October 2015

USING ‘REAL’ OBJECTS AND PICTURES

It is crucial that children throughout school use objects

and pictures to help them understand calculation

methods.

By using ’real’ objects and mathematical resources, the

children can see how the different methods work.

Even when children in Year 6 are starting to use Long

Division, mathematical objects are still really useful to let

the children see how the method works.

The aim is for children to start

out by using the ‘real’ objects.

Then move on to using a

picture of those objects.

Finally, they will be able to use

any object (or mathematical

resource) to represent the

object being calculated.

ADDITION

Children are taught to understand addition as combining

two sets and counting on.

Counting rhymes such as:

1,2,3,4,5 once I caught a fish alive..

5 fat sausages...

Sing rhymes and songs to develop

children’s number recognition and

counting skills.

2 + 3 =

At a party, I eat 2 cakes and my

friend eats 3. How many cakes did

we eat altogether?

Children could draw a picture to

help them work out the answer. You

can also use real objects in two

groups, to find the total.

7 + 4 =

7 people are on the bus. 4 more get

on at the next stop. How many

people are on the bus now?

xxxxxxx xxxx

Children could use dots or tally

marks to represent objects (this is

quicker than drawing a picture).

47 + 25 =

My sunflower is 47cm tall. It grows

another 25cm. How tall is it now?

Drawing an empty number line helps

children to record the steps they

have taken in a calculation (start on

47, + 20, then + 5). This is much

more efficient than counting on in

ones.

Expanded form of vertical addition In this method the children partition

(splitting) the numbers into

hundreds, tens and units. They then

add each part before recombining

to give the final answer, taking

children a step closer to a formal

compact method. Children should

be encouraged to add the units first,

then the tens and then the units.

These are then combined to find the

answer.

Semi-compact form of vertical

addition

236 + 158 =

This expanded method encourages

children to look at the value of each

digit as they work through the

calculation. The language used is

very important: “6 + 8 then 30 + 50

then 200 + 100”. The numbers in

brackets help children to think

through the place value and the

stages of the calculation.

Compact form of vertical addition Continue to talk through the place

value of each digit as children are

adding (e.g. “6 + 8, then 30 + 50,

then 200 + 100”). Do not use the

word ‘carrying’ but explain that 6 + 8

is 14 or one ten and four units, so

the ten needs to be placed in the

tens column.

The Bar Model This model allows chd to visualise

the question and understand the

‘maths’ involved. The Bar Model can

be used for all 4 operations

(addition, subtraction, multiplication

and division). For addition, the

‘bars’ are placed next to each other

to represent the values being added.

Word Problems When looking at word problems,

encourage children to underline the

key information and then write the

number sentences underneath.

(i.e. 12786 + 2568 =)

Continue to refer to the place value

of each digit (hundreds, tens, units,

etc) as they are being added

together.

Adding decimals When adding decimals, always

remember to keep all of the decimal

points lined up with each other.

Then start from the digit with the

smallest value, as with the previous

methods.

SUBTRACTION

Children are taught to understand subtraction as taking

away (counting back) and finding the difference (counting

up from the smallest to the largest number).

5 –2 =

I had five balloons. Two burst. How

many did I have left?

A teddy bear costs £5 and a doll

costs £2. How much more does the

bear cost?

Drawing a picture helps children

visualise the problem.

7—3 =

Mum baked 7 biscuits. How many

were left?

Lisa has 7 felt tip pens and Timothy

has 3. How many more does Lisa

have?

Using dots or tally marks is quicker

than drawing a detailed picture.

Use a number line to count back.

Crossing out or knocking down

objects or pictures

Use a number line to find the

difference by counting up from the

smallest to the largest number.

Jump to the nearest ten first.

Expanded form of vertical

subtraction

For this method children partition

(splitting) the numbers into

hundreds, tens and units. They then

subtract each part before

recombining the answers to give the

final answer. This method moves the

children closer to a formal written

method.

Expanded form of subtraction

with exchange

Start with the least significant digit.

7 - 9 cannot be done without using

negative numbers so we need to

exchange (not ‘borrow’) one of the

tens for 10 units. This leaves us with

50 and 17. Now we can do 17 - 9,

which equals 8 etc.

The Bar Model This model allows children to

visualise the question and

understand the ‘maths’ involved.

For subtraction, the largest number

is the total and the answer is

represented by an empty ‘bar’. This

allows children to visualise the

difference.

Compact subtraction Avoid using the phrases “borrow” or

“take from” - use the term

“exchange”.

Draw attention to the place value

each time.

E.g. 300 - 200, not 3 - 2.

23.05 - 18.24 =

A game costs £23.05. Emma has

saved £18.24. How much more

does she need?

When working with decimals,

ensure that the decimal points

always stay lined up and then use

the compact method as normal,

continuing to draw attention to the

place value of each digit.

MULTIPLICATION

Children are taught multiplication through various methods

including repeated addition, arrays and partitioning. It is very

important that children work on their times tables daily to

support them with their calculations (we do this at school

through CLIC). By the end of Year 4, children should know all

the table facts to 12 x 12.

Repeated Addition

4 x 2 =

Each child has 2 eyes. How many

eyes do four children have?

Again, a picture or real objects can

be useful to help children

understand what they are

calculating.

5 x 3 =

There are 5 cakes in a pack.

How many cakes in 3 packs?

Dots or tally marks are often drawn

in groups.

This shows 3 groups of 5.

Arrays

6 x 3 =

A chew cost 6p.

How much do 3 chews cost?

Drawing an array (3 rows of 6) gives

children an image of the answer. It

also helps develop the understanding

that 6 x 3 is the same as 3 x 6.

Repeated Addition

5 x 4 = 20

There are 5 cats. Each cat has 4

kittens. How many kittens are there

altogether?

Children could count on in equal

steps, recording each jump on an

empty number line. This shows 5

jumps of 4. It is very important to

emphasise that we count the jumps

to find our answer.

Grid Method

26 x 7 =

There are 26 biscuits in a packet. If

there are 7 packets, how many bis-

cuits are there in total?

For this method 26 is partitioned

into parts (20 and 6) and each of

these is then multiplied by 7 (7 x 20

and 7 x 6). The two answers are

then added together.

3.2 x 7 =

In exactly the same way as the previ-

ous method, the grid method can be

used to multiply a decimal number

by a whole number. Remind chil-

dren of the value of the digits in the

decimal number.

The Bar Model This model allows children to

visualise the question and

understand the ‘maths’ involved.

For multiplication, each bar

represents a ‘group’ (eg. 4 groups of

30).

346 x 9 = Children progress from the grid

method to this more formal,

expanded method. The

calculations at the side (in

brackets) are important because

they remind children of the process

they are working through in order to

complete the calculation.

346 x 9 = Some children may be ready to

progress to this compact method.

Continue to remind children of the

place value of the numbers so that

they understand that 6 x 9 = 54, so

the 5 needs to be placed in the 10s

column.

72 x 38 = Children may encounter this

method of long multiplication.

Once again, the calculations in

brackets are very important as an

aid to understanding the method

fully.

DIVISION

Children are taught to understand division as sharing and

grouping. It is essential that children have a secure knowledge

of multiplication facts in order to work on these methods.

6 ÷ 2 =

6 Easter eggs are shared between 2

children. How many eggs do they

each get?

There are 6 Easter eggs. How many

children can have two each?

Drawing objects often gives

children a way into solving the

problem. ‘Real’ objects can also be

used to share between groups.

This first picture shows division as

“sharing”.

This example shows division as

“grouping” into 2s.

Children are taught both sharing

and grouping in school and they are

both equally important.

15 ÷ 5 =

5 apples are packed in a basket.

How many baskets can you fill up

with 15 apples?

Dots or tally marks can either be

shared out one at a time or split up

into groups.

This example shows grouping

because the question has told us

that we are working with groups of

5 apples each time.

Arrays

15 ÷ 5 =

Children use faces to show the

number they are dividing by (i.e the

smaller number) and then share the

large number between the faces.

They find the answer by counting

the marks under one face, in this

example, the answer in 3.

20 ÷ 4 =

“How many groups of 4 are there in

20?”

Draw jumps of 4 along a number

line. This shows you need 5 jumps

of 4 to reach 20. Emphasise the

importance of counting the jumps to

find the answer.

92 ÷ 6 =

It would take a long time to jump

back in sixes from 92, so children

can jump back in bigger ‘chunks’. A

jump of 10 groups of 6 takes you

back to 32. Then another 5 groups

of 6 reaches 2. We then have a

remainder of 2. Count the jumps

(underlined in the brackets) to find

the answer.

The Bar Model This model allows children to

visualise the question and

understand the ‘maths’ involved.

For division, the whole is

represented by a line across the top

and each ‘group’ is represented by a

small bar.

Compact Method

427 ÷ 5 =

This method start with the largest

value number (e.g. How many 5s are

in 4?). Any remainders are ‘carried

over’ to the next number. Continue

to remind children of the place value

of the numbers so that they

understand that when the 4 is

exchanged over, it is no longer 400

but 40 tens. Any remainders at the

end are written with the letter ’r’

before them.

Long Division

4456 ÷ 25 =

The Long Division method is used

when you need to divide by a

2-digit number rather than a 1-digit

number.

For self-help calculation videos and links to useful websites, please

visit our school website:

www.foresttownprimaryschool.co.uk