Progression In Calculations at Lyndhurst First School. Addition and Subtraction Mathematical Calculations in School Today. The phases indicate progression in the development of children’s jottings to support their calculations. The phases do not represent any specific Year Group or National Curriculum level. Remember each child progresses at their own pace.

Phase 1 - Understanding Addition and Subtraction. Understanding addition as combining two groups Children need to experience counting out a set of objects and combining them with another set of objects to make a total amount. Eg. 4 plus 2 makes 6 Count out a set of 4 and another set of 2. Then count them altogether to reach a total of 6. Understanding subtraction as taking away Children need to experience counting out a set number of objects and then removing/taking away a certain number from that group. Eg. 7 take away 3 leaves 4 Count out 7 objects. Then remove 4 from that set and count the objects that are left over. Vocabulary/Language More, more than, one more Less, less than, one less Add, plus, count Take away, subtract, Leaves, equals, makes Explanation The physical act of counting out a set number of objects, and combining two groups or taking some away, is an important step for children to explore. This is best done in a ‘real life/play’ context as much as possible.

Phase 1 - Using a Number track. 1 2 3 4 5 6 7 8 9 10 One more than four is five 1 2 3 4 5 6 7 8 9 10 One less than nine is eight Vocabulary/Language Find the number that is one more than one less than Count on from/count back from How many…? Explanation Number tracks can be used for children to locate a number, learn the order of numbers, and to begin to find out one more and one less.

Phase 1 – Simple jottings/mark making for addition 3 + 2 = 5 Vocabulary/Language Count, count on More, add, plus Sum, altogether, total, equals How many altogether? Explanation Simple mark making is the first stage of children’s independent jottings to help them solve additions. They draw or make the appropriate number of marks under each number then count them up to reach the total.

Phase 1 – Simple jottings/mark making for subtraction 5 - 2 = 3 Vocabulary/Language Count, count back Subtract, take away, cross off Difference between Leaves, equals How many are left over? Explanation Simple mark making is the first stage of children’s independent jottings to help them solve subtractions. They draw the initial number of objects and then cross off the number it says to take away and count the ones left over.

Phase 1 – Using a blank number line for addition. At phase 1, children learn to use ones jumps, adding single digit numbers and working within a range up to about 20. It is only necessary to record where they start and where they end up after adding on. They can then progress to using this method of single jumps when adding ‘teen’ numbers and working up to about 30. 5 + 3 = 8 4 + 7 = 11 5 8 7 11 Vocabulary/Language Count on, count on in ones Add, plus, more than, total Equals, makes Explanation Blank number lines are used to enable children to count on and back with more than one jump. Children are taught to draw their own number line and start with the biggest number. There is no need to write +1 in each jump.

Phase 1 – Using a blank number line for subtraction. At phase 1, children learn to use ones jumps, subtracting single digit numbers and working within a range up to about 20. Their recording methods should be the same as for addition, except that with subtraction they start at the right hand end of the line and jump back. They can progress to using larger numbers when they are secure in this method. 9 – 4 = 5 14 – 6 = 8 5 9 8 14 Vocabulary/Language Count back, count back in ones Less than, take away, subtract Leaves, equals Explanation Blank number lines are used to enable children to count on and back with more than one jump. Children are taught to draw their own blank number lines, enabling them to do calculations within any range of numbers. There is no need to write -1 in each jump.

Phase 2 – Using a hundred square for addition and subtraction 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Count on and back in tens Count on and back in ones Vocabulary/Language One more, one less, ten more, ten less Count on in ones/tens Count back in ones/tens Add, plus, more than Subtract, take away, minus, less than Explanation Working with a hundred square helps develop a childs’ understanding of the number system. Looking at the patterns in numbers along the rows, and up and down the columns, will support their counting in ones and tens.

Phase 2 – Developing use of number lines, adding tens. Children need to be able to work out ten more than any 2 digit number in order to progress to this stage. +10 26 + 10 = 36 26 36 33p + 20p = 53p +10 +10 33p 43p 53p Vocabulary/Language Addition, add, plus, more, more than Subtract, take away, minus, less than Ten more, ten less, count in tens Difference between Leaves, equals, altogether Explanation Once children can count on in ones jumps on a number line, they learn to do tens jumps. To begin with they add 10, then multiples of ten, to 2-digit numbers. We write +10 inside the jumps. Begin to use these strategies in context, such as money and measures.

Phase 2 – Developing use of number lines, subtracting tens. Children need to be able to work out ten less than any 2 digit number in order to progress to this stage. -10 43 – 10 = 33 33 43 61cm– 20cm = 41cm -10 -10 41cm 51cm 61cm Vocabulary/Language Addition, add, plus, more, more than Subtract, take away, minus, less than Ten more, ten less, count in tens Difference between Leaves, equals, altogether Explanation Once children can count back in ones accurately on a number line, they learn to jump back in tens. To begin with they take away 10, then multiples of ten, to 2-digit numbers. We write -10 inside the jumps.

Phase 2 – Developing use of number lines, adding tens and ones. Children need to understand place value in order to partition 2-digit numbers into tens and ones at this phase. +10 26p + 12p = 38p 26p 36p 38p 33 + 21 = 54 +10 +10 33 43 53 54 Vocabulary/Language Addition, add, plus, more, more than Subtract, take away, minus, less than Ten more, ten less, count in tens One more, one less, count in ones Difference between Leaves, equals, altogether Explanation When children understand that 15 is made up of one ten and five ones, they can learn a more efficient method of using a number line than just doing 15 single jumps. We do the number of tens jumps first, followed by the number of ones jumps.

Phase 2 – Developing use of number lines, subtracting tens and ones. Children need to understand place value in order to partition 2-digit numbers into tens and ones at this phase. -10 48g – 13g = 35g 35g 38g 48g 116 – 24 = 92 -10 -10 92 96 106 116 Vocabulary/Language Addition, add, plus, more, more than Subtract, take away, minus, less than Ten more, ten less, count in tens One more, one less, count in ones Difference between Leaves, equals, altogether Explanation Subtracting tens and ones is the same as for addition. Jottings are set out as shown, with a record of where you have reached kept under the line and the jumps done recorded over the line. There is no need to write +1 or -1 in the small jumps, this would be inefficient.

Phase 2 – Partition and recombine To add 23 and 35, First add the number of tens, so 20 + 30 = 50 Then add the number of ones, So 3 + 5 = 8 Finally combine the answers to give the total, so 50 + 8 = 58 23 + 35 50 + 8 = 58 Vocabulary/Language Tens digit, ones digit Partition, split, recombine How many tens? How many ones? How many altogether? Explanation Some children find this strategy a quick and easy method for addition, that they soon are able to do mentally. Initially it is important to give calculations where the ones digits do not total more than 10, (we say they don’t cross the tens boundary).

Phase 2 – Add or subtract 9, by compensation. To add 9, + 10 then -1 To take away 9, -10 and then +1 25 + 9 = 34 +10 25 34 35 46 – 9 = 37 -10 36 37 46 Vocabulary/Language Add, plus, more than, sum Subtract, take away, minus, less than Equals, leaves, totals adjust Explanation When adding or taking away 9, children are taught that it as quicker to add/subtract ten and then adjust by one accordingly.

Phase 3 – Develop efficient use of number lines. 74 + 43 = 117 +40 +3 74 114 117 158 – 62 = 96 -60 -2 96 98 158 Vocabulary/Language Addition, add, plus, more, more than Subtract, take away, minus, less than Ten more, ten less, count in tens One more, one less, count in ones Difference between Equals, leaves, altogether Explanation Once children are confident and accurate in the use of tens and ones jumps, they can progress to using multiple of tens jumps. This obviously requires them to be able to add/subtract multiples of tens in their head.

Phase 3 – Partition and recombine 46 + 38 125 + 349 70 + 14 400 + 60 + 14 80 + 4 = 84 400 + 70 + 4 = 474 Vocabulary/Language Hundreds digit, tens digit, ones digit Partition, split, recombine How many hundreds? tens? ones? How many altogether? Explanation At this phase. Children can partition and recombine numbers that may cross the tens boundary. They will also be able to use this method with 3-digit numbers.

Phase 3 - Compensation 25 + 19 = 44 (+20, then -1) +100 25 44 45 754 850 854 63 – 28 = 35 (-30, then + 2) -30 33 34 35 63 Vocabulary/Language Add, plus, more than, sum Subtract, take away, minus, less than Equals, leaves, totals Multiples of tens, adjust Explanation To add, select the next multiple of 10 (or 100), add that number and then subtract the difference to compensate/adjust. To subtract, select the next multiple of 10 (or 100), subtract that number and then add to compensate.

Phase 3 – Subtraction as complementary addition Complementary addition is often used in the context of money and finding change. It is also useful when finding a small difference between two numbers. £1.00 – 63p = 37p +10 +10 +10 +7 63 70 80 90 100 106 – 98 = 8 +6 +2 98 100 106 Vocabulary/Language Subtract, take away, minus, less than Difference between Equals, leaves Count on, count up Explanation Start at the number being subtracted and then count on to the biggest number. When using this method, it is important to remember that the answer is now found above the line in the total that was added on.