Addition: Counting Mental maths strategies Rapid recall Written calculation and appropriate models and images to support conceptual understanding Stage 1: Count in ones to and across 100 forwards and backwards starting from 0, 1 and other numbers. Count in multiples of Pupils use apparatus to explore addition as the inverse of subtraction. . 4 add 1 is 5 Rapid recall of all pairs of numbers totalling numbers up to 20. Use structured apparatus – i.e. Numicon, Combining two groups: Children develop a mental picture of the number system for use with calculation. A range of key models and images support this, alongside practical equipment. Teachers model use of number tracks to count on or line up counters/objects along the number track. This is a precursor 3 + 2 = 5 ‘eight add two more makes ten’ ‘one more than four is five’
two, five and ten. 5 subtract 4 leaves 1 tens frames, abaci, etc. to use of a fully numbered number-line. Stage 2: Continue practicing above skills. Count in steps of 2, 3 and 5 forwards and backwards to and from zero. Count in tens from any number – link to coins in a piggy bank as well as a number square. Reorder numbers when adding, i.e. start with largest number, find bonds, etc. Add doubles and derive near doubles. Round numbers to the nearest 10. Recall addition facts for all numbers to 20. Counting on from any number: Children begin to use numbered lines to support their own calculations, initially counting in ones before beginning to work more efficiently. Counting on from the largest number: Children reorder calculations to start with the largest number. Number line with all numbers labelled 0 1 2 3 4 5 6 7 8 9 10 11 12 18 + 5 +1 +1 +1 +1 +1 18 19 20 21 22 23 24 …to… +2 +3 18 19 20 21 22 23 Use of questions such as: ‘How might I rearrange these to find the total?’ Stage 3: Continue practicing above skills. Count from 0 in multiples of 4, 8, 50 and 100. Count on by 10 or 100 Partitioning by bridging through 10 and multiples of 10 when adding. Adjusting when adding Connect pairs totalling ten to pairs of multiples of 10 totalling 100. Expanded horizontal addition: Add numbers using structured apparatus to support understanding of place value. Make connections between partitioning both numbers using structured apparatus and partition the second number only using a number line. Add… …and… By partitioning and recombining
from any two digit number. Link to counting stick: counting forwards and backwards flexibly. Count up and down in tenths – linking to visual image. 11 or 9 to a number. Relating inverse number operations – using structured apparatus to explore and understand that subtraction undoes addition. Use 10ps in tens frame. Recall pairs of two-digit numbers with a total of 100, i.e. 32 + ? = 100. 30+ 40 = 70 5 + 7 = 12 70 + 12 = 82 Stage 4: Continue practicing previous skills. Count forwards and backwards from 0 in multiples of 6, 7, 9, 25 and 1000 using counting sticks, number lines, number squares, etc. Count up and down in tenths, hundredths and simple Bridging through 60 for time, i.e. 70 minutes = 1 hour and 10 minutes. Rounding any number to the nearest 10, 100 or 1000. Rounding numbers with one decimal place to nearest whole number. Explore inverse as a way to derive new and As Use known facts and place value to derive new ones, i.e. ‘If I know 8 + 3 = 11, I also know 0.8 + 0.3 = 1.1 and 8/100 + 3/100 = 11/100.’ Sums and differences of pairs of multiples of 10, 100 or 1000. Expanded horizontal method, leading to columnar addition: Written recording should follow teacher modelling around the size of numbers and place value using a variety of concrete materials, e.g. straws, Numicon, Dienes and place-value cards. Teachers model how numbers can be partitioned into tens and ones, as well as in different ways, e.g. 20 + 5 10 + 15 As children move towards using a columnar method, links continue to be made with earlier models and images, including the number line. It is crucial that empty number lines are kept as well as using more formal written calculation methods.
fractions using models and images, i.e. Dienes equipment, counting stick, ITPs. to check accuracy of answers. Addition doubles of numbers to 100. Pairs of fractions totalling 1. Illustration of how to use Dienes equipment to ensure children have an understanding of place value when using columnar addition. 5: Count forwards and backwards in steps of powers of 10 for any given number up to one million. Continue to count forwards and backwards in simple fractions. Count forward and Use apparatus and knowledge of place value to add decimals, i.e. 3.8 + 2.5 = 5 + 1.3 Reorder increasingly complex calculations, i.e. 1.7 + 2.8 + 0.3 = 1.7 + 0.3 + 2.8 Compensating – i.e. 405 + Continue to practice previous and make links between known facts and addition pairs for fractions, percentages and decimals Doubles and halves of decimals, Expanded vertical method, leading to columnar addition: Teachers model a column method that records and explains partial mental methods. There remains an emphasis on the language of calculation, e.g. ‘Forty plus seventy equals and ten.’… ‘Seven add six equals thirteen.’ …before recombining numbers. Teachers also model the language of: ‘Four tens add seven tens total eleven tens or 110.’ Teachers similarly advance to model the addition of two 3-digit Informal columnar: Adding the tens first: Adding the ones
backwards in appropriate decimals and percentages. 399 → add 400 and then subtract 1. i.e. half of 5.6, double 3.4. Sums and differences of decimals, i.e. 6.5 + 2.7 numbers with the expectation that as children’s knowledge of place value is secured, they become ready to approach a formal compact method. Stage 6: Continue to practice previous skills. Count forwards and backwards in simple fractions, decimals and percentages. Bridging through decimals, i.e. 0.8 + 0.35 = 0.8 + 0.2 + 0.15 empty number lines. Partitioning using near doubles, i.e. 2.5 + 2.6 = 5 + 0.1 Reorder decimals, i.e. 4.7 + 5.6 – 0.7 …as… 4.7 – 0.7 + 5.6 = 4 + 5.6. Ensure all children are confident recalling basic facts to 20 and deriving facts using place value. Make links between decimals, fractions and percentages. Columnar addition (formal written method): The concept of exchange is introduced through continued use of practical equipment (manipulatives). Teachers model: “I have two tens and five ones, need adding to four tens and seven ones.” “I add five ones to seven ones, which gives me twelve ones.” “I exchange ten of my twelve ones for a ten counter.” “I add my three tens and four tens to make seven tens.” “Altogether, I have tens and two ones.” Teachers similarly advance to model the addition of two 3-digit numbers, e.g. Pupils to be encouraged to consider mental strategies first. Formal columnar: 25 +47 25 +47 2 1 25 +47 2 1 25 +47
Stage 1: in ones to and across 100, forwards and backwards starting from 0, 1 and other numbers. in multiples of two, five and ten. Pupils use apparatus to explore addition as the inverse of subtraction: . ‘four add one is five.’ ‘five subtract four leaves one’ Rapid recall of subtraction facts for numbers up to 10. Use structured apparatus, i.e. Numicon, tens frames, abaci etc. Subtraction as taking away from a group: Children develop a mental picture of the number system for use with calculation. A range of key models and images support this, alongside practical equipment. Teachers model use of number tracks to count back or remove counters/objects from the number track or set. This is a precursor to use of a fully numbered number-line. 5 - 2 = 3 ‘six take away two leaves four’ ‘one less than six is five’ Stage 2: Continue practicing above skills. Count in steps of 2, 3 and 5, forwards and backwards to Bridging through two digit numbers, i.e. 24 – 19 = 19 + 1 + 4 using number lines. Subtracting 11 by subtracting 10 and then 1 more. Move to subtracting 9 by subtracting 10 and Recall subtraction (and addition) facts for all numbers to 20. Subtracting by counting back and on: Children begin to use numbered lines to support their own calculations, Number line with all numbers labelled 0 1 2 3 4 5 6 7 8 9 10 11 12 13 – 5 = 8 -1 -1 -1 -1 -1 8 9 10 11 12 13
and from zero. Count in tens from any number – link to coins in a piggy bank as well as a number square. adding 1 using apparatus. initially counting back in ones before beginning to work more efficiently. 13 – 5 = 8 -2 -3 8 9 10 11 12 13 Stage 3: Continue practicing above skills. Count from 0 in multiples of 4, 8, 50 and 100. Count on and back by 10 or 100 from any two digit number. Link to counting stick counting forwards and backwards flexibly. Count up and down in tenths – linking to visual image. Partitioning by bridging through 10 and multiples of 10 when subtracting. Continue to practice adjusting when subtracting 11 or 9 from a number. Relating inverse number operations – use structured apparatus to explore and understand that subtraction undoes addition. Connect subtractions from ten to subtractions from multiples of 10 totalling 100. Use 10ps in tens frame. Subtract two digit numbers from 100 i.e. ? = 100 - 78 Finding the difference: Teachers model how to find the difference when two numbers are relatively ‘close together.’ Initially children compare two sets before moving on to a number line comparison. Pupils are taught to choose whether to count on or back depending on which is more efficient. Comparing two sets: comparison or difference. Finding the difference on a number line. Note: Finding the difference is often the most efficient way of solving a subtraction problem, e.g. 61 – 59 2,003 – 1,997
Stage 4: Continue practicing of previous skills. Count forwards and backwards from 0 in multiples of 6, 7, 9, 25 and 1000 using counting sticks, number lines, number squares, etc. up and down in tenths, hundredths and simple fractions using models and images, i.e. Dienes equipment, counting stick, ITPs. Bridging through 60 for time, i.e. 70 minutes = 1 hour and 10 minutes Rounding any number to the nearest 10, 100 or 1000. Rounding numbers with one decimal place to nearest whole number. Explore inverse as a way to derive new and to check accuracy of answers. As above. Use known facts and place value to derive new ones, i.e. ‘If I know 11 - 3 = 8, I also know 1.1 - 0.3 = 0.8 and 8/100 - 3/100 = 5/100.’ Sums and differences of pairs of multiples of 10, 100 or 1000. Subtraction of fractions totalling 1, i.e. 1 – 0.3 = 0.7 Subtracting TU – U and TU – TU: Use empty number lines to find the difference by bridging through multiples of ten. Subtract by starting with the first number and partitioning the second, i.e. 74 - 27 74 – 20 = 54 54 – 4 = 50 50 – 3 = 47 Stage 5: Count forwards and backwards in steps of powers of 10 for any given Use apparatus and knowledge of place value to subtract decimals, i.e. 3.8 - 2.5 = 1.3 Continue to practice previous stage and make links between stage of column method, including expanded method: Written recording should follow Children should continue to use number lines and use more formal written methods when numbers become too big or complex. Counting back in tens and ones to solve an addition calculation:
number up to one million. Continue to count forwards and backwards in simple fractions. Count forward and backwards in appropriate decimals and percentages. Reorder increasingly complex calculations, i.e. 1.7 – 5 – 0.7 = 1.7 – 0.7 – 5. Compensating – i.e. 405 - 399 → subtract 400 and then add 1. known facts and addition pairs for fractions, percentages and decimals. Doubles and halves of decimals, i.e. half of 5.6, double 3.4. Sums and differences of decimals, i.e. 6.5 + 2.7 teacher modelling around the size of numbers and place value using a variety of concrete materials, e.g. straws, Numicon, Dienes and place-value cards. Counting back more efficiently: Stage 6: Continue to practice Bridging through decimals, i.e. 1.5 – 0.8 = 1.5 – 0.5 then Ensure all children are confident Second stage of column method: Formal columnar:
previous skills. Count forwards and backwards in simple fractions, decimals and percentages. using empty number line. recalling basic facts to 20 and deriving using place value. Make links between decimals, fractions and percentages. The concept of exchange is introduced through continued use of practical equipment (manipulatives). Teachers model: “I have seven tens and two ones. I need to subtract four tens and seven ones.” “At the moment, I cannot subtract seven ones from two ones, so I need to transfer one ten to become ten ones.” “Now I can take away seven ones from twelve ones, so that
in 2s, 3s, 4s, 5s, 7s, 8s, 10s, 25s and 1000s from zero. Count up and down in tenths and hundredths. Know what happens when a number is multiplied by zero or one. Use reordering to multiply three numbers. up to 12 X 12. using long/short multiplication: 7 X 13 = 91 7 X 10 = 70 7 X 3 = 21 _____ = 91 At this stage, the non-statutory guidance in the national curriculum suggests teaching short multiplication; however, the team feel that an expanded form of calculation (as set out above) is be a better lead into long/short multiplication. Stage 5: Counting forwards and backwards in 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, 10s, 25s and 1000s from zero. Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers. Recall & use multiplication facts for all times-tables up to 12 X 12. Relate multiplying a 4/3/2-digit by 1/2-digit number with grid to using long multiplication: 18 X13 24 30 80 100 234 Stage 6: Consolidate all previous counting, including forwards and backwards in fractions. Perform mental calculations, including with mixed numbers and operations. Recall & use multiplication facts for all times-tables up to 12 X 12. In addition, use facts confidently to make larger calculations. Relate multiplying a 4/3/2-digit by 1/2-digit number with grid to using short multiplication: 18 X13 54 2 180 234 Division:
Counting Mental strategies Rapid recall Written calculation and appropriate models and images to support conceptual understanding Stage 1: Count forwards and backwards in 2s, 5s and 10s Doubling up to six and then ten whilst using related models and images. Derive/recall doubles up to five and derive/recall halves up to ten. Recall odd and even numbers to 10 in reference to structured apparatus. Developing early conceptual understanding of division as grouping and sharing: Use objects, pictorial representations and arrays to show the concept of division as grouping and sharing. “Two children share six pencils between them” “Six children are asked to get into three equal groups” 2: Count forwards and backwards in 2s, 3s, 5s and 10s from zero. Begin to understand and use inverse number operations. 15 3 5 Stories are used alongside a triad to help children understand links between number operations, e.g. “15 children are Derive/recall doubles up to ten and derive/recall halves up to twenty. Recall odd and even numbers to 20 in reference to structured apparatus. Understanding division as repeated subtraction: Investigate division as repeated subtraction. Through teacher modelling, children need to know that Number lines and arrays: 12 ÷ 3 = 4 15 5 = 3 0 5 10 15
- 64 (8 X 8) 8 X 8 = 64 0 Stage 5: Counting forwards and backwards in 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, 10s, 25s and 1000s from zero. Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers. Recall & use multiplication facts for all times-tables up to 12 X 12. Dividing a by 1-digit number, in relation to long division: By this stage, there is a statutory requirement that children can use a formal written calculation method, such as long division. division may begin to be taught alongside long division, but still with use of visual As schools have autonomy to decide children’s progression in learning between long and short division in Years 5 and 6, the maths team suggest beginning with long division. Remainders should be interpreted in the following ways when long division is used: as whole numbers as fractions through rounding in an appropriate way to the context Long division: 415 9 = 46 and 1/9 46 and 1/9 9 415 - 360 (9 X 40) 55 - 54 (9 X 6) 1 Stage 6: Consolidate all previous counting, including forwards and backwards in fractions. Perform mental calculations, including with mixed numbers and different number operations. Recall & use multiplication facts for all times-tables up to 12 X 12. In addition, use facts Dividing a by 2/1-digit number, in relation to long and then short division: By this stage, there As schools have autonomy to decide children’s progression in learning between long and division in Years 5 and 6, the maths team suggest moving from long division to short division. Remainders should be interpreted in the following way when short division is used: