Using visual models and representations

Use of visual models and representations to expose mathematical structure is one of the five ‘big ideas’ of teaching for mastery.  Paying attention to underlying structure helps learners to develop conceptual understanding, make connections between topics, and can help learners access problem solving questions.

In CfEM, a variety of models and representations have been used in colleges with very promising results – including the use of manipulatives such as algebra tiles.  However, the three representations that have proved most popular and consistently successful are:

• Ratio tables
• Double-number lines
• Bar models

Proportional reasoning

Watch this short introduction by a CfEM centre lead:

Ratio tables and double-number lines (DNLs) are both useful in helping learners to understand the key topic of proportional reasoning – which relates to large areas of the GCSE and Functional Skills curricula.  Ratio tables have proved particularly popular in helping learners to develop basic understanding of the processes of scaling up and down, and rates of change in a more informal and intuitive way, which can also develop learners’ number sense.  DNLs develop the approach further through the visualisation of a multiplier, and have been found particularly useful in tackling questions where a calculator is required.

Proportional reasoning intro screencast

The following set of videos relate specifically to the use of ratio tables have been produced by Weston College and Leyton Sixth Form College.

The first video proves a short introduction to ratio tables, and why learners find them helpful.

Ratio tables Screencast 1 – Rationale

The next two videos demonstrate how ratio table have been used successfully for questions relating to recipes and best buys.

Ratio Tables Screencast 2 - Recipes

Ratio Tables Screencast 3 – Best buys

The final video provides an interesting case study about how ratio tables have been used successfully within vocational areas.

Ratio Tables Screencast 4 – vocational area

The next set of videos, from Leyton Sixth Form College, reveal how double numbers lines can be used successfully across a number of different GCSE topics.  The first two explain what a DNL is, why it is helpful and how to introduce it to learners.

DNL Screencast 1

DNL Screencast 2

The third video explains how DNLs can be used across a range of topics that have a proportional structure.

DNL Screencast 3

The final video explains the use of DNLs to support learners’ understanding of percentages, by using a common structure for different types of question.

DNL Screencast4

Bar models

The bar model is another representation that has supported FE learners to ‘see’ mathematical structure.  It can be used to represent a range of mathematical concepts and processes, including the four operations, fractions, ratio and algebra.  They are particularly useful in representing discrete (rather than continuous) units or measures.

Below is a set of three videos that have been adapted from an online CPD session delivered by Christ the King Sixth Form College.  The first introduces bar models and provides a rationale for their use.

Bar models 1

The second video demonstrates how bar models can be introduced through the MathsBot app, and their use in answering fraction and percentages questions.

Bar models 2

The final video explains the use of bar models to answer tricky GCSE exam questions – specifically related to fractions and ratio.

Bar models 3