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Training slides for session 1. Please feel free to look in advance or afterwards, but please don't do any of the activities before the session.

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Maths ST 1 - Developing good subject knowledge

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Having excellent subject knowledge means we can scaffold, chunk and meet pupils where they are in their thinking, rather than try and bend pupils to the narrow minded thinking and methods of the teacher. It also builds a toolbox that teachers can reach into when they need to explain using different methods to aid understanding. This is good for SEND students, but good for neurotypical students also.

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This is a really interesting video that shows the different pupils add numbers, depending on how they are categorised as 'above, below or at average' in their maths scores.

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Research Paper - the importance of subject knowledge

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Research Paper - the importance of subject knowledge

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This study aimed to identify distinct profiles of mathematics teachers based on their professional competence and to examine how these profiles are associated with instructional quality. Four distinct competence profiles were identified, ranging from low to high across the various competence dimensions. Teachers with high levels of both content and pedagogical content knowledge CK, alongside strong motivational and self-regulatory traits, were associated with higher instructional quality. Instructional quality was measured through classroom observations and student feedback, focusing on cognitive activation, classroom management, and student support.

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Your ITTECF framework that governs what you learn on your ITT course and for your next 2 years as an ECT also.

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National curriculum for maths

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Purpose of study

Mathematics is a creative and highly interconnected discipline that has been developed over centuries, providing the solution to some of history's most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.

Aims

The national curriculum for mathematics aims to ensure that all pupils:

Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programmes of study are, by necessity, organised into apparently distinct domains, but pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects.

The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils' understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.

Information and communication technology (ICT)

Calculators should not be used as a substitute for good written and mental arithmetic. They should therefore only be introduced near the end of key stage 2 to support pupils' conceptual understanding and exploration of more complex number problems, if written and mental arithmetic are secure. In both primary and secondary schools, teachers should use their judgement about when ICT tools should be used.

Spoken language

The national curriculum for mathematics reflects the importance of spoken language in pupils' development across the whole curriculum - cognitively, socially and linguistically. The quality and variety of language that pupils hear and speak are key factors in developing their mathematical vocabulary and presenting a mathematical justification, argument or proof. They must be assisted in making their thinking clear to themselves as well as others, and teachers should ensure that pupils build secure foundations by using discussion to probe and remedy their misconceptions.

School curriculum

The programmes of study for mathematics are set out year-by-year for key stages 1 and 2. Schools are, however, only required to teach the relevant programme of study by the end of the key stage. Within each key stage, schools therefore have the flexibility to introduce content earlier or later than set out in the programme of study. In addition, schools can introduce key stage content during an earlier key stage, if appropriate. All schools are also required to set out their school curriculum for mathematics on a year-by-year basis and make this information available online.

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.

Schools are not required by law to teach the example content in [square brackets] or the content indicated as being 'non-statutory'.

Key stage 1 - years 1 and 2

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 4 operations, including with practical resources [for example, concrete objects and measuring tools].

At this stage, pupils should develop their ability to recognise, 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.

Year 1 programme of study

Number - number and place value

Pupils should be taught to:

Number - addition and subtraction

Pupils should be taught to:

Number - multiplication and division

Pupils should be taught to:

solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher

Number - fractions

Pupils should be taught to:

recognise, find and name a half as 1 of 2 equal parts of an object, shape or quantity

recognise, find and name a quarter as 1 of 4 equal parts of an object, shape or quantity

Notes and guidance (non-statutory)

Pupils are taught half and quarter as 'fractions of' discrete and continuous quantities by solving problems using shapes, objects and quantities. For example, they could recognise and find half a length, quantity, set of objects or shape. Pupils connect halves and quarters to the equal sharing and grouping of sets of objects and to measures, as well as recognising and combining halves and quarters as parts of a whole.

Measurement

Pupils should be taught to:

compare, describe and solve practical problems for:

lengths and heights [for example, long/short, longer/shorter, tall/short, double/half]

mass/weight [for example, heavy/light, heavier than, lighter than]

capacity and volume [for example, full/empty, more than, less than, half, half full, quarter]

time [for example, quicker, slower, earlier, later]

measure and begin to record the following:

lengths and heights

mass/weight

capacity and volume

time (hours, minutes, seconds)

recognise and know the value of different denominations of coins and notes

sequence events in chronological order using language [for example, before and after, next, first, today, yesterday, tomorrow, morning, afternoon and evening]

recognise and use language relating to dates, including days of the week, weeks, months and years

tell the time to the hour and half past the hour and draw the hands on a clock face to show these times

Geometry - properties of shapes

Pupils should be taught to:

recognise and name common 2-D and 3-D shapes, including:

2-D shapes [for example, rectangles (including squares), circles and triangles]

3-D shapes [for example, cuboids (including cubes), pyramids and spheres]

Geometry - position and direction

Pupils should be taught to:

describe position, direction and movement, including whole, half, quarter and three-quarter turns

Year 2 programme of study

Number - number and place value

Pupils should be taught to:

Number - addition and subtraction

Pupils should be taught to:

solve problems with addition and subtraction:

using concrete objects and pictorial representations, including those involving numbers, quantities and measures

applying their increasing knowledge of mental and written methods

recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100

add and subtract numbers using concrete objects, pictorial representations, and mentally, including:

a two-digit number and 1s

a two-digit number and 10s

2 two-digit numbers

adding 3 one-digit numbers

show that addition of 2 numbers can be done in any order (commutative) and subtraction of 1 number from another cannot

recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems

Number - multiplication and division

Pupils should be taught to:

Number - fractions

Pupils should be taught to:

recognise, find, name and write fractions , , and of a length, shape, set of objects or quantity

write simple fractions, for example of 6 = 3 and recognise the equivalence of and

Notes and guidance (non-statutory)

Pupils use fractions as 'fractions of' discrete and continuous quantities by solving problems using shapes, objects and quantities. They connect unit fractions to equal sharing and grouping, to numbers when they can be calculated, and to measures, finding fractions of lengths, quantities, sets of objects or shapes. They meet as the first example of a non-unit fraction.

Pupils should count in fractions up to 10, starting from any number and using the and equivalence on the number line (for example, 1 , 1 (or 1 ), 1 , 2). This reinforces the concept of fractions as numbers and that they can add up to more than 1.

Measurement

Pupils should be taught to:

choose and use appropriate standard units to estimate and measure length/height in any direction (m/cm); mass (kg/g); temperature (°C); capacity (litres/ml) to the nearest appropriate unit, using rulers, scales, thermometers and measuring vessels

compare and order lengths, mass, volume/capacity and record the results using >, < and =

recognise and use symbols for pounds (£) and pence (p); combine amounts to make a particular value

find different combinations of coins that equal the same amounts of money

solve simple problems in a practical context involving addition and subtraction of money of the same unit, including giving change

compare and sequence intervals of time

tell and write the time to five minutes, including quarter past/to the hour and draw the hands on a clock face to show these times

know the number of minutes in an hour and the number of hours in a day

Geometry - properties of shapes

Pupils should be taught to:

Geometry - position and direction

Pupils should be taught to:

order and arrange combinations of mathematical objects in patterns and sequences

use mathematical vocabulary to describe position, direction and movement, including movement in a straight line and distinguishing between rotation as a turn and in terms of right angles for quarter, half and three-quarter turns (clockwise and anti-clockwise)

Statistics

Pupils should be taught to:

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, 5,10).

Lower key stage 2 - years 3 and 4

The principal focus of mathematics teaching in lower key stage 2 is to ensure that pupils become increasingly fluent with whole numbers and the 4 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 with simple fractions and decimal place value. Teaching should also ensure that pupils draw with increasing accuracy and develop mathematical reasoning so they can analyse 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 measure and number.

By the end of year 4, pupils should have memorised their multiplication tables up to and including the 12 multiplication table 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.

Year 3 programme of study

Number - number and place value

Pupils should be taught to:

Number - addition and subtraction

Pupils should be taught to:

add and subtract numbers mentally, including:

a three-digit number and 1s

a three-digit number and 10s

a three-digit number and 100s

add and subtract numbers with up to 3 digits, using formal written methods of columnar addition and subtraction

estimate the answer to a calculation and use inverse operations to check answers

solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction

Notes and guidance (non-statutory)

Pupils practise solving varied addition and subtraction questions. For mental calculations with two-digit numbers, the answers could exceed 100.

Pupils use their understanding of place value and partitioning, and practise using columnar addition and subtraction with increasingly large numbers up to 3 digits to become fluent (see Mathematics appendix 1 (PDF, 248KB)).

Number - multiplication and division

Pupils should be taught to:

Number - fractions

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 fractions and non-unit fractions with small denominators

recognise and use fractions as numbers: unit fractions 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, + = ]

compare and order unit fractions, and fractions 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 operators (fractions of), 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.

Measurement

Pupils should be taught to:

Geometry - properties of shapes

Pupils should be taught to:

Statistics

Pupils should be taught to:

interpret and present data using bar charts, pictograms and tables

solve one-step and two-step questions [for example 'How many more?' and 'How many fewer?'] using information presented in scaled bar charts and pictograms and tables

Year 4 programme of study

Number - number and place value

Pupils should be taught to:

count in multiples of 6, 7, 9, 25 and 1,000

find 1,000 more or less than a given number

count backwards through 0 to include negative numbers

recognise the place value of each digit in a four-digit number (1,000s, 100s, 10s, and 1s)

order and compare numbers beyond 1,000

identify, represent and estimate numbers using different representations

round any number to the nearest 10, 100 or 1,000

solve number and practical problems that involve all of the above and with increasingly large positive numbers

read Roman numerals to 100 (I to C) and know that over time, the numeral system changed to include the concept of 0 and place value

Number - addition and subtraction

Pupils should be taught to:

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 (PDF, 248KB)).

Number - multiplication and division

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency.

Pupils practise mental methods and extend this to 3-digit numbers to derive facts, (for example 600 ÷ 3 = 200 can be derived from 2 x 3 = 6).

Pupils practise to become fluent in the formal written method of short multiplication and short division with exact answers (see Mathematics appendix 1 (PDF, 248KB)).

Pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations for example, 2 x 6 x 5 = 10 x 6 = 60.

Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as the numbers of choices of a meal on a menu, or 3 cakes shared equally between 10 children.

Number - fractions (including decimals)

Pupils should be taught to:

Measurement

Pupils should be taught to:

Geometry - properties of shapes

Pupils should be taught to:

Geometry - position and direction

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils draw a pair of axes in one quadrant, with equal scales and integer labels. They read, write and use pairs of co-ordinates, for example (2, 5), including using co-ordinate-plotting ICT tools.

Statistics

Pupils should be taught to:

interpret and present discrete and continuous data using appropriate graphical methods, including bar charts and time graphs

solve comparison, sum and difference problems using information presented in bar charts, pictograms, tables and other graphs

Upper key stage 2 - years 5 and 6

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 ratio.

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 number. 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 4 operations, including long multiplication and division, and in working with fractions, decimals and percentages.

Pupils should read, spell and pronounce mathematical vocabulary correctly.

Year 5 programme of study

Number - number and place value

Pupils should be taught to:

Number - addition and subtraction

Pupils should be taught to:

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 (PDF, 248KB)).

They practise mental calculations with increasingly large numbers to aid fluency (for example, 12,462 - 2,300 = 10,162).

Number - multiplication and division

Pupils should be taught to:

identify multiples and factors, including finding all factor pairs of a number, and common factors of 2 numbers

know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers

establish whether a number up to 100 is prime and recall prime numbers up to 19

multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers

multiply and divide numbers mentally, drawing upon known facts

divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context

multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000

recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)

solve problems involving multiplication and division, including using their knowledge of factors and multiples, squares and cubes

solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign

solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates

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 (PDF, 248KB)). They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations.

They use and understand the terms factor, multiple and prime, square and cube numbers.

Pupils interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding (for example, 98 ÷ 4 = = 24 r 2 = 24 = 24.5 ≈ 25).

Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1,000 in converting between units such as kilometres and metres.

Distributivity can be expressed as a(b + c) = ab + ac.

They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 9² x 10).

Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example 13 + 24 = 12 + 25; 33 = 5 x ?).

Number - fractions (including decimals and percentages)

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, + = = 1 ]

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 = ]

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 , , , , 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 , 50% is , 25% is ) and relate this to finding 'fractions of'.

Measurement

Pupils should be taught to:

Geometry - properties of shapes

Pupils should be taught to:

identify 3-D shapes, including cubes and other cuboids, from 2-D representations

know angles are measured in degrees: estimate and compare acute, obtuse and reflex angles

draw given angles, and measure them in degrees (°)

identify:

angles at a point and 1 whole turn (total 360°)

angles at a point on a straight line and half a turn (total 180°)

other multiples of 90°

use the properties of rectangles to deduce related facts and find missing lengths and angles

distinguish between regular and irregular polygons based on reasoning about equal sides and angles

Geometry - position and direction

Pupils should be taught to:

identify, describe and represent the position of a shape following a reflection or translation, using the appropriate language, and know that the shape has not changed

Notes and guidance (non-statutory)

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.

Statistics

Pupils should be taught to:

solve comparison, sum and difference problems using information presented in a line graph

complete, read and interpret information in tables, including timetables

Year 6 programme of study

Number - number and place value

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils use the whole number system, including saying, reading and writing numbers accurately.

Number - addition, subtraction, multiplication and division

Pupils should be taught to:

multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication

divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context

divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context

perform mental calculations, including with mixed operations and large numbers

identify common factors, common multiples and prime numbers

use their knowledge of the order of operations to carry out calculations involving the 4 operations

solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

solve problems involving addition, subtraction, multiplication and division

use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy

Notes and guidance (non-statutory)

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 (PDF, 248KB)).

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.

Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9.

Common factors can be related to finding equivalent fractions.

Number - Fractions (including decimals and percentages)

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, × = ]

divide proper fractions by whole numbers [for example, ÷ 2 = ]

associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, ]

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, + = ] 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 2 decimal places by one-digit and two-digit whole numbers. Pupils multiply decimals by whole numbers, starting with the simplest cases, such as 0.4 × 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.

Ratio and proportion

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (for example, similar shapes and recipes).

Pupils link percentages or 360° to calculating angles of pie charts.

Pupils should consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems. They might use the notation a:b to record their work.

Pupils solve problems involving unequal quantities, for example, 'for every egg you need 3 spoonfuls of flour', ' of the class are boys'. These problems are the foundation for later formal approaches to ratio and proportion.

Algebra

Pupils should be taught to:

Notes and guidance (non-statutory)

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:

missing numbers, lengths, coordinates and angles

formulae in mathematics and science

equivalent expressions (for example, a + b = b + a)

generalisations of number patterns

number puzzles (for example, what 2 numbers can add up to)

Measurement

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils connect conversion (for example, from kilometres to miles) to a graphical representation as preparation for understanding linear/proportional graphs.

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.

Geometry - properties of shapes

Pupils should be taught to:

Geometry - position and direction

Pupils should be taught to:

describe positions on the full coordinate grid (all 4 quadrants)

draw and translate simple shapes on the coordinate plane, and reflect them in the axes

Statistics

Pupils should be taught to:

interpret and construct pie charts and line graphs and use these to solve problems

calculate and interpret the mean as an average

Key stage 3

Introduction

Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. 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.

Decisions about progression should be based on the security of pupils' understanding and their readiness to progress to the next stage. 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.

Working mathematically

Through the mathematics content, pupils should be taught to:

Develop fluency

Reason mathematically

Solve problems

Subject content

Number

Pupils should be taught to:

understand and use place value for decimals, measures and integers of any size

order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥

use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property

use the 4 operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals

recognise and use relationships between operations including inverse operations

use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations

interpret and compare numbers in standard form A x 10n 1≤A<10, where n is a positive or negative integer or 0

work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and or 0.375 and )

define percentage as 'number of parts per hundred', interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express 1 quantity as a percentage of another, compare 2 quantities using percentages, and work with percentages greater than 100%

interpret fractions and percentages as operators

use standard units of mass, length, time, money and other measures, including with decimal quantities

round numbers and measures to an appropriate degree of accuracy [for example, to a number of decimal places or significant figures]

use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x≤b

use a calculator and other technologies to calculate results accurately and then interpret them appropriately

appreciate the infinite nature of the sets of integers, real and rational numbers

Algebra

Pupils should be taught to:

use and interpret algebraic notation, including:

ab in place of a × b

3y in place of y + y + y and 3 × y

a² in place of a × a, a³ in place of a × a × a; a²b in place of a × a × b

in place of a ÷ b

coefficients written as fractions rather than as decimals

brackets

substitute numerical values into formulae and expressions, including scientific formulae

understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors

simplify and manipulate algebraic expressions to maintain equivalence by:

collecting like terms

multiplying a single term over a bracket

taking out common factors

expanding products of 2 or more binomials

understand and use standard mathematical formulae; rearrange formulae to change the subject

model situations or procedures by translating them into algebraic expressions or formulae and by using graphs

use algebraic methods to solve linear equations in 1 variable (including all forms that require rearrangement)

work with coordinates in all 4 quadrants

recognise, sketch and produce graphs of linear and quadratic functions of 1 variable with appropriate scaling, using equations in x and y and the Cartesian plane

interpret mathematical relationships both algebraically and graphically

reduce a given linear equation in 2 variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically

use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations

find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs

generate terms of a sequence from either a term-to-term or a position-to-term rule

recognise arithmetic sequences and find the nth term

recognise geometric sequences and appreciate other sequences that arise

Ratio, proportion and rates of change

Pupils should be taught to:

change freely between related standard units [for example time, length, area, volume/capacity, mass]

use scale factors, scale diagrams and maps

express 1 quantity as a fraction of another, where the fraction is less than 1 and greater than 1

use ratio notation, including reduction to simplest form

divide a given quantity into 2 parts in a given part:part or part:whole ratio; express the division of a quantity into 2 parts as a ratio

understand that a multiplicative relationship between 2 quantities can be expressed as a ratio or a fraction

relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions

solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics

solve problems involving direct and inverse proportion, including graphical and algebraic representations

use compound units such as speed, unit pricing and density to solve problems

Geometry and measures

Pupils should be taught to:

derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)

calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes

draw and measure line segments and angles in geometric figures, including interpreting scale drawings

derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line

describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric

use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles

derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies

identify properties of, and describe the results of, translations, rotations and reflections applied to given figures

identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids

apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles

understand and use the relationship between parallel lines and alternate and corresponding angles

derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons

apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras' Theorem, and use known results to obtain simple proofs

use Pythagoras' Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles

use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D

interpret mathematical relationships both algebraically and geometrically

Probability

Pupils should be taught to:

Statistics

Pupils should be taught to:

Key stage 4

This programme of study specifies:

the mathematical content that should be taught to all pupils, in standard type

additional mathematical content to be taught to more highly attaining pupils, in braces { }

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. Wherever it is appropriate, given pupils' security of understanding and readiness to progress, pupils should be taught the full content set out in this programme of study.

Working mathematically

Through the mathematics content pupils should be taught to:

Develop fluency

Reason mathematically

Solve problems

Subject content

Number

In addition to consolidating subject content from key stage 3, pupils should be taught to:

apply systematic listing strategies, {including use of the product rule for counting}

{estimate powers and roots of any given positive number}

calculate with roots, and with integer {and fractional} indices

calculate exactly with fractions, {surds} and multiples of π {simplify surd expressions involving squares [for example √12 = √(4 × 3) = √4 × √3 = 2√3] and rationalise denominators}

calculate with numbers in standard form A × 10n, where 1 ≤ A < 10 and n is an integer

{change recurring decimals into their corresponding fractions and vice versa}

identify and work with fractions in ratio problems

apply and interpret limits of accuracy when rounding or truncating, {including upper and lower bounds}

Algebra

In addition to consolidating subject content from key stage 3, pupils should be taught to:

simplify and manipulate algebraic expressions (including those involving surds {and algebraic fractions}) by:

factorising quadratic expressions of the form x² + bx + c, including the difference of 2 squares; {factorising quadratic expressions of the form ax² + bx + c}

simplifying expressions involving sums, products and powers, including the laws of indices

know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments {and proofs}

where appropriate, interpret simple expressions as functions with inputs and outputs; {interpret the reverse process as the 'inverse function'; interpret the succession of 2 functions as a 'composite function'}

use the form y = mx + c to identify parallel {and perpendicular} lines; find the equation of the line through 2 given points, or through 1 point with a given gradient

identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically {and turning points by completing the square}

recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = with x ≠ 0, {the exponential function y = kx for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size}

{sketch translations and reflections of the graph of a given function}

plot and interpret graphs (including reciprocal graphs {and exponential graphs}) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration

{calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts}

{recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point}

solve quadratic equations {including those that require rearrangement} algebraically by factorising, {by completing the square and by using the quadratic formula}; find approximate solutions using a graph

solve 2 simultaneous equations in 2 variables (linear/linear {or linear/quadratic}) algebraically; find approximate solutions using a graph

{find approximate solutions to equations numerically using iteration}

translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or 2 simultaneous equations), solve the equation(s) and interpret the solution

solve linear inequalities in 1 {or 2} variable {s}, {and quadratic inequalities in 1 variable}; represent the solution set on a number line, {using set notation and on a graph}

recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rn where n is an integer, and r is a positive rational number {or a surd}) {and other sequences}

deduce expressions to calculate the nth term of linear {and quadratic} sequences.

Ratio, proportion and rates of change

In addition to consolidating subject content from key stage 3, pupils should be taught to:

compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios)

convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts

understand that X is inversely proportional to Y is equivalent to X is proportional to ; {construct and} interpret equations that describe direct and inverse proportion

interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion

{interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of instantaneous and average rate of change (gradients of tangents and chords) in numerical, algebraic and graphical contexts}

set up, solve and interpret the answers in growth and decay problems, including compound interest {and work with general iterative processes}

Geometry and measures

In addition to consolidating subject content from key stage 3, pupils should be taught to:

interpret and use fractional {and negative} scale factors for enlargements

{describe the changes and invariance achieved by combinations of rotations, reflections and translations}

identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment

{apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results}

construct and interpret plans and elevations of 3D shapes

interpret and use bearings

calculate arc lengths, angles and areas of sectors of circles

calculate surface areas and volumes of spheres, pyramids, cones and composite solids

apply the concepts of congruence and similarity, including the relationships between lengths, {areas and volumes} in similar figures

apply Pythagoras' Theorem and trigonometric ratios to find angles and lengths in right-angled triangles {and, where possible, general triangles} in 2 {and 3} dimensional figures

know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45°, 60°

{know and apply the sine rule, = = , and cosine rule, a² = b² + c² - 2bc cos A, to find unknown lengths and angles}

{know and apply Area = ab sin C to calculate the area, sides or angles of any triangle}

describe translations as 2D vectors

apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; {use vectors to construct geometric arguments and proofs}

Probability

In addition to consolidating subject content from key stage 3, pupils should be taught to:

Statistics

In addition to consolidating subject content from key stage 3, pupils should be taught to:

infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling

interpret and construct tables and line graphs for time series data

{construct and interpret diagrams for grouped discrete data and continuous data, ie, histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use}

interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:

appropriate graphical representation involving discrete, continuous and grouped data, {including box plots}

appropriate measures of central tendency (including modal class) and spread {including quartiles and inter-quartile range}

apply statistics to describe a population

use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing.

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National curriculum in England: mathematics programmes of study

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Mathematics is a creative and highly interconnected discipline that has been developed over centuries, providing the solution to some of history's most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment.

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A whole suite of pre made diagnostic questions

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We're in the business of helping teachers. We know how tight departmental budgets are, we know the pain of change and we know how frustrating it is when a product you've come to depend upon starts charging. Well, we're giving you, the teacher in the classroom, a promise that Diagnostic Questions will always remain free. It's a small gesture, but we hope it helps.

Supported Subjects

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A continuous system of misconception identification provides teachers with the foresight they need to intervene effectively in students' learning.

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Better results, more confidence, less time

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A quiz to test your knowledge of SATs and GCSE maths questions

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Discover how Moya Weighill, Head of Maths at Ecclesbourne School, uses Eedi's diagnostic and intervention tools to drive real progress in her department. Find out how Eedi has helped students resolve 92.5% of their misconceptions and provided over 659 hours of extra learning time. Read more here.
Last week, I challenged secondary maths teachers' knowledge of the primary maths programmes of study. It went down well. So, never one to miss an opportunity to milk a good idea, this week we are going to delve more into the differences and - more pertinently - the similarities between the maths students are exposed to at primary and secondary.

And yes, I have another game for you to play.

Below are a series of questions. Each one is taken from either a SATs exam (students sit this during their final year of primary school, aged 11), or a GCSE exam (students sit this during their final year of secondary school, aged 16).

Your job is to decide which question belongs to which exam.

There are 10 questions in all. Here is Question 1:

The rest can be found on this Google Form where you can anonymously record your answers.

Please share this with your colleagues so we can get a large sample of responses.

I will reveal the answers next week!

1. ... checked out our incredible, brand-new, free resources from Eedi?

2. ... read my latest Tips for Teachers newsletter about attention contagion?

3. ... listened to my most recent podcast with Ollie Lovell where we discuss the Do Now?

4. ... considered booking some CPD, coaching, or maths departmental support?

Thanks so much for reading and have a great week!

Craig

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At its most basic, a really useful graphing calculator. Geogebra also has loads of funky lesson resources that we will look at as the course progresses

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Investigates how the curriculum in Initial Teacher Education (ITE) can be evaluated and improved to better prepare new teachers.

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The wasted years report laments the loss of progress in years 7 and 8

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Ofsted survey report investigating whether key stage 3 is providing pupils with sufficient breadth and challenge.

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Details

The survey aims to:

identify any weaknesses in how the Key stage 3 curriculum is being developed and delivered

provide a baseline from which any future improvement can be measured and make recommendations as to changes required

identify and share examples of good practice

Thank you for your feedback

Report a problem with this page

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The blog emphasizes simplicity and clarity in curriculum thinking. It encourages schools to avoid overcomplicating their preparation for Ofsted inspections and instead focus on ensuring that The curriculum clearly identifies the knowledge pupils need to achieve educational goals, and that pupils are actually learning that knowledge.

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Despite the challenges of the last two years, staff are reviewing and renewing their curriculums. Subject leaders are working closely with teachers to shape a clear journey of the knowledge they want pupils to know.

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Nice rich surds activities

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Surds - what a fabulous topic to teach! I love the abundance of resources available and I've listed some of my favourites below.

Two things I always do when teaching surds:

Pupils list the first twelve square numbers in their books for reference throughout the lesson. They need to readily recognise square numbers in order to simplify surds so the more they practise listing them, the better.

When manipulating expressions containing surds (eg expanding brackets) I make comparisons to what they already know about algebra. For example √2 and √3 can be thought of in the same way as x and y (ie not 'like terms') whereas 2√3 and 5√3 can be thought of in the same way as 2x and 5x (ie we can add them to get 7√3).

The NCETM suggests a couple of nice 'hooks' for getting started teaching surds:

Ask pupils to find a way of drawing a line with a length of exactly √5 units (the hypotenuse of a right angled triangle with sides 2cm and 1cm)

Ask pupils to divide the length of an A4 piece of paper by its width. Repeat for A3 and A5. What do they notice? (The answer is always √2)

This blog post has some more ideas for introducing the concept of surds. And I like this on the history of surds.

Resources
Here's some great teaching resources (there's hundreds more online). Most of these work equally well at GCSE and A level:

For a lovely set of surds problems, see my Problem Sets page.

Finally, if you're looking for something really creative, check out this Wheel of Theodorus Art Project.

Edit 29/12/14: Thanks to@runningtstitch for suggesting an alternative method for simplifying surds:

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Resourceaholic - free resources and ideas for teaching secondary school maths

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Some lovely resources from the Don Steward site with many rich questions - in training 1 we look at the surds materials

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A look into why engagement in maths falls off so sharply in Y7

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As used in the session.

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As used in the session

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As used in the session

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As used in the session

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Ensure you sign up to you class (code 3xckfm9pe4) USING YOUR BPN EMAIL ADDRESS.

Ensure you sign up as a STUDENT - this account is for you to practice your own maths and develop your subject knowledge.

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World’s First Accelerated Learning Platform

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In here is a copy of your SKA, which you will need to download and keep updated (please write in the DATE of when you achieve each strand) and upload to mosaic at intervals throughout the year.

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High- Building relationships with the staff and kids! Getting to know my form!

Low- No log in, thrown in the deep end, however thats not always a bad thing!