Points to consider The key focus in pre-counting is an understanding of the concept more, less and the same and an appreciation of how these are related. Children at the early stages of Early level develop these concepts by comparison and no counting is involved. Children often have some concept of ‘more’; this needs to be extended and refined. ‘Less’ is a more difficult concept and understanding can be developed by pairing the terms less and more to help develop an understanding of the relationship between the two. The number ranges e.g. 1 – 5 are guides only and should be adapted to suit the needs / ability levels of the pupils in your class. Order-irrelevance principle: When counting the number of objects in a set, the order they are counted in is irrelevant, as long as each object is counted and pupils know and understand that the last number is ‘the count’. Sometimes an item is counted more than once by pupils or two items are counted for one number, e.g.... se-ven so lots of practice is required. Can at any animal – the count stays the same. Once children become very familiar with counting collections they realise that numbers can be used to count anything. The teen numbers are often the most difficult for pupils. The oral language pattern of teen numbers is the reverse of the usual pattern of ‘tens first and then ones’, e.g....13 is ‘thir’ ‘teen’ and 31 is ‘thirty’ ‘one’. There is often difficulty distinguishing between 12 and 21. Confusion can also arise when saying ty and teen endings.
Suggested Written Recording WHILST THE MAIN EMPHASIS IN THIS NUMBER PROGRESSION IS ON MENTAL STRATEGIES TO SOLVE NUMERICAL PROBLEMS, IT IS IMPORTANT THAT WE REALISE THAT WRITTEN RECORDING IS AN IMPORTANT PART OF NUMERACY. The following points are important when considering what, when and where pupils should be recording. Pupils use of standard symbols and conventions, for example numbers 0-9, equals sign and operation will become more precise when recording their own explanations about a calculation. paper and pencil for informal jottings can help keep a track of calculations if the numbers are large. TU + TU possibly moving into HTU + TU or HTU+HTU Example of horizontal expanded method, splitting A jotting area can be created by dividing the page in the maths jotter rather than using a separate book. This helps the pupil and teacher understand how the calculation was undertaken. Recording their own ideas on paper with words, numbers, symbols, diagrams and pictures is part of pupils’ exploration of number. See think boards in ‘Points to Consider’ It is important for children to establish connections between practical experiences, symbols, language and patterns. 5 x 3 = 3 + 3 + 3 + 3 + 3 3 x 5= 5 +5 + 5 Introducing standard algorithms before pupils have fully established partitioning strategies can slow down the development of mental problem solving skills and number sense. 3 + 3 + 3 = 9 3 x 3 = 9 Children at this level begin to understand that diagrams and equations often provide a more efficient means of communicating ideas than textual explanations.
Invent their own pattern with numbers and communicate the ‘rule’ so that someone else can continue it. e.g. 1, 3, 5, 7 ‘Add on 2 each time’ Confidently describe a ‘rule’ for continuing a more complex number sequence. e.g. 4,8,16 ‘The rule is going up by 4 or adding on 4’ Describe, predict and record rule for spatial patterns. Use materials to compare and order e.g. make 3 different sized towers of cubes and then place towers in ascending order. Classify odd and even numbers up to 20 and beyond as appropriate and be able to describe the rule – can be shared equally whereas odds can’t. Classify odd and even numbers up to 100 and beyond and is able to describe the rule (move to linking the equal sharing to being the same as the two times table). Experiences and outcomes
supports the ‘contracted form’ to the calculations mainly involve 1-digit numbers. Using this method requires pupils to follow a precise procedure, working from right to left, and so on. It also involves a precise layout of the writing, aligning the digits in columns. Suggested Written Recording Semi-formal Strategies for multiplication These are well- organised, standardised, written strategies. Children can begin to use the ‘empty number lines’ efficiently by making bigger jumps. Or use the grid method (provide children with opportunities to create arrays for two-digit x one-digit multiplication as this leads them into recording the ‘grid method’ effectively). E.g.14 x 3 = e.g. 127 x 4 Semi-formal Strategies for division Chunking - pupil repeatedly takes away ‘chunks’ of the large number, where each ‘chunk’ is an easy multiple of the divisor, until the large number has been reduced to zero or the remainder is less than the divisor. At the same time the pupil keeps track of the total amount of ‘chunks’ subtracted, which eventually becomes the final result of the sum. Ensure pupils have a clear layout and that they understand the process for chunking to divide a single-digit number before moving into dividing by a two-digit number.
Suggested Written Recording Order of Operations Where a calculation contains only addition and subtraction, the operations should be carried out in the order they appear. a calculation contains only multiplication and division, the operations should be carried out in the order they appear. Where a calculation contains a combination of the 4 operations, multiplication and division take priority over addition and subtraction. Pupils should also set out their working correctly. When a calculation requires more than one operation, there are rules for ensuring that the operations are done in the correct Example 1 This calculation contains only addition and subtraction. In this case, the operations are carried out in the order they appear: = = = = = 8 Example 2 This calculation contains only multiplication and division. In this case, the operations are carried out in the order they appear: = = = = Example 3 In a calculation containing a mixture of the 4 operations the rule is that multiplication and division should be carried out before addition and subtraction. = ( is done before even though the subtraction appears = = Example 4 (Problem solving) Which operations should go in the boxes to make the equation true? 24 3 4 2 = 0 Example 5 (Problem solving) How many ways can you find to make the number 42? Each calculation should contain at one multiplication/division and one addition/subtraction.
to solve a mix of calculations in the form of equations and word problems children choose the strategy from * and ** and discuss confidently the most efficient method chosen. e.g. 19 x 3 = (10 x 3) + (9x 3) = 30 + 27 = 57 e.g. 3 x 18 = e.g. 268 x 8 = (200 x 8) + (60 x 8) + (8 x 8) = = 1,600 + 480 + 64 = 2,144 Understand that multiplication is commutative and use this to solve a problem by changing the order of the factors. a known fact to derive a new fact, e.g. use their ten times facts to work out the 9 times table 8 x 9 is the same as 9 x 8. This is just 10 x 8 – 1 x 8 = 80 – 8 = 72 46 x 9 is the same as 9 x 46. So 10 x 46 – 1 x 46 = 460 – 46 = 414 68 x 9 is the same as 9 x 68. So 10 x 68 – 1 x 68 = 680 – 68 = 612 * Progress will be evident when pupils can: ** Progress will be evident when pupils can: *** Progress will be evident when pupils can: Strategies to Mentally solve Multiplication and Division Problems Strategies to Mentally Multiplication and Division Problems