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: Mathematical understanding
Learning in this area should include an appropriate balance of focused subject teaching and well-planned opportunities to use, apply and develop knowledge and skills across the whole curriculum.
Curriculum aims
This area of learning contributes to the achievement of the curriculum aims for all young people to become:
successful learners who enjoy learning, make progress and achieve
confident individuals who are able to live safe, healthy and fulfilling lives
responsible citizens who make a positive contribution to society.
Why this area of learning is important
Mathematics introduces children to concepts, skills and thinking strategies that are essential in everyday life and support learning across the curriculum. It helps children make sense of the numbers, patterns and shapes they see in the world around them, offers ways of handling data in an increasingly digital world and makes a crucial contribution to their development as successful learners.
Children delight in using mathematics to solve a problem, especially when it leads them to an unexpected discovery or new connections. As their confidence grows, they look for patterns, use logical reasoning, suggest solutions and try out different approaches to problems.
Mathematics offers children a powerful way of communicating. They learn to explore and explain their ideas using symbols, diagrams and spoken and written language. They start to discover how mathematics has developed over time and contributes to our economy, society and culture. Studying mathematics stimulates curiosity, fosters creativity and equips children with the skills they need in life beyond school.
1. Essential knowledge
Children should build secure knowledge of the following:
the range of ways mathematics can be used to solve practical problems, model situations, make sense of data and inform decision making
different types of numbers REF _Ref242106337 \r \h \* MERGEFORMAT 1 and what they represent
how numbers can be used for measurement, quantification and comparison and applied in different contexts
how to use geometry to explore, understand and represent shape and space
how likelihood and risk can be understood, quantified and used in everyday life.
2. Key skills
These are the skills that children need to learn to make progress:
generate and explore ideas and strategies, pursue lines of mathematical enquiry and apply logic and reasoning to mathematical problems
make and test generalisations, identify patterns and appreciate equivalences and relationships REF _Ref242106168 \r \h 2
develop, select and apply a range of mental, written and ICT-based methods and models to estimate, approximate, calculate, classify, quantify, order and compare
communicate ideas and justify arguments using mathematical symbols, diagrams, images and language
interpret findings, evaluate methods and check outcomes.
3. Cross-curricular studies REF _Ref240440341 \r \h \* MERGEFORMAT 3
This area of learning should provide opportunities for:
Children to develop and apply their literacy, numeracy and ICT skills
Personal, emotional and social development
Enhancing children's mathematical understanding through making links to other areas of learning and to wider issues of interest and importance.
4. Breadth of learning
When experiencing mathematics as a creative activity and being introduced to its role in the world around them children should:
be taught to work logically and critically as they undertake focused, practical, problem-solving activities REF _Ref240440303 \r \h 4 in mathematical, cross-curricular and real-world contexts
visualise quantities, patterns and shapes and develop strategies for working things out in their head as well as on paper and using ICT
work individually and collaboratively to explore ideas and pursue lines of mathematical enquiry
articulate their thinking in discussions and make choices about the strategies they use to solve problems, based on what they know about the efficiency and effectiveness of different approaches
use mathematics to manage money, make sense of information, assess likelihood and risk, predict outcomes and construct reasoned arguments
meet with people who use mathematics in their work
use a wide range of practical resources, including ICT
use mathematical language to explain, refine and evaluate their own and others' work.
Teachers will continue to find the Primary Framework for teaching mathematics a significant basis for planning teaching.
This includes natural numbers, integers (positive and negative whole numbers) and rational numbers (fractions and decimals)
This includes families of equivalent fractions; the inverse relationship between addition and subtraction
Further guidance and case studies to provide teachers with help to plan for cross-curricular studies are available on the National Curriculum Website (curriculum.qcda.gov.uk) from early 2010
Problem-solving skills should be developed across the primary phase by providing more substantial and increasingly open questions or tasks
5. Curriculum progression
The overall breadth of learning should be used when planning curriculum progression. Children should be taught:
Early REF _Ref226448741 \r \h \* MERGEFORMAT 5MiddleLaterNumber and the number systemto estimate the number of objects and count them, recognising conservation of number
to read, write and order numbers to 100 and beyond using a range of representations REF _Ref225592118 \r \h \* MERGEFORMAT 6
to explore and explain patterns REF _Ref225070818 \r \h \* MERGEFORMAT 7 , including number sequences in the counting system
to group, match, sort, partition and recombine numbers, developing an understanding of place value
to understand and interpret negative numbers, simple fractions REF _Ref223429391 \r \h \* MERGEFORMAT 19, large numbers and tenths, written as decimals, in practical and everyday contexts
to generate and explore a range of number patterns, including multiples REF _Ref225062430 \r \h \* MERGEFORMAT 20
to make and test general statements about numbers, sort and classify numbers and explain methods and findings
to approximate numbers, including rounding REF _Ref224974458 \r \h \* MERGEFORMAT 21 , and understand when that can be useful
about the representation of number in different contemporary cultures REF _Ref223429430 \r \h \* MERGEFORMAT 22to use decimals up to three decimal places in measurement contexts
to understand and use the equivalence of families of fractions and their decimal representation when ordering and comparing
to explore number patterns and properties REF _Ref225062540 \r \h \* MERGEFORMAT 31,and represent them using graphs, simple formulae and ICT REF _Ref225064559 \r \h \* MERGEFORMAT 32
about the development of the number system REF _Ref223429874 \r \h \* MERGEFORMAT 33
to interpret computer and calculator displays and round to an appropriate level of accuracy
Number operations and calculationa range of strategies for combining, partitioning, grouping and sharing (including doubling and halving) and increasing and decreasing numbers, to solve practical problems REF _Ref226427352 \r \h \* MERGEFORMAT 8.
to use number bonds to ten to add and subtract mentally REF _Ref223429077 \r \h \* MERGEFORMAT 9 whole numbers with one or two significant figures
to represent addition and subtraction as number sentences including finding missing numbers and understanding the equals sign REF _Ref225592269 \r \h \* MERGEFORMAT 10
to compare two numbers by finding the difference between them REF _Ref225592552 \r \h \* MERGEFORMAT 23
to use the relationship between addition and subtraction REF _Ref225592866 \r \h \* MERGEFORMAT 24 and addition and multiplication to understand and generate equivalent expressions REF _Ref225593017 \r \h \* MERGEFORMAT 25
to use simple fractions to find fractional parts and express proportions
to select from a range of mental strategies for the addition and subtraction of numbers with two significant figures
to understand division as grouping and as sharing and solve division problems using multiplication facts REF _Ref225592743 \r \h \* MERGEFORMAT 26
to visualise and understand multiplication represented as an array, record multiplication as number sentences and solve problems using multiplication facts
to use estimation to find approximate answers to calculations REF _Ref225593176 \r \h \* MERGEFORMAT 27,to record calculations and check answers and methodsto use proportional reasoning REF _Ref225594360 \r \h \* MERGEFORMAT 34 to compare numbers and quantities and solve problems
to extend their knowledge of multiplication facts to 1010 and use them to solve multiplication and division problems
to understand and use different models of division, including interpreting the outcome of a division calculation, in relation to the context, where the answer is not a whole number
to recognise and use the relationship between fractions and division and represent division as number sentences REF _Ref225594734 \r \h \* MERGEFORMAT 35
to recognise and use the relationships between addition, subtraction, multiplication and division
to develop a range of strategies REF _Ref241567619 \r \h \* MERGEFORMAT 36 including mental and written ones, for calculating and checking, including using a calculator or computer efficiently
to solve multi-step problems involving more than one operation
Moneyto use coins of different values and recognise the equivalence of different combinations of coins REF _Ref225592370 \r \h \* MERGEFORMAT 11
to compare and order costs of different items
to record amounts of money using pounds and/or pence, converting between them as appropriate
how to handle amounts of money in the contexts of shopping, saving up and enterprise activities REF _Ref225069047 \r \h \* MERGEFORMAT 28to solve problems related to borrowing, spending and saving REF _Ref223429945 \r \h \* MERGEFORMAT 37
to understand and convert between different currencies
how to manage money REF _Ref223429972 \r \h \* MERGEFORMAT 38 and prepare budgets for events, including using spreadsheets
Measuresto compare and order objects and events REF _Ref226365789 \r \h \* MERGEFORMAT 12
to create and use whole number scales REF _Ref223429104 \r \h \* MERGEFORMAT 13 to measure
to recognise when length and capacity are conserved
to use standard units to estimate measures and to measure with appropriate accuracy
to recognise and use equivalent representations of time
to measure angles using fractions of turn and right angles
to explore the development of different measuring systems, including metric and imperial measures
to recognise when area, volume and mass are conserved
to convert between units within the metric system
to use an angle measurer to measure angles in degrees
to solve problems involving time and time intervals, including time represented by the 24 hour clock
to use decimal calculations to solve problems with measures
Geometryto identify, group, match, sort and compare common shapes REF _Ref223429139 \r \h \* MERGEFORMAT 14 using geometric properties REF _Ref226365973 \r \h \* MERGEFORMAT 15
to identify, reproduce and generate geometric patterns including the use of practical resources and ICT
to generate instructions for straight and turning movement REF _Ref223429184 \r \h \* MERGEFORMAT 16
to recognise symmetry properties of 2D shapes and patterns
to make simple scalings REF _Ref223429559 \r \h \* MERGEFORMAT 29 of objects and drawings
to understand and use angle as the measure of turn
to understand perimeter as a length and to find the perimeter of rectangles and other shapes
to create sequences of instructions using ICT, including generating symmetric and repeating geometric patternsto use and make maps, scale models and diagrams for a purpose
to understand area as the space enclosed by a perimeter on a plane, and find areas of rectangles and related shapes REF _Ref226448887 \r \h \* MERGEFORMAT 39
to solve practical problems involving 3D objects REF _Ref223430074 \r \h \* MERGEFORMAT 40
to visualise geometric objects REF _Ref223430108 \r \h \* MERGEFORMAT 41 and to recognise and make 2D representations of 3D shapes
to create and refine sequences of instructions, using ICT to construct and explore geometric patterns and problems REF _Ref225064800 \r \h \* MERGEFORMAT 42
to explore aspects of geometry to find out about its origins REF _Ref223430163 \r \h \* MERGEFORMAT 43, and its use in different cultures, religions, art and architecture REF _Ref223430210 \r \h \* MERGEFORMAT 44
Statisticsto generate and explore questions that require the collection and analysis of information
to collect, group, match, sort, record and represent information REF _Ref223429201 \r \h \* MERGEFORMAT 17 for a purpose and store it using ICT REF _Ref225064192 \r \h 18
to interpret and draw conclusions from information they have collected REF _Ref225064192 \r \h \* MERGEFORMAT 18to collect and structure information using ICT so that it can be searched and analysed, including using appropriate field headings and data types REF _Ref225064345 \r \h \* MERGEFORMAT 30
to use frequency diagrams and bar charts to represent and record information
to interpret their own and others' datahow statistics are used in society today REF _Ref225777817 \r \h \* MERGEFORMAT 45
to use different kinds of averages and range to summarise and compare data sets
to use data to assess likelihood and risk and develop an understanding of probability through computer simulations, games and consideration of outcomes of everyday situations
to discuss, sort and order events according to their likelihood of occurring
to answer questions or test hypotheses by using ICT to collect, store, analyse and present data REF _Ref225595054 \r \h \* MERGEFORMAT 46
to use ICT to represent data REF _Ref223430247 \r \h \* MERGEFORMAT 47 on a scattergraph, and proportional data REF _Ref223430278 \r \h \* MERGEFORMAT 48 in a pie chart in order to explore possible relationships and interpret the findings REF _Ref225778040 \r \h \* MERGEFORMAT 49
Explanatory text
Early stage
Each area of learning should build on children's experiences and development in the Early Years Foundation Stage to ensure continuity of curriculum provision and their continuing progress
For example, number lines, number squares, structural apparatus
This includes additive number sequences, such as counting in groups of e.g. 2, 5 or 10, odds and evens; and relationships between numbers, e.g. the sum of two odd numbers is always even. Using calculators to explore number patterns and properties is important here
This lays the foundations for understanding number operations
For example 700+300=; 60+(=100; 57+33=; 57-8=; this develops their understanding of the inverse relationship between addition and subtraction
For example 3+1=1+3; 3+1=(+2; 3+1=5-(
Including in the context of buying and selling involving role play
This includes mass, time and length, for example answering questions such as 'which is heaviest?' 'which takes longer?' or 'which is longest?'
Number scales include standard and non-standard units
Common shapes include triangle, square, rhombus, rectangle, kite, parallelogram, circle, cube, prism, pyramid, cylinder, cone, and sphere
Geometric properties include edges, vertices, faces, right-angles, straight, curved, closed and open
For example using a programmable toy or describing a familiar journey including change of direction/angle of turn
This includes using Venn and Carroll diagrams, simple frequency diagrams and simple data-handling software to create tables and graphs
Including outcomes from using simple data-handling software
Middle stage
Simple fractions include half, third, quarter, fifth, tenth, two-thirds and three-quarters
using ICT for changing values and exploring in a spreadsheet model
For example rounding to the nearest ten, hundred and thousand
For example Arabic, Chinese and Indian numerals
For example finding how much the temperature changed
For example since 54+37=91, 91-37=54 and 91-54=37
For example 313=310+33; 519=520-51
Multiplication facts should include 2, 3, 4, 5 and 10
For example to estimate the cost of an apple sold in a pack of four or to recognise that 296+735 will be approximately 1,000
For example to find and compare unit costs of items that are sold in multiple unit quantities
Simple scales include half, twice and ten times
Analysis should include discussion about 'reasonableness' of outcomes
Later stage
This includes factors, primes and square numbers
Changing variables and rules in spreadsheet models; using graphing software
For example the number system we use today is Hindu-Arabic, the Roman and Egyptian number systems do not use a place value; Babylonian numbers and Mayan numbers use base 60 and base 20 respectively; Greeks explored square and triangle numbers
Including simple ratio and percentages - for example 45 is three times greater than 15, they are in the ratio 3:1
For example 3255=(300+25)5=3005+255 or 325/5=300/5+25/5
This includes mental methods, informal and formal written methods and using technology
This includes using and interpreting information from external sources and making decimal calculations
This includes using the context of enterprise activities where children need to work out a range of budgetary options, developing awareness of profit and loss
This includes triangles and shapes that are made up of triangles and rectangles including the surface area of 3D objects
This includes developing understanding of the volume of cuboids by solving problems such as what is the smallest possible box to hold six smaller boxes?
This includes imagining what something will look like in different orientations
This should include use of procedures to improve efficiency of sequences
For example Greek architecture and discoveries, stone circles and pyramids
For example Islamic patterns, Japanese temple art, Rangoli patterns, modern art and ancient and modern architecture
For example statistics are used to inform the public about how the local council spend their money, to monitor safety in factories, to inform decisions about whether to install traffic lights, or to decide what stock to order
For example using data types including text, number, date, currency, yes/no and error checking through inspecting outcomes
For example height and weight for a chart on a child's development
Proportional data means data where fractions of the population are represented, such as how a council spends its budget, or how all the children in a class travel to school
This should include understanding how these diagrams work and choosing the appropriate representation to present the data
Explanatory text
Mathematical Understanding Page PAGE 1 of NUMPAGES 8
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