Chapter

Equal Groups
9

Animal Jumps

Fill in the blank spaces with the appropriate numbers.

Find how many jumps the animal needs to take to reach
its food.
1. The frog jumps 3 steps at time. Which numbers will the frog touch?
Will it touch 67?
These numbers are multiples of 3.

6 12
3 18
0

2. The squirrel jumps 4 steps at a time. Which numbers will the

squirrel touch? How many times should the squirrel jump to reach
60?
These numbers are multiples of ___

0
4
8
12

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 3. The rabbit jumps 6 steps at a time.
Which numbers will the rabbit touch?
What is the smallest 3-digit number on which the rabbit will land?
How many times did the rabbit jump to reach this number?

These numbers are multiples of ___

18
12
6
0

4. The kangaroo jumps 8 steps at a time. Which numbers will the

kangaroo touch?

These numbers are multiples of ___

0
8
16

Are there numbers that both the rabbit and the kangaroo will
touch?
5. To reach 48, how many times did the rabbit jump? _______

How many times did the Kangaroo jump to reach the same number?
_______
What did you observe? Share your thoughts.
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 6. To reach 60, how many times did the frog jump? _______
How many times did the rabbit jump to reach the same number?
__________
What do you observe? Share your thoughts.

Common Multiples

1. Which numbers do both the frog and the squirrel touch? A few
common multiples of 3 and 4 are _____________________________.

2. Which numbers do both the rabbit and the kangaroo touch?

A few common multiples of 6 and 8 are ________________________.

7. If the cat and the rat land on the same

number, the cat will catch the rat.
The cat is now on 6 and the rat on 12.
When the cat jumps 3 steps forward,
the rat jumps 2 steps forward. Will
the cat catch the rat? If yes, at which
number?

8. Find multiplication and division sentences below.

Shade the sentences.
How many can you 3 4 2 7 4 9 8 2
find?
4 2 10 20 5 2 2 4
Two examples are 12 8 0 6 4 8 8 1
done for you.
3 2 6 2 2 6 16 2
2 3 6 18 6 5 3 1
10 3 4 1 12 2 7 14
2 0 2 2 6 10 7 2
20 5 8 2 2 5 10 2

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 Gulabo’s Garden

1. Gulabo’s garden has lily flowers. Each lily flower has 3 petals. How
many petals are there in 12 flowers? Show how you found your
answer.

Gulabo will have 12 × 3 petals.

Petals in 10 lilies 10 × 3 petals = 30 petals
Petals in 2 lilies __________________
Petals in 12 lilies _________________

No. of groups (Multiplier)

12 × 3 = 36 (Product)
Group size (Multiplicand)

Multiplication statement

2. In a hibiscus flower there are 5 petals. Gulabo counted all the

petals and found them to be 80. How many flowers did she have?

Gulabo has 80 ÷ 5 flowers.

5 petals is 1 flower.
10 petals are 2 flowers.
50 petals are 10 flowers.
Then, 80 petals are _______ flowers.

Note for Teachers: In this chapter, the focus is on multiplying 1-, 2-, and 3-digit
numbers by 1-digit numbers, with group sizes less than 10. Children should be
encouraged to break down the ‘multiplier (no. of groups)’ into multiples of 10 to
simplify calculations. They can also use strategies like doubling and halving.

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 3. Gulabo plants some marigold saplings in a box as shown in the
picture.
There are ______ saplings in each row.
There are ______ rows.
How many saplings has she planted?
How did you calculate it?
Mathematical Statement _________________
4. “Dailyfresh” supermarket has kept boxes of strawberries in a big
tray.
How many boxes of strawberries does the supermarket have?
Show how you found them.

There are _______ columns of strawberry boxes.

There are _______ boxes in each column.
There are _______ boxes in all.
Mathematical Statement _________________

5. Radha runs a bakery shop. She bakes 18 cupcakes in one tray of

the size shown below.
a) Complete arranging the cupcakes in the two trays given below.

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 b) She can use two such trays in her oven at a time. How many
cupcakes can she make in one attempt? _______
c) Today she has received a special order. She has made 108
cupcakes. How many trays has she baked?
d) She has another square baking tray. She can bake 36 mini
cupcakes in such a tray. Complete the arrangement below.

Number of columns: _______

Number of cupcakes in each column: _______
Multiplication statement _______

Find different ways of arranging the following numbers of cupcakes in

rows and columns in your notebook.
36, 8, 12, and 24

The Doubling Magic

Magician Anvi came one day,

To Gulabo’s house, ready to play.
From her coat, with a grand display,

Note for Teachers: Encourage learners to identify different ways of finding the
answers. Children can skip count, count in rows and columns and think in terms
of equal groups. The idea is to make children notice arrays as a way of representing
multiplication.

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 She pulled out 23 flowers, bright and gay! Abracadabra!!

She smiled and said, "Now watch and see,

How many flowers will there be?"

How many flowers are there now? ______________

What magic did
Anvi do?

Flowers 23 10 51 95 150 199 425 500

before magic
Flowers after 46 222 410 500
magic

a) Double of 32 =_____ b) Double of 14 =_____ c) Double of 26 =_____

d) Double of 17 =_____ e) Double of 39 =_____ f) Double of 45 =_____

1. Guess what will be the ones digit of the following

numbers when doubled.
Write the ones digit in the space provided.
a) 28 ______ b) 56 ______ c) 45 ______ d) 17 ______

2. Give examples of numbers that when doubled give the following

digits in the ones place.
Can we get 3,
a) 0 _______ b) 2 _______ c) 4 _______ 5, 7, 9 as the
d) 6 _______ e) 8 _______ ones digit after
doubling?
What do we notice about the numbers that we get after doubling?
Even or Odd?

Note for Teachers: Encourage children to use Diene’s blocks or a ganit mala
to double or half, especially for big numbers. Doubling and halving are useful
strategies to include when teaching multiplication and division. Teacher can
systematically change the numbers to include different digits in the ones place.

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 Fill each square in the chart by multiplying the row number by
the column number.

1 2 3 4 5 6 7 8 9 10

10

What do you notice about the numbers shaded in green? Why is this
happening?
1. Share the patterns that you notice in the table.
2. Are the numbers in row 7 the same as the numbers in column
7? In general, are the numbers in a given row the same as the
numbers in the corresponding column? Why does this happen?
3. Is there a row where all answers (products) are even numbers?
Which rows have this property?
4. Is there a row having only odd numbers as products?
5. Are there rows that have both even and odd numbers? What do
you notice? Why is it so?
6. Are there more even numbers in the chart or odd numbers? How
do you know?

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 7. Colour the common multiples of the following numbers. Use
different colours for each item.

a) 2 and 3 b) 4 and 8 c) 7 and 9

Share your observations regarding the numbers that are common
multiples in each case.
8. Observe the pattern in the ones digits of the products in row 5?
Observe the ones digit of the products in other rows also. What
patterns do you notice?
9. Here is row 8 of the chart: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
The ones digit of the products are: 8, 6, 4, 2, 0, 8, 6, 4, 2, 0
Do you see a repeating pattern here?
Guess the ones digit of the following products. Verify your answer
by multiplying. Write the digit in the space given.

11 × 8 _____ 12 × 8 _____ 13 × 8 _____

10. In row 8 of the chart, there is no number whose ones digit is 1.

What other digits do not appear as the ones digit?
11. Is there a row in which all the digits from 0 to 9 appear as the ones
digit? Which rows have this property?
12. It can be seen in row 8 that 0 appears as the ones digit two times.

× 8 gives 0 as the ones digit.

What numbers can go in the box? Give 5 examples of such numbers.

13. Is there a row in which 0 appears as the ones digit only once?
Which rows have this property?
14. What do you notice about the answers for Questions 11 and 13?
Share in the grade .

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 Multiples of Tens

1. Let us count the number of wheels in tricycles.

Number of wheels in 10 tricycles with 3 wheels in each is 10 × 3

wheels = ______ wheels.
Number of wheels in 10 more tricycles with 3 wheels in each is
10 × 3 wheels = _______ wheels.
Number of wheels in 20 tricycles with 3 wheels in each is 20 × 3
wheels = ______ + _____ = ______ wheels.
10 × 3 = _____
20 × 3 = _____
2. Let us count the number of wheels in cars.

Number of wheels in 10 cars with 4 wheels in each is 10 × 4 wheels

= _______ wheels.
Number of wheels in 30 cars with 4 wheels in each is 30 × 4 wheels
= _____ + _____ + _____ = _____ wheels.
10 × 4 = _____ What happens
30 × 4 = _____ when the number
of groups is a
Solve the following in a similar way. Share how multiple of 10?
you found the answers.
a) 10 × 6 = ____ c) 10 × 8 = ____
b) 40 × 6 = ____ d) 60 × 8 = ____ e) 6 × 8 = ____ g) 4 × 6 = ____
f) 60 × 8 = ____ h) 40 × 8 = ____

Note for Teachers: Encourage children to identify the relationship between

products like above. Ten-times is a good way of articulating this relationship
between products 6 × 8 and 60 × 8. Three-times could be a way of describing the
relationship between 10 × 4 and 30 × 4.

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 Multiplying Using 10s
1. Radha is packing cupcakes in boxes of 4. She has packed 18
boxes. How many cupcakes are there in the packed boxes?

18 boxes have 4 cupcakes in each.

So, there are 18 × 4 cupcakes.
10
Boxes
10 boxes with 4 cupcakes in each contain
10 × 4 cupcakes = ______ cupcakes.

8 boxes with 4 cupcakes in each contain

8 8 × 4 cupcakes = _________ cupcakes.
Boxes
18 boxes with 4 cupcakes in each
contain ____ + ____ cupcakes
= _________ cupcakes.

× 4
10 10 × 4 = 40
8 8 × 4 = 32
72

2. 8 bamboo rods are needed to make a bullock cart. How many

bamboo rods are needed for 23 carts?

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 One cart needs 8 bamboo rods. 23 carts need 23 × 8 rods.
20 carts with 8 rods in each need
× 8
20 × 8 rods = ________ rods.
20 20 × 8 = 160
3 carts with 8 rods in each need
3 3 × 8 = 24
3 × 8 rods = _________ rods.
184

Let Us Solve

1. A flock of 25 geese and 12 sheep have

gathered around a pond. Chippi the
lizard sees many legs.
How many legs does it see?

2. In an auditorium, 8 children are sitting in each row. There are 15

such rows in the school auditorium. How many children are in the
auditorium?

3. A book shop has kept 9 books in each pile.

There are 14 such piles. How many books does
the shop have?

4. Surya is making a patch work with beads of

two colours as shown in the picture. How many
beads has he used? How many each of golden
colour beads and white colour beads has he
used in making this patch work?

5. For each of the following multiplication problems, make your own

stories as above. Then find out the product.

a) 34 × 3 b) 75 × 5

c) 46 × 6 d) 50 × 9

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 Division

1. A factory has ordered 58 wheels for the small tempos that they
make. Each tempo has 3 wheels.
In how many tempos can they fix the wheels?
Discuss your thinking in each step.

Number of tempos is 58 ÷ 3
30 wheels are needed for 10 tempos. _______wheels are left.

15 wheels are needed for _______ tempos.

_______wheels are left.

9 wheels are needed for_______ tempos.

_______ wheels are left.

_______ wheels are needed for_______ tempos.

_______ wheels are left.

Can we make another tempo?

How many total tempos can the factory make using the 58 wheels?
______
3) 58 ( 10+5+3+1
- 30
_________
28
- 15
_________
13
- 9
_________
4
With 58 wheels, we
- 3
_________
can make 19 tempos.
1 1 wheel is left.

Note for Teachers: The division performed here is by partial quotient method. It is
carried out by taking away groups of 10s, 5s or any other small multiples, which are
easily available to children. Children can choose multiples of their own choice to solve the
problems. Encourage them to gradually shift to taking away 10s and multiples of 10s.

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 2. A dairy farm has many cows. Chippi the lizard is surprised to see
88 legs.
How many cows are there in the farm? Write appropriate sentences
as above to show your thinking.

Number of legs of a cow: ________

Number of cows is 88 ÷ ________

Show your work using the table below.

Hint: Taking out groups of 10s is easy.

No. of legs No. of No. of legs

cows remaining 4) 88 ( 10+_______
88 _____
– 40
40 10 48 48
_____ ______ ______ –_____
_____ ______ ______

Total number of cows = ______

Let Us Solve

1. In a big aquarium, Jolly fish sees 72 legs

of octopuses. How many octopuses are
there in the aquarium?
2. A sports store packs 4 shuttlecocks in a
bigger box. They have 50 shuttlecocks.
How many boxes will they need to pack all of them? Can they pack
all the shuttlecocks in the boxes? How many are left?
3. Rakul Chachi uses a part of her farm to grow flowering plants
for the upcoming festive season. She has planted 75 saplings of
roses. Each row has 5 saplings. How many rows of saplings has
she planted?
4. Make stories for the following problems and solve them:
a) 70 ÷ 5 c) 69 ÷ 3
b) 84 ÷ 7 d) 93 ÷ 6
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 Multiples of 100

2 people on each bike

100 bikes with 2 people on each have
100 × 2 people = _____ people.

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What do you
notice about
500 × 4 = _____ 5×4 = _____
multiplying by
100 × 4 = _____ 50 × 4 = _____
multiples
of 100s?

Note for Teachers: Encourage children to work out the answers in different ways. Also
help them notice the relationship between single digit multiplication and multiples of 100s
of the same group size. Expressing the relationship as ‘hundred-times’ is appropriate.

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 Observe the pattern and complete the answers.

1×3= 2×3= 4×3=

10 × 3 = 20 × 3 = 40 × 3 =
100 × 3 = 200 × 3 = 400 × 3 =

2×4= 4×2= 8×1=

20 × 4 = 40 × 2 = 80 × 1=
200 × 4 = 400 × 2 = 800 × 1 =

3×4= 3×5= 3×9=

30 × 4 = 30 × 5 = 30 × 9 =
300 × 4 = 300 × 5 = 300 × 9 =

More Multiplication

1. Big electric autorickshaws run in small

towns of India and can carry 8 passengers.
How many people can travel in 125 such
autos in a single round?

8 8 8 8 8 8 8 8 8 8
8 8 8 8 8 8 8 8 8 8
8 8 8 8 8 8 8 8 8 8
8 8 8 8 8 8 8 8 8 8
8 8 8 8 8 8 8 8 8 8
100 autorickshaws with
8 passengers each
8 8 8 8 8 8 8 8 8 8
8 8 8 8 8 8 8 8 8 8
8 8 8 8 8 8 8 8 8 8
8 8 8 8 8 8 8 8 8 8
8 8 8 8 8 8 8 8 8 8
20 autorickshaws with
8 8 8 8 8 8 8 8 8 8 8 passengers each
8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 5 autorickshaws
with 8 passengers each

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 The total number of passengers 125 × 8. × 8
100 autorickshaws with 8 passengers in 100 100 × 8 = 800
each have 100 × 8 passengers = ______ 20 20 × 8 = 160
passengers. 5 5 × 8 = 40
1000
20 autorickshaws with 8 passengers in
each have 20 × 8 passengers = _______
passengers.

5 autorickshaws with 8 passengers in each have 5 × 8 passengers =

__________ passengers.

125 autorickshaws with 8 passengers in each have ____ + ____ +

______= _________ passengers.

2. Kahlu and Rabia are potters and make earthen pots

(kulhad) for trains.
They pack 6 kulhads in a box and have packed 174 boxes for delivery.
How many kulhads have they made?
The total number of kulhads is ________.

100 boxes with 6

kulhads each
× 6
100 100 × 6 = ____
70 boxes with 6 174
kulhads each 70 70 × 6 = ____
4 4 ×6 = ____
4 boxes with 6
kulhads each

Let Us Solve

1. BP Girl’s school has decided to give all its students two pencils on
the first day of school.
It has 465 students.
How many pencils does the school need to buy?
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 2. 234 children of a school have decided to organise a school mela.
Each child contributes ` 5 for the organisation of the mela.
How much money do they collect?

3. Make stories for the following problems and solve them.

a) 439 × 4 b) 514 × 8
c) 356 × 5 d) 623 × 7

More Division

9 boats have to ferry 108 people waiting along the banks of the Cauvery
River. Every boat carries the same number of people. How many people
should be ferried in each boat?

108 people are to be ferried in 9 boats.

In 1 boat, the number of people ferried is 108 ÷ 9.

If 5 people sit in each of the boats, then 45 people can be ferried in 9

boats.
If 5 more people sit along with them in each of the boats (total 10),
then 90 people can be ferried
in the 9 boats. ________
9 ) 108 ( 5+5+2
– 45
______
The remaining 18 people have 63
to be adjusted in the 9 boats. 2 – 45
_______
more people will have to sit in 18
each of the boats. 9 boats need to
– 18
________ take 12 people
So, the 9 boats need to take 12 0 each.
people each.

Note for Teachers: Division problems are of 2 types—share and measure. In

share problems, the number of equal groups is given (i.e the multiplier) leading to
opportunities to share objects equally. The example above is a share problem. In
measure problems, the size of each group is given (i.e, the multiplicand), like when
we ask how many ants are there if the number of legs is 60. Including both kinds
of problems is helpful for children.

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 Patterns in Division
How much money will each get? Draw arrows linking the money and
the children to answer the questions.
en
1. ` 30 shared equally among 3
m

children ______________
en
i
ec

m
Sp

i
ec

en
Sp

2. ` 900 shared equally among 3

mi
ec
Sp
children ______________

en

en
im

im
ec

en
ec
Sp

im
Sp

ec
Sp
en

en
im

im
ec

en
ec
Sp

im
Sp

ec
Sp
en

en
im

im
ec

en
ec
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im
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ec
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Using the above way of thinking, solve the following problems.
Observe and explain the patterns that you notice below.

A B C

3 ÷ 3 = ______ 9 ÷ 3 = _____ 15 ÷ 3 = _____

30 ÷ 3 = _____ 90 ÷ 3 = ____ 150 ÷ 3 = ____
300 ÷ 3 = ____ 900 ÷ 3 = ___ 300 ÷ 3 = _____

5 ÷ 5 = _____ 8 ÷ 4 = ______ 4 ÷ 2 = ______

50 ÷ 5 = ____ 80 ÷ 4 = _____ 20 ÷ 2 = _____
500 ÷ 5 = ____ 800 ÷ 4 = ____ 100 ÷ 2 = ____

1. A load carrying truck has 6 tyres. Chippi the lizard sees 60 tyres.
How many trucks are there?
2. Chippi sees 80 wheels in a car parking space. How many cars are
standing in the parking space?
3. Chippi sees 600 legs of ants walking towards the anthill. How
many ants are there?
4. A fancy shop has packed 800 rubber bands in several packets.
Each packet has 4 rubber bands.
How many packets of rubber bands does the shop have?
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 Let Us Solve
1. A school bus hires 7 buses to take 245 children to the transport
museum. Each bus carry the same number of children. How many
children are traveling in each bus?
2. The Darjeeling Himalayan Railway
is fondly called the “Toy Train”. This
toy train ride is also a UNESCO
World Heritage Site.
This amazing train runs between
New Jalpaiguri and Darjeeling
and it also passes through one of
the highest stations in the world,
namely, Ghum. It runs 88 km
daily. How much distance does it
travel in a week?
3. The 16 Km river rafting from Shivpuri to Rishikesh in the Ganga
provides the most interesting rafting opportunity. In the summer
months, VentureOut company took 259 people for rafting. Each
raft can take 7 people. How many rafts did it take?
4. Anu saves `45 every month by putting it in her piggy bank.
a) How much money will Anu save in 6 months?
b) She distributes the total money saved after 6 months among 6
of her friends. How much does each friend get?
c)If she decides to distribute the saved money among 3 friends
after 6 months, how much money will each get?
5. Raju drives an auto in his village and takes people to the
bus stand. He makes 8 trips in a day. Which of the following
questions can be exactly calculated with the above statement?
a) How much money does he make in a day?
b) How many trips does he make in 7 days?
Note for Teachers: Encourage the children to observe relationships between
divisor, dividend and quotient, understand the relationship between quotients
when the dividend is changed by 5 times/10 times, and the relationship between
multiplication and division.

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 c) How much time does one trip take?
d) How many trips does he make in 4 weeks?

6. Solve
a) 45 × 9 b) 507 × 7

c) 94 ÷ 4 d) 778 ÷ 6

e) 94 × 5 f) 396 × 4

g) 83 ÷ 3 h) 635 ÷ 5

7. In mathematics, some statements are always true, some are

sometimes true, and some are never true.
Tick (√) in the appropriate column.

Statement Always Never Sometimes

True True True
Multiplying by 10 gives 0 in the ones
a)
digit of the number.

Multiplying a number by 2 gives an odd

b)
number.

Multiplying a number by 5 gives a

c)
number with 5 in the ones digit.
The number immediately after an odd
d)
number is an even number.
Halving any number gives an even
e)
number.
Adding 0 to a number increases the
f)
number by 1.

Note for Teachers: The “always true, never true, or sometimes true” type of
questions in math are designed to help students understand and evaluate the
validity of mathematical statements under different conditions. They encourage
critical thinking, testing conceptual understanding and encouraging students
to reason logically with counter examples. They help students to move beyond
rote memorisation to a better understanding of mathematical principles.

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