National Curriculum 2014

Year 3 programme of study

Number

Pupils should be taught to:

  • count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and in dividing one-digit numbers or quantities by 10
  • recognise, find and write fractions of a discrete set of objects: unit and non-unit fractions with small denominators
  • recognise and use fractions as numbers: unit and non-unit fractions with small denominators
  • recognise and show, using diagrams, equivalent fractions with small denominators
  • add and subtract fractions with the same denominator within one whole [for example,5/7 + 1/7 = 6/7 ]
  • compare and order unit fractions, and those with the same denominators
  • solve problems that involve all of the above

Notes and guidance (non-statutory)

Pupils connect tenths to place value, decimal measures and to division by 10.

They begin to understand unit and non-unit fractions as numbers on the number line, and deduce relations between them, such as size and equivalence. They should go beyond the [0, 1] interval, including relating this to measure.

Pupils understand the relation between unit fractions as operations and division by integers.

They continue to recognise fractions in the context of parts of a whole, numbers, measurements, a shape, and unit fractions as a division of a quantity.

Pupils practise adding and subtracting fractions with the same denominator through a variety of increasingly complex problems to improve fluency.

Year 4 programme of study

Number

Pupils should be taught to:

  • recognise and show, using diagrams, families of common equivalent fractions
  • count up and down in hundredths; recognise that hundredths arise when dividing an object by 100 and dividing tenths by 10
  • solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number
  • add and subtract fractions with the same denominator
  • recognise and write decimal equivalents of any number of tenths or hundreds
  • recognise and write decimal equivalents to 1/4 , 1/2 , 3/4
  • find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits in the answer as ones, tenths and hundredths
  • round decimals with 1 decimal place to the nearest whole number
  • compare numbers with the same number of decimal places up to 2 decimal places
  • solve simple measure and money problems involving fractions and decimals to 2 decimal places

Notes and guidance (non-statutory)

Pupils should connect hundredths to tenths and place value and decimal measure.

They extend the use of the number line to connect fractions, numbers and measures.

Pupils understand the relation between non-unit fractions and multiplication and division of quantities, with particular emphasis on tenths and hundredths.

Pupils make connections between fractions of a length, of a shape and as a representation of one whole or set of quantities. Pupils use factors and multiples to recognise equivalent fractions and simplify where appropriate (for example, 6/9 = 2/3 or 1/4 = 2/8 ).

Pupils continue to practise adding and subtracting fractions with the same denominator, to become fluent through a variety of increasingly complex problems beyond one whole. Pupils are taught throughout that decimals and fractions are different ways of expressing numbers and proportions.

Pupils’ understanding of the number system and decimal place value is extended at this stage to tenths and then hundredths. This includes relating the decimal notation to division of whole number by 10 and later 100.

They practise counting using simple fractions and decimals, both forwards and backwards.

Pupils learn decimal notation and the language associated with it, including in the context of measurements. They make comparisons and order decimal amounts and quantities that are expressed to the same number of decimal places. They should be able to represent numbers with 1 or 2 decimal places in several ways, such as on number lines.

Year 5 programme of study

Number

Pupils should be taught to:

  • compare and order fractions whose denominators are all multiples of the same number
  • identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths
  • recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number [for example, 2/5 + 4/5 = 6/5= 1 1/5 ]
  • add and subtract fractions with the same denominator, and denominators that are multiples of the same number
  • multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams
  • read and write decimal numbers as fractions [for example, 0.71 = 71/100 ]
  • recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents
  • round decimals with 2 decimal places to the nearest whole number and to 1 decimal place
  • read, write, order and compare numbers with up to 3 decimal places
  • solve problems involving number up to 3 decimal places
  • recognise the per cent symbol (%) and understand that per cent relates to ‘number of parts per 100’, and write percentages as a fraction with denominator 100, and as a decimal fraction
  • solve problems which require knowing percentage and decimal equivalents of 1/2 , 1/4 , 1/5 ,2/5 , 4/5 and those fractions with a denominator of a multiple of 10 or 25

Notes and guidance (non-statutory)

Pupils should be taught throughout that percentages, decimals and fractions are different ways of expressing proportions.

They extend their knowledge of fractions to thousandths and connect to decimals and measures.

Pupils connect equivalent fractions > 1 that simplify to integers with division and other fractions > 1 to division with remainders, using the number line and other models, and hence move from these to improper and mixed fractions.

Pupils connect multiplication by a fraction to using fractions as operators (fractions of), and to division, building on work from previous years. This relates to scaling by simple fractions, including fractions > 1.

Pupils practise adding and subtracting fractions to become fluent through a variety of increasingly complex problems. They extend their understanding of adding and subtracting fractions to calculations that exceed 1 as a mixed number.

Pupils continue to practise counting forwards and backwards in simple fractions.

Pupils continue to develop their understanding of fractions as numbers, measures and operators by finding fractions of numbers and quantities. Pupils extend counting from year 4, using decimals and fractions including bridging 0, for example on a number line.

Pupils say, read and write decimal fractions and related tenths, hundredths and thousandths accurately and are confident in checking the reasonableness of their answers to problems.

They mentally add and subtract tenths, and one-digit whole numbers and tenths.

They practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (for example, 0.83 + 0.17 = 1).

Pupils should go beyond the measurement and money models of decimals, for example, by solving puzzles involving decimals.

Pupils should make connections between percentages, fractions and decimals (for example, 100% represents a whole quantity and 1% is 1/100 , 50% is 50/100 , 25% is25/100 ) and relate this to finding ‘fractions of’.

Year 6 programme of study

Number

Pupils should be taught to:

  • use common factors to simplify fractions; use common multiples to express fractions in the same denomination
  • compare and order fractions, including fractions >1
  • add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions
  • multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, 1/4 × 1/2 = 1/8 ]
  • divide proper fractions by whole numbers [for example, 1/3 ÷ 2 = 1/6 ]
  • associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3/8 ]
  • identify the value of each digit in numbers given to 3 decimal places and multiply and divide numbers by 10, 100 and 1,000 giving answers up to 3 decimal places
  • multiply one-digit numbers with up to 2 decimal places by whole numbers
  • use written division methods in cases where the answer has up to 2 decimal places
  • solve problems which require answers to be rounded to specified degrees of accuracy
  • recall and use equivalences between simple fractions, decimals and percentages, including in different contexts

Notes and guidance (non-statutory)

Pupils should practise, use and understand the addition and subtraction of fractions with different denominators by identifying equivalent fractions with the same denominator. They should start with fractions where the denominator of one fraction is a multiple of the other (for example, 1/2 + 1/8 = 5/8 ] and progress to varied and increasingly complex problems.

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.

Pupils use their understanding of the relationship between unit fractions and division to work backwards by multiplying a quantity that represents a unit fraction to find the whole quantity (for example, if quarter of a length is 36cm, then the whole length is 36 × 4 = 144cm).

They practise calculations with simple fractions and decimal fraction equivalents to aid fluency, including listing equivalent fractions to identify fractions with common denominators.

Pupils can explore and make conjectures about converting a simple fraction to a decimal fraction (for example, 3 ÷ 8 = 0.375). 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 two decimal places by one-digit and two-digit whole numbers. Pupils multiply decimals by whole numbers, starting with the simplest cases, such as 0.4 × 2 = 0.8, and in practical contexts, such as measures and money.

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.