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Guiding Principles for Learning and Teaching of NumberInquiry –based learning Learning is inquiry based. Children are routinely engaged in thinking hard to solve arithmetical problems

Initial and on-going assessment Learning is informed by ongoing assessment through teaching.

Zone of Proximal Development Learning is focused just beyond the cutting edge of children’s current knowledge.

Quality crafted lessons Learning approaches are carefully selected to meet the needs of this child.

Developing more sophisticated strategies The teacher understands the children’s numerical strategies and deliberately engenders the development of more sophisticated strategies through mathematisation.

Observing and fine-tuning learning Observation of learning informs micro-adjustments within lessons

Symbolising and notating Children talk about their strategies and over time learn to notate their thinking and formalise this.

Sustained thinking and reflection Wait time is valued and given

Intrinsic Satisfaction Through distancing settings and self-checking their thinking children understand they are making progress.

Mathematisation Theme Addition and Subtraction Multiplication and Division Fractions, Decimals and Percentages

Structuring number Understanding the underlying connections between numbers (e.g. 16=10+6,8+8, etc.)

Understanding the fairness and equality of multiplication and divisionLinking and applying knowledge of addition and subtraction to multiplication and divisionUnderstand composite and unitary aspects of a quantity

Linking and applying knowledge of Multiplication and Division to Fractions, Decimals and PercentagesUnderstand composite and unitary aspects of a quantity

Extending the range of numbers Progressively introducing a wider range of numbers to calculate with

Introducing multiples and sequences in steps to ensure secure understanding

Building on knowledge of fair share and equal parts, sequentially introduce fractions in steps to ensure secure understanding. This leads to a depth of understanding within fractions and this knowledge enables learners to develop a meaningful understanding of decimals and percentages.

Decimalising towards Base-ten thinking

Developing base ten thinking that exploits using ten as a unit (e.g. conceptual place value and counting in tens)

Applying conceptual place value to multiplication and division Applying conceptual place value to Fractions, Decimals and Percentages

Unitising and not counting by ones Regarding a number larger than one as a unit and use this unit to solve a task.

Adopting appropriate settings to explore the composite and unitary aspects of a quantity

Unitising, Partitioning, Disembedding and Iterating

Distancing the setting of materials Progressively reducing the role of materials Distancing the Setting (Manipulate it →See it→Flash it→Screen It→Check It→Express It and Explain It)

Distancing the Setting (Manipulate it →See it→ Flash it→ Screen It→ Check It→ Express It and Explain It)

Notating Progressively formalising mathematical thought in a structured way

Notation (Informal Jottings→Semi-Formal Written Strategies→Formal Written Algorithms)

Notation (Informal Jottings→ Semi-Formal Written Strategies→ Formal Written Algorithms)

Formalising and Generalising Developing arithmetic to involve more formal notation and more formal procedures.Reasoning that involves proceeding from a few cases to many cases

Increasing the range of tasks to which children apply their strategies

Increasing the range of tasks to which children apply their strategies

Numeracy progression at a glanceAddition and Subtraction Multiplication and Division Fractions, Decimals and Percentages

Understanding Numbers and Numerals

Number Structuring for Addition and Subtraction Understanding of Multiples and

Sequences of MultiplesGrouping and Sharing for Multiplication and

Division Fractions, Decimals and Percentages

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On track in the Nursery Developing number word sequence Developing perceptual counting. Understanding fairness in equal groups and shares Developing an understanding of fair sharing

On track in P1 Developing number word sequence to at least 30

Developing number structures for addition and subtraction to 10 Making equal groups and shares Understanding wholes, halves and quarters

On track in P2 Developing number word sequence to at least 120

Further developing knowledge of addition and subtraction to 10

Understanding multiples of 2 Understanding, grouping and sharing in 2s Understanding unit and composite fractions

On track in P3 Sequencing and Place Value to 100

Developing a secure knowledge of addition and subtraction to 20

Incrementing and decrementing to, from and through decuples to 120

Adding and subtracting 10 within 120

Understanding multiples of 2 and 3 Understanding, grouping and sharing in 2s and 3s Understanding and comparing whole, unit and composite fractions

On track in P4 Sequencing and Place Value to 1000Incrementing and decrementing on and off the decuple

by 10s and 1s to 120Adding and subtracting from and to a decuple when the

part is within 10

Understanding multiples of 2, 3, 5 and 10 Understanding grouping and sharing in 2s, 3s, 5s and 10s and recall of facts

Understanding part whole reasoning and applying number to this

Applying number to part whole reasoningUnderstanding how to measure with unit fractions and use

reversible reasoning

On track in P5Incrementing and decrementing on and off the decuple by 10s and 1s beyond 120

Adding and subtracting from and to a decuple when the part is within 10 and then beyond 10Understanding conceptual place valul

Understanding multiples of 2, 4, 8, 3, 5, 6, 9, 10 and lucky 7

Understanding grouping and sharing in 2s, 4s, 8s, 3s, 5s, 6s, 9s, 10s and 7s and recall of facts

Understanding reversible multiplicative reasoningUnderstanding fractions as numbers

Understanding equal sharing of multiple items

On track in P6Understanding addition and subtraction to at least 1000 and beyond and working towards formal

algorithms Understanding 2-digit multiplication and division and working towards formal algorithmsUnderstanding multiplication of fractions

Understanding percentages and decimal fractions (including Place Value)

On track in P7 Making efficient use of mental strategies and formal algorithms within the four operationsUnderstanding addition, subtraction and division of

fractionsUnderstanding equivalent fractions

Addition and SubtractionMultiplication and Division Fractions, decimals and percentages

On track in S1Demonstrating confidence in the use of mental strategies and formal algorithms within the four operations

Multiples Factors and PrimesPowers, Roots and Scientific Notation

Mean, Median, Mode and Range of Data

Mixed Numbers and Improper fractionsAdding and subtracting fractions

Multiplying Fractions within and beyond the wholeDividing Fractions

Application of Fractions as NumbersApplication of equivalent fractions, decimals and percentages

On track in S2Demonstrating confidence in the use of mental strategies and formal algorithms within the four operations to solve multistep

problemsApplication of direct and indirect proportion

RoundingDemonstrating confidence to interpret and solve multistep problems involving Fractions, decimals and percentages, and the four operations

Use prior knowledge to interpret and solve fractions, decimals and percentages in real life, multi-step problems.Use prior knowledge and understand to calculate, compare and predict probabilities

On track in S3 Further Explore relationship between powers roots and irrational numbers ToleranceApply probability to connected events in a real life context.

On track in the NurseryExperiences and OutcomesI am developing a sense of size and amount by observing, exploring, using and communicating with others about things in the world around me. MNU 0-01aI have explored numbers, understanding that they represent quantities, and I can use them to count, create sequences and describe order. MNU 0-02aI use practical materials and can ‘count on and back’ to help me understand addition and subtraction, recording my ideas and solutions in different ways. MNU 0-03aI have spotted and explored patterns in my own and the wider environment and can copy and continue these and create my own patterns. MTH 0-13a

Experiences and OutcomesI can share out a group of items by making smaller groups and can split a whole object into smaller parts. MNU 0-07a

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Addition and Subtraction Multiplication and Division Fractions, Decimals and Percentages

Understanding numbers and numerals Number Structuring for Addition and Subtraction Understanding of multiples and sequences of multiples Grouping and Sharing for Multiplication and Division

DEVELOPING NUMBER WORD SEQUENCELearning Outcomes● I can say forward number word sequences to at least 20● I can say backward number word sequences from at least 20● I am beginning to work out the number word after and number word

before● I can read numeral sequences to at least 10● I can sequence numerals to at least 10● I can identify and recognise numerals to at least 10.● I can work out the missing number or numeral in a sequence to at least

10

Language● Forward, backward, before, after, in-between, first,

second……, greater or more than rather than bigger and smaller

● Use correct terminology for place value …14 is ten and four rather than one and four

● Make the distinction between a numeral and the number it represents clear

Settings:● Embedded in daily routines● Number Line● Numeral Track● Numeral Cards

Mathematisation – Check it, Prove it, Explain it Manipulation of materials Visual representation

Notation Mark making to represent numbers and numerals Reading numeral cards

Activities should progress from the organisation of thinking to generalising

DEVELOPING PERCEPTUAL COUNTINGLearning Outcomes● I can count perceived items presented in different ways● I can count items in one collection● I can count out a requested number of items● I can count items in a row forwards and backwards● I can count items in two collections● I can estimate within my number range


second……, greater, more than or less than rather than bigger and smaller, groups, altogether, add

Settings:● Items that can be manipulated and counted (strong

emphasis on play experiences)● Items displayed in different colours, types,

arrangements (rows, dominos, random arrays)


Notation Mark making


From Phase 3, please ensure that this column is a clear focus before the right hand column

UNDERSTANDING FAIRNESS IN EQUAL GROUPS AND SHARESLearning Outcomes● I can describe, organise and make equal groups● I can describe, organise and partition equal shares● I understand that for shares to be equal, a quantity may

remain● I can break a whole into parts (amounts and items)● I can estimate within my number range

Settings● Items that can be counted, grouped and shared

(strong emphasis on play experiences)● Items displayed in different colours, types,

arrangements, random arrays● Materials that can be broken up and shared e.g.

playdough, cake mixture etc.

Language● Parts, whole, share, group, the same, fair share, left

over




Please note that early experiences of grouping and sharing are essential and although it is not ‘formal’ times tables, it is still preparatory work for multiplication and division

DEVELOPING AN UNDERSTANDING OF FAIR SHARINGLearning Outcomes● I can break a whole into parts and can describe how I made it fair● I can estimate how many fair shares there are

Settings● Materials that can be broken up and shared e.g. playdough, cake mixture etc● Construction paper, String/Ribbon – use manipulatives as a unit of measure

Language● Parts, whole, share, the same, fair




5.1.1- 5.2.6 5.3.1 – 5.6.46.3.1 – 6.3.4

6.6.1 – 6.6.6

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On track in Primary 1Experiences and OutcomesI am developing a sense of size and amount by observing, exploring, using and communicating with others about things in the world around me. MNU 0-01aI have explored numbers, understanding that they represent quantities, and I can use them to count, create sequences and describe order. MNU 0-02aI use practical materials and can ‘count on and back’ to help me understand addition and subtraction, recording my ideas and solutions in different ways. MNU 0-03aI have spotted and explored patterns in my own and the wider environment and can copy and continue these and create my own patterns. MTH 0-13a

Experiences and OutcomesI can share out a group of items by making smaller groups and can split a whole object into smaller parts. MNU 0-07a


Understanding numbers and numerals Number Structuring for Addition and Subtraction Understanding of multiples and sequences of multiples

Grouping and Sharing for Multiplication and Division

DEVELOPING NUMBER WORD SEQUENCE TO AT LEAST 30Learning Outcomes● I can say and read forward number word sequences to at least 30● I can say and read backward number word sequences from at least 30● I can recall the number word after and number word before to at least

30● I can say the next 2,3 or 4 numbers in a forward and backward number

sequence to at least 30● I can sequence numerals to at least 30● I can identify, recognise and place numerals to at least 30● I can work out the missing number or numeral in a sequence to at least

30● I can count on or back from a given number to find or locate another

number in a sequence to at least 30


second……, greater or more than rather than bigger and smaller

● Use correct terminology for place value ….14 is ten and four rather than one and four


Settings● Embedded in daily routines● Number Line● Numeral Track● Numeral Cards

Mathematisation – Check it, Prove it, Explain it Manipulation of materials Visual representation Screening and flashing numerals on a numeral track

Notation Writing numerals correctly Reading and writing a numeral sequence


DEVELOPING NUMBER STRUCTURES FOR ADDITION AND SUBTRACTION TO 10Learning Outcomes● I understand that numbers can be expressed as parts that can

make up a whole using numerals and symbols● I understand the concept of zero● I can add and subtract using small doubles● I can add and subtract using partitions of 5● I can add and subtract using “5 plus” facts to 10● I understand the commutative relationship within addition● I understand the distributive and inverse relationship within

addition and subtraction● I can estimate within my number range and explain my thinking● I can solve the following tasks within 10; additive, removed item,

missing addend, missing subtrahend, missing minuend and use flexible strategies

Using all of the above● quickly derive addition and subtraction tasks

Language● Before, after, in-between, first, second……, greater, more than or

less than rather than bigger and smaller, groups, altogether, add, strategy

Settings● Items that can be manipulated, counted and screened● Items displayed in different colours, types, arrangements (rows,

dominos, random arrays)● Finger Patterns● Five Frames● Ten Frames (arranged in 5 plus)`● http://catalog.mathlearningcenter.org/apps/number-frames

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials to screened and flashed number

problems● Visual representation to screened and flashed number problems● Fully screened number problems● Number problems presented as number sentences

Notation● Reading number sentences that represent a problem● Creating number sentences● Reading and solving number sentences


From Phase 3, please ensure that this column is a clear focus before the right hand column

MAKING EQUAL GROUPS AND SHARESLearning Outcomes

I understand the term equal I can combine and count equal groups I can partition a collection into equal shares and

establish the number of shares I can partition a collection into equal shares and

establish the number in each share I can identify two equal number bonds e.g. 2+3 = 1+4 I understand that for shares to be equal, a quantity

may remain I can estimate within my number range and explain

my thinking

Settings Items that can be counted, grouped and shared Items displayed in different colours, types,

arrangements Materials that can be broken up and shared e.g.

playdough, cake mixture etc

Language Parts, wholes, share, shares, group, groups , the

same, fair share, equal, half way, half of, collection, left over, groups of, equal shares, partition

Equal

Mathematisation – Check it, Prove it, Explain it Manipulation of materials Visual representation Manipulation of materials to screened and flashed

number problems Visual representation to screened and flashed

number problems Fully screened number problems Number problems presented as number sentences

Notation Mark making to represent thinking


Please note that early experiences of grouping and sharing are essential and although it is not ‘formal’ times tables, it is still preparatory work for multiplication and division

UNDERSTANDING WHOLES, HALVES AND QUARTERSLearning Outcomes● I can break a whole into two equal parts (items)● I understand the term half way and half of● I understand that a quarter can be made halving the half● I understand the notation for a fraction● I can estimate one fair share and show this within practical application

Settings● Linear materials to be broken up and shared e.g. bars, paper strips, rods● String and ribbon● Construction paper

Language● Parts, whole, share, the same, fair, half, half way, half of, quarter and quarter of, unit fraction,

equal, comparative language for size and length




National BenchmarksEstimating and rounding Recognises the number of objects in a group, without counting (subitising) and uses this information to estimate the

number of objects in other groups. Checks estimates by counting. Demonstrates skills of estimation in the contexts of number and measure using relevant vocabulary, including less

than, longer than, more than and the same.Number and number processes Explains that zero means there is none of a particular quantity and is represented by the numeral 0. Recalls the number sequence forwards within the range 0 - 30, from any given number. Recalls the number sequence backwards from 20. Identifies and recognises numbers from 0 to 20. Orders all numbers forwards and backwards within the range 0 - 20. Identifies the number before, the number after and missing numbers in a sequence within 20.

Uses one-to-one correspondence to count a given number of objects to 20. Identifies ‘how many?’ in regular dot patterns, for example, arrays, five frames, ten frames, dice and irregular dot

patterns, without having to count (subitising). Groups items recognising that the appearance of the group has no effect on the overall total (conservation of number). Uses ordinal numbers in real life contexts, for example, ‘I am third in the line’. Uses the language of before, after and in-between. Counts on and back in ones to add and subtract. Doubles numbers to a total of 10 mentally. When counting objects, understands that the number name of the last object counted is the name given to the total

number of objects in the group. Partitions quantities to 10 into two or more parts and recognises that this does not affect the total. Adds and subtracts mentally to 10. Uses appropriately the mathematical symbols +, − and =. Solves simple missing number problems.

National BenchmarksFractions, decimal fractions and percentages Splits a whole into smaller parts and explains that ‘equal parts’ are the same size. Uses appropriate vocabulary to describe each part, at least halves and quarters. Shares out a group of items equally into smaller groups.

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http://catalog.mathlearningcenter.org/apps/number-frames

On track in Primary 2

Experiences and OutcomesI can share ideas with others to develop ways of estimating the answer to a calculation or problem, work out the actual answer, then check my solution by comparing it with the estimate. MNU 1-01aI have investigated how whole numbers are constructed, can understand the importance of zero within the system and can use my knowledge to explain the link between a digit, its place and its value. MNU 1-02aI can use addition, subtraction, multiplication and division when solving problems, making best use of the mental strategies and written skills I have developed. MNU 1-03aI can continue and devise more involved repeating patterns or designs, using a variety of media. MTH 1-13aThrough exploring number patterns, I can recognise and continue simple number sequences and can explain the rule I have applied. MTH 1-13bI can compare, describe and show number relationships, using appropriate vocabulary and the symbols for equals, not equal to, less than and greater than. MTH 1-15aWhen a picture or symbol is used to replace a number in a number statement, I can find its value using my knowledge of number facts and explain my thinking to others. MTH 1-15b

Experiences and OutcomesHaving explored fractions by taking part in practical activities, I can show my understanding of: how a single item can be shared equally the notation and vocabulary associated with fractions where simple fractions lie on the number line. MNU 1-07a

Through exploring how groups of items can be shared equally, I can find a fraction of an amount by applying my knowledge of division. MNU 1-07bThrough taking part in practical activities including use of pictorial representations, I can demonstrate my understanding of simple fractions which are equivalent. MTH 1-07c


Understanding numbers and numerals Number Structuring for Addition and Subtraction Understanding of multiples and sequences of multiples Grouping and Sharing for Multiplication and Division

DEVELOPING NUMBER WORD SEQUENCE TO AT LEAST 120Learning Outcomes

● I can recite the decuples from 0-120 forwards and backwards

● I can say and read forward number word sequences to at least 120

● I can say and read backward number word sequences from at least 120

● I can recall the number word after and number word before to at least 120

● I can track numerals● I can say the next 2,3 or 4 numbers in a forward and

backward number sequence to at least 120● I can sequence numerals to at least 120● I can work out the missing number or numeral in a

sequence to at least 120● I can identify, recognise and place numerals to at least

100● I can count on or back from a given number to find, locate

another number in a sequence to at least 100● I can calculate the number of backwards or forward jumps

from a to b

Language● Decuple, decade● Forward, backward, before, after, in-between, first,

second……, greater or more than, less than rather than bigger and smaller



Settings● Embedded in daily routines● Number Line● Numeral Track● Hundred Square● Empty Number Line

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials● Visual representation● Justifying, prove and explain the next number in a number

sequence● Screening and flashing numerals in various settings

Notation● Writing numerals correctly● Reading and writing a numeral sequence● Completing number squares


FURTHER DEVELOPING KNOWLEDGE OF ADDITION AND SUBTRACTION TO 10Learning Outcomes● I understand that numbers can be expressed as parts that

can make up a whole using numerals and symbols● I can add and subtract using partitions of 10.● I can add parts in the range 0 to 5● I understand the commutative relationship within addition● I understand the distributive and inverse relationship within

addition and subtraction● I can estimate within my number range and use a variety of

strategies● I can check and compare my estimate● I can solve the following tasks within 10; additive, removed

item, missing addend, missing subtrahend, missing minuend and use flexible strategies

SECURE KNOWLEDGE OF ADDITION AND SUBTRACTION TO 10Learning Outcomes● I can add when the whole is in the range 0 to 10● I can subtract parts in the range 0 to 5● understand the commutative relationship within addition● understand the distributive and inverse relationship within

addition and subtraction● I can solve the following tasks within 10; additive, removed

item, missing addend, missing subtrahend, missing minuend and use flexible strategies

Using all of the above● quickly derive and recall addition and subtraction tasks

Language● Before, after, in-between, first, second……, greater, more

than or less than rather than bigger and smaller, groups, altogether, add, subtract, remove, missing, addend, subtrahend, minuend, strategy

Settings● Items that can be manipulated, counted and screened● Items displayed in different colours, types, arrangements

(rows, dominos, random arrays)● Finger Patterns● Five Frames● Ten Frames (arranged in 5 plus)● Arithmetic rack● http://catalog.mathlearningcenter.org/apps/number-frames


problems● Visual representation to screened and flashed number

problems● Fully screened number problems● Number problems presented as number sentencesNotation● Reading number sentences that represent a problem● Creating number sentences● Reading, writing and solving number sentences


UNDERSTANDING MULTIPLES OF 2Learning Outcomes● I can say the forward number word sequences in

multiples of 2s and keep track of the counts on my fingers

● I can say the backward number word sequences in multiples of 2s and keep track of the counts on my fingers

● I can say the next number word before and after in a multiple number sequence in 2s

● I can identify the placement of numerals in sequences of 2s

Settings● Numeral Lines● Numeral Tracks

Language● Multiples, sequence, increment, decrement,

keeping track

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials● Visual representation● Justifying, prove and explain the next

number in a number sequence● Screening and flashing numerals in various

settings

Notation● Reading and writing a numeral sequence

Activities should progress from the organisation of thinking to generalising.

Please ensure that this column is a clear focus before the right hand column

UNDERSTING, GROUPING AND SHARING IN 2sLearning OutcomesI know that 2 can be regarded as a composite group of 1 and can use this to:

● describe, build and count simple arrays of 2● count and describe items grouped in 2s● increment and decrement in groups of 2● half numbers● link groups of 2 and 2 equal groups to doubles● partition a collection into equal shares and establish

the number of shares● partition a collection into equal shares and establish

the number in each share● demonstrate the commutative relationship within

multiplication using manipulatives● demonstrate that for shares to be equal, a quantity

may remain● estimate within my number range and use a variety

of strategies● check and compare my estimate

Settings● n-tiles● Arithmetic Rack● Dot Arrays → Square Arrays (Base 5 and 10 Grids)

Language● Parts, wholes, share, shares, group, the same, fair

share, equal, half way, half of, collection, groups of, equal shares, partition, half, double, rows, columns, arrays, increment and decrement, multiply and divide , rows and columns, commutative, remainder, horizontal and vertical


problems● Visual representation to screened and flashed number

problems● Fully screened number problems● Number problems presented as number sentences

Notation● Reading number sentences that represent a problem● Creating number sentences● Reading and solving number sentences


UNDERSTANDING UNIT AND COMPOSITE FRACTIONSLearning Outcomes● I can partition a whole into equal parts (halves, , quarters)●Using linear representation, I can verify fractions of a whole● I can iterate a unit fraction part to reform the whole● I understand the notation for a fraction● I understand the difference between a unit fraction and a composite fraction●When shown a part and a whole I can work out the size of the part

Settings●Linear materials to be broken up and shared e.g. paper strips. The children should be given the

opportunity to make the strips and not just be given them.●Children should be able to see a whole strip and a fraction of the strip without partitions and relate them to

each other.e.g.

This is half of the whole strip (you do not need to see two halves to know because you care comparing it to the whole.

Language●Out of e.g. - a third should be referred to as ‘one part out of three parts’ at this stage●Parts, whole, share, the same, fair, half, half way, half of, equal, reform, comparative language for size and

length, partition, unit fraction (1/2, 1/3, 1/4 etc), composite fraction (2/3, 3/4, 2/5 etc)

Mathematisation – Check it, Prove it, Explain it● Manipulation of material● Visual representation● Mental Actions- unitizing, fragmenting and partitioning, iterating, disembedding, unit coordination● * Do not iterate beyond the whole at this stage*

Notation● Visual representations● Reading, writing and interpreting fractional notation


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On track in Primary 3Experiences and OutcomesI can share ideas with others to develop ways of estimating the answer to a calculation or problem, work out the actual answer, then check my solution by comparing it with the estimate. MNU 1-01aI have investigated how whole numbers are constructed, can understand the importance of zero within the system and can use my knowledge to explain the link between a digit, its place and its value. MNU 1-02aI can use addition, subtraction, multiplication and division when solving problems, making best use of the mental strategies and written skills I have developed. MNU 1-03aI can continue and devise more involved repeating patterns or designs, using a variety of media. MTH 1-13aThrough exploring number patterns, I can recognise and continue simple number sequences and can explain the rule I have applied. MTH 1-13bI can compare, describe and show number relationships, using appropriate vocabulary and the symbols for equals, not equal to, less than and greater than. MTH 1-15aWhen a picture or symbol is used to replace a number in a number statement, I can find its value using my knowledge of number facts and explain my thinking to others. MTH 1-15b






SEQUENCING AND PLACE VALUE TO 100● I can round whole numbers to the nearest 10● I can reads, write, order and recite whole numbers to 100, starting from any

number in the sequence.● Identifies the value of each digit in a whole number with two digits, for

example, 67 = 60 + 7 (Canonical form)● I can begin to find non-canonical forms of 2 digit numbers e.g 67 = 50 + 17

or 67 = 30 + 37

INCREMENTING AND DECREMENTING TO, FROM AND THROUGH DECUPLES TO 120 Learning Outcome

● I can increment and decrement by 10s on and off the decuple● I can increment and decrement by 10s from a given number to find or

locate another number in a sequence to 100● I can calculate the number of backwards or forward increments from a to b● I can round numbers to the nearest ten within 100 and beyond● I can name a numeral and describe the value of each digit to at least 100,

including zero as a place holder

Settings● Numeral Lines● Numeral Tracks● Word Problems● Identifying patterns

Language Canonical, Non-canonical, sequence, increment, decrement, keeping track,

rounding, Place value, before, after

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials● Visual representation● Justify, prove and explain the next number in a number sequence● Screening and flashing numerals in various settings

DEVELOPING KNOWLEDGE OF ADDITION AND SUBTRACTION TO 20Learning Outcomes

● I understand that numbers can be expressed as parts that can make up a whole using numerals and symbols

● I can add and subtract using 10 plus facts● I can add and subtract using doubles when the whole is in the range

10 to 20● I can add and subtract using near doubles when the whole is in the

range 0 to 20● I can add parts in the range 0 to 10● I can subtract when the whole is in the range 0 to10● I can solve the following tasks within 20; additive, removed item,


FURTHER DEVELOPING KNOWLEDGE OF ADDITION AND SUBTRACTION TO 20Learning Outcomes

● I can add and subtract using partitions of 20● I can add when the whole is in the range 0-20● I can subtract when the part is in the range 0 to 10● I understand the commutative relationship within addition● I understand the distributive and inverse relationship within addition

and subtraction● I can solve the following tasks within 20; additive, removed item,


DEVELOPING A SECURE KNOWLEDGE OF ADDITION AND SUBTRACTION TO 20Learning Outcomes

● I can add and subtract when the whole is in the range 0 to 20● I understand the commutative relationship within addition● I understand the distributive and inverse relationship within addition

and subtraction● I can estimate within my number range and use a variety of strategies● I can check and compare my estimate● I can estimate where to place given numerals● I can solve the following tasks within 20; additive, removed item,


PLEASE NOTE – this continues on the next page

ADDING AND SUBTRACTING 10 WITHIN 120Learning Outcome

● I can describe how I use my knowledge of number structures to add from a decuple number when the part is within 10

● I can describe how I use my knowledge of number structures to subtract to a decuple number within when the part is within 10

UNDERSTANDING MULTIPLES OF 2 and 3Learning Outcomes● I can say the forward number word sequences in

multiples of 2s and 3s keep track of the counts on my fingers

● I can say the backward number word sequences in multiples of 2s and 3s and keep track of the counts on my fingers

● I can say the next number word before and after in a multiple number sequence in 2s and 3s

● I can identify the placement of numerals in sequences of 2s and 3s

Settings● Numeral Lines● Numeral Tracks● Word Problems

Language● Multiples, sequence, increment,

decrement, keeping track, before, after

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials● Visual representation● Justify, prove and explain the next

number in a number sequence● Screening and flashing numerals in

various settings

Notation● Reading and writing a numeral

sequence



UNDERSTANDING GROUPING AND SHARING IN 2s and 3sLearning OutcomesI know that 2 and 3 can be regarded as composite groups of 1 and use this to:●describe, build and count simple arrays of 2s and 3s●count and describe items grouped in 2s and 3s● increment and decrement in groups of 2s and 3s●half and third numbers within relevant tables●partition a collection into equal shares and establish the

number of shares●partition a collection into equal shares and establish the

number in each share●demonstrate that for shares to be equal, a quantity may

remain. This is called the remainder.● relate area and perimeter markers to operations of

multiplication and division●demonstrate knowledge of perimeter markers and area

to solve multiplication and division facts in different ways

●demonstrate the commutative, associative and distributive relationship within multiplication

●estimate the number of groups or shares●check and compare my estimate

Settings● n-tiles● Arithmetic Rack● Dot arrays → square arrays (Base 5 and 10

Grids), perimeter markers and empty arrays● Word problems

Language●Parts, wholes, share, shares, group, the same, fair

share, equal, half way, half of, collection, groups of, equal shares, partition, half, double, thirds, rows, columns, arrays, increment and decrement, multiply and divide, remainder, horizontal and vertical, perimeter and area, commutative, associative and distributive,

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials to screened and flashed

number problems● Visual representation to screened and flashed

number problems● Fully screened number problems● Number problems presented as number sentences

Notation● Reading number sentences that represent a

problem● Creating number sentences● Reading, writing and solving number sentences


UNDERSTANDING AND COMPARING WHOLE, UNIT AND COMPOSITE FRACTIONSLearning Outcomes

● I can compare unit fractions with each other● I can compare unit fractions as part of a whole and describe the

relationship between them (e.g. a quarter of whole is smaller than a third of the whole)

● When shown a part and a whole I can work out the size of the part

Whole quarter

third

● I can iterate a unit fraction to reform the whole● I can iterate a unit fraction part and create a unit beyond the whole and

describe it● I can use unit fractions as a unit of measure to create composite fraction

measure e.g. use a 1/3 measure to create a 2/3 measure● I know that the bottom number in a fraction is called a denominator and

it tells me how many parts there are in a whole● I know that the top number in a fraction is called a numerator and it tells

me how many parts of the whole should be considered● I can use symbols to represent greater than or less than

Working with linear bars:● I can identify what fraction of the whole a given part is● I can identify and create proper fractions using my knowledge of parts

and wholes

e.g. What fraction of the longer bar is the smaller bar?

Settings● Linear materials that can be broken up and shared e.g. bars, paper

strips, rods

Language● Parts, whole, share, the same, fair, half, half way, half of, equal, iterate,

reform, comparative language for size and length, partition, unit fraction (1/2, 1/3, 1/4 etc), composite fraction (2/3, 3/4, 2/5 etc), disembed, denominator, numerator, composite fraction, proper fraction

Mathematisation – Check it, Prove it, Explain it● Manipulation of material● Visual representation● Mental Actions- unitizing, fragmenting and partitioning, iterating,

disembedding, unit coordination● * Do not iterate beyond the whole at this stage*



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● I can describe how I use my knowledge of number structures to add to a decuple number when the part is within 10

● I can describe how I use my knowledge of number structures to subtract from a decuple number when the part is within 10

● I can describe how I use my knowledge of number structures to add through a decuple number when the part is within 10

● I can describe how I use my knowledge of number structures to subtract through a decuple when the part is within 10

● I understand the commutative relationship within addition● I understand the distributive and inverse relationship within addition

and subtraction● I can estimate within my number range and use a variety of strategies● I can check and compare my estimate● I can solve the following tasks within 120; additive, removed item,


Language● Decade, decuple, before, after, in-between, first, second……, greater,

more than or less than rather than bigger and smaller, groups, altogether, add, subtract, remove, missing, addend, subtrahend, minuend, strategy



Settings● Number Lines, tracks, hundred squares● Arithmetic rack, 100 bead strings● Arrow Cards● Empty Number Line● Ten Frames● http://catalog.mathlearningcenter.org/apps/number-frames ● http://catalog.mathlearningcenter.org/apps/number-pieces ● http://catalog.mathlearningcenter.org/apps/number-line ● Word problems

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials to screened and flashed number problems● Visual representation to screened and flashed number problems● Fully screened number problems● Justifying, prove and explain the next number in a number sequence● Screening and flashing numerals in various settings● Number problems presented as number sentences

Notation● Representing strategies on an empty number line● Reading number sentences that represent a problem● Creating number sentences● Reading, writing and solving number sentences

● Activities should progress from the organisation of thinking to generalising

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http://catalog.mathlearningcenter.org/apps/number-line

http://catalog.mathlearningcenter.org/apps/number-pieces



Experiences and OutcomesHaving explored fractions by taking part in practical activities, I can show my understanding of: how a single item can be shared equally the notation and vocabulary associated with fractions where simple fractions lie on the number line. MNU 1-07aThrough exploring how groups of items can be shared equally, I can find a fraction of an amount by applying my knowledge of division. MNU 1-07bThrough taking part in practical activities including use of pictorial representations, I can demonstrate my understanding of simple fractions which are equivalent. MTH 1-07c




SEQUENCING AND PLACE VALUE TO 1000 I can rounds whole numbers to the nearest 10 and

100 and use this routinely to estimate and check the reasonableness of a solution.

I can reads, write, order and recite whole numbers to 1000, starting from any number in the sequence.

I can demonstrate my understanding of zero as a placeholder in whole numbers to 1000.

I can identify the value of each digit in a whole number with three digits, for example, 867 = 800 + 60 + 7 (Canonical form)

I can find non-canonical forms of three digit numbers e.g 867 = 700 + 167, 867 = 500 + 367


Language Canonical, Non-canonical, sequence,

increment, decrement, keeping track, rounding, Place value, before, after

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials● Visual representation● Justify, prove and explain the next number in

a number sequence● Screening and flashing numerals in various

settings

INCREMENTING AND DECREMENTING ON AND OFF THE DECUPLE BY 10S AND 1S TO 120Learning Outcome

● I can increment and decrement by 10s and by 10s and 1s on and off the decuple

● I can increment and decrement by 10s and by 10s and 1s from a given number to find or locate another number in a sequence to 120

● I can calculate the number of backwards or forward increments from a to b

● I can round numbers to the nearest ten within 120 and beyond

● I can name a numeral and describe the value of each digit to at least 120, including zero as a place holder

ADDING AND SUBTRACTING FROM AND TO A DECUPLE WHEN THE PART IS WITHIN 10Learning Outcome

● I can describe how I use my knowledge of number structures to add from a decuple number when the part is within 10

● I can describe how I use my knowledge of number structures to subtract to a decuple number within when the part is within 10

● I can describe how I use my knowledge of number structures to add to a decuple number when the part is within 10

● I can describe how I use my knowledge of number structures to subtract from a decuple number when the part is within 10

● I can describe how I use my knowledge of number structures to add through a decuple number when the part is within 10

● I can describe how I use my knowledge of number structures to subtract through a decuple when the part is within 10

● I can explain the commutative relationship within addition

● I can demonstrate my understanding of the distributive and inverse relationship within addition and subtraction

● I can estimate within my number range and use a variety of strategies

● I can check and compare my estimate● I can solve the following tasks within 120;

additive, removed item, missing addend, missing subtrahend, missing minuend and use flexible strategies

Language● Decade, decuple, to, from, before, after, in-

between, first, second……, greater, more than or less than rather than bigger and smaller, groups, altogether, add, subtract, remove, missing, addend, subtrahend, minuend, strategy, commutative, distributive

● Use correct terminology for place value ….14 is ten and four rather than one and four, 24 is 2 tens and four ones


● Settings● Number Lines, tracks, hundred squares● Arithmetic rack, 100 bead strings● Arrow Cards● Empty Number Line● Mini Ten Frames / Ten Frames● Bundles and sticks● http://catalog.mathlearningcenter.org/apps/

number-frames

UNDERSTANDING MULTIPLES OF 2, 3, 5 and 10Learning Outcome● I can say the forward number word

sequences in multiples of 2s, 3s, 5s and 10s keep track of the counts on my fingers

● I can say the backward number word sequences in multiples of 2s, 3s, 5s and 10s and keep track of the counts on my fingers

● I can say the next number word before and after in a multiple number sequence in 2s, 3s, 5s and 10s

● I can identify hidden numerals and locate their position within sequences of 2s, 3s, 5s and 10s


Language● Multiples, sequence, increment,

decrement, keeping track, before, after

Mathematisation – Check it, Prove it, Explain it

● Manipulation of materials● Visual representation● Justify, prove and explain the

next number in a number sequence

● Screening and flashing numerals in various settings

Notation● Reading and writing a numeral

sequence



UNDERSTANDING GROUPING AND SHARING IN 2s, 3s, 5s and 10s AND RECALL OF FACTSLearning OutcomesI know that 2, 3, 5 and 10 can be regarded as composite groups of 1 and use this to:

● describe, build and count simple arrays of 2s, 3s, 5s and 10s within 100

● count and describe items grouped in 2s, 3s, 5s and 10s

● increment and decrement in groups of 2s, 3s, 5s and 10s within 100

● half, third, fifth and tenth numbers within relevant tables

● partition a collection into equal shares and establish the number of shares

● partition a collection into equal shares and establish the number in each share

● demonstrate that for shares to be equal, a quantity may remain. This is called the remainder.

● demonstrate the commutative, associative and distributive relationship within multiplication

Using all of the above● quickly derive and recall

multiplication and division facts● identify the multiples and factors

of numbers from familiar times tables. (e.g. 10 is a multiple of 5 and 2 and 5 and 2 are factors of 10)

● I understand the commutative, associative and distributive relationship within multiplication


● I can check and compare my estimate

Settings● n-tiles● Arithmetic Rack● Dot arrays → square arrays

(Base 5 and 10 Grids), perimeter markers and empty arrays

● Teacher modelling Commutative, Associative and Distributive properties

● Word problems● http://

catalog.mathlearningcenter.org/apps/number-pieces

Language● Parts, wholes, share, shares,

group, the same, fair share, equal, half way, half of, collection, groups of, equal shares, partition, half, double, thirds, rows, columns, arrays, increment and decrement, multiply and divide, remainder, horizontal and vertical, perimeter and area, commutative,

UNDERSTANDING PART WHOLE REASONING AND APPLYING NUMBER TO THISLearning Outcomes● I can estimate the size of an equal share and justify this by iteration.● I can use a unit fractions to make one whole and then use this knowledge to make a different unit fraction and can justify why it is longer or shorter

If this is 13

, what would the whole look like?

What would 12

of the whole look like?

What can you tell me about the difference between the two unit fractions? How do you know?

● I can predict and compare the difference in size between unit fractions of the same whole. E.g.14

is larger than and 16

This is the whole. Ask child A to estimate ¼ and Child B to estimate 16

. .

Ask the children to estimate the size of their unit fraction, predict which one is the largest and justify their reasoning using the whole measure.

APPLYING NUMBER TO PART WHOLE REASONING● I can apply my whole number knowledge and partitioning knowledge to make unit fractions partitioned bars● I can identify a unit fraction of a partitioned bar and describe how many partitions make up that unit fraction● I can represent these unit fractions in notation form, making reference to numerator and denominator

e.g Give the children the following partitioned bars and ask them to find 1 out of 2 equal parts of the six-bar (12¿ and 1 out of 3 equal parts of the six-bar

bar (13¿ . How many partitions can you see within each unit fraction?

Repeat with other unit bars.

UNDERSTANDING HOW TO MEASURE WITH UNIT FRACTIONS AND USE REVERSIBLE REASONINGMeasuring with Unit Fractions● I can partition and iterate to make fractions of specific sizes

e.g Child says, “My bar is a half of the brown bar. What colour is my bar?” or, “My bar is three times the size of the red bar. What colour is my bar?”

Page | 8











● http://catalog.mathlearningcenter.org/apps/ number-pieces

● http://catalog.mathlearningcenter.org/apps/ number-line

● Word problems

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials to screened and

flashed number problems● Visual representation to screened and

flashed number problems● Fully screened number problems● Justify, prove and explain your solution● Screening and flashing numerals in various

settings● Number problems presented as number

sentences

Notation● Representing strategies on an empty number

line● Reading number sentences that represent a

problem● Creating number sentences● Reading, writing and solving number

sentences


associative and distributive,


● Manipulation of materials to screened and flashed number problems

● Visual representation to screened and flashed number problems

● Fully screened number problems● Number problems presented as

number sentences

Notation● Reading number sentences that

represent a problem● Creating number sentences● Reading, writing and solving

number sentences


●Using an empty number line I can position fractions and explain my reasoning

● I can produce composite fractions from unit fractions through partitioning and iteration

e.g. Use the whole bar to create 13

. What happens if we partition this fraction into 2? What fraction is this of the whole? (26¿

Reversible Reasoning● I can identify the unit fraction and the whole when given a composite fraction

If the yellow bar is ¾, what colour is the whole and what colour is the unit fraction?

● I can use whole number knowledge and partitioning to produce a specified length of the whole bare.g. My bar is 8 measures long. Can you show me six measures without using a ruler?● I can create a composite fraction of the whole when presented with a different composite fraction of the same wholee.g. This strip is 2/3 of the whole. Can you make me ¾ of the whole?● I can create a unit fraction of a given unit fraction and describe this new unit fraction in relation to the original wholee.g. This is the whole. Here is 1

/3 of this whole. Can you make 1/5 of the 1/3 and tell me what fraction of the original whole this is? I can solve fraction problems using visual representation I can use notation to demonstrate my understanding

Settings● Linear materials that can be broken up and shared e.g. bars, paper strips, rods● Empty Number Line

Language● Parts, whole, share, the same, fair, half, half way, half of, equal, appropriate language to describe different fractions e.g. a sixth, iterate, reform,

comparative language for size and length, partition, unit fraction (1/2, 1/3, 1/4 etc), composite fraction (2/3, 3/4, 2/5 etc), disembed, denominator, numerator, composite fraction, proper fraction

Mathematisation – Check it, Prove it, Explain it● Manipulation of material● Visual representation● Mental Actions- unitizing, fragmenting and partitioning, iterating, disembedding, unit coordination● * Do not iterate beyond the whole at this stage*

Notation● Visual representations● Reading, writing and interpreting fractional notationActivities should progress from the organisation of thinking to generalising

National BenchmarksEstimation and rounding Uses strategies to estimate an answer to a calculation or problem, for example, doubling and rounding. Rounds whole numbers to the nearest 10 and 100 and uses this routinely to estimate and check the reasonableness of a solution.Number and number processes Reads, writes, orders and recites whole numbers to 1000, starting from any number in the sequence. Demonstrates understanding of zero as a placeholder in whole numbers to 1000. Uses correct mathematical vocabulary when discussing the four operations including, subtract, add, sum of, total, multiply, product, divide and shared equally. Identifies the value of each digit in a whole number with three digits, for example, 867 = 800 + 60 + 7. Counts forwards and backwards in 2s, 5s, 10s and 100s. Demonstrates understanding of the commutative law, for example, 6 + 3 = 3 + 6 or 2 × 4 = 4 × 2. Applies strategies to determine multiplication facts, for example, repeated addition, grouping, arrays and multiplication facts. Solves addition and subtraction problems with three digit whole numbers.

National BenchmarksFractions, decimal fractions and percentages Explains what a fraction is using concrete materials, pictorial representations and appropriate mathematical vocabulary. Demonstrates understanding that the greater the number of parts, the smaller the size of each share. Uses the correct notation for common fractions to tenths, for example, ½ , 2/3 and 5/8. Compares the size of fractions and places simple fractions in order on a number line. Uses pictorial representations and other models to demonstrate understanding of simple equivalent fractions, for example,

½ = 2/4 = 3/6. Explains the role of the numerator and denominator. Uses known multiplication and division facts and other strategies to find unit fractions of whole numbers, for example, 𝟏/𝟐 or 𝟏/𝟒.

Page | 9

½ ¾0 1










Adds and subtracts multiples of 10 or 100 to or from any whole number to 1000. Applies strategies to determine division facts, for example, repeated subtraction, equal groups, sharing equally, arrays and multiplication facts. Uses multiplication and division facts to solve problems within the number range 0 to 1000. Multiplies and divides whole numbers by 10 and 100 (whole number answers only). Applies knowledge of inverse operations (addition and subtraction; multiplication and division). Solves two step problems.

Page | 10

On track in Primary 5Experiences and OutcomesI can use my knowledge of rounding to routinely estimate the answer to a problem then, after calculating, decide if my answer is reasonable, sharing my solution with others.MNU 2-01aI have extended the range of whole numbers I can work with and having explored how decimal fractions are constructed, can explain the link between a digit, its place and its value.MNU 2-02aHaving determined which calculations are needed, I can solve problems involving whole numbers using a range of methods, sharing my approaches and solutions with others.MNU 2-03aI have explored the contexts in which problems involving decimal fractions occur and can solve related problems using a variety of methods. MNU 2-03bHaving explored the need for rules for the order of operations in number calculations, I can apply them correctly when solving simple problems. MTH 2-03cI can show my understanding of how the number line extends to include numbers less than zero and have investigated how these numbers occur and are used. MNU 2-04aHaving explored the patterns and relationships in multiplication and division, I can investigate and identify the multiples and factors of numbers. MTH 2-05aHaving explored more complex number sequences, including well-known named number patterns, I can explain the rule used to generate the sequence, and apply it to extend the pattern. MTH 2-13aI can apply my knowledge of number facts to solve problems where an unknown value is represented by a symbol or letter. MTH 2-15a

Experiences and OutcomesI have investigated the everyday contexts in which simple fractions, percentages or decimal fractions are used and can carry out the necessary calculations to solve related problems. MNU 2-07aI can show the equivalent forms of simple fractions, decimal fractions and percentages and can choose my preferred form when solving a problem, explaining my choice of method. MNU 2-07bI have investigated how a set of equivalent fractions can be created, understanding the meaning of simplest form, and can apply my knowledge to compare and order the most commonly used fractions. MTH 2-07c


Understanding of multiples and sequences of multiples Grouping and Sharing for Multiplication and Division

INCREMENTING AND DECREMENTING ON AND OFF THE DECUPLE BY 10S AND 1S BEYOND 120 to at least 1000Learning Outcome● I can increment and decrement by 10s and by 10s and 1s on and off

the decuple● I can increment and decrement by 10s and by 10s and 1s from a

given number to find or locate another number in a sequence beyond 120

● I can calculate the number of backwards or forward increments from a to b

● I can round numbers to the nearest ten beyond 120● I can name a numeral and describe the value of each digit to beyond

120, including zero as a place holder

ADDING AND SUBTRACTING FROM AND TO A DECUPLE WHEN THE PART IS WITHIN 10 AND THEN BEYOND 10Learning Outcome● I can describe how I use my knowledge of number structures to add

from a decuple number when the part is within 10 and then beyond 10● I can describe how I use my knowledge of number structures to

subtract to a decuple number when the part is within 10 and then beyond 10

● I can describe how I use my knowledge of number structures to add to a decuple number when the part is within 10 and then beyond 10

● I can describe how I use my knowledge of number structures to subtract from a decuple number when the part is within 10 and then beyond 10

● I can describe how I use my knowledge of number structures to add through a decuple number when the part is within 10 and then beyond 10

● I can describe how I use my knowledge of number structures to subtract through a decuple when the part is within 10 and then beyond 10

● I can explain the commutative relationship within addition● I can demonstrate my understanding of the distributive and inverse

relationship within addition and subtraction● I can estimate within my number range and use a variety of strategies● I can check and compare my estimate● I can solve the following tasks within 120; additive, removed item,


UNDERSTANDING CONCEPTUAL PLACE VALUELearning Outcomes:● I can identify, recognise, order and sequence a range of numerals

including negative numerals● I can estimate where to place given numerals to at least 1000● I can increment and decrement by 100s, 10s and 1s on and off the

decuple and centuple to at least 1000.● I can describe and compare how I solve a variety of addition and

subtraction tasks using my knowledge of tens and ones and number structures; within 100, within 1000

● I can explain the commutative relationship within addition● I can demonstrate my understanding of the distributive and inverse

relationship within addition and subtraction

Language●Decade, decuple, centuple, to, from, before, after, in-between, first,

second……, greater, more than or less than rather than bigger and smaller, groups, altogether, add, subtract, remove, missing, addend, subtrahend, minuend, strategy, commutative, distributive

UNDERSTANDING MULTIPLES OF 2, 4, 8, 3, 5, 6, 9, 10 AND LUCKY 7Learning OutcomeIn the following clusters - 2s, 4s and 8s, 3s, 5s and 6s, 9s and 10s then lucky 7s:● I can say the forward number word sequences in multiples and keep track of the counts on my

fingers● I can say the backward number word sequences in multiples and keep track of the counts on

my fingers● I can say the next number word before and after in a multiple number sequence● I can identify hidden numerals and locate their position within sequences


Language● Multiples, sequence, increment, decrement, keeping track, before, after


Notation● Reading and writing a numeral sequence



UNDERSTANDING GROUPING AND SHARING IN 2s, 4s, 8s, 3s, 5s, 6s, 9s, 10s AND 7s AND RECALL OF FACTSLearning OutcomesIn the following clusters - 2s, 4s and 8s, 3s, 5s and 6s, 9s and 10s then lucky 7s:

● describe, build and count simple arrays● count and describe items grouped● increment and decrement in groups● partition a collection into equal shares and

establish the number of shares● partition a collection into equal shares and

establish the number in each share● demonstrate that for shares to be equal, a

quantity may remain. This is called the remainder.

● demonstrate the commutative, associative and distributive relationship within multiplication

Using all of the above● quickly derive and recall multiplication and

division facts● identify the multiples and factors of

numbers from familiar times tables. (e.g. 10 is a multiple of 5 and 2 and 5 and 2 are factors of 10)

● I can use the commutative and inverse relationships between multiplication and division.

Settings● n-tiles● Arithmetic Rack● Dot arrays → square arrays (Base 5 and

10 Grids), perimeter markers and empty arrays

● Teacher modelling Commutative, Associative and Distributive properties

● Word problems● http://catalog.mathlearningcenter.org/

apps/number-piecesLanguage

● Parts, wholes, share, shares, group, the same, fair share, equal, half way, half of, collection, groups of, equal shares, partition, half, double, thirds, rows, columns, arrays, increment and decrement, multiply and divide, remainder, horizontal and vertical, perimeter and area, commutative, associative and distributive,

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials to screened and

flashed number problems● Visual representation to screened and

flashed number problems● Fully screened number problems● Number problems presented as number

sentencesNotation

● Reading number sentences that represent a problem

● Creating number sentences● Reading, writing and solving number

sentences

UNDERSTANDING REVERSIBLE MULTIPLICATIVE REASONINGLearning Outcomes● I can use my knowledge of unit fractions and iterating to help me solve related problemse.g If the pink is 81, what value is the red? Can you use this to work out the value of the green?If the white bar is 45 what is the value of the green?

● I can visually represent my solution to number problems using my knowledge of number and unit fractionse.g. Make a linear representation of this problem. If Sam has to walk 36 miles and has travelled 1/3 of the distance, how far has he travelled?● I can visually represent my solution to number problems using my knowledge of number and composite fractionse.g. Sam has to raise £42 and has raised 3/7 of this amount. How much has Sam raised so far?● I can use the denominator and the numerator to calculate the fraction of an amount e.g. two thirds of 27 is 27 divided by 3 and multiplied by 2● I can identify unit fractions, composite fractions and improper fractions when comparing linear bars

UNDERSTANDING FRACTIONS AS NUMBERSLearning Outcomes●When given the whole bar, I can produce a composite fraction or an improper fraction of the whole.e.g. Here is the whole. Give me 7/8. Give me 7/6

● I can predict and compare the difference in size between composite fractions of the same whole. E.g.45

is larger than and 36

This is the whole. Ask child A to estimate 4/5 and Child B to estimate 36

. .

Ask the children to estimate the size of their unit fraction, predict which one is the largest and justify their reasoning using the whole measure.

● I can predict and compare how different improper fractions determine the length beyond whole. E.g.9/8 will be larger than 10/9

This is the whole. Ask child A to estimate 9/8 and Child B to estimate 10/9. .

Ask the children to estimate the size of their improper fraction, predict which one is the largest and the impact on the whole.

● I can create a fraction number sequence and compare it to whole number sequences.Here is whole. Can you draw ¼ of the whole. Use this ¼ to make a number track in quarters

e.g.¼ 2/4

3/44/4

5/46/4

7/48/4

9/410/4

.

How many numbers are in this fraction number sequence of quarters?How is this fraction number sequence the same and/or different to the whole number sequence?How many fraction number sequences are there? (infinite)

Using prior knowledge I can demonstrate how to simplify fractions

UNDERSTANDING EQUAL SHARING OF MULTIPLE ITEMSLearning Outcomes● I can begin to use distributive reasoning to equally share multiple items of the same size and explain the different strategies I can use.

Page | 11



On track in Primary 5Experiences and OutcomesI can use my knowledge of rounding to routinely estimate the answer to a problem then, after calculating, decide if my answer is reasonable, sharing my solution with others.MNU 2-01aI have extended the range of whole numbers I can work with and having explored how decimal fractions are constructed, can explain the link between a digit, its place and its value.MNU 2-02aHaving determined which calculations are needed, I can solve problems involving whole numbers using a range of methods, sharing my approaches and solutions with others.MNU 2-03aI have explored the contexts in which problems involving decimal fractions occur and can solve related problems using a variety of methods. MNU 2-03bHaving explored the need for rules for the order of operations in number calculations, I can apply them correctly when solving simple problems. MTH 2-03cI can show my understanding of how the number line extends to include numbers less than zero and have investigated how these numbers occur and are used. MNU 2-04aHaving explored the patterns and relationships in multiplication and division, I can investigate and identify the multiples and factors of numbers. MTH 2-05aHaving explored more complex number sequences, including well-known named number patterns, I can explain the rule used to generate the sequence, and apply it to extend the pattern. MTH 2-13aI can apply my knowledge of number facts to solve problems where an unknown value is represented by a symbol or letter. MTH 2-15a



Understanding of multiples and sequences of multiples Grouping and Sharing for Multiplication and Division

●Use correct terminology for place value ….14 is ten and four rather than one and four, 124 is 1 hundred, 2 tens and four ones

●Make the distinction between a numeral and the number it represents clear

Settings●Number lines, tracks, hundred squares●Arithmetic rack, 100 bead strings●Arrow Cards●Empty Number Line●Mini Ten Frames / Ten Frames●Bundles and sticks●http://catalog.mathlearningcenter.org/apps/number-frames ●http://catalog.mathlearningcenter.org/apps/number-pieces ●http://catalog.mathlearningcenter.org/apps/number-line ●Word problems

Mathematisation – Check it, Prove it, Explain it●Manipulation of materials to screened and flashed number problems●Visual representation to screened and flashed number problems●Fully screened number problems●Justify, prove and explain your solution●Screening and flashing numerals in various settings●Number problems presented as number sentences

Notation●Representing strategies on an empty number line●Reading number sentences that represent a problem●Creating number sentences●Reading, writing and solving number sentences



e.g. Share these 3 bars between 4 people. What fraction of the bar does each person get?

Share these 12 bars between 8 people. What fraction of the bar does each person get?

● I can use my knowledge of distributive reasoning to solve practical problems with differently sized wholes

e.g. Here are three cakes. Make a 1/7 of each cake and explain what you notice about each 1/7.Make 1/7 of all 3 cakes and justify how this is 1/7 of all the cakes

Settings● Linear materials that can be broken up and shared e.g. bars, paper strips, rods● Numeral Tracks● Manipulatives● Empty Number Line● Candy factory● Candybot http://ltrg.centers.vt.edu/projects/games/apps/candyBot.htm

Language● Parts, whole, share, the same, fair, half, half way, half of, equal, appropriate language to describe different fractions e.g. a sixth, iterate, reform,

comparative language for size and length, sequences, partition, unit fraction (1/2, 1/3, 1/4 etc), composite fraction (2/3, 3/4, 2/5 etc), improper fraction (7/4, 10/8, 9/5, etc), disembed, denominator, numerator, composite fraction, proper fraction, improper fraction

Mathematisation – Check it, Prove it, Explain it● Manipulation of material● Visual representation● Mental Actions- unitizing, fragmenting and partitioning, iterating, disembedding, unit coordination● Moving beyond the whole



Page | 12

http://ltrg.centers.vt.edu/projects/games/apps/candyBot.htm





Experiences and OutcomesI can use my knowledge of rounding to routinely estimate the answer to a problem then, after calculating, decide if my answer is reasonable, sharing my solution with others.MNU 2-01aI have extended the range of whole numbers I can work with and having explored how decimal fractions are constructed, can explain the link between a digit, its place and its value.MNU 2-02aHaving determined which calculations are needed, I can solve problems involving whole numbers using a range of methods, sharing my approaches and solutions with others.MNU 2-03aI have explored the contexts in which problems involving decimal fractions occur and can solve related problems using a variety of methods. MNU 2-03bHaving explored the need for rules for the order of operations in number calculations, I can apply them correctly when solving simple problems. MTH 2-03cI can show my understanding of how the number line extends to include numbers less than zero and have investigated how these numbers occur and are used. MNU 2-04aHaving explored the patterns and relationships in multiplication and division, I can investigate and identify the multiples and factors of numbers. MTH 2-05aHaving explored more complex number sequences, including well-known named number patterns, I can explain the rule used to generate the sequence, and apply it to extend the pattern. MTH 2-13aI can apply my knowledge of number facts to solve problems where an unknown value is represented by a symbol or letter. MTH 2-15a



UNDERSTANDING ADDITION AND SUBTRACTION TO AT LEAST 1000 AND BEYOND AND WORKING TOWARDS FORMAL ALGORITHMSLearning Outcomes:

● I can read, write and sequence numbers forwards and backwards in the range 0 to 1 000 000

● I can describe the value of each digit in a numeral to at least 1000 and up to 1 000 000

● I can read, write and order negative numbers● I can estimate where to place given numerals● I can understand the commutative relationship within

addition and subtraction● I can understand the distributive and inverse

relationship within addition and subtraction● I can use jottings to solve addition and subtraction tasks

within one thousand● I understand and use formal algorithms to solve

addition and subtraction tasks within one thousand● I can use both mental strategies and formal algorithms

to solve a variety of addition and subtraction tasks. I can choose the most efficient method for the problem given.


● I can check and compare my estimate● I can solve the following tasks; additive, removed item,


Language● Addition, subtraction, forwards, backwards, to, from,

before, after, in-between, first, second……, greater, more than or less than rather than bigger and smaller, groups, negative numbers, jottings, algorithms, altogether, remove, missing, addend, subtrahend, minuend, strategy, commutative, distributive

● Use correct terminology for place value ….tens, hundreds, thousands etc


Settings● Number lines, tracks, hundred squares/ mini hundred

squares● Arrow Cards● Empty Number Line● Mini Ten Frames / Ten Frames● Conceptual Place Value materials● http://catalog.mathlearningcenter.org/apps/number-

frames● http://catalog.mathlearningcenter.org/apps/number-

pieces● http://catalog.mathlearningcenter.org/apps/number-line ● Word problems

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials to screened and flashed

number problems● Visual representation to screened and flashed number

problems● Fully screened number problems

UNDERSTANDING 2-DIGIT MULTIPLICATION AND DIVISION AND WORKING TOWARDS FORMAL ALGORITHMSLearning Outcomes

● I can represent and describe how I use my understanding of number structures and equal groups to multiply and divide tens and ones by a single digit. (e.g. 32x4, 56÷4, 34÷7)

● I can multiply and divide by 100 up to at least 1000● I can divide whole numbers and identify when there is a remainder.● I can use jottings to solve multiplication and division tasks● I can demonstrate my understanding and use formal algorithms to solve multiplication and division tasks● I can use both mental strategies and formal algorithms to solve a variety of multiplication and division tasks. I can choose the most

efficient method for the problem given.● I can demonstrate the commutative, associative and distributive relationship within multiplication

Settings● Dot Arrays → Square Arrays (Base 5 and 10 Grids)→Perimeter Markers→Empty Arrays● Numeral Lines, Empty Number Lines● http://catalog.mathlearningcenter.org/apps/number-line ● Numeral Tracks● Word Problems● http://catalog.mathlearningcenter.org/apps/number-pieces

Language● Parts, wholes, share, shares, group, groups , the same, fair share, equal, half way, half of, collection, composite groups, groups of,

equal shares, partition, rows, columns, arrays, increment and decrement, appropriate multiplicative and divisional language, multiples sequence, keeping track, commutative and inverse relationships, multiples and factors, multiply and divide, algorithm, efficiency, rows and columns, perimeter and area, commutative, associative and distributive , remainder, horizontal and vertical


Notation● Reading and writing a numeral sequence● Jotting→Semi-Formal →Formal Algorithms


UNDERSTANDING MULTIPLICATION OF FRACTIONSLearning Outcomes● I can share shares and name the result in relation to the designated whole or share.● I can present and explain my solutionse.g.Share this whole equally between yourself and 4 other people. Demonstrate the fraction that each person gets.

2 more people arrive. You share your share with these 2 people. Demonstrate the fraction that each person gets. What fraction is this in relation to your share and the original whole.

● I can show a unit fraction of a unit fraction and use the appropriate notation to show my thinking.e.g.Show me 1/3 of 1/5 of this whole.

Now show me 1/5 of a 1/3 of this whole.How did you solve this using visual representation?How can we use notation to demonstrate your thinking? 1/5 of 1/3 3x5 = 15 equal parts of the whole so, 1/5 of 1/3 is 1/15 of the whole.Look at a number of examples so children begin to generalise their thinking

● I can solve a composite fraction of a unit fraction and use the appropriate notation to show my thinking.e.g.Show me 2/3 of 1/5 of this whole.

How did you solve this using visual representation?How can we use notation to demonstrate your thinking2/3 of 1/5 = 2x(1/3 of 1/5) 5x3 = 15 equal parts of the whole so each part is 1/15 of the whole so 2 x 1/15 = 2/15

Look at a number of examples so children begin to generalise their thinking

● I can solve a unit fraction of a composite fraction and use the appropriate notation to show my thinking.

e.g.Show me 1/5 of 2/3 of this whole.

How did you solve this using visual representation?How can we use notation to demonstrate your thinking ?


● I can solve a composite fraction of a composite fraction and use the appropriate notation to show my thinking.e.g.Show me 3/7 of 2/3 of this whole.

How did you solve this using visual representation?How can we use notation to demonstrate your thinking


UNDERSTANDING PERCENTAGES AND DECIMAL FRACTIONS (including Place Value)●Using my knowledge that percent means per hundred, I can partition a whole into a hundred parts● I can use my knowledge of the unit fraction which is 1% and iterate this a given number of times to create percentages between 1-100%● I can use my knowledge of the unit fraction which is 1% and iterate this a given number of times to create percentages beyond 100%● I can partition a unit into 1/10 and understand its significance to conceptual place value.

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Experiences and OutcomesI can use my knowledge of rounding to routinely estimate the answer to a problem then, after calculating, decide if my answer is reasonable, sharing my solution with others.MNU 2-01aI have extended the range of whole numbers I can work with and having explored how decimal fractions are constructed, can explain the link between a digit, its place and its value.MNU 2-02aHaving determined which calculations are needed, I can solve problems involving whole numbers using a range of methods, sharing my approaches and solutions with others.MNU 2-03aI have explored the contexts in which problems involving decimal fractions occur and can solve related problems using a variety of methods. MNU 2-03bHaving explored the need for rules for the order of operations in number calculations, I can apply them correctly when solving simple problems. MTH 2-03cI can show my understanding of how the number line extends to include numbers less than zero and have investigated how these numbers occur and are used. MNU 2-04aHaving explored the patterns and relationships in multiplication and division, I can investigate and identify the multiples and factors of numbers. MTH 2-05aHaving explored more complex number sequences, including well-known named number patterns, I can explain the rule used to generate the sequence, and apply it to extend the pattern. MTH 2-13aI can apply my knowledge of number facts to solve problems where an unknown value is represented by a symbol or letter. MTH 2-15a



● Justify, prove and explain your solution● Screening and flashing numerals in various settings● Number problems presented as number sentences

Notation● Representing strategies on an empty number line● Reading number sentences that represent a problem● Creating number sentences● Reading, writing and solving number sentences● Using jottings and working towards formal algorithms


● I can partition 1/10 of a unit into ten parts recognising this is 1/100 and understand its significance to conceptual place value.● I can use my knowledge of 1, 1/10 , 1/100 to construct and order various decimal fractions● I can understand the equivalence of a fraction, a decimal and a percentage (e.g. 7/100, 0.07 and 7%)● I can treat 100 square as a whole to demonstrate my understanding of percentages and decimal fractions.● I can use my knowledge of simplifying fractions, decimals and percentages to solve numerical tasks presented through a setting (e.g. 35% of £12)e.g. Regard the 100 grid as a one. Colour in 35% of the grid. If the grid represented £12 what does the shaded 35% represent?

Settings● Linear materials that can be broken up and shared e.g. bars, paper strips, rods● Hundred Squares/ Grids● Manipulatives● Empty Number Line● http://www.visnos.com/demos/percentage-fraction-decimals-grid ● Word Problems

Language● Fraction, percentage, decimal fraction, parts, whole, share, the same, fair, half, half way, half of, equal, appropriate language to describe different

fractions/percentages and decimal fractions e.g. a sixth, iterate, reform, comparative language for size and length, sequences, partition, unit fraction (1/2, 1/3, ¼, 1/10 etc), composite fraction (2/3, 3/4, 2/5, 3/10 etc), improper fraction (7/4, 10/8, 9/5, 12/10 etc), disembed, denominator, numerator, composite fraction, proper fraction, improper fraction

Mathematisation – Check it, Prove it, Explain it● Manipulation of material● Visual representation● Screening and flashing● Mental Actions- unitizing, fragmenting and partitioning, iterating, disembedding, unit coordination● Moving beyond the whole

Notation● Visual representations● Reading, writing and interpreting percentage, decimal and fractional notation


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http://www.visnos.com/demos/percentage-fraction-decimals-grid

On track in Primary 7Experiences and OutcomesI can use my knowledge of rounding to routinely estimate the answer to a problem then, after calculating, decide if my answer is reasonable, sharing my solution with others.MNU 2-01aI have extended the range of whole numbers I can work with and having explored how decimal fractions are constructed, can explain the link between a digit, its place and its value.MNU 2-02aHaving determined which calculations are needed, I can solve problems involving whole numbers using a range of methods, sharing my approaches and solutions with others.MNU 2-03a I have explored the contexts in which problems involving decimal fractions occur and can solve related problems using a variety of methods. MNU 2-03b

Having explored the need for rules for the order of operations in number calculations, I can apply them correctly when solving simple problems. MTH 2-03cI can show my understanding of how the number line extends to include numbers less than zero and have investigated how these numbers occur and are used. MNU 2-04aHaving explored the patterns and relationships in multiplication and division, I can investigate and identify the multiples and factors of numbers. MTH 2-05aHaving explored more complex number sequences, including well-known named number patterns, I can explain the rule used to generate the sequence, and apply it to extend the pattern. MTH 2-13aI can apply my knowledge of number facts to solve problems where an unknown value is represented by a symbol or letter. MTH 2-15a


Addition and SubtractionMultiplication and Division

Fractions, Decimals and Percentages

MAKING EFFICIENT USE OF MENTAL STRATEGIES AND FORMAL ALGORITHMS WITHIN THE FOUR OPERATIONSLearning Outcomes● I can use both mental strategies and formal algorithms to solve multi-step addition and subtraction tasks beyond 100.● I can apply the correct order of operations in number calculations when solving multi-step tasks● I can represent and describe how I use my understanding of number structures and equal groups to multiply and divide tens and ones by two digit numbers. (e.g. 32x43, 56÷14)● I understand and use formal algorithms to solve multiplication and division tasks● I can demonstrate my knowledge of remainders within problem solving tasks● I can use both mental strategies and formal algorithms to solve a variety of multiplication and division tasks. I can choose the most efficient method for the problem given.● I can demonstrate my ability to use the order of operations to solve equations●Within a variety of applications I can demonstrate the commutative, inverse, associative and distributive relationship within multiplication●Within a variety of applications I can demonstrate the commutative, distributive and inverse relationship within addition and subtraction●Within a variety of applications I can demonstrate the inverse relationship within division● I can interpret brackets (Parentheses) to solve equations e.g. Example: (3 + 2) × (6 − 4) The parentheses groups the 3 and 2 together and the 6 and 4 together, so they get done first; (3 + 2) × (6 − 4) = (5) × (2) = 5×2

= 10. Without the parentheses the multiplication is done first; 3 + 2 × 6 − 4 = 3 + 12 − 4 = 11 (not 10)

Language● Addition, subtraction, multiplication, division, greater than, more than or less than rather than bigger and smaller, groups, jottings, algorithms, altogether, remove, missing, addend, subtrahend, minuend,

strategy, commutative, distributive, inverse, brackets, four operations,● Use correct terminology for place value ….tens, hundreds, thousands etc● Make the distinction between a numeral and the number it represents clear● BIDMAS (order of operation) Brackets, Indices (eg squared), Division, Multiplication, Addition, Subtraction

Settings● Number lines, tracks, hundred squares/ mini hundred squares● Empty Number Line● Mini Ten Frames / Ten Frames● Conceptual Place Value materials● http://catalog.mathlearningcenter.org/apps/number-frames ● http://catalog.mathlearningcenter.org/apps/number-pieces ● http://catalog.mathlearningcenter.org/apps/number-line ● Word problems

Mathematisation – Check it, Prove it, Explain it● Manipulation of materials to screened and flashed number problems● Visual representation to screened and flashed number problems● Fully screened number problems● Justify, prove and explain your solution● Screening and flashing numerals in various settings● Number problems presented as number sentences

Notation● Representing strategies on an empty number line● Reading number sentences that represent a problem● Creating number sentences● Reading, writing and solving number sentences● Using jottings and working towards formal algorithms


Settings●Dot Arrays → Square Arrays (Base 5 and 10 Grids)→Perimeter Markers→Empty Arrays●http://catalog.mathlearningcenter.org/apps/number-pieces ●Word Problems●Empty Number Line●http://catalog.mathlearningcenter.org/apps/number-line ●Jotting→Semi-Formal →Formal Algorithms●Conceptual Place Value Materials (For addition and subtraction)

UNDERSTANDING ADDITION, SUBTRACTION AND DIVISION OF FRACTIONSLearning OutcomesPreparing for addition and subtraction of fractions

● I can partition a given partitioned bar in order to show a different fraction within the bar (when these share a common factor).e.g Using the 5/5 bar, can you find 1/10 Explain how you solved this.

How many 1/10 are required to make the unit fraction (1/5) - Repeat with 1/15, 1/20, 1/25 - Repeat with other fraction bars e.g to find 1/9 on a 3/3 bar, 1/12 on a 4/4 bar.

● I can partition a given partitioned bar in order to show a different fraction within the bar (when there is no common factor). e.g Here is a 2/2 bar, find 1/3 without erasing the third partitions. Can you identify at least two ways to solve this task. - Repeat with other fraction bars e.g to find 1/4 on a 3/3 bar, 1/4 on a 5/5 bar.

Adding and Subtracting Fractions I can find the common fraction to add and subtract two unit fractions. e.g. Using a 3/3 bar. Add together 1/3 and 1/12. Now subtract 1/12 from 1/3.How did you solve this using visual representation? , How can we use notation to demonstrate your thinking?What were the similarities and differences in the strategies you used to add and subtract?

Repeat with other fractions (Can be more challenging by using smaller fractions e.g. 1/6 + 1/24).Repeat without providing visual bar (children required to make own visual representation)

I can find the common fraction to add and subtract two composite fractions.e.g. Using a 3/3 bar. Add together 2/3 and 3/12. Now Subtract 3/12 from 2/3.How did you solve this using visual representation? - How can we use notation to demonstrate your thinking?What were the similarities and differences in the strategies you used to add and subtract?

Repeat with other fractions (Can be more challenging by using smaller fractions e.g. 5/6 + 9/24)Repeat without providing visual bar (children required to make own visual representation) I can add and subtract fractions which involve fractions within and beyond the wholee.g. 2/3 + 5/12, 6/5-2/3How did you solve this using visual representation? - How can we use notation to demonstrate your thinking?

I can add and/or subtract fractions to solve multi step word problems.e.g. Katy has 4/5 of a litre of petrol in her lawnmower. She buys ¼ litre more. Then she uses 3/8 of a litre. How much of a litre does she have now?e.g. David has 10 metres of wood to make a table. He uses 4/7 of this to make the tabletop and 3/8 to make the legs. How much wood does David have left?

Dividing Fractions

I can distinguish between two meanings of division- measurement meaning and sharing meaning.e.g. ask children the children to make a visual representation to show the difference between these two division tasks:The ribbon is 39 cm long. A single serving is 3cm/ How many people can get a serving? (measurement meaning)The ribbon is 39 cm long and it is to be shared between 3 people. How much does each person get? (sharing meaning)

I can create a visual representation to demonstrate why dividing by a fraction is different from diving by a whole number..e.g. ask children the children to make a visual representation to show the difference between these two division tasks:There is 3 kg of flour. A cake needs a 1/3 kg to bake a cake. How many cakes can they make? (3kg ÷ 1/3 kg)= 9 cakesThere is 3kg of flour. This is used to make 3 cakes. How much flour goes in each cake? (3kg ÷ 3cakes)= 1kg per cake

I can create word problems to demonstrate my understanding of why dividing by a fraction is different from diving by a whole number.e.g. give the children 8÷⅓ and 8÷3. Ask them to create word problems to demonstrate the difference between the two

I can solve word division problems involving fractions.e.g. provide word problems and ask how did you solve this using visual representation? How can we use notation to demonstrate your thinking?A grocer gets 7kg of rice delivered. She has to package the rice into bags that contain ¼ kg: How many bags can she make?A baker uses 3/8 of a cup of sugar for each batch of cookies he makes. He has 5 cups of sugar. How many batches of cookies can he make, including fractions of a batch?At camp a jug of lemonade contains 4 litres. The camp leader will give out 2/3 litres servings of the lemonade. How many servings can be given out?

Example of linking annotation and notation:At camp a jug of lemonade contains 4 litres. The camp leader will give out 2/3 litres servings of the lemonade. How many servings can be given out?The bar represents 4 litres

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On track in Primary 7Experiences and OutcomesI can use my knowledge of rounding to routinely estimate the answer to a problem then, after calculating, decide if my answer is reasonable, sharing my solution with others.MNU 2-01aI have extended the range of whole numbers I can work with and having explored how decimal fractions are constructed, can explain the link between a digit, its place and its value.MNU 2-02aHaving determined which calculations are needed, I can solve problems involving whole numbers using a range of methods, sharing my approaches and solutions with others.MNU 2-03a I have explored the contexts in which problems involving decimal fractions occur and can solve related problems using a variety of methods. MNU 2-03b

Having explored the need for rules for the order of operations in number calculations, I can apply them correctly when solving simple problems. MTH 2-03cI can show my understanding of how the number line extends to include numbers less than zero and have investigated how these numbers occur and are used. MNU 2-04aHaving explored the patterns and relationships in multiplication and division, I can investigate and identify the multiples and factors of numbers. MTH 2-05aHaving explored more complex number sequences, including well-known named number patterns, I can explain the rule used to generate the sequence, and apply it to extend the pattern. MTH 2-13aI can apply my knowledge of number facts to solve problems where an unknown value is represented by a symbol or letter. MTH 2-15a




The bar now represents 4x1 litre

We need to split each litre into thirds

4 multiplied by the denominator (3) gives us the total number of thirds available (12)4 ÷ ⅓= 12

We need to group the parts into 2/3 to calculate how many servings we can have

12 divided by the numerator (2) gives us the total number of servings (6)4 ÷ ⅔= 6

UNDERSTANDING EQUIVALENT FRACTIONS● I can use my knowledge of simplifying and equivalent factions to put fractions in order● I can use my knowledge of equivalent fractions to order various fractions with different denominators● I can explain the link between a digit, its place and its value to at least 3 decimal places● I can understand the equivalence of a fraction, a decimal and a percentage to solve problems, justifying choice of method used● I can round decimal fractions to at least 3 decimal places● I can use the denominator and numerator to calculate the fraction of an amount e.g. two thirds of 27 is 27 divided by 3 and multiplied by 2● I can calculate simple fractions and percentages of a quantity, with and without a calculator, and use this knowledge to solve problems in every day contexts● I can multiply and divide decimal fractions with at least 3 decimal places mentally by 10, 100 and 1000Settings

● Linear materials that can be broken up and shared e.g. bars, paper strips, rods● Manipulatives● Empty Number Line● http://www.visnos.com/demos/percentage-fraction-decimals-grid ● Word Problems

Language● Fraction, percentage, decimal fraction, parts, whole, share, the same, fair, half, half way, half of, equal, appropriate language to describe different

fractions/percentages and decimal fractions e.g. a sixth, iterate, reform, comparative language for size and length, sequences, partition, unit fraction (1/2, 1/3, ¼, 1/10 etc), composite fraction (2/3, 3/4, 2/5, 3/10 etc), improper fraction (7/4, 10/8, 9/5, 12/10 etc), disembed, denominator, numerator, composite fraction, proper fraction, improper fraction, equivalent fraction, measurement fraction and sharing fraction, common factor, simplifying

Mathematisation – Check it, Prove it, Explain it● Manipulation of material● Visual representation● Screening and flashing● Mental Actions- unitizing, fragmenting and partitioning, iterating, disembedding, unit coordinationNotation● Visual representations● Reading, writing and interpreting percentage, decimal and fractional notation


National Benchmarks – 2nd Level

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Estimation and Rounding

Rounds whole numbers to the nearest 1000, 10 000 and 100 000. Rounds decimal fractions to the nearest whole number, to one decimal place and two decimal places. Applies knowledge of rounding to give an estimate to a calculation appropriate to the context.

Number and Number Processes

Reads, writes and orders whole numbers to 1 000 000, starting from any number in the sequence. Explains the link between a digit, its place and its value for whole numbers to 1 000 000. Reads, writes and orders sets of decimal fractions to three decimal places. Explains the link between a digit, its place and its value for numbers to three decimal places. Partitions a wide range of whole numbers and decimal fractions to three decimal places, for example, 3∙6 = 3 ones and 6 tenths = 36 tenths. Adds and subtracts multiples of 10, 100 and 1000 to and from whole numbers and decimal fractions to two decimal places. Adds and subtracts whole numbers and decimal fractions to two decimal places, within the number range 0 to 1 000 000. Uses multiplication and division facts to the 10th multiplication table. Multiplies and divides whole numbers by multiples of 10, 100 and 1000. Multiplies and divides decimal fractions to two decimal places by 10, 100 and 1000. Multiplies whole numbers by two digit numbers. Multiplies decimal fractions to two decimal places by a single digit. Divides whole numbers and decimal fractions to two decimal places, by a single digit, including answers expressed as decimal fractions, for example, 43 ÷ 5 =

8∙6. Applies the correct order of operations in number calculations when solving multi-step problems. Identifies familiar contexts in which negative numbers are used. Orders numbers less than zero and locates them on a number line.

Multiples, factors and primes

Identifies multiples and factors of whole numbers and applies knowledge and understanding of these when solving relevant problems in number, money and measurement.

Fractions, decimal fractions and percentages

Uses knowledge of equivalent forms of common fractions, decimal fractions and percentages, for example, 𝟑/𝟒 =𝟎∙𝟕5 = 𝟕5%, to solve problems, justifying choice of method used.

Calculates simple percentages of a quantity, and uses this knowledge to solve problems in everyday contexts, for example, calculates the sale price of an item with a discount of 𝟏5%.

Calculates simple fractions of a quantity and uses this knowledge to solve problems, for example, find 𝟑/𝟓 of 𝟔0. Creates equivalent fractions and uses this knowledge to put a set of the most commonly used fractions in order. Expresses fractions in their simplest form.

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On track in Secondary 1Experiences and OutcomesI can use a variety of methods to solve number problems in familiar contexts, clearly communicating my processes and solutions. MNU 3-03aI can continue to recall number facts quickly and use them accurately when making calculations. MNU 3-03bI have investigated strategies for identifying common multiples and common factors, explaining my ideas to others, and can apply my understandingto solve related problems. MTH 3-05aI can apply my understanding of factors to investigate and identify when a number is prime. MTH 3-05bHaving explored the notation and vocabulary associated with whole number powers and the advantages of writing numbers in this form, I can evaluate powers of whole numbers mentally or using technology. MTH 3-06aI can work collaboratively, making appropriate use of technology, to source information presented in a range of ways, interpret what it conveys and discuss whetherI believe the information to be robust, vague or misleading. MNU 3-20aWhen analysing information or collecting data ofmy own, I can use my understanding of how bias may arise and how sample size can affect precision,to ensure that the data allows for fair conclusions to be drawn. MTH 3-20bI can display data in a clear way using a suitable scale, by choosing appropriately from an extended range of tables, charts, diagrams and graphs, making effective use of technology. MTH 2-21a / MTH 3-21aIn order to compare numerical information in real- life contexts, I can find the mean, median, mode and range of sets of numbers, decide which type of average is most appropriate to use and discuss how using an alternative type of average could be misleading. MTH 4-20b

Experiences and OutcomesI can solve problems by carrying out calculations with a wide range of fractions, decimal fractions and percentages, using my answers to make comparisons and informed choices for real-life situations. MNU 3-07aBy applying my knowledge of equivalent fractions and common multiples, I can add and subtract commonly used fractions.MTH 3-07bHaving used practical, pictorial and written methods to develop my understanding, I can convert between whole or mixed numbers and fractions.MTH 3-07c



DEMONSTRATING CONFIDENCE IN THE USE OF MENTAL STRATEGIES AND FORMAL ALGORITHMS WITHIN THE FOUR OPERATIONSLearning Outcomes

I can use both mental strategies and formal algorithms to accurately solve multi-step addition and subtraction tasks, regardless of number range and can communicate and justify my strategies

I can use both mental strategies and formal algorithms to solve multiplication and division tasks, regardless of number range and can communicate and justify my strategies

Activities should support generalising

MULTIPLES,FACTORS AND PRIMESLearning Outcomes

I can apply my knowledge of multiplication facts to allow me to find the multiples of any given number I can find the lowest common multiple when presented with two or more given numbers I can apply my knowledge of multiplication or division facts to find the factors of any given number I can find the highest common factor when presented with two or more given numbers I can identify all prime numbers between 1 and 200 I can show that any given number can be shown as the product of its prime factors

POWERS, ROOTS AND SCIENTIFIC NOTATIONLearning Outcomes

I can identify all square numbers between 1 and 400 Using my knowledge of square numbers, I can find the square root of a given number Using my knowledge of multiplication, I can evaluate simple whole number powers Using my knowledge of multiplication, I can write a given whole number using powers I can use scientific notation to express large or small numbers in a more efficient way and can understand and work with numbers written in this form.

DATA HANDLINGLearning Outcomes

I can apply my understanding of addition, subtraction, division and number finding the Mean, Median, Mode and Range of any given data set I can apply my knowledge of number operations to successfully collect and display data using the most appropriate form.

Language Addition, subtraction, multiplication, division, groups, jottings, algorithms, altogether, remove, missing, addend, subtrahend, minuend, strategy , four operations, Use correct terminology for place value e.g. tens, hundreds, thousands etc Make the distinction between a numeral and the number it represents clear

Multiple, Lowest Common Multiple (LCM), Factor, Highest Common Factor, Prime Number

Mean, Median, Mode and Range, Axis, Scale, Label, Title, Bar Graph, Line Graph

Square Numbers, Cube Numbers, Square Root, Powers (Indices), Base, Index, Scientific Notation (Standard Form)

Settings Hundred squares/ mini hundred squares Empty Number Line Conceptual Place Value materials Base 10 Grid and SMARTPAL Show Me Board Tarsia http://catalog.mathlearningcenter.org/apps/number-frames http://catalog.mathlearningcenter.org/apps/number-pieces http://catalog.mathlearningcenter.org/apps/number-line Word problems

Prime Factor Tree Square Tiles

MIXED NUMBERS AND IMPROPER FRACTIONSLearning Outcomes I can convert between mixed numbers and improper fractions and can represent this visually I can convert 5/4 to a mixed numberThis represents 4/4

This represents 5/4

This represents 1 whole and 1/4

I can convert 17/12 to a mixed number

How can I show 12/12 as an array?e.g. 3 by 4, 2 by 6 or 1 by 12

As an example we will use 3 by 4

This represents 12/12 which is equivalent to one whole.

How many more /12 do we need to make 17/12 and where can we place these.

This represents 1 whole and 5/12

Convert can 2 2/3 into an Improper Fraction

We have 8/3.ADDING AND SUBTRACTING FRACTIONSLearning Outcomes I can add, subtract fractions which involve fractions within and beyond the whole and can represent this visually

e.g. 2/3 + 13/12,

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Factor Rainbow Sieve of Eratothsenes -> 10 x 20 Number Grid (up to 200)

Dot Arrays → Square Arrays (Base 5 and 10 Grids)→Perimeter Markers→Empty Arrays

Survey

Mathematisation – Check it, Prove it, Explain it Visual representation Justify, prove and explain your solution Number problems presented as number sentences

Notation Representing strategies on an empty number line Reading, writing and solving number sentences Using jottings and working towards formal algorithms


How did you solve this using visual representation?How can we use notation to demonstrate your thinking?What were the similarities and differences in the strategies you used to add and subtract?

This bar represents 1 third

This bar represents the link between the thirds and twelfthsOne third is equivalent to 4 twelfths.

2/3 + 13/12 is the same as 8/12 + 13/12 = 21/12 or 1 9/12

Look only at the 9/12 how can this be simplified to ¾ so the answer is 1 ¾ in its simplest form

I can add and/or subtract fractions to solve multi step word problems.e.g. Katy has 4/5 of a litre of petrol in her lawnmower. She buys ¼ litre more. Then she uses 3/8 of a litre. How much of a litre does she have now?e.g. David has 10 metres of wood to make a table. He uses 4/7 of this to make the tabletop and 3/8 to make the legs. How much wood does David have left?

MULTIPLYING FRACTIONSLearning Outcomes I can multiply fractions which involve fractions within and beyond the wholeFind 1/2 of 5/4 of the whole (Revising P6 knowledge)

This represents 4/4

This represents 5/4

This represents halving the 5/4

This represents ½ of 5/4

i.e ½ of 5/4 = 5/8Note that the denominator is 8 not 10 because we are relating it back to the whole

How did you solve this using visual representation?How can we use notation to demonstrate your thinking?

Find 2/3 of 5/4

This represents 4/4

This represents 5/4

This represents dividing the 5/4 into thirds

This represents 2/3 of 5/4 (Remember to shade within the whole before extending out)

i.e. 2/3 of 5/4 is 10/12 = 5/6…note the denominator is 12 not 15 because we are relating it back to the whole

Find 4/3 of 5/4 of this whole

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This represents 4/4

This represents 5/4

This represents the extension beyond the whole to include the 4/3 and the 5/4

This represents 4/3 of 5/4 which equals 20/12 because it extends beyond the whole.NB – the denominator is /12 because the whole is now represented as 12/12NB – this can be extended to consider the mixed number i.e. 20/12 = 1 whole and 8/12

DIVIDING FRACTIONSLearning Outcomes I can distinguish between two meanings of division- measurement meaning and sharing meaning.e.g. ask the children to make a visual representation to show the difference between these two division tasks:The ribbon is 39 cm long. A single serving is 3cm/ How many people can get a serving? (measurement meaning)The ribbon is 39 cm long and it is to be shared between 3 people. How much does each person get? (sharing meaning)

I can create word problems to demonstrate my understanding of why dividing by a fraction is different from diving by a whole number.e.g. give the children 8÷⅓ and 8÷3. Ask them to create word problems to demonstrate the difference between the two.

I can solve word division problems involving fractions.e.g. see examples below

At camp a jug of lemonade contains 2 1/2 litres. The camp leader will give out 1/4 litre servings of the lemonade. How many servings can be given out?Example of linking annotation and notation:The bar represents 2 1/2 litres

The bar now represents 5 half litres – 2 litres and a ½

2 ½ = 5/2We need to split each litre into quarters - this is 10 quarters

5/2 ( 2 and a half) multiplied by the denominator of 4 gives us 20/2 = 10 i.e. the total number of servings available5/2 ÷ ¼ = 10

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A lawnmower fuel tank can hold 4 and a half litres of petrol. Each time the grass is cut the lawnmower uses ¾ of a litre of petrol. How many times can you cut the grass before needing to refuel?Example of linking annotation and notation:The bar represents 4 1/2 litres

The bar now represents 9 half litres – 4 litres and a ½

4 ½ = 9/2We need to split tank into quarters

9/2 ( 4 and a half) split in quarters is 18 quarters

We need to establish how many ¾ of a litre we haveNote: ¾ of each litre is the volume of fuel required to cut the grass (the number of shares)

18 parts shared in groups of 3 = 6

NB - Activities should progress from the visual representation to the algorithm and learners should be able to demonstrate how the algorithm works

FRACTIONS AS NUMBERSLearning Outcomes I can round to any given number of decimal places I fully understand the commutative, associative, distributive and inverse relationships within number operations which include decimals I can convert any fraction, decimal fraction or percentage into an equivalent fraction, decimal fraction or percentage I can apply my knowledge of the equivalence of a fraction, a decimal and a percentage to solve problems I can convert a percentage into a simplified fraction (e.g. 35% is 7/20) e.g. Regard the 100 grid as a one.

Colour in 30% of the grid and notice that the grid can be grouped into lines of 10 i.e. 3/10.

Shade 35% and ‘spot’ the link to splitting the square into 20 parts ie 7/20

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Link to simplifying fractions when learners are confident with the visualisation

I can use my knowledge of simplifying fractions, decimals and percentages to solve numerical tasks (e.g. 35% of £12) I can convert between a whole or mixed number, improper fractions and decimal fractions where appropriate I can calculate percentages of an amount where the percentage given is greater than 100%

Example:The takings at the school show were 154% in comparison with the previous year.Last year, the takings came to £1500. How much did they make this year?

100% = £150050% = £7501% = £154% = £60

154% = 100% + 50% + 4% = £1500 + £750 + £60154% = £2850

I can calculate simple fractions and percentages of a quantity, with and without a calculator, and use this knowledge to solve problems in every day contexts, including percentage increase or decreaseExample:The price of a train ticket has increased by 17%It was £6 last year, how much is it now?

10% = 60p5% = 30p (half of 10%)1% = 6p2% = 12pTherefore increase is 60p + 30p + 12p = 102pTotal cost = £6 + £1.02 = £7.02 (original price is 100% new total is 117%)

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I can express given quantities as a ratio and where appropriate I can simplify these to their simplest formExample:A smoothy is made from 2 bananas and 8 strawberries

Can you simplify this ratio?

2 : 8 1 : 4

What if you had 5 bananas, how many strawberries would you need?

1: 4 5 : (4 x 5) 5: 20

Ensure each learner is visualizing the grouping of columns

UNDERSTANDING EQUIVALENT FRACTIONSLearning Outcomes● I can use my knowledge of simplifying and equivalent fractions to put fractions in order● I can use my knowledge of equivalent fractions to order various fractions with different denominators● I can explain the link between a digit, its place and its value to at least 3 decimal places● I can understand the equivalence of a fraction, a decimal and a percentage to solve problems, justifying choice of method used● I can round decimal fractions to at least 3 decimal places● I can use the denominator and numerator to calculate the fraction of an amount e.g. two thirds of 27 is 27 divided by 3 and multiplied by 2● I can calculate simple fractions and percentages of a quantity, with and without a calculator, and use this knowledge to solve problems in every day contexts● I can multiply and divide decimal fractions with at least 3 decimal places mentally by 10, 100 and 1000

Increasing/ decreasing of place value, not moving the decimal point

Settings● Linear materials that can be broken up and shared e.g. bars, paper strips, rods● Manipulatives● Empty Number Line● http://www.visnos.com/demos/percentage-fraction-decimals-grid ● Word Problems

Language● Fraction, percentage, decimal fraction, parts, whole, share, the same, fair, half, half way, half of, equal, appropriate language to describe different fractions/percentages and decimal fractions

e.g. a sixth, iterate, reform, comparative language for size and length, sequences, partition, unit fraction (1/2, 1/3, ¼, 1/10 etc), composite fraction (2/3, 3/4, 2/5, 3/10 etc), improper fraction (7/4, 10/8, 9/5, 12/10 etc), disembed, denominator, numerator, composite fraction, proper fraction, improper fraction, equivalent fraction, measurement fraction and sharing fraction, common factor, simplifying

Mathematisation – Check it, Prove it, Explain it● Manipulation of material● Visual representation● Screening and flashing● Mental Actions- unitizing, fragmenting and partitioning, iterating, disembedding, unit coordination

Notation

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● Visual representations● Reading, writing and interpreting percentage, decimal and fractional notation


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On track in Secondary 2

Experiences and OutcomesI can use a variety of methods to solve number problems in familiar contexts, clearly communicating my processes and solutions. MNU 3-03aI can continue to recall number facts quickly and use them accurately when making calculations. MNU 3-03bHaving explored the notation and vocabulary associated with whole number powers and the advantages of writing numbers in this form, I can evaluate powers of whole numbers mentally or using technology. MTH 3-06aI can use my understanding of numbers less than zero to solve simple problems in context. MNU 3-04aUsing simple time periods, I can work out how long a journey will take, the speed travelled at or distance covered, using my knowledge of the link between time, speed and distance. MNU 3-10a

Experiences and OutcomesI can round a number using an appropriate degree of accuracy, having taken into account the context of the problem. MNU 3-01aI can solve problems by carrying out calculations with a wide range of fractions, decimal fractions and percentages, using my answers to make comparisons and informed choices for real- life situations. MNU 3-07aI can show how quantities that are related can be increased or decreased proportionally and apply this to solve problems in everyday contexts. MNU 3-08aI can find the probability of a simple event happening and explain why the consequences of the event, as well as its probability, should be considered when making choices. MNU 3-22aBy applying my understanding of probability, I can determine how many times I expect an event to occur, and use this information to make predictions, risk assessment, informed choices and decisions. MNU 4-22a



DEMONSTRATING CONFIDENCE IN THE USE OF MENTAL STRATEGIES AND FORMAL ALGORITHMS WITHIN THE FOUR OPERATIONSLearning Outcomes

I can interpret and solve multi-step problems involving negative numbers using the four operations I can apply the correct order of operations in calculations involving negatives, powers and roots explaining the process and my solution

DIRECT AND INDIRECT PROPORTIONLearning Outcomes

I can use my knowledge of both direct and indirect proportion to solve problems in real-life which involve changes in related quantities I can calculate time durations across hours and days. I can identify when given quantities are related (e.g. Speed, Distance and Time) and can increase or decrease these proportionally as appropriate

Language

Multi-Step, Order of Operations, Powers, Roots, Brackets, Proportion, Time Duration BIDMAS (order of operation) Brackets, Indices (e.g. squared), Division, Multiplication, Addition, Subtraction

Settings Active learning e.g. this is the answer, what is the question? ‘Broken calculator’ puzzles Whiteboards & pens to promote trial and error Table of proportionality:

Time Distance15 mins 12 km

60 mins (1 hr) 48 km

Mathematisation – Check it, Prove it, Explain itInterpreting problems from a wordy settingJustify, prove and explain a solution

NotationConstant of proportionality not requiredRepresenting strategies in tabular form


ROUNDINGLearning Outcomes I can round any given number to any number of decimal places or significant figures

e.g.The attendance at Murrayfield for the rugby was 67106.What is this to 2 significant figures? Why would this be of interest?Compare the capacity of Murrayfield with other venues.

I can use my knowledge of rounding to estimate answers to calculations and determine the reasonableness of a solution within specific contextsTry to include estimation and reasoning within the question for example circle work, money, time intervals or cutting sections of woode.g. a skeleton model of a cube of sides 6.4cm is to be built. How long must the dowling need to be in total? Rounding too early causes error with your calculation!Consider calculator work

FRACTIONS, DECIMALS AND PERCENTAGESLearning Outcomes

I can interpret and solve multi-step problems using the four operations involving fractions and decimals and can communicate and justify my strategiese.g. A farmer needs to split a field into three equal sections. The field is 32 ½ m by 18 ¼ m. What is the area of one of these sections?Encourage learners to discuss the strategies they have used and other ways in which they might come to the same solution.

I can apply my knowledge of the equivalence of a fraction, a decimal and a percentage to solve problems, justifying choice of method usede.g. Probability:Do you have a better chance of picking a blue smartie in a tube when there are 6 blue sweets, 4 red sweets and 10 green sweetsOR from a tube where 25% of the sweets are blue?

e.g. Two companies have damaged goods. One company says 32% of their stock is fault whereas the other one says only 7 out of 20 items is faulty. Which company has a better success rate?Explain your answer

I can express one value as a percentage of anotherAPPLICATIONS OF FRACTIONS, DECIMALS AND PERCENTAGESLearning Outcomes Multiplying a decimal by a decimal

See previous levels on using arrays for multiplication and developing understanding of fractions using a hundred square, then extend into decimals from this prior knowledgee.g. Start with regarding the 100 grid as a one. Colour in 0.14 of the grid. Then colour in 0.14 three times. So 0.14 x 3 = 0.42

Explore this further beyond the whole, and consider the number families ie 0.42 ÷3 = 0.14Link from the visual that 0.14 x 3 is the same as 14 x 3 then ÷ 100

I can find the original amount in a percentage calculation, when given the solution to the original probleme.g.A family pass for the zoo has increased by 20% to £240 a year. How much was it last year?

120% = £2401% = £240 / 120 = £2100% = £200 (100% is original price)NB encouraging checking of answer by adding 20% back on to see if you get back to current price

I can apply my understanding of percentages to perform calculations in a real-life context (e.g. Wages, Salaries, Bonuses, Commission) I can use the equivalence between a percentage and a decimal fraction to create a decimal multiplier for any given percentage

Eg If there is a 20% increase in the cost of electricity bills, how much would the bill be if it was £360 last year?Explore the ways of getting to a solution – eg 100% + 20% of £360 or 1.2 x £360 or 6/5 of £360 and visual representation of why these methods are all the same

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On track in Secondary 2

Experiences and OutcomesI can use a variety of methods to solve number problems in familiar contexts, clearly communicating my processes and solutions. MNU 3-03aI can continue to recall number facts quickly and use them accurately when making calculations. MNU 3-03bHaving explored the notation and vocabulary associated with whole number powers and the advantages of writing numbers in this form, I can evaluate powers of whole numbers mentally or using technology. MTH 3-06aI can use my understanding of numbers less than zero to solve simple problems in context. MNU 3-04aUsing simple time periods, I can work out how long a journey will take, the speed travelled at or distance covered, using my knowledge of the link between time, speed and distance. MNU 3-10a

Experiences and OutcomesI can round a number using an appropriate degree of accuracy, having taken into account the context of the problem. MNU 3-01aI can solve problems by carrying out calculations with a wide range of fractions, decimal fractions and percentages, using my answers to make comparisons and informed choices for real- life situations. MNU 3-07aI can show how quantities that are related can be increased or decreased proportionally and apply this to solve problems in everyday contexts. MNU 3-08aI can find the probability of a simple event happening and explain why the consequences of the event, as well as its probability, should be considered when making choices. MNU 3-22aBy applying my understanding of probability, I can determine how many times I expect an event to occur, and use this information to make predictions, risk assessment, informed choices and decisions. MNU 4-22a



CALCULATING PROBABILITIESLearning Outcomes

I can use the probability scale of 0 to 1 to show probability as a fraction, decimal fraction or percentage

I can use my knowledge of fractions and probability, to calculate and compare the probabilities of different eventsEg cards (suits, faces, aces, red/black, higher/ lower etc)

I can link expected occurrences to make calculated predictions and assess risks to make informed decisionse.g. 30% of High School pupils get the bus to school each day. The cost per journey is £1.80. How much does this cost the council each week?HINT: look up school role in your establishment/ councilDiscussion points: How much would the saving be if this service wasn’t available for over 16s?

Language

Probability Scale, Probability, Expected Occurrences, Risk, Certain, Impossible, Likelihood, Decimal Multiplier, Mutually exclusive events Multi-step, Operations, Fractions, Decimals, One Value as a Percentage of Another, Compound interest, Multiplier Decimal Places, Significant Figures, Estimate, Reasonableness, Realistic, Probability, Likelihood, Percentage chance, Ratio, Data,

Setting Empty number line for rounding (including with decimals) Napier’s bones for multiplying decimals by decimals https://www.engageny.org/resource/math-studio-talk-common-core-instruction-5nbt (watch from 21 mins) Venn diagrams – including links to mutually exclusive events

e.g. Right/ Left handed and Love marmite

Mathematisation – Check it, Prove it, Explain itVisual representationsMental actions - unitising/ fragmenting and partitioning/ parts of a whole

NotationVisual representations


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6 on a dice1/6

0 1Heads on a coin50% = 0.5 = ½

80% chance of rain

Red cardBlack card

Picture card

Example:What fraction of cards are red/ black/ picture cards?What is the probability of a black card, a red card, a picture card?What is the probability of a black, non-picture card?Why do all these probabilities add to 1?How could you change the diagram to identify the probability of black picture cards?

On track in Secondary 3Experiences and OutcomesI have developed my understanding of the relationship between powers and roots and can carry out calculations mentally or using technology to evaluate whole number powers and roots, of any appropriate number. MTH 4-06a

Experiences and OutcomesHaving investigated the practical impact of inaccuracy and error, I can use my knowledge of tolerance when choosing the required degree of accuracy to make real-life calculations.MNU 4-01aBy applying my understanding of probability, I can determine how many times I expect an event to occur, and use this information to make predictions, risk assessment, informed choices and decisions. MNU 4-22



POWERS AND ROOTSLearning Outcomes I can discuss and recognise sets of numbers I can illustrate the links between powers and roots, simplifying where appropriate

e.g. 23 x 22 = = (2x2x2) x (2x2) = 25

2x2x2x2 = 16 4√16 = 2 I can recognise the multiplication properties of roots I can apply my understanding of square roots and factors to simplify surds

Any number to be simplified needs to be arranged into perfect squares of whole numberse.g. Simplify √75

√75=√25×√3=5√3

These remaining squares can’t be arranged into a square, hence √3 cannot be further simplified

I can apply my understanding of simplifying surds to rationalise the denominator in any given fraction with an irrational denominatorLanguage

Irrational, Rational, Infinity, Negative, Integers, Whole numbers, Natural numbers, Fractional Power, Power, Root, Simplify, Surds, Rationalise the DenominatorSettings

Venn diagrams for discussion of sets of numbersMathematisation – Check it, Prove it, Explain it

Notation – Expressions, Equations and visual representationsActivities should progress from the organisation of thinking to generalising

TOLERANCELearning Outcomes I can use a given tolerance to decide if there is an allowable amount of variation of a specified quantity, (e.g. dimensions of a machine part)

E.g. Find the maximum area of a rabbit hutch if the dimensions are 120 x 60 (± 2) cm

APPLICATIONS OF FRACTIONS, DECIMALS AND PERCENTAGESLearning Outcomes

I can apply my understanding of powers and decimal multipliers to calculate compound interest

PROBABILITYLearning Outcome When given a series of connected events, I can calculate the combined probability of the connected event occurring and relate this to the real-life context

Settings – See Examples above

Language Equivalence, Percentage, Decimal Fraction, Decimal Multiplier, Powers, Compound Interest, Connected Events, Combined Probability Tolerance, Quantity, Expectation


Notation - Expressions, Equations and visual representations


National Benchmarks – 3rd levelEstimation and Rounding Rounds decimal fractions to three decimal places. Uses rounding to routinely estimate the answers to calculations.Number and number processes Recalls quickly multiplication and division facts to the 10th multiplication

table. Uses multiplication and division facts to the 12th multiplication table. Solves addition and subtraction problems working with whole numbers

and decimal fractions to three decimal places. Solves addition and subtraction problems working with integers. Solves multiplication and division problems working with whole numbers

and decimal fractions to three decimal places. Solves multiplication and division problems working with integers.Multiples, factors and primes Identifies common multiples, including the lowest common multiple for

whole numbers and can explain method used.

Identifies common factors, including the highest common factor for whole numbers and can explain method used.

Identifies prime numbers to 100 and can explain method used. Solves problems using multiples and factors. Writes a given number as a product of its prime factors.Powers and roots Explains the notation and uses associated vocabulary appropriately, for

example, index, exponent and power

Evaluates whole number powers, for example, 24 = 16. Expresses whole numbers as powers, for example, 27 = 33.

National Benchmarks – 4th levelEstimation and Rounding Rounds answers to a specified significant figure. Demonstrates that the context of the question needs to be considered

when rounding. Demonstrates the impact of inaccuracy and error, for example, the impact

of rounding an answer before the final step in a multi-step calculation. Uses a given tolerance to decide if there is an allowable amount of

variation of a specified quantity, for example, dimensions of a machine part, 235 mm ± 1 mm.

Number and number processes Interprets and solves multi-step problems using the four operations. Applies the correct order of operations in all calculations, including those

with brackets.Powers and roots Shows understanding that square roots of whole numbers can have

positive and negative values, for example, Uses knowledge of the inverse relationship between powers and roots to

evaluate whole number roots of any appropriate number, Uses knowledge of mathematical notation to express numbers in scientific

notation.

National Benchmarks – 3 rd level Fractions, decimal fractions and percentages Converts fractions, decimal fractions or percentages into equivalent fractions, decimal fractions or percentages. Adds and subtracts whole numbers and fractions, including when changing a denominator. Converts between whole or mixed numbers, improper fractions and decimal fractions. Uses knowledge of fractions, decimal fractions and percentages to carry out calculations with and without a calculator. Solves problems in which related quantities are increased or decreased proportionally. Expresses quantities as a ratio and where appropriate simplifies, for example,

‘if there are 6 teachers and 60 children in a school find the ratio of the number of teachers to the total amount of teachers and children’.

National Benchmarks – 4th levelFractions, decimal fractions and percentages Chooses the most efficient form of fractions, decimal fractions or percentages when making calculations. Uses calculations to support comparisons, decisions and choices. Calculates the percentage increase or decrease of a value. Applies addition, subtraction and multiplication skills to solve problems involving fractions and mixed numbers. Uses knowledge of proportin to solve problems in real-life which involve changes in related quantities.

GLOSSARY

Addend. A number to be added. In 7 + 4 = 11, 7 and 4 are addends, and 11 is the sum.

Algorithm. A step-wise procedure for carrying out a task.

Arithmetic rack. A setting used to explore the structures up to 20 and beyond (Abacus).

Array. Spots organised in columns and rows to show equal grouping and sharing

Associate principle. The operation of addition is associative because, when any three numbers are added: 7 + 4 + 5, the order of performing the two addition operations does not affect the sum, for example, (7 + 4) + 5 = 7 + (4 + 5). Multiplication is also associative whereas subtraction and division are not associative.

Automaticity. A capacity to quickly recall or figure out the answer to basic (e.g. 7 + 9, 4 x 8).

Base ten. A characteristic of numeration systems and number naming systems whereby numbers are expressed in a form that involves grouping by tens, tens of tens, and larger powers of ten (1000, 10,000 etc).

Base-ten materials. A generic name for instructional settings consisting of materials organized into ones, tens, hundreds and so on, such as bundling sticks and base-ten dot materials.

Base-ten thinking. Thinking of numbers in terms of one, tens, hundreds, and so on, for example thinking of 257 as consisting of 200 + 50 + 7.

Basic facts. Combinations or number bonds of the form a + b = c (basic facts for addition) or a x b = c (basic facts for multiplication) where a and b are numbers in the range 0 to 10. Also, corresponding combinations involving subtraction or division.

Canonical. The number 64 can be expressed in the form of 60 + 4. This form is referred to as a canonical (standard place-value) form of 64, whereas 50 + 14 is a non-canonical form of 64. Knowledge of canonical and non-canonical forms are useful in addition, subtraction, and so on.

Centuple. A Multiple of 100 (e.g. 100, 200, 300, 1300, 2500). This is distinguished from century which means a sequence of 100 numbers, for example from 267 to 366 or a period of 100 years.

Century. See Centuple.

Combinations. An alternative name for number bonds or basic facts, for example, 5 + 3 = 8, 8 – 2 = 6, 7 x 9 = 63, 48 ÷ by 8 = 6.

Combining. An arithmetical strategy involving combining (i.e. in a sense adding) two numbers whose sum is in the range one to 10, without counting, for example, 4 and 3, 7 and 2.

Commutative principle. The operation of addition is commutative because, for any two numbers, their sum is unchanged when the numbers are commuted, for example 7 + 4 = 4 + 7. Multiplication is also commutative whereas subtraction and division are not commutative.

Compensation strategy. An arithmetical strategy that involves first changing one number to make an easier calculation and then compensating for the change, for example 17 + 38: calculate 17 + 40 and subtract 2.

Complementary addition. A strategy for subtracting based on adding up, for example 82 – 77 is solved as 77 + ? = 80, 80 + ? = 82, answer 5.

Conceptual place value (CPV). An instructional topic focusing on developing students facility to increment and decrement flexibly by one, tens, hundreds, and so on. CPV is distinguished from conventional place value, that is, the conventional instructional topic that is intended to provide a basis for learning formal written algorithms.

Conventional place value. See Conceptual place value.

Coordinating units. Conceiving of a unit fraction and a whole simultaneously. This involves knowing how to iterate a unit fraction such as one-third to form a whole.

Counting by ones. A range of strategies used to solve arithmetical tasks. Some examples are, counting-on, counting-back and counting from one. Contrasted with non-counting (non-count-by-ones) strategies, that is, arithmetical strategies which do not involve counting by ones such as adding through ten, using a double, using a five-structure.

Decade. See Decuple.

Decrementing. See Incrementing.

Decimalising. Developing base-ten thinking, that is, approaches to arithmetic that exploit the decimal (base-ten) numeration system, such as using 10 as a unit, and organising calculations into 1s, 10s and 100s.

Decuple. A multiple of ten (e.g. 10, 20, 30, 180, 240). Distinguished from decade which means a sequence of 10 numbers, for example, from 27 to 36 or a period of 10 years.

Difference. See Minuend.

Digit. The digits are the ten basic symbols in the modern numeration system, that is ‘0’, ‘1’, …’9’.

Disembedding. Taking a part out of a whole without destroying the whole.

Distancing materials. An instructional technique involving progressively reducing the role of materials, for example, materials are unscreened, then flashed and screened, then screened without flashing and used only to check, and so on.

Distributive principle. The principle that multiplication distributes over addition and subtraction as does division, for example (7 – 5) x 3 = (7 x 3) – (5 x 3).

Dividend. In a division equation such as 29 + 4 = 7 r 1, 29 is the dividend, 4 is the divisor, 7 is the quotient and 1 is the remainder.

Divisor. See Dividend.

Doubles. The addition basic facts that involve adding a number to itself: 1 + 1, 2 + 2,…10 + 10.

Empty number line (ENL). A number line with no numbers or markers for recording and sharing students thinking strategies during mental computation

Equal sharing. Creating same-sized parts from a whole (a continuous whole or a collection of items).

Equivalent fractions. An equivalence class of fractions, where all of the elements of the class are represented by a single fraction to which the other fractions simplify.

Facile. Used in the sense of having good facility, that is, fluent or dexterous, for example, facile with a compensation strategy, or facile with the backword number word sequence.

Factor. If a number F, when multiplied by a whole number gives a number M, we call F a factor of M and M a multiple of F, for example, 3 is a factor of 27 and 27 is a multiple of 3, because 3 x 9 = 27.

Finger patterns. Arrangements of fingers used by students when calculating.

Five-wise pattern. A spatial pattern on a ten-frame for a number in the range 1 to 10. Five-wise patterns are made by filling first one row, then the second. For example, a five-wise pattern for 4 has a row of 4 and a row of 0, a five-wise pattern for 7 has a row of 5 and a row of 2. This is contrasted with pair-wise patterns which are made by progressively filling the columns. For example, a pair-wise pattern for 4 has two pairs, a

Flashing. A technique which involves briefly displaying (typically for half a second) some part of an instructional setting. For example, a ten-frame with 8 red and 2 black dots is flashed.

Formal algorithm. A standard written procedure for calculating with multi-digit numbers that relies on the conventions of formal place value; for example, the column-based procedures for adding, subtracting, and so on, contrasted with an informal strategy, for example, solving 58 + 25 by adding 58 and 20, and then 78 and 5.

Formalizing. Developing an approach to arithmetic that involves more formal notation or more formal procedures. The term ‘formal’ is used to indicate adult-level, abstract mathematics.

Fraction number sequence. A set of lengths created by taking multiples of a unit fraction, such as 1/5, 2/5, 3/5.

Fraction strips. Long, thin rectangular pieces of paper

Fractional unit. A unit fraction treated a unit of measure, relative to the whole.

Fragmenting. Breaking a whole into parts

Generalizing. Reasoning that involves proceeding from a few cases to many cases.

Improper fraction. A fraction whose measure is greater than the whole.

Incrementing. Increasing a number, typically by one or more ones, tens, hundreds or some combination of these. Similarly, decreasing a number in this way is called decrementing.

Inquiry-based teaching. An approach to teaching that emphasizes the inquiry mode. Thus tasks are designed to be at the cutting-edge of students current levels of knowledge.

Inverse relationship. Commencing with a number N, if another number, for example 6, is added to N and then subtracted from the sum obtained, then the result will be N. Thus addition and subtraction have an inverse relationship – each is the inverse of the other. Similarly, multiplication and division have an inverse relationship.

Iterating. Making connected copies of a part by repeating it.

Jump strategy. A category of mental strategies for 2-digit addition and subtraction. Strategies in this category involve starting from one number and imcrementing or decrementing that number by first tens and then ones (or first ones then tens). Is also used with 3-digit numbers.

Jump to the decuple. A variation of the jump strategy where the first step is to add up to the next decuple, for example 37 + 25 as 37 + 3, 40 + 20, 60 + 2. Similarly for subtraction, for example, 73 – 35 as 73 – 3, 70 – 30, 40 – 2.

Mathematisation. See Progressive mathematisation.

Mental computation. Typically refers to doing whole number arithmetic with multi-digit numbers, and without any writing. Contrasted with written computation that is computation that involves writing.

Minuend. The number from which another number is subtracted, for example 12 – 3 = 9, 12 is the minuend, 3 is the subtrahend, that is the number subtracted, and 9 is the difference, that is the number obtained.

Missing addend task. An arithmetical task where one addend and the sum are given, for example, 9 + ? = 13.

Missing subtrahend task. An arithmetical task where the minuend and the difference are given, for example, 11 - ? = 8.

Mixed number. A number with an integer part and a fractional part.

Multiplicative reasoning. Working with multiples of a composite unit on at least two levels simultaneously; the multiplicity of the composite units and the multiplicity of the units it contains.

Multi-digit. Involving numbers with two or more digits.

Multiple. See Factor.

Non-canonical. The number 64 can be expressed in the form of 50 + 14. This form is referred to as a non-canonical (non-standard) form of 64, whereas 60 + 4 is the canonical form of 64. Knowledge of non-canonical forms is useful in addition, subtraction, and so on.

Non-regrouping task. See Regrouping task.

Non-unit fraction. A fraction that is not a unit fraction. In other words, its numerator is not 1

Notating. Purposeful writing in an arithmetical situation, for example, notating a jump strategy on an empty number line.

Number. A number is the idea or concept associated with, for example, how many items in a collection. We distinguish among the number 24 – that is, the concept – the spoken or heard number word ‘twenty-four’, the numerical ‘24’ and also the read or written word ‘twenty-four’. These distinctions are important in understanding students numerical strategies.

Number word. Number words are names or words for numbers. In most cases the term ‘number word’ refers to the spoken and heard names for numbers rather than the read or written names.

Number word sequence (NWS). A regular sequence of number words, typically but not necessarily by ones, for example the NWS from 97 to 112, the NWS from 82 back to 75, the NWS by tens from 24, the NWS by threes to 30.

Numeral. Numerals are symbols for numbers, for example ‘5’, ‘27’, ‘307’.

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Numeral identification. Stating the name of a displayed numeral. The term is used similarly to the term ‘letter identification’ in early literacy. When assessisng numeral identification, numerals are not displayed in numerical sequence.

Numeral track. A sequence of numbers that can be tracked and can also be covered and uncovered.

Operation. A mental action (internalized action) that has been organized with other mental actions.

Partitioning. The operation of creating equal parts within a continuous whole

Pair-wise pattern. See Five-wise pattern.

Parts Out of Wholes Fraction Scheme. A way of conceptualizing and working with proper fractions in which the fraction is taken out of (disembedded from) and compared to the whole, by way of the number of equal parts in the fraction and the whole (see Chapter 5).

Parts Within Wholes Fraction Scheme. A way of conceptualizing and working with proper fractions in which the fraction is compared to the whole, by way of the number of equal parts in the fraction and the whole in which the fraction is contained (see Chapter 5).

Part-whole construction of number. The ability to conceive simultaneously of a whole and two parts. For example conceiving of 10 and also of the parts 6 and 4. This is characteristic of students who have progressed beyong a reliance on counting-by-ones to add and subtract.

Partitioned fraction. A fraction regarded as being in a context or setting, for example, half an apple, two fifths of a rectangle. This is contrasted with quantity fraction, that is, a fraction regarded as a number (rational number) in abstract mathematics.

Partitioning. An arithmetical strategy involving partitioning a number into two parts without counting, for example, when solving 8 + 5, 5 is partitioned into 2 and 3.

Partitions of a number. The ways a number can be expressed as a sum of two numbers, for example, the partitions of 6 are 1 and 5, 2 and 4, 3 and 3, 4 and 2, 5 and 1.

Partitive division. A division equation such as 15 ÷ 3 is interpreted as distributing 15 items into three groups, that is, three partitions. This is contrasted with Quotitive division where 15 ÷ 3 is interpreted as distributing 15 items into groups of three, that is, groups with a quota of three.

Product. See Multiplicand.

Progressive mathematisation. The development over time, of the mathematical sophistication of students’ knowledge and reasoning, with respect to a specific topic, for example, addition.

Proper fraction. A fraction whose measure does not exceed the whole.

Quantity fraction. See Partitioned fraction.

Quotient. See Dividend.

Quotitive division. . A measurement meaning for division in which a known quantity is to be shared into shares of a known size, and the number of equal shares is unknown

Regrouping task. In the case of addition of two 2-digit numbers, a task where the sum of the numbers in the ones column exceeds 9, for example, in 37 + 48, 7 + 8 exceeds 9. In the case of subtraction involving two 2-digit numbers, a task where, in the ones column, the subtrahend exceeds the minuend, for example in 95 – 48, 8 exceeds 5. This is similarly applied to addition and subtraction with numbers with three or more digits.

Remainder. See Dividend.

Relational thinking. Reasoning explicitly or implicitly with the structure of one’s number systems, such as with mathematical properties (see Chapter 13).

Scaffolding. Actions on the part of the teacher to provide support for students to reason about or solve a task beyond what they could manage on their own.

Screening. A technique used in the presentation of instructional tasks which involves placing a small screen or cover over all or part of an instructional setting (for example, screening a collection of 6 bundles of sticks).

Segmenting. Using a smaller length to partition off segments within a larger length.

Semi-formal written strategy. A well-organized, standardized, written strategy – less formal than a formal written algorithm – for performing an arithmetical calculation.

Setting. Materials used as a standard context for posing arithmetical tasks, for example, Two-colour ten-frames, Numeral roll and multi-lid frame, Bundling sticks, Arrow cards. See Appendix for more examples.

Split strategy. A category of mental strategies for 2-digit addition and subtraction. Strategies in this category involve splitting the numbers into tens and ones and working separately with the tens and ones before recombining them. Split strategies can also be used with 3-digit and larger numbers.

Splitting. The simultaneous coordination of partitioning and iterating.

Split-jump strategy. A hybrid strategy, for example, 47 + 25 as 40 + 20, 60 + 7, 67 + 5.

Strategy. A generic label for a method by which a student solves an arithmetical task, for example, an add through ten strategy: 8 + 6 as 8 + 2, and 10 + 4. The term procedure is used with similar meaning.

Structuring numbers. Coming to know numbers through organizing numbers in terms of networks of relationships, and applying that knowledge to computation. For example, thinking of 16 as 10 + 5 + 1 or as a double 8.

Subtrahend. See Minuend.

Sum. See Addend.

Ten-frame. A 2x5 or 5x2 grid representing ten with dots that can be arranged in pairs or five-plus which shows relationships within 10

Ten-wise pattern. See Five-wise pattern.

Unit. A thing that is countable and therefore is regarded as a single item. For example, when one counts how many 3s in 18: one 3, two 3s, three 3s….six 3s; the 3s are regarded as units

Unit fraction. Names 1/n, a unit fraction is a part (or unit) that results from partitioning a whole into n equal parts; conversely, this is the part that can be iterated n times to reproduce the whole.

Unitising. A conceptual procedure that involves regarding a number larger than one, as a unit, for example, three is regarded as a unit of three rather than three ones, or 10 is regarded as a unit of 10. Unitising enables students to focus on the unitary rather than the composite aspect of the number. Whole unit. The unit to which fractional units refer.

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