Year 3 Teaching Sequence Spring 11 - Multiplication and division (including commutativity and finding remainders)
(four days)
Prerequisites:
 Know by heart multiplication facts for 2, 3, 5, and 10 times tables (see oral and mental starter banks 10 and 11)
 Begin to know by heart multiplication facts for the 4 times table (see teaching sequence 10 and oral and mental
starter bank 11)
 Multiply single-digit numbers by 2, 3, 4, 5 and 10, and divide two-digit numbers by the same (answers not greater than
10) (see autumn teaching sequence 11)
 Understand how multiplication is commutative (see autumn teaching sequence 11 and spring oral and mental starter
bank 11)
 Understand that division can leave a remainder (initially as ‘some left over’) (see autumn teaching sequence 11)
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, or make the counting on easier. Children use
division and their knowledge of multiplication facts to find mystery numbers in multiplication and division sentences. They
use grouping to divide numbers by 2, 3, 4, 5, 9 and 10 including those which leave a remainder and make up their own
divisions which leave a remainder of 1. Remainders are discussed in the context of word problems.
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 whichever
way round we write a multiplication. Thus 4 lots of 5 is equivalent to 5 lots of 4. Understanding commutativity helps children
to multiply efficiently.
Note that the grouping model of division is used in this sequence to relate multiplication and division. There is a more
natural link between grouping and multiplication facts (e.g. four 10s are 40, how many 10s in 40?) that between sharing and
multiplication. For this reason it is important that the ÷ sign is not read as ‘sharing’, which would confuse the two models.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
Y3 Maths TS11 – Spr – 4days
Watch out for children who try to share between groups rather than using grouping, as once learnt this will be easier to
visualise and more efficient when dividing larger numbers too cumbersome to share.
Watch out for children who do not understand the link between multiplication and division, and so cannot use one to solve
the other, particularly in missing number problems.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
Y3 Maths TS11 – Spr – 4days
Objectives:

Understand the vocabulary ‘multiplied by’ and the concept of commutativity

Multiply single digits by 2, 3, 4, 5, 9 and 10

Divide two-digit numbers by 2, 3, 4, 5, 9 and 10 (answers not greater than 10, but including remainders)

Begin to decide whether to round up or down after division depending on the context
Whole class
Group activities
Paired/indiv practice
Resources
Write 3 × 6 on the board. Explain that this can be read
as ‘three lots of six’, or ‘three multiplied by six’. Which
way would you work this one out? Three lots of six or six
lots of three? We haven’t learned our sixes yet so
perhaps six lots of three might be easier! Draw six jumps
of three above a 0-100 beaded line and three jumps of
six below it, to show that they come to the same number.
What divisions could we write to go with these hops?
Write 18 ÷ 3 = 6 above the line to go with the six hops of
three, and 18 ÷ 6 = 3 below the line.
Also show an array of six rows of three:
Group of 4-5 children
Write 5 × 4 on the flipchart.
What pictures could we draw to go with
this multiplication? Sketch them on
your whiteboards. Share children's
sketches and agree that we could draw
an array or hops on a number line. What
are five 4s? And four 5s? What
pictures show each of these? Draw the
pairs of arrays and number line jottings
on the flipchart. Write the complete
multiplications on the flipchart by the
side of the relevant number line jotting.
If we know five 4s are 20, what division
can we write? What number line jotting
goes with this? What division can we
write to go with the other number line
jotting?
Repeat with 4 × 6.
Easier: Use 3 × 4 and 2 × 10.
Harder: Use 4 × 6, and then ask
children to work in pairs to choose
their own multiplication and to write
the other three related multiplication
and division facts. Share children’s
responses.
Write the following multiplications on
the board: 10 × 5, 2 × 8, 3 × 4, 7 × 2, 5
× 7, 4 × 6, 4 × 10, 7 × 9, 9 × 5, 8 × 4.
Children work through them to
practise multiplying single digits
together. They write how they
decided to find the answer, so for
example for 5 × 7 they write 7 lots of
5 if they choose to count on in 5s or
use their tables facts for 5 rather
than counting on in 7s. Children 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 9).
Easier: Children will probably need to
use a 0-100 beaded line to support
counting on more frequently.
Harder: Ask children to write the
division sentence to go with the way
they solved the multiplication, e.g. if
they worked out 4 × 6 by working out
six fours, they record 24 ÷ 4 = 6.
 Large 0100 beaded
line
 0-100
beaded
lines (see
resources)
Discuss how we can find the total by counting in threes
down the rows, or else in sixes across the columns.
Rotate the array to show three rows of six, and discuss
how we can count in threes or sixes to find the same
total.
Write 6 × 9. How can we read this? We don’t yet know
our 6 or 9 times table, would you prefer to count on in 6s
or 9s? Count along the 100 bead bar slowly in sixes, and
then in nines. Discuss that adding nine on each time is
actually quite easy as we can think of adding ten, and
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
Y3 Maths TS11 – Spr – 4days
then subtract one. Count in nines to 99 along the bead
bar to practise this. Also show recording hops of 6 and 9
above and below the 0-100 beaded line to help children
see that the answer is the same. What division sentences
could we write?
Write 5 × 6 = 30 on a card, and use a ‘slidy box’ to cover
up the 6.
5 ×
-
= 30
What number do you think might be hiding? Why do you
think that? We could read this as five lots of something
equals thirty, or five multiplied by something equals
thirty. How many fives are in thirty? What division
number sentence could we write? Reveal the mystery
number.
Show the following:
20 ÷ 5 = 4
What number do you think is hiding? Why do you think
that? We can read this as something divided into groups
of five, gives four groups. What are four 5s? What
multiplication sentence could we write? Reveal the
mystery number.
Repeat with slidy box cards for other multiplication
facts e.g. 3 ×
= 18,
× 4 = 24,
÷ 5 = 3,
÷ 3 = 4, ÷ 4 = 3.
Group of 4-5 children
Write the following number story on
the flipchart:
There are
children. They get into
groups of
. There are
groups.
Ask children to discuss what numbers
could go in this story to make it true.
Which will be the biggest number? Why
do you think that? What size groups
might they get in? What multiplication
facts could we use to help us? Take
feedback, and try out children's
suggestions. Write them on Post-its™,
put them in the story and discuss if
they make sense.
What other numbers could we put in
this story?
Repeat for the following story:
Matthew has
coins. They are all
p
pieces. Altogether he has
p.
Easier: You may need to make a sketch
to show if children’s ideas work, for
example sketching four groups of 5
smiley faces to show 20 children
altogether.
Harder: Say that one number must be
greater than 20.
Children work in pairs to each fill in
the missing numbers in multiplication
and division sentences (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: Children use 0-100 beaded
lines to help them to work out the
mystery multiplications.
Harder: Ask children 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
 Post-its™
 Activity
sheet of
multiplicati
on and
division
sentences
with
missing
numbers
(see
resources)
Y3 Maths TS11 – Spr – 4days
Write 17 ÷ 5 on the board. What does this mean? Read it
both as ‘seventeen divided by five’, and ‘how many fives in
seventeen?’ How many fives do you think there might be
in 17? Do you think there will be a remainder? Why?
Launch the ITP Grouping, and choose 17 ÷ 5 as the
calculation.
Click on 17 to show 17 objects, and then click on five to
show one group. Can we make another five? Repeat
clicking on fives until there are only two left. What's
happened? Click to show the number sentence. The little
‘r’ stands for remainder, the number left over.
What do you think would happen if we had 16 divided by
5? Draw three hops of five on a large 0-100 beaded line,
and point out the one left over. What if we had 18 ÷ 5,
how many would be left then? Write the division on your
whiteboards. And 19 ÷ 5? And what happens when we
have 20 ÷ 5? Discuss how we have an extra one, and so
we can make another group of five.
Group of 4-5 children
Write the following word problem on
the flipchart:
Jess has 17 stickers. She puts five
stickers on each page of her
sticker album. How many pages can
she fill?
Ask children to discuss this. What do
we need to do to solve it? Agree that
we need to find out how many 5s there
are in 17 as this will tell us how many
pages she will fill. We found that there
are three fives in 17, and two left over.
So what’s the answer to the number
story? Agree that the answer to the
question is three. What do you think
Jess will do with the other two?
Write the following number story on
the flipchart:
Jess has 17 stickers. She puts five
stickers on each page of her
sticker album. How many pages will
she need to put them all in her
album?
Ask children to discuss this problem.
What do we need to work out this time?
We know 17 divided into groups of five
is three and 2 left over. So how many
pages will she need? So the answer to
this problem is four!
Repeat with similar word problems,
discussing the necessary divisions, but
then how this is not necessarily the
Write the following divisions on the
board:
32 ÷ 5, 13 ÷ 2, 14 ÷ 4, 33 ÷ 10, 20 ÷ 9,
22 ÷ 3, 51 ÷ 5, 45 ÷ 10, 25 ÷ 4, 17 ÷ 3.
Children work through them to
practise dividing by 2, 3, 4, 5, 9 and
10. Children 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 9).
Easier: Children will probably need to
use a 0-100 beaded line to support
counting on more frequently.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
 Large 0100 beaded
line
 0-100
beaded
lines (see
resources)
Y3 Maths TS11 – Spr – 4days
Write the following numbers on the board:
21, 25, 13, 26 and 31.
Which of these numbers do you think would have a
remainder of 1 when divided by 10? Why? Take children’s
suggestions and test them out by drawing hops on the 0100 beaded line. Work with a partner to come up with
another division by 10 that will give a remainder of 1.
Which do you think will have a remainder of 1 when
divided by 5? Take children’s suggestions and test them
out by drawing hops on the 0-100 beaded line.
And by 3? And by 4? Test out chn’s suggestions.
answer to the ‘real-life’ problem.
Easier: You may need to represent the
objects in the stories with cubes to
ensure children's understanding of
what to do with the objects in the
story.
Group of 4-5 children
What remainders do you think are
possible when dividing by 5? Ask chn to
work in pairs to divide numbers 10 to 20
by 5 and discuss what they find. When
you divide by 5, could you get a
remainder of 6? Why not? Agree that
there would be another group of five.
What remainders do you think you can
get when dividing by 4? Take chn’s
suggestions and ask them to try them
out by dividing numbers 10 to 20 by 4.
Easier: Discuss what remainders are
possible when dividing by 2, then 5,
using counters to show the remainders.
Ask chn to make up ten of their own
divisions that would give a remainder
of 1.
Easier: Children will probably need to
use a 0-100 beaded line. Suggest that
they only divide by 10 and 5.
Harder: Children think of divisions
that will give a remainder of 2.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
 Large 0100 beaded
line
 0-100
beaded
lines (see
resources)
Y3 Maths TS11 – Spr – 4days