surface area of the cone = Area of the sector

Result
We have derived the formula for calculating the curved surface area of cone.

Applications
This result is useful in

  1. estimation of canvas required to make a conical tent.
  2. estimation of material required to make joker’s cap, ice-cream cone, etc.

Viva-Voce

Question 1.
What is the sector of a circle?
Answer:
The sector of a circle is the portion which is enclosed by two radii and an arc.

Question 2.
How will you define a cone?
Answer:
A cone is a three dimensional geometrical shape that has one circular base and one vertex.

Question 3.
What is the formula for finding the curved surface area of a cone of radius r and slant height l?
Answer:
Curved surface area of a cone = πrl

Question 4.
Do you know about any formula for finding the area of base of a cone?
Answer:
Yes, we know that area of base of a cone can be calculated with help of the formula for finding the area of a circle, i.e. πr².

Question 5.
What is the slant height of a cone having radius r and height h?
Answer:
Slant height, l = √(h² + r²)

Suggested Activity
After forming a cone from the sector of a circle, verify experimentally that curved surface area of cone is equal to the area of the sector.

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NCERT Class 9 Maths Lab Manual 

Form a Cube and Find the Formula for its Surface Experiment Class 9 Maths Practical NCERT

The experiment to determine Form a Cube and Find the Formula for its Surface are part of the Class 9 Maths Lab Manual provides practical activities and experiments to help students understand mathematical concepts effectively. It encourages interactive learning by linking theoretical knowledge to real-life applications, making mathematics enjoyable and meaningful.

Maths Lab Manual Class 9 CBSE Form a Cube and Find the Formula for its Surface Experiment

Determine Form a Cube and Find the Formula for its Surface Class 9 Practical

OBJECTIVE

To form a cube and find the formula for its surface area experimentally.

Materials Required

  1. Cardboard
  2. Ruler
  3. Cutter/Scissors
  4. Adhesive tape
  5. Pen/Pencil

Prerequisite Knowledge

  1. Knowledge about basic properties of a cube.
  2. Surface area, lateral surface area and diagonal of a cube.

Theory

  1. For basic properties of a cube refer to Activity 7.
  2. Surface Area of a Cube
    Let each edge of the cube be a units. Then,

    1. Total Surface Area (TSA) of the cube = Sum of areas of six faces
      = 6 (area of each faces)
      = 6 (a x a)
      => TSA = 6a² sq units
    2. Lateral surface area of the cube
      = Sum of areas of four faces only leaving the top and bottom faces = 4(area of each face) = 4 (a x a)
      => Lateral surface area = 4a² sq units
    3. Diagonal of the cube = \(\sqrt { { a }^{ 2 }+{ a }^{ 2 }+{ a }^{ 2 } }\)  = \(\sqrt { { 3a }^{ 2 } }\)
      => Diagonal of the cube = √3a units

Procedure

  1. Take a cardboard of a suitable size, using cardboard make six identical squares each of side x units.
  2. Now, arrange all six squares and join all of them with the help of tape, (see Fig. 26.1)
    NCERT Class 9 Maths Lab Manual - Form a Cube and Find the Formula for its Surface 1
  3. Fold the square along the dotted markings (or along tape) (see Fig. 26.1) and form a cube using tape, (see Fig. 26.2)
    NCERT Class 9 Maths Lab Manual - Form a Cube and Find the Formula for its Surface 2

Demonstration
Surface area of a cube of side x units =6x (area of each face of cube of side x units)
= 6x² [since, each face of the cube is a square, so area of square = (side)² = (x)² ]
Note:
A net of a cube be directly prepared on the cardboard itself instead of making six squares separately.

Observation
By actual measurement,
Length of side x = ……….. ,
Area of one square/one face = x² = ……….. ,
So, sum of areas of all the squares = ……….. + ………..  + ……….. + ……….. + ……….. + ………..
Hence, surface area of the cube is 6x².

Result
We have verified the formula for calculating the surface area of the cube.

Application
This activity may be used in estimation of materials necessary in production of cubical boxes used for packing.

Viva Voce
Question 1:
How many edges are there in a cube?
Answer:
12

Question 2:
If each edge of a cube is 3 cm, then what is the area of each face of a cube?
Answer:
9 cm².

Question 3:
If a room is in the shape of cube of length x units, then what will be the area of room which has to be painted by painter including gate?
Answer:
5x² (= 6x² – x²), because one face will be floor of the room.

Question 4:
What is the required condition for a room to be in the shape of cube?
Answer:
The length, breadth and height of the room should be equal.

Question 5:
How will you define a cube?
Answer:
A cube is a symmetric three dimensional solid shape bounded by six square faces.

Question 6:
Find the maximum length of the rod that can be placed in a cube of side x.
Answer:
The maximum length of the rod that can be placed in a cube is equal to the diagonal of the cube, i.e.√3x.

Question 7:
How many lateral surfaces a cube have?
Answer:
Their are 4 lateral surfaces of a cube.

Question 8:
Write three real life examples of cube.
Answer:
Dice, sugar free cube, rubiks cube.

Question 9:
How many dimensions of a cube?
Answer:
A cube has three dimensions.

Question 10:
“If we take one surface of a folding paper cube, then it is a two dimensional plane”. Is this statement true?
Answer:
Yes

Suggested Activity
Using the above activity, find the area of a cube, if each edge of a cube is 6 cm.

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NCERT Class 9 Maths Lab Manual