Stage 1: Count in ones to and across 100 forwards and backwards starting from 0, 1 and other numbers. Count in multiples of two, five and ten. Pupils use apparatus to explore addition as the inverse of subtraction. . 4 add 1 is 5 5 subtract 4 leaves 1 Rapid recall of all pairs of numbers totalling numbers up to 10. Recall addition facts for all numbers to 20. Use structured apparatus – i.e. Numicon, tens frames, abaci, etc. 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 to use of a fully numbered number-line. 3 + 2 = 5 ‘eight add two more makes ten’ ‘one more than four is five’ Stage 2: Continue practicing above skills. Count in steps of 2, 3 and 5, 10 forwards and backwards to and from any number. Reorder numbers when adding, i.e. with largest number, find bonds, etc. Add doubles and derive near doubles. Embed addition facts for all numbers to 20. Number bonds to 100. Double and half numbers below 100. Counting on from any number: Children embed 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
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 using empty number lines. Partitioning 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, which need adding to four tens and seven ones.” “I add five ones to 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 seven tens and two ones.” Teachers similarly advance to model the addition of two 3-digit numbers, e.g. 587 475 1 1 __ 1062 Pupils to be encouraged to consider mental strategies first. Formal columnar: 25 +47 25 +47 2 1 25 +47 2 1 25 +47 72 1
Subtraction: Counting Mental 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 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, etc. Subtraction as taking away from a group: Children develop a mental picture of the number system for use with calculation. A 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 Bridging through two numbers, i.e. 24 – 19 = 19 + 1 + 4 using number lines. Recall subtraction (and addition) for all Subtracting by counting back and on: Children begin to use Number line with all numbers labelled 0 1 2 3 4 5 6 7 8 9 10 11 12
and 5, forwards and backwards to and from zero. Count in tens from any number – link to counting teddies as well as a number square. Subtracting 11 by subtracting 10 and then 1 more. Move to subtracting 9 by subtracting 10 and adding 1 using apparatus. numbers to 20. numbered lines to support their own calculations, initially counting back in ones before beginning to work more efficiently. 13 – 5 = 8 -1 -1 -1 -1 -1 8 9 10 11 12 13 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. up and down in 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 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 Comparing two sets: comparison or difference. Finding the difference on a number line Counting on to subtract: Note: Finding the difference is often the most efficient way of solving a subtraction problem, e.g. 61 – 59 2,003 – 1,997
tenths – linking to visual image. on which is more efficient. 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. Count 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 facts 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 HTU –HT U and ThHTU – ThHTU: 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 Use apparatus and knowledge of value to subtract Continue to practice previous stage of column method, including expanded method: Children should continue to use empty number lines and use more formal written methods when numbers become too big or complex.
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 backwards in appropriate decimals and percentages. decimals, i.e. 3.8 - 2.5 = 1.3 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. and make links between 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 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. Counting back in tens and ones to solve an addition calculation: Counting back more efficiently:
6: Continue to practice previous skills. Count forwards and backwards in simple fractions, decimals and percentages. Bridging through decimals, i.e. 1.5 – 0.8 = 1.5 – 0.5 then -0.3 using empty number line. Ensure all children are confident recalling basic facts to 20 and deriving using place value. Make links between decimals, fractions and percentages. Second stage of column method: 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 Formal columnar:
from twelve ones, so that I have fives ones left. “I can now subtract four tens from six tens, which leaves me with two tens.” “I recombine two tens and fives ones to understand that I am left with Teachers similarly advance to model the subtraction of one 3-digit number from another, e.g.
Multiplication: Counting Mental strategies Rapid recall Written calculation and appropriate models and images to support conceptual understanding Stage 1: 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 conceptual understanding of multiplication: Use objects, pictorial representations and arrays to show the concept of multiplication: Stage 2: Count forwards and backwards in 2s, 3s, 5s and 10s from zero. Begin to understand and use inverse number operations: 10 2 5 Derive/recall doubles up to ten and derive/recall halves up to twenty. Recall odd and even numbers to Understanding multiplication as repeated addition: Investigate multiplication as repeated addition, so that the law of Arrays: 5 X 3 3 X 5 and Number lines: 6 X 4 = 24
Stories are used alongside a triad to help children understand links between number operations, e.g. “There are five pencils in two packs, which means that there are ten pencils altogether.” 20 in reference to structured apparatus. Recall & use multiplication facts for the 2X, 5X and 10X-tables. is understood. Whilst arrays are also modelled explicitly at this stage, it is important to note that they will continue to be a key model at later stages, alongside more formal methods of calculation. So: ‘Six taken four times” Stage 3: Counting forwards and backwards in 2s, 3s, 4s, 5s, 8s and 10s from zero. Count up and down in tenths. Use doubling to make connections between the 2X, 4X and 8X-tables. Understand that multiplication can be undertaken by partitioning numbers, e.g. 12 X 4 = 10 X 4 + 2 X 4 Introduce the structure of scaling: e.g. Find a ribbon that is 4 as long as the blue ribbon 2cm 8cm Recall odd and even numbers to 100 in reference to structured apparatus. Recall and use multiplication facts for the 2X, 3X, 4X, 5X, 8X and 10X tables. Relate multiplying a by 1-digit number using repeated addition and arrays to represent: Children use an number line to chunk efficiently: 4 X 12 = 48 4 X 10 = 40 4 X 2 = 8 3 X 13 = 39 7 X 13 = 91
digit Stage 4: Counting forwards and backwards in 2s, 3s, 4s, 5s, 7s, 8s, 10s, 25s and 1000s from zero. Count up and down in tenths and hundredths. Derive factor of numbers using models and images, e.g. Know what happens when a number is multiplied by zero or one. Use reordering to multiply three numbers. Recall & use multiplication facts for all times-tables up to 12 X 12. Relate multiplying a 3/2-digit by number with arrays towards using long/short multiplication: Relate multiplying a 3/2-digit by 1-digit number, now also setting it out as 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 Relate multiplying a by number with grid to using short multiplication: 18 X13 54 2 180 234
3 5 Stories are used alongside a triad to help children understand links between number operations, e.g. “15 children are asked to get into three groups and find out that there are five people in each group.” numbers to 20 in reference to structured apparatus. Recall and use multiplication facts for the 2X, 5X and 10X-tables. Through teacher modelling, children need to know that division is not 0 5 10 15 Stage 3: Counting forwards and backwards in 2s, 3s, 4s, 5s, 8s and 10s from zero. Use doubling to make connections between the 2X, 4X and 8X-tables. Understand that multiplication can be undertaken by partitioning numbers, e.g. 12 X 4 = 10 X 4 + 2 X 4 Introduce the structure of scaling: e.g. Find a ribbon that is 4 times as long as the blue ribbon. 2cm 8cm Recall odd and even numbers to 100 in reference to structured apparatus. Recall & use multiplication facts for the 2X, 3X, 4X, 5X, 8X and 10X tables. Dividing a 2-digit by 1-digit number, representing this efficiently on a number line: Children use an empty number line to chunk efficiently. 96 6 = 16 6 x 6 = 36 10 x 6 = 60 0 36 96 Stage 4: Counting forwards and backwards in 2s, 3s, 4s, 5s, 7s, 8s, 10s, 25s Derive factor of numbers using models and images. Know what happens when a number is multiplied by zero or one. Recall & use multiplication facts for all times-tables up to 12 X 12. Dividing a by 1-digit number, representing this efficiently on a number line, also in relation to long division: Children use an number line to chunk efficiently. 224 ÷ 8 = 28 8 x 8 = 64 20 x 8 = 160
and 1000s from zero. Use reordering to multiply three numbers. At this no remainders are present unless in a practical context. 0 64 224 28 28 8 224 8 224 - 160 (8 X 20) 20 X 8 = 160 64 …or… 64 - 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. Short 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
digit Stage 6: Consolidate all previous counting, including forwards and backwards in fractions. Perform mental calculations, including with numbers and different number operations. Recall & use multiplication facts for all times-tables up to 12 X 12. In addition, use facts confidently to make larger calculations. Dividing a by 2/1-digit number, in relation to long and then short division: By this stage, there is a statutory requirement that children can use formal written calculation methods, including long and short division. Use of visual – like the ones opposite – remain important. 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 moving from long division to short division. Remainders should be interpreted in the following way when short division is used: through rounding in an appropriate way to the context Long division: 432 ÷ 15 = 28 4/5 Short division: 138 ÷ 6 = 23