Transcript Progression in Calculations

÷
Introduction
Written methods of calculations are based on mental strategies. Each of the
four operations builds on secure mental skills which provide the foundation for
jottings and informal written methods of recording. Skills need to be taught,
practised and reviewed constantly. These skills lead on to more formal written
methods of calculation.
Strategies for calculation must be supported by familiar models and images.
When approaching a new strategy it is important to start with numbers that
the child can easily manipulate so that they have every opportunity to fully
grasp each concept.
The transition between stages should not be hurried as not all children will be
ready to move on to the next stage at the same time, therefore the
progression in this document is outlined in stages. Previous stages may need to
be revisited to consolidate understanding before progressing. Failure to secure
understanding can lead to misconceptions later so it is essential learning is
personalised for every child to ensure solid mathematical foundations are laid
which can be built upon in the future
.
A sound understanding of the number system and the patterns within it is
essential for children to carry out calculations efficiently and accurately.
Mathematics is NOT just a memory game
Children need a level of understanding
Children MUST be encouraged to think
for themselves and to reason
Head first.
Can the calculation be done mentally more efficiently?
49 + 1
49
+ 1
£5.00 – £4.99
£5.00
- £4.99
Children need to develop understanding of number
Children need to understand the position of
numbers and how they relate to one another
Children need to understand
the value of numbers
(Place Value)
Children need to be
able to partition and
recombine numbers
Learn
number
bonds
Add/subtract
one to/from any
number
Add/subtract
ten to/from any
number
Learn
multiplication
tables and related
division facts
Learn how to tell
the time on an
anologue clock
Mental
Calculation
Learn facts about
measures e.g. 24
hours in a day,
100cm in a metre
Number Bonds
A number bond is an addition sum
with two numbers.
Knowing bonds to 10 then 20 then 100
helps with addition, both mental and
written.
e.g. 18 + 7 = 18 + 2 + 5 = 20
Partition 7 into 2 and 5 so that 2 can be
added to 18 to make 20 and then add 5
Year 1 – recognise and reason bonds up to 10 and then up to 20 and related subtraction facts
Year 2 – practise addition and subtraction bonds up to 20 to become increasingly fluent, use knowledge of
bonds to calculate and use related bonds to 100 using multiples of 10 e.g. 70 + 30
Year 3 – consolidate previous learning then investigate bonds of larger numbers, bonds to 1 using tenths,
fractions and decimals e.g. 0.1 + 0.9, 1/10 + 9/10
Year 4 – consolidate previous learning, decimal and fraction bonds bonds using hundredths 0.99 + 0.01 – link
to money and measures
Year 5 – practise fluency with bonds with one-, two- and three-decimal places, including links with money and
measures
Year 6 – consolidate understanding of bonds to three-decimal places to achieve fluency
Number
Bond
Practice
Progression in methods for addition
Number Track
0
1
2
3
4
5
6
7
8
9
10
Number Line
Expanded method
(partitioning and
recombining)
4 3
+ 2 8
7 1
1
4 0
+
3
2 0
+
8
6 0
+
1 1
7 0
+
1
= 7 1
Compact
Method
Stage 1 – Understanding Addition & Number Track
What happens
if we start at
7 and
add/count on
3?
Use a puppet to practise counting on. Practise
counting on/adding small numbers. If the puppet
makes a ‘mistake’ can the child spot it?
0
1
and
2
3
4
5
6
Combine two (or more)
sets of objects and
find out how many
there all together
7
8
9
Remember to use
the different
words linked to
‘addition’
10
Stage 2 – Introducing the number line – counting on
0
1
2
3
4
5
6
7
8
9
10
Use a puppet to reinforce counting forwards. Link to
number track. Start with a fully numbered number line
and then progress to encouraging the children to sketch
their own to help with calculation.
Ensure children
understand place value
e.g. 11 is one ten and
one unit or one
13 + 11
+ 10
13
Start on the
largest number
+ 1
23
Add the tens
24
… and then the units
Stage 3 – The Expanded Method (partitioning & recombining)
40
3
20
8
Use place value cards and place
value apparatus alongside
written jottings. Partition the
numbers into tens and units, add,
and then recombine.
4 0
+
3
2 0
+
8
7 0
+
1 = 7 1
1 0
Stage 4 – Compact Method
40
20
3
8
4 0
+
3
2 0
+
8
6 0
+
1 1
7 0
+
1 =
4 3
Link the expanded
method to the
compact method
+
2 8
7 1
1
7 1
Progression in methods for subtraction
Number Track
0
1
2
3
4
5
6
7
8
9
10
Number Line
Expanded method
(partitioning and
recombining)
3
-
4 3
30
40
3
10 +
- 20
10
7
and
6
1
2 7
1 6
Compact
Method
Stage 1 – Number Track (counting back) & taking away
What happens if
we start at 7
and take
away/count back
3?
Use a puppet to practise counting backwards.
Practise taking away small numbers. If the
puppet makes a ‘mistake’ can the child spot it?
0
1
2
3
4
5
6
Take away
objects from a
group and count
how many are left
7
8
9
Remember to use
the different
words linked to
‘subtraction’
10
Stage 2 – Introducing the number line
0
1
2
3
4
5
6
7
Use a puppet to reinforce counting backwards. Link to
number track. Start with a fully numbered number line
and then progress to encouraging the children to sketch
their own to help with calculation.
33 - 19
14
- 6
- 3
23
20
… and then the units
- 10
Count back
the tens
8
9
10
Start counting back
in ones and then
progress to larger
jumps
33
Start on the
largest number
Stage 3 – Expanded Method
43 - 27 = 16
to subtract 7 units we
need to exchange a ten
for ten units
30
-
40
10 +
20
10
3
7
and
6
Use place value apparatus
alongside written jottings.
Partition the numbers into
tens and units, subtract
and then recombine
Stage 4 – Compact Method
30
10 +
40
- 20
10
3
7
and
Is the
answer
sensible?
Link the expanded
method to the
compact method
6
3
1
4 3
-
2 7
1 6
Progression in methods for multiplication
Repeated addition
Arrays
Grid method
10
Compact method
4
5 16
× 2 7
1 1 2 0
3 9 2
1 5 1 2
1
(56 × 20)
(56 × 7)
10
100
3
30
2
20
6
100 + 30 + 20 + 6 = 156
Stage 1 – Repeated addition & …
Children need to understand
that multiplication is the same
as repeated addition. Find
opportunities to count in
groups e.g. socks, ‘fingers’ on 4
hand prints.
… arrays
Children need to be able
to see numbers as
arrays. An array is an
arrangement of a number
visually in rows and
columns
Stage 2 – The grid method
4 x 13
3
10
When learning the grid
method use place value
equipment to help see
the numbers.
4
4
10
3
40
12
Partition the numbers into tens and
units. Draw a grid and place the
partitioned numbers across the top
and down the side of the grid.
Multiply each of the part of the
partitioned numbers and write the
answers in the sections of the grid.
Lastly add together the answers to
find the final total.
40 + 12 = 52
12 x 13
10
10
100
3
30
2
20
6
100 + 30 + 20 + 6 = 156
Stage 3 – Long multiplication
Because you are
multiplying by ‘tens’
you must put a zero in
the units column
Then multiply
the two tens
by the units
(6) and then
the tens (5)
5 6
× 2 7
1 1 2 0
3 9 2
1 5 1 2
Next multiply the seven units by
the units (6) and then the tens (5).
Finally add the two totals together
to get a final answer
14
1
(56 × 20)
(56 × 7)
Progression in methods for division
… and grouping
Sharing …
Chunking
Compact
method
÷
560 ÷ 24
2 3 r 8
2 4
5 6 0
- 4 8 0
8 0
-
7 2
8
96 ÷ 5 = 19 r 1
96
- 50
( 10 lots of 5 )
46
- 25
21
- 20
1
( 5 lots of 5 )
Fact Box
1 x 5 = 5
5 x 5 = 25
10 x 5 = 50
Stage 1 - Sharing …
4 shared
by 2
Share objects practically
one at a time. Draw a
picture to show this. The
objects do not need to be
drawn these could just be
crosses.
8 divided
into equal
groups of
2
Divide objects practically into
equal groups. Draw a picture to
show this. The objects do not
need to be drawn these could
just be crosses.
… and grouping
Stage 2 – Using multiplication and division facts.
Children can use a number line to count up in the divided number. E.g. 30 ÷ 5.
Count up in 5s until you reach 30. How many jumps have you done?
0
5
10
96  5
Using times tables knowledge to inverse
division questions.
15
30
What basic facts
do I know about
the
5 times-table?
12 x 5 = 60
7 x 5 = 35
Remainder 1
96  5 = 19 r 1
Fact Box
2 x 5 = 10
5 x 5 = 25
10 x 5 = 50
Stage 3 – Short division and long division
96  7 = 13 r 5
10 + 3
7
r 5
70 + 26
1 3
7
9
560 ÷ 24
2 3 r 8
2 4
5 6 0
- 4 8 0
8 0
-
7 2
8
Is the
answer
sensible?
2
6
r 5
Progression in Calculations – by magnitude
Year 1 – U + U, U + multiple of 10, TU + multiple of 10, U – U, TU – U, TU – multiple of 10, counting
groups of objects in ones, twos, fives and tens, sharing objects in equal groups
Year 2 - U + U, TU + U, TU + TU , U - U, TU - U, TU – TU, simple multiplication, simple division
including with remainders
Year 3 - TU + TU, HTU + TU, HTU + HTU, TU - TU, HTU - TU, HTU – HTU, TU x U, TU ÷ U including
with remainders
Year 4 - TU + TU, HTU + TU, HTU + HTU, TU - TU, HTU - TU, HTU – HTU, TU x U, TU ÷ U including
with remainders
Year 5 – Add whole numbers and decimals to two decimal places, subtract whole numbers and
decimals to two decimal places, HTU x TU, TU x TU, U x decimal, TU ÷ U, HTU ÷ U
Year 6 - Add whole numbers and decimals to two decimal places, subtract whole numbers and
decimals to two decimal places, TU x U, HTU x U, decimal x U, TU x TU, HTU x TU, TU ÷ U, HTU ÷ U,
decimal ÷ U
Mathematical Language
Number sentence
e.g. 2 + 4, 5 – 3, 6 x 3, 12 ÷ 3
Partition
splitting a number up e.g. 123 … 100 + 20 + 3
Recombine
putting a number back together e.g. 100 + 20 + 3 … 123
Bridging
crossing over 10/100 etc
Exchanging
e.g. swapping a 10 for 10 ones
Place value
the value of each digit in a number e.g. hundreds, tens and ones (units)
Remember there are different words for +, -, x and ÷ to learn
in order to help solve mathematical word problems