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# Calculation Progression

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Calculation Progression
Ysgol Llangatwg

Purpose of the Meeting To make parents aware of the requirements of the Foundation Phase and the National Curriculum. To demonstrate some strategies used in school, so parents can support their children at home. To suggest ways parents can help their children at home. To make parents aware of the Literacy and Numeracy Framework. To share examples of National Numeracy tests.

Number & Calculations Foundation Phase Outcomes 1-6
Children use mathematics as an integral part of classroom activities. They count, order, add and subtract numbers when solving problems involving up to 10 objects. They count on and back in steps of different sizes and from different numbers. They can read and write numbers up to 10. Outcome 5 Children choose the appropriate operation when solving addition or subtraction problems. They use mental calculation strategies to solve number, money and measures problems. They count sets of objects reliably, and use mental recall of number facts to 10 to add or subtract larger numbers. They order numbers to 100. Outcome 6 Children organise their work and try different approaches. They use place value in numbers up to 1000 to make approximations. They develop further mental strategies for adding and subtracting numbers with at least two digits. They use mental recall of the 2, 3, 4,5 and 10 multiplication tables in solving whole-number problems involving multiplication and division, including those giving rise to remainders. Outcome 1 Children participate, follow, respond to and join in with familiar number rhymes, stories, songs, activities and games. They show an awareness of number activities, recite, sign or indicate one or more numbers to five and count or indicate two objects. Outcome 2 Children use mathematics in day-to-day activities and in their play, responding appropriately to key vocabulary and questions. They join in rote counting of numbers from They recognise and name numbers 1 to 3, and count up to three objects reliably. They record numbers initially by making marks or drawing pictures. They begin to develop an understanding of one-to-one correspondence by matching pairs of different objects or pictures. They understand the concept of ‘one more’. Outcome 3 Children use familiar words in practical situations. They rote count to beyond 10, and onwards from a given small number. They carry out simple addition using 1-5 and understand that zero means none. They recognise and try to record numerals from one to nine. They understand the concept of ‘one less’. These are the outcomes in number children are expected to achieve during the Foundation Phase. By the end of year 2 most children should be at outcome 5 (equivalent to the old National Curriculum level 2). As you can see, the progress from start to finish in the Foundation Phase is huge. There is a big focus on children being able to USE and apply maths in various situations and to choose appropriate strategies

Number & Calculations Progression
Level 1-3 are the same as Foundation Phase Outcomes 4, 5 and 6. LEVEL 3 Pupils organise their work, check results, and try different approaches. They talk about and explain their work. They use and interpret mathematical symbols and diagrams. They find particular examples that satisfy a general statement. They develop further mental strategies for adding and subtracting numbers with at least two digits. They use mental recall of the 2, 3, 4,5 and 10 multiplication tables in solving whole-number problems involving multiplication and division, including those giving rise to remainders. LEVEL 4 Pupils develop their own strategies for solving problems, and present information and results systematically. They search for a solution by trying out ideas of their own. They use their understanding of place value to multiply and divide whole numbers by 10 and 100. They use a variety of mental and written methods for computation, including recall of multiplication facts up to 10x10. They add and subtract decimal to two places. They check their results are reasonable by considering the context or the size of the numbers. They use simple fractions and percentages to describe approximate parts of a whole. They recognise and describe number patterns and relationships and use simple formulae expressed in words. LEVEL 5 Pupils identify and obtain information to solve problems, and check whether their results are sensible in the context of the problem. They describe situations mathematically using symbols, words and diagrams and draw their own conclusions, explaining their reasoning. They make general statements of their own, based on available evidence. They use their understanding of place value to multiply and divide whole numbers and decimals. They order, add and subtract negative numbers. They check their solutions by applying inverse operations or estimating using approximations. They calculate fractional or percentage parts or quantities and measurements. They construct and use formulae involving one or two operations. LEVEL 6 Pupils solve complex problems by breaking them down into smaller tasks, and give some mathematical justifications to support their methods, arguments or conclusions. They interpret, discuss and synthesise information presented in a variety of mathematical forms. They use trial-and improvement methods involving approximating and ordering decimals. They calculate one number as a fraction or percentage of another. They use the equivalences between fractions, decimals and percentages and calculate using ratios in appropriate situations. They find and describe in words the rule for the next term or nth term of a sequence where the rule is linear, and they formulate and solve a variety of simple linear equations. They represent mappings expressed algebraically. These are the national curriculum levels for number and calculations. A level 4 at year 6 would be a good achievement. Once again the focus is on developing their own strategies and solving problems. Describing/explaining what they have done and why. As you can see what they are expected to learn from start to finish is huge!

Progression in Number/Calculation Foundation Phase
Singing songs and counting games Finding numbers around the classroom Chanting numbers-to 5 then 10. Count two objects. Identify 1 more than a number to 5. Use mathematics in their daily play, both inside and out and in roleplay. Recognise and name numbers 1-3 and reliably count three objects. Begin to record numbers by making marks or drawing. Begin to make one to one correspondence and match pairs of different objects together. Use ‘one more then’ and ‘one less than’ in their play. Develop mathematical language and use it in practical situations eg add, together. Rote count to beyond 10 from various numbers. Understand zero means none. Use simple addition using numbers to 5. Use mathematics across all classroom activities. (Role play) Count, order, add and subtract numbers when solving problems. They count on/back in steps of different sizes and from different numbers. Read/write numbers up to 10. Choose the appropriate ‘sum’ when solving addition/subtraction problems. Use mental calculation strategies to solve problems. Count sets of objects reliably. Use mental recall of number facts to 10 to add/subtract. Order numbers up to 100. Use halves and quarters in practical situations. Organise their work and try different approaches. Use place value to 1000 to make approximations. Develop more strategies for adding/subtracting numbers with at least 2 digits. Use mental recall of 2,3,5 and 10 times tables in solving whole number. Use multiplication and division Use decimal notation problems and those giving rise to remainders. This shows the sorts of activities children would engage in, in the Foundation Phase. It is generally progressive, but lots of these things would be ongoing and overlap.

Number and Calculation in the Foundation Phase
RANGE Develop an interest in number. Recognise importance of certain numbers and numerals to them selves eg. Birthdays, house numbers. Use number names accurately, matching the symbol to the sound. Count, read, write, compare and order numbers, and appreciate the conservation of number. Use numbers naturally in their play and daily activities, including number rhymes, songs, stories and counting activities from Wales and around the World. Experiment with numbers, observe numbers and patterns in the environment and everyday life. Begin to develop their mental calculation strategies, during counting and grouping activities, games and through day-to-day classroom activities. Progress from counting on or back in steps, to mental mathematics involving all four operations with small numbers, using their own methods to record their calculations. Explore patterns in numbers, tables and sequences. Begin to understand the relationships between addition and subtraction, between multiplication and division, between halving and doubling. Match pairs of objects in practical contexts, leading to an understanding of one-to-one correspondence. SKILLS Select and use appropriate mathematical ideas, equipment and materials to solve practical problems. Develop a variety of mental approaches and strategies. Estimate solutions to calculations, check their answers in various ways. Develop their mathematical language. Present their work orally, pictorially and in written form, moving on to using more formal methods of recording when they developmentally ready. Devise and refine informal, personal methods of recording mental calculations, gradually moving to using words and symbols in number sentences. Develop a variety of mental and written strategies of computation. Interpret solutions to calculations within the context of the given problem. Recognise patterns, sequences and relationships through practical activities and discussion. Investigate repeating patterns and relationships and make simple predictions. This is taken from the Foundation Phase Curriculum document and shows the skills and range expected in number in the Foundation phase.

Progression in Calculations
Y3 TU ± U TU ± near multiple of 10 HTU – HTU (small difference) U / TU x 10 / 100 TU x U (partition) TU ÷ U (linked to recall facts) Fractions of amounts (eg ¼ of £24) Rounding for estimation Y4 TU ± TU +/– multiples of 10 / 100 ThHTU – ThHTU (small difference) x / ÷ by 10 / (whole no. answer) M of 10 x U (including near multiples) Divide using factors (eg 2400 ÷ 20) Solve problems involving money, measures & time Fractions of quantities (eg 2/3 of 24 cm) This is a summary of what would be taught in each year group and is taken from the National Curriculum.

Progression in Calculations
Y5 +/– decimals: apply known facts & bridge through landmarks ThHTU – ThHTU (eg 7012 – 3984) x / ÷ whole nos. & decimals by 10 / 100 / 1000 Rounding for estimation (integers & decimals) Integers x / ÷ U Fractions of amounts (eg ¾ of £220) Y6 Mental multiplication and division with decimals (U.t x / U) Efficient written methods: - add and subtract whole numbers and decimals; - multiply and divide whole numbers and decimals by a one-digit number, TU x TU and HTU x TU And this is what would be taught in years 5 and 6.

Support for Calculations
Number lines Numicon 100 squares Jottings Partitioning Tables and division facts ( trios) Adjusting Estimating Rounding Inverse operations Mutilink EASE Numberline Key Resources : Piece of string, bead strings, Ruler Number Lines These are some of the resources we use to support the teaching and learning of maths. Some of these resources are available for you to look at. There are many more!

Important Things For Children To Know
Make one to one correspondence Recognise numbers Number bonds to 10 and 20 Symbols and equations Tables Counting in various ways eg2s, 5s and 10s Tens and units Understand place value-including using zero eg 102 Important mathematical language eg different ways of saying add/subtract More than/less than Estimating Before we have a look at some of the strategies we use, here are some concepts which are really important and useful for children to know, before they start attempting calculations. When we look at the strategies, you’ll be able to see why these concepts are so important and how they help children with more sophisticated and difficult problems and calculations.

Addition/Subtraction in the Foundation Phase
Counting , grouping, combining and subtracting objects Lots of practical experiences through play Introducing the language of maths through practical every day activities The use of the outdoors and role play Demonstrating how to organise/write calculations Use of OWN numberlines Starts of by being very practical-using sets of objects such as themselves, toys, and often takes place through roleplay. What is always noticeable when children start to do ‘sums’ is that they are very good at adding and combining sets of objects but find it more difficult to take away. They can find1 more, but finding 1 less is much more difficult and the language of subtraction needs to be specifically taught. We don’t very often take things away from children or suggest they have 1 less biscuit than their brother or sister! But often we tell them ‘1 more’. It is important we model how to write formal sums, because one of the things they have to be able to do is record their findings.

Addition Written Methods- Number line Compact vertical 233
+416 649 Number line Partitioning = = 50 = 7 =57 Here are the strategies we, as a school have decided to teach. They are progressive, but children may use more than one of these strategies. Some children like to stick to one strategy, others are happy to use a variety. Next step would be bridging over 10. The number line is the next step form combining two sets of objects. We usually use number lines for numbers to 10 and 20. Children need to be taught to jump first. We encourage children to always use the biggest/highest number first and count on the smallest. An extension of this would be to use a 100 square, particularly for larger numbers. Partitioning is another method-probably not one you learnt in school! Partitioning means you separate the number into 10s and units. Then add the tens together, add the units and then add the answers together. You can see why it is important children know about place value. This would progress to bridging through 10. For example = =30 7+8=15 30+15= 30+10=40 =5 Compact Method-probably more familiar. Always start with the units. Vertical addition without regrouping first. Followed by regrouping. We might teach these strategies alongside each other. Children will decide on a method they feel happy with. Exchange rather than carry. 297 +165 462 11

Subtraction-1 Number Line 20 – 5 = 15 20 - 6 = 14 Partitioning
98 – 62 = 36 = 36 Find the difference by counting on/back depending on the numbers. (Estimating) = Count back from 20 = Count on from 18 Number line again. Jumping back-remembering to jump first. Simple algebra-Start at 20. Where am I trying to get to? How many jumps do I need to do? Finding the difference-children choose whether to count on or back depending on the numbers they are using. This is where estimating and number knowledge becomes really important. Partitioning-take away the tens first, then the units. Place value and concept of tens and units is really important.

Written Method– Decomposition
Subtraction-2 Written Method– Decomposition Without having to regroup tens/units 378 -254 124 With regrouping 3 5 8 -1 7 9 1 7 9 2 14 1 This is the formal, written method, sometimes known as decomposition. The language we use here is really important-We are exchanging or changing –NOT BORROWING and PAYING BACK. This is where it is really important children understand place value.

Multiplication Repeated Addition 5 x 3 or 3 x 5
Partition and Combine Method 17 x 4 10 x 4 = X 4 = 28 = 68 Expanded vertical 43 x 6 18 ( 3 x 6) (40 x 6) 258 Multiplication begins with repeated addition. 2 times 5 is the same as 5 and 5. Children have lots of practise at grouping sets of objects. We talk about ‘lots of’ and ‘groups of’. We count in sets of etc. When children understand the concept of multiplying, then we concentrate on tables-LEARNING THEM OFF BY HEART/CHANTING THEM. PARTITION METHOD-multiply tens , then units then add them together. With higher numbers -17x12= 17 x 10= x2= =204 1 2 3 4 5 o 3 6 9 12 15

Division Sharing-practically -10 sweets shared between 2.
Counting in 2s, 5s,10s etc. Chanting tables eg 1 x 2 =2 Learning Times Tables 2 x 5 = 10 How many 5’s in 10? 5 = 2 Division Signs / Language = 2 With bigger numbers, multiply up eg. 540 divided by 24. 2 Tables become really important when children are solving division sums. We don’t usually divide larger numbers, we use our tables knowledge. How many...in? Estimating is important-could that be the right number? Is it too high or too low? 2 10

Resources We Use Fingers! Mutilink Numberlines 100 squares Numicon
Multiplication grids Counters/cubes Base 10 These are just some-you will see others being used in the classroom

Lessons – What Do They Look Like ?
Rec -Y6 – children have a daily numeracy lesson ( mental/oral, main input, differentiated group work or independent task, plenary) Foundation Phase – maths happens throughout the day, children explore maths through a range of child initiated and adult led activities The Numeracy Framework places a firm emphasis on maths being used across the curriculum

Useful Websites http://www.bbc.co.uk/schools/bitesize/
Wide variety of apps for tablets etc. If you find a useful website, please let us know and we’ll add it our list

How You Can Help Your Child At Home
Talk to you child about his/her maths homework. Encourage them to talk through what they are doing and explain why they have chosen to do it that way. Counting and making one-to-one correspondence. Importance of adding on and taking away 1, 2 etc. Help your child to learn number bonds to 10/20. Help your child to learn tables-in the car, out for a walk. Anywhere! Use money with your child.

PROBLEM SOLVING PROBLEM SOLVING
These steps may help your child to solve the problem. Read the question carefully Check you understand what it means. Work out the problem What if I..? Calculate Do the sum. Have you done all the steps? What is the answer? Is it close to your estimate? Highlight all the important information Underline all the important words. What are you trying to find out? Is there more than one step? What information don’t you need? How did you do it? Think about, or tell someone else, how you did it. Why did you do it like that? What do I need to do to solve the problem? How are you going to work it out? Have a sensible guess/estimate. What sort of sum is it? Where shall we start? Is there a pattern? Is there anything that might help? Check Is your answer reasonable? Have you answered the question? Does it make sense? PROBLEM SOLVING These steps may help your child to solve the problem.

USEFUL MATHEMATICAL LANGUAGE
+ - plus altogether total makes and sum of more combine add increase count on take away minus count back leaves decrease less difference how many left? X repeated addition groups of lots of times multiply product pair double share divide group split share equally divisible divided by What fraction of? What percentage of? = is the same as totals equals balances

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