Year 3 Teaching Sequence 11 - Multiplication and division (including commutativity, grouping and finding a remainder)
(four days)
Prerequisites:
 Know by heart multiplication facts for 2, 5, and 10 times tables (see oral and mental starter banks 10 and 11)
 Begin to know by heart multiplication facts for the 3 times table (see teaching sequence 10 and oral and mental
starter bank 11)
 Count on in 3s and 4s (see teaching sequence 10 and oral and mental starter bank 11)
 Understand grouping as one model of division (see Year 2 Summer teaching sequence 11 and Year 3 Autumn teaching
sequence 10)
Overview of progression:
Children revise the concept of commutativity (a x b  b x a) and use this to solve multiplication of single-digit numbers
choosing which order will help them to make use of their multiplication facts. They use grouping to divide numbers by 2, 3,
4, 5 and 10 including those which leave a remainder and investigate which numbers between 10 and 20 leave remainders when
divided by 2, 3, 5 or 10. They use division and their knowledge of multiplication facts to find mystery numbers in
multiplication sentences. They write division sentences to go with multiplication sentences.
Note that multiplication sentences can be read in two ways; 4 × 5 can be read as four fives (4 lots of 5) or alternatively as 4
multiplied by 5, i.e. 4, five times (5 lots of 4). Children need to be aware of the different terminology used (lots of, times,
multiplied by), but more importantly understand that multiplication is commutative and so the answer is the same which ever
way round we write a multiplication. Thus 4 lots of 5 is equivalent to 5 lots of 4. Understanding commutativity helps chn to
multiply efficiently.
Note that the grouping model of division is used in this sequence to relate multiplication and division. It is not useful to use
a sharing model. There is a more natural link between grouping and multiplication facts (e.g. four 10s are 40, how many 10s in
40?). For this reason it is important that the ÷ sign is not read as ‘sharing’, which would confuse the two models.
Watch out for children who try to share between groups rather than using grouping, as once learnt this will be more
efficient, easier to visualise and needs to be learned for when they divide larger numbers too cumbersome to share.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
Y3 Maths TS11 – Aut – 4days
Objectives:
 Multiply single digit numbers by 2, 3, 4, 5 and 10, and divide two-digit numbers by the same (answers not greater than 10)
 Understand how multiplication is commutative
 Understand that division can leave a remainder (initially as ‘some left over’)
Whole class
Group activities
Paired/indiv practice
Resources
Write 20 + 54 on the board. How would you
work this out? Which number would you start
with? Why? Would the answer still be the
same? Explain that this is because addition
works in any order and that multiplication is
the same. Write 5 × 7 on the board. Explain
that this can be read as ‘five lots of seven’, or
‘five multiplied by seven’ (seven lots of five).
Which way would you work this one out? Five
lots of seven or seven lots of five? We haven’t
learned our sevens yet so perhaps seven lots
of five might be easier! Re-write the
calculation as 7 x 5 and read as seven lots of
five. This is how we have decided to do this!
Draw seven jumps of five above a 0-100
beaded line and five jumps of seven below it,
to show that they come to the same number.
Show an array of seven rows of five:
Group of 4-5 children
Write 30 in the middle of the flipchart.
What multiplication facts do we know
with an answer of 30? How many tens
are in 30? If we know that there are 3
tens in 30, what else do we know? Write
10 × 3 = 30, 3 × 10 = 30, 30 ÷ 10 = 3 and
30 ÷ 10 = 3 in a new bubble joined to 30
in the middle.
What other numbers go into 30 without
leaving any over? Build up other facts
including using 15 × 2, 5 × 6 and other
related facts. Discuss double 15 and
how we can write this as a
multiplication.
Easier: Give chn 30 cubes to put into
equal groups to help.
Write the following
multiplications on the board:
4 × 3, 10 × 3, 7 × 5, 2 × 4, 9 × 10, 7
× 2, 5 × 6, 2 × 9, 8 × 5, 5 × 4, 3 × 6.
Chn work through them to
practise multiplying single digits
together. They write how they
decided to find the answer, so for
example for 5 × 8 they write 8
lots of 5 if they choose to count
on in 5s or use their tables facts
for 5 rather than counting on in
8s. Chn use a 0-100 beaded line
(see resources) to help them with
any for which they don’t yet know
the multiplication facts
(particularly multiples of 4).
Easier: Chn will probably need to
use a 0-100 beaded line to
support counting on more
frequently.
Harder: They also choose one
multiplication to show both ways,
drawing two empty number lines.
 Arrays as
opposite
 A large 0-100
beaded number
line
 Cubes
 0-100 beaded
lines (see
resources)
Discuss how we can find the total by counting
in fives down the rows, or else in sevens across
the columns. Rotate the array to show five
rows of seven, and discuss how we can count in
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
Y3 Maths TS11 – Aut – 4days
fives or seven to find the same total.
Write 5 × 3 on the board. Would you prefer to
work out five threes or three fives? Draw
hops to show three jumps of five and five
jumps of three. Show an array of three rows
or five and discuss as before.
Write 2 × 8 on the board. How would you work
this out? Discuss how we could count in twos,
or use our knowledge of doubles, in this case
double 8, and so we are thinking of two groups
of eight.
Write 4 × 6. We don’t yet know our 4 or 6
times table, would you prefer to count on in 4s
or 6s? Show recording hops of 4 on a 0-100
beaded line to help.
Write 20 ÷ 5 on the board. What does this
mean? Read it both as ‘twenty divided by five’,
and ‘we have 20, how many fives?’ Write ? x 5
= 20. How many fives are in 20?
Write 21 ÷ 5. What if it was 21 instead of 20?
What would happen? Launch the ITP Grouping,
and choose 21 ÷ 5 as the calculation.
Click on 21 to show 21 objects, and then click
on five to show one group. Can we make
another five? Repeat clicking on fives until
there is only one left. What's happened?
Write 21 ÷ 5 = 4 and 1 left over. Say that we
call this one left over a remainder.
Group of 4-5 children
What remainders do you think are
possible when dividing by five? Take
chn’s suggestions. Take 11 cubes and
divide them into groups of five. What
happens? What do you think will happen
if you divided 12 into groups of five?
Ask chn to try dividing 13, 14 and 15
into groups of five. What happened
when you divided 14 by 5? What
happened when you added another
cube? Discuss how this made another
group of five, and so remainders of 1, 2,
3 and 4 are possible when dividing by 5.
What remainders do you think are
possible when dividing by 3? Use cubes
and test out your ideas.
Ask chn to work in pairs to
investigate which numbers less
than 50 will leave a remainder of 1
when divided by 5. They record
the divisions, e.g. 11 ÷ 5 = 2 and 1
left over. They can use 0-50
beaded lines to help where
necessary.
Easier: Chn use cubes to support
the division.
Harder: Chn also find at least
four numbers less than 30 will
leave a reminder of 1 when
divided by 3.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
 ITP Grouping
 0-50 beaded
lines
 Cubes
Y3 Maths TS11 – Aut – 4days
Harder: Also ask what remainders are
possible when dividing by 4 and 10. Can
they generalise the biggest possible
size reminder when dividing by any
number?
What do you think would happen if we had 22
divided by 5? And 23?
Write 21 ÷ 5 on the board. What does this
mean? Read it as 21 divided into groups of 5.
How many groups will we have? And left over?
Show this on the Grouping ITP to confirm.
What do you think will happen if we divide 21
into groups of 3? Use the ITPs to confirm
that there will not be a remainder. And what if
we divide 21 into groups of 10? And if divide
21 into groups of 2? Agree that there will be
one left over each time and show this on the
Grouping ITP.
Repeat for 22 ÷ 5, 22 ÷ 2, 22 ÷ 3 and 22 ÷ 10.
Group of 4-5 children
Write 23 in the middle of the flipchart.
Do you think we would get any left over
if we divided 23 by 2? Is 23 a multiple
of 2? How do you know? Ask chn to
check this by working in pairs to put 23
into group of 2. Write 23 ÷ 2 = 11 and 1
left over. Repeat, this time dividing 23
by 3, then 4, 5 and 10, building up a fact
web.
Ask chn to work in pairs to
investigate which numbers from
10 to 20 will leave a remainder
when divided by 2, 3, 5 and 10.
They record their results on a
table (see resources) and discuss
why they think some numbers
leave remainders. They use 0-20
beaded lines to support their
calculations.
Easier: Chn use cubes to support
the division.
Harder: Chn also find the next
two numbers which will leave some
left over when divided by 2, 3, 5
and 10.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
 ITP Grouping
 Activity sheet
of table on
which to
record results
of division
investigation
 0-20 beaded
lines
 Cubes
Y3 Maths TS11 – Aut – 4days
23 ÷ 2
= 11 and
1 left
over
23 ÷ 10
= 2 and
3 left
over
23 ÷ 5 =
5 and 3
left
over
23
cubes
23 ÷ 3 =
7 and 2
left
over
23 ÷ 10
= 2 and
3 left
over
23 always leaves some over when
divided by 2, 3, 4, 5 or 10. Why do you
think this is? Discuss how it is not in any
of their times tables, and how actually
the only numbers that go into it without
leaving some left over are 1 and 23! This
is a very special number; it is called a
prime number.
Easier: Use 13 cubes.
Harder: Also challenge chn to think of a
number that wouldn't leave any left
over when divided by 2, 5 or 10.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
Y3 Maths TS11 – Aut – 4days
Write 5 × 3 = 15 on a card, and use a ‘slidy’
box’ to cover up the 3.
5 ×
-
= 15
What number do you think might be hiding?
Why do you think that? We could read this as
five lots of something equals fifteen, or five
multiplied by something equals fifteen. How
many fives are in fifteen? What division
number sentence could we write? Reveal the
mystery number.
Show the following:
364 × 5 = 20
What number do you think is hiding? Why do
you think that? We can read this as something
multiplied by 5 equals 20, or five lots of
something equals 20. How many 5s are in 20?
What division number sentence could we
write? Reveal the mystery number.
Repeat with slidy box cards for other
multiplication facts e.g. 5 × □ = 15, □× 4 =
40, □× 5 = 40, 4 × □= 12, 3 × □= 21.
Group of 4-5 children
Record □× □= 40 on the flipchart.
What do you think the mystery numbers
might be? Can you think of a number
that goes into 40 exactly? Take
suggestions. And how many 10s are in
40? What two multiplication sentences
could we write? Record 10 × 4 = 40 and
4 × 10 = 40. We know that ten 4s are
40, but this also means four 10s are 40,
so 4 goes into 40 too. What division
sentences could we write to go with
these? Record 40 ÷ 4 = 10 and 40 ÷10 =
4.
Repeat with other suggestions until you
have families of four facts including
those derived from 20 × 2 and 8 × 5.
Repeat with 24.
Easier: Use □ × □ = 20 instead, and try
and make arrays with a total of 20
counters/cubes. Rpt with 30.
Chn work in pairs to each fill in
the empty boxes on an activity
sheet (see resources). Encourage
them to use multiplication to
check their answers, counting in
fours, for example, on their
fingers if they don’t know the
appropriate multiplication fact.
Easier: Chn use 0-100 beaded
lines to help them to work out the
mystery multiplications.
Harder: Ask chn to also write
division sentences to go with at
least four of the multiplications.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
 Slidy box cards
as opposite
 Activity sheet
of
multiplication
sentences with
missing
numbers (see
resources)
Y3 Maths TS11 – Aut – 4days