# Abercrombie Primary School Progression in Calculation 2014

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1 Abercrombie Primary School Progression in Calculation 204

3 Mental Methods First Children should always be encouraged to consider if a mental calculation would be appropriate before using written methods. These are covered in the first part of each section. When working out mental calculations, children will usually be taught to work with the numbers with the largest value first. E.g could be worked out by: = = 08 or = 00, = 8 so = 08 Children will be taught both methods and they will choose the best for them. They may also use their own methods for solving, as long as they are efficient. Methods will be modelled to others and encouraged to be used. Why do children need to do written calculations? To represent work that has been done practically. To support, record and explain mental calculation. To keep track of steps in a longer task. To work out calculations that are too difficult to do mentally. Children should be taught when it is appropriate to do an approximate or estimate for the calculation using a known mental method. Children will be taught that addition is the inverse of subtraction and that multiplication is the inverse of division. They will use the inverses to check their calculations are accurate. By upper Key Stage 2, children should be confident in choosing and using a strategy that they know will get them to the correct answer as efficiently as possible. Please note: If, after much practise, some children find the written method too difficult to understand, alternative methods will be taught to these children so that all children have the same opportunities to be fluent in number.

5 Glossary 2-digit: A number with 2 digits like 23, 45, 2 or digit: A number with 3 digits like 23, 542 or 903. Addition facts: Knowing that + = 2 and + 3 = 4 and = 7. Normally we only talk about number facts with totals of 20 and under. Array: An array is an arrangement of a set of numbers or objects in rows and columns it is mostly used to show how you can group objects for repeated addition. Bridge to ten: A strategy when using numberlines. Adding a number that takes you to the next tens number. Bus Stop Method: Traditional method for division with a single or two digit divisor. Concrete apparatus: Objects to help children count these are most often cubes (multilink) but can be anything they can hold and move. Dienes (hundreds, tens and units blocks), Numicon, Cuisenaire rods are also referred to as concrete apparatus. Decimal number: A number with a decimal point. Divisor: The smaller number in a division calculation. Double: Multiply a number by 2. Exchanging and regrouping: Moving a ten or a hundred from its column into the next column and splitting it up into ten ones (or units ) or ten tens and putting it into a different column. Expanded Multiplication: A method for multiplication where each stage is written down and then added up at the end in a column. Find the difference: A method for subtraction involving counting up from the smaller to the larger number. Grid method: A method for multiplying two numbers together involving partitioning. Grouping: Putting objects into groups of a certain size. Half: A number, shape or quantity divided into 2 equal parts. Halve: Divide a number by 2. Integer: A number with no decimal point (whole number). Inverse: The opposite operation. Addition is the inverse of subtraction; multiplication is the inverse of division. Long Multiplication: Column multiplication where only the significant figures are noted. Number bonds to ten: 2 numbers that add together to make ten, like 2 and 8, or 6 and 4.

6 Number bonds to 00: 2 numbers that add together to make 00 like 20 and 80, or 45 and 65 or 2 and 88. Numberline: A line either with numbers or without (a blank numberline). Children use this tool to help them count on for addition of subtraction and also in multiplication and division. Numberline Chunking: Method of division involving taking chunks or groups or the divisor away from the larger number. Number sentence: Writing out a calculation with just the numbers in a line E.g = 6 or 35 7 = 5 or 2 x 3 = 36 or 32 5 = 27 Partition: Split up a larger number into the hundreds, tens and units. E.g. 342 is 300 and 40 and 2. Significant digit: The digit in a number with the largest value. E.g. in 34: the most significant digit is the 3, as it has a value of 30 and the 4 only has a value of 4. Single digit: A number with only one digit. These are always less than 0. Taking away: A method for subtraction involving counting backwards from the larger to the smaller number. Tens number: A number in the ten times tables 0, 20, 30 etc. Unit: Another term for single digit numbers. The right hand column in column methods is the units column or the ones column. Place Value: Knowing that in the number 342 the 3 means 3 hundreds, the 4 means 4 tens and the 2 means 2. Quarter: A number, shape or quantity divided into 4 equal parts Recombine: For addition, once you have partitioned numbers into hundreds, tens and units then you have to add the hundreds together, and then add the tens to that total, and then add the units to that total. Remainder: A whole number left over after a division calculation. Repeated addition: Repeatedly adding groups of the same size for multiplication (e.g = 8 this would link to 6 X 3 = 8).

7 Place Value/Arrow Cards Resources that your children will use to help with calculation These help children when partitioning numbers and working out the place value of numbers. Dienes Cuisenaire Rods and a number track Numicon Although it has been used in schools for years, it is still a crucial step in knowing what a one (unit), a ten, a hundred and a thousand look like and how they can be added together and split up to form smaller and larger numbers. This concrete apparatus will be used alongside written calculations to support children s understanding. Numicon is an especially useful resource as it can be used for teaching all four operations as well as fractions, decimals, percentages and a range of other aspects of maths. Each piece represents an integer from to 0. The children love using it as it is colourful and tactile.

8 Cuisenaire Rods and Number Rods These little rods usually represent integers from to 0 but they can also be used to represent other numbers. They are really useful for all the number operations as well as looking at fraction and decimal equivalents. Numberlines Hundred Square Multiplication Square Numberlines are used for teaching calculations. We have pre - numbered and blank numberlines in school that children can write on, or they can draw their own as appropriate for the calculation. 3 + = 4 Age Appropriate Counters It is important that all children have access to age appropriate counters to help them with their maths calculations.

10 Progression through Addition Skills Non-standard written methods. Count up to 0 objects reliably (using apparatus) and then up to = 9 2. Find one more than a number. E.g. when given a number, they can count on to find one more e.g. one more than 3 is Add two or more groups of objects together to find a total of less than 0. These may be concrete apparatus or pictures. + + = 7 4. Use the + and = signs to record mental calculations in a number sentence. E.g = Count along a numberline to add single digit numbers together to find a total of less than 0. E.g = 9. Move onto numbers up to 0 in the same way Add single digit numbers that bridge to 0 using a numberline. This involves partitioning the smaller number in to 2 parts, one of which will add to the larger number to make 0 e.g = Add a 2-digit and a single digit number using a numberline e.g =

11 8. Add two 2-digit numbers bridging to 0 using a numberline. E.g = 36 and show the partitioning e.g (0+2) = Add two 2-digit numbers adding the most significant digit first using a blank numberline. E.g = 88 and show the partitioning e.g ( ) = Add two 2-digit numbers adding the most significant digit first using a blank numberline. E.g =

13 Progression through Subtraction Vocabulary Subtract Take Away Minus Less than Less Fewer Difference Decrease Subtraction can be seen in two ways: as taking away/counting back or as finding the difference/counting on. Knowledge Taking away is usually used when subtracting a small number from a much larger one; usually 2-digit subtract a single digit like This is sometimes called counting back. Must know number bonds to 0 and the reverse, e.g. + 9 = 0, , = 0 etc and 0 = 9, 0 2 = 8, 0 3 = 7 etc. Must know number bonds to 00 (sometimes called complements to 00) e.g = 00, = 00, = 57, etc. Understand the numberline as a continuum. A numberline is just a tool that helps us count forwards and backwards it has no official starting or ending point. Unlike with addition, subtraction cannot be calculated in any order, e.g. 9 4 = 5 is not the same as 4 9 = -5. Understand place value, e.g. knows that in the number 327, the 3 means 3 hundreds, the 2 means 20 and the 7 means. Addition and subtraction inverses e.g = = 6 = 5 5 = 6. In mental work, when subtracting 9, subtract 0 and add (This could be shown on a numberline). e.g =

14 Progression through Subtraction Skills Non-standard written methods. Use concrete apparatus to physically take away from numbers up to 0 and then up to 20. Imagine you have 7 teddies and you want to take away 3 of them. For this calculation you would have 7 teddies and physically take 3 away from them. 2. Find one less than a number, e.g. when given a number, they can count back to find one less e.g. one less than 3 is Use the and = signs to record mental calculations in a number sentence. e.g. 6 2 = When counting back, count underneath the numberline Count back on a numberline to subtract single digit numbers less than 0 E.g. 9 2 = Count back on a numberline to subtract single digit numbers less than 20 e.g. 7 5 = Subtract single digit numbers that bridge to 0 using a numberline. This involves partitioning the smaller number into 2 parts, one of which will be subtracted from the larger number to make 0 e.g. 3 5= 8 (3 3 2 = 8)

15 7. Use counting back with a blank numberline for larger two digit numbers e.g = 66 and show the partitioning. e.g. 78 (0 + 2) = Use counting back with a blank numberline for larger two digit numbers e.g = 22 and show the partitioning. e.g. 56 ( ) = Use counting back with a blank numberline for larger two digit numbers e.g = Finding the difference by counting on. By using a numberline, count from a smaller number to a larger one. E.g Start at 74 and count on to 00. The difference is the answer. When counting on, count above the line = =

17 Progression through Multiplication Vocabulary X Lots of Groups of Times Multiply Multiplication Multiple Product Double Twice Three Times Repeated Addition (Array column, row) Knowledge Understand place value, e.g. knows that in the number 327, the 3 means 3 hundreds, the 2 means 20 and the 7 means 7. Recognise simple sequences of numbers, e.g. 5, 0, 5, 20 (add five each time or count in 5s) 2, 4, 6, 8 (add 2 each time or count in 2s). Be able to use a method for adding and subtraction (see previous sections). Know that multiplication can be calculated in any order, e.g. 3 x 4 = 2 and 4 x 3 = 2. That multiplication and division are inverse of each other, e.g. 2 x 6 = 2 and 2 6 = 2. Can double and halve numbers from to 00, e.g. Double 4 is 8, 4 x 2 = 8; half of 8 is 4, 8 2 = 4. Multiplication is repeated addition, e.g. to find 4 x 3, you add 4 groups of 3, or you add 3 four times: = 2.

18 Progression through Multiplication Skills. Counting out loud in jumps. 2. Counting on in multiples using a hundred square to colour in jumps of 2s, 0s, 5s etc. 3. Use resources such as bead strings and unifix to show grouping. 4. Pictorial e.g. there are 3 sweets in one bag. How many sweets are there in 5 bags? 5. Use Cuisenaire rods on a numberline or number track or Numicon to multiply using repeated addition and the link between multiplication, e.g = Use a numbered numberline to count in jumps to show repeated addition and the link between multiplications. group of 6 group of 6 group of 6 = 3 group of Be able to show multiplication facts using arrays. You can show a number, e.g. 6, in several ways using pictures or objects. 2 rows of 3 are 6 3 rows of 2 are 6

19 8. Arrays can be linked to the use of Cuisenaire rods where children will see that 2 rows of 3 equal the same amount as 3 rows of 2. 3 x 2 = 6 2 x 3 = 6 9. Use times tables facts to make more efficient jumps on a blank numberline e.g. for 4 x 5, you could partition and add 0 x 5 to 4 x 5. 0 groups of 5 4 groups of Link the use of a numberline (above) to the use of grid method (below). Support with the use of Dienes equipment and use place value chart as necessary to record calculation. E.g. x x

20 Abercrombie Primary School CALCULATION POLICY ~ PROGRESSION THROUGH MULTPLICATION The more formal written examples below are in year groups to match in with the objectives from the National Curriculum (written in bold) Please Note: Calculations should always be written alongside the use of concrete apparatus using the place value sheets as appropriate Concrete apparatus and pictorial representations should be used as appropriate Y3 Y4 Y5 Y6 Use steps from previous page to teach the objectives above e.g. steps, 6, 7, 8 Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables. Write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental, grid method and progressing to formal written methods. Solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects. Recall multiplication facts up to 2 x 2. Multiply two-digit and three-digit numbers by a onedigit number using formal written layout, supported by grid method. E.g. 23 x 6 through grid method x to formal written methods ( Expanded columns) x 6 x 6 8 (6 x 3) (6 x 20) 38 Move to compact method above (only if children are ready they must understand what the method is showing and be able to explain it. Do not teach as a method only) Numbers carried when adding should go below the answer line. Solve problems involving multiplying and adding, including using the distributive law to multiply two digit numbers by one digit. E.g. a knife costs 70p and a fork cost 30p, how much for 5 sets of knives and forks? (70p + 30p) x 5 = (70 x 5) + (30 x 5) Pupils write statements about the equality of expressions (for example, use the distributive law 39 7 = and associative law (2 3) 4 = 2 (3 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations for example, 2 x 6 x 5 = 0 x 6 = 60. Multiply numbers up to 4 digits by a one or two-digit number using a formal method, including long multiplication for two-digit numbers. Use Grid method if needed for up to 2 digit x 2 digit and 3 digit x digit. (See Y4 opposite) Then move onto expanded columns for up to 2 digit x 2 or 3 digit numbers 23 X 35 5 (5x3) 00 (5 x 20) 90 (30x3) (30x20) 805 Next, move onto formal long multiplication (showing calculations in brackets) x (5 x 243) (0 x 243) NB: Multiplying units first. Place 0 before multiplying by 0 and explaining why this is done Numbers carried when multiplying should be crossed off when added on so there is no confusion later in the calculation Multiply numbers mentally drawing upon known facts Multiply whole numbers and those involving decimals by 0, 00 & 000. **For interactive representations try mathsframe.co.uk ~ Resources ~ New Mathematics Curriculum (204) / Addition & Subtraction Solve problems involving + x and a combination of these, including understanding meaning of = sign Multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication (including decimals) E.g. 326 x (8 x 326) (20 x 326) 3728 Solve problems involving addition, subtraction, multiplication and division Use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy.

21 Progression through Division Vocabulary Lots of Groups of Share Knowledge Understand place value, e.g. knows that in the number 327, the 3 means 3 hundreds, the 2 means 20 and the 7 means 7. Put objects into groups of the same number. Recognise simple sequences of numbers. e.g. 5, 0, 5, 20 (add five each time or count in 5s) 2, 4, 6, 8 (add 2 each time or count in 2s). Be able to use a method for adding and subtraction (see previous sections). Recall multiplication facts up to 2 x 2 and derive division facts. E.g. 5 x 4 = 20, so 20 5 = 4 and 20 4 = 5. That multiplication and division are the inverse of each other. E.g. 2 x 6 = 2 and 2 6 = 2. Know that division cannot be calculated in any order e.g. 2 4 = 3 is not the same as 2 3 = 4. Know that grouping and sharing are not the same. To begin with children will group, rather than share.

22 Progression through Division Skills. Grouping using apparatus e.g. 2 teddies split into 3 groups of Sharing using apparatus e.g. If 0 boats are shared equally into 2 groups, how many in each group? You would physically share out the boats one for me, one for you 3. Use dots/pictures and circles on paper e.g = Division on a numberline, e.g. 2 4 = 3. This shows that these are 3 lots of 4 in 2. This will be taught alongside step 3 above to begin with to show the link

23 5. Use arrays to show the link between multiplication and division and write multiplication and division sentences which show the link. E.g. 3 x 2 = 6, 2 x 3 = 6, 2 6 = 2, 2 2 = 6. Arrays can be linked to the use of Cuisenaire rods where children will see that 2 rows of 3 equals the same amount as 3 rows of 2. They can then divide these up to write corresponding division facts with arrays and Cuisenaire. 3 x 2 = 6 2 x 3 = 6 Examples of how apparatus can be used with bus stop method (at any stage) to help with division = =

26 Even though the laws below look confusing, they are the basic rules which govern our teaching at Mathematics. The children won t be taught these laws as shown below but they will be taught the application of these laws through daily maths learning. It is worth noting how the laws work for multiplication and addition but not always for division or subtraction. Commutative Laws: Associative Laws: Distributive Law: a + b = b + a a b = b a (a + b) + c = a + (b + c) (a b) c = a (b c) a (b + c) = a b + a c Commutative Laws The "Commutative Laws" say we can swap numbers over and still get the same answer......when we add: a + b = b + a Example:... or when we multiply: a b = b a Example:

27 Associative Laws The "Associative Laws" say that it doesn't matter how we group the numbers (i.e. which we calculate first)... when we add: (a + b) + c = a + (b + c)... or when we multiply: (a + b) + c = a + (b + c) This: Has the same answer as this: (2 + 4) + 5 = = 2 + (4 + 5) = = Examples: This: (3 4) 5 = 2 5 = 60 Has the same answer as this: 3 (4 5) = 3 20 = 60 Uses: Sometimes it is easier to add or multiply in a different order: What is ? = 9 + (36 + 4) = = 59 Or to rearrange a little: What is 2 6 5? = (2 5) 6 = 0 6 = 60

28 Distributive Law The "Distributive Law" needs careful attention. This is what it lets us do: 3 lots of (2 + 4) is the same as 3 lots of 2 plus 3 lots of 4 So, the 3 can be "distributed" across the 2 + 4, into 3 2 and 3 4 And we write it like this: a (b + c) = a b + a c Uses: Sometimes it is easier to break up a difficult multiplication: Example: What is 6 204? = =, =,224 Or to combine: Example: What is ? = 6 (6 + 4) = 6 0 = 60 We can use it in subtraction too: Example: = (26 24) 3 = 2 3 = 6 We could use it for a long list of additions, too: Example: = ( ) 7 = 20 7 = 40

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