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Calculation guide Flipbook PDF
Calculation guide
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Introduction The principal focus of mathematics teaching at Someries Infant and Nursery School is to ensure that pupils develop confidence and mental fluency with whole numbers, counting and place value. This involves working with numerals, words and the four operations, including with practical resources. Children also develop their ability to recognise, describe, draw, compare and sort different shapes and use the related vocabulary. We also teach children how to use a range of measures to describe and compare different quantities such as length, mass, capacity/ volume, time and money.
What happens at school? The seven areas of mathematics taught in school are:
Number - number and place value Number - addition and subtraction Number - multiplication and division Measurement Geometry - properties of shapes Geometry - position and direction Statistics
Children are taught mathematics on a daily basis. This guide 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 strategies your child will be taught in the school.
ADDITION In developing a written method for addition, it is important that children understand the concept of addition, in that it is: combining two or more groups to give a total or sum increasing an amount Children 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
Stage one Anticipated learning outcome: Using quantities and objects, children add two single-digit numbers and count on to find the answer Children 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. Counting all method Children 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.
1
2
3
4
5
6
By touch counting and dragging in this way, it allows children to keep track of what they have already counted to ensure they don’t count the same item twice.
Counting on method To support children in moving from a counting all strategy to one involving counting on, children 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.
4 For most children, it is beneficial to place the digit card on top of the cloth to remind the children 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 children who are ready may record their own calculations. Children should also be provided with opportunities to engage in play using Numicon Shapes as much as possible, for example:
Those children who are ready may record this as:
What does this look like in practice? This example demonstrates how a pupil has used counting objects to add two single-digit numbers:
Children 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
Stage two Anticipated learning outcome: Add one-digit and two-digit numbers to 20, including zero (using concrete and pictorial representations) Children will continue to use practical equipment, combining groups of objects to find the total by counting all or counting on. 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, children should use two different colours of base 10 equipment so that the initial amounts can still be seen. Children then begin to use number lines to support their own calculations using a structured number line to count on in ones:
8 + 5 = 13 +1
0
1
2
3
4
5
6
7
8
+1
9
+1
+1
+1
10 11 12 13 14 15
Bead strings or bead bars can be used to illustrate addition including bridging through ten by counting on two before counting on three:
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 Children 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, children should be encouraged to exchange 10 units/ones for 1 ten. This is the start of children 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:
Children 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:
With exchange, for example: exchanged 10 28 + 36 =
will become
so 28 + 36 = 64 It is important that children 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.
SUBTRACTION In developing a written method for subtraction, it is important that children understand the concept of subtraction, in that it is: removal of an amount from a larger group (take away) comparison of two amounts (difference) Children 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
Stage one Anticipated learning outcome: Using quantities and objects, children subtract two single-digit numbers and count on or back to find the answer Children 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 Children 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 e.g. 9 – 4.
For illustration purposes, the amount being taken away are show crossed out. Children would be encouraged to physically remove these using touch counting.
4
3
2
1
By touch counting and dragging in this way, it allows children 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.
Children 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 Children 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.
Stage two Anticipated learning outcome: Subtract one-digit and two-digit numbers to 20, including zero (using concrete objects and pictorial representations)
Children 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, e.g. 13 - 4
Touch count and remove the number to be taken away, in this case 4.
4
3
2
1
Touch count to find the number that remains.
1
2
3
4
5
6
7
8
9
Children 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
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 Children 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, children 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
Children 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), e.g. to calculate 39 – 17 children 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 children to identify how many remain.
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.
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):
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 children’s 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 children 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, children would cross out a ten and exchange for ten units. Drawing them in a vertical line, as in Step 2, ensures that children create ten ones and do not get them confused with the units that were already in place. Step 1
Step 2
Circling the tens and units that remain will help children to identify how many remain.
Step 3
MULTIPLICATION In developing a written method for multiplication, it is important that children understand the concept of multiplication, in that it is: repeated addition Children should also be familiar with the fact that it can be represented as an array. Children 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, children 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:
Stage one Anticipated learning outcome: Children solve problems, including doubling Children 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. Children 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.
Children should also be provided with opportunities to engage with Numicon Shapes as much as possible to support their understanding of doubling, for example:
What does this look like in practice? This example demonstrates how a pupil has used Numicon to calculate double 7:
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, children will continue to solve multiplication problems using practical equipment and jottings. They may use the equipment to make groups of objects. Children 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 Children 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:
Children 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, children should be encouraged to use practical apparatus and jottings to support their understanding, e.g. 5 x 3* can be represented as an array in two forms (as it has commutativity):
5 + 5 + 5 = 15 3 + 3 + 3 + 3 + 3 = 15 *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’, children are more familiar with the initial notation. Once children understand the commutative order of multiplication the order is irrelevant).
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 Children 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 3
2
3
4 3
5
6
7
5 8
3
9
10 11 12 13 14 15 3
3
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:
DIVISION In developing a written method for division, it is important that children understand the concept of division, in that it is:
repeated subtraction sharing into equal amounts
Children 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: Using quantities and objects, children subtract two single-digit numbers and count on or back to find the answer Children 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. Children 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.
Stage two 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 In year one, children 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?
Children will utilise Numicon to represent division calculations as grouping (repeated subtraction) and use jottings to support their calculations:
12 ÷ 4 =
12 ÷ 4 = 3
Children need to understand that this calculation reads as ‘How many groups of 4 are there in 12?’ Children 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’. Children will utilise practical equipment to represent division calculations as grouping (repeated subtraction) and use jottings to support their calculation, e.g. 12 ÷ 3 =
Children need to understand that this calculation reads as ‘How many groups of 3 are there in 12?’
What does this look like in practice? This example demonstrates how a pupil has calculated a division problem using the whole-school strategy:
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 Children develop their knowledge of division with remainders, for example: 13 ÷ 4 =
13 ÷ 4 = 3 remainder 1 Children 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, children will also continue to use: Repeated subtraction using a number line Children 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:
FRACTIONS 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 3
,
1 4
,
2 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
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:
What does this look like in practice? These examples demonstrate how a pupil has used the whole-school strategies to compare two fractions:
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: