÷. written methods of calculations are based on mental strategies. each of the four operations...
TRANSCRIPT
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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.
Introduction
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?
£5.00 – £4.99
£5.00 - £4.99
49 + 1
49 + 1
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
Learn multiplication
tables and related division
facts
Learn facts about measures e.g. 24 hours in a day, 100cm in
a metre
Learn how to tell the time on
an anologue clock
Add/subtract one to/from any number
Add/subtract ten to/from any
number
Mental Mental CalculationCalculation
Number Bonds
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 + 30Year 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 + 9/10Year 4 – consolidate previous learning, decimal and fraction bonds bonds using hundredths 0.99 + 0.01 – link to money and measuresYear 5 – practise fluency with bonds with one-, two- and three-decimal places, including links with money and measuresYear 6 – consolidate understanding of bonds to three-decimal places to achieve fluency
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
Number Bond
Practice
Progression in methods for addition
Compact Method
1 2 3 4 5 876 1090
Number Track
Number Line
Expanded method (partitioning and
recombining)
4 3
+ 2 8
1
7 1
4 0 + 3
2 0 + 8
6 0 + 1 1
7 0 + 1 = 7 1
Stage 1 – Understanding Addition & Number Track
1 2 3 4 5 876 1090
and
Use a puppet to practise counting on. Practise counting on/adding small numbers. If the
puppet makes a ‘mistake’ can the child spot it?
What happens if we start at 7 and add/count
on 3?
Combine two (or more) sets of objects and find out how many there all
together
Remember to use the different
words linked to ‘addition’
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.
+ 1+ 10
13 23 24
13 + 11
Ensure children understand place value e.g. 11 is one ten and
one unit or one
Start on the largest number
Add the tens … and then the units
Stage 3 – The Expanded Method (partitioning & recombining)
20 8
40 3
4 0 + 3
2 0 + 8
7 0 + 1 = 7 1
Use place value cards and place value apparatus
alongside written jottings. Partition the numbers into tens
and units, add, and then recombine.
1 0
4 3
+ 2 8
7 1
1
20 8
4 0 + 3
2 0 + 8
6 0 + 1 1
7 0 + 1 = 7 1
40 3
Link the expanded
method to the compact method
Stage 4 – Compact Method
Progression in methods for subtraction
Compact Method
1 2 3 4 5 876 1090
Number Track
Number Line
Expanded method (partitioning and
recombining)
4 3
- 2 7
1 6
13
40 3
- 20 7
10 and 6
10 +
30
Stage 1 – Number Track (counting back) & taking away
1 2 3 4 5 876 1090
Use a puppet to practise counting backwards. Practise taking away small numbers. If the
puppet makes a ‘mistake’ can the child spot it?
What happens if we start at 7 and take away/count
back 3?
Take away objects from a
group and count how many are
left
Remember to use the different words linked to
‘subtraction’
Stage 2 – Introducing the number line
0 1 2 3 4 5 6 7 8 9 10
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.
- 3 - 10
14 23 33
33 - 19
Start counting back in ones and then progress to
larger jumps
Start on the largest number
Count back the tens… and then the units
- 6
20
Stage 3 – Expanded Method
40 3
- 20 7
10 and 6
10 +30
to subtract 7 units we need to exchange
a ten for ten units
43 - 27 = 16
Use place value apparatus alongside written jottings. Partition the numbers into tens and units, subtract
and then recombine
Stage 4 – Compact Method
40 3
- 20 7
10 and 6
10 +30
4 3
- 2 7
1 6
13
Is the answer
sensible?
Link the expanded
method to the compact method
Progression in methods for multiplication
Compact method
Repeated addition
Arrays
Grid method
10 2
3
10 100 20
630
100 + 30 + 20 + 6 = 156
5 6 × 2 7 1 1 2 0 (56 × 20) 3 9 2 (56 × 7) 1 5 1 2
14
1
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.
… arraysChildren need to be
able to see numbers as arrays. An array is an
arrangement of a number visually in rows and columns
4 x 13
4
10 3
40 + 12 = 524
10 3
40 12
Stage 2 – The grid method When learning the grid method use place
value equipment to help see the numbers.
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.
10 2
3
10 100 20
630
100 + 30 + 20 + 6 = 156
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.
12 x 13
Stage 3 – Long multiplication
5 6 × 2 7 1 1 2 0 (56 × 20) 3 9 2 (56 × 7) 1 5 1 2
4
1
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)
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
1
Progression in methods for division
Compact method
Sharing …
Chunking
… and grouping
÷96 ÷ 5 = 19 r 1
96
- 50 ( 10 lots of 5 )
46
- 25 ( 5 lots of 5 )
21
- 20
1
560 ÷ 24
2 3 r 8
2 4 5 6 0
- 4 8 0
8 0
- 7 2
8
Fact Box1 x 5 = 5
5 x 5 = 2510 x 5 = 50
Stage 1 - Sharing …
… and grouping
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.
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.
4 shared by 2
8 divided into equal
groups of 2
Fact Box
2 x 5 = 10
5 x 5 = 25
10 x 5 = 50
Stage 2 – Using multiplication and division facts.
96 5
Using times tables knowledge to inverse division questions.
12 x 5 = 60
7 x 5 = 35
Remainder 1
96 5 = 19 r 1
What basic facts do I know about
the 5 times-table?
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 3010 15
560 ÷ 24
2 3 r 8
2 4 5 6 0
- 4 8 0
8 0
- 7 2
8
Stage 3 – Short division and long division
10 + 3 r 5
7 70 + 26
96 7 = 13 r 5
7 9 6
1 3 r 52
Is the answer
sensible?
Progression in Calculations – by magnitudeYear 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 groupsYear 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 remaindersYear 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