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10 challenging open-ended mathematical investigations with a low threshold and high ceiling. Aimed at KS2 and extendable to KS3 levels, for pupils aged 8 to 11. We are delighted to be publishing these activities that exploit the mathematical potential of Numicon Imagery. All the activities start at a low threshold and use the Numicon Shapes and Cuisenaire Rods as a starting point to allow the majority of pupils to be involved at a basic level. The activities have also been designed as open-ended investigations and give plenty of opportunity for gifted and talented children to develop their skills and the take the challenges to higher levels. Among other ideas the challenges focus on the relative values of numbers that are shown clearly by the Numicon Shapes, in ways not provided by numerals. By using the Numicon Shapes and Cuiseniare Rods investigation is dynamic, helping children to recognize that numbers form a highly organized system full of many patterns. Each challenge can be taken to higher levels by the most able children in Key Stage 2, and some of the challenges can be extended into KS3 Levels. The investigations give children opportunities to use and apply their number understanding and to extend their mathematical thinking. There are opportunities for conjecture, trial, improvement, logical thinking about numbers and careful mathematical deduction. Children will also see the advantage of setting out their work systematically to reveal emerging patterns and demonstrate their thinking.