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# Numeracy

## Next steps in addition, subtraction, multiplication and division

**Progression and sequenced learning**

Our maths curriculum and provision is carefully sequenced so that new learning is always based on secure prior learning. This is closely linked to assessment for learning and how teachers plan and differentiate within their classroom settings. This document outlines the knowledge, skills and understanding children need before progressing.

**Recognition of early numeracy and mastery of numbers up to 10**

All early number work is rooted in getting children confident in working and manipulating numbers up to 10 in various contexts before moving on to larger numbers. This includes a secure understanding of how numbers work, conservation of number, number patterns and how numbers up to 10 can be combined.

**Early numeracy and mathematical progression**

Children progress through the following stages. A more detailed sequence and progression can be seen in the Mathematics Bands we have developed which outline the skills, knowledge and understanding expected for each year group.

**Stage 1** : **Pre-numeracy and mathematical skills**

Early numeracy and mathematical skills begin to be developed during childhood at home and in the nursery/preschool environment through play, including sorting, sequencing, matching and pattern making activities.

We call this indirect preparation for learning that is to come.

Children are also likely to develop an awareness of the names of numbers e.g. one, two, three through rhymes such as *One, two, three, four, five, once I caught a fish alive*, reading books that involve numbers and referring to numbers in context in the environment.

Developing an awareness of number through songs and rhymes

Children who have either missed out on these experiences or who need further reinforcement, these experiences will need to continue with these experiences when they start school.

At this stage, children are not expected to recognise the digits 1-9, or connect the names of numbers with a quantity, or understand that they come in a fixed order. Instead, they have an awareness and curiosity of number, pattern and order.

Developing pre-numeracy skills through sorting and matching activities.

Sorting and matching activities in various contexts.

Instilling the concept of pattern and order in the classroom environment.

Pattern and ordering activity – indirect learning for counting, early number skills and mathematical language for space, shape and measure.

Pattern and ordering activity – indirect learning for counting, early number skills and mathematical language for space, shape and measure.

Pattern and ordering activity – indirect learning for counting, early number skills.

Repeating patterns combined with developing fine motor control.

Repeating patterns with everyday objects help children identify patterns and develop ordering and sequencing skills.

Conceptual variation is where the same concept is presented in a different way.

Reinforcing repeating patterns across the curriculum builds on and deepens learning**.**

The Pink Tower is used as indirect learning for counting and early number skills, pattern and order.

Developing children’s sense for pattern, order, early numeracy skills and spatial awareness.

Ordering the long rods is an example of indirect learning for counting and early number skills, pattern and length.

**Stage 2 : Rote counting to 10**

Children learn the order of numbers 1 to 10 through practise and rote counting. They need this knowledge before they can accurately count a set of objects.

Children only need to orally recite the correct number sequence at this stage. They do not need to recognise the digit although they are likely to have awareness of number digits displayed in and around the classroom and may begin to make this association.

Whilst some children may have the ability to remember numbers in sequence up to 100 or even more, this does not necessarily mean that they are aware each number is representative of a quantity or are ready to work with large numbers.

All initial number work should focus on ensuring children are secure with numbers up to 10.

**Stage 3 : Counting objects up to 10 (one to one correspondence)**

To count accurately, a child needs to combine their knowledge of the sequence of numbers with the skill of 'one to one correspondence'.

One to one correspondence is the ability to match one object to one other object or person so that when they count, they do not count the same object twice.

One to one correspondence : Pre-counting skills.

Touch is really important in developing this skill and conceptual understanding.

At this stage, the adult may model counting as the child puts the apples into the tray one at a time and counting to 6 until the carton is full.

One to one correspondence – filling an egg box.

Counting different objects in different contexts and in different arrangements is an example of contextual variation and is important in deepening learning so it can be applied in different contexts.

One to one correspondence – counting spots on a ladybug.

Soon the child is able to accurately count a set of objects independently.

Using the skill of one to one correspondence to accurately count.

Counting requires careful hand-eye coordination. The children are not expected to recognise the symbol/digit that represents the quantity they are counting. The number digit is presented here to help children associate the number with the symbol.

Early counting skills and one to one correspondence.

Many children may have the ability to say numbers in sequence (sometimes up to 100 or even more!). This does not necessarily mean that they are aware each number is representative of a quantity.

This can easily be assessed by putting out a small number of objects and asking the child :

How many?

Activities that involve one to one correspondence that reinforce the idea that numbers represent a quantity are crucial.

Counting teddy bears using domino templates to support one to one correspondence.

When asked, ‘show me 2’, the child correctly identifies the set of 2 cubes using one to one correspondence.

Dice and domino games are useful activities as counting spots on a domino requires accurate one to one correspondence and reinforces the concept that a number represents a quantity.

Dice and domino activities can support children’s early number skills and helps them understand that numbers

represent a quantity.

Children will soon develop the ability to subitise when they see a small number of objects.

2 1 3

Subitising is the ability to look at a small number of objects and instantly recognise how many objects there are without needing to count. This means when a child is shown two objects they know there are two without having to count them. Our brains generally can subitise up to 5 objects. Try it and see!

The spindle box reinforces the idea of quantity through a kinaesthetic approach as the child soon realises the feel of a bundle of 9 spindles is much greater than preceding bundles. This activity also introduces number symbols/digits, although the primary purpose of the activity is accurate counting.

The Spindle Box helps children learn to associate quantity increasing.

Children need to learn to count a set of objects systematically to avoid counting the same number twice. This can be achieved by teaching them to place the objects in a line and counting left to right.

This is indirect preparation for future number line work. Children count out loud as they physically move their finger left to right:

Children learn to arrange objects in a straight line and count left to right as indirect preparation for later number line work and to avoid double counting.

**Stage 4 : Conservation of number**

There is a stage where children will count out the objects in a group, but when you ask them how many there are or rearrange them, they either go back and count again, or say a completely different number.

Conservation of number is the concept whereby a child is able to understand that the number of objects remains the regardless of how they are arranged or order they are counted.

Repeated exposure of counting the same number of objects but in a different order or arrangements helps a child internalise this concept.

Children learn through planned experiences that even if you re-arrange the 5 objects differently or count them in a different order, there are still the same number of objects.** **

**Stage 5** : More, less, the same

Before children can develop the ability to find one more and one less, they need to understand the terms more and less

Children need lots of repeated experiences to compare two sets of objects using their ability to count.

They need to be able to say which is more, which is less and which is the same using their skills of counting.

**Stage 6 :** Counting on and back

Counting on is the ability to verbally count on from a number other than one, e.g. when asked to count on from 5, the child continues 6,7,8 etc.

Children only need to verbally count on and back. They do not have to recognise the digits and count on as number recognition comes later, although some children may recognise numbers.

Both the ability to verbally count on (and back) and understand the last number they count stands for the number in the group is fundamental for the next stage and the ability to calculate one more and one less.** **

**Stage 7 : One more/one less**

This is the ability to say what is one more or one less than a given number. For example, if the teacher asks what is one more than 6, the child says 7. If the teacher asks what is one less than 9, the child says 8.

This should be explored practically first to ensure the child is referring to quantity rather than just saying what number comes next or before. This is because not all children realise that if you add one more or take one away, they do not need to recount all the objects again.

If they are secure with the concept of conservation of number, they will count on from the known number.

The child knows there is 5 in the first group and uses this knowledge and understanding to say there is 6 when one is added.

If a child is not secure, they will count all the objects from the beginning which indicates that they need further work at Stage 4 and conservation of number.

These children will need the teacher to show them repeatedly that the number does not change even if another object is added, time has elapsed or the objects are rearranged or counted in a different order.

A simple number line with or without the digits can be used to reinforce the concept of one more/less.

Number rods can also be used to develop a child’s understanding of one more and one less. This contextual variation is important in developing flexibility of thought and deepens and broadens children’s understanding of number.

Whilst to the untrained, it may be perceived to confuse children, it is important that the context is varied only when the child has fully understood the previous context.

Finding one less uses the same processes however children will need practise counting backwards to support them in finding one less and be familiar with the concept of less or fewer.

At this stage, children only work with numbers up to 10.

Once a child is secure in their understanding that they do not need to count the entire set from the beginning and has memorised the order and sequence of numbers, they will be able to verbally say which number is one more or one less than a given number by counting on in their head.

**Stage 8 : Number recognition and ordering up to 10**

This is where children learn to match the number to the physical symbol. It is an abstract concept similar to matching sounds with their corresponding graphemes because the symbol often bears little resemblance to what it actually represents unlike the spots on dice.

Number rods increase in length to show quantity.

It is likely that many children will have already learnt some of this indirectly though the previous stages.

Children learn this recognition through repeated exposure until it is firmly embedded into their visual memory. This will include daily practise using flash cards and the ‘look and say’ approach as well as repeated exposure and labelling in various contexts.

Contextual variation : Number recognition and ordering in different contexts embeds learning .

Contextual variation : Number recognition and ordering numbers.

Contextual variation : Number recognition and ordering numbers.

Contextual variation : Number recognition.

Contextual variation : Number recognition.

Number rods increase in length and can be used to underpin the concept that abstract number represents a fixed quantity. Each section on the rod represents one unit.

Number rods are used alongside number names.

Children are taught how to count and arrange the counters as indirect preparation for teaching odds and evens.

To show a deep understanding, children should be able to order a randomly arranged set of numerals as well as identify any missing numbers.

Identifying a missing digit from a sequence.

**Stage 9 : Adding and subtracting numbers to 10**

*Addition*

The child begins with combining two sets of objects together to find out how many altogether in various contexts.

It begins with showing the child two sets of objects, counting the total of each set before physically moving them together and counting how many altogether.

The teacher models the language of addition for the child or class to repeat.

*‘Five add four is the same as (or equals) nine’.*

Symbols can be introduced when the child is able to identify each numeral. The teacher models the language whilst pointing to or placing the symbols.

Children use moveable numbers, symbols, counters or rods and practise saying out loud the number sentence.

*‘Two add two equals (is the same as) four’.*

Number rods provide contextual variation and reinforce the concept that when two or more numbers are combined they equal a larger number.

This dual approach of using objects and number rods is an example of conceptual variation and is important to deepen children’s understanding of number.

The child is shown that a three rod and a two rod is the same as and equals a five rod. They see it is the same and equals the five rod. The teacher models the language at the same time.

Children are taught that the order the two numbers are combined makes no difference.

It is important that the children do not fall into the habit of always forming a number sentence with the equals symbol at the end.

The teacher models the language whilst pointing to the symbols.

The number rods provide indirect preparation for the addition board and Diennes Cubes when children are familiar with the symbols for add and equals.

Children practise addition using an addition board.

*Subtraction*

Subtraction should be taught alongside addition because it helps children to see the important relationship between the two. This is typical practice in the Far East.

This relationship is introduced by physically combining two known sets of objects as an addition and then taking away the last set to show we end up with the number we started with.

*‘Five add four equals nine. *

*Nine take away four equals five’.*

Children learn that we arrive back at the original number if we add a number and then remove the same number.

By physically showing the child movement of counters by combining them to find the total, then removing the second number set, they see we get back to the first number. Using two different coloured sets of objects assists with this.

It is important the teacher physically moves the counters whilst modelling the language of subtraction.

This can be reinforced with the number rods. The child is reminded that three add two equals five. As with the counters, the teacher shows the child that five take away two equals three whilst modelling the language of subtraction.

*‘Three add two equals (or is the same as) five. *

*Five take away two is three’.*

Numbers rods provide a meaningful transition to the subtraction board.

The child starts with the number that is to be subtracted from (in this case a blue 8 rod) and places it on the subtraction board.

The child finds the number they are subtracting. In this case the red 2 rod.

By placing them alongside each other, the child can physically see how much of the 8 rod would remain if 2 was subtracted from it.

*Equations*

It is important that children see an equation representing a number value rather than a problem to solve, process or sum.

The children will have been introduced to the formal presentation of equations but they need to understand that 3 + 4 is the same as 10 - 3.

This demands a deep level of understanding as the two equations do not look the same. They need to understand the equation represents a specific value.

A child who is secure in this will be able to match equivalent addition and subtraction equations.

**Stage 10: Working with numbers up to 10**

This stage deepens and consolidates children’s understanding of number relationships up to and including the number 10. Children work with, and manipulate only numbers up to 10.

*Patterns and number facts*

Children learn different ways to make 10 until they have a secure recall of these number facts.

During this stage, the concept and value of 10 is explored in different ways using different equipment to explore patterns.

Numicon is used as it provides a tactile experience as well as the addition board and number rods.

*Doubling and halving*

Children learn that to double something is repeated addition; that is adding the same amount again.

Counters, rods and Numicon provide contextual variation to develop and deepen the understanding of doubling. Here counters are introduced to show the number 3 is doubled (repeated addition, making the number twice as big, adding on the same number).

Number rods provide conceptual variation to develop and deepen children’s understanding of doubling.

Children will be familiar at this stage with simple equations and can be shown how an equation can be used to find a double.

*Equivalence*

Through contextual variation, children learn to represent equivalent values in different ways. It is important that children understand that the equals sign refers to equivalence rather than a symbol for an answer to a sum or equation.

**Beyond Stage 10: Place value and the 4 operations**

This is where children order and sequence numbers up to 20 and learn the value of each digit in a two digit number. Once secure, this understanding can be transferred to two digit numbers beyond 20.

Children learn that 10 units is the equivalent and can be exchanged for a 10 rod. They physically can see they are equivalent. Either Diennes cubes or Montessori coloured beads are used.

Children are then taught that when they count out a set of objects greater than ten, they exchange 10 units for a 10 rod.

Place value cards are used to help children understand the value of each digit.

The child is shown the number 10 and number 4 using the place value cards to show 14 is made up of 10 and 4. This shows them how the number is represented and written.

It is important that the teacher emphasises that the 1 represents 10 units and the 4 is 4 units.

The Seguin board can be used to further reinforce this concept. Here beads are used but these can be substituted for Diennes.

Once children have a secure understanding of ordering, sequencing and the value of each digit in numbers between 10 and 20, they can transfer this skill to numbers up to 100.

They will need lots of practise rote counting and recognising two digit numbers. Children also need to learn to count in 10s. The hundred square grid as well as other activities can assist with this.

Once they can do this, they will be able to make any 2 digit number using the Diennes cubes or Coloured Beads.

**Number bonds and tables fluency**

There are certain facts that children will need to learn and memorise as they progress through the maths curriculum. This is particularly important as children embark on more complicated processes where the lack of knowledge of certain facts may hinder their ability to accurately calculate.

In Year 1, children begin to memorise pairs of numbers that make 10 e.g. 3+7 =10 before moving on to single digit facts such as 7+6 =13.

Whilst children initially find the solutions to these equations using practical equipment, this would not be reliable or time efficient for solving equations such as 26 +45.

The progression is as follows :

Standard 1 : Addition number facts to 10

e.g. 6 + 3 = 9

Standard 2 : Subtraction number facts from 10.

e.g. 9 - 6 =3

Standard 3: Doubles up to 20.

Standard 4 : Additions of 10’s up to 100

e.g. 60 + 20 = 80

Standard 5 : Subtraction of 10’s from 100

e.g. 80 - 60 = 20

Standard 6 : Addition up to 20

e.g. 13 + 6 =19

Standard 7 : Subtraction up to 20

e.g. 19 - 6 =13

Standard 8: Counting in multiples forward and back in 2s, 3s,5s and 10s from

different points

Standard 9: Know multiplication and division facts for x2, x3, x5 and x10

e.g. 8 x 3 = 24

24 ÷3 = 8

Standard 10: Counting in multiples forward and back in 4s, 8s, 50s and 100s

Standard 11: Know multiplication and division facts for x4 and x8 tables

Standard 12: Counting in multiples forward and back in 6s, 7s, 9s,

25s and 1000s

Standard 13: Know multiplication and division facts up to x12 tables (Yr 4)

e.g. 12 x 9 = 108

108 ÷9 = 12

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