Overview of the Concept framework
Approaches to Calculation and Number Concepts for Year 7 and Beyond
Purpose of this framework
This sets out a coherent approach to the teaching of number in Key Stage 3, beginning in Year 7 and continuing beyond. Its purpose is to help pupils build a secure understanding of calculation and number structure so that algebra is experienced not as a new and separate topic, but as a generalisation of number.
The central principle of this framework is:
Algebra makes visible the structure that is already present in arithmetic.
If pupils understand how numbers behave, how operations are related, and how calculations can be represented and reorganised, then algebra becomes a natural next step, not a leap into unfamiliar territory, but a shift from working with specific numbers to expressing what is always true.
What this framework is for
This framework establishes three things:
A shared language. When teachers in different schools and different year groups talk about multiplication, equivalence, or the distributive law, they mean the same thing. Pupils who move between classes or schools encounter consistency rather than contradiction.
A shared set of representations. The models teachers use, number lines, area models, two-coloured counters, bar models, are chosen because they reveal mathematical structure, not because they are fashionable. Pupils meet these representations across topics and across years, building fluency with them rather than starting from scratch each time.
A shared understanding of why. The framework does not list what to teach. It explains the mathematical reasoning behind each approach, so that when teachers adapt for their classes, as they should, they do so within a coherent framework rather than in isolation.
What this framework is not
This is not a curriculum or curriculum sequence. It is an attempt to organise concepts to help teachers understand their connections better. The hope is that whatever curriculum sequence they and their school follow, they can support their pupils to experience mathematics as a connected discipline, not as a series of unrelated topics that happen to appear in the same timetable slot.
The approaches in this framework are not new inventions; they draw on well-established mathematics and widely respected pedagogy. Many teachers will recognise much of what is here as consistent with what they already do.
How this is organised
The framework has two layers, designed to serve different purposes.
This overview layer, the section you are reading now. It sets out the framework’s purpose, its unifying principles, and the concept map showing how the 17 concept areas connect to each other and to later curriculum topics.
The detailed reference layer is the bulk of the content. It contains one section for each of the 17 concept areas, all following a consistent template. Each section includes the core mathematical idea, key representations, worked examples, the explicit bridge to algebra, key vocabulary, common misconceptions, diagnostic questions, and a progression spine.
Next page: Unifying principles
- Introduction
- Overview
- Unifying principles
- Concept map
Concept Reference
- Place Value and Unitising
- Laws of Arithmetic
- Equality Equivalence and the Equals Sign
- Zero and One as Identities
- Directed Number
- Additive and Multiplicative Reasoning
- The Area Model
- Division
- Scaled Multiplication Tables
- Structural Approaches to Calculation
- Inverse Operations and Fact Families
- Fractions Decimals and Percentages
- Factors Multiples Primes
- Generalising from Patterns
- Variable as a Concept
- Expressions Equations and Identities
- Estimation Approximation and Sense-Checking