Maths methods for blogs

Do your children start talking a strange language whilst doing their maths homework?
Do words such as ‘chunking’, ‘grid method’, or ‘partitioning’ baffle you?
The aim of this PowerPoint is to enable parents and carers to understand more about the
methods used in our school and to assist them in supporting children with maths.
The traditional, formal methods of teaching children mathematics are now being used
less during the stages whilst children develop and acquire number skills and
understanding.
This PowerPoint demonstrates the methods in the progressive order that pupils will
experience them at Minchinhampton. Methods are taught in a way that will aid the
pupils’ conceptual understanding and support their mental calculations. Although pupils
may be able to master more advanced methods, they are taught methods appropriate to
their understanding of number: it is essential that they fully comprehend every stage of
the process.
Some Year 5/6 pupils are in the process of creating short videos to demonstrate these
methods being used. To see these videos, please visit your child’s class blog.

Contents
Click on the operation that you would like to find out about,
or use the forward arrow key to view the PowerPoint from
start to finish.
• Addition
• Subtraction
• Multiplication
• Division

Addition Section Contents
• Pictures/objects/symbols
• Number lines/tracks – jumps of 1
• Number lines – efficient jumps
• Vertical (column) addition – expanded
• Vertical (column) addition – compact
Click here to return to main contents.

Pictures/objects/symbols
Pupils use concrete resources and drawings/symbols to solve the
problem. Objects may include real items (such as small plastic toys,
money), counting cubes (Unifix, Multilink) or Numicon. As they
progress, symbols, and eventually numbers will be used to show
working and answers.
I eat 3 cherries and my friend
eats 2 cherries. How many
cherries do we have altogether?
Numicon will be used to support calculations.
2 cherries and 3 cherries make 5
cherries altogether.
II III
2 + 3 = 5
9 + 1 = 10
20 + 3 = 23
Click here to return to contents for addition section. Click here to return to main contents.

Pupils show and calculate addition problems on a number line or
number track. They begin by counting in ones and progress to larger,
more efficient, jumps. When counting on, pupils always count from the
larger number.
18 + 5 = 23 (Counting in ones)
18 19 20 21 22 23
+1 +1 +1 +1 +1
Number Lines/Tracks

Before moving on to more traditional methods, pupils use the number
line method with more complex numbers (including decimals). This
reinforces mental calculations and understanding of number.
Number Lines – efficient jumps
807747
+30
+3
47 + 35 = 82
+2
82
18 2320
+2
+318 + 5 = 23
18 38
+20
18 + 19 = 37
-137

This method links the number line with the more traditional column
method. It encourages children to consider the value of the digits and
aids conceptual understanding. This method will be revisited as
numbers become more complex, for example as pupils solve written
addition problems with decimals.
H T O
3 3 6
+ 1 8 7
1 3 Add the ones (6 + 7)
1 1 0 Add the tens (30 + 80)
4 0 0 Add the hundreds (300 + 100)
5 2 3 The new columns are added,
starting with the digit of least value.
Vertical (Column) Addition - Expanded

When confident with using the expanded method, pupils learn the
compact method. To calculate in this way, digits are added in order of
value (least to greatest). For this sum, the order of value is: ones, tens,
hundreds. Tens and hundreds are carried by writing a digit below the
line.
H T O
3 3 6
+ 1 8 7
5 2 3
1 1
Vertical (Column) Addition - Compact

Subtraction Section Contents
• Number lines/tracks – counting back
• Number lines – counting on
• Vertical (column) subtraction – no exchanging
• Vertical (column) subtraction – with
exchanging

problem. Objects may include real items (such as small plastic toys,
money), counting cubes (Unifix, Multilink) or Numicon. As they
progress, symbols, and eventually numbers will be used to show
working and answers.
I have 5 cherries and eat 2 of
them. How many do I have left?
Numicon can be used for subtraction by arranging the shapes on a
number line or laying pieces on top of each other.
5 cherries take away 2 cherries
leaves 3 cherries.
III II
5 – 2 = 3
Click here to return to contents for subtraction section. Click here to return to main contents.

Pupils begin to show and calculate subtraction problems on a pre-drawn
number line or number track by counting back. They begin by counting back in
ones and progress to larger, more efficient, steps. For subtraction, the jumps
are drawn below the line (this is a very recent change to our previous policy and
older pupils will still record subtraction jumps above the line). The larger
number is always written on the right. As they progress, pupils draw their own
number lines, including only the relevant numbers. They also begin to take
more efficient jumps as in the bottom example. Once competent with efficient
jumps, children apply this method to find the difference between any numbers of
any size, including decimals.
Number Lines/Tracks
13 – 5 = 8
(Jumps of one) 1312111098
-1 -1 -1-1-1
13108
-3-2
13 – 5 = 8
(Efficient jumps)

When numbers are close together, pupils may choose to find the
difference by counting on. As always, the number of least value is
recorded at the left of the line. Firstly, they count onto the next multiple
of ten, then count on to the multiple of ten before the higher number.
Finally they count on to the higher number. Then they add together the
jumps that they have taken (40 + 6 + 1 = 47).
Some children may split the jump of 40 into a ten and a 30 to ease
going over the hundred; others may complete it in two jumps from 389
to 400 and then from 400 to 436. (This method is also taught and used
with smaller numbers.)
Number Line – counting on to find the difference between numbers
430 436
+1
+40
+6
390389
486 – 389 = 47

Initially children will be taught column subtraction with numbers that
do not involve exchanging (sometimes referred to as borrowing,
decomposition or carrying) tens for ones, hundreds for tens and so
on. This method will be developed, with the aid of concrete
resources, to use exchanging.
Digits are arranged in columns, with the
larger number on top. Starting with the
digit of least value (in this sum, the
ones), children subtract vertically:
Ones: 4 – 3 = 1
Tens: 70 – 20 = 50
Hundreds: 800 – 500 = 300
The answer to each part of the
calculation is recorded in the
appropriate column below the line.
H T O
8 7 4
- 5 2 3
3 5 1
Vertical (Column) Subtraction – no exchanging

In the first example below, you cannot do 2 – 7 (for this method), so a
ten needs to be exchanged for ten ones, thus making 12 – 7 = 5. Due to
the exchanging, there are now 2 tens and not 3 on the top number.
20 – 50 cannot be done so (for this method), so a hundred is exchanged
for ten tens, thus making 120 – 50 = 70.
The second example has been included to show an example with a
zero.
Vertical (Column) Subtraction - with exchanging
8 1 9
-

Multiplication Section Contents
• Arrays
• Number lines/tracks
• Partitioning and the grid method
• Using the grid method to multiply 2 digit
numbers
• Vertical (column) multiplication – expanded
• Vertical (column) multiplication - compact

problem.
How many socks in three
pairs?
There are five cakes in each
bag.
How many cakes are there
in three bags?
3 pairs, 2 socks in each
pair, 6 socks altogether.
II II II
3 bags, 5 cakes in each
bag, 15 cakes altogether.
I I I I I
I I I I I
I I I I I
Click here to return to contents for multiplication section. Click here to return to main contents.

Multiplication problems can be represented in arrays.
2 x 5 or 5 x 2
6 x 4 or 4 x 6
Arrays

Pupils show and calculate multiplication problems on a number line.
This is sometimes referred to as repeated addition.
5 x 3 =15 or 3 x 5 =15
0 15105
0 3 6 9 12 15
+3 +3 +3 +3 +3
+5 +5 +5
Number Lines/Tracks

In all areas of maths, pupils are taught to recognise the value of digits
and to partition numbers. In the grid method, pupils partition numbers
according to their value and multiply each part separately. Even when
pupils have progressed to more efficient methods, this method is
revisited as numbers become more complex. This method will reinforce
mental calculations, including doubling.
36 x 4 = 144
X 30 6
4 120 24
120 + 24 = 144
136 x 4 = 544
X 100 30 6
4 400 120 24
400 + 120 + 24 = 544
X 10 3 0.6
4 40 12 2.4
13.6 x 4 = 54.4
40 + 12 + 2.4 = 54.4
Partitioning and the grid method

The grid method can be used to multiply pairs of numbers that have
more than one digit, this also includes decimal numbers.
Using the grid method to multiply 2 digit numbers
X 30 4
(Total
s)
20 600 80 680
7 210 28 238
918
34 X 27 = 918

As with the grid method, pupils need to be able to partition the digits and
understand their value. Even when pupils have progressed to more
efficient methods, this method is revisited as numbers become more
complex.
H T O
3 6
X 4
2 4 (4 X 6) Digits are multiplied, starting
1 2 0 (4 X 30) with the lowest value digit.
1 4 4 (24 + 120) The new columns are added,
starting with the digit of least value.
Vertical (Column) Multiplication - Expanded
Click here for more examples.

Multiply a 2 digit number by a
2 digit number
T H T O
3 6
X 7 4
2 4 (4 X 6)
1 2 0 (4 X 30)
4 2 0 (70 x 6)
2 1 0 0 (70 x 30)
2 6 6 4
Multiply a 2 digit decimal
number by a 1 digit number
T O . t
3 . 6
X 4
2 . 4 (4 X 0.6)
1 2 . 0 (4 X 3)
1 4 . 4 (2.4 + 12)
Examples of more complex vertical (column) multiplication - expanded

When children fully understand the maths behind the expanded vertical
method, they will be taught a compact vertical method. Initially this will
be for multiplying by a 1 digit number and, when ready, for multiplying
larger numbers.
T O
3 6
X 4
1 4 4
2
4 x 6 is 24. Pupils write the 4
in the ones column and carry
the 2 tens below the line in the
tens column. Then they do 4 x
30 which is 120, add on the 2
tens and make 140.
H T O
2 4
X 1 6
1 4 4
2
2 4 0
3 8 4
The method for this more
advanced calculation is the
same as the first one, but first
the 24 is multiplied by 6 and then
by 10. Then the two totals are
added together.
Vertical (Column) Multiplication - Compact

Division Section Contents
• Arrays
• Number lines/tracks
• Number line - chunking
• Bus stop method – short division
• Vertical chunking
• Bus stop method – long division

problem, either by sharing or grouping.
6 cakes shared between 2
people. How many cakes
each?
6 pencils are put into packs of
2. How many packs are there?
(Grouping into twos)
II II II
How many apples in each bowl
if I share 12 apples between 3
bowls?
Four eggs fit into a box. How
many boxes would you need to
pack 20 eggs? (Grouping into
fours)
Click here to return to contents for division section. Click here to return to main contents.

Arrays can be used to solve division problems, for example:
10 ÷ 5 or 10 ÷ 2
24 ÷ 4 or 24 ÷ 6
Arrays
Links can easily be made to fractions. For example, the first array could
be used to calculate ½ of 10 and the second one to calculate ¼ of 24.

Pupils use number lines to count up and see how many lots of a
number are in a given number. They count up in one ‘lot’ (jump of the
multiple) each time to see how many ‘lots’ there are. Pupils will also be
given problems with remainders.
0 15105
+5 +5 +5
Number Lines/Tracks
16 ÷ 3 = 5 r1 or 16 ÷ 5 = 3 r1
0 3 6 9 12 15
+3 +3 +3 +3 +3
r1
r1

Number Line - Chunking
As pupils become more confident in the concept of division and their
recall speed of times tables increases, they begin to use larger jumps
(chunks) or more ‘lots’ of the multiple. This is called chunking.
In this example, the first jump is of 20 lots of 4. The second jump is of 4
lots of 4. Therefore there are 24 (20 + 4) lots of 4 altogether in 96.
98 ÷ 4 = 24 r2
0 80 96
20 x
4 x
r2
Chunking on the number line is also used for dividing by 2 digit
numbers.
0 240 264
10 x
1 x
283 ÷ 24 = 11 r19
r19

Bus Stop – Short Division (Dividing by numbers with only one digit)
When pupils are secure with the concept of division, they are taught the
‘bus stop’ method.
353 ÷ 4 = 88 r1
To work out this sum, divide 353 by 4, one digit at a time, starting from
the left. The remainder of each part of the calculation is carried on to
the next digit.
4 3 5 3
0 8 8
3 3
R 1
Before moving on to divide by two digit numbers, pupils will use this
method for larger numbers and decimals.
Pupils will also be taught how to calculate the remainder as a
fraction: 88 ¼ or 88.25

Vertical Chunking with numbers of more than one digit
For this method, children solve division problems by repeated
subtraction. They are taking away chunks (lots of) a number at a time.
The size of the chunk (number of lots) is recorded to the left of the sum
and the subtraction carried down. At the end the children count up and
see how many lots of the number have been taken away and what the
remainder is. The subtraction part of the sum is completed as the
vertical compact method.
56 13 2 ÷ 1 7
(20 X 17) - 3 4 0
2 9 2
(10 X 17) - 1 7 0
1 112 12
(5 X 17) - 8 5
3 7
(2 X 17) - 3 4
Remainder 3
632 ÷ 17 = 37 r3

Bus Stop – Long Division (dividing by numbers with more than one digit)
2 7
3 6 9 7 2
- 7 2 0 (20x36)
This is the biggest chunk (lot) of 36 that can be taken away. A 2 is recorded on
the bus stop (tens column) and the remainder to be divided (252) is brought
down.
2 5 2 (7x36) 252 is 7 lots of 36 so this is recorded on the bus stop in the ones column.
Answer: 972 ÷ 36 = 27
During their primary education, some pupils may learn long division. This
is a method that is taught.

Maths methods for blogs

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