The Parkland Federation. February 2016


 Randolph Hardy
 4 years ago
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1 The Parkland Federation February 206
2 EYFS/KS Calculations: Recording Addition (page of ). Aggregation/combining 2. Augmentation/counting on 3. Counting Contexts: Pupils physically combining groups Pupils counting objects moving each object in time to their verbal counting Combining (adding together) two or more quantities and finding the total. Combining groups of objects Buying 3 bags of crisps ( ) Pouring two cans of drink into a jug (ml) EG: A pile of 20 bananas Increase a given number/amount/quantity. Start at 5 and count on 4. Increase the quantity by a half. Ensure pupils meet the different structures within a range of different contexts. Moving objects onto a number line or recognised pattern. How much/many all together? What is the total when you add the amounts together? In this example, the pupil will end up with a pattern/group of 7 coins. The pattern will have a seveness about it. Recognising a group of objects has a sense of five or a threeness about it enables a pupil to use this as a known fact when combining or counting on. For other calculation methods, e.g: partitioning, see the KS2 section of the calculations policy as teaching methods should be similar.
3 KS/KS2 Calculations: Recording Addition (page of 2). Adding single digits (units or ones) Using a number line: Example: = Adding two digits (tens & units) Using a number line: A number line helps to break the number being added into manageable steps. There are many ways to jump along the line (three are shown below): Example: = Partitioning & Expanded Addition Partition (split) numbers into tens and units, add each part separately then recombine to find the total. This can be extended into larger numbers and is the first step towards using a column based method. Example: Using a number square: Example: = or (add all tens) = 50 (add all units) = 8 (recombine answers) = 58 This method of partitioning can then be used in simple column calculations where numbers are added vertically (downwards). Using a number square: Example: = 84 Count on in tens (3 tens) and then units (6) Example: = 58 Example: = 9 Example: = 592
4 KS/KS2 Calculations: Recording Addition (page 2 of 2) 4. Moving onto Column methods To build on the understanding of number and place value, developed through partitioning, it is important to continue to use the correct language that shows the value of each digit. However, column methods rely on adding the smallest numbers first so it is useful to do this from the start. Expanded column method: Example: = Add the units (or ones) seven add six Add the tens by saying forty add seventy is one hundred and ten 2 3 Total the numbers Even with column methods, it is important to use the correct language that shows the value of each digit. Whilst learning this method, some children find it useful to have the columns labelled: Example: (U) 0 0 (T) (H) 8 Example: Th (U) 5 0 (T) (H) (Th) Example: T U. t h (t). 0 (h) (U) (T) 4. 6 t = tenths h = hundredths Column Addition Example : Add the units: seven add six = The 3 stays in the units column. The one ten moves to the tens column 7 3 Add the tens: forty add twenty add ten (under the line) = 70 The 70 (or 7 tens) stay in the tens column Add the hundreds: Two hundred add nothing else = 200 The 200 (or 2 hundreds) stay in the hundreds column. Column Addition Example 2: Add the units as above (7 + 6 = 3) Add the tens: forty add seventy add ten + _ 7 6 (under the line) = The 20 stays in the tens column, the one hundred moves to the hundreds column Add the hundreds: Two hundred add one hundred (under the line) = 300. The 300 (3 hundreds) stay in the hundreds column.
5 EYFS/KS/KS2 Calculations: Recording Subtraction (page of 3). Subtracting single digits (units or ones) Using a labelled number line: Count back Example:  4 = Subtracting using a blank number line Counting back A blank number line helps to break subtraction questions into manageable steps. There are many ways to jump along the line but children find it easier to work around their knowledge of counting to tens, hundreds and thousands. Example: 57 = 8 3. Subtracting using a blank number line Counting forward to find the difference A blank number line can also be used to subtract by counting up from the smaller number to the larger. This is called finding the difference. Again, there are many ways to jump along the line but children find it easier to work around their knowledge of counting to tens, hundreds and thousands. Example: 227 = Mark zero and the two numbers on the number line. Example: 57 = 8 Using a number square: Count back In the example above, the child has counted back to the nearest multiple of 0 using a jump of 5. They then needed to count back 2 more to make a total of 7 in their jumps We want to take away 7, so we scribble away 7. Example: = 47 Worked out by counting back 20 then 4 then 3 (total =27) How many do we have left? Count up from 7 to the next multiple of ten, which is 20. Record the number landed on below the line. Count up from 20 to Jumps may be recorded in a different order. Here, the first jump lands of a multiple of ten. (Total of jumps = 27) The final landing place is Find the total of the jumps to give the answer = 5 So = 47 So = 5
6 KS/KS2 Calculations: Recording Subtraction (page 2 of 3) 4. Subtracting using a blank number line 5. Partitioning & Expanded Column method Partitioning & Expanded Column method Counting forward to find the difference Example: = 47 0 Larger numbers = 47 So = 47 As children become more confident, they will no longer need to mark on zero or scribble out the part being subtracted. They will record just the values on the number line that they need. Example: = = 48 As confidence grows, adding of jumps should become more efficient. Eg. Mentally adding hundreds, tens, units then recombining eg = 48 Some children will also be able to make more efficient jumps along the number line = 48 So = 48 Partition (split) numbers into tens and units and write each part underneath each other. This mirrors the column method but supports understanding of the value of each number. Answers are recombined to find the final answer. This can be extended into larger numbers and is the first step towards using a column based method. Example: 453 = = 32 Example: = = = 30 It is not possible to do 47 so a ten is borrowed from the is left behind in the tens column the 4 already in the units column means there is now 4 in the units column The answer can now be completed = 47 Example: = Can t calculate so borrow 0, leaving 30 behind Units now total 0 + = 307 = Can t calculate so borrow 00, leaving 600 behind Tens now total = = = 374 Example: = Where a question is so close to a place value boundary eg. 500, it is often more efficient to use a number line or mental method though a column method can be used Can t calculate Need to borrow a ten but there are none. The tens will need to borrow a hundred before being able to lend 90 a ten to the units = 225
7 KS/KS2 Calculations: Recording Subtraction (page 3 of 3) 6. Column Subtraction Whilst column subtraction is the last step in written calculations, it is essential to realise the importance of all the strategies that have come before it. Often a mental strategy, using an informal number line or finding the difference are more efficient methods to use than a formal written method. It is vital that the correct vocabulary is used when using column methods so that children understand the value of the number they are working with. Eg. If subtracting hundreds, say 6 hundreds subtract 4 hundreds, rather than 6 subtract 4 Example: 453 = = Example: = = Example: = = Example: = =
8 KS/KS2 Calculations: Recording Multiplication (page of 5) To be able to multiply successfully, children need to be able to: * Recall times tables facts speedily and accurately * Recognise that multiplication is commutative (can be done in any order) eg. 3 x 5 is the same as 5 x 3 * Use times table facts and place value to quickly calculate larger values eg. If 7 x 5 = 70 x 5 = 350 and 700 x 5 = then we can work out * Partition numbers into hundreds, tens & units It is important that children's mental strategies and accuracy of recalling times tables are continually practised alongside written calculations. Children need to be able to count times tables facts in order but also recall immediately when in a random order (without having to go through all facts to get to the answer). National Curriculum Expectations Pupils should, by the end of: Year * Be able to count in 2s, 5s, 0s Year 2 * Be able to count in 2s, 3s, 5s, 0s * Know by heart all x and facts for x2, x5 and x0 eg. 4 x 5 = 20 and 20 5 = 4 * Relate counting in 5s to the markings on a clock face. Year 3 * Know by heart all x and facts for x2, x5, x0, x3 and through doubling x2, know by heart all x4 and x8 facts and related facts * Understand how to use times table to get larger values eg. Use 4 x 3 to get to 40 x 3 Year 4 * Know by heart ALL times tables up to 2 x 2 and the related division facts. * Be able to multiple three numbers eg. 2 x 3 x 5 Year 5 * Multiply and divide whole numbers and decimals by 0/00 and 000 * Recognise prime numbers, square and cubed numbers Year 6 * Understand the order of calculations when using brackets. Early Stages of Multiplication Handson experience
9 KS/KS2 Calculations: Recording Multiplication (page 2 of 5). Early Stages of Multiplication Handson experience continued Understanding arrays 2. Early Stages of Multiplication Using times table facts and place value Any number multiplied by ten, moves up to the next column. Eg. 3 x 0 = 30 or 0 x 3 = 30 H The 3 moves up a column to the tens. A place holder (zero) is needed to show that there are no units left behind. Knowing how place value works is useful when using times tables facts and multiplying by multiples of ten. Eg. If we know 7 x 5 = 35, T We can use multiplying the question and the answer by 0 to help us work out 70 x 5 U It is important to avoid saying Add a zero in multiplication as this is NOT mathematically true. In this example 3 add 0 does not equal still equals 3 or = 3 Using times table facts and place value Any number multiplied by multiples of a hundred works in a similar way to that of x by 0 except that values move up by two columns. Example: If we know 3 x 6 = 8 we already know that 30 x 6 = 80 now we know 300 x 6 = 800 Th Th A place holder (zero) is needed again to show that there are no units left behind. To multiply by multiples of a thousand, move all values up by three columns. This method also works with decimals. Examples: 3 x 6 = x 6 = x 6 = x 0 = x 00 = Children need to recognise and understand that multiplication answers can be worked out using different arrays. In this example, 2 rows of 4 mean the same as 4 rows of 2. Both equal 8. H T 7 U H 7 x 5 = x x 5 5 = = So 70 x 5 = 350 (multiples of ten  any number in 0 times tables) T U The decimal point does NOT move. The numbers move.
10 KS/KS2 Calculations: Recording Multiplication (page 3 of 5) 3. Multiplying using partitioning Numbers are partitioned in the same way as addition and subtraction. Example : 2 4 x 3 Both 20 and 4 need to be 20 4 multiplied by 3 Step : 20 x 3 = 60 4 x 3 = 2 Final step is to add both answers. Step 2: = 72 Example 2: x 4 Step : 300 x 4 = x 4 = x 4 = and 50 and 2 need to be multiplied by 4 Step 2: Recombine or use column method to add answers = 408 Th (U) 0 (T) (H) _ (Th) This method uses two separate steps: Step = Working out multiplication answers. Step 2 = Separate addition calculation to work out final answer. 4. Moving on to column methods Expanded column multiplication This method relies on a secure understanding of place value, partitioning and how to add answers in columns. If any one of these skills is insecure, further work is needed before this method is used. As with addition, to build on the understanding of place value developed through partitioning, it is important to continue to use the correct language that shows the value of each digit. However, column methods rely on adding the smallest numbers first so it is useful to do this from the start. Example : 2 4 x 3 T U Partitioned x (U) (4 x 3 = 2) (T) (20 x 3 = 60) So 24 x 3 = 72 Example 2: x x (U) (6 x 4 = 24) (T) (30 x 4 = 20) (H) (200 x 4 = 800) So 236 x 4 = Column multiplication using partitioning (showing working out) Many children will quickly want to move on from expanded recording. Partitioning is still used verbally to support recording and the value of each digit should continue to be emphasised throughout. Example. 2 4 x 3 T U 2 4 x 3 2 (U) (4 x 3 = 2) 6 0 (T) (20 x 3 = 60) 7 2 So 24 x 3 = 72 Example 2: x x (U) (6 x 4 = 24) 2 0 (T) (30 x 4 = 20) (H) (200 x 4 = 800) So 236 x 4 = 944 As children become more confident with this method, they stop recording their calculations alongside the columns. It is vital that they still say all numbers correctly to avoid place value errors. eg. 30 x 4 (not three 4s) and 200 x 4 (not two 4s).
11 KS/KS2 Calculations: Recording Multiplication (page 4 of 5) 6. Column multiplication using Example. 2 4 x 3 T U 2 4 x 3 2 (U) 6 0 (T) 7 2 Example 2: x x (U) 2 0 (T) (H) Partitioning 7. Short Multiplication This method for multiplying by one digit, builds on all previous methods but answers are recorded on one line. Example. 2 4 x 3 T U 2 4 x 3 Calculation: four x three = The 2 stays in the units column. The one ten moves to the tens column. T U 2 4 Calculation: twenty x three = 60 x 3 or 6 tens. 7 2 There is one ten under the line that needs to be added to the 6 tens = 70 (6 tens + ten = 7 tens) The 70 (7 tens) stay in the tens column Example 2: x Step : 6 x 4 = 2 4 x or 2 tens Step 2: 30 x 4 = 20 x (2 tens) from Step = or 4 tens 00 or hundred Step 3: 200 x 4 = 800 x ( hundred) from Step 2 = or 9 hundreds Example 3: x 7 Example 3: x 7 Ten Th Th x 7 7 (U) (T) (H) (Th) Ten Th Th x
12 KS2 Calculations: Recording Multiplication (page 5 of 5) 8. Long Multiplication Multiplying by 2 digits T U x T U 9. Long Multiplication Multiplying by 2 digits x T U Example. 2 4 x x _ (Multiplying 24 by 6) (Multiplying 24 by 0) Explanation: Multiplying 24 by 6 ( 4 x 6 ) + ( 20 x 6 ) = 44 Once calculated 20 x 6, add the two tens from 24 to the total for the tens column Explanation: Multiplying 24 by 0 ( 4 x0 ) + ( 20 x 0 ) = 240 OR Use place value for multiplying by 0 Example x 4 6 Numbers moved or Th carried into next 3 2 column while x _ 4 6 multiplying 9 2 ( 2 x 6 ) + ( 30 x 6 ) ( 2 x 40 ) + (30 x 40 ) Example. 2 4 x 2 6 Numbers moved or carried into next 2 column while multiplying 2 4 x _ (Multiplying 24 by 6) (Multiplying 24 by 20) Explanation: Multiplying 24 by 6 ( 4 x 6 ) + ( 20 x 6 ) + (00 x 6 ) = 744 Explanation: Multiplying 24 by 20 ( 4 x 20 ) + ( 20 x 20 ) + (00 x 20 ) = 2480 OR Use place value for multiplying by 20 Numbers moved or carried into next column while adding final answers
13 KS/KS2 Calculations: Recording Division (page of 4) To be able to divide successfully, children need to be able to: * Know and be able to recall times tables * Recall division facts speedily and accurately eg. If we know 2 x 6 = 2 then 2 6 = 3 and if we know 6 x 2 = 2 then 2 2 = 6 * Recognise that division can only be done in the order that the question says eg. 5 3 is NOT the same as 3 5 * Use known division facts and place value to quickly calculate answers with larger numbers It is important that children's mental strategies and accuracy of recalling division facts are continually practised alongside written calculations. Children need to be able to use times tables facts in order to calculate division facts but also recall immediately when in a random order (without having to go through all facts to get to the answer). National Curriculum Expectations Pupils should, by the end of: Year * Be able to recognise that halving is the same as sharing between two and quarters mean sharing between 4 Year 2 * Be able to count backwards in 2s, 3s, 5s,0s * Know by heart all facts for 2s, 5s and 0s eg. 4 x 5 = 20 and 20 5 = 4 Year 3 * Know by heart all facts for 2s, 3s, 5s, 0s, 4s and 8s * Understand how to use division facts to calculate larger values eg. Use 9 3 to work out 90 3 Year 4 * Know by heart ALL division facts linked to times tables up to 2 x 2 Year 5 * Divide whole numbers and decimals by 0/00 and 000 Year 6 * Understand the order of calculations when using brackets. Early Stages of Division Handson experience Practically sharing objects between a number of groups 2 3 = 4 Using different strategies to work out 5 3 and linking it to 5 5
14 KS/KS2 Calculations: Recording Division (page 2 of 4) 2. Division using a number line As division is repeated subtraction, like subtraction a number line can be used to count back repeatedly: 0 2 (Count back in jumps of 2 to zero) There are 5 jumps so 0 2 = or count forward repeatedly until the total is reached: 42 3 is worked out by counting on in 3s until 42 is reached. 4 jumps of 3 are made. So 42 3 = Division using division facts (and applying knowledge of times tables) Before moving onto more formal methods, children need to have an understanding of number size so that they have an idea of how big answers should be. This will help them detect if any final answers are unreasonable. One informal method for division is drawing groups to represent the number being divided by (divisor) or by using the division spider. They then use division facts (or times tables) to share out the largest amounts they can until they have no more to share. Example: 46 4 = r (0 given to each group) 6 left to share  4 ( given to each group) 2 left to share There is not enough left to share equally so 2 becomes the remainder. Each group has with a remainder of 2. Division spider: Using a number line method can also be used to see where any division questions have remainders or objects/ numbers left over that can not be shared equally between the groups. Example: 27 fish shared between 6 nets. Counting on by 6 has been used but after 4 jumps there are only 3 fish left which is not enough for another jump of 6. They are left over or remainders. So 27 6 = 4 fish in each net with a remainder of 3 fish 27 6 = 4 r 3 eg Amount to share Number of groups (divisor) Using division facts (times tables) we know that each group will have 7 in it = 7 (because 7 x 5 = 35) We can also use this idea and a knowledge of place value to work out: = Using place value to divide by 0/00/000 As in multiplication, dividing any number by 0, 00, 000 moves all digits to new columns. Dividing has the opposite effect to multiplication. Dividing moves all digits into lower value places (moves digits to right) t 0 t The decimal point does not move Digits jump over it. Eg = 3 All digits move one column smaller Eg = All digits move two columns smaller Dividing by 000 = move three columns smaller 5 h 3 2
15 KS2 Calculations: Recording Division (page 3 of 4) 4. Short Division This method is the quickest method for division but relies entirely on a secure knowledge of division facts, times tables and, as long as answers are recorded in the correct place, can be calculated without using place value. For every calculation, How many..? needs to be asked is written as: T U How many 3s are there in 30? There are 0. (How many 3s fit in the first column?) There is ten. T U This digit has been dealt with so is crossed out. T U 3_ How many 3s are there in 9? There are 3. (How many 3s fit in the second column?) There are 3. So = 3 Example: is written as: Short Division 2 Continue to ask How many..? These examples show what to do if numbers do not fit exactly (are not divisible exactly). Example: is written as How many 7s are there in 9? There is with 2 left over. The 2 is moved on to the next column. Cross through the 9. _ 4_ How many 7s are there in 28? There are 4. So = 28 Example: is written as How many 5s in 4? Zero. Move the 4 to next column. How many 5s in 45? 9 but nothing left over to move on. How many 5s in? Zero. Move the to next column. How many 5s in 0? 2 but nothing left so finished. So = Short Division 3 (remainders) There are sometimes amounts left over with nowhere to put them. These become left overs or remainders. It has not been possible to divide them equally. In a worded question, children will need to make sure they know what these remainders mean. Eg. If sharing sweets, the remainders are left over sweets. If dividing children into teams, these are left over children who have not been put into a team. They will need to decide what to do with remainders in the context of the question. Example: 8 9 sweets 7 children = 2 remainder 5 _ 2_ remainder 5 or 2 r How many 7s in 8? with left over. Move on to next column. How many 7s in 9? 2 with 5 left over. There is nowhere to move the 5 to so it is left over. It is called a remainder. In this question, the left overs are sweets so a decision needs to be made as to what to do with them. Example: = remainder remainder 3 or r There is no worded question so the remainder can be left as a number value. So = 2 0
16 KS2 Calculations: Recording Division (page 4 of 4) 5. Long Division When dividing by 2 digits, a long division method is needed. It works in a similar way to short division but is based more on the place value understanding of earlier methods. Example: is written as Explanation: How many 5s in 400? x 5 = = 32 2_ 8_ Explanation: How many 5s in 32? x 5 = = 2 2 So = 28 remainder 2 or 28 r 2 5. Long Division Recording remainders as fractions When dividing by whole numbers, the remainder can be left as a whole number (as in all previous examples shown) or shown as a fraction. Example: _ 8_ Explanation: How many 5s in 32? x 5 = = 2 2 So = 28 r 2 The remainder can be recorded as: 2 5 So = The fraction can be further simplified: So = 28 5 Division terminology = 28 remainder 2 Dividend 4 Divisor 2 3 = Quotient Remainder 5. Long Division Recording remainders as decimals When dividing by whole numbers, the remainder can also be recorded as a decimal (full name = decimal fraction). This is useful when working with money and measures. Example: Explanation: 3 0 How many 5s in 43 tens? x 5 tens = 30 tens 43 tens  30 tens = 3 tens The 2 units are added to the 3 tens to make 32. t 2_ Explanation: 3 0 How many 5s in 32? x 5 = A decimal point is used 2.0 to show that whole numbers and decimals are being recorded. A place holder (0) is recorded to show that there are zero tenths in the dividend. This is added to the remainder to show 20 tenths are left. So = t 2_ 8. 8_ Explanation: x 5 =
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