The principal focus of mathematics teaching in key stage 1 is to ensure that pupils develop confidence and mental fluency with whole numbers, counting, and place value. This should involve working with numerals, words, and the four operations, including practical resources [for example, concrete objects and measuring tools]. At this stage, pupils should develop their ability to recognize, describe, draw, compare, and sort different shapes and use the related vocabulary. Teaching should also involve using a range of measures to describe and compare different quantities such as length, mass, capacity/volume, time, and money. By the end of year 2, pupils should know the number bonds to 20 and be precise in using and understanding place value. An emphasis on practice at this early stage will aid fluency. Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling knowledge at key stage 1. The principal focus of mathematics teaching in lower key stage 2 is to ensure that pupils become increasingly fluent with whole numbers and the four operations, including number facts and the concept of place value. This should ensure that pupils develop efficient written and mental methods and perform calculations accurately with increasingly large whole numbers. At this stage, pupils should develop their ability to solve a range of problems, including simple fractions and decimal place values. Teaching should also ensure that pupils draw with increasing accuracy and develop mathematical reasoning so they can analyze shapes and their properties, and confidently describe the relationships between them. It should ensure that they can use measuring instruments with accuracy and make connections between measurements and numbers. By the end of year 4, pupils should have memorized their multiplication tables up to and including the 12 multiplication tables and show precision and fluency in their work. Pupils should read and spell mathematical vocabulary correctly and confidently, using their growing word reading knowledge and their knowledge of spelling. The principal focus of mathematics teaching in upper key stage 2 is to ensure that pupils extend their understanding of the number system and place value to include larger integers. This should develop the connections that pupils make between multiplication and division with fractions, decimals, percentages, and ratios. At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems. Teaching in geometry and measures should consolidate and extend knowledge developed in numbers. Teaching should also ensure that pupils classify shapes with increasingly complex geometric properties and that they learn the vocabulary they need to describe them. By the end of year 6, pupils should be fluent in written methods for all four operations, including long multiplication and division, and in working with fractions, decimals and percentages. Pupils should read, spell, and pronounce mathematical vocabulary correctly. KS3 Place value Positive and negative numbers Operations - Calculations Rounding and estimating Powers and roots Prime numbers, factors, and multiples Decimals Fractions Percentages Standard index form Distance, speed, and time Ratio Direct and inverse proportion Expressions and formulae Inequalities Equations Graphs Patterns and sequences Perimeter, Area, Volume Angles Shapes Symmetry and transformations Measurement Pythagoras and trigonometry Calculating probabilities Sets and Venn diagrams Collecting and recording data Representing data Averages