Representation and Structure
Key Messages:

The representation needs to clearly show the concept being taught, and in particular the key difficulty point. It exposes the structure.

In the end, the students need to be able to do the maths without the representation

A stem sentence describes the representation and helps the students move to working in the abstract (“ten tenths is equivalent to one whole”) and could be seen as a representation in itself

There will be some key representations which the students will meet time and again

Pattern and structure are related but different: Students may have seen a pattern without understanding the structure which causes that pattern
Representing quadratic sequences with Dienes blocks in different bases.
NCETM have more examples and videos here
Representation and Structure for Secondary Schools Handbook
This handbook contains all the models above and more. It describes how to introduce and use manipulatives and with loads of examples to use in the classroom.
All sections show a coherent journey from familiar KS2 representation to secondary mathematics concepts.
Where possible, ideas are research and evidence based and have all been tried and tested in real classrooms.
Dienes Blocks and Algebra Tiles
Dienes blocks are an excellent visual representation of base 10 that allow students to see the connections with base x (algebra tiles). You can use Dienes blocks to address common misconceptions with standard algorithms.
Download the full handbook here
Dynamic Representations
https://www.enigmadynamicrepresentations.com/dynamicrepresentations
This site contains static and dynamic representations that are freely available online. They can be used as a teacher modelling tool on a whiteboard or by students on tablets or laptops.
The Desmos representations are best viewed in Desmos. Click on the link below the representation or click “edit in Desmos” on the bottom right of the image. When you open in Desmos you can adapt and save the representations.
Language and Definitions
This example is a common definition that leads to misconceptions. When defining angles on a straight line as angles that add up to 180 students often misinterpret this as angles anywhere on a straight line sum to 180 as in a+b+c+d=180 in the second example.
A full definition gives clarity and avoids misinterpretation and misconceptions.