select and use suitable problem-solving strategies and efficient techniques to solve numerical and algebraic problems
identify what further information may be required in order to pursue a particular line of enquiry and give reasons for following or rejecting particular approaches
use algebra to formulate and solve a simple problem − identifying the variable, setting up an equation, solving the equation and interpreting the solution in the context of the problem
make mental estimates of the answers to calculations; use checking procedures, including use of inverse operations; work to stated levels of accuracy
select and use appropriate and efficient techniques and strategies to solve problems of increasing complexity, involving numerical and algebraic manipulation
break down a complex calculation into simpler steps before attempting to solve it and justify their choice of methods
make mental estimates of the answers to calculations; present answers to sensible levels of accuracy; understand how errors are compounded in certain calculations
use a range of strategies to create numerical, algebraic or graphical representations of a problem and its solution; move from one form of representation to another to get different perspectives on the problem
present and interpret solutions in the context of the original problem
review and justify their choice of mathematical presentation
discuss their work and explain their reasoning using an increasing range of mathematical language and notation
use a variety of strategies and diagrams for establishing algebraic or graphical representations of a problem and its solution; move from one form of representation to another to get different perspectives on the problem
examine critically, improve, then justify their choice of mathematical presentation, present a concise, reasoned argument
inverse operations and the relationships between them; simple integer powers and their corresponding roots; methods of simplification in order to select and use suitable strategies and techniques to solve problems
select appropriate operations, methods and strategies to solve number problems, including trial and improvement where a more efficient method to find the solution is not obvious
estimate answers to problems; use a variety of checking procedures, including working the problem backwards, and considering whether a result is of the right order of magnitude
give solutions in the context of the problem to an appropriate degree of accuracy, interpreting the solution shown on a calculator display, and recognising limitations on the accuracy of data and measurements
knowledge of operations and inverse operations (including powers and roots) and of methods of simplification (including factorisation and the use of the commutative, associative and distributive laws of addition, multiplication and factorisation)
check and estimate answers to problems; select and justify appropriate degrees of accuracy for answers to problems; recognise limitations on the accuracy of data and measurements
use index notation for simple integer powers, and simple instances ofindex laws; substitute positive and negative numbers into expressions such as 3x^2 + 4 and 2x^3
use index notation for simple integer powers, and simple instances of index laws; substitute positive and negative numbers into expressions such as 3x^2 + 4 and 2x^3
solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns, one of which is linear in each unknown, and the other is linear in one unknown and quadratic in the other, or
where the second is of the form x^2 + y^2 = r^2
graphs representing the linear equations; find approximate solutions of a point of intersection of the graphs of a linear and of simple quadratic functions
select problem-solving strategies and resources, including ICT tools, to use in geometrical work, and monitor their effectiveness; consider and explain the extent to which the selections they made were
appropriate
select and combine known facts and problem-solving strategies to solve complex problems
identify what further information is needed to solve a geometrical problem; break complex problems down into a series of tasks; develop and follow alternative lines of enquiry
select the problem-solving strategies to use in geometrical work, and consider and explain the extent to which the selections they made were appropriate
select and combine known facts and problem-solving strategies to solve more complex geometrical problems
develop and follow alternative lines of enquiry, justifying their decisions to follow or reject particular approaches
Communicating
interpret, discuss and synthesise geometrical information presented in a variety of forms
communicate mathematically with emphasis on a critical examination of the presentation and organisation of results, and on effective use of symbols and geometrical diagrams
use geometrical language appropriately
review and justify their choices of mathematics presentation
communicate mathematically, with emphasis on a critical examination of the presentation and organisation of results, and on effective use of symbols and geometrical diagrams
use precise formal language and exact methods for analysing geometrical configurations
explore the geometry of cuboids (including cubes), and shapes made from cuboids
use 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, including plan and elevation; solve problems involving surface areas and volumes of prisms and cylinders
use 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections; solve problems involving surface areas and volumes of prisms, pyramids, cylinders, cones and spheres
carry out each of the four aspects of the handling data cycle to solve problems:
(i) specify the problem and plan
(ii) collect data from a variety of suitable sources
(iii) process and represent the data
(iv) interpret and discuss the data
identify what further information is needed to pursue a particular line of enquiry; select the problem-solving strategies to use in statistical work, and monitor their effectiveness
select and organise the appropriate mathematics and resources to use for a task
review progress while working; check and evaluate solutions
find loci, both by reasoning and by using ICT to produce shapes and paths
select the problem-solving strategies to use in statistical work, and monitor their effectiveness
Communicating
interpret, discuss and synthesise information presented in a variety of forms
communicate mathematically, including using ICT, making use of diagrams and related explanatory text
examine critically, and justify, their choices of mathematical presentation of problems involving data
communicate mathematically, with emphasis on the use of an increasing range of diagrams and related explanatory text, on the selection of their mathematical presentation, explaining its purpose, and on the use of symbols to convey statistical meaning
Reasoning
apply mathematical reasoning, explaining and justifying inferences and deductions
identify exceptional or unexpected cases when solving statistical problems
explore connections in mathematics and look for relationships between variables when analysing data
recognise the limitations of any assumptions and the effects that varying the assumptions could have on the conclusions drawn from data analysis
apply mathematical reasoning, explaining and justifying inferences and deductions, justifying arguments and solutions
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