This practice material is suitable for any learner who does not yet have a deep, confident grasp of what a fraction is. It consists of practice-items (a mixture of tasks and questions) that have been sequenced and varied to help pupils develop deep understanding of the nature of fractions by 'drawing-out' for themselves some basic 'observations'. For example, they should ‘see’ that not only are fractions used to denote various particular numbers of other particular numbers of equal parts of 'wholes', but they are very useful 'tools' when we want to express comparisons between numbers and quantities (‘that number/length/weight/etc is half/two-thirds/one tenth/etc of this number/length/weight/etc’). This kind of practice should help them, later, understand fundamental links between the concepts of ‘fraction’ and ‘ratio’. The various ways of seing the same multiplicative relationships should also help them grasp what is meant by ‘equivalent fractions’. This material is intended to be used flexibly; it is easily-extended by pupils making-up their own examples. Pupils who 'fall-behind' in mathematics often make un-anticipated progress when they are encouraged to make choices, as they have to do when they set themselves their own tasks. Also, the possibility of using physical objects (in this case actual Cuisenaire rods) to represent and explore number-relationships in their own (concrete) ways can significantly improve pupils' attitudes to their learning.

These are the first two of a set of original mathematical puzzles that appear to be solely spatial, but that are related to, and can be created from, numerical 'Magic Squares'. A variety of other similar puzzles will be published soon. Each puzzle is presented on a single page, as is a solution of it. Another page provides, for the first puzzle only, a set of shapes for pupils to cut-out and manipulate. A solution of a third puzzle is included on the introductory page. The last page offers examples of the kinds of mathematical question that pupils can ask themselves in order to 'spark-off' productive chains of reasoning. In learning how appropriate questioning enables them to persevere to solutions, pupils have opportunities to develop, and to appreciate, their natural reasoning abilities. They may also become more aware of how geometric intuition can support insights that are confirmed through reasoning. Some implications that they 'see' and use will be derived from existing geometrical and numerical knowledge, and that knowledge is likely thereby to be deepened , and possibly extended.