Calculations Policy Multiplication and Division · As larger 2 digit numbers are used, the size of jumps should continue to grow, with jumps of 10 lots of being very helpful. The

1

Calculations Policy

Multiplication and Division

Harbinger Primary School September 2016

2

Our aim is for each child to have an efficient strategy for each operation so that they are able to calculate independently. This will be helped by all staff having a

shared understanding and common approach when teaching the methods.

The requirements of the new Maths curriculum for the 4 operations have been set out for each year group, and ideas for areas of application have been included.

Teaching of mental maths, place value and number bonds is also vital, and regular reference to the inverse operation will strengthen understanding. This can be particularly useful for checking that results are feasible, therefore, always ask children to write the calculation and answer horizontally

in order that they consider whether it makes sense as they finish their calculation. This also makes sure children are aware that a written calculation is not always necessary.

With each new stage, revert to numbers that the children are comfortable with, and use ‘old’ and ‘new’ strategies side by side. When children are feeling completely confident, calculations can become more complex and the new strategy used alone.

3

MULTIPLICATION Notes DIVISION Notes

Pictorial representation: (repeated addition) Arrays:

Language: groups of, lots of Areas for application: Measurements Geometry—how many sides altogether? Dots/pictures/any representations which children count to get the answer Writing of number sentence follows as child understands x, / and = symbols Arrays – More formal organisation/accurate pictorial representation of operation. Either array is OK as long as the child’s explanation represents the sum ie. 3 x 2 is 3 rows of 2 columns or 3 x 2 is 3 lots of 2 in a column. This also match-es the spoken number sentence ie 3 lots of 2

Pictorial representation: i) (Draw circles shown by the second number in the sum then ‘share’ all of the dots shown by the first number) ii) (Draw the number of dots shown by the first number in the sum then circle ‘groups’ of the second number)

Language: groups of, lots of, share Areas for application: Measurements Geometry—8 squares, how many sides? Dots/pictures/any representations which children count to get the Answer. This will follow on from practical expe-rience of sharing. It is necessary from this early stage to discuss the two different ways of understanding divi-sion ie i) 6 divided by 2 can be 6 divided into two groups (a logical pro-gression from practical sharing) OR ii) 6 divided into groups of 2 (most helpful for future calculation strate-gies, esp no line) “Two groups with three in each” OR “Three groups of two” Both are valid and should be recog-nised, with weighting on the latter being increased to prepare the children for moving forward

Year 1

Curriculum learning objectives

• solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher.

3 X 2 = 6

2 + 2 + 2 = 6

3 X 2 = 6

6 ÷ 2 = 3

7 ÷ 2 = 3 r 1

6 ÷ 2 = 3

4


Pictorial representation Arrays Completed number line:

Language: groups of, lots of, times, mul-tiply, multiplied by, multiple of, repeat-ed addition Areas for application: Measurements Geometry—how many sides altogether? Fractions Number lines always have to have the ‘how many lots of’ written above and the size of the jumps inside. Number lines are used as an aid to counting the units inside the jumps of ‘lots of’ Start from 0 At first do the same number of jumps as the 1st number in the sum. Do the jumps the size of the 2nd num-ber in the calculation . The answer is the number the child lands on

Pictorial representation Completed number line:

Language: groups of, lots of, share, divide by, divided by, remainder Areas for application: Measurements Geometry—8 squares, how many sides? Fractions Number lines always have to have the ‘how many lots of’ written above and the size of the jumps inside. Use the first number in the sum to mark where you will stop and jump in lots of the second number. Count the number of ‘lots of’ noted above the jumps to reach the answer Children may begin to use jumps of more than one ‘lot of’ if they feel con-fident and can use their times table knowledge to help them. This will be helpful for the empty number line later on.

Year 2


• recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers

• calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs

• show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot

• solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts.

5


Empty number line: Number lines always to how many ‘lots of’ written above, the running total

kept along the bottom and the size of the jump inside.

Children begin to realise they can start with either number. This can be useful in being more efficient ie in doing less jumps or using known multiplication facts. “If I actually know what 3 lots of 4 are, do I need to do a jump for each group or could I do one big jump of the whole amount?” Children become more confident in choosing the size of their jumps and begin to use partitioning when ready Children working at a lower level or with SEN may use this strategy with a multiplication square to support them Children may do jumps within jumps if they need a calculation to be simplified Some children may use compensation, particularly for calculation close to a multiple of 10. This does not need to be explicitly taught, but will be interesting to recognise and discuss.

Empty number line: Number lines always to how many ‘lots of’ written above, the running total kept along the bottom and the

size of the jump inside. Write the first number in the calcula-tion on the right, zero on the left At first jump in one ‘lot of’ until you reach the right hand side of the num-ber line The answer is reached by counting how many ‘lots of’ the second number were jumped Children then begin to use known times table facts to do larger jumps ie more than one ’lot of’ at a time. This is im-proving efficiency “If I know that 2 lots of 3 are 6, can I do this bigger jump? It will mean I can write less and finish the sum quicker!” “So 2 lots of 3 are 6, I think I can fit in another 2 lots which will take me to 12. I don't think I can do another jump this big, so if I try 1 lot of 3, where does that take me to? I’ve landed on 15 which is the first number in my sum, so if I count how many lots of I have jumped altogether, there are 2, plus another 2 is 4 and then add 1 more group of 3 so my answer is 5.”

6


Empty number line: Language: groups of, lots of, times, mul-tiply, multiplied by, multiple of, repeat-ed addition, product of Areas for application: Measurements Geometry—how many sides altogether? Fractions Discuss methods of multiplying by 10/100 accurately ie 70 x 8 can be thought of as 7 x 8 = 56, then make 10x bigger = 560 or we can count in tens of 7s: 70, 140, 210, 280, 350, 420, 480, 560 This strategy provides consolidates understanding of the number system and place value and allows children to stretch themselves in these areas

Empty number line: Language: groups of, lots of, share, divide by, divided by, remainder Areas for application: Measurements Geometry—8 squares, how many sides? Fractions As larger 2 digit numbers are used, the size of jumps should continue to grow, with jumps of 10 lots of being very helpful. The empty number line can be used for 3 and 4 digit numbers and beyond, us-ing 10x, 100x and 1000x. Calculations of this type are excellent for consoli-dating understanding of the number system and place value, and are good evidence that this is embedded. Meth-ods of calculating 10x, 100x etc can be discussed as noted in ‘Multiplication’ “I’d like to be able to do a big jump to start my calculation off - I know that 7 lots of 6 are 42, 420 is ten times big-ger so 70 lots of 6 equal 420. With one more lot of 6 I land on 426, so the answer is 71 with a remainder of 1” Sums including remainders should be used throughout

Year 3


• 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 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.

7


Grid Method: This provides a more formal way of organising partitioning and builds on existing place value knowledge

Emphasise need to present the calculation carefully and consistently Partition any number with 2 digits or more. Organise in grid, number with most digits positioned horizontally. Starting with the units, or the ‘outermost square’, work through the individual calculations To begin with, write the individual calculations in the appropriate square Add the totals together using their current/preferred addition strategy - ref to Addition & Subtraction Calculation Policy Children may use a number line or other jottings alongside this strategy to keep track of parts of the calculation

8


Grid Method: Language: groups of, lots of, times, mul-tiply, multiplied by, multiple of, repeat-ed addition, product of Areas for application: Measurements Geometry—how many sides altogether? Fractions Most errors occur in the top left-hand box and therefore discussion of place value and the aforementioned methods of multiplying accurately by 10/100/1000 remain very important. 2 digit x 2 digit is the most complex of these calculations and therefore most open to error

Empty number line: Language: groups of, lots of, share, divide by, divided by, remainder, fac-tor, divisible by Areas for application: Measurements Geometry—8 squares, how many sides? Fractions The empty number line should be used until a child demonstrates un-derstanding and ability with the full range of calculation shown in this stage, after which they will move onto compact division.

Year 4


• recall multiplication and division facts for multiplication tables up to 12 × 12

• use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together three numbers

• recognise and use factor pairs and commutativity in mental calculations

• multiply two-digit and three-digit numbers by a one-digit number using formal written layout (short division towards the end of Year 4)

• solve problems involving multiplying and adding, including using the distributive law to multiply two digit numbers by one digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects.

9


Expanded columns: Numbers begin to be arranged in columns, multiplication sign is to the right and individual calculations are

written on the right. Emphasise the importance of vertical organisation and place value Begin with the units “Does it matter which number I start with? We know in multiplication sums the answer will be the same no matter which way the number sentence is or-dered. It will really help our strategy to begin with the number we are multiply-ing by, and make sure we are multiplying it by every digit in the other number.” Read as “4 x 3, 4 x 10” “3 x 4, 3 x 10, 3 x 200” Crossing boundaries and moving into other columns is good preparation for compact method: “4 x 3 is 12. Where should I write this? Should both digits be under the 4 or do I need to move into a different col-umn?” “6 x 20 is 120 so I will need to think carefully about where I write that. How will I know?” etc Values to be carried across the columns when adding the totals are noted below the calculation, as in the Addition and Subtraction Calculation Policy

Short division: (Summer of Y4?) “I have 3 Tens that need to be shared equally into 3 groups. How many Tens does each group get? They get 1 each—I’ll write that in the space above, in the Tens column. I have 6 Units that need to be shared equally into 3 groups. How many Units does each group have? They get 2 each—I’ll write that in the Units col-umn above. So 36 shared into 3 groups is 12.”

Introducing this method using dienes supports understanding and is good provision for kinaesthetic and visual learners. It will also be very helpful when ‘moving’, ’taking’ or ’carrying’. Individual digits should continue to be spoken of in line with their true value, with the emphasis being placed on partitioning the first number in the calculation in order to make the calcu-lation manageable.

10


Expanded columns: 2 digit x 2 digit is the most complex of these calculations and there-fore most open to error. Reverting to the grid method to check the answer is an option, and it provides an opportunity for children to ensure they have includ-ed each of the 4 multiplications involved

“I need to share 72 into 6 groups. I can put a rod of Ten in each group and I can show this by writing 1 in the Tens column. I have one Ten left over which I can’t share equally as it is, so I will need to break it into individual dienes or ’ones’ or Units. One Ten is the same as 10 Units. So now I have 12 Units altogether which I’m going to make a note of in the Units column. I can then put 2 Units into each group and I will write this answer in the Units column. So 72 shared into 6 groups is 12.” “I know that that the 8 means 8 Hun-dreds as I’m confident with place value, so I’ll just think about it as an ‘8’. How many groups of 4’s in 8? There are 2. How many 4’s in 9? There are 2, but one is left. I will move this to the Units column—so there are now 12 Units (because 1 ten makes 10 units). How many 4’s in 12? There are 3. So my answer is 223.”

11

12


Short multiplication: Language: groups of, lots of, times, mul-tiply, multiplied by, multiple of, repeat-ed addition, product of, square Areas for application: Measurements Geometry Fractions, decimals, percentages Values being carried across columns are noted below the calculation, with

the x sign on the right

Focus on becoming more efficient in recording eg “This feels like I’m doing a lot of writing, I wonder if I can cut out some steps now that I am more confi-dent and use what I know about carrying into other columns from my addition strategies?” For an example explanation of the strategy to the children, see over.

Short division: (see Year 4 for the introduction of short division)

Language: groups of, lots of, share, divide by, divided by, remainder, fac-tor, divisible by Areas for application: Measurements Geometry Fractions, decimals, percentages “I know that the 4 means 4 Hundreds as I’m confident with place value, so I’ll just think about it as a ‘4’. How many groups of 4’s in 4? There is only 1. How many 4’s in 8? There are 2. How many 4’s in 3? There aren’t any, so I write 0 above. But I MUST note down how many remain. As there aren’t any groups of 4 in 3, it means that 3 re-main / are left. I write this as r.3. So my answer is 120 with 3 remaining.”

Year 5


• identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers

• know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers

• establish whether a number up to 100 is prime and recall prime numbers up to 19

• multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers

• multiply and divide numbers mentally drawing upon known facts

• divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context

• multiply and divide whole numbers and those involving decimals by 10, 100 and 1000

13


Short multiplication: Begin with examples such as 13 x 6 that give an answer below 100. Progressing onto larger values such as 28 x 6 then requires a move into the hundreds col-umn. “6 x 3 is 18 so if I put the 8 where it belongs in the units column, where could I place the 10 to make sure it is includ-ed? I could move it underneath the total line just like when I am doing compact addition (if children are using compact multiplication it can be expected that they will be using compact addition) Then is 6 x 10 is 60, how should I write this? Do I need the 0? Have I got just 1 tens or are there any other tens I need to include in this column?” etc. Revert to previous strategies to consoli-date/ ’prove’ if necessary. This is also helpful for checking an an-swer as, with calculations of this com-plexity, using the inverse to check is not feasible eg 1242 ÷ 27! Discussion of 0 as a placeholder is very important here as children may feel it can be ignored eg “3 x 0 is 0, so can I just move straight on? Do I need to write 0? What happens to my answer if I ignore it? 3 x 300 is 900. If I put this next to the 6 it is worth 90 which is not correct. 96 is not a reasonable answer when multiplying 300 and something by 3. If I put it in the hundreds column then there is a gap in my answer, so that 0 is important to hold in the tens column to make sure my answer is 906.” Accurate vertical alignment of decimal point as part of place value is crucial here. 1 decimal place is sufficient but 2 decimal places may be used if working within the context of money. Long multiplication—children can check answers using the grid method.

Short division: “How many 7’s in 6? There are none, so we use both the 6 AND the next digit 0. Therefore my calculation becomes: how many 7’s in 60? There are 8 with 4 remaining. How many 7’s in 42? There are 6. Therefore my answer is 86.

14


Long multiplication: For calculations x 2-digits It is VERY important to discuss 0 as a placeholder. “We know that the calculation is 30 x 3 and 30 x 40. But this can be tricky, to think about Tens. So let’s put a 0 in as a placeholder straight away (represents the Ten) and now we just need to think about 3 x 2 and 3 x 4.”

Short division:

15


Long multiplication: Short division: 1 2 0.7 5 4 4 8 3 0²0

483 ÷ 4 = 120 r.3

120 ¾

Interpreting remainders as a decimal Write in the decimal point in the correct place in the answer. There are 0 tenths—include this in the remaining 3, which makes 30. How many 4’s in 30? There are 7. This ‘7’ is 7 tenths. But 2 tenths remain. We add this to the 0 hundredths. How many 4’s in 20? There are 5. So the answer as a decimal is 120.75 Interpreting remainders as a fraction 3 remain—3 out of 4—this can be rep-resented by a fraction. Can the frac-tion be simplified? Interpreting remainders by rounding eg. 483 sweets are shared equally between 4 friends. How many sweets do they get each? (answer = 120)

16


Short & Long multiplication:

(see Year 5)

Areas for application: Measurements Geometry Fractions, decimals, percentages Ratio and Proportion Algebra

Short division Long division:

Areas for application: Measurements Geometry Fractions, decimals, percentages Ratio and Proportion Algebra Like short division, we think about each place value column: How many lots of 14 in ‘5’? (no need for ‘500’ as Year 6 children should have a secure knowledge of place value!) How many lots of 14 in 53? (=3) This makes 42—how much of 53 remains? 11…but don’t forget the ‘2 Units’! How many lots of 14 in 112? (children can note the times table down the side to keep track—14, 28, 42…) There are 8, which makes the answer 38. Long division using ‘chunking’ can also be used. See over...

Year 6


• multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication

• divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, frac-tions, or by rounding, as appropriate for the context

• divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context

• perform mental calculations, including with mixed operations and large numbers

• use their knowledge of the order of operations to carry out calculations involving the four operations

• solve problems involving addition, subtraction, multiplication and division

• use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy.

532 ÷ 14 =

17



(see Year 5)

Long division: ‘Chunking’ - thinking about the number as a whole rather than the individual columns / place value. I know that 10 ‘lots of’ 14 make 140. Let’s subtract this ‘chunk’ of 140 from 532 = 392. I know that 20 ‘lots of’ 14 make 280. Subtract this from 392 = 112. I know that there are 8 ‘lots of’ 14 in 112. So...I have calculated that there are 38 ‘lots of’ 14 in 532. Therefore 532 ÷ 14 = 38.”

532 ÷ 14 =

18



(see Year 5)

Long division: Interpreting remainders as a decimal Write in the decimal point in the correct place in the answer. There are 0 tenths—include this in the remaining 12, which makes 120. How many 15’s in 120? There are 8. This ‘8’ is 8 tenths. Therefore the answer is 28.8. Interpreting remainders as a fraction 12 remain—12 out of 15—this can be represented by a fraction. Can the fraction be simplified?

Documents

Calculations Policy Multiplication and Division · As larger 2 digit numbers are used, the size of jumps should continue to grow, with jumps of 10 lots of being very helpful. The