6 Be My Multiple, I'll be Your Factor

The Mouse and the Cat

The hungry cat is trying to ١

catch Kunjan the mouse. ٢
Kunjan is now on the 14th ٣
step and it can jump 2 steps ٤
at a time. The cat is on the ٥
third step. She can jump ٦
3 steps at a time. If the mouse
٧
reaches 28 it can hide in the
hole. Find out whether the ٨
mouse can get away safely! ٩
١٠
a) The steps on which the ١١
mouse jumps — ١٢
١٣
١٤
b) The steps on which
the cat jumps — ١٥
١٦
١٧
c) The steps on which both the ١٨
cat and the mouse jump — ١٩
٢٠
٢١
d) Can the mouse get away?
٢٢
Find out ٢٣
٢٤
If the cat starts from the 5th step and jumps ٢٥
five steps at a time and the mouse starts ٢٦
from the 8th step and jumps four steps at a ٢٧
time, can the mouse get away? ٢٨ ٢٩
Children should be encouraged to make similar questions
with different multiples and ask each other to solve.
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Who is Monto waiting for?

Monto cat is waiting for somebody. Do

you know for whom he is waiting?
There is a trick to find out.

D X
1 2 3 4 5 6 7 8 9 10
M P I
11 12 13 14 15 16 17 18 19 20
O
21 22 23 24 25 26 27 28 29 30
R N U
31 32 33 34 35 36 37 38 39 40
B W S
41 42 43 44 45 46 47 48 49 50
J H E
51 52 53 54 55 56 57 58 59 60

Mark with a red dot all the numbers which can be divided by 2.
Mark a yellow dot on the numbers which can be divided by 3 and
a blue dot on the numbers which can be divided by 4.
Which are the boxes which have dots of all three colours?

What are the letters on top of those boxes?

Write those letters below in order.

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Meow Game

To play this game, everyone

stands in a circle. One player
calls out ‘one’. The next player
says ‘two’ and so on. A player
who has to call out 3 or a
number which can be divided by
3 has to say ‘Meow’ instead of
the number. One who forgets to
say ‘Meow’ is out of the game.
The last player left is the winner.
Which numbers did you replace
with ‘Meow’?
3, 6, 9...............................

We say these numbers are the multiples of 3.

Play the game by changing the number to 4.

Now, which numbers did you replace with ‘Meow’?

These numbers are the multiples of 4.
❈ Write any ten multiples of 5.

Make children play this game several times with multiples of different numbers.

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Dice Game
Throw two dice together. What are the
numbers that turn up on the faces of the
dice? Make a two-digit number using them.
If it is a multiple of any of the numbers
written next to the circles, you can write it
in that circle. Then it is your friend’s turn.
The one who can write more numbers in 10
rounds is the winner.

I have 3 and 2 on my dice.

If I make 23, it is not the
multiple of any of the
numbers. So I will make 32,
which is a multiple of 4, and
write it in the red circle.

6 4

5 7

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Common Multiples

Think of a number. If it is a multiple of 3 write it in the red circle.

If it is a multiple of 5 write it in the blue circle.

3 5

ite
do I wr le
Where
I t i s a multip
15 ? .
b o t h 3 and 5
of

Some numbers are multiples of both 3 and 5.

Sowecansaythattheyare common toboth3and5.

Think! If you write the multiples common to

3 and 5 in the purple part, then will they still be
in both the red and the blue circles?
❈ Which is the smallest among these common
multiples? _________
Repeat the game using the numbers 2 and 7.

❈ Write the common multiples of 2 and 7.

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Repeat the game by putting the multiples of 4, 6 and 5 in the circles.

4
6

5
❈ What common multiples of 5 and 6 did
you write in the green part?

❈ What common multiples of 4 and 6

are written in the orange part?
❈ In which coloured part did you write
the common multiples of 4,6 and 5?
❈ What is the smallest common multiple
of 4, 6 and 5? ________
Puzzle
Tamarind seeds
Sunita took some tamarind (imli)seeds. She
made groups of five with them, and found
that one seed was left over. She tried making
groups of six and groups of four. Each time
one seed was left over. What is the smallest
number of seeds that Sunita had?
Encourage children to try out themselves such activities using seeds, pebbles etc.

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More tamarind seeds
Ammini is arranging 12 tamarind seeds in
the form of different rectangles. Try to make
more rectangles like this using 12 tamarind
seeds. How many different rectangles can
you make?
If there are 15 tamarind seeds how many
rectangles can you make?
Colouring the Grid

In the grid here, a rectangle made of 20 boxes is drawn.

The width of this rectangle is 2 boxes.
❈ What is its length?
❈ Colour a rectangle made of 20 boxes in some other way.
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❈ What is the length and width of the rectangle you coloured?
❈ In how many ways can you colour a rectangle of 20 boxes?
Colour them all in the grid, and write the length and width of
each rectangle you have coloured.

Bangles
There are 18 bangles on
the rod. Meena is trying to
group them. She can put
them in groups of 2, 3, 6, 9
and 18 — without any
bangle being left.
❈ How many groups will
she have if she makes
groups of 1 bangle
each? ____
Now complete the table, Number Different groups we can make
for different numbers of of bangles
bangles. For each number 18 1, 2, 3, 6, 9, 18
see what different groups
can be made. 24 1, 2, ...............
5
9
7
2
10
1
20
13
21

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Fill the Chart
Complete the multiplication chart given here.

× 1 2 3 4 5 6 7 8 9 10 11 12

1 12

2 12

3 12 21

4 12 40

5 20

6 12

8 72

10

11 66

12 12

Look at the green boxes in the chart. These show how

we can get 12 by multiplying different numbers. 12
4×3
12 = 4 × 3, so 12 is a multiple of both 4 and 3. 12 is also
6×2
a multiple of 6 and 2, as well as 12 and 1. We say 1, 2,
1×12
3, 4, 6, 12 are factors of 12.

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❈ What are the factors of 10? ___________
10
Can you do this from the chart? 5×2
❈ What are the factors of 36 ? ___________ ___

❈ Find out all the factors of 36 from the multiplication chart.

❈ What is the biggest number for which you can find the factors
from this chart?
❈ What can you do for numbers bigger than that?

Common factors

Write the factors of 25 in the red circle and the factors of 35 in the
blue circle.

Which are the factors you have written

in the common part (purple) of both
circles? These are common factors of
25 and 35.

Now write the factors of 40 in the red

circle and 60 in the blue circle.

What are the factors written in

the common (purple) part of the
circle? Which is the biggest
common factor of 40 and 60?

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Factor Tree
Look at the factor tree. Now can you make another tree like this?

3 3
2
9

18 18

❈ In how many ways can you draw a factor tree for 24? Draw
three of them below.

24 24 24

❈ Try drawing the factor tree using other numbers also.

Tiling Problems

1) There is a garden in Anu’s

house. In the middle of the garden
there is a path. They decided to tile
the path using tiles of length 2 feet,
3 feet and 5 feet.

The mason tiled the first row with 2

feet tiles, the second row with 3 feet
tiles and the third row with 5 feet
tiles. The mason has not cut any of
the tiles. Then what is the shortest
length of the path?

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 2) Manoj has made a new house. He wants to lay
tiles on the floor. The size of the room is 9 feet × 12
feet. In the market, there are three kinds of square
tiles: 1 foot × 1 foot, 2 feet × 2 feet and 3 feet × 3 feet.
Which size of tile should he buy for his room, so that
he can lay it without cutting?

Geetha's House Naseema's House

3) Rani's House
90

90

90
fee

fee

fee
t

Road

Rani, Geetha and Naseema live near each other. The distance
from their houses to the road is 90 feet. They decided to tile the
path to the road. They all bought tiles of different designs and
length. Rani bought the shortest tile,Geetha bought the middle
sized one and Naseema bought the longest one. If they could
tile the path without cutting any of the tiles, what is the size of
the tiles each has bought? Suggest 3 different solutions.
Explain how you get this answer.
It will be useful to have a discussion about a ‘foot’ and how we use it often to talk about our
own heights. Children can use their cm scale to get idea about how long a foot is.

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 7 Can You See the Pattern?
Isha, your skirt My mother
made this
is beautiful!
pattern
I have seen the same
block making a different
pattern on a kurta.

How was it.

different?

In your skirt, the rule

of the pattern is: one
up, one down. Then
this is repeated. But in my brother’s kurta,
it is once up, then takes a
1 turn every time.
4
The rule is to repeat it
with a 41 clockwise turn.

Now you use these two rules to make patterns with this block.

Also make your own rule.

In Math-Magic Class IV (page 107- 108) , children have seen how one motif is used in 3
different ways and in Class III (page 145), the same sequence of motifs is repeated. Discuss
how the motif here turns clockwise.

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Turns and Patterns

Look at this block . We make three different rules to turn it

clockwise and see the patterns.

Rule 1: Repeat it with a one-fourth turn.

Rule 2: Repeat it with a half turn.

Rule 3: Repeat it with a three-fourth turn.

Practice time

1) What should come next?

a)

N N
N

b)

Encourage children to think of other alternatives. Answers obtained by anticlockwise turns

should also be accepted and discussed.

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c)

d)

2) See this pattern

F F
F
a)
F

The rule of the pattern is — turning by 45º each time. Which will
be the next? Tick (✓) the right one.
F
F
F

( ) ( )
( )

Using the same rule take it forward till you get back to what you
started with.

b)
L
L

c)
P
P

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3) Some patterns are given below on the left side of the red line.
For each pattern, write the rule. Then choose what comes
next from the right side of the line and tick (✓) it.
a)

Rule:

b)

Rule:

c)

Rule:

d)

Rule:

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Look for a Pattern

Mark that picture which is breaking the rule. Also correct it.
a)

b)

c)

d)

Magic Squares

Do you remember magic triangles? Come now, let’s make some

magic squares.
49
❈ Fill this square using all the numbers
from 46 to 54.
46
Rule: The total of each line is 150.

52 47

25 ❈ Fill this square using all the

numbers from 21 to 29.

Rule: The total of each side is 75.

You can see Math-Magic Class IV (page 11) for similar magic patterns.

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Magic Hexagons
Look at the patterns of numbers in hexagons.
Each side has 2 circles and 1 box. 7
70
98
You get the
number in each 14 10
box by multiplying
the numbers in
the circles next 70 20
to it.

5 2
65
26
13
5 13 = 65 Look at the number 65 in the box.
Which are the circles next to it?
= 70 Can you see how the rule works?

❈ Use the same rule to fill the hexagons below.

a) 9 b)
108 104
78
11 8

6 7 4 8
64
17
Now you also make your own magic hexagons.
You can discuss that a hexagon is a six-sided closed figure, but this is not to be evaluated.

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Numbers and Numbers

24 + 19 + 37 = 37 + 24 + 19

215 + 120 + 600 = 600 + 215 + 120

❈ Are they equal?

❈ Fill in the blank spaces in the same way.

a) 14 + + = 34 + 14 + 20

b) + 42 + = 65 + + 80

c) 200 + 300 + = + 400 +

d) + + = + +
❈ Now, look at this — 48 × 13 = 13 × 48
Check if it is true or not.

Left Right — Same to Same

Can you see something 121

special about 121? What, it’s just
a number!

See it is the
Oh, yes! It is
same forward as
1,2,1 from right
well as backward.
to left also!

Discuss with students that changing the order of numbers does not make any difference to the sum.

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Come, let’s Take a number, say 43
see how to
get such Now turn it back to front 34
numbers. You have
Then add them together 77 reversed the
number by
77 is one such special number. writing it back
There are many such numbers. to front.

Take another number 48

Now turn it back to front 84
Then add them together 132
Is this a special number? No! Why not?
OK, carry on with the number 132
Again turn it back to front 231
Then add the two together 363
Ah! 363 is a special number.

So we see that to get special numbers we sometimes need more steps.

❈ Now you try and change these numbers into special numbers —
a) 28 b) 132 c) 273
Now let’s use words in a special way.

N O L E M O N S N O M E L O N

S T E P N O T O N P E T S
Did you notice that it reads the same from both sides — right to
left and left to right?
Now try and use words in a special way.

Special words/numbers which read the same both ways are called palindromes. Help
children to read them from both the ends.

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Calendar Magic

Look at the calendar below.

Let us mark a 3 × 3 box (9 dates) on the calendar and see some magic.

I can quickly find

s m t w th f s the total of these
numbers in the box.
1 2 3 4 5 6 7

8 9 10 11 12 13 14

Won't that
15 16 17 18 19 20 21
take some
time?
22 23 24 25 26 27 28
The total
29 30 31 is 99.

Take the smallest number 3

Add 8 to it +8

= 11

Multiply it by 9 ×9

Total 99 Hey! Just take the middle number

and multiply it by 9. See you can
get the answer even faster.

Now you choose any 3 × 3 box from a calendar and find the total
in the same way. Play this game with your family.

You can see Math-Magic Class III (page 105 -106 ) for other calendar tricks.

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Some more Number Patterns
❈ Take any number. Now multiply it by 2, 3, 4 ............... at every
step. Also add 3 to it at each step. Look at the difference in the
answer. Is it the same at every step?

12 2 + 3 = 27
12 3 + 3 = 39
12 4 + 3 = 51
12 5 + 3 = 63
12 + 3 =
7 + 3 =
+ 3 =
+ =
Now try doing it with some other number and also take a
different number to add at each step .
❈ Look at the numbers below. Look for the pattern. Can you
take it forward?

(9 – 1) ÷ 8 = 1
(98 – 2) ÷ 8 = 12
(987 – 3) ÷ 8 = 123
(9876 – 4) ÷ 8 = ____
(98765 – 5) ÷ 8 = ____
( ________–__ ) ÷ 8 = ____
( __________–__ ) ÷ 8 = ____
Encourage children to read aloud the numbers on the left hand side, even if they can not read
them correctly. Some of the numbers are large. To help children read them, recall the concept
of 1 lakh or 100 thousand.

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Smart Adding What if someone
gives you to add ten
numbers together?

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +10 = 55
Oh! I can find
it quickly. 11+12 + + + + + + + +20 = 155
21 + + + + + + + + +30 =
Smart! How
can you do 31 + + + + + + + + +40 =
that?
41 + + + + + + + + +50 =
I can get the
sum without 51 + + + + + + + + +60 = 555
adding.
61 + + + + + + + + +70 =
❈ Did you notice some pattern in the answers?

Fun with Odd Numbers

Take the first two odd numbers. Now add them, see what you get.
Now, at every step, add the next odd number.
1 + 3 = 4 = 2 2
1 + 3 + 5 = 9 = 3 3
1 + 3 + 5 + 7 = 16 = 4 4
1 + 3 + 5 + 7 + 9 = =
1 + 3 + 5 + 7 + 9 + 11 = =
1 + 3 + 5 + 7 + 9 + 11 + 13 = =
How far can you go on?
When we add the first n odd numbers, we will get the sum as n × n . Children should be left free
to add the numbers.

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Secret Numbers

Banno and Binod were playing a guessing game by writing clues

about a secret number. Each tried to guess the other’s secret
number from the clues.

Can you guess their secret numbers?

It is larger than half of 100

It is more than 6 tens and What is
less than 7 tens my secret
number?
The tens digit is one more
than the ones digit
Together the digits have a
sum of 11
It is smaller than half of 100
It is more than 4 tens and
less than 5 tens

What is The tens digit is two more

my secret than the ones digit
number? Together the digits have a
sum of 6

❈ Write a set of clues for a secret number of your own. Then give
it to a friend to guess your secret number.
Number Surprises
a) Ask your friend — Write down your age. Add 5 to it. Multiply
the sum by 2. Subtract 10 from it. Next divide it by 2. What
do you get?
Is your friend surprised?

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b) Take a number

Double it 2 =

Multiply by 5 5 =

Divide your answer by 10 ÷ 10 =

c)
Take a number

Double it 2 =

Again double it 2 =
Add the number you took
+ =
first to the answer

Now again double it 2 =

Divide by 10 ÷ 10 =

d) Look at this pattern of numbers and take it forward.

1 = 1 × 1

121 = 11 × 11

12321 = 111 × 111

1234321 = ?

❈ Now make your own number surprises.

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 8 Mapping Your Way
Ashi is going to India Gate to see the Republic Day Parade with
the other children of her school. As the children settle down, they
hear something about India Gate on the loudspeaker. “To the
right of the President is the India Gate. This was built in memory of the
Indian soldiers who died in the First World War.”

There are lots of people sitting on both sides of Rajpath, the main
road along which the parade passes. Children are talking about
the buildings they can see around them.

Look Aditi, INDIA

this is
Sir told us that Rashtrapati India Gate.
.
Bhawan faces India Gate. So
that last building on our
right side must be
Rashtrapati Bhawan.

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Here is a photograph taken from a helicopter. You can see
Rajpath — the road which joins India Gate to Rashtrapati
Bhawan. Mark where on Rajpath will Aditi be.
Map 1

Rashtrapati Bhavan

Vijay Chowk

Rafi Marg Rafi Marg

Rajpath
Janpath Janpath

Rajpath
rg
Ma
dhi
Man Singh Rd Gan
tu rba
Kas

Tila
Rd k
in Ma
sa rg
Hus
kir
Za National Stadium
Dr.

Match the map and the photo

1) Have you seen a map of a city? Look at Map 1. Match it with the
photo and find out where India Gate is. Draw it on the map.

2) Some roads are shown in this part of the map. Look for them in
the photo.

3) Name roads that you will cross on your way from Rashtrapati
Bhawan to India Gate.

4) LookfortheNationalStadiuminMap1.Canyouseeitinthephoto?

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The Central Hexagon

If we ‘zoom in’ to look more closely at one part of the map, it looks
like this.
Janpath
Map 2

Ak

Rajpath
ba

Rd
rR

ka
d

ho
As
Man Singh Rd Man Singh Rd
Sh rg
ah
jah i Ma
an India
dh
Rd an
rbaG
stu
Ka
India Gate

Pandara Rd Copernicus Marg

Children's
Park Til
rg a
Ma kM
a in arg
ss
Rd

Hu
kir Pu
Za
ah

ran
Dr
Sh

aQ
er

ila
Sh

Rd
National Stadium

Look at the shape of the yellow area. Have you seen this shape
before? How many sides does it have?
This place is called the Central Hexagon.
Find out from the map
1) If you are walking on Rajpath then after India Gate on which
side would Children’s Park be?
2) Which of these roads make the biggest angle between them?
a) Man Singh Road and Shahjahan Road
b) Ashoka Road and Man Singh Road (the angle away from
India Gate)
c) Janpath and Rajpath
3) Which of the above pairs of roads cut at right angles?

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 Waiting for the Parade
While waiting for the parade, Kancha and some of his friends
wonder where this parade ends. Kancha is carrying a newspaper
in which the route of the parade is written —
Vijay Chowk — Rajpath — India Gate — Tilak Marg — B.S. Zafar
Marg — Subhash Marg — Red Fort.
NORTH
Map 3
The children look at this map
to see the parade route. Red Fort

Chandni Chowk Rd

Marg
sh
ha b
Jama Masjid

Su
Marg
Subhash

Raj Ghat

Yamuna River
Za far Ma rg

Medical College

E AST
W EST

Connaught
Place
Ba ha du r .Sh ah

Ring Rd

Doll's
Museum
I.G. Stadium
Rin
gR

Pragati
d
rg
Ma

Maidan
Rashtrapati
k
Tila

Bhavan
Rafi Marg

Janpath

Vijay
Chowk
National
Rin

Stadium
gR
d
Rd

Mathura Rd

11 Murti
an
jah
ah
Sh

S OUTH Scale: 2 cm = 1 km

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 Hey! See, that is
And the long
India Gate and this is
road between
Rashtrapati Bhawan. Let's guess how
these is Rajpath.
many kilometres
long Rajpath is.
My mother told me it is
2 km from Vijay Chowk On this map, it is about
to India Gate. 4 cm. So 4 cm on the
map is the same as You are right!
2 km on the ground. See, it is written
at the bottom of
this map. Scale:
2 cm = 1 km

Mark the route

1) Trace the route of the
parade in Map 3 and
mark India Gate and
Rajpath.

2) Look at the map carefully and find out:

a) Which of these is the longest road?
● B.S. Zafar Marg ● Subhash Marg ● Tilak Marg
b) If Rubia is coming from Jama Masjid to join the parade,
guess about how far she will have to walk.
c) The total route of the parade is about how long?
● 3 km ● 16 km ● 25 km ● 8 km
As the parade passes by, they see some children coming on an
elephant. These children have got bravery awards. They also
enjoy the colourful dances and aerobics by school children.
They want to follow the parade to Red Fort. Gappu has seen
Red Fort before and tells them about his trip.
Children should understand the need for a scale. We need to discuss that when we show a
big area on paper, we have to reduce it by a fixed ratio everywhere, so that the relative
distances and positions remain the same.

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Trip to Red Fort
"When we reached Red Fort, there was a
long queue for tickets. The main entrance
is called Lahori Gate. After entering it,
we turned left into a long corridor with
little shops on both sides. This is called
Meena Bazar. I bought some lovely
bangles from there for my sister".

Meena Bazar Lahori Gate

E AST
Map 4
Yamuna Ring Road
River Hammam
Aram Gah
Moti Masjid Rang Mahal

Moti Mahal

Diwan-e-Khaas

Diwan-e-Aam
N ORTH

S OUTH

Red Fort Naqqar Khana

Meena Bazar

Lahori Gate

Subhash Marg
W EST
Scale: 1 cm = 100 m

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 "You can go straight through Naqqar
Naqqar Khana —
where drums were Khana and reach Diwan-e-Aam. This
beaten to shout out the
king's messages is where the king used to meet the
common people.

Walking straight from Diwan-

e-Aam, we saw Rang Mahal. It
is a beautiful building! There
were three more buildings on
our left side. Look for these on
the map.

We walked left from Rang Mahal. Diwan-e-Khaas was

where the king used to meet his ministers and other
important(khaas)people."

From the right – Rang Mahal,

Aaram Gah and Diwan-e-Khas

Inside Rang Mahal Inside Diwan-e-Khaas

Find out from Map 4

a) Which of these is nearer to river Yamuna? — the Diwan-e-Aam
or the Diwan-e-Khaas?
b) Between which two buildings is Aaram Gah?
c) Which buildings do you pass while going from Rang Mahal to
the Hammam?
d) Which building on this map is farthest from Meena Bazar?
e) About how far is Lahori Gate from Diwan-e-Khaas?

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Make It Bigger, Make It Smaller.

Here are some pictures drawn on a

1 cm square grid. Try making the same
pictures on a 2 cm
grid and also on a
1 cm grid. One picture
2
isalreadydone. 1 cm
1
2 cm grid
grid

2 cm
grid

The side of the square was made two times bigger. Does its area
also become two times bigger?

Enlarging or reducing of pictures and maps can be done on the classroom floor, the mud ground
etc. This should be related to the use of scale in maps, which keeps the shape the same.

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Now try this —
This is a part of the parade-route Map 3.
1) Can you see which part of the route-
map it is?
2) Now try to make it bigger in this 2 cm 1 cm grid
grid. Remember that the ‘shape’ of
the map should not change.
3) If the parade route map is smaller,
and the distance between India Gate
and Vijay Chowk becomes 2 cm,
what would be its scale?
● 1 cm on map = 1 km on ground
1
● 2 cm on map = 1 km on ground
● 2 cm on map = 1 km on ground
2 cm grid
Dancers from Different States
The children saw many floats (jhankis) and dancers in the parade.

Dancers from
Karnataka
were the best.
I liked the Tripura dance. From
All these people have Tripura and Sikkim they have to
to travel so much to travel far, but Haryana and
come to Delhi! Uttarakhand are closer to Delhi.

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Look at the map of India below and find the states these children
are talking about. Answer the questions:

This map is made on an approximate scale, so that children

can easily estimate distance, area and directions

1) The Karnataka team starts from Bangalore and moves in the

north direction. Which states does it cross to reach Delhi?
As the children are being introduced to directions for the first time, many activities need to be done
to use terms like 'towards north', 'southwards', 'in the east direction', 'to the west of Madhya
Pradesh' etc. One can draw maps on the floor and get children themselves to stand on the map and
say things like Venkat is to the south of Shanti', 'Maharashtra is to the east of Gujarat' etc.

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2) Jammu and Kashmir is to the north of Delhi so the team
from there travels towards south to reach Delhi. Which
states does it cross?
3) Nonu lives in Gujarat. Nonu's friend Javed lives in West Bengal.
Nonu wants to visit his friend. In which direction will he travel?
a) Towards west
b) Towards east
c) Towards south
d) Towards north
4) Is there any state which is to the north of Jammu and Kashmir?
5) Is there any state which is to the west of Gujarat?
6) If 1 cm on the map shows 200 km on the ground, use this
scale to find out:
A) About how far is Delhi from Jaipur?
a) 50 km b) 500 km c) 250 km
B) Estimate, how far is Jaipur from Bhopal?
On the map = _______ cm.
On the ground = _______ km.
7) Look at the map and tell:
a) Which state is surrounded by four other states?
b) Which state has the largest area? If its name is not in the
map, find it from your teacher or parents.
Explain how you got your answer.
c) Which state is about 8 times bigger in area than Sikkim?
● Uttar Pradesh
● Tripura
● Maharashtra
● Himachal Pradesh
d) About how many times of Punjab is the area of Rajasthan?
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The Sea
Bala is standing on the sea-coast and looking at the vast sea. The
sea looks endless .

Have you seen the sea? In the picture where is the sea? Now look for
the sea in the map of India. What colour is used to show the sea?

❈ Mark those states which have the sea on one side.

❈ Name one state which does not have the sea on any side.

Find out
Look for different maps. Compare the different scales used in a
local area map, a map of India and a world map etc.
Lines between the States
Sabu is confused about the lines shown between the states.
I travelled from Delhi to Haryana, but
I never saw any lines on the ground.
How do we see them on the map?

No, there are no lines painted

on the ground! The map shows
us where one state ends and
the next begins. O yes! We paid the toll-tax at
the border. I saw a big board
saying — Welcome to Haryana.

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 Distances between Towns Map 6
Bhelpur
These are five towns. Find out:
Dhoklabad
1) How many cm away is Idlipur Chholaghat

from Barfinagar on the map?

2) How many kilometres will you Barfinaga r
Idlipur
have to travel if you go from
Idlipur to Barfinagar? Scale: 1 cm on the paper = 10 km
3) There is a place called Thukpagram midway between Idlipur
and Barfinagar. Mark it with a 'T'.
4) A town called Jalebipur is 35 kms away from both Chholaghat
and Dhoklabad. Where do you think it can be? Mark 'J' for it.
5) Measure the length of the route between Bhelpur and
Chholaghat. (You can use a thread)
Ashi's School
Ashi’s school looks like this from the top.
Use the squares to find out: Map 7

Hall
Garden

Assembly Ground

Window
Scale: 2 cm = 5 m

III A V II IX VII VI A

Door

III B IV I X VIII VI B

Playground
Office Main Gate

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1) How many times bigger is the area of the Assembly ground than
that of the office?
2) How much is the length and width of each classroom?
a) length 5 m, width 4 m b) length 2 m, width 1 m
c) length 12 m, width 10 m d) length 5 m, width 5 m
3) All the classrooms in Ashi’s school look like this.
Blackboard Map 8

Window Door
Benches

Notice board
Window
Almirah

Display board

Look carefully and answer.

a) Which of these is exactly opposite to the blackboard?
❈ Almirah, windows, notice board, display board
b) Now look at the school-map again. Guess and mark where
would these be:
❈ Blackboard in III A and VII
❈ Almirah in IV and X
❈ Notice board in V and VI B
❈ Last seat of middle-row in II
❈ Display board in I.
c) Can a child sitting in III A see the playground?

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 9 Boxes and Sketches
Sweet Box
Ramya went to buy sweets. The shopkeeper took a paper cut-out
and quickly made a lovely pink box for the sweets!

❈ Look at the photo and make your own

box. Use the cut-out on page 201. How
fast can you fold it?
After coming home Ramya unfolded the
box. She removed the extra flaps so the
cut-out looked like this.

This shape makes

a box. Let me see
what other shapes
will make a box.

❈ She made four more shapes. Each is to be folded along the

dotted lines. You have to find out which of these can be made
into a box.

a) c)

b) d)

This chapter focuses on visualisation of 3-dimensional shapes and how they can be
represented on paper (in 2 dimensions). The representation used here are nets (like the
ones above), layout plans for a house, and perspective drawings.

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Shapes that Fold into a Cube
A. Buddha wants to make a paper
cube using a squared sheet. He
knows that all the faces of a
How many faces
cube are squares.
does the cube
have? _____
He draws two different shapes.

❈ Will both these shapes fold into a cube?

❈ Draw at least one more shape which can fold into a cube.
❈ What will be the area of each face of the cube?
❈ Draw one shape which will not fold into a cube.
❈ Look around and discuss which things around you look like
a cube. List a few.
Shapes for an Open Box
Remember the puzzles with five squares in chapter 3? You saw
12 different shapes made with five squares (page 46).
If you cut those shapes and fold them, some of those will fold into
an open box (box without a top).

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 I can make
open boxes
with both
these.
But with these
I cannot make
open boxes.

❈ Find out which of the other 8 shapes (on page 46) can be folded
to make an open box.
❈ Draw more shapes which will not fold to make an open box.
Boxes and Boxes
All boxes are not cubes. Here are some different kinds of boxes.
Match the shape on the left with a box into which it will fold.

Making mental images of shapes is an important mathematical ability. Children will need
many exercises to visualise the net of a box, to think of how it looks when flattened, and also
to check which nets (like those on page 126) do not make a box.

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Floor Maps
Window
For making a house a floor map
Window is first made. Have you ever
seen a floor map? Here is a floor
map of Vibha’s house. It shows
where the windows and the
Door
Window doors are in the house.
Window

❈ Which is the front side of her house? How many windows are
there on the front side?

From the floor map we cannot

make out what her house
really looks like or how high
the windows are. So we look a)
for a special way of drawing the
house which is deep — to show
the length, width and height.

Here are four deep drawings

of houses.
❈ Which one is Vibha’s house?
b)

d) c)
❈ Why do the other three deep drawings not match the floor
map? Discuss.
A 3-dimensional perspective drawing has been called a 'deep drawing' so that children get a
sense of the need to represent depth. They should be able to see the difference between
deep drawings and layout plans.

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Practice time
1. Look at this floor map of a house. Make doors and windows on
the deep drawing of this house.

Window Window

Door
Window Window

❈ Are there any windows you couldn’t show on the deep

drawing? Circle them on the floor map.
2. Try to make a floor map of your own house.

A Deep Drawing of a Cube

Soumitro and his friends made deep drawings of a cube.
These are their drawings.

a) b)

c) d) e) f) g)
❈ Which of the drawings look correct to you? Discuss.
❈ Can you add some lines to make drawing f ) into a deep
drawing of the cube?

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Puzzle
This cut-out is folded to make a cube.

Which of these are the correct deep drawings of that cube?

a) b) c) d) e)

A Simple Way to Draw a Cube

Chanda wants to make a deep drawing of this cube.

She draws the cube like this.

I drew two squares like
this to show the front I joined the corners of the
face and the back face. squares like this to make the
deep drawing of the box.

❈ In the same way make a deep

drawing of a box which looks
like this.

The 2D representation of 3D objects is a matter of convention and is learnt by children

through experience. Here the conventional way of drawing the cube is given.

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Matchbox Play
Navin, Bhaskar and Pratigya made this bridge using matchboxes.

Navin and Pratigya made drawings of the bridge.

The bridge looks

like this to me
from where I am
standing.

From your drawing I can

make out how long and
The bridge looks how high the bridge is.
like this to me. My But I cannot make out
drawing shows how how wide it is.
high our bridge is
and how wide it is.

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❈ If you look at the bridge from the top, how will it look? Choose
the right drawing below:
a)

b)

❈ Look at the photo and try to make a deep drawing of this bridge.

Practice time
1) Make drawings to show how this bridge will look
❈ From the top
❈ From the front

❈ From the side

2) Make a matchbox model which looks like this.

From the top
From the front From the side

❈ Also make a deep drawing of the model in your notebook.

3) How many cubes are needed to make

this interesting model?
❈ Here are some drawings of the model.
Mark the correct top view drawing
with 'T' and the correct side view
drawing with 'S'.

a) b) c) d)

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 11 Area and its Boundary

Whose Slice is Bigger?

Parth and Gini bought aam paapad (dried
mango slice) from a shop.
Their pieces looked like these.
6 cm

Both could not make out whose

5 cm piece was bigger.
❈ Suggest some ways to find out
whose piece is bigger. Discuss.

A friend of Parth and Gini showed

Piece A one way, using small squares.

11 cm

3 cm

Piece B

The length of piece A is 6 cm.

So 6 squares of side 1 cm can be arranged along its length.

The width of piece A is 5 cm.

So 5 squares can be arranged along its width.
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❈ Altogether how many squares can be arranged on it? ________

❈ So the area of piece A = ________ square cm

It's silly to
count them all!
Just multiply!

Piece A

❈ In the same way find the area of piece B.

❈ Who had the bigger piece? How much bigger?

Cover with Stamps

This stamp has an area of 4 square cm. Guess how many such
stamps will cover this big rectangle.

25 Hkkjr India

Encourage children to first discuss different strategies for comparing the area of things by
using different tokens, stamps, etc. In Class IV they have compared irregular shapes by
counting squares. In the case of rectangles they can measure the sides to see how many
squares of 1 cm side will fit in the whole shape.

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Check your guess
a) Measure the yellow rectangle. It is ________ cm long.

b) How many stamps can be placed along its length? ________

c) How wide is the rectangle? ________ cm

d) How many stamps can be placed along its width? ________

e) How many stamps are needed to cover the rectangle? ________

f) How close was your earlier guess? Discuss.

g) What is the area of the rectangle? ________ square cm

h) What is the perimeter of the rectangle? ________ cm

Practice time
a) Arbaz plans to tile his kitchen floor with
green square tiles. Each side of the tile is 10
cm. His kitchen is 220 cm in length and 180
cm wide. How many tiles will he need?

b) The fencing of a square garden is 20 m in length. How long is

one side of the garden?

c) A thin wire 20 centimetres long is formed into a

rectangle. If the width of this rectangle is
4 centimetres, what is its length?

This ‘Guess and check’ activity can be done in the class by making use of other things
present. For example: how many postcards can be placed on the top of the mathematics
book, how many charts will cover the classroom walls, etc? Children can be asked to check
their guesses by tiling things wherever possible. Once they are able to make close guesses,
this work can be further extended by asking them to guess the area in terms of square cm.

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d) A square carrom board has a perimeter of 320 cm.
How much is its area?
e) How many tiles like the triangle given here will fit
in the white design?
This triangle Area of design = ________ square cm
is half of the
cm square

❈ Make your own designs of area 4 and 6 square cm.

f ) Sanya, Aarushi, Manav and Kabir made greeting

cards. Complete the table for their cards:

Whose card Length Width Perimeter Area

Sanya 10 cm 8 cm
Manav 11 cm 44 cm
Aarushi 8cm 80 square cm
Kabir 40 cm 100 square cm

My Belt is Longest!
Take a thick paper sheet of length 14 cm and width 9 cm. You
can also use an old postcard.
❈ What is its area? What is its perimeter?
❈ Now cut strips of equal sizes out of it.

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Using tape join the strips, end to
end, to make a belt.
❈ How long is your belt?_____
❈ What is its perimeter _____
❈ Whose belt is the longest in the
class? _____
Discuss
❈ Why did some of your friends get longer belts than others?
❈ Is the area of your belt the same as the area of the postcard?
Why or why not?
❈ What will you do to get a longer belt next time?

This belt is for

the elephant.
ha
pass throug
Look! I can
o s t c a r d . I made a loop
p ips.
ting the str
without cut

Puzzle: Pass through a Postcard

Can you think of how to cut a postcard
so that you can pass through it? (See
photo.) If you have tried hard enough
and still not got it… look for the answer
somewhere ahead.

The aim of the belt activity is to understand that things with the same area can take different
forms and also have very different perimeters. While measuring sides, lengths in mm can be
rounded off for this activity.

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People People Everywhere With four Math-Magic
books in a line you can
A) You can play this game in a ground. get the length of around
one metre 9 cm.
Make two squares of one square metre each.
Divide your class in two teams. Ready to play!

Try these in your teams —

❈ How many of you can sit in one square
metre? _______
❈ How many of you can stand in it? _______

❈ Which team could make more children

stand in their square? How many? _______
❈ Which team could make more children sit
in their square? How many?

B) Measure the length of the floor of your classroom

in metres. Also measure the width.
❈ What is the area of the floor of your classroom
in square metres? _______
❈ Howmanychildrenarethereinyourclass?______

❈ So how many children can sit in one square

metre? _______
❈ If you want to move around easily then how
many children do you think should be there
in one square metre? _______

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 Can you imagine how big a
square of side 1 km is! It has
In West Bengal there are
an area of ______ square km.
about 900 people living in
Guess how many people can live
a square km.
on that.
But in Arunachal Pradesh
it feels very lonely! There
are less than 15 people
living in a square km!

Share the Land

Nasreena is a farmer who wants to divide her land equally among
her three children — Chumki, Jhumri and Imran. She wants to
divide the land so that each piece of land has one tree. Her land
looks like this.

❈ Can you divide the land equally? Show how you will divide it.
Remember each person has to get a tree. Colour each person’s
piece of land differently.
Children are not expected to do conversion of sq m into sq km or vice-versa. The aim of
exercise B is to develop a sense of how big or small the units of sq m and sq km are.

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❈ If each square on this page is equal to 1 square
metre of land, how much land will each of her
children get? ________ square m
Chumki, Jhumri and Imran need wire to make
a fence.
❈ Who will need the longest wire for fencing? _________

❈ How much wire in all will the three need? ___________

Practice time
A. Look at the table. If you were to write the area of each of these
which column would you choose? Make a ( ✔).

Square Square Square

cm metre km
Handkerchief
Sari
Page of your book
School land
Total land of a city
Door of your classroom
Chair seat
Blackboard
Indian flag
Land over which a river flows

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 B. Draw a square of 9 square cm. Write A on it.
Draw another square with double the side.
Write B on it.
Answer these —
1. The perimeter of square A is __________ cm.
2. The side of square B is __________ cm.
3. The area of square B is __________ square cm.
4. The area of square B is __________ times the
area of square A.
5. The perimeter of square B is __________ cm.
6. The perimeter of square B is __________ times the perimeter of
square A.

Answer — Pass Through a Postcard (page 152)

1. 2.

Make lines on a postcard like this.

Cut the postcard only on the lines.
3.

So, can you pass through it!

❈ You know the area of the loop,
don’t you? It is ___________.

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Thread Play

Take a 15 cm long thread. Make different shapes by joining its

ends on this sheet.

A) Which shape has the biggest area? How much? _______

What is the perimeter of this shape? _______

B) Which shape has the smallest area? How much? _______

What is the perimeter of this shape? ________

Also make a triangle, a square, a rectangle and a circle. Find

which shape has biggest area and which has the smallest.

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Save the Birds

There are two beautiful lakes near a village. People come for
boating and picnics in both the lakes. The village Panchayat is
worried that with the noise of the boats the birds will stop
coming. The Panchayat wants motor boats in only one lake. The
other lake will be saved for the birds to make their nests.

A B

1 cm on this drawing = 1 km on the ground

a) How many cm is the length of the boundary of lake A in the

drawing? _____________ (use thread to find out)

b) What is the length of the boundary of lake B in the drawing?

c) How many kilometres long is the actual boundary of lake A ?

d) How many kilometres long is the actual boundary of lake B?

e) A longer boundary around the lake will help more birds to lay
their eggs. So which lake should be kept for birds? Which lake
should be used for boats?
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 f) Find the area of lake B on the drawing in square cm. What is its
actual area in square km?
King's Story

The King was very happy with carpenters Cheggu and Anar. They
had made a very big and beautiful bed for him. So as gifts the
king wanted to give some land to Cheggu, and some gold to Anar.

take as
Cheggu, s what Cheggu was happy. He took 100
nd a
much la thin 100 metres of wire and tried to make
i
comes w f wire.
s o different rectangles.
meter
He made a 10 m × 40 m rectangle.
Its area was 400 square metres.
So he next made a 30 m × 20 m
rectangle.
❈ What is its area? Is it more than
the first rectangle?

❈ What other rectangles can he make

Ah! I want this piece of
with 100 metres of wire? Discuss land. It covers an area
which of these rectangles will have the of 800 square metres.
biggest area.
Cheggu’s wife asked him to make a circle
with the wire. She knew it had an area of
800 square metres.
❈ Why did Cheggu not choose a
rectangle? Explain.

Ok. Cheggu has taken 800 square metres

of land. Anar! Now I will give you as much
gold wire which can make a boundary for
land with area 800 square metres.

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So Anar also tried many different ways to make a
boundary for 800 square metres of land.
❈ He made rectangles A, B and C of different sizes.
Find out the length of the boundary of each. How
much gold wire will he get for these rectangles?

A 40 m × 20 m
Gold wire for A = _________ metres

B 80 m × 10 m Gold wire for B = _________ metres

C 800 m × 1 m

Gold wire for C = ____________ metres

But then Anar made an even longer rectangle.... See how long!

D
8000 m × 0.1 m
So he will get ____________ metres of gold wire!!

Now do you understand why the

ow
Gosh! H o king fainted!!!
ive s
can I g
old?
much g
Can you make a rectangle with a
still longer boundary? I made a
rectangle 1 cm wide and 80000 m
long. Imagine how long that
boundary will be!!! With that much
gold wire I can become the king!

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 12 Smart Charts
Chi-Chi, Meow-Meow

Yamini did a project ‘Animals and Birds’. She asked each child of
her class about one favourite pet animal.

She used tally marks to record each answer. For example

if someone said ‘cat’ she put one line in front of ‘cats’. When someone
said ‘cat’ again,sheaddedaline.So meanstwocatsand means 5
cats. In all 24 children said ‘cat’ was their favourite animal. Help
Yamini complete the table.

Animal Tally Marks Number

24
Cats

Dogs

Rabbits

Cows

Parrots

Goats

Squirrel

✤ Look at the tally marks and write the number for each animal
in the table. How many children in all did Yamini talk to?
' Which is the most favourite pet animal in this table?
✤
'
✤ Which pet will you like to have? What will you name it? Which
other animals can be kept at home? Discuss.

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Making Tally Marks on the Road
Sumita stood on the road for half an
hour and counted the number of
vehicles passing by. She made a tally
mark for each vehicle. This helped her
in counting quickly the total number
of vehicles in each group.

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 Tally Marks Number

Cycle

Car

Auto rickshaw

Bus

Cycle rickshaw

Truck

✤ Write the number of each vehicle in the table.

✤ How many vehicles in all did Sumita see on the road in half
an hour?

✤ Auto rickshaws are thrice the number of trucks — true/false?

✤ Make tally marks for 7 more buses, and 2 more trucks.

Try yourself
✤ Take a round in your colony. Find out how many types of
trees you can see there. Do you know their names? You can
make drawings. Use tally marks to note the number of
different trees.

Children should be encouraged to use tally marks to simultaneously record data of a variety of
things with larger numbers.

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Helping Hands
In the EVS period, the teacher
asked children whether they
help their parents at home.
There were different answers.
Children named the work in
which they help their parents
the most. The teacher collected
their answers and made a table.

Help most in house work Number of children

Going to the market 47

Washing utensils 15

Washing clothes 3
Making, serving food 25

Cleaning the house 10

Total children who said

they help their parents

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Now you can fill the chapati chart to show the numbers given in the
table.
1) Look and find out
Children who help in making or serving food are

a) One-third of the total children

b) Half of the total children
c) One-fourth of the total children
ng
ani
Cle house
the 10
2) Practice time: After school
Ask 10 of your friends about what they like to do most after school.
What they like to do Number of children
after school
Watching TV

Playing football

Reading story books

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Ad Mad!!
Ragini loves to watch cartoons on television. One day she
thought of counting the number of ads during the breaks. She
found that in each break there were 14 advertisements. In 10 of
those ads there were children as actors.

✤ Why do you think that children are used in so many ads?

✤ Use tally marks to count the number of ads during a short
break in a programme.
Were there ads during the news programme?

Try yourself
? ✤ Next time when you watch your favourite
TV programme, count the number of
' advertisements during each break. Use
tally marks. Put a dot below the tally
when you find children in any
advertisement.
✤ Compare with your friends. Do you get
' different answers?

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Hot and Cold

Have you seen the weather report on TV or in a newspaper? These

are two bar charts. These show the highest temperature (in degrees
Celsius) in four cities, on two different days. The cities are Delhi,
Shimla, Bangalore and Jaisalmer.

40°C 38° 40°C

33°
30°C 28° 30°C
23° 24° 25°
22°
20°C 20°C

Bangalore
Bangalore

Jaisalmer
Jaisalmer
10°

Shimla
10°C 10°C

Shimla
Delhi
Delhi

1 June 1 December

Find out from the bar chart —

✤ Which city is the hottest on 1 June?
✤ Which city is the coldest on
1 December?
✤ Which city shows little change in
temperature on the two days —
1 June and 1 December.

Try yourself
On any one day, choose any three cities and record their
temperature from the TV or newspaper.

✤ Make a bar chart in your notebook and ask your friends a few
questions about it. See if they understand your chart!

Encourage children to look at the map of India to locate different cities. They can try to relate
the temperature variations in a city to get an idea of the climate there.

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Rabbits in Australia
Earlier there were no rabbits in Australia. Rabbits were brought
to Australia around the year 1780. At that time there were no
animals in Australia which ate rabbits. So the rabbits began to
multiply at a very fast rate. Imagine what they did to the crops!

The table shows how rabbits grew every year.

Time Number of rabbits

Start 10

1 year 18

2 year 32

3 year 58

4 year 108

5 year

6 year

1) After each year the number of rabbits was —

a) a little less than double the number of rabbits in the last year.

b) double the number in the last year.

c) 8 more than the number in the last year.

d) more than double the number of rabbits in the last year.

2) At the end of year 6, the number of rabbits was close to
400 600 800

3) After which year did the number of rabbits cross 1000?

More such examples should be done in class. It is important for children to get a sense of
approximation.

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 Family Tree
Madhav went to a wedding along
with his parents. He met many
relatives there. But he didn’t know everyone. He met his
mother’s grandfather, but found that her grandmother is not
alive. He also found that her Dadi’s mother (grandmother’s
mother) is still alive, and is more than a hundred years old.

Madhav got confused. He couldn’t imagine his mother’s

grandmother’s mother! So, Madhav’s mother made a family
tree for him —

Shobna's Shobna's Shobna's Shobna's Shobna's Great great

Shobna's Shobna's grand parents
Dada's Dada's Dadi's Nana's Nana's Nani's Nani's
Father Mother Father Father Mother Father Mother
Shobna's V Generation
Dadi's Mother

Great
Shobna's grand parents
Dadi
Shobna's Shobna's
Shobna's IV Generation
Dada Nani
Nana

Grand parents

Shobna's Shobna's
Father III Generation
Mother

Madhav's
Parents

Madhav's
Father Madhav's Mother
II Generation
Shobna

Madhav

Madhav
I Generation

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 Madhav’s mother helped him understand her family with
the help of this drawing. You can also find out about your
older generations using such a family tree.

Answer these questions:

1) How many grand parents in all does Shobna have?

2) How many great, great grand parents in all does Madhav have?

3) How many elders will be in the VII generation of his family?

4) If he takes his family tree forward in which generation will he

find 128 elders?

Day Length of the plant

Growth Chart of a Plant (in cm)

Amit sowed a few seeds of moong dal 0 0

in the ground. The height of the plant 4 1.4
grew to 1.4 cm in the first four days.
8 5.3
After that it started growing faster.
12 9.5
Amit measured the height of the 16 10.2
plant after every four days and put a 20 10.9
dot on the chart. For example if you
look at the dot marked on the fourth
day, you can see on the left side scale
that it is 1.4 cm high.
Now look at the height of each dot in
cm and check from the table if he has
marked the dots correctly.

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 12

11

10
Length of plant (in cm)

0 4 8 12 16 20
Days
Find out from the growth chart

a) Between which days did the length of the plant change the most?

i) 0-4 ii) 4-8 iii) 8-12 iv) 12-16 v) 16-20

b) What could be the length of this plant on the 14th day? Guess.

i) 8.7 cm ii) 9.9 cm iii) 10.2 cm iv) 10.5 cm

c) Will the plant keep growing all the time? What will be its length
on the 100th day? Make a guess!

There should be some discussion on the last question. Children should be encouraged to
observe growth patterns of many other plants and animals.

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