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Well graded series of Activity Books for Schools

By Subodh Narvekar1. VIKAS DRAWING

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These books teach –Colouring, Drawing, Shading, Painting techniques, Creative work, Mosaic painting, Spray painting, Stippling, Chinese painting, Mask making, Clay modelling, etc.

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A Laboratory Manual

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Explore and Learn Mathematics Laboratory manual is a set of five books from Classes 6 to 10. This series is based on syllabus prescribed by the NCERT and latest guidelines issued by CBSE. We have tried to incorporate the vision of the NCF that suggests significant strategies to make education more relevant to the present day and future needs. We understand that students have different ways of learning and they use different intelligences to understand concepts in their daily lives. Some learn very well by reading and writing, others understand better by mathematical logic-based learning, while some learn by doing activities with hands.

A Mathematics Laboratory is a place where students learn and explore mathematical concepts and verify mathematical facts and theorems through a variety of activities using different materials. The Lab Manual collects the various activities together in a useful companion. These activities may be carried out under the guidance of the teacher or by students on their own to explore, to learn, stimulate interest and develop a positive and empowered attitude towards Mathematics. Mathematics activities enable the teachers to demonstrate, explain and reinforce abstract mathematical ideas by using concrete objects, models, charts, graphs, pictures, posters, etc. These allow and encourage the students to think, discuss with each other and the teacher and assimilate the concepts in a more effective manner. Keeping in mind the needs of different students, activities on different concepts have been given to lend a strong base to their mathematical awareness.

This series has been developed and reviewed by experts who are actively engaged in the field of education. We are confident that Mathematics Lab Manual will prove to be of immense value for the teachers, students and parents. Every care has been taken to keep the book error free. We look forward to feedback and constructive criticism from our users. This would definitely help us to improve further.

The Publisher

Features of the series

Learning objective: Describes the concept that will be investigated

Basic knowledge required: Includes basic concepts/words required to do the activity

Materials required: Items required to do the activity

Procedure: Actual stepwise process with diagrams of the activities leading to the final outcome

Observations and Results: Observation points listed after the activity is performed and the inference drawn

Learning outcome: Skills achieved after performing the activity

Remember: Includes important information on the concepts learnt

Did you know?: Interesting snippets of information based on the concepts learnt

Let’s explore!: More questions based on the concept for practice

Everyday maths: Includes real-life applications of the concepts learnt

Know more!: Includes amazing facts about the concepts discussed

Viva voce: Includes oral questions for practice

For Teachers: Suggestions for teachers that help in teaching-learning process

Group activities and projects: Includes additional activities, games and projects based on the concepts learnt

Appendix: Includes ICT activities, case-based study questions, chapter-wise worksheets and special page on India in Maths

Mathematics Laboratory provides an excellent opportunity for the students to understand mathematical concepts and verify mathematical facts and theorems through a variety of activities using different materials. These activities help students to visualise, manipulate and reason. These also help students to explore and learn while stimulating interest towards Mathematics.

Need and Purpose of Mathematics Laboratory

Enables students to verify or discover geometrical properties and facts using models or different methods such as paper cutting and folding.

Helps students to develop interest in the subject and gain confidence in learning the subject.

Provides an opportunity to relate mathematical concepts with daily life.

Enables teachers to demonstrate and explain abstract mathematical concepts using concrete objects such as models, charts, graphs, pictures, posters, dice, cutouts, counters, etc.

Provides opportunity for individual participation in the process of learning and help in self-learning.

Encourages students to think and discuss with peers and teachers that helps in assimilating the concepts in a more effective manner.

General Design and Physical Infrastructure

A Mathematics laboratory should be able to accommodate 25 to 30 students. The laboratory should have suitable furniture, all essential equipment/tools and other necessary things to carry out the activities included in the manual effectively. The quantity of different materials may vary from one school to another.

Human Resources

The person in-charge of Mathematics laboratory should have a minimum qualification of graduation (with Mathematics as one of the subjects) and a professional qualification of Bachelor in Education. He/she is expected to have an interest in the subject and have suitable skills to help students perform practical work. Mathematics teacher should help the students to perform the laboratory activities. A laboratory attendant or laboratory assistant with suitable qualifications, knowledge and experience can be a helping hand.

Time Allocation for Activities

About 15%–20% of the total available time should be devoted to performing activities in Mathematics lab. The total available time should be divided judiciously between theory classes and practical work. Periods should be allotted for laboratory activities in the time table as well.

Otherstationeries

Broomsticks

Sketch pens

Unit cubes

Beads of one colour(50 pieces)

Counters of different colours (8 pieces)

Dice numbered 1 to 6

Cutouts of various shapes

Paper nets of various shapes

Whitesheets

of paperDrawing sheets/

Chart papers

Glazed papers

Cardboards

Graph paper

Geoboard

Pencils

Rubber bands

A pair of scissors

Pushpins and threadsGlue/

Sticky tape

Transparency sheets

Some of the methods used

1. Paper cutting and pasting

2. Paper folding

Mathematical instruments

Ruler, divider, compasses, protractor, set squares

Models for verifying included identities, models for verifying

included geometrical results, etc.

Date Parent's SignatureTeacher's Signature

This is to certify that the activities written in

index have been performed by

CERTIFICATE

Satisfactorily during the academic year

Class

School

Section

ExcellentInitiative, Co-operationand Participation

Aesthetic presentation, Visual appeal, Expression and Neatness

Accuracy, Creativity, Originality and Analysis

Good Average

: Number systemuNitActivity 1. To represent an irrational number on a number line .......................................................................................................................................9Activity 2. To represent the construction of a square root spiral .................................................................................................................................... 11

: PolyNomialsuNit Activity 3. To verify the following algebraic identity using geometric shapes: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac .............................................................................................................................................................15Activity 4. To verify the following algebraic identities using geometrical solid shapes: A. (a + b)3 = a3 + b3 + 3ab (a + b) B. (a – b)3 = a3 – b3 – 3ab (a – b) C. (a3 + b3) = (a + b) (a2 + b2 – ab) D. (a3 – b3) = (a – b) (a2 + b2 + ab) .............................................................................................................................................................................. 17Activity 5. To interpret the factors of a quadratic expression of type x2 + Bx + C using square grids and square strips .....23

: CoordiNate GeometryuNitActivity 6. To find the values of abscissae and ordinates of various given points in a Cartesian plane ............................................28Activity 7. To find a hidden picture by plotting and joining the coordinates of given points in a Cartesian plane ..................30Activity 8. To obtain the mirror image of a given geometrical figure with respect to the x-axis or y-axis ......................................32

: liNear equatioNs iN two VariablesuNitActivity 9. To plot a graph for a given linear equation in two variables. ....................................................................................................................35Activity 10. To understand the concept that a linear equation represents a straight line ...............................................................................36

: liNes aNd aNGlesuNitActivity 11. To verify that the sum of angles of a triangle is 180° .....................................................................................................................................40Activity 12. To verify the exterior angle property of a triangle ............................................................................................................................................41

: triaNGlesuNitActivity 13. To verify experimentally the different criteria for congruency of triangles using triangle cutouts ............................ 44Activity 14. To verify experimentally that in a triangle, the longer side has the greater angle opposite to it ..................................47

: quadrilateralsuNitActivity 15. To verify experimentally that the sum of the angles of a quadrilateral is 360º ...........................................................................51Activity 16. To explore the similarities and differences in the properties with respect to diagonals

of different quadrilaterals – a parallelogram, a square, a rectangle and a rhombus ...............................................................52

: area of ParalleloGrams aNd triaNGlesuNitActivity 17. To verify the midpoint theorem for a triangle using cutting and pasting method ...................................................................54Activity 18. To find a formula for the area of a parallelogram .............................................................................................................................................57

Preface

List of materials and methods used in Mathematics Laboratory

Activity 19. A. To obtain a parallelogram by paper folding B. To verify that the quadrilateral obtained by joining the midpoints of a quadrilateral is a parallelogram .......58

: CirClesUnitActivity 20. To verify experimentally that the angle subtended by an arc of a circle at the centre is

double the angle subtended by it at any point on the remaining part of the circle ................................................................62Activity 21. To verify that the angles in the same segment of a circle are equal ...................................................................................................63Activity 22. To verify that the opposite angles of a cyclic quadrilateral are supplementary ........................................................................64Activity 23. To verify the following using paper cuttings, pasting and folding method: A. Angle in a semicircle is a right angle B. Angle in a major segment is an acute angle C. Angle in a minor segment is an obtuse angle ..............................................................................................................................................65

Unit 10 : ConstrUCtionsActivity 24. To find the following by paper folding method: A. The midpoint of a line segment B. The perpendicular bisector of a line segment C. The bisector of an angle D. The perpendicular to a line from a point given outside it E. The perpendicular to a line at a point given on it F. The median of a triangle ..............................................................................................................................................................................................70

Unit 11 : MensUrationActivity 25. To show that the area of a triangle is half of the product of the base and the height of the triangle in following

cases using paper cutting and pasting method A. Right-angled triangle B. Acute-angled triangle C. Obtuse-angled triangle .................................................................................................................................................................................................. 74Activity 26. To find the formula of total surface area of cuboid experimentally....................................................................................................76

Unit 12 : statistiCs and ProBaBilitYActivity 27. To draw histograms for classes of equal widths and varying widths by using data

provided by the teacher .......................................................................................................................................................................................................80Activity 28. To find the experimental probability of unit place digit of telephone numbers listed on a page (selected at

random) of a telephone directory ................................................................................................................................................................................82Activity 29. To find the experimental probability of each outcome of a die when it is thrown for a

large number of times............................................................................................................................................................................................................84

GroUP aCtivities1. To establish the relationship between measurement of different parts of the body and to find out the

Body Mass Index (BMI) .......................................................................................................................................................................................................................................882. To find the percentage of the students, in a group of students, who write faster with their left/right hand .......................................90

ProjeCts1. To explore the changes in behaviour of surface areas and volumes of cuboids with respect to each other by taking A. different measurement cuboids with same volumes B. different measurement cuboids but with same surface areas ...........................................................................................................................................922. To develop a project on any topic (here, Pythagoras theorem) from the history of Mathematics ..............................................................943. To develop the chronology of Indian Mathematicians with their contributions .........................................................................................................96

aPPendiXICT Connect .........................................................................................................................................................................................................................................................................98

Case-Based Study Questions ....................................................................................................................................................................................................................................99

WORKSHEETS: Chapters 1 to 15 .......................................................................................................................................................................................................................... 101

Special page: India in Maths .................................................................................................................................................................................................................................... 121

Worksheets Answers .....................................................................................................................................................................................................................................................122

Number system

Activity 1Learning ObjectiveTo represent an irrational number on a number line

Basic Knowledge Required1. Rational numbers: The numbers which can be expressed in the form p

q , where p and q are integers and q ≠ 0, are called rational numbers.

2. Irrational numbers: The numbers which cannot be written in the form pq , where p and q both are integers

and q ≠ 0, are called irrational numbers. For example, √2, √3, 3.02142141414…, p, etc. are irrational numbers.

3. Real numbers: A collection of all rational numbers and irrational numbers is called real numbers.

4. Pythagoras theorem: According to Pythagoras theorem, ‘a2 = b2 + c2’, where ‘a’ is the length of the hypotenuse, ‘b’ and ‘c’ are the lengths of the other two sides containing the right angle in a right-angled triangle.

5. Rationalisation: A process of converting a fraction with a radical number in denominator to an equivalent expression, whose denominator is a rational number, is called rationalisation.

Materials RequiredTwo rulers, a wooden board, white sheet of paper, some iron nails, thread, a wooden block and a marker

ProcedureLet us represent irrational number √2 on the number line.

1. Fix a white sheet of paper on a wooden board. Take a ruler and fix it on the wooden board horizontally.

2. Take second ruler and fix it perpendicular to first ruler with the help of supports (Fig. 1.1), so that it can move freely in the both the directions.

3. Keep the vertical ruler on the 0 mark of horizontal ruler (say point O) as shown in Figure 1.1 and slide the vertical ruler 1 unit distance (1 cm) starting from 0 on horizontal ruler and fix it at point P as shown in Figure 1.2.

Corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number.

R. Dedekind, G. Cantor

Topic: Irrational Numbers

Fig. 1.1

O

9

O

Fig. 1.2

P

4. Now, fix a nail at point O on 0 mark of horizontal ruler and one more nail at 1 unit distance (1 cm) on a point (say Q) on the vertical ruler. Also, tie one end of a thread to the nail at point O and its other end to another nail at point Q on vertical ruler as shown in Figure 1.3.

OQ

P

Fig. 1.3

P

5. In Figure 1.3, points O, P and Q together form a right-angled triangle. So, according to the Pythagoras theorem, the length of the thread OQ will be the square root of the sum of the squares of length of sides OP and PQ.

i.e., OQ2 = OP2 + PQ2

OQ2 = 12 + 12

OQ = √1+1

OQ = √2

So, the length of the thread OQ will be equal to √2. O

P

Fig. 1.4

P Q

1.4 cm

10

6. Now, remove the thread from the point Q on vertical ruler and place it on the horizontal ruler. Now, point Q represents the point corresponding to √2 on the number line (Fig. 1.4).

Observation and ResultOn measuring the length of the thread OQ on horizontal ruler, i.e., OQ = ............................ cm.

So, √2 = ............................ cm.

Learning OutcomeAfter performing this activity, students will be able to represent √3, √5, √7, … on the number line and will find their respective values.

Activity 2Learning ObjectiveTo represent the construction of a square root spiral

Materials Required A wooden board, white sheet of paper, nails, thread and a marker

Procedure1. Take a wooden board and fix a white sheet of paper on it. Now, fix a nail on the board and mark it as point O.

Now, fix another nail to the right side of first nail and mark it as point P1 as shown in Figure 2.1.

2. Tie a thread between both the nails (Fig. 2.1). Consider the length of the thread OP1 as 1 unit length.

Note: Here, 1 unit length = 1 cm.

Fig. 2.1

1 unit

O

P1

3. Using set squares, fix another nail at the distance equal to the length OP1 = 1 unit perpendicular to the P1, marking the point as P2. Now, tie the thread between P1 and P2 representing P1P2 = 1 unit length. Join OP2 with a dotted line using a marker as shown in Figure 2.2.

Since, OP1P2 is right-angled triangle, right angle at P1. According to Pythagoras theorem,

OP22 = OP1

2 + P1P22

OP22 = 12 + 12

OP2 = √1 + 1

OP2 = √2 units

Therefore, OP2 will represent √2.

Fig. 2.2

1 unit

1 unitO

P2

P1√2

11

4. Now, fix another nail at the distance equal to the length P1P2 = 1 unit perpendicular to the P2, marking the point as P3. Now, tie the thread between P2 and P3 representing P2P3 = 1 unit length. Join OP3 with a dotted line using the marker as shown in Figure 2.3.

Here, OP2P3 is right-angled triangle, right angle at P2.

According to Pythagoras theorem,

OP32 = P2P3

2 + OP22

OP32 = 12 + (√2)2

OP3 = √1 + 2

OP3 = √3 units

Therefore, OP3 will represent √3.

5. By repeating the above steps, you can get P3P4, P4P5,… by fixing the nails and tightening the thread along them at 1 unit length. Then, by joining the points P4, P5, … with point O, you can get OP4, OP5, … which will represent the values √4, √5, … and so on. In this manner, you can continue making square root spiral as shown in Figure 2.4.

Fig. 2.4

1 unit1 unit

1 unit

1 un

it

1 unit

1 unitO

P1

P2P3

P4

P5

P6

√2√3

√4

√5

√6

Observation and Result Since there is no end point of square root spiral, therefore, by continuing the above steps, the values of OP4, OP5, OP6, ... can be found out by students.

Thus,

OP4 = ....................., OP5 = ....................., OP6 = .....................

Learning OutcomeAfter performing this activity, students will be able to find out the value of square root of any number using square root spiral.

Let's expLore!We know that √2 is an irrational number and we can also expand it in decimals as

√2 = 1.4142135623730950488016887242096... (which is a non-terminating non-recurring).

Fig. 2.3

1 unit1 unit

1 unitO

P1

P2P3

√2√3

12

But in Vedic period (800 BC–500 BC), in the Sulbasutras, a mathematical technique has been given to find the approximate value of √2. We can find an approximation of √2 as follows:

√2 = 1 + 13 + 14 × 13 – 1

34 × 14 × 13 = 1.4142156

Can you find the value of √5 and √6 using above method?

everyday MathsThe concept of rational numbers are used when we want to divide particular item among different number of people. For example, if there are four cousins and they want to divide a cake equally among themselves, then the

quantity of cake that each of four cousins will get, will be one-fourth of the cake, that is a rational number 14.

If the whole cake is divided among 7 cousins, then what part of the cake will each cousin get?

Know More!Ancient Greek mathematicians thought that all things could be measured using rational numbers. So, when the Pythagoras theorem came into play and showed that some lengths could not be written as a rational number, their whole idea of numbers was changed.

Viva VoceQ.1. What is a rational number?Ans. A number which can be written in the form of pq, where p and q both are integers and q ≠ 0 is called a

rational number.

Q.2. Define real numbers.Ans. A collection of all rational numbers and irrational numbers is called real numbers.

Q.3. What is rationalisation?Ans. A process of converting a fraction with a radical number in denominator to an equivalent expression,

whose denominator is a rational number, is called rationalisation.

Q.4. What is the difference between rational and irrational numbers?Ans. A rational number is either terminating or non-terminating recurring decimal, while an irrational number is

non-terminating non-recurring decimal.

Q.5. What do you mean by spiral?Ans. A spiral is any plane curve formed by a point, winding around a fixed point on an even increasing distance

from it.

Q.6. What is Pythagoras theorem?Ans. According to Pythagoras theorem, in a right-angled triangle, the sum of the squares of two sides is equal

to the square of hypotenuse.

Q.7. Write three numbers whose decimal expansions are non-terminating non-recurring.Ans. Since, an irrational number is non-terminating non-recurring. So, √3, √7 and √11 are three numbers whose

decimal expansions are non-terminating non-recurring.

13

For Teachers:

Since √2 can be written as (2)12 in exponential form and by using exponential law (am)n = amn, we can calculate (√2)2

= 2 12 = 2

12 × 2 = 2. You may ask the students to find the square root of any number more than 8 by using spiral

method. Also, cross verify the results by measuring the length of hypotenuse and by using calculator.

Q.8. For constructing a square root spiral, which kind of triangle is used?Ans. For constructing a square root spiral, we need to draw a right-angled triangle, where the length of

hypotenuse of the triangle represents the square root of a number.

Q.9. How many rational numbers are there in between two rational numbers?Ans. There are infinite rational numbers in between two rational numbers.

Q.10. Define non-terminating non-repeating decimal expansion.Ans. While converting rational or irrational numbers in decimal form, the decimal expansion continues

endlessly, with no group of digits repeating endlessly is called non-terminating non-repeating expansion.

14

PolyNomials

Activity 3Learning ObjectiveTo verify the following algebraic identity using geometric shapes:

(a + b + c)2 = a2 + b2 + c 2 + 2ab + 2bc + 2ac

Basic Knowledge Required1. Polynomial: An algebraic expression containing numerals and variables that involve only fundamental

operations (addition, subtraction, multiplication or division) is called a polynomial.

2. Algebraic identity: An algebraic identity is an equality which is true for all values of the variables in the equality.

3. Linear polynomial: A polynomial of degree one is called a linear polynomial.

4. Quadratic polynomial: A polynomial of degree two is called a quadratic polynomial.

5. Cubic polynomial: A polynomial of degree three is called a cubic polynomial.

6. Area of a rectangle = (Length × breadth) sq. units

7. Area of a square = (side × side) = (side)2 sq. units

Materials Required Different coloured drawing sheets, sticky tape, a pair of scissors, a ruler, a sketch pen or pencil and a glue stick

Procedure1. Draw three squares of sides a units, b units and c units, respectively (let a = 4 cm, b = 3 cm and c = 2 cm) on

three different coloured sheets and cut them out from the sheets as shown in Figure 3.1.

a2b2

c2

c

c

b

ba

aFig. 3.1

2. Now, draw two rectangles of dimensions a × b sq. units, two rectangles of dimensions b × c sq. units and two rectangles of dimensions a × c sq. units on different coloured sheets and cut them out from the sheets as shown in Figure 3.2.

If a polynomial equation a0xn

+ a1xn–1 + a2x

n–2 + … + an = 0 has integer coefficients, then it is possible to make a complete list of all possible rational roots. This list consists of all possible numbers of the form cd , where c is any integer that divides evenly into the constant term an and d is any integer that divides evenly into the leading term a0.

Rene Descartes

Topic: Algebraic Identities

15

Fig. 3.2

ab ab ac acbc bc

a a

b b

b b

c c c c

aa

3. Arrange all squares and rectangles in such a manner to form a square and paste them on a cardboard sheet as shown in Figure 3.3.

a + b + c

a + b + c

Fig. 3.3

P

S

Q

a2 ab ac

ab bc

ac bc

b2

c2

R

Observations and ResultsIn Figure 3.3, the side of square PQRS will be (a + b + c) units.

So, area of the square PQRS = (a + b + c)2 … (i)

Also, area of the square PQRS

= Sum of the areas of all squares and rectangles in Figure 3.3 = a2 + b2 + c2 + ab + ab + bc + bc + ac + ac = a2 + b2 + c2 + 2ab + 2bc + 2ac … (ii)

From (i) and (ii), we get

(a + b + c) 2 = a2 + b2 + c2 + 2ab + 2bc + 2ac

By taking actual measurement,

a = ............................, b = ............................, c = ............................; a + b + c = ............................

a2 = ............................, b2 = ............................, c2 = ............................; (a + b + c)2 = ............................

2ab = ............................, 2bc = ............................, 2ac = ............................

a2 + b2 + c2 + 2ab + 2bc + 2ac = ............................

Learning OutcomeAfter performing this activity, students will be able to find the square of the sum of any three given numbers.

16

Activity 4Learning ObjectivesTo verify the following algebraic identities using geometrical solid shapes:

A. (a + b)3 = a3 + b3+ 3ab (a + b) C. (a3 + b3) = (a + b) (a2 + b2 – ab)

B. (a – b)3 = a3 – b3 – 3ab (a – b) D. (a3 – b3) = (a – b) (a2 + b2 + ab)

Basic Knowledge Required1. Volume of a cuboid = (length × breadth × height) cu. units

2. Volume of a cube = (side × side × side) = (side)3 cubic units

Materials Required Different coloured cardboard sheets, sticky tape, a pair of scissors, a ruler, a sketch pen and a glue stick

A. Verification of algebraic identity: (a + b)3 = a3 + b3 + 3ab (a + b)Procedure1. Take some different coloured cardboard sheets and make a cube of side a

units and another cube of side b units by making first their nets (let a = 3 cm and b = 2 cm) as shown in the Figure 4.1.

2. Again, take few more different coloured cardboard sheets and make three cuboids of dimensions a × b × a cubic units (say 3 cm × 2 cm × 3 cm) as shown in Figure 4.2 (a) and three cuboids of dimensions b × a × b cubic units (say 2 cm × 3 cm × 2 cm) as shown in Figure 4.2 (b).

a

ba

a

ba

a

ba

b

a b

b

a b

b

a b

(a) (b)Fig. 4.2

3. Now, arrange all the cubes and the cuboids in such a manner to get a large cube as shown in Figure 4.3.

a b

a

a

b

b

Fig. 4.3

b

bb

a

aa

Fig. 4.1

17

Observations and ResultsIn Figure 4.3,

volume of the cube of side a units = a3 cu. units.volume of the cube of side b units = b3 cu. units.volume of the cuboid of dimensions (a × b × a) cu. units = a2b cu. units.volume of the three such cuboids = 3 × a2b cu. units = 3 a2b cu. units.volume of the cuboid of dimensions (b × a × b) cu. units = ab2 cu. units.volume of the such three cuboids = 3 × ab2 cu. units = 3ab2 cu. units.

Here, the volume of the larger cube will be the sum of the volumes of all cubes and cuboids.

So, the volume of the larger cube = Volume of cube of side a units

+ Volume of cube of side b units

+ Volume of three cuboids of dimensions (a × b × a) cu. units

+ Volume of three cuboids of dimensions (b × a × b) cu. units

= a3 + b3 + 3 a2b + 3ab2

= a3 + b3 + 3ab (a + b) … (i)

Also, side of large cube is (a + b) units.

So, the volume of larger cube = (a + b)3 cu. units … (ii)

From (i) and (ii), we get

(a + b)3 = a3 + b3 + 3ab (a + b)

By taking actual measurement,

a = ....................., b = .....................; a + b = .....................

a3 = ....................., b3 = .....................; (a + b)3 = .....................

3a2b = ....................., 3ab2 = .....................; a3 + b3 + 3a2b + 3ab2 = .....................

B. Verification of algebraic identity: (a – b)3 = a3 – b3– 3ab (a – b)Procedure1. Take a coloured cardboard sheet and make a cube of side (a – b) units

(a > b) as shown in Figure 4.4.

2. Make three cuboids of dimensions a × (a – b) × b cu. units using different coloured cardboard sheets as shown in Figure 4.5.

b

a – b

a

b

a – b

a

b

a – b

a

Fig. 4.5

a – b

a – b

a – b

Fig. 4.4

18

3. Make another cube of side b units (say b = 2 cm) using different coloured cardboard sheet as shown in Figure 4.6.

a – b

a – b

a – b

a – b b

b

b b

a

Fig. 4.7

b

b

b

Fig. 4.6

4. Arrange all the cubes and the cuboids as shown in Figure 4.7

Observations and ResultsIn Figure 4.7,

volume of the cube of side (a – b) units = (a – b)3 cu. units.

volume of the cube of side b units = b3 cu. units.

volume of the cuboid of dimensions {a × (a – b) × b} cu. units = ab (a – b) cu. units.

volume of three such cuboids = 3 × ab (a – b) cu. units = 3ab (a – b) cu. units.

Here, the volume of the combined cube will be the sum of the volumes of all cubes and cuboids.

So, the volume of the combined cube = Volume of the cube of side (a – b) units

+ Volume of the cube of side b units

+ Volume of three cuboids of dimensions {a × (a – b)× b} cu. units

= (a – b)3 + 3ab (a –b) + b3 … (i)

However, the side of combined cube is (a – b + b) = a units

So, the volume of combined cube = a3 cu. units … (ii)

From (i) and (ii), we get

a3 = (a – b)3 + 3ab (a – b) + b3

(a – b)3 = a3 – b3 – 3ab (a – b)

By taking actual measurement,

a = .........................., b = .......................... (a > b); a – b = ..........................

a3 = .........................., b3 = .......................... (a – b)3 = ..........................

3a2b = .........................., 3ab2 = .......................... a3 – b3 – 3ab (a – b) = ..........................

Learning OutcomeAfter performing this activity, students will be able to calculate the cube of difference of any two numbers.

19

C. Verification of algebraic identity: a3 + b3 = (a + b) (a2 + b2 – ab)Procedure1. Make two cubes of sides a units and b units, respectively (say a = 5 cm and b = 3 cm) using different

coloured cardboard sheets and sticky tape as shown in Figure 4.8.

a

a

b

b

Fig. 4.8

2. Make two cuboids of dimensions a × b × a cu. units (say 5 cm × 3 cm × 5 cm) and b × b × a cu. units (say 3 cm × 3 cm × 5 cm), respectively using different coloured cardboard sheets as shown in Figure 4.9.

b

b

aa

b Fig. 4.9

3. Now, arrange all cubes and cuboids as shown in Figure 4.10.

a

a

a

ab

b

b

Fig. 4.10

Observations and ResultsIn Figure 4.10,

volume of the cube of side a units = a3 cu. units.

volume of the cube of side b units = b3 cu. units.

volume of the cuboid of dimensions (a × b × a) cu. units = a2b cu. units.

volume of the cuboid of dimensions (b × b × a) cu. units = ab2 cu. units.

a

20

Here, the volume of the combined figure will be the sum of the volumes of all cubes and cuboids.

So, the volume of the combined figure = Volume of the cube of side a units

+ Volume of the cube of side b units

+ Volume of cuboids of dimensions (a × b× a) cu. units

+ Volume of cuboids of dimensions (b × b× a) cu. units

= a3 + b3 + a2b + ab2

= a2(a + b) + b2(a + b) = (a + b) (a2 + b2)

If we remove both cuboids from Figure 4.10, then the volume of rest figure thus obtained will be

= (a + b) (a2 + b2) – (a2b + ab2) cu. units

= (a + b) (a2 + b2 – ab) … (i)

Also, the adjacent figure is the combination of the cubes of side a units and b units, respectively.

So, volume of the adjacent figure

= a3 + b3 cu. units … (ii)

Thus, from (i) and (ii), we get

a3 + b3 = (a + b) (a2 + b2 – ab)

By taking actual measurement,

a = ....................., b = .....................; a + b = .....................

a3 = ....................., b3 = .....................; a3 + b3 = .....................; (a + b) (a2 + b2 – ab) = .....................

Learning OutcomeAfter performing this activity, students will be able to find the sum of the cubes of any two numbers.

D. Verification of algebraic identity: a3 – b3 = (a – b) (a2 + b2 + ab)Procedure1. Take a coloured cardboard sheet and make a cuboid of dimensions (a – b) × a × a cu. units (say a = 5 cm

and b = 2 cm, where a > b) by making first a net of cuboid of considered measurements using sticky tape as shown in Figure 4.11.

(a – b)

a

a

Fig. 4.11

I

21

2. Choose some coloured cardboard sheets and make another cuboid of dimensions (a – b) × a × b cu. units (say a = 5 cm and b = 2 cm, where a > b) by making first a net of cuboid of considered measurements using a sticky tape as shown in Figure 4.12.

Fig. 4.12(a – b)

ab II

3. Again, take some different coloured cardboard sheets and make one more cuboid of dimensions (a – b) × b × b cu. units (say a = 5 cm and b = 2 cm, where a > b) by making first a net of cuboid of considered measurements using a sticky tape as shown in Figure 4.13.

Fig. 4.13(a – b) b

bIII

4. Now, make a cube of side b units (let b = 2 cm) using different colour cardboard sheet as shown in Figure 4.14.

bb

b

IV

5. Now, arrange all cubes and cuboids as shown in Figure 4.15.

b

(a – b)

(a – b)

a

ab

a

I

Fig. 4.15

IV

III

II

Observations and ResultsIn Figure 4.15,

volume of the cuboid of dimensions {(a – b) × a × a} cu. units = (a – b) a2 cu. units.

volume of the cuboid of dimensions {(a – b) × a × b} cu. units = (a – b) ab cu. units.

volume of the cuboid of dimensions {(a – b) × b × b} cu. units = (a – b) b2 cu. units.

Fig. 4.14

22

volume of cube of side b units = b3 cu. units.

volume of the combined figure will be the sum of the volumes of all cubes and cuboids.

So, volume of the combined figure

= Volume of cuboid of dimensions {(a – b) × a × a} cu. units

+ Volume of cuboid of dimensions {(a – b) × a × b} cu. units

+ Volume of cuboid of dimensions {(a – b) × b × b} cu. units

+ Volume of a cube of side b units

= (a – b) a2 + (a – b) ab + (a – b) b2 + b3

= (a – b) (a2 + b2 + ab) + b3 … (i)

Here, the above combined Figure 4.15 is a cube of side a units.

So, volume of the above combined figure = a3 cu. units … (ii)

Thus, from (i) and (ii), we get

a3 = (a – b) (a2 + b2 + ab) + b3

a3 – b3 = (a – b) (a2 + b2 + ab)

ObservationsBy taking actual measurement,

a = ....................., b = ..................... (a > b); a – b = .....................

a3 = ....................., b3 = .....................; a3 – b3 = .....................; (a – b) (a2 + b2 + ab) = .....................

Learning OutcomeAfter performing this activity, students will be able to calculate the difference of the cubes of any two numbers.

Activity 5Learning ObjectiveTo interpret the factors of a quadratic expression of type x2 + Bx + C using square grids and square strips

Basic Knowledge Required1. Splitting the middle term of a quadratic polynomial

2. Area of a rectangle = Length × Breadth

Materials RequiredA square grid, some square strips of similar size of square grid, a glue stick, a sketch pen and a pair of scissors

ProcedureTo interpret the factors of a quadratic expression of type x2 + Bx + C, let us take an example of quadratic polynomial x2 + 5x + 6.

23

Let us find two numbers whose sum is 5 and the product is 6, i.e., 3 and 2.

Therefore, x2 + 5x + 6 = x2 + 3x + 2x + 6

To interpret the factors of the above quadratic expression, we will have to follow the following steps.

1. Take a square grid of dimensions 10 × 10 and paste it on a cardboard sheet. Represent it by x × x = x2 as shown in Figure 5.1.

2. Add three rectangular strips of dimensions 10 × 1, i.e., x × 1 each as shown in Figure 5.2. The area of the above figure thus formed is x × (x + 3) = (x2 + 3x) sq. units.

3. Add two more square strips of dimensions 10 × 1, i.e., x × 1 each as shown in Figure 5.3.

x + 3

3

x

x

Fig. 5.2

3

x + 3

x

x 2

Fig. 5.3

The area of the above figure thus formed = (x2 + 3x) + 2 × x sq. units

= x2 + 3x + 2x sq. units

= x2 + 5x sq. units

4. Now, paste a square grid of 6 squares (2 × 3) as shown in Figure 5.4.

3

x + 3

x

x 2

Fig. 5.4

x

xFig. 5.1

24

The area of Figure 5.4 = Area of Figure 5.3

+ Area of square grid of six squares (2 × 3)

= x2 + 5x + (2 × 3) sq. units

= x2 + 5x + 6 sq. units

Since, the above figure is a rectangle of dimensions (x + 2) × (x + 3) sq. units = (x + 2)(x + 3) sq. units

Thus, x2 + 5x + 6 = (x + 2)(x + 3)

Observations and ResultsHence, x2 + Bx + C = (x + a)(x + b) is the general quadratic expression, B = a + b (splitting into two terms) and C = ab.

By taking suitable values of a and b,

a = ....................., b = ....................., B = (a + b) = ..................... and C = ab = .....................

(x + a) = ..................... and (x + b) = ..................... then (x + a)(x + b) = .....................

Now, let us take b = 1 and c = – 6 for the quadratic expression x2 + Bx + C.

Therefore, the polynomial will be x2 + x – 6 = x2 + 3x – 2x – 6

To interpret the factors of the above quadratic expression, we have to follow the following procedure.

1. Take a square grid of dimensions 10 × 10 and paste it on a cardboard sheet as shown in Figure 5.5. Represent it by x × x = x2.

x

xFig. 5.5

3

x

x

Fig. 5.6

2. Add three rectangular strips of dimensions 10 × 1, i.e., x × 1 each as shown in Figure 5.6.

The area of Figure 5.6 is x × (x + 3) = x2 + 3x sq. units.

3. Remove two rectangular strips of dimensions 1 × 10, i.e., 1 × x each from Figure 5.6.

Here, area of Figure 5.7 = x2 + 3x sq. units – 2× (x × 1) sq. units = x2 + 3x – 2x sq. units.

4. Now, remove a piece of six squares (2 × 3) of the square grid from Figure 5.6.

Now, area of Figure 5.8 = Area of Figure 5.7 – 6 sq. units

= x2 + 3x – 2x – 6 sq. units

= x2 + x – 6 sq. units25

3

x – 2

x

Fig. 5.7

3

x – 2

x

Fig. 5.8

Observations and ResultsSince, the Figure 5.7 is a rectangle of dimensions (x + 3) × (x – 2) sq. units= (x + 3) (x – 2) sq. unitsThus, x2 + x – 6 = (x + 3) (x – 2)Hence, x2 + Bx – C = (x + a)(x – b) Here, B = a – b, where a > b (splitting into two terms) and C = – abBy taking suitable values of a and b,a = ....................., b = ....................., B = (a – b) = ..................... and C = ab = .....................(x + a) = ..................... and (x – b) = ..................... then (x + a)(x – b) = .....................

Learning OutcomeAfter performing this activity, students will be able to find out the factors of a quadratic polynomial using middle term splitting method.

Let's expLore!1. Find the value of the polynomial 2x – 3x2 + 1: a. at x = 1 b. at x = 0 c. at x = – 12. Factorise x2 – 5x + 6

everyday MathsPolynomials are used for modelling of various buildings and objects, in industries, in construction.

Professionals who need to make complex calculations on daily basis use polynomials. For example, a civil engineer uses polynomials in designing the layout or structure of roads. Medical researchers take the help of polynomials to describe the behaviour of bacterial colonies.

Know More!The word ‘polynomial’ was first used in the 17th century. Polynomials are used to describe various types graph curves. Rene Descartes is one of the people who introduced the concept of the graph of a polynomial equation, in La geometric in 1637.

26

Viva VoceQ.1. What is a polynomial?Ans. A polynomial is an algebraic expression consisting of variables and coefficients that involves only the

fundamental operations, and non-negative integer exponents of variables.

Or,

An algebraic expression of the form a0xn + a1x

n–1 + a2xn–2 + … + an is a polynomial, where a0, a1, …, an are

real numbers and n is non-negative integer.

Q.2. How can we distinguish between a polynomial and an equation?Ans. A polynomial is an algebraic expression consisting of variables and coefficients that involves only the

fundamental operations, and non-negative integer exponents of variables, while an equation is a statement of an equality containing one or more variables.

Q.3. Give an example of biquadratic polynomial.Ans. A polynomial of degree 4 is called a biquadratic polynomial. For examples: x4 – 2x3 + 3 or x2y2 – 2xy + 1.

Q.4. How can we identify the degree of a polynomial?Ans. The highest non-negative integer exponent of variable in a term of a polynomial is called the degree of the

polynomial. For example, in the polynomial 4x4 + 3x2 + 2x + 7, the highest non-negative integer exponent of variable in term 4x4 is 4. So, the degree of the polynomial is 4.

Q.5. What do you mean by a quadratic polynomial?Ans. A polynomial whose degree is two is called a quadratic polynomial.

Q.6. What is an algebraic identity?Ans. An algebraic identity is an algebraic equation that is true for all the variables present in the equation.

Q.7. How many zeroes does a trinomial polynomial have?Ans. Three.

Q.8. Define algebraic factor of a polynomial.Ans. An algebraic factor of a polynomial is any polynomial which divides the polynomial evenly.

Q.9. How many zeroes does a biquadratic polynomial have?Ans. Four.

Q.10. What is the zero polynomial? Give an example.Ans. A constant polynomial 0 is called the zero polynomial. In general, in the polynomial a0x

n + a1xn–1 + a2x

n–2 + … + an, if a0 = a1 = a2 = a3 = . . . = an = 0, then we get zero polynomial and it is denoted by 0.

For Teachers:You may give students different examples of quadratic polynomials and tell them to factorise each one of them by factor theorem and middle term splitting method. Then ask them which method takes more time to get the required factors.

27

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This series is based on NCERT syllabus following NCF guidelines. This course has been designed for schools

teaching a basic course in Mathematics at middle and secondary levels, and provides various tools for learning,

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The books are designed to help students understand mathematical concepts through concrete materials and

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learning by doing approach.

Strictly as per NCERT syllabus and latest CBSE guidelines

A wide variety of activities/experiments to lend a strong base to mathematical awareness

Simple and lucid language to explain each activity

Step-wise description to perform the activities

Activities supported with suitable diagrams and pictures

Viva voce questions after activities give practice for answering oral questions

Important terms and brief theory included for quick recapitulation

Group activities and projects to facilitate scope of applying knowledge of the textbook in real-life

Everyday Maths encourage applying knowledge to real-life situations

Did you know and Know more sections cover interesting information on the concepts learnt

Chapter-wise worksheet to practice important concepts

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