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Progression In Calculations at Lyndhurst First School. Addition and Subtraction Mathematical Calculations in School Today. This document is designed to help you to understand the calculation methods your child will be taught in school. When supporting your child at home with Maths work it would be helpful if you could reinforce these methods rather than teach them the way that you were taught. The methods are levelled according to ability and you would need to speak to you child’s teacher to find out which methods would currently be the most appropriate for your child to practice at home. Remember each child progresses at their own pace. Understanding Addition. (EY/1C/1B) Explanation The physical act of counting out a set number of objects, and combining two groups, is an important step for children to explore. This is best done in a practical play based context as much as possible. Use a Numicon shape and add on the 1 shape... What number do you have now? Find the new Numicon shape to cover over the top. 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. Initially this needs to be adding 1 more. Explore 1 more through simple songs and rhymes, for example ‘1 man went to mo’ or ‘1, 2, 3, 4, 5, once I caught a fish alive’. Eg. 4 plus 1 makes 5. Count out a set of 4 and another set of 1. Then count them altogether to reach a total of 5 Key Questions/Vocabulary More, more than, one more, after, add, plus, count, total, equals, makes What is the number after 6? How will you find out how many there are in total? Can you show me how you worked out 1 more than....? Success criteria •I can add one to a small group of objects up to a total of 10 and explain what I am doing. •I can count out 1p coins to pay for 2 items totalling up to 10p. Understanding Subtraction. (EY/1C/1B) Explanation The physical act of counting out a set number of objects and taking some away is an important step for children to explore. This is best done in a practical play based context as much as possible. Count out a given number of objects and take away 1 of them... How many do you have now? 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. Initially this needs to be taking away 1 from a set. Eg. Explore 1 less through simple songs and rhymes, for example ’10 green bottles’ or ‘5 little speckled frogs’. 7 take away 1 leaves 6 Count out 7 objects. Then remove 1 from that set and count the objects that are left over. Key Questions/Vocabulary Less, less than, one less, before, take away, subtract, leaves, equals What is the number before 5? How many different ways can you show me that 8 subtract 1 is 7? Can you show me how you worked out 1 less than.....? Success criteria •I can use objects to take away 1 from any number up to 10. • I can work out how much I have left from an amount up to 10p when I buy a sweet costing 1p (using 1p coins to work practically). Using a Number Track for Addition. (1B/1A) 1 2 3 4 5 6 7 8 9 10 One more than four is five Explanation Number tracks can be used for children to locate a number, learn the order of numbers, and to begin to add one and then more than one to a given number. Children need to be able to understand the order of numbers remains the same and that as we count on the numbers get bigger by 1. They need plenty of practise in counting objects and by rote... Count the stairs as you go up to bed... Count your footsteps as you walk across a room.... Write the numbers to 10 on separate pieces of paper and get your child to put them into order, counting to check., then progress to ordering numbers to 20. Begin to recall all pairs of numbers that total 10. Key Questions/Vocabulary Begin to know doubles up to Order Numicon double 5. Count on, add, more, more than, equals, totals, 1 – 10 set, place a makes shape on top of Success criteria Find the number that is one more than...? 4 the next one, • I can use a number track to more than...? noticing how count on and find the answer to an there is a Count on 3 places from 15, where do you land? addition sum up to 10 and then to difference of 1. How many is 5 more than 8? 20. Simple Jottings/Mark Making for Addition. (1B/1A) 13 + 2 = 15 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. It is not necessary to draw the number of marks under the answer. Children can also use objects, such as counters, sweets, beads, to create groups to combine to find the total of an addition. Key Questions/Vocabulary Count, count on, more, add, plus, sum, altogether, total, equals How many altogether? Find the number that is five more than...? Count on 6 more from 13, what number do you get to? Children need to begin to see that addition produces the same answer which ever way round it is solved eg 8 + 2=10 and 2+8=10. It is COMMUTATIVE. Lay the 8 and 2 Numicon shapes together and place another 2 then 8 on top to see it is the same. Encourage children to see that it is easier and quicker to count on from the largest number. Success criteria • I can use simple jottings to support the addition of two numbers up to a total of 10, then 20. • I can begin to select appropriate apparatus to support addition. Using a Number Track for Subtraction. (1B/1A) 1 2 3 4 5 Explanation Number tracks can be used for children to locate a number, learn the order of numbers, and to begin to find out one less and then a few less than a given number. Key Questions/Vocabulary Count back, take away, subtract, less than, leaves, equals Find the number that is one less than ....? Five less than....? Count back 4 places from 8, where do you land? How many is 5 less than 9? 6 7 8 9 10 One less than nine is eight Children need to be able to understand the order of numbers remains the same and that when counting back the numbers get smaller by 1. They need lots of practice in counting backwards... Count back as you go down the stairs.... Do a countdown from 10 or 20 before you leave the house....Write the numbers to 10 on separate pieces of paper and get your child to order them in reverse, then try from 20. Order Numicon 1 – 10 set, note the difference between two shapes next to each other is 1. Success criteria • I can use a number track to count back and find the answer to a given question starting from numbers up to 10, then up to 20. Simple Jottings/Mark Making for Subtraction (1B/1A) 12 - 3 = 9 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. Children can also use objects, such as counters, sweets, beads, to create the initial group and then physically take away the right number to find the answer to the subtraction. Key Question/Vocabulary Count, count back, subtract, take away, cross off Difference between, leaves, equals How many are left over? Find the number that is 6 less than...? Count back 5 from 16, what number do you get to? At this level, children need to see that when doing subtraction the biggest number needs to be first and you take away the smaller number. Success criteria • I can use simple jottings to support the subtraction of two numbers starting from numbers up to 10, then 20. • I can begin to select appropriate apparatus to support subtraction. Using a Blank Number Line for Addition. (1A/2C) 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. 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 within numbers to about 30. Remember to jump on from the biggest number! 15 + 3 = 18 15 4 + 17 = 21 18 17 Rapidly recall all pairs of numbers that have a total of 10. Know doubles to double 5 and begin to know doubles up to double 10. Key Questions/Vocabulary Count on, count on in ones, add, plus, more than, total, equals, makes Which number are you going to start your line with? How many ones jumps do you need to do? What number have you reached? Children can use Deines (one blocks) to place in the jumps and support the visual image of how many they need to add on. 21 Success criteria • I can use number lines to support the addition of two numbers starting from numbers up to 20, then 30. • I can use a number line to help me solve addition problems involving money , up to 30p, and measures, up to a similar amount. Using a Blank Number Line for Subtraction. (1A/2C) 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. Children learn to use ones jumps, subtracting single digit numbers and working within a range up to about 20, then 30. 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. 9–4= 5 5 14 – 6 = 8 9 Key Questions/Vocabulary Count back, count back in ones, less than, take away, Subtract, leaves, equals Where are you going to start your number line? Which number are you starting with? How many jumps back do you need to do? What number have you reached? 8 Children can use Deines (one blocks) to place in the jumps and support the visual image of how many they need to count back. 14 Success criteria • I can use number lines to support the subtraction of two numbers starting from numbers up to 20, then 30. • I can use a number line to help me solve subtraction problems involving money , up to 30p, and measures, up to a similar amount. Developing use of Number Lines, Adding Tens and Ones. (2B) Children need to understand the place value of each digit in order to partition 2-digit numbers into tens and ones. 26p + 12p = 38p Recall all + and – facts for each number to at least 10. Know doubles up to double 10 and their corresponding halves. +10 26p 36p 38p 33 + 21 = 54 +10 33 Key Questions/Vocabulary Addition, add, plus, more, more than, ten more, count in tens, one more, count in ones, total, equals, altogether How many tens jumps do you need to do? How many ones jumps? Inverse (children need to know at this stage that addition and subtraction are inverse operations, they undo each other) +10 43 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. When confident adding on ‘teen’ numbers, progress to adding numbers with more than one ten. 53 54 Success criteria • I can use number lines more efficiently to add on 2-digit numbers by adding on the tens and then the ones. Developing use of Number Lines, Subtracting Tens and Ones. (2B) Children need to understand the place value of each digit in order to partition 2-digit numbers into tens and ones. 48g – 13g = 35g -10 35g 38g 48g 116 – 24 = 92 -10 92 Key Questions/Vocabulary Subtract, take away, minus, less than, ten less, count back in tens, one less, count back in ones, leaves, equals, Difference between (we find the difference between when the numbers are close together and it is easier and quicker to count up than back. Eg 53 – 47, it is easier to count on from 47 to find the difference between the two numbers than count back 47 places.) 96 -10 106 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. When confident adding on ‘teen’ numbers, progress to adding numbers with more than one ten. 116 Success criteria • I can use number lines more efficiently to subtract a 2-digit number by counting back the tens jumps and then the ones jumps. Add or subtract 9 or 11 by Compensation (2B/2A) To subtract 9, -10 and then +1 To add 9, + 10 then -1 25 + 9 = 34 146 – 9 = 137 +10 25 -10 34 35 136 137 To subtract 11, -10 then - 1 To add 11, + 10 then + 1 85 – 11 = 74 117 + 11 = 128 -10 +10 117 146 127 128 74 75 Practical exploration with money, using 10p and 1p coins can help to Explanation support the understanding of When adding or taking away 9, children which way to compensate. are taught that it as quicker to add Key Questions/Vocabulary /subtract ten and then adjust by one Add, plus, more than, sum accordingly. This is why it is important Subtract, take away, that children recognise number minus, less than patterns to count on and back in tens Equals, leaves, totals from any number. adjust 85 Success Criteria • I can quickly add or subtract 9 from any 2 or 3 digit number by adding or subtracting 10 first and then adjusting by 1. • I can quickly add or subtract 11 from any 2 or 3 digit number by adding or subtracting 10 first and then adjusting by 1. Partition and Recombine. (2B/2A) To add 23 and 35....First add the number of tens, so 20 + 30 = 50 23 + 35 Then add the number of ones, so 3 + 5 = 8 Finally combine the answers to give the total, so 50 + 8 = 58 50 + 8 = 58 Explanation Some children find this strategy a quick and easy method for addition, that they soon are able to do it 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). Once they are confident in this method they can progress to partition and recombining 2-digit numbers where the ones do cross the tens boundary, eg 37 + 26. So 30+20=50, 7+6=13 Then 50 +13=63 Key Questions/Vocabulary Tens digit, ones digit, units Partition, split, recombine How many tens? How many ones? How many altogether? Confidently recall + and – facts to 10 then 20 (eg, 9 + 6, 13 – 7, 15 + 4) Know all + and – facts for multiples of 10 to 100 (eg, 30 + 50, 90 – 20) Success Criteria • I can do addition more efficiently by partitioning numbers into tens and ones and then recombining them. Develop Efficient Use of Number Lines. (3C/3B) 74 + 43 = 117 +40 +3 74 114 152 – 68 = 96 -60 -6 84 117 90 -2 150 152 Explanation Once children are confident and accurate in the use of tens and ones jumps, they can progress to using multiple of tens jumps. Encourage children to use their knowledge of number bonds to bridge to the nearest multiple of 10 to make counting easier (as in the second eg). Make sure they keep a record in their jumps of what they are doing so that they can check they have + or – the correct number. Key Questions/Vocabulary 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, inverse Equals, leaves, altogether Use mental recall of + and – facts to 20 and apply to problems. Know all the + and –facts for multiples of 5 to 100 (eg 35 + 45, 80 – 55). Derive rapidly all number pairs that total 100 (eg 61 + 39, 22 + 78) Success Criteria • I can add and subtract chunks of tens and ones to make my calculations more efficient. • I can bridge through the nearest multiple of ten. Partition and Recombine. (3C/3B) 346 + 38 300 + 70 + 14 300 + 80 + 4 = 384 Key Questions/Vocabulary Hundreds digit, tens digit, ones digit, Partition, split, recombine How many hundreds? tens? ones? How many altogether? 137 + 400 + 110 389 + 500 + 20 + Explanation At this phase. Children can partition and recombine numbers that may cross the tens or hundreds boundary. They will also be able to use this method with 3-digit numbers. 16 6 Deines can be used to create a clear visual image of the place value of each digit and supports the understanding when the tens or hundreds boundaries are crossed. = 526 Success Criteria • I can add 3 digit numbers by partitioning them into hundreds, tens and ones, adding them and then recombining to reach the total.