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Pupils calculate the area from scale drawings using given measurements. Pupils use all 4 operations in problems involving time and money, including conversions for example, days to weeks, expressing the answer as weeks and days.

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Pupils become accurate in drawing lines with a ruler to the nearest millimetre, and measuring with a protractor. They use conventional markings for parallel lines and right angles. Pupils use the term diagonal and make conjectures about the angles formed between sides, and between diagonals and parallel sides, and other properties of quadrilaterals, for example using dynamic geometry ICT tools.

Pupils use angle sum facts and other properties to make deductions about missing angles and relate these to missing number problems. Pupils recognise and use reflection and translation in a variety of diagrams, including continuing to use a 2-D grid and coordinates in the first quadrant. Reflection should be in lines that are parallel to the axes. Pupils connect their work on coordinates and scales to their interpretation of time graphs. They begin to decide which representations of data are most appropriate and why. Pupils practise addition, subtraction, multiplication and division for larger numbers, using the formal written methods of columnar addition and subtraction, short and long multiplication, and short and long division see Mathematics appendix 1.

They undertake mental calculations with increasingly large numbers and more complex calculations. Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency. Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50, etc, but not to a specified number of significant figures.

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Common factors can be related to finding equivalent fractions. Pupils should practise, use and understand the addition and subtraction of fractions with different denominators by identifying equivalent fractions with the same denominator. Pupils should use a variety of images to support their understanding of multiplication with fractions. This follows earlier work about fractions as operators fractions of , as numbers, and as equal parts of objects, for example as parts of a rectangle. They practise calculations with simple fractions and decimal fraction equivalents to aid fluency, including listing equivalent fractions to identify fractions with common denominators.

For simple fractions with recurring decimal equivalents, pupils learn about rounding the decimal to three decimal places, or other appropriate approximations depending on the context. Pupils multiply and divide numbers with up to 2 decimal places by one-digit and two-digit whole numbers. Pupils multiply decimals by whole numbers, starting with the simplest cases, such as 0.

Pupils are introduced to the division of decimal numbers by one-digit whole numbers, initially, in practical contexts involving measures and money. They recognise division calculations as the inverse of multiplication. Pupils also develop their skills of rounding and estimating as a means of predicting and checking the order of magnitude of their answers to decimal calculations.

This includes rounding answers to a specified degree of accuracy and checking the reasonableness of their answers. Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio for example, similar shapes and recipes. Pupils should consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems.

1. The Real Numbers Include Integers, Rational, and Irrational Numbers

They might use the notation a:b to record their work. These problems are the foundation for later formal approaches to ratio and proportion. Pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:. They know approximate conversions and are able to tell if an answer is sensible. Using the number line, pupils use, add and subtract positive and negative integers for measures such as temperature. They relate the area of rectangles to parallelograms and triangles, for example, by dissection, and calculate their areas, understanding and using the formulae in words or symbols to do this.

Pupils could be introduced to compound units for speed, such as miles per hour, and apply their knowledge in science or other subjects as appropriate. Pupils draw shapes and nets accurately, using measuring tools and conventional markings and labels for lines and angles. Pupils describe the properties of shapes and explain how unknown angles and lengths can be derived from known measurements. Pupils draw and label a pair of axes in all 4 quadrants with equal scaling.

This extends their knowledge of one quadrant to all 4 quadrants, including the use of negative numbers.


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Pupils draw and label rectangles including squares , parallelograms and rhombuses, specified by coordinates in the four quadrants, predicting missing coordinates using the properties of shapes. Pupils connect their work on angles, fractions and percentages to the interpretation of pie charts. Pupils both encounter and draw graphs relating 2 variables, arising from their own enquiry and in other subjects. They should connect conversion from kilometres to miles in measurement to its graphical representation.

Pupils know when it is appropriate to find the mean of a data set. The programme of study for key stage 3 is organised into apparently distinct domains, but pupils should build on key stage 2 and connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge in science, geography, computing and other subjects. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content in preparation for key stage 4.

Those who are not sufficiently fluent should consolidate their understanding, including through additional practice, before moving on. Together, the mathematical content set out in the key stage 3 and key stage 4 programmes of study covers the full range of material contained in the GCSE Mathematics qualification. To help us improve GOV.

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Department for Education. Contents Key stage 1 - years 1 and 2 Year 1 programme of study Year 2 programme of study Lower key stage 2 - years 3 and 4 Year 3 programme of study Year 4 programme of study Upper key stage 2 - years 5 and 6 Year 5 programme of study Year 6 programme of study Key stage 3 Key stage 4. Aims The national curriculum for mathematics aims to ensure that all pupils: become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas.

Information and communication technology ICT Calculators should not be used as a substitute for good written and mental arithmetic. School curriculum The programmes of study for mathematics are set out year-by-year for key stages 1 and 2. Attainment targets By the end of each key stage, pupils are expected to know, apply and understand the matters, skills and processes specified in the relevant programme of study.

Notes and guidance non-statutory Pupils practise counting 1, 2, 3… , ordering for example, first, second, third… , and to indicate a quantity for example, 3 apples, 2 centimetres , including solving simple concrete problems, until they are fluent. Notes and guidance non-statutory Through grouping and sharing small quantities, pupils begin to understand: multiplication and division; doubling numbers and quantities; and finding simple fractions of objects, numbers and quantities.


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Notes and guidance non-statutory The pairs of terms: mass and weight, volume and capacity, are used interchangeably at this stage. Notes and guidance non-statutory Pupils handle common 2-D and 3-D shapes, naming these and related everyday objects fluently.

Fibonacci Number Patterns

Notes and guidance non-statutory Pupils use the language of position, direction and motion, including: left and right, top, middle and bottom, on top of, in front of, above, between, around, near, close and far, up and down, forwards and backwards, inside and outside. Notes and guidance non-statutory Using materials and a range of representations, pupils practise counting, reading, writing and comparing numbers to at least and solving a variety of related problems to develop fluency.

Notes and guidance non-statutory Pupils extend their understanding of the language of addition and subtraction to include sum and difference. Notes and guidance non-statutory Pupils use a variety of language to describe multiplication and division. Notes and guidance non-statutory Pupils use standard units of measurement with increasing accuracy, using their knowledge of the number system. Notes and guidance non-statutory Pupils handle and name a wide variety of common 2-D and 3-D shapes including: quadrilaterals and polygons and cuboids, prisms and cones, and identify the properties of each shape for example, number of sides, number of faces.

Notes and guidance non-statutory Pupils should work with patterns of shapes, including those in different orientations. Notes and guidance non-statutory Pupils record, interpret, collate, organise and compare information for example, using many-to-one correspondence in pictograms with simple ratios 2, 5, Notes and guidance non-statutory Pupils now use multiples of 2, 3, 4, 5, 8, 10, 50 and Notes and guidance non-statutory Pupils practise solving varied addition and subtraction questions.

Notes and guidance non-statutory Pupils continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order to improve fluency. Notes and guidance non-statutory Pupils connect tenths to place value, decimal measures and to division by Notes and guidance non-statutory Pupils understand and use simple scales for example, 2, 5, 10 units per cm in pictograms and bar charts with increasing accuracy.

2. The Number Line Contains all the Real Numbers (and Nothing Else)

Notes and guidance non-statutory Using a variety of representations, including measures, pupils become fluent in the order and place value of numbers beyond 1,, including counting in 10s and s, and maintaining fluency in other multiples through varied and frequent practice. Notes and guidance non-statutory Pupils continue to practise both mental methods and columnar addition and subtraction with increasingly large numbers to aid fluency see Mathematics appendix 1. Notes and guidance non-statutory Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency.

Notes and guidance non-statutory Pupils should connect hundredths to tenths and place value and decimal measure.

5 Things You Should Know About Real Numbers in Math

Notes and guidance non-statutory Pupils build on their understanding of place value and decimal notation to record metric measures, including money. Notes and guidance non-statutory Pupils continue to classify shapes using geometrical properties, extending to classifying different triangles for example, isosceles, equilateral, scalene and quadrilaterals for example, parallelogram, rhombus, trapezium. Notes and guidance non-statutory Pupils draw a pair of axes in one quadrant, with equal scales and integer labels.

Notes and guidance non-statutory Pupils understand and use a greater range of scales in their representations. Notes and guidance non-statutory Pupils identify the place value in large whole numbers. Notes and guidance non-statutory Pupils practise using the formal written methods of columnar addition and subtraction with increasingly large numbers to aid fluency see Mathematics appendix 1. Notes and guidance non-statutory Pupils practise and extend their use of the formal written methods of short multiplication and short division see Mathematics appendix 1.

Notes and guidance non-statutory Pupils should be taught throughout that percentages, decimals and fractions are different ways of expressing proportions. Notes and guidance non-statutory Pupils use their knowledge of place value and multiplication and division to convert between standard units.