Oasis Math 6

Steps:

• Draw BC = 4 cm.
• Taking B and C as centres, draw an arc of 4 cm, which intersect each other at A.
• Join AB and AC

Hence, ABC is the required equilateral triangle.

Alternative method: E
A
F

A Rough

B 60° 4 cm60° C

B 60° 4 cm 60° C

An equilateral triangle has all its sides equal and each angle is equal to 60°.
Steps:

• Draw BC = 4 cm.
• At B, draw BE, making 60° with BC
• At C, draw CF, making 60° with BC which intersect at A.

Hence, ABC is the required equilateral triangle.

Construction of a square

Construct a square ABCD in which the side AB = 5.2cm

K

D 5.2 cm C

5.2 cm
5.2 cm
5.2cm
5.2cm
D 5.2cm C Rough

A B A 5.2cm B

5.2 cm

Oasis School Mathematics – 6 245

Steps:
• Draw AB = 5.2 cm
• Construct ∠KAB = 90°.
• Cut of AD = 5.2 cm from AK.
• With centre B and radius 5.2 cm draw an arc.
• With center D and radius 5.2 cm and arc cutting the first arc at C.
• Join BC and CD.

Hence, ABCD is the required square.

Exercise 14.4

1. Construct an equilateral triangle ABC in which each side (i.e. AB=BC=AC) is
equal in length.

(a) 4.5 cm (b) 5 cm (c) 3.5 cm (d) 6.1 cm

e) 7.5cm f) 5cm

2. Construct squares of given length of sides: (d) 6 cm
(a) 4.5 cm (b) 3 cm (c) 5.5 cm

e) 6.5cm (f) 7.2cm

Answers
Consult your teacher.

246 Oasis School Mathematics – 6

14.4 Circle

Lets observe the following objects:

► What is the shape of above objects?
Above objects are circular in shape.

This is a figure of a circle.

SP

A circle is the path of all the points that are equidistant from a O T
Q
fixed point in a given plane. R
Fixed point is the centre of the circle. In the above figure,

OP = OQ = OR = OS = OT.

If we take a stone, tie it at one end of a string and swing it in the air holding the other
end of the string, the path carved by the stone is a circle.

Different parts of circle

Circumference:

The length of outer boundary of a circle is its circumference.
The perimeter of a circle is its circumference.

How to find the circumference of a circular object?

Lets take a bangle.

Fixed a string on its outer part and open it and make it straight.

Take the measurement of straight string.

Its length represents the circumference of circle.

Centre: P Q
R O
In the given figure, 'O' is the centre of the circle. Hence the centre
of a circle is point in a plane from which all the points on the
circumference are equidistant.

If 'P', 'Q' and 'R' are the points on the circumference, then

OP = OQ = OR.

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Radius: P O

In the given figure, OP is the radius of the circle. Where 'O' is B
the centre and 'P' is any point on the circumference. Hence, the
radius is the line joining the centre of the circle to a point on the Q
circumference. B

Chord: A
In the given figure, AB is a chord. It is a line joining the two P
points on the circumference of the circle. Hence, a chord is a
line joining two points on the circumference of the circle.

Diameter:
In the given figure, PQ is a chord which passes through the
centre of the circle. Here, PQ is a diameter. Hence, a diameter is
a chord passing through the centre.

Arc: A

In the given figure, AB is an arc. It is a part a circle included
between any two-points on the circumference. Hence, an arc is
a portion between any two points on the circumference. Arc AB
is denoted by AB.

Construction of circle with the help of compass

Take a pencil compass.

OA

Keep the tip of the compass at any position in your copy and rotate it as shown in the

figure.

The position of the tip of the compass is the centre of the circle. P

How to draw a radius?

• Draw a circle with the help of compass. O

• Take any point P on the circle. Join OP

248 Oasis School Mathematics – 6

Then, OP is a radius. O B B
How to draw a chord? A
Take any two points A and B on a circle. O
Join AB. Then AB is a chord. A

How to draw a diameter?
• Draw a circle with the help of compass.
• Locate the centre 'O'
• Draw a line AOB passing through 'O'.
• AOB is a diameter.

Exercise 14.5

1. Identify the following parts of the circle.

(a) Point O (b) Q PQ.
P
O
O

(c) AB (d) OP
AO B OP

(e) A AB

B

2. Draw a circle in each case and show the following parts.

(a) Chord AB (b) Diameter PQ

(c) Radius OP (d) Arc AB.

3. Draw the circle in each of the following cases using compass and draw the

following parts. BQ

(a) radius (b) diameter (c) chord (d) arc

4. In the given figure, name the following parts. O R

(a) AB (b) PQ (c) OR (d) AB A
P

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Objective Questions

Choose the correct alternatives:

1. If all three sides of a triangle are equal then the triangle is

(i) scalene triangle (ii) isosceles triangle (iii) equilateral triangle

2. Which of the following statement is not true?

(i) Sum of three angles of a triangle is 360º

(ii) Sum of three angles of a triangle is 180º.

(iii) Sum of four angles of a quadrilateral is 360º.

3. Who am I ? I am a quadrilateral, my opposite sides and angles are equal and all my
angles are 90º.

(i) Parallelogram (ii) Rectangle (iii) Rhombus

4. If one angle of a triangle is 1120, then the triangle is

(i) acute angled triangle (ii) obtuse angled triangle (iii) right angled triangle

5. Which of the following statement is not true?

(i) opposite angles of a parallelogram are equal.

(ii) all sides of rhombus are equal

(iii) all sides of rectangle are equal

6. What is the value of each angle of an equilateral triangle?

(i) 90º (ii) 180º (iii) 60º

7. If all sides of a parallelogram are equal and none of its angle is 90º,

(i) 90º (ii) 180º (iii) 60º

8. If two opposite angles of a parallelogram are 2x0 and (x+30)0, what is the value of
the angle.

(i) 60º (ii) 120º (iii) 30º

9. A triangle having two ocute angles and an obtuse angle is

(i) acute angled triangle (ii) right angled triangle (iii) obtuse angled triangle

10. I am a parallelogram none of my angles is a right angle and all my sides are equal.
Then I am a

(i) rhombus (ii) square (iii) rectangle

11. In the given figure, AB is a AB
(i) radius (ii) diameter (iii) chord

12. In the given figure, AB is
(i) a radius (ii) an arc (iii) a chord

13. Which of the following statement is not true? AB
(i) diameter is twice the radius
(ii) longest chord is the diameter
(iii) radius is longer than diameter

250 Oasis School Mathematics – 6

Assessment Test Paper

Attempt all the questions. Group 'A' [5 × 1 = 5] D C
B
1. (a) In the given figure, ABCD is a rhombus, If CD = 4cm,
find the length of other three sides.

A

(b) I am a quadrilateral, my one pair of opposite sides are parallel, who am I?

PQ
x

(c) In the given figure, find the value of x.

(d) What does the line AOB called in A 650 R
the given figure? S
O
B

(e) If one angle of a triangle is obtuse, what type of triangle is that?

Group 'B' [5 × 2 = 10]

2. (a) i. If three sides of a triangles are 5cm, 5 cm and 7 cm, what type of triangle is this?
Give reason.

ii. If ∠A = 900, ∠B = 300, ∠C = 600, what type of triangle is ∆ABC?

A

(b) i. In ABC, if ∠A = ∠B, which two sides of P
∆ABC are equal?

B C
R
ii. In ∆PQR, if PQ = PR, which two sides of O B
∆PQR are equal?

Q

3. (a) i. What does the shaded part of given figure
represent?

A

ii. What does OP present? PQ

(b) In the given figure, ABCD is a parallelogram. Ax B
Find the value of x, y and z.
y 800
z 4 cm

D C
5 cm

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Unit

15 Solid Figures

15.1 Introduction

In our daily life, we use many objects like box, ball, pipe, etc. They are made up of

the combination of different planes. According to their shape, those regular objects

can be categorized as follows:

Geometrical Shapes Name Examples

Cuboid

Book

Cylinder

Cup

Sphere

Volleyball

Cone

Ice cream

Cube

Dice


252 Oasis School Mathematics – 6

Introduction of cone, sphere and cylinder
Cone:

vertex
Above objects are the example of cone. base
Its geometrical shape is shown in the figure.
There is a circle on the base of the cone.

Sphere

Above objects are the examples of sphere.
Its geometrical shape is as shown in the figure.
A sphere has no plane surface.

Cylinder

Above objects are the examples of cylinder. base
Its geometrical shape is as shown in the figure.
There is a circle on the base of the cylinder.


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Vertices, edges and faces of solid objects

an edge

a vertex a vertex

a face a face an edge

Look at the above figures and identify the vertices, edges and faces of solid objects.
Faces:
A face is the flat surface of a solid object.
Edge:
An edge is the a line segment where two faces meet.
Vertex:
A vertex is the meeting point of two or more than two edges.

Vertices, edges and faces of a cuboid

Given figure is a cuboid. D A B
Lets find the edges of this figure. E C
AB, BC, CD, AD, DH, AE, BF, CG, EF, FG, GH and
HE are the edges of cuboid. F
Hence, the a cuboid has 12 edges. G

Lets discuss about the faces of cuboid. H

Faces of above cuboid are ABCD, EFGH, ABFE,
DCGH, ADHE and BCGF.

Hence, a cuboid has 6 faces.

Again, lets count the vertices of cuboid.

Its vertices are: A, B, C,D, E, F, G and H.
Hence, a cuboid has 8 vertices.

254 Oasis School Mathematics – 6

Vertices, Edges and Faces of Different Solid Object

Lets study the given figures and try understand the concept of vertices, edges and
faces of solid objects.

A A BO O

B C DC AB P
D F HG D Pyramid C Q

E E Cube F R Tetrahedron

Edges: AB, AC, BC, Edges: AB, AD, DC, Edges: OA, OB Edges: OP, OQ, OR,

DE, EF, DF, BE, AD BC, DE, AH, CF, , OC, OD, AB, BC, PQ, QR and PR

and CF. BG, EH, EF, FG and CD and AD Vertices: O, P, Q
Vertices: A, HG Vertices: O, A, B, and R
B, C,
Vertices: A, B, C,
D, E and F. ABED, D, E, F, G, and H C, and D Faces: OPQ, OPR,
Faces: Faces: OAB, OAD, OQR and PQR

ADFC, BCFE, ABC Faces: ABCD, OBC, OCD and

and DEF. ADEH, EFGH, ABCD

BCFG, DCFE and

ABGH.

Lets count the edges, vertices and faces of above figures.

Name of solid Number of faces Number of Number of edges
object (F) vertices (V) (E)

Cube 68 12
8
Pyramid 55 6
9
Tetrahedron 4 4

Prism 56

In each solid object we can see that F + V = E + 2.

This formula is called Eulers formula.

Exercise 15.1 (c) (d)

1. Identify the following solid figures.
(a) (b)

(e) (f) (g) (h)
Oasis School Mathematics – 6 255

2. Count the number of edges (E), vertices (V) and faces of given solid objects.

(a) (b)

(c) (d)

A D
E
3. Observe the given cuboid and name its B C
(i) 6 faces (ii) 8 vertices (iii) 12 edges H F

G E
F
4. (a) In the adjoining prism, write down the number of D

(a) faces (b) vertices (c) edges C
A
Hence, show that;
Number of faces + numbers of vertices

= number of edges + 2

B

(b) In the adjoining cube, find: BF
(a) the number of faces (F) AE

(b) the number of vertices (V) D C G
(c) the number of edges (E) H

Hence, show that F + V = E + 2

5. Draw the following solid figures. Write down the number of faces (F), vertices
(V) and number of edges (E) and show that F + V = E + 2

(a) cuboid (b) pyramid (c) tetrahedron.
6. Collect the following objects and count their faces, edges and vertices.

(a) Duster (b) Book

Answers
Consult your teacher.



256 Oasis School Mathematics – 6

Construction of some models of solid

By folding the paper we can construct the model of different figures. To construct
such a model, we have to draw nets in the paper. We have to cut its outline and we
have to fold and paste in faces along the dotted line.
• Draw a figure of net in your copy.
• Cut its outline from your copy.
• Fold this net along the dotted line.
• Name the structure so formed.
Activity I
Draw the following nets on a paper and cut the outlines of the net. Fold along the
dotted lines and paste the edges of the folded faces. Name the solid so formed.
(a) (b) (c) (d)

(e) (f)

Activity II

Construction of Skeleton of solids

We can construct the skeleton of solids by using match stick or juice pipe or pieces
of straws.
• Cut some pieces of straws or juice pipes.

• Join these pipes with help of needle and thread.

• Fold this to form a structure of rectangle.

This is the skeleton of a rectangle.

Oasis School Mathematics – 6 257

Similarly we can make the skeleton of a triangle.
We can use these to make a face of solid figures. Join such structure to form the
skeleton of solid figure.

Activity III

• Using match stick or juice pipe or pieces of straws make the structure of a
triangle and a rectangle.

• Use these structures to form a skeleton of cuboid, pyramid and tetrahedron.

Net and skeleton of side figure.

Name of Solids Figures Net Skeleton

Cube

Tetrahedron

Octahedron

Pyramid

258 Oasis School Mathematics – 6

Objective Questions

1. Given figure is a (ii) cone (iii) sphere
(i) cylinder

2. Give figure is a (ii) cube (iii) cuboid
(i) tetrahedron

3. How many vertices does the given figure have?

(i) 6 (ii) 8 (iii) 12

4. Which geometrical shape is there on the base of the cone?

(i) square (ii) circle (iii) rectangle

5. Which one of the following is Euler's formula?

(i) F + V = E + 2 (ii) F + V = E – 2 (iii) F – V = E + 2

6. A geometrical object has 7 vertices and 7 faces, how many edges does it have?

(i) 14 (ii) 7 (iii) 12

7. Which of the following figure does not have vertex?

(i) cone (ii) sphere (iii) cube

Unit Test Full Marks : 14

Attempt all the questions.
Group-A [10 × 1= 10]

1. Identify the following figures:

2. Observe the given figure and identify whether the D A B
given are vertex, edge or face. E C

(i) AE (ii) ABCD (iii) B F
G
(iv) ABFE (v) EF (vi) G H

Group-B

3. Count the number of vertices, edges and faces of given
object and verify Euler's formula.

Oasis School Mathematics – 6 259

Co-ordinates

7Estimated Teaching Hours

Contents
• Co-ordinates System
• Plotting the points
Expected Learning Outcomes
At the end of this unit, students will be able to develop the following
competencies:
• To identify axes and quadrants.
• To identify the co-ordinates of given points
• To plot the given points on the graph
Teaching Materials
• Graph paper, pencils, etc.

260 Oasis School Mathematics – 6

Unit

16 Co-ordinates

The co-ordinate plane –4 –3 –2 –1 0 1 2 3 4 5 5
4
Look at the given number line 3
2
In this line the numbers right from '0' are positive and the numbers 1
left from the from '0' are negative. 0
–1
Again, in the vertical line the –2
numbers above 0 are positive –3
and the numbers below 0 –4
are negative. Keeping these
two lines in the same place,
we can get the figure like
this.

16.1 Co-ordinates System Second quadrant 5
4
In co-ordinates system, the horizontal
line is the X–axis. It is denoted 3
by XX'. The vertical line is the 2 First quadrant
Y–axis. It is denoted by YY'. Point of 1
intersection of horizontal and vertical line
is the origin. –5 –4 –3 –2 –1 O –11 2 3 4 5
Third quadrant –2 Fourth quadrant
X–axis and Y–axis divide the plane into –3
four parts which are quadrants named first,
second, third and fourth quadrants. –4

–5

Starting from right top and moving on anti-clockwise
direction, first, second, third and fourth quadrants
are determined as shown in the figure.

Oasis School Mathematics – 6 261

Origin

In the given figure, XOX' is a X-axis and YOY'
is Y-axis. The point of intersection of X-axis and
Y-axis is point O. Point 'O' is the origin.

Hence, the point of intersection of X-axis and Y-axis
is the origin.

Co-ordinates of a point

Co-ordinates of a point is its position with reference
to co-ordinate axes

6
5

4

3

2

O1

–7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 67
–1

–2

–3

–4

–5
–6

Lets find the co-ordinates of given points. Lets go to point P. Lets check the position of
point P. Starting from origin, move 2 units right along X-axis and 3 units up from it.
Then, co-ordinates of P is (2, 3).
2 represent X-co-ordinates and
3. represents Y-co-ordinates.
Lets find the position of Q.
For Q, move 6 units right from origin along X-axis and 2 units up from that position.
Hence, co-ordinates of Q is (6,2).
For R, move 1 unit right from origin along X-axis 3 units down from that position.
Hence, co-ordinates of R is (1, –3).
Similarly, we can find the co-ordinates of other points.


Remember! Co-ordinates of origin is (0, 0)

262 Oasis School Mathematics – 6

How to plot the given points?
Lets take some points (3, 4), (5, –2), (–3, –5) and (–4, 5) and plot these points.

Remember!
• X-co-ordinates of any point on Y-axis is 0.
• Y-co-ordinates of any point on X-axis is 0.

• Co-ordinates of point on Y-axis, which is 4 units up from origin is (0, 4).
• Co-ordinates of a point on Y-axis which 4 units below from origin is (0, –4).
• Co-ordinates of a point on X-axis which is 3 units right from the origin is (3, 0).
• Co-ordinates of a point on X-axis which is 3 units left from the origin is (–3, 0).

Note:
• X-co-ordinates is also known as abscissa.
• Y-co-ordinates is also known as ordinate.

Y

(–4, 5)

(3, 4)

X' O X

(5, –2)

(–3, –5)

Y'
To plot point (3, 4)
Starting from origin, move 3 units right laong X-axis and 4 units up from that points. The
final point is (3, 4).
Again, for (5, –2), starting from origin, move 5 units right along X-axis and 2 units down
from that point.
Similarly, we can plot other points also.

Oasis School Mathematics – 6 263

Exercise 16.1

1. Answer the following questions.
(a) What is the notation of X-axis?
(b) What is the notation of Y-axis?
(c) What is a point called where X-axis and Y-axis meet?
(d) How many quadrants are there in co-ordinates?
(e) What is the another name of X-co-ordinates?
(f) What is the another name of Y-co-ordinates?
2. Study the given figures and write the co-ordinates of given points.

Y

C
I

D B X
H A J
X' O

F G
E

Y'

3. Plot the following points on the squared paper. Join the points one after the
other and write the names of the figures formed.

(a) A (3, 3), (b) B (1, 1) (c) C (5,1) (d) P(–2, 3)

(e) Q (3, 0), (f) R (1, 7), (g) D (7, 3), (h) E (–3, –4)

264 Oasis School Mathematics – 6

4. Write the vertices of each figure: X
Y
P
A

Q
RW

DE

V
BC

DZ

E
F

F

5. Answer the following questions:

(a) What is the co-ordinate of origin?
(b) What is the x co-ordinate of a point on y-axis?
(c) What is the y co-ordinate of a point on x-axis?
(d) A point lies on x-axis at a distance of 4 units to the right of y-axis. Find the co-

ordinates of the point.
(e) A point lies on y-axis at a distance of 7 units lying above x-axis. Find the

co-ordinates of the point.
(f) A point lines on X-axis at a distance of 5 units left from origin. What is the co-

ordinates of that point?
(g) A point is 5 units right from the origin along X–axis and 4 units up along y-axis. What

is the co–ordinate of that point?
(h) A point lies on Y-axis at a distance of 3 units below the origin. Find the co-ordinates

of that point.
(i) A point is 3 units right from the origin along X–axis and 5 units up along

Y–axis, what is the co–ordinate of that point?
(j) The co–ordinate of a point on X–axis is 5 units left from the origin. Find the co–

ordinates of that point.
(k) The co–ordinate of a point on Y–axis is 5 units below from the origin. What is the

co–ordinate of that point.

õõõ

Answers

Consult your teacher.

Oasis School Mathematics – 6 265

Objective Questions

1. The point where X-axis and Y-axis meet is called:

(i) horizontal line (ii) vertical line (iii) origin

2. Co-ordinates of origin is: Y
B
(i) (1, 0) (ii) (0, 0) (iii) (0, 1)

3. Co-ordinates of point A is:

(i) (0, 0) (ii) (2, 0) (iii) (0, 2)

4. Co-ordinates of point P is: O A X
Y X
(i) (0, 3) (ii) (2, 3) (iii) (2, 4)

5. Co-ordinates of point Q is:

(i) (1, 3) (ii) (3, 1) (iii) (3, 3) P

6. On which quadrant does a point (–3, 2) lie? Q
O
(i) Second (ii) Third (iii) Fourth

7. On which quadrant does a point (–5, –7) lie?

(i) Second (ii) Third (iii) Fourth

Assessment Test Paper

Attempt all the questions Full Marks –10

Group 'A' [3 × 1 = 3] AB
1. (a) What is the co-ordinates of the point origin? C

(b) What is the X – co–ordinates of a point on the Y – axis?
(c) What is the Y- co-ordinates of a point on the X-axis?

Group 'B' [3×2=6] Y

2. Write the co-ordinates of the points

A, B, C, and D, in the given figure.

3. (a) Point on Y-axis is 5 units above the origin, D X
what is the co-ordinates of that point?

(b) A point lies on X-axis. It is 7 units
above the origin. What is the co-ordinatess of that point?

4. Plot the following points in the graph paper
A(6, 3), B(1, 2), C(2, 7), D(5, 9)

266 Oasis School Mathematics – 6

Symmetry, Tessellation,
Pattern and Designs
9Estimated Teaching Hours

Contents

• Symmetry
• Tessellation
• Pattern and design

Expected Learning Outcomes
At the end of this unit, students will be able to develop the following
competencies:
• To identify whether the given objects are symmetrical or

not
• To draw the line of symmetry of symmetrical objects
• To tessellate the surface with given shape
• To make a design using circle and polygons
Teaching Materials
• symmetrical objects on the surrounding, A4 size paper,

crayons, etc.

Oasis School Mathematics – 6 267

Unit Symmetry, Tessellation,
Pattern and Designs
17

17.1 Symmetry

Let's observe the following figures:

What is the common feature of these figures?
We can see that, if we draw vertical line at the middle, the either sides of each figure
are identical. In nature we find many flowers, leaves, etc. that have two identical
sides if we draw a line through the middle of them.

Line Line

Two identical portions A Two identical portions
A
Line of symmetry
CD
Look at the following shapes B

A Figure (iii)

B B
Figure (i) Figure (ii)

268 Oasis School Mathematics – 6

In each of the above figures, the line AB divides each of these figures into two
identical parts. If the figure is folded along the line AB, one half of the figure
will coincide exactly with other half. Hence we say that the above figures are the
symmetrical figures and the line AB is the line of symmetry. In figure (iii) CD is also
the line of symmetry.

A figure may have more than one line of symmetry and some figures may have no
line of symmetry. Let's observe the line of symmetry of some geometrical figures.

Line of symmetry

Line of symmetry

Angle Isosceles triangle Scalane triangle Equilateral triangle
(No line of symmetry) (Three lines of symmetry)

Line of symmetry Line of symmetry

Kite Rectangle Arrow head
(Two lines of symmetry)

Note :
A figure may have more than one
line of symmetry like in the given
figures.

The letter 'O' has infinite line of
symmetry
]

Oasis School Mathematics – 6 269

Exercise 17.1

1. Identify whether the following figures are symmetrical or not.
(a) (b) (c)

(d) (e) (f)

2. Draw the line of symmetry of the following figures by using the dotted line.
(a) (b) (c)

(d) (e) (f)

3. Draw the line of symmetry in the given English letters:

(a) (b) (c)

(d) (e) (f)

270 Oasis School Mathematics – 6

4. Complete the following figures from the given lines of symmetry (dotted line):

(a) (b) (c)

(f)
(d) (e)

(g) (h) (i)

(j) (k) (l)

5. (a) Make a list of capital alphabets of English which have one line of symmetry.
(b) Make a list of capital alphabets of English which have two lines of symmetry.
(c) Make a list of capital alphabets of English which have no line of symmetry.
(d) Make a list of flowers which have one line of symmetry.

6. How many lines of symmetry do the following figures have?
(a) Parallelogram (b) Regular pentagon (c) Letter H (d) Arrow head (e) Square

7. Construct an angle of 60° and draw its line of symmetry.

õõõ

Answers
Consult your teacher.

Oasis School Mathematics – 6 271

Activity
• Take a rectangular sheet of paper.

fig (i)

• Fold it once in the middle as in figure (ii).

fig (ii)

• Draw the half heart in the middle as shown in figure (iii). fig (iii)

• Cut the sheet on the line of the shape as in figure (iv) fig (iv)

• The shape of the heart is obtained. fig (v)
The line of symmetry is the line of folding.

17.2 Tessellation

A tessellation is a method which is used to cover surface with congruent geometrical
shape in a repeating pattern without leaving any gaps and without overlapping each
other.

Tessellation is done on the surface of floor or wall or carpet to make them more
attractive.
Example: 1
Tessellate with a

Solution:

272 Oasis School Mathematics – 6

According to the given geometrical shape with the pattern,
we tessellate as shown in figure.

Exercise 17.2

1. Complete the following tessellation in a full page.
(a) (b) (c)

2. Tessellate with a (b)
(a)



õõõ

Answers
Consult your teacher.

Activity

Draw the square Tessellation in the given dot and shade with suitable colour.

Oasis School Mathematics – 6 273

Activity

Draw the rectangular Tessellation in the given dot and shade with suitable colour.

17.3 Designs using Circles and Polygons

Different designs can be made of circles, polygons, arc etc. They can be used to
decorate walls or floors of room or halls.

Look at the following designs, study them and try to draw them.

(a) Design using circle: (b) Design using circle and arc:

(c) Design using squares: (d) Design using circles and triangles.

274 Oasis School Mathematics – 6

(e) Design using pentagon. (f) Design using hexagon

(g) Design using hexagon

Some other different Patterns:

9 • Draw a vertical line and
8 horizontal line as shown in
given figure.
7
6 • Join point 1 of horizontal line
5 with 9 of vertical line, 2 of
4 horizontal line with 8 vertical
line and so on.
3
2 • Similarly, follow the same method
1 in other parts also.

–9 –8 –7 –6 –5 –4 –3 –2 –1–1 123456789
–2
–3
–4
–5
–6
–7
–8

–9

Exercise 17.3

1. Draw a circle of radius 4 cm and design as in (a).
2. Draw a square of side 4 cm and make a design as in (c)
3. Draw a regular pentagon of 5 cm and design as in (e).
4. Draw a regular hexagon having each side 6cm and design it as in (f).
5. Draw a circle of radius 5cm and design it as in (d).
6. Draw a regular hexagon of having each side 5cm and design it as in (g).

Answers
Consult your teacher.

Oasis School Mathematics – 6 275

Objective Questions

Chose the correct alternatives:

1. The letter has

(i) no line of symmetry.
(ii) one line of symmetry.
(iii) two lines of symmetry.
2. Which of the following letter has infinite lines of symmetry?

(i) (ii) (iii)

3. Which of the following number has only one line of symmetry?

(i) (ii) (iii)

4. An equilateral triangle has
(i) one line of symmetry.
(ii) two lines of symmetry
(iii) three lines of symmetry.

276 Oasis School Mathematics – 6

Assessment Test Paper

Full Marks : 11

Attempt all the questions.

Group "A" [5 × 1 = 5]

1. Draw the line of symmetry in the given figure:



(a) (b) (c)

2. Identify whether the given letters are symmetrical or not.

(a) (b)

Group "B" [3 × 2 = ]

3. (a) Draw the symmetrical shape about the given line

(i) (ii)


(b) Write the names of two English letters which have:
(i) one line of symmetry
(ii) two lines of symmetry
(c) Draw a square having each side 5cm and make a design using only squares.

Oasis School Mathematics – 6 277

Statistics

10Estimated Teaching Hours

Contents
• Collection of Data
• Construction of Frequency Distribution Table
• Bar Graph
Expected Learning Outcomes
At the end of this unit, students will be able to develop the follow-
ing competencies:
• To construct the frequency distribution table from the

given raw data
• To construct the bar graph from the given data
Teaching Materials
• Model of bar graph, graph sheet, etc.

278 Oasis School Mathematics – 6

Unit

18 Statistics

18.1 Collection of Data

The teacher decided to take class VI students in an educational tour to Bhaktapur
Durbar Square. He told the students to find out the following things:

• the number of temples.
• the average number of flow of visitors in a day.
• the number of statues.

The students collected the information and handed over to the teacher. The teacher
finally said that the information that they had gathered was simply a data.

Raw data

Marks obtained by 20 students of class VI in an examination out of 100 full marks
is given below:
72, 56, 88, 95, 72, 56, 95, 78, 72, 95, 88, 95, 56, 72, 95, 78, 56, 88, 95, 56.
From the above data, try to answer the following questions:
(i) How many students got 95 marks?
(ii) How many students got 88 marks?
(iii) What is the lowest marks of the class?

Above data are in original form and not organized so the answer of these questions
is difficult.

Such collection of data is raw data.

Organization of data

Let's organize the above data in another way:

• Let's arrange the marks in ascending order in one column.

• Count the number of students who have got the given marks.

Marks Tally marks Number of students.

56 5

72 4

78 2

88 3

95 6

From this organization, it is easier to answer the above questions.

Oasis School Mathematics – 6 279

I understand!
6 students have got 95 marks. 3 students got 88 marks.

This table is called frequency distribution table. The number of repetition of an item
is the frequency of the item.

Oh! I  understand!
frequency of 56 is 5 frequency of 72 is 4

78 repeated twice means
frequency of 78 is 2.

Construction of Frequency Table

Let's take an example to construct frequency distribution table. The following data
shows the number of children in 30 families.

2, 1, 3, 4, 1, 5, 2, 2, 4, 1, 3, 1, 1, 2, 3, 2, 2, 2, 5, 4 2, 3, 3, 2, 1, 1, 3, 2, 3, 3

Number of Tally Frequency Steps
children marks
1 7 • Construct the table of three columns showing
10 variables, tally marks and frequency.
2 8
3 • Arrange the variables in ascending order
3 2 • Put tally marks for counting of each item.
• Write the value obtained from tally marks in
4
third columns.
5

represents 1
represents 2
represents 3
represents 4
represents 5

Worked Out Examples

Example: 1

The following data is collected from 24 students of their favourite game.
Football, Volleyball, Football, Cricket, Tennis, Football, Basketball, Basketball,
Cricket, Tennis, Cricket, Football, Volleyball, Cricket, Basketball, Cricket,
Cricket, Football, Football, Volleyball, Football, Volleyball, Volleyball, Football.

280 Oasis School Mathematics – 6

Construct a frequency table and answer the questions given below:

(i) How many students like Tennis?
(ii) How many students like Volleyball?
(iii) Which is the most favourite game in the class?
(iv) Which is the least favourite game among these?
Solution:

Game Tally marks Frequency

Football 8

Volleyball 5

Cricket 6

Tennis 2

Basketball 3

(i) 2 students like Tennis.
(ii) 5 students like Volleyball.
(iii) The most favourite game in the class is football.
(iv) The least favourite game among these is Tennis.

Exercise 18.1

1. (a) The ages of 22 students in class VI are given below.
10, 10, 14, 12, 12, 10, 14, 11, 12, 10, 10, 12, 10, 13, 13, 11, 11, 12, 10, 12, 12, 13.
Prepare a frequency table from the above data.
(b) The weights (in kilograms) of 25 students are given as follows.
35, 38, 36, 37, 38, 35, 37, 36, 35, 38, 36, 36, 37, 37, 35, 38, 36, 35, 36, 37, 37, 38,
36, 38, 37.
Construct a frequency table using tally marks.
(c) The following data is the daily demand of milk in 25 household of Chhaimale village.
700 ml, 500 ml, 500 ml, 1000 ml, 2000 ml, 1000 ml, 1500 ml, 1500 ml, 500 ml, 500
ml, 2000 ml, 500 ml, 700 ml, 700 ml, 500 ml, 1000 ml, 500 ml, 700 ml, 1000 ml,
2000 ml, 1500 ml, 1000 ml, 500 ml, 700 ml, 1500 ml

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Represent this information in the form of a frequency table.

(d) Marks obtained by 26 students in class VI out of 50 full marks is given below.

45, 35, 25, 40, 45, 30, 35, 20, 25, 35, 40, 45, 45, 35, 40, 35, 25, 40, 30, 30, 20, 25,
25, 25, 20, 40

Construct a frequency table to represent the above information.
2. (a) Following data shows the number of children in 40 families.

1, 2, 6, 5, 1, 5, 1, 3, 2, 6, 2, 3, 4, 2, 0, 4, 4, 3, 2, 2,
0, 0, 1, 2, 2, 4, 3, 2, 1, 0, 5, 1, 2, 4, 3, 4, 1, 6, 2, 2,
Construct a frequency table representing the above information and answer the

following questions.
(i) How many families have no child?
(ii) How many families have one child?
(iii) How many families have 4 children?

(iv) How many families have 6 children?

(b) 30 students were interviewed by class teacher of VI about their favorite subjects
in school. The following information was obtained.

Maths, Maths, Science, English, Nepali, Nepali, English, Nepali, Maths, Science,
English, Maths, Nepali, Science, Science, Nepali, English, Maths, Science, Nepali,
Science, Science, Maths, Maths, English, English, Maths, English, Science and Maths.

Represent this information in the form of a frequency table and answer the
following questions.

(i) Which is the most favourite subject in the class?
(ii) Which one is the least favourite subject in the class?
(iii) How many students like Nepali?

(iv) How many students like Science?

Answers:

1. Consult your teacher. 2. (a) Consult your teacher for frequency table.

(i) 4 (ii) 7 (iii) 6 (iv) 3

(b) Consult your teacher for frequency table. (i) Maths (ii) Nepali (iii) 6 (iv) 8

Project work

Collect the data about the favourite subjects among the students of your class.
Construct the frequency table and draw out the conclusion from the table.

282 Oasis School Mathematics – 6

18.2 Bar Graph

Data can be represented by using different types of figure. A bar graph can represent
any numerical data.
When numerical data is presented as columns of a graph, this graphical representation
of a data is called a bar graph.

The marks obtained by 5 students in a class are represented by a bar graph.

Remember

• Each column (bar) should be of

equal width.

Y • There should be equal space
between two columns.

Marks

O Ansu Rashmi Prapti Himanka Zenith X

Study the above bar graph and answer I understand !
the questions given below:
• Rashmi got the highest
(a) Which student got the highest mark? marks.
(b) Which student got the lowest mark?
(c) How many marks did Prapti score? • Himanka got the
(d) How many marks did Zenith score? lowest marks.
(e) What is the highest mark of the class?
• Prapti scored 40
• Zenith scored 30
• 50 is the highest mark

of the class.

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Construction of a bar graph

The bar graph may be drawn in a graph paper or in a plane paper.

Let's learn to draw the bar graph with the help of given example.

The highest marks obtained by the students in different subjects in an examination
are given below:

Subjects English Nepali Mathematics Science Social studies

Marks 75 80 95 90 80

Let's present the information in bar graphs.

Y

100

90

80
70

Marks 60
50
40
30

20

10

O English Nepali Mathematics Science X

Social
Studies
–Subject–

Steps

• On the graph paper or on a plane paper, draw a horizontal line (X-axis)
and a vertical line (Y-axis).

• Mark the points at equal distances and write the name of items for which
data is to be represented.

• Choose the suitable scale and locate the height of different bars
according to the scale.

• Draw the bars of equal width and keep them at equal distances.

284 Oasis School Mathematics – 6

Worked Out Examples

Example 1Section

The following bar graph shows the number of sections of class VI to X of Bal Bikash
Boarding School. Read the bar graph and answer the following questions.

Y

7

6
5

4
3

2

1

O VI VII VIII IX XX

(i) How many sections are there in class VI?
(ii) How many sections are there in class IX?
(iii) Which class has highest number of sections?
(iv) Which class has only one section?
(v) How many sections are there altogether?

Solution:
From the bar graph it is clear that,

(i) There are 2 sections in class VI.

(ii) There are 4 sections in class IX.

(iii) Class VIII has highest number of sections.

(iv) Class X has only one section.

(v) Altogether there are (2 + 3 + 5 + 4 + 1) = 15 sections.

Example: 2

The following table gives the family budget of Pasang.

Item Food Rent Clothing Education Others Savings

Cost (in Rs.) 500 400 300 150 200 250

Draw the bar graph to represent the above informations.

Oasis School Mathematics – 6 285

Solution: Bar graph showing the family budget of Pasang

700 Cost in Rs.
600
500 Food Rent Clothing Education Others Saving
400 Items
300
200
100

0

Exercise 18.2

1. (a) Given bar graph shows the number of attendance of class VI from Sunday to
Friday.

Attendance Y

70 Monday Tuesday Wednesday Thursday Friday X
60
50
40
30
20
10

0 Sunday

Study the above bar graph, Complete the given frequency table. Friday

Days Sunday Monday Tuesday Wednesday Thursday
Attendance

286 Oasis School Mathematics – 6

(b) The following bar diagram shows the number of students in different classes
in Tara Chandra School, Ajirkot Gorkha.

Y

100

90

80

70

Number of Students 60

50

40

30

20

10

0I II III IV V X

VI

Class

Study the above bar graph and answer the questions given below.

(i) Which class has the highest number of students?
(ii) Which class has the lowest number of students?
(iii) How many students are there in class V ?
(iv) Find the total number of students from class IV to class VI?

(v) How many more students are there in class VI than in class V?

(c) StudentsinclassVIwereinterviewedbytheclassteacherabouttheirfavourite
picnic spot. The result thus obtained is represented by the given bar graph:

Y

Number of Students 140
120
100

80
60
40
20

0 Thankot Godawari Dhulikhel Kakam Nagarkot X
Picnic Spot

Oasis School Mathematics – 6 287

Study the above bar diagram and answer the questions given below.
(i) How many students are there in the class?
(ii) In which spot did the majority of students want to go for the picnic?
(iii) How many students voted for Nagarkot for the picnic?
(iv) Which spot was the second choice of the students?
(v) For which spot did the least number of students vote?

2. (a) The given table shows the marks obtained by Anasuya in different subjects. Draw

the bar graph to represent the given information.

Subjects English Nepali Maths Science Social studies
Marks 60 65 85 70 75

(b) The following table shows the number of children taking part in different games

in a school.

Games Table Tennis Badminton Chess Carrom Ludo

No. of children 5 10 10 30 25

Draw a bar graph to represent this information.

(c) The table below shows the number of periods in a week for different subjects.

Subjects English Nepali Science Maths Social Studies

No. of periods 8 6 10 12 4

Draw a bar graph to represent this information.

õõõ

Answers

1. (a) Sunday – 30, Monday – 35, Tuesday – 40,Wednesday – 25, Thursday – 30,

Friday – 45 (b) (i) Class VI (ii) Class V (iii) 30 (iv) 200 (v) 60.

(c) (i) 500 (ii) Dhulikhel (iii) 80 (iv) Kakani (v) Godawari.

2. Consult your teacher.

288 Oasis School Mathematics – 6

Objective Questions

Choose the correct alternatives.

1. In the data 5, 7, 18, 15, 13, 7, 13, 15, 13, 7, 5, 13, frequency of 13 is

(i) 13 (ii) 4 (iii) 3

2. What does // represent?

(i) 8 (ii) 7 (iii) 5

3. While drawing bar graph, which of the following conditions may not be applied?

(i) All bar should be of same height

(ii) All bar should be of same width

(iii) There should be same gap between two bars.

4. What number is represented by ?

(i) 13 (ii) 10 (iii) 5

Given bar graph shows the number of students in different classes.

Y

80

70

Number of Students 60

50

40

30

20

10

0 VI VII VIII IX X X

Class

5. Total number of student in class IX and X is:

(i) 50 (ii) 40 (iii) 90

6. Which class has highest number of students?

(i) IX (ii) X (iii) VIII

7. Which class has the lowest number of students?

(i) VI (ii) VII (iii) X

8. Which two classes have equal number of students?

(i) VI and VII (ii) VIII and X (iii) VI and VII

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Assessment Test Paper

Attempt all the questions Full marks – 12

[3×4=12]

1. Weight of 25 students of class VI is given below. Prepare the frequency
distribution table from given data:

35kg, 42kg, 42kg, 35kg, 37kg, 39kg, 40kg, 42kg, 45kg,
35kg, 37kg, 45kg, 39kg, 45kg, 39kg, 37kg, 35kg, 42kg,
42kg, 45kg, 37kg, 35kg, 39kg, 45kg, 35kg

2. Mark obtained by Sundar in 5 different subjects is given below represent this

data in bar graph.

Subjects Nepali English Mathematics Science Social studies

Marks 50 45 60 55 35

3. The given bar graph shows the marks obtained by 5 students in the maths test
paper. Answer the questions given below.

Marks Y

80 Bandana Kanchan Puran Prem X
70
60
50
40
30
20
10

0 Aman

(i) Write the marks obtained by each student.
(ii) Who has got the highest marks?
(iii) Which two students have got equal marks?
(iv) By how much Kanchan's marks is more than that of Puran?

4. Grade obtained by 20 students of VI are given below. Prepare a frequency
distribution table and answer the questions given below:

A, A+, B, A, B+, A, A, A+, B+, B, A, A+, B+ A, B+­ , B+, C+, C+, A, B++
(i) How many students get grade A+?

290 Oasis School Mathematics – 6

Model Test Paper

Group A [Very short questions] [18 × 1 = 18]

1. Choose the correct alternatives:

(a) How many line segments can be drawn from given two points?

(i) one (ii) two (iii) infinite

(b) 910 is

(i) an acute angle (ii) a reflex angle (iii) an obtuse angle

(c) Which one of the following statement in not true?

(i) vertically opposite angles are equal.

(ii) complementary angles are equal.

(iii) sum of linear pairs is 1800.

(d) Which one of the following statement is not true?

(i) every parallelogram is a rhombus.

(ii) every rhombus is a parallelogram.

(iii) every rectangle is a parallelogram.

(e) A point lies on x-axis, which is 5 units right from the origin, then its
co-ordinates is

(i) (-5, 0) (ii) (0.5) (iii) (5, 0)

(f) The formula for the perimeter of a rectangle is

(i) l × b (ii) 2 (l + b) (iii) l × b

(g) If the perimeter of a rhombus is 44cm, the length of each side is

(i) 11cm (ii) 22 cm (iii) 44 cm

(h) The cardinal number of null set is

(i) 0 (ii) 1 (iii) 2

(i) If B = {x : x is the letter of the word 'MATHEMATICS"}. Then, the value of n(B) is

(i) 11 (ii) 8 (iii) 9

(j) If A and B are equivalent sets, then

(i) n(A) = n(B) (ii) A = B (iii) n(A) ≠ n(B)

(k) If the digit in the place of ones is '0' or '5' then the number is divisible by

(i) 10 (ii) 2 (iii) 5

(l) 7 multiplied to the sum of 2 and 3 is

(i) 14 (ii) 21 (iii) 42

(m) Which one of the following statement is not true?

(i) 3 is a rational number.
4

(ii) 0 is an integer (iii) -2 is a whole number.

(n) 0.5432 ÷ 100 is equal to

(i) 0.0543 (ii) 0.00543 (iii) 54.32

Oasis School Mathematics – 6 291

(o) 1 – 1 is equal to: (i) 1 (ii) 1 (iii) 5
2 3 6 5 6

(p) The algebraic expression for the sum of 1 of the sum of x and y is
4
1 2 1
(i) 4 x + y (ii) x+ 4 y (iii) 4 (x + y)

(q) The inequality of given figure is

-3 -2 -1 0 1 2 3 4 5

(i) x ≥ 1 (ii) x > 1 (iii) x ≤ 1
(r) 10% of 60 is equal to (ii) 10 (iii) 60
(i) 6

Group B [Very short questions] [18 × 1 = 18]

2. (a) In the given figure, PQ is the perpendicular A P
bisector of AB then write the relation between
QB
AQ and BQ.

(b) Draw a line segment of length 7.3 cm. Y
(c) Classify the angles 650 and 1400.

3. In the given figure, what are the P
co-ordinates of P and Q? Q

4. (a) Find the perimeter of given triangle. O X

P
6.4 cm
3 cm

Q 7.2 cm R

(b) Find the area of a rectangle whose length and breadth are 6 cm and 5.2 cm.

5. (a) Write the name of transformation in which position and size of given object
changes?

(b) Draw the line of symmetry
in the given figure.

6. Describe the set A = {2, 3, 5, 7, 11, 13, 17, 19}.
7. (a) Write the smallest and the greatest number formed by the digits 2, 3, 7, 7, 0.
(b) F(4) = A set of factors of 4, find F(4).

292 Oasis School Mathematics – 6

(c) Find the square of 13.

8. (a) Subtract : 3 – 1 . (b) Find the product of: 0.06 × 12.
4 2 Y

9. (a) From the given bar graph,

find the number of students in 50
class IX and class X. 40

30
20

10

O IX X XI XII X
Class

10. (a) Separate power, base and co-efficient in the term 3x7.

(b) Write in the exponential form: 3 × a × a × a × b × b × b × b

(c) Multiply: 2x2y3 × 4x3y4 Y [19 × 2 = 38]

Group-C (Long questions)
11. (a) Measure the given angle with the help of
protractor and mention its type.

Z

(b) Construct an angle 450 with the help of ruler and compass.

CD

(c) Find the value of x in the given figure. x

A 600 400 B
Q

(d) Find the value of x in the given figure. P
x0

12. Plot the following points on the graph sheet. Q 300 720
R

(i) A (3,4) (ii) B(1,3) (iii) C(5,1) (iv) D (–4, 4)

13. (a) If the perimeter of a square is 18cm, find the length of each side.

(b) Find the volume of a cuboid whose three sides are 6cm, 5 cm and 4 cm.

14. A complete the given figure using the
given line of symmetry.

15. If A = a set of natural number less than 5 and B = a set of the letter of word
"APPLE". Identify whether A and B are equal sets or equivalent sets.

16. (a) Find the prime factors of 36.

Oasis School Mathematics – 6 293

(b) Find the square root of 144.

(c) Draw a number line and multiply : (+3) × (+2)

17. (a) Simplify : 1 2 + 1 – 1 .
3 2 6

(b) Round off the following decimals:

(i) 8.67 (to the nearest one decimal place)
(ii) 54.325 (to the nearest two decimals places)
18. (a) If C.P. = Rs. 450, profit = Rs. 150, find S.P.
(b) If the cost of a pen is 15, then find the cost of 12 pens.
(c) Calculate the value of 20% of Rs. 500.
20. (a) If a = 1, b = 2 and c = 3, find the value of a2 + b2 + c2.
(b) Simplify: 2x + 3y + 6y + 5x.
(c) What should be added to x + 3y – 5z to make 2x – y + 4z?

Group-D (Very long questions) 780 [8 × 3 = 24]

21. (a) Find the value of 'x', 'y' and 'z' from x
the given figure.
1010 z0
y0
940

(b) Construct an equilateral ∆ABC where AB = BC = AC = 6cm.

22. Find the area of given figure. 3cm
5cm

2cm 10cm

23. (a) Simplify: 25 + {3 + {16 ÷ 4 + 6}.

(b) Find the smallest number which is exactly divisible by 12, 15 and 18.

24. Simplify: 3 2 – {2 1 + ( 5 – 1 )}.
3 3 3 3

25. The number of family members of 25 students of Class VI is given below.

Prepare a frequency distribution table from the given information:

4, 5, 6, 3, 5, 4, 3, 6, 7, 6, 5, 4, 3, 5, 4, 6, 7, 4, 3, 5, 5, 5, 4, 3

26. Find the product of: (x–y) and (x2 + xy + y2).

–The End–

294 Oasis School Mathematics – 6