## What Matters in Mathematics and Numeracy

## Organise By

Numbers are the symbol system for describing and comparing quantities. This is the first abstract concept that learners meet in mathematics, and it helps to establish the principles of logical reasoning. In mathematics the number system provides a basis for algebraic, statistical, probabilistic and geometrical reasoning, as well as for financial calculation and decision-making.

Knowledge of, and competence in, number and quantities are fundamental to confident participation in the world, and provide a foundation for further study and for employment. Computational fluency is essential for problem-solving and progressing in all areas of learning and experience. Fluency is developed through using the four basic arithmetic operations and acquiring an understanding of the relationship between them. This leads to preparing the way for using algebraic symbolisation successfully.

Descriptions of learning based on progression within what matters statements and reflecting the four purposes of the curriculum.

Principles of progression are the basis on which the achievement outcomes have been developed and should guide the progression of learning within the area of learning and experience.

The following interdependent proficiencies have been used to formulate the achievement outcomes and are central to progression at each stage of mathematics learning.

Numeracy involves applying and connecting these proficiencies in a range of real-life contexts, across and beyond the curriculum.

**Conceptual understanding:**Mathematical concepts and ideas should be dwelt on, built on, and connected together as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding by explaining and expressing concepts, finding examples (or non-examples) and by representing a concept in a variety of ways, including verbal, concrete, visual, digital and abstract representations.**Communication with symbols:**Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility in their application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.**Strategic competence:**(i.e. formulating problems mathematically in order to solve them) Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to be able to solve them.**Logical reasoning:**As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal, visual or concrete explanations to representations involving symbols and conventions.**Fluency:**As learners experience, understand and apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow. As a result, facts, relationships and techniques learned previously should become firmly established, memorable and usable.#### Achievement outcomes

I have experienced and explored numbers in a variety of contexts and can express whole numbers in words, signs and symbols to convey size and order.

I can notice and read numbers in number-rich indoor and outdoor environments, experiencing cardinal, ordinal and nominal numbers.

I can use my experience of the counting sequence of numbers and of one-to-one correspondence to count sets reliably. I can count objects that I can touch, and ones that I cannot.

I can notice, read and write numbers in a range of media, through a multisensory approach, from zero to at least 20, ensuring they are correctly formed.

I have experienced the counting sequence of numbers in different ways, counting forwards and backwards, and starting at different points.

I have explored composing a quantity in different ways, using combinations of objects or quantities and using the corresponding mathematical language such as words like ‘more’, ‘less’, ‘most’ or ‘least’.

I can understand that a number can be partitioned in different ways.

I have experienced grouping and sharing with objects/quantities, and I can group or share small quantities into equal-sized groups.

I can use my visual sense of number to make estimates and comparisons. I can check estimates using counting or measuring.

I have used money in play and real-life situations and understand that I need to exchange money for items. I can use the language of money.

#### Achievement outcomes

I can read, write and interpret numbers, using figures and words up to at least 1,000.

I can order and sequence numbers, including odd and even numbers, in number-rich indoor and outdoor environments, and can count on and back in uniform steps of any size.

I have experienced numbers below zero in practical contexts, including temperature measurements.

I have explored composing numbers in a variety of ways and I can understand that number value can be determined by the position of the digits used.

I have explored additive relationships using a range of representations. I can use my understanding of additive relationships to add and subtract whole numbers, using a variety of written and mental methods. I can estimate and check the accuracy of my answers using inverse operations when appropriate.

I have demonstrated an understanding of, and can recall, multiplication tables including 2, 3, 4, 5 and 10, and use the term ‘multiples’.

I have explored multiplicative relationships, using a range of representations (including sharing/grouping and the array), and I can use my understanding of multiplicative relationships to multiply and divide whole numbers, using a variety of written and mental methods.

I have engaged in practical tasks and real-life problems to estimate and round numbers to the nearest 10 and 100.

I can reflect on how reasonable answers are in the light of estimations and I have verified calculations using inverse operations.

I have demonstrated an understanding that unit fractions represent equal parts of a whole and are a way of conveying quantities and relationships.

I have experienced fractions in practical situations, using a variety of representations.

I have explored equivalent fractions and understand equivalent fraction relationships.

I have demonstrated an understanding of when to count, when to measure and when to calculate to find quantities.

I can explain that money and ways to save and pay come in different forms. I can make sensible financial transactions in familiar role-play scenarios, including making informed decisions and choices about spending and saving.

#### Achievement outcomes

I have used a range of practical equipment to develop and secure my understanding of place value for positive and negative integers. I can read, write and interpret numbers, using figures and words up to at least one million.

I have extended my understanding of the number system, through a range of activities using non-digital and digital manipulatives, to include decimals and fractions, and I can confidently place whole numbers and fractional quantities on a number line. I have demonstrated my understanding that a fraction can be used as an operator, or to represent division. I can use place value for non‑integers.

I have demonstrated my understanding that fractions (including improper fractions) as well as mixed numbers, decimals and percentages, provide different ways of representing non-integer quantities.

I have explored number patterns, connections and calculations with manipulatives and digital technology, and can demonstrate my understanding of number facts and relationships. I have used my knowledge of number facts and relationships to solve problems in mathematical and real-life contexts.

I have developed, used and discussed efficient and accurate methods when applying all four arithmetic operations to integers and decimals. I can combine these operations, in both mathematical and real-life contexts of problem-solving.

I can verify calculations and statements about number by inverse reasoning and approximation methods.

I have explored the meaning of negative numbers in meaningful and authentic contexts. I can compare the sizes of negative numbers and I can calculate the differences between any two integers. I can check my answers.

I can fluently recall times tables up to at least 10 x 10 and demonstrate my understanding by using them appropriately in applications and in mental and written arithmetic.

I have demonstrated an understanding of rounding and can solve problems requiring rounding to the nearest unit, 10, 100 and 1,000.

I have demonstrated an understanding of the equivalence of simple fractions, decimals and percentages, and I can convert between representations. I can use my knowledge of equivalence to compare the sizes of fractions. I understand the inverse relation between the denominator of a fraction and its value.

I can use my knowledge of multiplication, division, fractions and percentages to calculate proportions of a number or quantity, and to divide a number or quantity in a given ratio. I have solved problems involving ratio and proportion in real-life contexts, and I have used my knowledge of estimation and rounding to predict and check my answers.

I have demonstrated an understanding of income and expenditure, and I can calculate profit and loss. I have created and evaluated budgets for activities and events.

#### Achievement outcomes

I can fluently and accurately apply the four arithmetic operations, in the correct order, on integers, decimals and fractions, using written, mental and digital methods. I can use my sense of number to predict and check my answers.

I have demonstrated an understanding of the equivalence of fractions, decimals and percentages and I can convert fluently between the different forms, using both written methods and a calculator. I have used my knowledge of percentages and ratio to solve problems that involve simple interest, compound interest, depreciation, and calculating bills and budgets that include basic taxation on goods and services.

I have derived and applied the rules of indices, with the exclusion of fractional indices, to calculate values and solve problems.

I have consolidated my understanding of reciprocals when dividing fractions.

I can use standard index form to represent small and large numbers and to perform calculations in appropriate real-life and mathematical contexts.

I can solve problems requiring rounding or significant figures at various stages of the calculation and give the answer, using both written and digital methods, and I can interpret the calculator outputs.

I have demonstrated an understanding of ratio and proportion and can solve numerical problems that involve direct and inverse proportion, including expressing one quantity as a proportion of another, proportional change and problems that involve foreign currencies and exchange rates.

I can use my knowledge of number to predict and check my answers.

I can justify choices based on value for money, personal well-being and global impact.

#### Achievement outcomes

I can recognise the difference between rational and irrational numbers, and apply all four arithmetic operations to them.

I have explored the relationship between powers, roots and fractional indices, and I can derive and apply the rules to simplify and decompose surds.

I have demonstrated fluency in moving between representations of numbers, including converting a given recurring decimal to a fraction.

I have demonstrated an understanding that measurements are not always accurate and are subject to tolerance and margins of error. I can solve problems involving upper and lower bounds, and justify the outcome. I can use my knowledge of tolerance when choosing the required degree of accuracy to make real-life calculations.

I can solve problems involving repeated and inverse proportional reasoning.

I can use my knowledge of annual equivalent rate (AER) and annual percentage rate (APR) to develop models to evaluate and compare financial products.

I can calculate income tax and understand the implications of taxation.

Supporting information to aid practitioners with the design and development of curricula in settings and schools.

This section suggests where learning can be enriched through drawing links between other what matters statements across the Mathematics and Numeracy Area of Learning and Experience. It also suggests where different elements of learning could be considered together in order to support more holistic learning.

Algebra uses symbol systems to express the structures of relationships between numbers, quantities and relations.

- The connections between arithmetic and algebra develop tools and skills for abstract reasoning from an early age.
- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

Geometry focuses on relationships involving properties of shape, space, and position, and measurement focuses on quantifying phenomena in the physical world.

- Number is used throughout geometry to quantify shape, size and movement.
- Measure is an aspect of geometrical thinking which is highly connected to number and much of the development of understanding of number can emerge through increasingly sophisticated measuring.
- Geometric thinking involves reasoning with proportion, which connects with development in number work.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

Statistics represent data, probability models chance, and both support informed inferences and decisions.

- Probability and statistics are described and manipulated by using number; they are represented using number.
- Probability is expressed through number in various ways, e.g. through the use of percentages, fractions and decimals, and the connections between the representations is necessary for effective expression of probability.
- Statistics involves manipulation and representation of data, which involves numerical thinking.

This section suggests where learning can be enriched through drawing links between other what matters statements across all the areas of learning and experience. It also suggests where different elements of learning could be considered together in order to support more holistic learning.

### Expressive Arts

- Scale, proportion and ratio.
- Fractions in music, rhythm.
- Songs and rhymes.

### Health and Well-being

- Understanding of estimation and rounding.
- Application to making real-life decisions including financial ones.
- Fractions, percentages and proportions, e.g. balanced diet.

### Humanities

- Ratio and scale.
- Finance.
- Rounding.
- Ordering.

### Languages, Literacy and Communication

- Songs and rhymes.

### Science and Technology

- Rounding and estimating.
- Laws of indices.
- Reading, writing and calculating in standard index form.
- Direct and inverse proportion.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- explore numbers and quantities through multisensory experiences both indoors and outdoors
- learn that the number system, and whole numbers in particular, allow us to convey size and quantity
- represent and communicate with whole numbers
- begin to understand that money is needed to pay for things
- use a variety of representations to explore number, including objects, visual representations and, where appropriate, digital representations.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- extend their understanding of the number system, through experiential learning, to include large numbers, zero, negative numbers and fractions
- explore composition and equivalence of number, and learn about place value
- learn about relationships in numbers
- understand the additive relationship and be introduced to the multiplicative relationship, including using the array
- use their understanding of the equivalence and value of coins and notes to make appropriate transactions.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- extend their understanding and use of the number system, through a broad range of experiences, to include negative numbers, decimals and fractions
- further develop understanding of place value
- explore the properties of numbers, including factors, multiples, primes, and the inverse relationship between squares and square roots
- become increasingly confident in using all four arithmetic operations in their calculations with whole numbers and decimals, and combine these, using distributive, associative and commutative laws where appropriate
- create and evaluate enterprise projects linked to their immediate and local environment.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- become increasingly fluent in their use of number in calculations, through a broad range of experiences, in order to interpret mathematical statements and describe quantities, both with and without the use of calculators
- further develop their understanding of equivalence, appreciating that any number may be represented in many ways
- use proportional reasoning to compare two quantities using multiplicative thinking, and then apply this to a new situation
- deepen their understanding of using and comparing very big and very small numbers
- select and use efficient mental, written and digital methods to perform calculations
- extend their understanding of finance to personal, local and global contexts.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- become increasingly fluent in their use of number, through a broad range of experiences, in order to describe, interpret and communicate size, scale and comparisons, both within and beyond mathematics
- become increasingly fluent in their calculations, both with and without the use of calculators, and deepen their understanding of how to use rational and irrational numbers
- become critical consumers in broader financial contexts.

**All our children and young people will be:****ambitious, capable learners who:**- set themselves high standards and seek and enjoy challenge
- are building up a body of knowledge and have the skills to connect and apply that knowledge in different contexts
- are questioning and enjoy solving problems
- can communicate effectively in different forms and settings, using both Welsh and English
- can explain the ideas and concepts they are learning about
- can use number effectively in different contexts – understand how to interpret data and apply mathematical concepts
- use digital technologies creatively to communicate, find and analyse information
- undertake research and evaluate critically what they find

and

**are ready to learn throughout their lives****enterprising, creative contributors who:**- connect and apply their knowledge and skills to create ideas and products
- think creatively to reframe and solve problems
- identify and grasp opportunities
- take measured risks
- lead and play different roles in teams effectively and responsibly
- express ideas and emotions through different media
- give of their energy and skills so that other people will benefit

and

**are ready to play a full part in life and work****ethical, informed citizens who:**- find, evaluate and use evidence in forming views
- engage with contemporary issues based upon their knowledge and values
- understand and exercise their human and democratic responsibilities and rights
- understand and consider the impact of their actions when making choices and acting
- are knowledgeable about their culture, community, society and the world, now and in the past
- respect the needs and rights of others, as a member of a diverse society
- show their commitment to the sustainability of the planet

and

**are ready to be citizens of Wales and the world****healthy, confident individuals who:**- have secure values and are establishing their spiritual and ethical beliefs
- are building their mental and emotional well-being by developing confidence, resilience and empathy
- apply knowledge about the impact of diet and exercise on physical and mental health in their daily lives
- know how to find the information and support to keep safe and well
- take part in physical activity
- take measured decisions about lifestyle and manage risk
- have the confidence to participate in performance
- form positive relationships based upon trust and mutual respect
- face and overcome challenge
- have the skills and knowledge to manage everyday life as independently as they can

and

**are ready to lead fulfilling lives as valued members of society.**Algebra is the study of structures abstracted from computations and relations, and provides a way to make generalisations. Algebraic thinking moves away from context to structure and relationships. This powerful approach provides the means to abstract important features and to detect and express mathematical structures of situations in order to solve problems. Algebra is a unifying thread running through the fabric of mathematics.

Algebraic thinking is essential for reasoning, modelling and solving problems in mathematics and in a wide range of real‑world contexts, including technology and finance. Making connections between arithmetic and algebra develops skills for abstract reasoning from an early age.

Descriptions of learning based on progression within what matters statements and reflecting the four purposes of the curriculum.

Principles of progression are the basis on which the achievement outcomes have been developed and should guide the progression of learning within the area of learning and experience.

The following interdependent proficiencies have been used to formulate the achievement outcomes and are central to progression at each stage of mathematics learning.

Numeracy involves applying and connecting these proficiencies in a range of real-life contexts, across and beyond the curriculum.

**Conceptual understanding:**Mathematical concepts and ideas should be dwelt on, built on, and connected together as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding by explaining and expressing concepts, finding examples (or non-examples) and by representing a concept in a variety of ways, including verbal, concrete, visual, digital and abstract representations.**Communication with symbols:**Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility in their application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.**Strategic competence:**(i.e. formulating problems mathematically in order to solve them) Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to be able to solve them.**Logical reasoning:**As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal, visual or concrete explanations to representations involving symbols and conventions.**Fluency:**As learners experience, understand and apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow. As a result, facts, relationships and techniques learned previously should become firmly established, memorable and usable.#### Achievement outcomes

I have explored patterns through a range of hands-on activities and by using a variety of concrete, visual and digital resources. I can recognise, copy and generalise patterns and sequences around me.

I have demonstrated an understanding of the concepts of equality and inequality, using objects.

I can talk about how sets of objects change when objects are added to and taken away from them.

#### Achievement outcomes

I have explored patterns of numbers and shape using concrete, paper-based and digital materials. I can recognise, copy and generate sequences of numbers and visual patterns.

I have demonstrated an understanding of the concepts of equality and inequality within a number equation. I can use the equals sign to indicate that both sides of a number sentence have the same value and I can use inequality signs when comparing quantities.

I have explored commutativity with addition and multiplication and I can recognise when two different numerical expressions describe the same situation but are written in different ways.

#### Achievement outcomes

I can recognise equalities, inequalities and equivalence of expressions, and also when commutativity, distributivity and associativity can be used to state a simple expression in a different way.

I have explored patterns of numbers and shapes, using digital and non-digital methods. I can interpret, explain in words and generalise numerical sequences and spatial patterns.

I can create equations to model problems, using symbols or words to represent unknown values. I can use inverse operations to find unknown values in simple equations using mental, written and digital methods, and manipulatives. I can check my answers.

I have explored the notion of function, including the use of digital function machines.

I have demonstrated an understanding of the idea of input, application of a rule (including inverse operations) and output, using a function machine or other appropriate methods.

#### Achievement outcomes

I have demonstrated an understanding of the concept of a variable and I have used letters to represent variables in forming linear algebraic expressions.

I can manipulate algebraic expressions fluently by simplifying, expanding and factorising by extracting a common factor. I can also substitute values and change the subject of a formula when the subject appears in one term.

I can distinguish between algebraic expressions, equations and inequalities. I have used a variety of methods, including trial and improvement where appropriate, to solve equations and inequalities in the first degree, which may include brackets and unknowns on one or both sides. I can check my answers using substitution.

I have used equations and inequalities in the first degree to represent and model real-life situations and solve problems. I can interpret my answers and check that they make sense in context.

I can recognise linear sequences and can generalise them using algebra. I can find, describe and use the

*n*th term. I can describe and use the term-to-term rule for simple iterative sequences. I can apply my knowledge of sequences to solve real-life and mathematical problems.I can recognise, draw, sketch and interpret linear graphs, and investigate graphs, using written and digital methods. I can demonstrate an understanding of each of the terms in the equation of a straight line. I can explore the effect on the line when the constant or coefficient is changed.

#### Achievement outcomes

I can fluently manipulate algebraic expressions by expanding double brackets, factorising quadratic expressions and changing the subject of a formula where the subject appears in more than one term. I can simplify and manipulate algebraic fractions.

I can solve a range of linear and higher-order equations, and inequalities, including simultaneous, quadratic and trigonometric equations, using numerical, graphical and algebraic methods where appropriate. I can then interpret the meaning of the answer, or answers, checking for reasonableness.

I have used equations and inequalities, and relevant graphs, to model and solve problems in real-life and mathematical contexts, including those which describe proportion and exponentiation, and I can use my knowledge of the real world and number sense to predict and check my interpretations of these.

I can recognise and generalise simple non-linear sequences using algebra.

I can understand the concept of an identity and can translate statements describing mathematical relationships into algebraic models, using expressions and equations.

I have investigated a variety of non-linear graphs (including quadratic, cubic and reciprocals), using written and digital methods. I can demonstrate an understanding of the effect of the coefficients, indices and constants on the shape of the graph. I can determine the gradient at a point and the area under a graph, and understand what these represent. I can use graphs to solve problems.

Supporting information to aid practitioners with the design and development of curricula in settings and schools.

This section suggests where learning can be enriched through drawing links between other what matters statements across the Mathematics and Numeracy Area of Learning and Experience. It also suggests where different elements of learning could be considered together in order to support more holistic learning.

The number system is used to represent and compare relationships between numbers and quantities.

- The connections between arithmetic and algebra develop tools and skills for abstract reasoning from an early age.
- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

Geometry focuses on relationships involving properties of shape, space, and position, and measurement focuses on quantifying phenomena in the physical world.

- Algebra and geometry are connected principally through the expression of shape, measure and movement through algebraic expressions, equations and formulae.
- An equation is an algebraic concept, which, when graphed, becomes a geometric concept. The variables within the equation refer to geometric concepts.
- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.
- Algebraic formulae and equations are used to connect the geometric concepts of triangles with measures of angles and sides.

Statistics represent data, probability models chance, and both support informed inferences and decisions.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In probability, algebra allows us to use the concept of a variable and we can apply this in probability by using a random variable; this is a parameter or event with a random outcome.
- In statistics, general formulae are written using algebra.
- Algebra and statistical analysis are interlinked; use of algorithms and formulae to calculate further statistical measures to analyse data.

This section suggests where learning can be enriched through drawing links between other what matters statements across all the areas of learning and experience. It also suggests where different elements of learning could be considered together in order to support more holistic learning.

### Expressive Arts

- Sequences and patterns.

### Science and Technology

- Concept of the variable.
- Equations and formulae.
- Direct and inverse proportion.
- Patterns and graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- explore, copy, repeat, continue and generate a range of simple patterns and sequences using a multisensory approach
- start to predict what will come next, through shared stories, pictures and rhymes that have a pattern element
- through active experience, demonstrate an understanding of equality and inequality, and how these can be preserved when numbers are changed, using the language of comparison and equality.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- express and use, through experiential learning, general laws and rules of arithmetic
- explore and describe sequences of whole numbers with a common difference between terms
- find missing terms and continue sequences
- understand that Robert Recorde’s equals sign is placed between two or more expressions that have the same value
- recognise commutativity of addition and multiplication, using objects, diagrams and numbers
- reason logically, using equality and the laws of arithmetic.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences, describe relations which depend on equality, equivalence and the commutative, associative and distributive laws
- describe and generate, in words, visual patterns and numerical sequences
- model simple problem situations, using words and symbols to create equations from which they can find an unknown value
- understand that a function can transform a set of numbers to a new set of numbers, according to a rule
- move between concrete, visual and abstract representations throughout their mathematical work.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences, appreciate the power of mathematics to express relationships concisely and in forms that are universally understood
- express the laws of arithmetic using algebraic notation, developing their conceptual understanding of a variable
- explore, generate, identify and represent linear patterns in a variety of contexts
- model real-life situations, using equations and inequalities to solve problems, checking the reasonableness of their solutions
- explore equations graphically, using digital technologies
- explore numerical and physical sequences, using written and digital methods
- investigate linear graphs in realistic situations.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences, create multiple representations of a mathematical concept
- explore connections between equations and expressions with geometric, numerical and graphical representations, and understand their equivalence
- explore, generate and identify non-linear patterns, and express them algebraically and graphically
- develop and apply their knowledge of a range of methods such as factorisation, simplification and inverse functions to change the subject of formulae involving two or more variables
- model real-life situations, identifying variables and constructing polynomials.

**All our children and young people will be:****ambitious, capable learners who:**- set themselves high standards and seek and enjoy challenge
- are building up a body of knowledge and have the skills to connect and apply that knowledge in different contexts
- are questioning and enjoy solving problems
- can communicate effectively in different forms and settings, using both Welsh and English
- can explain the ideas and concepts they are learning about
- can use number effectively in different contexts – understand how to interpret data and apply mathematical concepts
- use digital technologies creatively to communicate, find and analyse information
- undertake research and evaluate critically what they find

and

**are ready to learn throughout their lives****enterprising, creative contributors who:**- connect and apply their knowledge and skills to create ideas and products
- think creatively to reframe and solve problems
- identify and grasp opportunities
- take measured risks
- lead and play different roles in teams effectively and responsibly
- express ideas and emotions through different media
- give of their energy and skills so that other people will benefit

and

**are ready to play a full part in life and work****ethical, informed citizens who:**- find, evaluate and use evidence in forming views
- engage with contemporary issues based upon their knowledge and values
- understand and exercise their human and democratic responsibilities and rights
- understand and consider the impact of their actions when making choices and acting
- are knowledgeable about their culture, community, society and the world, now and in the past
- respect the needs and rights of others, as a member of a diverse society
- show their commitment to the sustainability of the planet

and

**are ready to be citizens of Wales and the world****healthy, confident individuals who:**- have secure values and are establishing their spiritual and ethical beliefs
- are building their mental and emotional well-being by developing confidence, resilience and empathy
- apply knowledge about the impact of diet and exercise on physical and mental health in their daily lives
- know how to find the information and support to keep safe and well
- take part in physical activity
- take measured decisions about lifestyle and manage risk
- have the confidence to participate in performance
- form positive relationships based upon trust and mutual respect
- face and overcome challenge
- have the skills and knowledge to manage everyday life as independently as they can

and

**are ready to lead fulfilling lives as valued members of society.**Geometry involves playing with, manipulating, comparing, naming and classifying shapes and structures. The study of geometry encourages the development and use of conjecture, deductive reasoning and proof. Measurement allows the magnitude of spatial and abstract features to be quantified, using a variety of standard and non-standard units, and can support the development of numerical reasoning.

Reasoning about the sizes and properties of shapes and their surrounding spaces helps us to make sense of the physical world and the world of mathematical shapes. Geometry and measurement have applications in many fields, including art, construction, science and technology, engineering, and astronomy.

Descriptions of learning based on progression within what matters statements and reflecting the four purposes of the curriculum.

Principles of progression are the basis on which the achievement outcomes have been developed and should guide the progression of learning within the area of learning and experience.

The following interdependent proficiencies have been used to formulate the achievement outcomes and are central to progression at each stage of mathematics learning.

Numeracy involves applying and connecting these proficiencies in a range of real-life contexts, across and beyond the curriculum.

**Conceptual understanding:**Mathematical concepts and ideas should be dwelt on, built on, and connected together as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding by explaining and expressing concepts, finding examples (or non-examples) and by representing a concept in a variety of ways, including verbal, concrete, visual, digital and abstract representations.**Communication with symbols:**Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility in their application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.**Strategic competence:**(i.e. formulating problems mathematically in order to solve them) Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to be able to solve them.**Logical reasoning:**As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal, visual or concrete explanations to representations involving symbols and conventions.**Fluency:**As learners experience, understand and apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow. As a result, facts, relationships and techniques learned previously should become firmly established, memorable and usable.#### Achievement outcomes

I can understand and apply the language of time in relation to my daily life and in relation to events that happen around me, including naming the days of the week, the months of the year and the seasons, all in meaningful contexts.

I have used a variety of objects to measure lengths, masses and capacities, and I can understand the need to repeat the same physical unit without any gaps when measuring.

I can make estimates and comparisons with measures. I can use language and non-standard units to discuss my sense of size.

I have explored shapes through investigative play, and I can categorise and sort shapes using their properties. I can use the language of shapes to describe objects.

I have explored movements and directions, both physically and by using digital technology, and I can use mathematical language to describe position.

#### Achievement outcomes

I can tell the time on an analogue clock (in at least 15-minute intervals) and I can connect this to time displayed digitally. I have explored and used different ways of showing the passing of time, including calendars, timelines, simple timetables and schedules.

I have estimated and measured length, volume, capacity, mass, temperature and time in practical situations, using non-standard units.

I have used a variety of measuring devices from different starting points.

I can apply standard units in practical situations to measure length, volume, capacity, mass, temperature and time accurately.

I have explored and named two-dimensional and three-dimensional shapes in a range of contexts. I can sort and categorise regular and irregular two-dimensional shapes in different ways, according to their properties. I can sketch two-dimensional shapes and make models of three-dimensional objects.

I can identify reflective symmetry in a range of contexts and can identify symmetry in two-dimensional shapes.

I have explored the concept of rotation, both physically and using digital technology, and can use simple fractions of a complete rotation to describe turns.

I have described and quantified the position of objects relative to other objects through active learning experiences, using appropriate language, including the words ‘left’ and ‘right’.

I can follow and create instructions related to movement, using a range of approaches and resources (including digital technologies) to demonstrate my understanding.

#### Achievement outcomes

I can read analogue and digital clocks accurately and I can make calculations involving the passing of time. I have used timetables and schedules to make calculations involving time.

I have estimated and measured length, capacity, mass, temperature and time, using appropriate standard units.

I can convert between metric units in mathematical and real-life problem-solving contexts, and I can check my answers to make sure they make sense.

I have explored properties of two-dimensional shapes, using concrete, paper-based and digital resources, and I can use mathematical language to describe and compare the properties (including number of sides and symmetry) of given shapes, and to classify them accordingly. I can name different types of triangles.

I have explored vertices, edges and faces of three-dimensional shapes and I can use these characteristics to describe a three-dimensional shape. I have explored the relationship between a three-dimensional shape and two-dimensional nets, using concrete, paper-based and digital resources, and I can recognise nets of common three-dimensional shapes.

I have used a range of hands-on activities to explore the perimeter and area of shapes. I have derived the respective formulae for finding the area of a rectangle and of a right-angled triangle, and have applied them in mathematical and real-life problem-solving contexts, using estimation to predict and check my answers.

I can demonstrate my understanding of angle as a measure of rotation and I can recognise, name and describe types of angles.

I have developed an understanding of why we need co-ordinates and I can use them to locate and plot points in the first quadrant of the Cartesian plane. I can use my knowledge of coordinates to solve problems involving shape, length, angle and position in mathematical and real-life contexts.

#### Achievement outcomes

I can represent and use compound measures, using standard units, and I can demonstrate an understanding of the relationship between a formula representing a measurement and the units used.

I can create and use conversion graphs to solve problems set in local and global contexts.

I have explored symmetries and other properties of regular and irregular two-dimensional and three-dimensional shapes. I can construct two-dimensional representations of three-dimensional shapes, in order to investigate properties further. I can classify two-dimensional and three-dimensional shapes according to their mathematical properties.

I have explored all four transformations of two-dimensional shapes, using a variety of approaches, including digital technology. I can use my understanding to predict and describe how shapes will change under a given transformation.

I can use co-ordinates to plot points in the four quadrants and deduce the location of additional points.

I can use a protractor to measure and draw angles. I have modelled and solved problems involving bearings. I can use reasoning to calculate the size of angles in triangles and quadrilaterals. I have explored angles formed by parallel lines and by a transversal, and I can use my understanding to calculate angles in these contexts.

I have calculated the areas or surface areas of two-dimensional and three-dimensional simple and compound shapes, including circles, and have demonstrated an understanding of pi (π) as the ratio of the circumference of a circle to its diameter. I have derived the formulae for the volume of simple three-dimensional prisms and I can calculate the volumes of three-dimensional shapes to solve problems.

I have demonstrated an understanding of the relationship between right-angled triangles and squares in the context of Pythagoras’ theorem, and I have used it to solve problems in mathematical and real-life contexts.

#### Achievement outcomes

I can explain why two or more shapes are similar, congruent, or neither. I have used my knowledge of congruency and similarity to solve problems involving angles and lengths, both in mathematical and real-life contexts.

I can calculate the perimeter, area or surface area and volume of compound two-dimensional and three‑dimensional shapes, and I can rearrange formulae to find missing lengths. I have demonstrated an understanding of the effect of scale when comparing measurements and shapes in all three dimensions, and I have used my knowledge of scale and ratio to calculate the lengths and areas of fractions of shapes, including arcs and segments of circles.

I have located and described the locus of points defined by a range of different criteria, using digital and non‑digital technologies.

I have demonstrated an understanding of trigonometric ratios in right-angled triangles and I have used, in mathematical and real-life contexts, my knowledge of the trigonometric ratios to solve problems involving lengths, angles and area of any triangle.

I have used reasoning and logical arguments, along with my knowledge of polygons, intersecting lines, angle and the circle theorems, to solve problems, deduce and calculate angles and lengths in diagrams that involve combinations of these.

Supporting information to aid practitioners with the design and development of curricula in settings and schools.

This section suggests where learning can be enriched through drawing links between other what matters statements across the Mathematics and Numeracy Area of Learning and Experience. It also suggests where different elements of learning could be considered together in order to support more holistic learning.

The number system is used to represent and compare relationships between numbers and quantities.

- Number is used throughout geometry to quantify shape, size and movement.
- Measure is an aspect of geometrical thinking which is highly connected to number and much of the development of understanding of number can emerge through increasingly sophisticated measuring.
- Geometric thinking involves reasoning with proportion, which connects with development in number work.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

Algebra uses symbol systems to express the structures of relationships between numbers, quantities and relations.

- Algebra and geometry are connected principally through the expression of shape, measure and movement through algebraic expressions, equations and formulae
- An equation is an algebraic concept, which, when graphed, becomes a geometric concept. The variables within the equation refer to geometric concepts.
- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.
- Algebraic formulae and equations are used to connect the geometric concepts of triangles with measures of angles and sides.

Statistics represent data, probability models chance, and both support informed inferences and decisions.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphs are used to convert from one value to another, and to represent statistical information including spread, quantity and central tendency.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

This section suggests where learning can be enriched through drawing links between other what matters statements across all the areas of learning and experience. It also suggests where different elements of learning could be considered together in order to support more holistic learning.

### Expressive Arts

- Shape symmetry in movement and artwork.
- Scale.
- Use of appropriate equipment to measure accurately.

### Health and Well-being

- Use of appropriate equipment to measure accurately.

### Humanities

- Use of appropriate equipment to measure accurately.
- Scale.
- Time and chronological ordering.

### Languages, Literacy and Communication

- Use of prepositions to describe the location of selves and objects.

### Science and Technology

- Use of appropriate equipment to measure accurately.
- Units – use of appropriate unit, converting between units, and links between units and formulae.
- Scale.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through exploring situations in their everyday life, develop an understanding of measurement, using non‑standard units with developing accuracy
- through play and discovery, categorise and recognise shapes
- use the language of direction, location and position.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through experiential learning and solving problems in real-life and more abstract contexts, develop an understanding of standard measurement and of the properties of shapes
- select and use appropriate equipment and units to measure accurately
- describe location, movement and position, using mathematical terminology.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences and through solving problems in real-life and abstract contexts, refine their understanding of standard measurement and the relationship between units of measure
- select and use appropriate equipment and units to measure accurately
- refine their understanding of the properties of shapes
- use the mathematical concepts of angle, co-ordinates and distance to describe location, movement and position.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences and through solving problems within real-life and abstract contexts, develop their understanding of standard and compound measurement, the properties of shapes and solids, and location, movement and position
- select and use appropriate equipment and units to measure accurately
- use digital technologies to explore shape and space, developing and testing conjectures
- use angle and shape facts to deduce further features and relationships
- recognise pi (π) as the ratio of the circumference of a circle to its diameter and appreciate the significance of William Jones’s contribution.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences, describe, represent, analyse and explain properties of shapes in two‑dimensional and three-dimensional space
- use digital technologies to explore shape and space, movement and position, developing and testing conjectures
- work in real-life and mathematical contexts, using local examples where possible.

**All our children and young people will be:****ambitious, capable learners who:**- set themselves high standards and seek and enjoy challenge
- are building up a body of knowledge and have the skills to connect and apply that knowledge in different contexts
- are questioning and enjoy solving problems
- can communicate effectively in different forms and settings, using both Welsh and English
- can explain the ideas and concepts they are learning about
- can use number effectively in different contexts – understand how to interpret data and apply mathematical concepts
- use digital technologies creatively to communicate, find and analyse information
- undertake research and evaluate critically what they find

and

**are ready to learn throughout their lives****enterprising, creative contributors who:**- connect and apply their knowledge and skills to create ideas and products
- think creatively to reframe and solve problems
- identify and grasp opportunities
- take measured risks
- lead and play different roles in teams effectively and responsibly
- express ideas and emotions through different media
- give of their energy and skills so that other people will benefit

and

**are ready to play a full part in life and work****ethical, informed citizens who:**- find, evaluate and use evidence in forming views
- engage with contemporary issues based upon their knowledge and values
- understand and exercise their human and democratic responsibilities and rights
- understand and consider the impact of their actions when making choices and acting
- are knowledgeable about their culture, community, society and the world, now and in the past
- respect the needs and rights of others, as a member of a diverse society
- show their commitment to the sustainability of the planet

and

**are ready to be citizens of Wales and the world****healthy, confident individuals who:**- have secure values and are establishing their spiritual and ethical beliefs
- are building their mental and emotional well-being by developing confidence, resilience and empathy
- apply knowledge about the impact of diet and exercise on physical and mental health in their daily lives
- know how to find the information and support to keep safe and well
- take part in physical activity
- take measured decisions about lifestyle and manage risk
- have the confidence to participate in performance
- form positive relationships based upon trust and mutual respect
- face and overcome challenge
- have the skills and knowledge to manage everyday life as independently as they can

and

**are ready to lead fulfilling lives as valued members of society.**Statistics is the practice of collecting, manipulating and analysing data, allowing representation and generalisation of information. Probability is the mathematical study of chance, enabling predictions of the likelihood of events occurring. Statistics and probability rely on the application and manipulation of number and algebra.

Managing data and representing information effectively provides the means to test hypotheses, draw conclusions and make predictions. Reasoning with statistics and probability, and evaluating their reliability, develops critical thinking and analytical skills that are fundamental to making ethical and informed decisions.

**Conceptual understanding:**Mathematical concepts and ideas should be dwelt on, built on, and connected together as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding by explaining and expressing concepts, finding examples (or non-examples) and by representing a concept in a variety of ways, including verbal, concrete, visual, digital and abstract representations.**Communication with symbols:**Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility in their application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.**Strategic competence:**(i.e. formulating problems mathematically in order to solve them) Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to be able to solve them.**Logical reasoning:**As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal, visual or concrete explanations to representations involving symbols and conventions.**Fluency:**As learners experience, understand and apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow. As a result, facts, relationships and techniques learned previously should become firmly established, memorable and usable.#### Achievement outcomes

I have investigated and collected data found in my environment. I can record data, giving meaning to marks which represent number and where each quantity corresponds to something in the real world, keeping simple records, which include tallying and pictograms.

I have grouped sets of objects or pictures into categories, through practical experiences, and I can classify by one criterion or more. I can talk about the rule(s) I have used and can reclassify according to new criteria.

I can present data using non-digital and digital methods. I have created simple charts and graphs and I have talked about what they mean.

#### Achievement outcomes

I can collect and organise data for posing and answering questions in relevant situations.

I can sort and classify using more than one criterion, using Venn diagrams and Carroll diagrams in practical situations.

I have used digital and non-digital methods to record and present data in a variety of ways, including the use of tally charts, frequency tables, and block graphs, when appropriate axes and scales are provided.

I can interpret and analyse graphs, charts and data.

I can explain what I have found out, justify my reasoning and I can evaluate how well my method worked.

#### Achievement outcomes

I have used a sequential data-handling cycle to ask and answer appropriate questions in meaningful situations.

I can pose and answer sensible questions and I have demonstrated an understanding of the importance of collecting relevant data that can be used to answer my questions.

I have demonstrated an understanding of the kinds of data I need, including discrete and continuous data.

I can find and use the mean of a simple set of data in meaningful contexts.

I have used appropriate methods to answer my questions by collecting, analysing and summarising my data and interpreting my results. I can evaluate my methods and suggest different or better ways to approach investigations in the future.

I have represented information by creating a variety of appropriate charts of increasing complexity, including tally charts, frequency tables, bar graphs, line graphs, with and without the use of digital technologies. I have created pie charts using digital technology.

I can use different scales on axes to extract and interpret information from a range of diagrams, tables (including databases) and graphs, including pie charts with simple fractions and proportions.

I have investigated simple statistics, presented in the media and elsewhere, to support an argument and I can explain how the statistics do, or do not, support the argument.

I can recognise validity and trends, and can discuss how anomalies may affect conclusions when evaluating results.

I have explored the possibility of given outcomes and have used the language of probability to describe the chance of an event occurring.

I have played games that involve flipping coins, rolling dice and using spinners in order to simulate, and discuss chance.

I have hypothesised and anticipated outcomes of chance experiments in a range of contexts, recording my findings in a systematic and appropriate way.

#### Achievement outcomes

I have used a sequential data-handling cycle to ask and answer appropriate questions.

I can choose a sensible hypothesis to investigate and I have demonstrated an understanding of the methods I can use to collect appropriate data.

I can design and have critiqued questionnaires to ensure that the data to be collected will enable the sequential data-handling cycle to be used.

I can calculate the mean, median, mode and range of a set of data and I have made comparisons between small sets of data using summary statistics.

I can select and justify an appropriate way to use my data to investigate my hypothesis. I have explored different ways to understand and summarise my data, including using averages to make comparisons between large data sets, with grouped frequency distributions for discrete and continuous data. I can use a scatter diagram to analyse two sets of variables and investigate correlation between them. I can make predictions and identify trends and anomalies in data sets.

I have presented my data in the form of appropriate graphs, charts and tables, and I have used digital technologies, taking into account the purpose of the data and nature of the audience. I have discussed the relative advantages and disadvantages of each presentation method, and I can justify my choice of method.

I have used my data to draw conclusions about my hypotheses and I have communicated my findings clearly. I can critique my own methods and findings, and consider what I may have done differently or better at each stage of the sequential data-handling cycle.

I have critically analysed statistics published in the media and elsewhere to consider what it means and how it does, or does not, support any findings reported. I can pose relevant questions to check the credibility of the findings.

I can explain randomness, and I have investigated chance by modelling and by comparing theoretical and experimental probabilities.

I have explored all the possible mutually exclusive outcomes of successive and combined events. I can work systematically making use of lists and sample space diagrams, and I have shown an understanding that the sum of probabilities of all mutually exclusive outcomes is 1. I have demonstrated an understanding of when it is appropriate to add or multiply probabilities.

I can make meaningful real-life judgements based on outcomes of experimental data and risk.

#### Achievement outcomes

I have used a sequential data-handling cycle to ask and answer appropriate questions.

I have explored different sampling methods and I have demonstrated an understanding of the need to select a sample when collecting data. I can evaluate, choose and use different sampling techniques, including random sampling, stratified sampling and systematic sampling.

I have experimented with different approaches to presenting data, including cumulative frequency, box and whisker, and histograms, to interpret measures of central tendency and measures of spread. I can select appropriate approaches, when comparing data sets, justifying and evaluating my choices.

I have critically analysed statistics in the media, considering how data is presented, its reliability, and whether and how the data has been manipulated to tell a particular story. I can make informed decisions based on statistical evidence, identifying bias and anomalies.

I can solve problems involving probabilities of mutually exclusive, independent and dependent events in real-life and mathematical contexts. I can use a variety of strategies, including using Venn and tree diagrams, to solve problems in local and wider contexts.

I have played and created games to understand the relationship between relative frequency and theoretical probabilities, making judgements on outcomes of experimental data and risk.

I can use probabilistic arguments, drawing on theory, information, research and experimentation to support my conclusion.

The number system is used to represent and compare relationships between numbers and quantities.

- Probability and statistics are described and manipulated by using number; they are represented using number.
- Probability is expressed through number in various ways, e.g. through the use of percentages, fractions and decimals, and the connections between the representations is necessary for effective expression of probability.
- Statistics involves manipulation and representation of data, which involve numerical thinking.

Algebra uses symbol systems to express the structures of relationships between numbers, quantities and relations.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In probability, algebra allows us to use the concept of a variable and we can apply this in probability by using a random variable; this is a parameter or event with a random outcome.
- In statistics, general formulae are written using algebra.
- Algebra and statistical analysis are interlinked; use of algorithms and formulae to calculate further statistical measures to analyse data.

Geometry focuses on relationships involving properties of shape, space, and position, and measurement focuses on quantifying phenomena in the physical world.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphs are used to convert from one value to another, and to represent statistical information including spread, quantity and central tendency.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Health and Well-being

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Humanities

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Science and Technology

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through play and discovery in stimulating and relevant local contexts, collect and sort a variety of simple data and answer questions
- use appropriate digital and non-digital methods to represent and interpret data
- talk about their results.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through experiential learning in stimulating and relevant local contexts, use a sequential data-handling cycle to pose questions, then collect and sort a variety of purposeful data to answer questions
- use appropriate digital and non-digital methods to analyse, represent and interpret data
- explain their results and evaluate their methods.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences in stimulating and relevant local and global contexts, use a sequential data-handling cycle to pose questions, collect and sort a variety of purposeful data
- use appropriate digital and non-digital methods to analyse, summarise, represent and interpret data
- through using different approaches, compare, check and evaluate their results, explaining what they have found
- experiment with chance.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences in stimulating and relevant local and global contexts, use a sequential data-handling cycle to pose questions, collect and sort a variety of purposeful data
- use appropriate digital and non-digital methods to analyse, summarise, represent and interpret data to draw conclusions
- check and evaluate their results and recognise anomalies and trends in data
- extend their understanding of chance by expressing theoretical probability numerically.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences in stimulating and relevant local and global contexts, use a sequential data-handling cycle to pose questions, select a sample, and collect and sort a variety of purposeful data
- use appropriate digital and non-digital methods to analyse, summarise, represent and interpret data to draw conclusions
- check and evaluate their results
- engage critically with statistics in the media
- calculate probabilities of combined events and consider probabilities of real-life events.

**All our children and young people will be:****ambitious, capable learners who:**- set themselves high standards and seek and enjoy challenge
- are questioning and enjoy solving problems
- can communicate effectively in different forms and settings, using both Welsh and English
- can explain the ideas and concepts they are learning about
- use digital technologies creatively to communicate, find and analyse information
- undertake research and evaluate critically what they find

and

**are ready to learn throughout their lives****enterprising, creative contributors who:**- connect and apply their knowledge and skills to create ideas and products
- think creatively to reframe and solve problems
- identify and grasp opportunities
- take measured risks
- lead and play different roles in teams effectively and responsibly
- express ideas and emotions through different media
- give of their energy and skills so that other people will benefit

and

**are ready to play a full part in life and work****ethical, informed citizens who:**- find, evaluate and use evidence in forming views
- engage with contemporary issues based upon their knowledge and values
- understand and exercise their human and democratic responsibilities and rights
- understand and consider the impact of their actions when making choices and acting
- are knowledgeable about their culture, community, society and the world, now and in the past
- respect the needs and rights of others, as a member of a diverse society
- show their commitment to the sustainability of the planet

and

**are ready to be citizens of Wales and the world****healthy, confident individuals who:**- have secure values and are establishing their spiritual and ethical beliefs
- are building their mental and emotional well-being by developing confidence, resilience and empathy
- know how to find the information and support to keep safe and well
- take part in physical activity
- take measured decisions about lifestyle and manage risk
- have the confidence to participate in performance
- form positive relationships based upon trust and mutual respect
- face and overcome challenge
- have the skills and knowledge to manage everyday life as independently as they can

and

**are ready to lead fulfilling lives as valued members of society.**

**Conceptual understanding:**Mathematical concepts and ideas should be dwelt on, built on, and connected together as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding by explaining and expressing concepts, finding examples (or non-examples) and by representing a concept in a variety of ways, including verbal, concrete, visual, digital and abstract representations.**Communication with symbols:**Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility in their application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.**Strategic competence:**(i.e. formulating problems mathematically in order to solve them) Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to be able to solve them.**Logical reasoning:**As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal, visual or concrete explanations to representations involving symbols and conventions.**Fluency:**As learners experience, understand and apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow. As a result, facts, relationships and techniques learned previously should become firmly established, memorable and usable.Numbers are the symbol system for describing and comparing quantities. This is the first abstract concept that learners meet in mathematics, and it helps to establish the principles of logical reasoning. In mathematics the number system provides a basis for algebraic, statistical, probabilistic and geometrical reasoning, as well as for financial calculation and decision-making.

Knowledge of, and competence in, number and quantities are fundamental to confident participation in the world, and provide a foundation for further study and for employment. Computational fluency is essential for problem-solving and progressing in all areas of learning and experience. Fluency is developed through using the four basic arithmetic operations and acquiring an understanding of the relationship between them. This leads to preparing the way for using algebraic symbolisation successfully.

#### Achievement outcomes

I have experienced and explored numbers in a variety of contexts and can express whole numbers in words, signs and symbols to convey size and order.

I can notice and read numbers in number-rich indoor and outdoor environments, experiencing cardinal, ordinal and nominal numbers.

I can use my experience of the counting sequence of numbers and of one-to-one correspondence to count sets reliably. I can count objects that I can touch, and ones that I cannot.

I can notice, read and write numbers in a range of media, through a multisensory approach, from zero to at least 20, ensuring they are correctly formed.

I have experienced the counting sequence of numbers in different ways, counting forwards and backwards, and starting at different points.

I have explored composing a quantity in different ways, using combinations of objects or quantities and using the corresponding mathematical language such as words like ‘more’, ‘less’, ‘most’ or ‘least’.

I can understand that a number can be partitioned in different ways.

I have experienced grouping and sharing with objects/quantities, and I can group or share small quantities into equal-sized groups.

I can use my visual sense of number to make estimates and comparisons. I can check estimates using counting or measuring.

I have used money in play and real-life situations and understand that I need to exchange money for items. I can use the language of money.

Algebra is the study of structures abstracted from computations and relations, and provides a way to make generalisations. Algebraic thinking moves away from context to structure and relationships. This powerful approach provides the means to abstract important features and to detect and express mathematical structures of situations in order to solve problems. Algebra is a unifying thread running through the fabric of mathematics.

Algebraic thinking is essential for reasoning, modelling and solving problems in mathematics and in a wide range of real‑world contexts, including technology and finance. Making connections between arithmetic and algebra develops skills for abstract reasoning from an early age.

#### Achievement outcomes

I have explored patterns through a range of hands-on activities and by using a variety of concrete, visual and digital resources. I can recognise, copy and generalise patterns and sequences around me.

I have demonstrated an understanding of the concepts of equality and inequality, using objects.

I can talk about how sets of objects change when objects are added to and taken away from them.

Geometry involves playing with, manipulating, comparing, naming and classifying shapes and structures. The study of geometry encourages the development and use of conjecture, deductive reasoning and proof. Measurement allows the magnitude of spatial and abstract features to be quantified, using a variety of standard and non-standard units, and can support the development of numerical reasoning.

Reasoning about the sizes and properties of shapes and their surrounding spaces helps us to make sense of the physical world and the world of mathematical shapes. Geometry and measurement have applications in many fields, including art, construction, science and technology, engineering, and astronomy.

#### Achievement outcomes

I can understand and apply the language of time in relation to my daily life and in relation to events that happen around me, including naming the days of the week, the months of the year and the seasons, all in meaningful contexts.

I have used a variety of objects to measure lengths, masses and capacities, and I can understand the need to repeat the same physical unit without any gaps when measuring.

I can make estimates and comparisons with measures. I can use language and non-standard units to discuss my sense of size.

I have explored shapes through investigative play, and I can categorise and sort shapes using their properties. I can use the language of shapes to describe objects.

I have explored movements and directions, both physically and by using digital technology, and I can use mathematical language to describe position.

Statistics is the practice of collecting, manipulating and analysing data, allowing representation and generalisation of information. Probability is the mathematical study of chance, enabling predictions of the likelihood of events occurring. Statistics and probability rely on the application and manipulation of number and algebra.

Managing data and representing information effectively provides the means to test hypotheses, draw conclusions and make predictions. Reasoning with statistics and probability, and evaluating their reliability, develops critical thinking and analytical skills that are fundamental to making ethical and informed decisions.

#### Achievement outcomes

I have investigated and collected data found in my environment. I can record data, giving meaning to marks which represent number and where each quantity corresponds to something in the real world, keeping simple records, which include tallying and pictograms.

I have grouped sets of objects or pictures into categories, through practical experiences, and I can classify by one criterion or more. I can talk about the rule(s) I have used and can reclassify according to new criteria.

I can present data using non-digital and digital methods. I have created simple charts and graphs and I have talked about what they mean.

Numbers are the symbol system for describing and comparing quantities. This is the first abstract concept that learners meet in mathematics, and it helps to establish the principles of logical reasoning. In mathematics the number system provides a basis for algebraic, statistical, probabilistic and geometrical reasoning, as well as for financial calculation and decision-making.

Knowledge of, and competence in, number and quantities are fundamental to confident participation in the world, and provide a foundation for further study and for employment. Computational fluency is essential for problem-solving and progressing in all areas of learning and experience. Fluency is developed through using the four basic arithmetic operations and acquiring an understanding of the relationship between them. This leads to preparing the way for using algebraic symbolisation successfully.

- The connections between arithmetic and algebra develop tools and skills for abstract reasoning from an early age.
- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

- Number is used throughout geometry to quantify shape, size and movement.
- Measure is an aspect of geometrical thinking which is highly connected to number and much of the development of understanding of number can emerge through increasingly sophisticated measuring.
- Geometric thinking involves reasoning with proportion, which connects with development in number work.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

- Probability and statistics are described and manipulated by using number; they are represented using number.
- Probability is expressed through number in various ways, e.g. through the use of percentages, fractions and decimals, and the connections between the representations is necessary for effective expression of probability.
- Statistics involves manipulation and representation of data, which involves numerical thinking.

### Expressive Arts

- Scale, proportion and ratio.
- Fractions in music, rhythm.
- Songs and rhymes.

### Health and Well-being

- Understanding of estimation and rounding.
- Application to making real-life decisions including financial ones.
- Fractions, percentages and proportions, e.g. balanced diet.

### Humanities

- Ratio and scale.
- Finance.
- Rounding.
- Ordering.

### Languages, Literacy and Communication

- Songs and rhymes.

### Science and Technology

- Rounding and estimating.
- Laws of indices.
- Reading, writing and calculating in standard index form.
- Direct and inverse proportion.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- explore numbers and quantities through multisensory experiences both indoors and outdoors
- learn that the number system, and whole numbers in particular, allow us to convey size and quantity
- represent and communicate with whole numbers
- begin to understand that money is needed to pay for things
- use a variety of representations to explore number, including objects, visual representations and, where appropriate, digital representations.

Algebra is the study of structures abstracted from computations and relations, and provides a way to make generalisations. Algebraic thinking moves away from context to structure and relationships. This powerful approach provides the means to abstract important features and to detect and express mathematical structures of situations in order to solve problems. Algebra is a unifying thread running through the fabric of mathematics.

Algebraic thinking is essential for reasoning, modelling and solving problems in mathematics and in a wide range of real‑world contexts, including technology and finance. Making connections between arithmetic and algebra develops skills for abstract reasoning from an early age.

The number system is used to represent and compare relationships between numbers and quantities.

- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

- Algebra and geometry are connected principally through the expression of shape, measure and movement through algebraic expressions, equations and formulae.
- An equation is an algebraic concept, which, when graphed, becomes a geometric concept. The variables within the equation refer to geometric concepts.
- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.
- Algebraic formulae and equations are used to connect the geometric concepts of triangles with measures of angles and sides.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In probability, algebra allows us to use the concept of a variable and we can apply this in probability by using a random variable; this is a parameter or event with a random outcome.
- In statistics, general formulae are written using algebra.
- Algebra and statistical analysis are interlinked; use of algorithms and formulae to calculate further statistical measures to analyse data.

### Expressive Arts

- Sequences and patterns.

### Science and Technology

- Concept of the variable.
- Equations and formulae.
- Direct and inverse proportion.
- Patterns and graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- explore, copy, repeat, continue and generate a range of simple patterns and sequences using a multisensory approach
- start to predict what will come next, through shared stories, pictures and rhymes that have a pattern element
- through active experience, demonstrate an understanding of equality and inequality, and how these can be preserved when numbers are changed, using the language of comparison and equality.

Geometry involves playing with, manipulating, comparing, naming and classifying shapes and structures. The study of geometry encourages the development and use of conjecture, deductive reasoning and proof. Measurement allows the magnitude of spatial and abstract features to be quantified, using a variety of standard and non-standard units, and can support the development of numerical reasoning.

Reasoning about the sizes and properties of shapes and their surrounding spaces helps us to make sense of the physical world and the world of mathematical shapes. Geometry and measurement have applications in many fields, including art, construction, science and technology, engineering, and astronomy.

The number system is used to represent and compare relationships between numbers and quantities.

- Number is used throughout geometry to quantify shape, size and movement.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

- Algebra and geometry are connected principally through the expression of shape, measure and movement through algebraic expressions, equations and formulae
- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphs are used to convert from one value to another, and to represent statistical information including spread, quantity and central tendency.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Expressive Arts

- Shape symmetry in movement and artwork.
- Scale.
- Use of appropriate equipment to measure accurately.

### Health and Well-being

- Use of appropriate equipment to measure accurately.

### Humanities

- Use of appropriate equipment to measure accurately.
- Scale.
- Time and chronological ordering.

### Languages, Literacy and Communication

- Use of prepositions to describe the location of selves and objects.

### Science and Technology

- Use of appropriate equipment to measure accurately.
- Units – use of appropriate unit, converting between units, and links between units and formulae.
- Scale.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through exploring situations in their everyday life, develop an understanding of measurement, using non‑standard units with developing accuracy
- through play and discovery, categorise and recognise shapes
- use the language of direction, location and position.

Statistics is the practice of collecting, manipulating and analysing data, allowing representation and generalisation of information. Probability is the mathematical study of chance, enabling predictions of the likelihood of events occurring. Statistics and probability rely on the application and manipulation of number and algebra.

Managing data and representing information effectively provides the means to test hypotheses, draw conclusions and make predictions. Reasoning with statistics and probability, and evaluating their reliability, develops critical thinking and analytical skills that are fundamental to making ethical and informed decisions.

The number system is used to represent and compare relationships between numbers and quantities.

- Statistics involves manipulation and representation of data, which involve numerical thinking.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In statistics, general formulae are written using algebra.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Health and Well-being

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Humanities

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Science and Technology

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through play and discovery in stimulating and relevant local contexts, collect and sort a variety of simple data and answer questions
- use appropriate digital and non-digital methods to represent and interpret data
- talk about their results.

**All our children and young people will be:****ambitious, capable learners who:**- set themselves high standards and seek and enjoy challenge
- are questioning and enjoy solving problems
- can communicate effectively in different forms and settings, using both Welsh and English
- can explain the ideas and concepts they are learning about
- use digital technologies creatively to communicate, find and analyse information
- undertake research and evaluate critically what they find

and

**are ready to learn throughout their lives****enterprising, creative contributors who:**- connect and apply their knowledge and skills to create ideas and products
- think creatively to reframe and solve problems
- identify and grasp opportunities
- take measured risks
- lead and play different roles in teams effectively and responsibly
- express ideas and emotions through different media
- give of their energy and skills so that other people will benefit

and

**are ready to play a full part in life and work****ethical, informed citizens who:**- find, evaluate and use evidence in forming views
- engage with contemporary issues based upon their knowledge and values
- understand and exercise their human and democratic responsibilities and rights
- understand and consider the impact of their actions when making choices and acting
- are knowledgeable about their culture, community, society and the world, now and in the past
- respect the needs and rights of others, as a member of a diverse society
- show their commitment to the sustainability of the planet

and

**are ready to be citizens of Wales and the world****healthy, confident individuals who:**- have secure values and are establishing their spiritual and ethical beliefs
- are building their mental and emotional well-being by developing confidence, resilience and empathy
- know how to find the information and support to keep safe and well
- take part in physical activity
- take measured decisions about lifestyle and manage risk
- have the confidence to participate in performance
- form positive relationships based upon trust and mutual respect
- face and overcome challenge
- have the skills and knowledge to manage everyday life as independently as they can

and

**are ready to lead fulfilling lives as valued members of society.****Conceptual understanding:**Mathematical concepts and ideas should be dwelt on, built on, and connected together as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding by explaining and expressing concepts, finding examples (or non-examples) and by representing a concept in a variety of ways, including verbal, concrete, visual, digital and abstract representations.**Communication with symbols:**Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility in their application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.**Strategic competence:**(i.e. formulating problems mathematically in order to solve them) Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to be able to solve them.**Logical reasoning:**As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal, visual or concrete explanations to representations involving symbols and conventions.**Fluency:**As learners experience, understand and apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow. As a result, facts, relationships and techniques learned previously should become firmly established, memorable and usable.#### Achievement outcomes

I can read, write and interpret numbers, using figures and words up to at least 1,000.

I can order and sequence numbers, including odd and even numbers, in number-rich indoor and outdoor environments, and can count on and back in uniform steps of any size.

I have experienced numbers below zero in practical contexts, including temperature measurements.

I have explored composing numbers in a variety of ways and I can understand that number value can be determined by the position of the digits used.

I have explored additive relationships using a range of representations. I can use my understanding of additive relationships to add and subtract whole numbers, using a variety of written and mental methods. I can estimate and check the accuracy of my answers using inverse operations when appropriate.

I have demonstrated an understanding of, and can recall, multiplication tables including 2, 3, 4, 5 and 10, and use the term ‘multiples’.

I have explored multiplicative relationships, using a range of representations (including sharing/grouping and the array), and I can use my understanding of multiplicative relationships to multiply and divide whole numbers, using a variety of written and mental methods.

I have engaged in practical tasks and real-life problems to estimate and round numbers to the nearest 10 and 100.

I can reflect on how reasonable answers are in the light of estimations and I have verified calculations using inverse operations.

I have demonstrated an understanding that unit fractions represent equal parts of a whole and are a way of conveying quantities and relationships.

I have experienced fractions in practical situations, using a variety of representations.

I have explored equivalent fractions and understand equivalent fraction relationships.

I have demonstrated an understanding of when to count, when to measure and when to calculate to find quantities.

I can explain that money and ways to save and pay come in different forms. I can make sensible financial transactions in familiar role-play scenarios, including making informed decisions and choices about spending and saving.

#### Achievement outcomes

I have explored patterns of numbers and shape using concrete, paper-based and digital materials. I can recognise, copy and generate sequences of numbers and visual patterns.

I have demonstrated an understanding of the concepts of equality and inequality within a number equation. I can use the equals sign to indicate that both sides of a number sentence have the same value and I can use inequality signs when comparing quantities.

I have explored commutativity with addition and multiplication and I can recognise when two different numerical expressions describe the same situation but are written in different ways.

#### Achievement outcomes

I can tell the time on an analogue clock (in at least 15-minute intervals) and I can connect this to time displayed digitally. I have explored and used different ways of showing the passing of time, including calendars, timelines, simple timetables and schedules.

I have estimated and measured length, volume, capacity, mass, temperature and time in practical situations, using non-standard units.

I have used a variety of measuring devices from different starting points.

I can apply standard units in practical situations to measure length, volume, capacity, mass, temperature and time accurately.

I have explored and named two-dimensional and three-dimensional shapes in a range of contexts. I can sort and categorise regular and irregular two-dimensional shapes in different ways, according to their properties. I can sketch two-dimensional shapes and make models of three-dimensional objects.

I can identify reflective symmetry in a range of contexts and can identify symmetry in two-dimensional shapes.

I have explored the concept of rotation, both physically and using digital technology, and can use simple fractions of a complete rotation to describe turns.

I have described and quantified the position of objects relative to other objects through active learning experiences, using appropriate language, including the words ‘left’ and ‘right’.

I can follow and create instructions related to movement, using a range of approaches and resources (including digital technologies) to demonstrate my understanding.

#### Achievement outcomes

I can collect and organise data for posing and answering questions in relevant situations.

I can sort and classify using more than one criterion, using Venn diagrams and Carroll diagrams in practical situations.

I have used digital and non-digital methods to record and present data in a variety of ways, including the use of tally charts, frequency tables, and block graphs, when appropriate axes and scales are provided.

I can interpret and analyse graphs, charts and data.

I can explain what I have found out, justify my reasoning and I can evaluate how well my method worked.

- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

- Number is used throughout geometry to quantify shape, size and movement.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

- Statistics involves manipulation and representation of data, which involves numerical thinking.

### Expressive Arts

- Scale, proportion and ratio.
- Fractions in music, rhythm.
- Songs and rhymes.

### Health and Well-being

- Understanding of estimation and rounding.
- Application to making real-life decisions including financial ones.
- Fractions, percentages and proportions, e.g. balanced diet.

### Humanities

- Ratio and scale.
- Finance.
- Rounding.
- Ordering.

### Languages, Literacy and Communication

- Songs and rhymes.

### Science and Technology

- Rounding and estimating.
- Laws of indices.
- Reading, writing and calculating in standard index form.
- Direct and inverse proportion.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- extend their understanding of the number system, through experiential learning, to include large numbers, zero, negative numbers and fractions
- explore composition and equivalence of number, and learn about place value
- learn about relationships in numbers
- understand the additive relationship and be introduced to the multiplicative relationship, including using the array
- use their understanding of the equivalence and value of coins and notes to make appropriate transactions.

The number system is used to represent and compare relationships between numbers and quantities.

- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

- Algebra and geometry are connected principally through the expression of shape, measure and movement through algebraic expressions, equations and formulae.
- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In statistics, general formulae are written using algebra.

### Expressive Arts

- Sequences and patterns.

### Science and Technology

- Concept of the variable.
- Equations and formulae.
- Direct and inverse proportion.
- Patterns and graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- express and use, through experiential learning, general laws and rules of arithmetic
- explore and describe sequences of whole numbers with a common difference between terms
- find missing terms and continue sequences
- understand that Robert Recorde’s equals sign is placed between two or more expressions that have the same value
- recognise commutativity of addition and multiplication, using objects, diagrams and numbers
- reason logically, using equality and the laws of arithmetic.

The number system is used to represent and compare relationships between numbers and quantities.

- Number is used throughout geometry to quantify shape, size and movement.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

- Algebra and geometry are connected principally through the expression of shape, measure and movement through algebraic expressions, equations and formulae
- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Expressive Arts

- Shape symmetry in movement and artwork.
- Scale.
- Use of appropriate equipment to measure accurately.

### Health and Well-being

- Use of appropriate equipment to measure accurately.

### Humanities

- Use of appropriate equipment to measure accurately.
- Scale.
- Time and chronological ordering.

### Languages, Literacy and Communication

- Use of prepositions to describe the location of selves and objects.

### Science and Technology

- Use of appropriate equipment to measure accurately.
- Units – use of appropriate unit, converting between units, and links between units and formulae.
- Scale.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through experiential learning and solving problems in real-life and more abstract contexts, develop an understanding of standard measurement and of the properties of shapes
- select and use appropriate equipment and units to measure accurately
- describe location, movement and position, using mathematical terminology.

The number system is used to represent and compare relationships between numbers and quantities.

- Statistics involves manipulation and representation of data, which involve numerical thinking.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In statistics, general formulae are written using algebra.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Health and Well-being

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Humanities

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Science and Technology

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through experiential learning in stimulating and relevant local contexts, use a sequential data-handling cycle to pose questions, then collect and sort a variety of purposeful data to answer questions
- use appropriate digital and non-digital methods to analyse, represent and interpret data
- explain their results and evaluate their methods.

**All our children and young people will be:****ambitious, capable learners who:**- set themselves high standards and seek and enjoy challenge
- are questioning and enjoy solving problems
- can communicate effectively in different forms and settings, using both Welsh and English
- can explain the ideas and concepts they are learning about
- use digital technologies creatively to communicate, find and analyse information
- undertake research and evaluate critically what they find

and

**are ready to learn throughout their lives****enterprising, creative contributors who:**- connect and apply their knowledge and skills to create ideas and products
- think creatively to reframe and solve problems
- identify and grasp opportunities
- take measured risks
- lead and play different roles in teams effectively and responsibly
- express ideas and emotions through different media
- give of their energy and skills so that other people will benefit

and

**are ready to play a full part in life and work****ethical, informed citizens who:**- find, evaluate and use evidence in forming views
- engage with contemporary issues based upon their knowledge and values
- understand and exercise their human and democratic responsibilities and rights
- understand and consider the impact of their actions when making choices and acting
- are knowledgeable about their culture, community, society and the world, now and in the past
- respect the needs and rights of others, as a member of a diverse society
- show their commitment to the sustainability of the planet

and

**are ready to be citizens of Wales and the world****healthy, confident individuals who:**- have secure values and are establishing their spiritual and ethical beliefs
- are building their mental and emotional well-being by developing confidence, resilience and empathy
- know how to find the information and support to keep safe and well
- take part in physical activity
- take measured decisions about lifestyle and manage risk
- have the confidence to participate in performance
- form positive relationships based upon trust and mutual respect
- face and overcome challenge
- have the skills and knowledge to manage everyday life as independently as they can

and

**are ready to lead fulfilling lives as valued members of society.****Conceptual understanding:**Mathematical concepts and ideas should be dwelt on, built on, and connected together as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding by explaining and expressing concepts, finding examples (or non-examples) and by representing a concept in a variety of ways, including verbal, concrete, visual, digital and abstract representations.**Communication with symbols:**Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility in their application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.**Strategic competence:**(i.e. formulating problems mathematically in order to solve them) Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to be able to solve them.**Logical reasoning:**As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal, visual or concrete explanations to representations involving symbols and conventions.**Fluency:**As learners experience, understand and apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow. As a result, facts, relationships and techniques learned previously should become firmly established, memorable and usable.#### Achievement outcomes

I have used a range of practical equipment to develop and secure my understanding of place value for positive and negative integers. I can read, write and interpret numbers, using figures and words up to at least one million.

I have extended my understanding of the number system, through a range of activities using non-digital and digital manipulatives, to include decimals and fractions, and I can confidently place whole numbers and fractional quantities on a number line. I have demonstrated my understanding that a fraction can be used as an operator, or to represent division. I can use place value for non‑integers.

I have demonstrated my understanding that fractions (including improper fractions) as well as mixed numbers, decimals and percentages, provide different ways of representing non-integer quantities.

I have explored number patterns, connections and calculations with manipulatives and digital technology, and can demonstrate my understanding of number facts and relationships. I have used my knowledge of number facts and relationships to solve problems in mathematical and real-life contexts.

I have developed, used and discussed efficient and accurate methods when applying all four arithmetic operations to integers and decimals. I can combine these operations, in both mathematical and real-life contexts of problem-solving.

I can verify calculations and statements about number by inverse reasoning and approximation methods.

I have explored the meaning of negative numbers in meaningful and authentic contexts. I can compare the sizes of negative numbers and I can calculate the differences between any two integers. I can check my answers.

I can fluently recall times tables up to at least 10 x 10 and demonstrate my understanding by using them appropriately in applications and in mental and written arithmetic.

I have demonstrated an understanding of rounding and can solve problems requiring rounding to the nearest unit, 10, 100 and 1,000.

I have demonstrated an understanding of the equivalence of simple fractions, decimals and percentages, and I can convert between representations. I can use my knowledge of equivalence to compare the sizes of fractions. I understand the inverse relation between the denominator of a fraction and its value.

I can use my knowledge of multiplication, division, fractions and percentages to calculate proportions of a number or quantity, and to divide a number or quantity in a given ratio. I have solved problems involving ratio and proportion in real-life contexts, and I have used my knowledge of estimation and rounding to predict and check my answers.

I have demonstrated an understanding of income and expenditure, and I can calculate profit and loss. I have created and evaluated budgets for activities and events.

#### Achievement outcomes

I can recognise equalities, inequalities and equivalence of expressions, and also when commutativity, distributivity and associativity can be used to state a simple expression in a different way.

I have explored patterns of numbers and shapes, using digital and non-digital methods. I can interpret, explain in words and generalise numerical sequences and spatial patterns.

I can create equations to model problems, using symbols or words to represent unknown values. I can use inverse operations to find unknown values in simple equations using mental, written and digital methods, and manipulatives. I can check my answers.

I have explored the notion of function, including the use of digital function machines.

I have demonstrated an understanding of the idea of input, application of a rule (including inverse operations) and output, using a function machine or other appropriate methods.

#### Achievement outcomes

I can read analogue and digital clocks accurately and I can make calculations involving the passing of time. I have used timetables and schedules to make calculations involving time.

I have estimated and measured length, capacity, mass, temperature and time, using appropriate standard units.

I can convert between metric units in mathematical and real-life problem-solving contexts, and I can check my answers to make sure they make sense.

I have explored properties of two-dimensional shapes, using concrete, paper-based and digital resources, and I can use mathematical language to describe and compare the properties (including number of sides and symmetry) of given shapes, and to classify them accordingly. I can name different types of triangles.

I have explored vertices, edges and faces of three-dimensional shapes and I can use these characteristics to describe a three-dimensional shape. I have explored the relationship between a three-dimensional shape and two-dimensional nets, using concrete, paper-based and digital resources, and I can recognise nets of common three-dimensional shapes.

I have used a range of hands-on activities to explore the perimeter and area of shapes. I have derived the respective formulae for finding the area of a rectangle and of a right-angled triangle, and have applied them in mathematical and real-life problem-solving contexts, using estimation to predict and check my answers.

I can demonstrate my understanding of angle as a measure of rotation and I can recognise, name and describe types of angles.

I have developed an understanding of why we need co-ordinates and I can use them to locate and plot points in the first quadrant of the Cartesian plane. I can use my knowledge of coordinates to solve problems involving shape, length, angle and position in mathematical and real-life contexts.

#### Achievement outcomes

I have used a sequential data-handling cycle to ask and answer appropriate questions in meaningful situations.

I can pose and answer sensible questions and I have demonstrated an understanding of the importance of collecting relevant data that can be used to answer my questions.

I have demonstrated an understanding of the kinds of data I need, including discrete and continuous data.

I can find and use the mean of a simple set of data in meaningful contexts.

I have used appropriate methods to answer my questions by collecting, analysing and summarising my data and interpreting my results. I can evaluate my methods and suggest different or better ways to approach investigations in the future.

I have represented information by creating a variety of appropriate charts of increasing complexity, including tally charts, frequency tables, bar graphs, line graphs, with and without the use of digital technologies. I have created pie charts using digital technology.

I can use different scales on axes to extract and interpret information from a range of diagrams, tables (including databases) and graphs, including pie charts with simple fractions and proportions.

I have investigated simple statistics, presented in the media and elsewhere, to support an argument and I can explain how the statistics do, or do not, support the argument.

I can recognise validity and trends, and can discuss how anomalies may affect conclusions when evaluating results.

I have explored the possibility of given outcomes and have used the language of probability to describe the chance of an event occurring.

I have played games that involve flipping coins, rolling dice and using spinners in order to simulate, and discuss chance.

I have hypothesised and anticipated outcomes of chance experiments in a range of contexts, recording my findings in a systematic and appropriate way.

- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

- Number is used throughout geometry to quantify shape, size and movement.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

- Statistics involves manipulation and representation of data, which involves numerical thinking.

### Expressive Arts

- Scale, proportion and ratio.
- Fractions in music, rhythm.
- Songs and rhymes.

### Health and Well-being

- Understanding of estimation and rounding.
- Application to making real-life decisions including financial ones.
- Fractions, percentages and proportions, e.g. balanced diet.

### Humanities

- Ratio and scale.
- Finance.
- Rounding.
- Ordering.

### Languages, Literacy and Communication

- Songs and rhymes.

### Science and Technology

- Rounding and estimating.
- Laws of indices.
- Reading, writing and calculating in standard index form.
- Direct and inverse proportion.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- extend their understanding and use of the number system, through a broad range of experiences, to include negative numbers, decimals and fractions
- further develop understanding of place value
- explore the properties of numbers, including factors, multiples, primes, and the inverse relationship between squares and square roots
- become increasingly confident in using all four arithmetic operations in their calculations with whole numbers and decimals, and combine these, using distributive, associative and commutative laws where appropriate
- create and evaluate enterprise projects linked to their immediate and local environment.

The number system is used to represent and compare relationships between numbers and quantities.

- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In statistics, general formulae are written using algebra.

### Expressive Arts

- Sequences and patterns.

### Science and Technology

- Concept of the variable.
- Equations and formulae.
- Direct and inverse proportion.
- Patterns and graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences, describe relations which depend on equality, equivalence and the commutative, associative and distributive laws
- describe and generate, in words, visual patterns and numerical sequences
- model simple problem situations, using words and symbols to create equations from which they can find an unknown value
- understand that a function can transform a set of numbers to a new set of numbers, according to a rule
- move between concrete, visual and abstract representations throughout their mathematical work.

The number system is used to represent and compare relationships between numbers and quantities.

- Number is used throughout geometry to quantify shape, size and movement.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Expressive Arts

- Shape symmetry in movement and artwork.
- Scale.
- Use of appropriate equipment to measure accurately.

### Health and Well-being

- Use of appropriate equipment to measure accurately.

### Humanities

- Use of appropriate equipment to measure accurately.
- Scale.
- Time and chronological ordering.

### Languages, Literacy and Communication

- Use of prepositions to describe the location of selves and objects.

### Science and Technology

- Use of appropriate equipment to measure accurately.
- Units – use of appropriate unit, converting between units, and links between units and formulae.
- Scale.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences and through solving problems in real-life and abstract contexts, refine their understanding of standard measurement and the relationship between units of measure
- select and use appropriate equipment and units to measure accurately
- refine their understanding of the properties of shapes
- use the mathematical concepts of angle, co-ordinates and distance to describe location, movement and position.

The number system is used to represent and compare relationships between numbers and quantities.

- Statistics involves manipulation and representation of data, which involve numerical thinking.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In statistics, general formulae are written using algebra.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Health and Well-being

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Humanities

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Science and Technology

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences in stimulating and relevant local and global contexts, use a sequential data-handling cycle to pose questions, collect and sort a variety of purposeful data
- use appropriate digital and non-digital methods to analyse, summarise, represent and interpret data
- through using different approaches, compare, check and evaluate their results, explaining what they have found
- experiment with chance.

**All our children and young people will be:****ambitious, capable learners who:**- set themselves high standards and seek and enjoy challenge
- are questioning and enjoy solving problems
- can communicate effectively in different forms and settings, using both Welsh and English
- can explain the ideas and concepts they are learning about
- use digital technologies creatively to communicate, find and analyse information
- undertake research and evaluate critically what they find

and

**are ready to learn throughout their lives****enterprising, creative contributors who:**- connect and apply their knowledge and skills to create ideas and products
- think creatively to reframe and solve problems
- identify and grasp opportunities
- take measured risks
- lead and play different roles in teams effectively and responsibly
- express ideas and emotions through different media
- give of their energy and skills so that other people will benefit

and

**are ready to play a full part in life and work****ethical, informed citizens who:**- find, evaluate and use evidence in forming views
- engage with contemporary issues based upon their knowledge and values
- understand and exercise their human and democratic responsibilities and rights
- understand and consider the impact of their actions when making choices and acting
- are knowledgeable about their culture, community, society and the world, now and in the past
- respect the needs and rights of others, as a member of a diverse society
- show their commitment to the sustainability of the planet

and

**are ready to be citizens of Wales and the world****healthy, confident individuals who:**- have secure values and are establishing their spiritual and ethical beliefs
- are building their mental and emotional well-being by developing confidence, resilience and empathy
- know how to find the information and support to keep safe and well
- take part in physical activity
- take measured decisions about lifestyle and manage risk
- have the confidence to participate in performance
- form positive relationships based upon trust and mutual respect
- face and overcome challenge
- have the skills and knowledge to manage everyday life as independently as they can

and

**are ready to lead fulfilling lives as valued members of society.****Conceptual understanding:**Mathematical concepts and ideas should be dwelt on, built on, and connected together as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding by explaining and expressing concepts, finding examples (or non-examples) and by representing a concept in a variety of ways, including verbal, concrete, visual, digital and abstract representations.**Communication with symbols:**Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility in their application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.**Strategic competence:**(i.e. formulating problems mathematically in order to solve them) Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to be able to solve them.**Logical reasoning:**As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal, visual or concrete explanations to representations involving symbols and conventions.**Fluency:**As learners experience, understand and apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow. As a result, facts, relationships and techniques learned previously should become firmly established, memorable and usable.#### Achievement outcomes

I can fluently and accurately apply the four arithmetic operations, in the correct order, on integers, decimals and fractions, using written, mental and digital methods. I can use my sense of number to predict and check my answers.

I have demonstrated an understanding of the equivalence of fractions, decimals and percentages and I can convert fluently between the different forms, using both written methods and a calculator. I have used my knowledge of percentages and ratio to solve problems that involve simple interest, compound interest, depreciation, and calculating bills and budgets that include basic taxation on goods and services.

I have derived and applied the rules of indices, with the exclusion of fractional indices, to calculate values and solve problems.

I have consolidated my understanding of reciprocals when dividing fractions.

I can use standard index form to represent small and large numbers and to perform calculations in appropriate real-life and mathematical contexts.

I can solve problems requiring rounding or significant figures at various stages of the calculation and give the answer, using both written and digital methods, and I can interpret the calculator outputs.

I have demonstrated an understanding of ratio and proportion and can solve numerical problems that involve direct and inverse proportion, including expressing one quantity as a proportion of another, proportional change and problems that involve foreign currencies and exchange rates.

I can use my knowledge of number to predict and check my answers.

I can justify choices based on value for money, personal well-being and global impact.

#### Achievement outcomes

I have demonstrated an understanding of the concept of a variable and I have used letters to represent variables in forming linear algebraic expressions.

I can manipulate algebraic expressions fluently by simplifying, expanding and factorising by extracting a common factor. I can also substitute values and change the subject of a formula when the subject appears in one term.

I can distinguish between algebraic expressions, equations and inequalities. I have used a variety of methods, including trial and improvement where appropriate, to solve equations and inequalities in the first degree, which may include brackets and unknowns on one or both sides. I can check my answers using substitution.

I have used equations and inequalities in the first degree to represent and model real-life situations and solve problems. I can interpret my answers and check that they make sense in context.

I can recognise linear sequences and can generalise them using algebra. I can find, describe and use the

*n*th term. I can describe and use the term-to-term rule for simple iterative sequences. I can apply my knowledge of sequences to solve real-life and mathematical problems.I can recognise, draw, sketch and interpret linear graphs, and investigate graphs, using written and digital methods. I can demonstrate an understanding of each of the terms in the equation of a straight line. I can explore the effect on the line when the constant or coefficient is changed.

#### Achievement outcomes

I can represent and use compound measures, using standard units, and I can demonstrate an understanding of the relationship between a formula representing a measurement and the units used.

I can create and use conversion graphs to solve problems set in local and global contexts.

I have explored symmetries and other properties of regular and irregular two-dimensional and three-dimensional shapes. I can construct two-dimensional representations of three-dimensional shapes, in order to investigate properties further. I can classify two-dimensional and three-dimensional shapes according to their mathematical properties.

I have explored all four transformations of two-dimensional shapes, using a variety of approaches, including digital technology. I can use my understanding to predict and describe how shapes will change under a given transformation.

I can use co-ordinates to plot points in the four quadrants and deduce the location of additional points.

I can use a protractor to measure and draw angles. I have modelled and solved problems involving bearings. I can use reasoning to calculate the size of angles in triangles and quadrilaterals. I have explored angles formed by parallel lines and by a transversal, and I can use my understanding to calculate angles in these contexts.

I have calculated the areas or surface areas of two-dimensional and three-dimensional simple and compound shapes, including circles, and have demonstrated an understanding of pi (π) as the ratio of the circumference of a circle to its diameter. I have derived the formulae for the volume of simple three-dimensional prisms and I can calculate the volumes of three-dimensional shapes to solve problems.

I have demonstrated an understanding of the relationship between right-angled triangles and squares in the context of Pythagoras’ theorem, and I have used it to solve problems in mathematical and real-life contexts.

#### Achievement outcomes

I have used a sequential data-handling cycle to ask and answer appropriate questions.

I can choose a sensible hypothesis to investigate and I have demonstrated an understanding of the methods I can use to collect appropriate data.

I can design and have critiqued questionnaires to ensure that the data to be collected will enable the sequential data-handling cycle to be used.

I can calculate the mean, median, mode and range of a set of data and I have made comparisons between small sets of data using summary statistics.

I can select and justify an appropriate way to use my data to investigate my hypothesis. I have explored different ways to understand and summarise my data, including using averages to make comparisons between large data sets, with grouped frequency distributions for discrete and continuous data. I can use a scatter diagram to analyse two sets of variables and investigate correlation between them. I can make predictions and identify trends and anomalies in data sets.

I have presented my data in the form of appropriate graphs, charts and tables, and I have used digital technologies, taking into account the purpose of the data and nature of the audience. I have discussed the relative advantages and disadvantages of each presentation method, and I can justify my choice of method.

I have used my data to draw conclusions about my hypotheses and I have communicated my findings clearly. I can critique my own methods and findings, and consider what I may have done differently or better at each stage of the sequential data-handling cycle.

I have critically analysed statistics published in the media and elsewhere to consider what it means and how it does, or does not, support any findings reported. I can pose relevant questions to check the credibility of the findings.

I can explain randomness, and I have investigated chance by modelling and by comparing theoretical and experimental probabilities.

I have explored all the possible mutually exclusive outcomes of successive and combined events. I can work systematically making use of lists and sample space diagrams, and I have shown an understanding that the sum of probabilities of all mutually exclusive outcomes is 1. I have demonstrated an understanding of when it is appropriate to add or multiply probabilities.

I can make meaningful real-life judgements based on outcomes of experimental data and risk.

- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

- Number is used throughout geometry to quantify shape, size and movement.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

- Statistics involves manipulation and representation of data, which involves numerical thinking.

### Expressive Arts

- Scale, proportion and ratio.
- Fractions in music, rhythm.
- Songs and rhymes.

### Health and Well-being

- Understanding of estimation and rounding.
- Application to making real-life decisions including financial ones.
- Fractions, percentages and proportions, e.g. balanced diet.

### Humanities

- Ratio and scale.
- Finance.
- Rounding.
- Ordering.

### Languages, Literacy and Communication

- Songs and rhymes.

### Science and Technology

- Rounding and estimating.
- Laws of indices.
- Reading, writing and calculating in standard index form.
- Direct and inverse proportion.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- become increasingly fluent in their use of number in calculations, through a broad range of experiences, in order to interpret mathematical statements and describe quantities, both with and without the use of calculators
- further develop their understanding of equivalence, appreciating that any number may be represented in many ways
- use proportional reasoning to compare two quantities using multiplicative thinking, and then apply this to a new situation
- deepen their understanding of using and comparing very big and very small numbers
- select and use efficient mental, written and digital methods to perform calculations
- extend their understanding of finance to personal, local and global contexts.

The number system is used to represent and compare relationships between numbers and quantities.

- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In statistics, general formulae are written using algebra.

### Expressive Arts

- Sequences and patterns.

### Science and Technology

- Concept of the variable.
- Equations and formulae.
- Direct and inverse proportion.
- Patterns and graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences, appreciate the power of mathematics to express relationships concisely and in forms that are universally understood
- express the laws of arithmetic using algebraic notation, developing their conceptual understanding of a variable
- explore, generate, identify and represent linear patterns in a variety of contexts
- model real-life situations, using equations and inequalities to solve problems, checking the reasonableness of their solutions
- explore equations graphically, using digital technologies
- explore numerical and physical sequences, using written and digital methods
- investigate linear graphs in realistic situations.

The number system is used to represent and compare relationships between numbers and quantities.

- Number is used throughout geometry to quantify shape, size and movement.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Expressive Arts

- Shape symmetry in movement and artwork.
- Scale.
- Use of appropriate equipment to measure accurately.

### Health and Well-being

- Use of appropriate equipment to measure accurately.

### Humanities

- Use of appropriate equipment to measure accurately.
- Scale.
- Time and chronological ordering.

### Languages, Literacy and Communication

- Use of prepositions to describe the location of selves and objects.

### Science and Technology

- Use of appropriate equipment to measure accurately.
- Units – use of appropriate unit, converting between units, and links between units and formulae.
- Scale.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences and through solving problems within real-life and abstract contexts, develop their understanding of standard and compound measurement, the properties of shapes and solids, and location, movement and position
- select and use appropriate equipment and units to measure accurately
- use digital technologies to explore shape and space, developing and testing conjectures
- use angle and shape facts to deduce further features and relationships
- recognise pi (π) as the ratio of the circumference of a circle to its diameter and appreciate the significance of William Jones’s contribution.

The number system is used to represent and compare relationships between numbers and quantities.

- Statistics involves manipulation and representation of data, which involve numerical thinking.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In statistics, general formulae are written using algebra.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Health and Well-being

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Humanities

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Science and Technology

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- use appropriate digital and non-digital methods to analyse, summarise, represent and interpret data to draw conclusions
- check and evaluate their results and recognise anomalies and trends in data
- extend their understanding of chance by expressing theoretical probability numerically.

**All our children and young people will be:****ambitious, capable learners who:**- set themselves high standards and seek and enjoy challenge
- are questioning and enjoy solving problems
- can communicate effectively in different forms and settings, using both Welsh and English
- can explain the ideas and concepts they are learning about
- use digital technologies creatively to communicate, find and analyse information
- undertake research and evaluate critically what they find

and

**are ready to learn throughout their lives****enterprising, creative contributors who:**- connect and apply their knowledge and skills to create ideas and products
- think creatively to reframe and solve problems
- identify and grasp opportunities
- take measured risks
- lead and play different roles in teams effectively and responsibly
- express ideas and emotions through different media
- give of their energy and skills so that other people will benefit

and

**are ready to play a full part in life and work****ethical, informed citizens who:**- find, evaluate and use evidence in forming views
- engage with contemporary issues based upon their knowledge and values
- understand and exercise their human and democratic responsibilities and rights
- understand and consider the impact of their actions when making choices and acting
- are knowledgeable about their culture, community, society and the world, now and in the past
- respect the needs and rights of others, as a member of a diverse society
- show their commitment to the sustainability of the planet

and

**are ready to be citizens of Wales and the world****healthy, confident individuals who:**- have secure values and are establishing their spiritual and ethical beliefs
- are building their mental and emotional well-being by developing confidence, resilience and empathy
- know how to find the information and support to keep safe and well
- take part in physical activity
- take measured decisions about lifestyle and manage risk
- have the confidence to participate in performance
- form positive relationships based upon trust and mutual respect
- face and overcome challenge
- have the skills and knowledge to manage everyday life as independently as they can

and

**are ready to lead fulfilling lives as valued members of society.****Conceptual understanding:**Mathematical concepts and ideas should be dwelt on, built on, and connected together as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding by explaining and expressing concepts, finding examples (or non-examples) and by representing a concept in a variety of ways, including verbal, concrete, visual, digital and abstract representations.**Communication with symbols:**Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility in their application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.**Strategic competence:**(i.e. formulating problems mathematically in order to solve them) Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to be able to solve them.**Logical reasoning:**As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal, visual or concrete explanations to representations involving symbols and conventions.**Fluency:**As learners experience, understand and apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow. As a result, facts, relationships and techniques learned previously should become firmly established, memorable and usable.#### Achievement outcomes

I can recognise the difference between rational and irrational numbers, and apply all four arithmetic operations to them.

I have explored the relationship between powers, roots and fractional indices, and I can derive and apply the rules to simplify and decompose surds.

I have demonstrated fluency in moving between representations of numbers, including converting a given recurring decimal to a fraction.

I have demonstrated an understanding that measurements are not always accurate and are subject to tolerance and margins of error. I can solve problems involving upper and lower bounds, and justify the outcome. I can use my knowledge of tolerance when choosing the required degree of accuracy to make real-life calculations.

I can solve problems involving repeated and inverse proportional reasoning.

I can use my knowledge of annual equivalent rate (AER) and annual percentage rate (APR) to develop models to evaluate and compare financial products.

I can calculate income tax and understand the implications of taxation.

#### Achievement outcomes

I can fluently manipulate algebraic expressions by expanding double brackets, factorising quadratic expressions and changing the subject of a formula where the subject appears in more than one term. I can simplify and manipulate algebraic fractions.

I can solve a range of linear and higher-order equations, and inequalities, including simultaneous, quadratic and trigonometric equations, using numerical, graphical and algebraic methods where appropriate. I can then interpret the meaning of the answer, or answers, checking for reasonableness.

I have used equations and inequalities, and relevant graphs, to model and solve problems in real-life and mathematical contexts, including those which describe proportion and exponentiation, and I can use my knowledge of the real world and number sense to predict and check my interpretations of these.

I can recognise and generalise simple non-linear sequences using algebra.

I can understand the concept of an identity and can translate statements describing mathematical relationships into algebraic models, using expressions and equations.

I have investigated a variety of non-linear graphs (including quadratic, cubic and reciprocals), using written and digital methods. I can demonstrate an understanding of the effect of the coefficients, indices and constants on the shape of the graph. I can determine the gradient at a point and the area under a graph, and understand what these represent. I can use graphs to solve problems.

#### Achievement outcomes

I can explain why two or more shapes are similar, congruent, or neither. I have used my knowledge of congruency and similarity to solve problems involving angles and lengths, both in mathematical and real-life contexts.

I can calculate the perimeter, area or surface area and volume of compound two-dimensional and three‑dimensional shapes, and I can rearrange formulae to find missing lengths. I have demonstrated an understanding of the effect of scale when comparing measurements and shapes in all three dimensions, and I have used my knowledge of scale and ratio to calculate the lengths and areas of fractions of shapes, including arcs and segments of circles.

I have located and described the locus of points defined by a range of different criteria, using digital and non‑digital technologies.

I have demonstrated an understanding of trigonometric ratios in right-angled triangles and I have used, in mathematical and real-life contexts, my knowledge of the trigonometric ratios to solve problems involving lengths, angles and area of any triangle.

I have used reasoning and logical arguments, along with my knowledge of polygons, intersecting lines, angle and the circle theorems, to solve problems, deduce and calculate angles and lengths in diagrams that involve combinations of these.

#### Achievement outcomes

I have used a sequential data-handling cycle to ask and answer appropriate questions.

I have explored different sampling methods and I have demonstrated an understanding of the need to select a sample when collecting data. I can evaluate, choose and use different sampling techniques, including random sampling, stratified sampling and systematic sampling.

I have experimented with different approaches to presenting data, including cumulative frequency, box and whisker, and histograms, to interpret measures of central tendency and measures of spread. I can select appropriate approaches, when comparing data sets, justifying and evaluating my choices.

I have critically analysed statistics in the media, considering how data is presented, its reliability, and whether and how the data has been manipulated to tell a particular story. I can make informed decisions based on statistical evidence, identifying bias and anomalies.

I can solve problems involving probabilities of mutually exclusive, independent and dependent events in real-life and mathematical contexts. I can use a variety of strategies, including using Venn and tree diagrams, to solve problems in local and wider contexts.

I have played and created games to understand the relationship between relative frequency and theoretical probabilities, making judgements on outcomes of experimental data and risk.

I can use probabilistic arguments, drawing on theory, information, research and experimentation to support my conclusion.

- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

- Number is used throughout geometry to quantify shape, size and movement.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

- Statistics involves manipulation and representation of data, which involves numerical thinking.

### Expressive Arts

- Scale, proportion and ratio.
- Fractions in music, rhythm.
- Songs and rhymes.

### Health and Well-being

- Understanding of estimation and rounding.
- Application to making real-life decisions including financial ones.
- Fractions, percentages and proportions, e.g. balanced diet.

### Humanities

- Ratio and scale.
- Finance.
- Rounding.
- Ordering.

### Languages, Literacy and Communication

- Songs and rhymes.

### Science and Technology

- Rounding and estimating.
- Laws of indices.
- Reading, writing and calculating in standard index form.
- Direct and inverse proportion.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- become increasingly fluent in their use of number, through a broad range of experiences, in order to describe, interpret and communicate size, scale and comparisons, both within and beyond mathematics
- become increasingly fluent in their calculations, both with and without the use of calculators, and deepen their understanding of how to use rational and irrational numbers
- become critical consumers in broader financial contexts.

The number system is used to represent and compare relationships between numbers and quantities.

- There is a strong relationship between the algorithms of arithmetic and the laws of algebra.
- The order of operations and laws of arithmetic are followed in algebra.

- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In statistics, general formulae are written using algebra.

### Expressive Arts

- Sequences and patterns.

### Science and Technology

- Concept of the variable.
- Equations and formulae.
- Direct and inverse proportion.
- Patterns and graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences, create multiple representations of a mathematical concept
- explore connections between equations and expressions with geometric, numerical and graphical representations, and understand their equivalence
- explore, generate and identify non-linear patterns, and express them algebraically and graphically
- develop and apply their knowledge of a range of methods such as factorisation, simplification and inverse functions to change the subject of formulae involving two or more variables
- model real-life situations, identifying variables and constructing polynomials.

The number system is used to represent and compare relationships between numbers and quantities.

- Number is used throughout geometry to quantify shape, size and movement.
- Geometry involves lengths, areas and volumes which are expressed as numerical quantities.
- Use of rules of number to calculate further values related to measurement and geometry.

- Co-ordinates, geometrically represented on the Cartesian plane, are defined by algebraic functions.
- Functions and mappings in algebra can be used to describe transformations.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Expressive Arts

- Shape symmetry in movement and artwork.
- Scale.
- Use of appropriate equipment to measure accurately.

### Health and Well-being

- Use of appropriate equipment to measure accurately.

### Humanities

- Use of appropriate equipment to measure accurately.
- Scale.
- Time and chronological ordering.

### Languages, Literacy and Communication

- Use of prepositions to describe the location of selves and objects.

### Science and Technology

- Use of appropriate equipment to measure accurately.
- Units – use of appropriate unit, converting between units, and links between units and formulae.
- Scale.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences, describe, represent, analyse and explain properties of shapes in two‑dimensional and three-dimensional space
- use digital technologies to explore shape and space, movement and position, developing and testing conjectures
- work in real-life and mathematical contexts, using local examples where possible.

The number system is used to represent and compare relationships between numbers and quantities.

- Statistics involves manipulation and representation of data, which involve numerical thinking.

- Algebra is used within probability and statistics to express generalities and develop formulae.
- In statistics, general formulae are written using algebra.

- Geometry involves graphical thinking which is central to statistical representations and analysis.
- Graphical techniques are used to make connections between different sets of data.
- Data generated by measure can be analysed using statistics.

### Health and Well-being

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Humanities

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

### Science and Technology

- Sorting and classifying.
- Lines of best fit.
- Analysing and inferring from data.
- Spotting trends and anomalies.
- Averages and range.
- Drawing and interpreting a range of graphs.

#### Experiences, knowledge and skills

Learners need to experience, to know or be able to:

- through a broad range of experiences in stimulating and relevant local and global contexts, use a sequential data-handling cycle to pose questions, select a sample, and collect and sort a variety of purposeful data
- check and evaluate their results
- engage critically with statistics in the media
- calculate probabilities of combined events and consider probabilities of real-life events.

**All our children and young people will be:****ambitious, capable learners who:**- set themselves high standards and seek and enjoy challenge
- are questioning and enjoy solving problems
- can communicate effectively in different forms and settings, using both Welsh and English
- can explain the ideas and concepts they are learning about
- use digital technologies creatively to communicate, find and analyse information
- undertake research and evaluate critically what they find

and

**are ready to learn throughout their lives****enterprising, creative contributors who:**- connect and apply their knowledge and skills to create ideas and products
- think creatively to reframe and solve problems
- identify and grasp opportunities
- take measured risks
- lead and play different roles in teams effectively and responsibly
- express ideas and emotions through different media
- give of their energy and skills so that other people will benefit

and

**are ready to play a full part in life and work****ethical, informed citizens who:**- find, evaluate and use evidence in forming views
- engage with contemporary issues based upon their knowledge and values
- understand and exercise their human and democratic responsibilities and rights
- understand and consider the impact of their actions when making choices and acting
- are knowledgeable about their culture, community, society and the world, now and in the past
- respect the needs and rights of others, as a member of a diverse society
- show their commitment to the sustainability of the planet

and

**are ready to be citizens of Wales and the world****healthy, confident individuals who:**- have secure values and are establishing their spiritual and ethical beliefs
- are building their mental and emotional well-being by developing confidence, resilience and empathy
- know how to find the information and support to keep safe and well
- take part in physical activity
- take measured decisions about lifestyle and manage risk
- have the confidence to participate in performance
- form positive relationships based upon trust and mutual respect
- face and overcome challenge
- have the skills and knowledge to manage everyday life as independently as they can

and

**are ready to lead fulfilling lives as valued members of society.**