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Calculation policy Flipbook PDF
Calculation policy
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Calculation policy
Calculation policy Introduction In its report entitled ‘Good Practice in Primary Mathematics: Evidence from 20 successful schools’ (2011: 7), Ofsted states that: A feature of strong practice in maintained schools is their clear, coherent calculation policies and guidance, which are tailored to the particular school’s context. They ensure consistent approaches and use of visual images and models that secure progression in pupils’ skills and knowledge lesson by lesson and year by year. We have developed this policy to provide consistency across the school and to enable teaching and learning staff to work together to ensure progression in skill development. A key role of this policy is to enable all teaching and learning staff to see how the methods in any year build on what went before and feed into what is learned later. Ultimately, we believe a clear progression in calculation will support the learning and teaching of Mathematics throughout the school, allow clarity and provide a secure foundation upon which to build and develop mathematical knowledge, skills and understanding. This policy outlines the end of year calculation expectations for each year group in each of the four operations (addition, subtraction, multiplication and division) and contains the key representational and written procedures that will be taught within the school. Although the focus of the policy is on the use of representations and written methods, it is important to recognise that the mental facility with numbers and the ability to calculate mentally lies at the heart of successful calculation methods. All methods should be taught with understanding rather than by rote, and should be put into real life contexts. Our approach to the teaching of calculation is guided by the following principles: -
practical, representational activities are essential in the development of the mathematical concepts needed for calculation
-
mental calculation is not at the exclusion of representational or written procedures and should be seen as complimentary to and not as separate from them
-
in every representational or written procedure there is an element of mental processing
-
sharing representational or written methods with teaching and learning staff and their peers encourages pupils to think about the mental strategies that underpin them and to develop new ideas
-
representational and written recording helps pupils to clarify their thinking and supports and extends the development of more fluent and sophisticated mental strategies
During their time at Someries Infant School, pupils will be encouraged to understand mathematics as both a written and spoken language. All teaching and learning staff will support and guide pupils through the following important stages: -
The use of equipment, pictures and a mixture of words and symbols to represent numerical activities
-
The use of standard symbols and conventions
-
The use of informal jottings to aid a mental strategy
Pupils will always be encouraged to look at a calculation or problem before deciding which is the best method to use. Our ultimate aim is for pupils to be able to select an efficient method of their choice that is appropriate for a given task. They will do this by asking themselves: -
Can I do this in my head?
-
Can I do this in my head using equipment, drawings or jottings?
-
Can I explain what I have done to somebody else?
page 2
Calculation policy At first, pupils’ recordings may not be easy for someone else to interpret, but they form an important stage in developing fluency. During Year One, pupils should begin to use the +, - and = signs to record their mental calculations. When pupils start to work with larger numbers and calculations with several steps, it will be harder for them to hold all the information in their head, but they need to be thoroughly secure in a range of mental strategies before they are ready to begin to use more formal written methods. As an infant school, we do not expect pupils to use, or to be taught, any formal written methods for addition, subtraction, multiplication or division. Instead, our focus is on recording calculations in horizontal form, so that the written record closely resembles the way in which pupils calculate mentally and would describe their working. Aims of this policy As a school, it is our aim for every child to develop efficient written methods for calculation for each of the four operations which they can use confidently, accurately and with an understanding of the mathematics involved. Pupils also need to be able to describe their calculation methods confidently and consistently using appropriate mathematical vocabulary. The purpose of this policy is to provide all teaching and learning staff with guidance for the teaching of calculation methods from nursery to year two. The policy attempts to ensure consistency of practice, and a smooth progression from mental to written calculation in which pupils’ written calculation methods build effectively upon their mental methods. Numicon Numicon is a quality-first teaching approach designed to give pupils the understanding of number ideas and number relationships that is essential for success in mathematics. As a school, we use a series of structured patterns – Numicon Shapes – to represent numbers, as part of a progressive mathematics teaching programme. Pupils are expected to have learned to recognise by sight the structured Numicon patterns by the end of the Early Years Foundation Stage. In all of the representative methods used in this policy, structured Numicon patterns are used to represent the units/ ones within a calculation in all year groups. Rekenrek Rekenrek is an interactive teaching resource which combines features of the number line, counters, base-10 and Numicon models to support children in developing secure foundations in number sense. Rekenreks are used alongside various other concrete and pictorial representations to aid learning in a range of mathematical concepts, such as subitising, addition, subtraction and doubling. Across our Early Childhood Education Centre, Rekenreks are an integrated part of mathematics teaching to ensure pupils are familiar with a wide range of representations and manipulatives to establish understanding and fluency of mathematical concepts.
page 3
Calculation policy The part-part-whole relationship Before beginning to add numbers and look at number bonds, pupils need to understand that a number can be partitioned into two or more parts. At Someries Infant School, we introduce the part whole model to show this concept clearly.
We encourage class teachers to teach both addition and subtraction alongside each other, as pupils will use this model to identify the inverse link between them. Pupils could place ten on top of the whole as well as writing it down. The parts could also be written in alongside the concrete representation. This model begins to develop the understanding of the commutativity of addition, as pupils become aware that the parts will make the whole in any order. Once pupils have developed a secure understanding that a number can be partitioned into two or more parts (and are familiar with the part whole model), they can be gradually introduced to the addition and subtraction calculation strategies outlined within this policy to support them in solving mathematical problems in the context of the part whole model, for example:
page 4
Calculation policy Progression towards a written method for addition In developing a written method for addition, it is important that pupils understand the concept of addition, in that it is: -
combining two or more groups to give a total or sum increasing an amount
Pupils also need to understand and work with certain principles, i.e. that it is: -
the inverse of subtraction commutative, i.e. 5+3 = 3 + 5
The fact that it is commutative and associative means that calculations can be rearranged, for example 4 + 13 = 17 is the same as 13 + 4 = 17 Before beginning to add numbers and look at number bonds, pupils need to understand that a number can be partitioned into two or more parts. At Someries Infant School, we introduce the part whole model to show this concept clearly.
page 5
Calculation policy Stage one Anticipated learning outcome: Using quantities and objects, pupils add two single-digit numbers and count on to find the answer Pupils are encouraged to develop a mental picture of the number system in their heads to use for calculation. They should experience practical calculation opportunities using a wide variety of practical equipment, including small world play, Rekenrek, counters, cubes, etc. Counting all method Pupils will begin to develop their ability to add by using practical equipment to count out the correct amount for each number in the calculation and then combine them to find the total. For example, when calculating 4 + 2, they are encouraged to count out four counters and count out two counters.
To find how many altogether, touch and drag them into a line one at a time whilst counting.
By touch counting and dragging in this way, it allows pupils to keep track of what they have already counted to ensure they don’t count the same item twice. Pupils should also use Rekenrek alongside other methods of addition to support them in building a rich mental picture of how numbers are made up and manipulated. Pupils use Rekenrek for addition by counting and pushing the correct number of beads until they are familiar enough with the resource to visually identify the amount of beads they need to push.
Pupils explore 10 + 3 = 13 as shown using the Rekenrek.
‘Push 10, push 1, 2, 3 makes 13’ ‘10 add 3 equals 13’
page 6
Calculation policy Counting on method To support pupils in moving from a counting all strategy to one involving counting on, pupils should still have two groups of objects but one should be covered so that it cannot be counted. For example, when calculating 4 + 2, count out the two groups of counters as before.
then cover up the larger group with a cloth.
For most pupils, it is beneficial to place the digit card on top of the cloth to remind the pupils of the number of counters underneath. They can then start their count at 4, and touch count 5 and 6 in the same way as before, rather than having to count all of the counters separately as before. Those pupils who are ready may record their own calculations. Pupils should also be provided with opportunities to engage in play using Numicon Shapes as much as possible, for example:
Those pupils who are ready may record this as:
page 7
Calculation policy What does this look like in practice? This example demonstrates how a pupil has used counting objects to add two single-digit numbers:
Pupils then begin to use number lines to support their own calculations using a structured number line to count on in ones:
8+1=9 +1
0
1
2
3
4
5
6
7
8
9
10
11
12 13
14 15
page 8
Calculation policy Stage two Anticipated learning outcome: Add one-digit and two-digit numbers to 20, including zero (using concrete and pictorial representations) Pupils will continue to use practical equipment, combining groups of objects to find the total by counting all or counting on. Pupils begin by using tens frames to support their addition of single digits, for example:
4+5=9 And progress to using tens frames to support their addition of two-digit numbers, for example:
14 + 5 = 19
page 9
Calculation policy Using their developing understanding of place value, they will move on to be able to use base-10 equipment to make teens numbers using separate tens and units. For example, when adding 11 and 5, they can make the 11 using a ten rod and a unit.
The units can then be combined to aid with seeing the final total, for example:
So 11 + 5 = 16. If possible, pupils should use two different colours of base-10 equipment so that the initial amounts can still be seen. Pupils then begin to use number lines to support their own calculations using a structured number line to count on in ones:
8 + 5 = 13
Bead strings or bead bars can be used to illustrate addition including bridging through ten by counting on two before counting on three:
page 10
Calculation policy Stage three Anticipated learning outcome: Add numbers using concrete objects, pictorial representations, and mentally, including: a two-digit number and ones; a two-digit number and tens; two two-digit numbers; three one-digit numbers Pupils also continue to use the base-10 equipment to support their calculations. For example, to calculate 32 + 21, they can make the individual amounts, counting the tens first and then count on the units.
53
When the units total more than 10, pupils should be encouraged to exchange 10 units/ones for 1 ten. This is the start of pupils understanding ‘carrying’ in vertical addition. For example, when calculating 35 + 27, they can represent the amounts using base-10 as shown:
Then, identifying the fact that there are enough units/ones to exchange for a ten, they can carry out this exchange:
to leave:
page 11
Calculation policy Pupils can also record the calculations using their own drawings of the base-10 equipment (as slanted lines for the 10 rods and dots for the unit blocks), for example: 34 + 23 =
What does this look like in practice? This example demonstrates how a pupil has recorded calculation using their own drawings of the base-10 equipment (as slanted lines for the 10 rods and dots for the unit blocks) to add two two-digit numbers:
page 12
Calculation policy With exchange, for example:
exchanged 10
28 + 36 =
will become
so 28 + 36 = 64 It is important that pupils circle the remaining tens and units after exchange to identify the amount remaining. This method can also be used with adding three digit numbers, for example 122 + 217 using a square as the representation of 100.
page 13
Calculation policy Progression towards a written method for subtraction In developing a written method for subtraction, it is important that pupils understand the concept of subtraction, in that it is: -
removal of an amount from a larger group (take away) comparison of two amounts (difference)
Pupils also need to understand and work with certain principles, i.e. that it is: -
the inverse of addition not commutative, i.e. 5 – 3 is not the same as 3 – 5
Before beginning to subtract numbers, pupils need to understand that a number can be partitioned into two or more parts. At Someries Infant School, we introduce the part whole model to show this concept clearly.
Stage one Anticipated learning outcome: Using quantities and objects, pupils subtract two single-digit numbers and count on or back to find the answer Pupils are encouraged to develop a mental picture of the number system in their heads to use for calculation. They should experience practical calculation opportunities using a wide variety of practical equipment, including small world play, role play, counters, cubes etc. Taking away Pupils will begin to develop their ability to subtract by using practical equipment to count out the first number and then remove or take away the second number to find the solution by counting how many are left, for example 9 - 4:
page 14
Calculation policy For illustration purposes, the amount being taken away are show crossed out. Pupils would be encouraged to physically remove these using touch counting.
By touch counting and dragging in this way, it allows pupils to keep track of how many they are removing so they don’t have to keep recounting. They will then touch count the amount that are left to find the answer. Pupils continue to use number lines as well as practical resources to support calculation. Teachers continue to demonstrate the use of the number line:
6–3=3 -1
-1
-1
_____________________________ 0
1
2
3
4
5
6
7
8
9
10
Pupils should also be provided with opportunities to engage with Numicon Shapes as much as possible to support their understanding of subtraction, for example:
Those who are ready may record their own calculations.
page 15
Calculation policy What does this look like in practice? This example illustrates how Early Years pupils may use pictorial representations to solve subtraction problems before or alongside recording the abstract calculation:
page 16
Calculation policy Stage two Anticipated learning outcome: Subtract one-digit and two-digit numbers to 20, including zero (using concrete objects and pictorial representations) Pupils will continue to use practical equipment and taking away strategies. To avoid the need to exchange for subtraction at this stage, it is advisable to continue to use equipment such as counters, cubes and the units from the base-10 equipment, but not the tens, for example: 13 – 4:
Touch count and remove the number to be taken away, in this case 4.
Touch count to find the number that remains.
Pupils continue to use number lines as well as practical resources to support calculation. Teachers continue to demonstrate the use of the number line:
6–3=3
page 17
Calculation policy Pupils then use tens frames to support their subtraction of single digits, for example:
9–2=7 And progress to using tens frames to support their subtraction of two-digit numbers, for example:
19 - 2 = 17
page 18
Calculation policy Stage three Anticipated learning outcome: Subtract numbers using concrete objects, pictorial representations, and mentally, including: a two-digit number and ones; a two-digit number and tens; two two-digit numbers Pupils will begin to use the base-10 equipment to support their calculations, still using a take away, or removal, method. They need to understand that the number being subtracted does not appear as an amount on its own, but rather as part of the larger amount. For example, to calculate 54 - 23, pupils would count out 54 using the base-10 equipment (5 tens and 4 units). They need to consider whether there are enough units/ones to remove 3, in this case there are, so they would remove 3 units and then two tens, counting up the answer of 3 tens and 1 unit to give 31.
which leaves
so 54 – 23 = 31
Pupils can also record the calculations using their own drawings of the base-10 equipment (as slanted lines for the 10 rods and dots for the unit blocks); for example, to calculate 39 – 17, pupils would draw 39 as 3 tens (lines) and 4 units (dots) and would cross out 7 units and then one ten, counting up the answer of 2 tens and 2 units to give 22.
Circling the tens and units that remain will help pupils to identify how many remain.
page 19
Calculation policy What does this look like in practice? This example demonstrates how a pupil has calculated one two-digit number from another by recording the calculations using their own drawings of the base-10 equipment (as slanted lines for the 10 rods and dots for the unit blocks). Circling the tens and units that remain helped the child to identify how many remain.
page 20
Calculation policy What does this look like in practice? This example demonstrates how a pupil has calculated the difference between two temperatures by recording the calculations using their own drawings of the base-10 equipment (as slanted lines for the 10 rods and dots for the unit blocks):
page 21
Calculation policy When the amount of units to be subtracted is greater than the units in the original number, an exchange method is required. This relies on pupils’ understanding of ten units being an equivalent amount to one ten. To calculate 53 – 26, by using practical equipment, they would count out 53 using the tens and units, as in Step 1. They need to consider whether there are enough units/ ones to remove 6. In this case there are not so they need to exchange a ten into ten ones to make sure that there are enough, as in step 2. Step 1
Step 2
becomes
The pupils can now see the 53 represented as 40 and 13, still the same total, but partitioned in a different way, as in step 3 and can go on to take away the 26 from the calculation to leave 27 remaining, as in Step 4. Step 3
Step 4
When recording their own drawings, when calculating 37 – 19, pupils would cross out a ten and exchange for ten units. Drawing them in a vertical line, as in Step 2, ensures that pupils create ten ones and do not get them confused with the units that were already in place. Step 1
Step 2
Step 3
Circling the tens and units that remain will help pupils to identify how many remain.
page 22
Calculation policy Progression towards a written method for multiplication In developing a written method for multiplication, it is important that pupils understand the concept of multiplication, in that it is: -
repeated addition
Pupils should also be familiar with the fact that it can be represented as an array. Pupils also need to understand and work with certain principles, i.e. that it is: -
the inverse of division commutative, i.e. 5 × 3 is the same as 3 × 5
Through their introduction to Numicon, pupils are encouraged to develop a mental picture of the number system in their heads to use for calculation. Numicon is used to visualise the repeated addition of the same number:
page 23
Calculation policy Stage one Anticipated learning outcome: Pupils solve problems, including doubling Pupils are encouraged to develop a mental picture of the number system in their heads to use for calculation. They should experience practical calculation opportunities using a wide variety of equipment, including small world play, role play, counters, Number blocks programmes and games etc. Pupils may also investigate putting items into resources such as egg boxes, ice cube trays and baking tins which are arrays.
They may develop ways of recording calculations using pictures, etc. A child’s jotting showing the fingers on each hand as a double.
A child’s jotting showing double three as three cookies on each plate.
Pupils should also be provided with opportunities to engage with Numicon Shapes as much as possible to support their understanding of doubling, for example:
page 24
Calculation policy What does this look like in practice? This example demonstrates how a pupil has used Numicon to calculate double 7:
page 25
Calculation policy Across our Early Childhood Education Centre, pupils continue to use Rekenrek as an additional method to develop their understanding of doubling as repeated addition. Pupils secure in using this resource will be able to fluently move beads and visually relate to their previous experience of using Numicon and tens frames to solve doubling calculations.
Pupils should be provided with opportunities to explore various doubles within 10 and explain how they know the beads on their Rekenrek represent a double calculation.
What does this look like in practice? This example illustrates a pupil doubling 4 using the Rekenrek:
page 26
Calculation policy Stage two Anticipated learning outcome: Solve one-step problems involving multiplication by calculating the answer using concrete objects, pictorial representations and arrays with the support of their teacher In year one, pupils will continue to solve multiplication problems using practical equipment and jottings. They may use the equipment to make groups of objects. Pupils should see everyday versions of arrays, for example egg boxes, baking trays, ice cube trays, wrapping paper, etc., and use this in their learning, answering questions such as ‘How many eggs would we need to fill the egg box? How do you know?’
Stage three Anticipated learning outcome: Calculate mathematical statements for multiplication (using repeated addition) and write them using the multiplication (×) and equals (=) signs Pupils should understand and be able to calculate multiplication as repeated addition, supported by the use of practical apparatus such as counters or cubes, for example: 5 x 3 can be shown as five groups of three with counters, either grouped in a random pattern, as below:
or in a more ordered pattern, with the groups of three indicated by the border outline:
Pupils should then develop this knowledge to show how multiplication calculations can be represented by an array, (this knowledge will support with the development of the grid method in the future). Again, pupils should be encouraged to use practical apparatus and jottings to support their understanding, for example: 5 x 31 can be represented as an array in two forms (as it has commutativity):
3 + 3 + 3 + 3 + 3 = 15
5 + 5 + 5 = 15
1
For mathematical accuracy 5 x 3 is represented by the second example above, rather than the first as it is five, three times. However, because we use terms such as ‘groups of’ or ‘lots of’, pupils are more familiar with the initial notation. Once pupils understand the commutative order of multiplication the order is irrelevant).
page 27
Calculation policy Repeated addition 3 times 5 is 5 + 5 + 5 or 3 lots of 5 or 5 × 3 Repeated addition can be shown easily on a number line:
5x3=5+5+5
0
1
2
5
5
5
3
4
5
6
7
8
9
10
11
12
13
14
15
As well as on a bead bar:
5x3=5+5+5 5
5
5
Commutativity Pupils should know that 3 x 5 has the same answer as 5 x 3 and how this can be represented on a number line: 5
5
0
1
2 3
3
4
5 3
6
7
5
8 3
9
10
11 3
12
13
14
15
3
page 28
Calculation policy What does this look like in practice? These examples demonstrate how a pupil has calculated the answer to a multiplication problem using an array and repeated addition:
page 29
Calculation policy Progression towards a written method for division In developing a written method for division, it is important that pupils understand the concept of division, in that it is: -
repeated subtraction sharing into equal amounts
Pupils also need to understand and work with certain principles, i.e. that it is: -
the inverse of multiplication not commutative, i.e. 15 ÷ 3 is not the same as 3 ÷ 15
Stage one Anticipated learning outcome: Pupils solve problems, including halving Pupils are encouraged to develop a mental picture of the number system in their heads to use for calculation. They should experience practical calculation opportunities using a wide variety of equipment, including small world play, role play, counters, cubes etc. Pupils may also investigate sharing items or putting items into groups using items such as egg boxes, ice cube trays and baking tins which are arrays.
They may develop ways of recording calculations using pictures, etc.
A child’s jotting showing halving six spots between two sides of a ladybird.
A child’s jotting showing how they shared the apples at snack time between two groups.
page 30
Calculation policy What does this look like in practice? This example demonstrates how a pupil explores early practical division by sharing objects into groups. Pupils may use mathematical vocabulary to discuss how they know they have shared equally and what to do with any remaining objects:
page 31
Calculation policy Stage two Anticipated learning outcome: Solve one-step problems involving division by calculating the answer using concrete objects, pictorial representations with the support of their teacher In year one, pupils will continue to solve division problems using practical equipment and jottings. They should use the equipment to share objects and separate them into groups, answering questions such as ‘If we share these six apples between the three of you, how many will you each have? How do you know?’ or ‘If six football stickers are shared between two people, how many do they each get?’ They may solve both of these types of question by using a ‘one for you, one for me’ strategy until all of the objects have been given out.
If 6 sweets are shared between 2 people, how many do they each get?
Grouping or repeated subtraction If there are 6 sweets, how many people can have 2 sweets each?
page 32
Calculation policy Pupils will utilise Numicon to represent division calculations as grouping (repeated subtraction) and use jottings to support their calculations:
12 ÷ 4 =
12 ÷ 4 = 3
Pupils need to understand that this calculation reads as ‘How many groups of 4 are there in 12?’ Pupils should be introduced to the concept of simple remainders in their calculations at this practical stage, being able to identify that the groups are not equal and should refer to the remainder as ‘… left over’. Pupils will utilise practical equipment to represent division calculations as grouping (repeated subtraction) and use jottings to support their calculation, for example:
12 ÷ 3 =
Pupils need to understand that this calculation reads as ‘How many groups of 3 are there in 12?’
page 33
Calculation policy What does this look like in practice? This example demonstrates how a pupil has calculated a division problem using the whole-school strategy:
page 34
Calculation policy Stage three Anticipated learning outcome: Solve problems involving division, using materials, mental methods and division facts, including problems in contexts Pupils develop their knowledge of division with remainders, for example:
13 ÷ 4 =
13 ÷ 4 = 3 remainder 1 Pupils need to be able to make decisions about what to do with remainders after division and round up or down accordingly. In the calculation 13 ÷ 4, the answer is 3 remainder 1, but whether the answer should be rounded up to 4 or rounded down to 3 depends on the context, as in the examples below: I have £13. Books are £4 each. How many can I buy? Answer: 3 (the remaining £1 is not enough to buy another book) Apples are packed into boxes of 4. There are 13 apples. How many boxes are needed? Answer: 4 (the remaining 1 apple still needs to be placed into a box) Numicon can also be used to represent division problems involving remainders:
9÷4=
9 ÷ 4 = 2 remainder 1 Ensure that the emphasis is on grouping rather than sharing. As well as Numicon, pupils will also continue to use:
page 35
Calculation policy Repeated subtraction using a number line Pupils will use an empty number line to support their calculation.
13 ÷ 4 = 3 r. 1 4 0 1
4 5
4 9
13
What does this look like in practice? This example demonstrates how a pupil has used a structured number line to support their calculation of a division problem involving remainders:
page 36
Calculation policy Progression towards mastering fractions throughout Key Stage One Stage one Anticipated learning outcome: Recognise, find and name a half as one or two equal parts of an object, shape or quantity; recognise, find and name a quarter as one of four equal parts of an object, shape or quantity
Stage two Anticipated learning outcome: Recognise, find, name and write fractions simple fractions for example,
1 2
1 1 2 3, 4, 4
and
3 4
of a length, shape, set of objects or quantity; write
of 6 = 3 and recognise the equivalence of
2 4
and
1 . 2
The agreed whole-school strategy for finding a fraction of a set of objects, number or quantity is the following jotting:
1 4
of 16 = 4
2 4
of 16 = 8
page 37
Calculation policy What does this look like in practice? These examples demonstrate how a pupil has calculated a fraction of a number using the whole-school strategy:
page 38
Calculation policy What does this look like in practice? These examples demonstrate how a pupil has used the whole-school strategies to compare two fractions:
page 39
Calculation policy What does this look like in practice? This example demonstrates how a pupil has used the whole-school strategy to solve a series of problems involving finding a fraction of a quantity:
page 40