Mathematics learning objectives
The 'example' links below objectives
connect to a supplement of examples that illustrate what pupils should know and
be able to do within a particular strand by the end of the year.
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Year
7(Class6) |
Year
8(Class7) |
Year
9(Class8) |
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1 Mathematical processes and applications |
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1.1
Representing |
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1.2
Analysing – use mathematical reasoning |
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1.3
Analysing – use appropriate mathematical procedures |
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1.4
Interpreting and evaluating |
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1.5
Communicating and reflecting |
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2 Number |
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2.1
Place value, ordering and rounding |
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understand and use decimal
notation and place value; multiply and divide integers and decimals by 10,
100, 1000, and explain the effect |
read and write positive integer
powers of 10; multiply and divide integers and decimals by 0.1, 0.01 |
extend knowledge of integer powers
of 10; recognise the equivalence of 0.1, |
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compare and order decimals in
different contexts; know that when comparing measurements the units must be
the same |
order decimals |
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round positive whole numbers to
the nearest 10, 100 or 1000, and decimals to the nearest whole number or one
decimal place |
round positive numbers to any
given power of 10; round decimals to the nearest whole number or to one or
two decimal places |
use rounding to make estimates and
to give solutions to problems to an appropriate degree of accuracy |
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2.2
Integers, powers and roots |
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understand negative numbers as
positions on a number line; order, add and subtract integers in context |
add, subtract, multiply and divide
integers |
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recognise and use multiples,
factors, primes (less than 100), common factors, highest common factors and
lowest common multiples in simple cases; use simple tests of divisibility |
use multiples, factors, common
factors, highest common factors, lowest common multiples and primes; find the
prime factor decomposition of a number, e.g. 8000 = 26 × 53
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use the prime factor decomposition
of a number |
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recognise the first few triangular
numbers; recognise the squares of numbers to at least 12 × 12 and
the corresponding roots |
use squares, positive and negative
square roots, cubes and cube roots, and index notation for small positive
integer powers |
use ICT to estimate square roots
and cube roots |
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use index notation for integer
powers; know and use the index laws for multiplication and division of
positive integer powers |
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2.3
Fractions, decimals, percentages, ratio and proportion |
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express a smaller whole number as
a fraction of a larger one; simplify fractions by cancelling all common
factors and identify equivalent fractions; convert terminating decimals to
fractions, e.g. 0.23 = |
recognise that a recurring decimal
is a fraction; use division to convert a fraction to a decimal; order
fractions by writing them with a common denominator or by converting them to
decimals |
understand the equivalence of
simple algebraic fractions; know that a recurring decimal is an exact
fraction |
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add and subtract simple fractions
and those with common denominators; calculate simple fractions of quantities
and measurements (whole-number answers); multiply a fraction by an integer |
add and subtract fractions by
writing them with a common denominator; calculate fractions of quantities
(fraction answers); multiply and divide an integer by a fraction |
use efficient methods to add,
subtract, multiply and divide fractions, interpreting division as a
multiplicative inverse; cancel common factors before multiplying or dividing |
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understand percentage as the
'number of parts per 100'; calculate simple percentages and use percentages
to compare simple proportions |
interpret percentage as the
operator 'so many hundredths of' and express one given number as a percentage
of another; calculate percentages and find the outcome of a given percentage
increase or decrease |
recognise when fractions or
percentages are needed to compare proportions; solve problems involving
percentage changes |
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recognise the equivalence of
percentages, fractions and decimals |
use the equivalence of fractions,
decimals and percentages to compare proportions |
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understand the relationship
between ratio and proportion; use direct proportion in simple contexts; use
ratio notation, simplify ratios and divide a quantity into two parts in a
given ratio; solve simple problems involving ratio and proportion using
informal strategies |
apply understanding of the
relationship between ratio and proportion; simplify ratios, including those
expressed in different units, recognising links with fraction notation;
divide a quantity into two or more parts in a given ratio; use the unitary
method to solve simple problems involving ratio and direct proportion |
use proportional reasoning to
solve problems, choosing the correct numbers to take as 100%, or as a whole;
compare two ratios; interpret and use ratio in a range of contexts |
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2.4
Number operations |
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understand and use the rules of
arithmetic and inverse operations in the context of positive integers and
decimals |
understand and use the rules of
arithmetic and inverse operations in the context of integers and fractions |
understand the effects of
multiplying and dividing by numbers between 0 and 1; consolidate use of the
rules of arithmetic and inverse operations |
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use the order of operations,
including brackets |
use the order of operations,
including brackets, with more complex calculations |
understand the order of precedence
of operations, including powers |
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2.5
Mental calculation methods |
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recall number facts, including
positive integer complements to 100 and multiplication facts to
10 × 10, and quickly derive associated division facts |
recall equivalent fractions,
decimals and percentages; use known facts to derive unknown facts, including
products involving numbers such as 0.7 and 6, and 0.03 and 8 |
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strengthen and extend mental
methods of calculation to include decimals, fractions and percentages,
accompanied where appropriate by suitable jottings; solve simple problems
mentally |
strengthen and extend mental
methods of calculation, working with decimals, fractions, percentages,
squares and square roots, cubes and cube roots; solve problems mentally |
use known facts to derive unknown
facts; extend mental methods of calculation, working with decimals,
fractions, percentages, factors, powers and roots; solve problems mentally |
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make and justify estimates and
approximations of calculations |
make and justify estimates and
approximations of calculations |
make and justify estimates and
approximations of calculations |
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2.6
Written calculation methods |
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use efficient written methods to
add and subtract whole numbers and decimals with up to two places |
use efficient written methods to
add and subtract integers and decimals of any size, including numbers with
differing numbers of decimal places |
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multiply and divide three-digit by
two-digit whole numbers; extend to multiplying and dividing decimals with one
or two places by single-digit whole numbers |
use efficient written methods for
multiplication and division of integers and decimals, including by decimals
such as 0.6 or 0.06; understand where to position the decimal point by
considering equivalent calculations |
use efficient written methods to
add and subtract integers and decimals of any size; multiply by decimals;
divide by decimals by transforming to division by an integer |
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2.7
Calculator methods |
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carry out calculations with more
than one step using brackets and the memory; use the square root and sign
change keys |
carry out more difficult
calculations effectively and efficiently using the function keys for sign
change, powers, roots and fractions; use brackets and the memory |
use a calculator efficiently and
appropriately to perform complex calculations with numbers of any size, knowing
not to round during intermediate steps of a calculation; use the constant, π
and sign change keys; use the function keys for powers, roots and fractions;
use brackets and the memory |
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enter numbers and interpret the
display in different contexts (decimals, percentages, money, metric measures)
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enter numbers and interpret the
display in different contexts (extend to negative numbers, fractions, time) |
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2.8
Checking results |
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check results by considering
whether they are of the right order of magnitude and by working problems
backwards |
select from a range of checking
methods, including estimating in context and using inverse operations |
check results using appropriate
methods |
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3 Algebra |
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3.1
Equations, formulae, expressions and identities |
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use letter symbols to represent
unknown numbers or variables; know the meanings of the words term, expression
and equation |
recognise that letter symbols play
different roles in equations, formulae and functions; know the meanings of
the words formula and function |
distinguish the different roles
played by letter symbols in equations, identities, formulae and functions |
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understand that algebraic
operations follow the rules of arithmetic |
understand that algebraic
operations, including the use of brackets, follow the rules of arithmetic;
use index notation for small positive integer powers |
use index notation for integer
powers and simple instances of the index laws |
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simplify linear algebraic
expressions by collecting like terms; multiply a single term over a bracket
(integer coefficients) |
simplify or transform linear
expressions by collecting like terms; multiply a single term over a bracket |
simplify or transform algebraic
expressions by taking out single-term common factors; add simple algebraic
fractions |
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construct and solve simple linear
equations with integer coefficients (unknown on one side only) using an
appropriate method (e.g. inverse operations) |
construct and solve linear
equations with integer coefficients (unknown on either or both sides, without
and with brackets) using appropriate methods (e.g. inverse operations,
transforming both sides in same way) |
construct and solve linear
equations with integer coefficients (with and without brackets, negative
signs anywhere in the equation, positive or negative solution) |
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use graphs and set up equations to
solve simple problems involving direct proportion |
use algebraic methods to solve
problems involving direct proportion; relate algebraic solutions to graphs of
the equations; use ICT as appropriate |
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use systematic trial and
improvement methods and ICT tools to find approximate solutions to equations
such as x2 + x = 20 |
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explore ways of constructing
models of real-life situations by drawing graphs and constructing algebraic
equations and inequalities |
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use simple formulae from
mathematics and other subjects; substitute positive integers into linear
expressions and formulae and, in simple cases, derive a formula |
use formulae from mathematics and
other subjects; substitute integers into simple formulae, including examples
that lead to an equation to solve; substitute positive integers into
expressions involving small powers, e.g. 3x2 + 4
or 2x3; derive simple formulae |
use formulae from mathematics and
other subjects; substitute numbers into expressions and formulae; derive a
formula and, in simple cases, change its subject |
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3.2
Sequences, functions and graphs |
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describe integer sequences;
generate terms of a simple sequence, given a rule (e.g. finding a term from
the previous term, finding a term given its position in the sequence) |
generate terms of a linear
sequence using term-to-term and position-to-term rules, on paper and using a
spreadsheet or graphics calculator |
generate terms of a sequence using
term-to-term and position-to-term rules, on paper and using ICT |
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generate sequences from patterns
or practical contexts and describe the general term in simple cases |
use linear expressions to describe
the nth term of a simple arithmetic sequence, justifying its form by
referring to the activity or practical context from which it was generated |
generate sequences from practical
contexts and write and justify an expression to describe the nth term
of an arithmetic sequence |
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express simple functions in words,
then using symbols; represent them in mappings |
express simple functions
algebraically and represent them in mappings or on a spreadsheet |
find the inverse of a linear
function |
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generate coordinate pairs that satisfy
a simple linear rule; plot the graphs of simple linear functions, where y
is given explicitly in terms of x, on paper and using ICT; recognise
straight-line graphs parallel to the x-axis or y-axis |
generate points in all four
quadrants and plot the graphs of linear functions, where y is given
explicitly in terms of x, on paper and using ICT; recognise that
equations of the form y = mx + c
correspond to straight-line graphs |
generate points and plot graphs of
linear functions, where y is given implicitly in terms of x
(e.g. ay + bx = 0, y + bx + c = 0),
on paper and using ICT; find the gradient of lines given by equations of the
form y = mx + c, given values for m
and c |
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plot and interpret the graphs of
simple linear functions arising from real-life situations, e.g. conversion
graphs |
construct linear functions arising
from real-life problems and plot their corresponding graphs; discuss and
interpret graphs arising from real situations, e.g. distance–time graphs |
construct functions arising from
real-life problems and plot their corresponding graphs; interpret graphs
arising from real situations, e.g. time series graphs |
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use ICT to explore the graphical
representation of algebraic equations and interpret how properties of the
graph are related to features of the equation, e.g. parallel and
perpendicular lines |
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interpret the meaning of various
points and sections of straight-line graphs, including intercepts and
intersection, e.g. solving simultaneous linear equations |
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4 Geometry and measures |
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4.1
Geometrical reasoning |
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use correctly the vocabulary,
notation and labelling conventions for lines, angles and shapes |
distinguish between conventions,
definitions and derived properties |
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identify parallel and
perpendicular lines; know the sum of angles at a point, on a straight line
and in a triangle; recognise vertically opposite angles |
identify alternate angles and
corresponding angles; understand a proof that: • the angle sum of a triangle is
180° and of a quadrilateral is 360° |
explain how to find, calculate and
use: • the sums of the interior and
exterior angles of quadrilaterals, pentagons and hexagons |
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know the definition of a circle
and the names of its parts; explain why inscribed regular polygons can be
constructed by equal divisions of a circle |
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identify and use angle, side and
symmetry properties of triangles and quadrilaterals; explore geometrical
problems involving these properties, explaining reasoning orally, using
step-by-step deduction supported by diagrams |
solve geometrical problems using
side and angle properties of equilateral, isosceles and right-angled
triangles and special quadrilaterals, explaining reasoning with diagrams and
text; classify quadrilaterals by their geometrical properties |
solve problems using properties of
angles, of parallel and intersecting lines, and of triangles and other
polygons, justifying inferences and explaining reasoning with diagrams and
text |
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know that if two 2-D shapes are
congruent, corresponding sides and angles are equal |
understand congruence and explore
similarity |
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investigate Pythagoras' theorem,
using a variety of media, through its historical and cultural roots,
including 'picture' proofs |
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use 2-D representations to
visualise 3-D shapes and deduce some of their properties |
visualise 3-D shapes from their
nets; use geometric properties of cuboids and shapes made from cuboids; use
simple plans and elevations |
visualise and use 2-D
representations of 3-D objects; analyse 3-D shapes through 2-D projections,
including plans and elevations |
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4.2
Transformations and coordinates |
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understand and use the language
and notation associated with reflections, translations and rotations |
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recognise and visualise the
symmetries of a 2-D shape |
identify all the symmetries of 2-D
shapes |
identify reflection symmetry in
3-D shapes |
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transform 2-D shapes by: • reflecting in given mirror lines |
transform 2-D shapes by rotation,
reflection and translation, on paper and using ICT |
recognise that translations,
rotations and reflections preserve length and angle, and map objects on to
congruent images |
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explore these transformations and
symmetries using ICT |
try out mathematical
representations of simple combinations of these transformations |
explore and compare mathematical
representations of combinations of translations, rotations and reflections of
2-D shapes, on paper and using ICT |
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devise instructions for a computer
to generate and transform shapes |
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understand and use the language
and notation associated with enlargement; enlarge 2-D shapes, given a centre
of enlargement and a positive integer scale factor; explore enlargement using
ICT |
enlarge 2-D shapes, given a centre
of enlargement and a positive integer scale factor, on paper and using ICT;
identify the scale factor of an enlargement as the ratio of the lengths of
any two corresponding line segments; recognise that enlargements preserve
angle but not length, and understand the implications of enlargement for
perimeter |
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make scale drawings |
use and interpret maps and scale
drawings in the context of mathematics and other subjects |
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use conventions and notation for
2-D coordinates in all four quadrants; find coordinates of points determined
by geometric information |
find the midpoint of the line
segment AB, given the coordinates of points A and B |
use the coordinate grid to solve
problems involving translations, rotations, reflections and enlargements |
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4.3
Construction and loci |
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use a ruler and protractor to: • measure and draw lines to the
nearest millimetre and angles, including reflex angles, to the nearest degree |
use straight edge and compasses to
construct: • the midpoint and perpendicular
bisector of a line segment |
use straight edge and compasses to
construct triangles, given right angle, hypotenuse and side (RHS) |
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use ICT to explore constructions |
use ICT to explore these
constructions |
use ICT to explore constructions
of triangles and other 2-D shapes |
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use ruler and protractor to
construct simple nets of 3-D shapes, e.g. cuboid, regular tetrahedron,
square-based pyramid, triangular prism |
find simple loci, both by
reasoning and by using ICT, to produce shapes and paths, e.g. an equilateral
triangle |
find the locus of a point that
moves according to a simple rule, both by reasoning and by using ICT |
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4.4
Measures and mensuration |
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choose and use units of
measurement to measure, estimate, calculate and solve problems in everyday
contexts; convert one metric unit to another, e.g. grams to kilograms; read
and interpret scales on a range of measuring instruments |
choose and use units of
measurement to measure, estimate, calculate and solve problems in a range of
contexts; know rough metric equivalents of imperial measures in common use,
such as miles, pounds (lb) and pints |
solve problems involving
measurements in a variety of contexts; convert between area measures (e.g. mm2
to cm2, cm2 to m2, and vice versa) and
between volume measures (e.g. mm3 to cm3, cm3
to m3, and vice versa) |
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distinguish between and estimate
the size of acute, obtuse and reflex angles |
use bearings to specify direction |
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interpret and explore combining
measures into rates of change in everyday contexts (e.g. km per hour, pence
per metre); use compound measures to compare in real-life contexts (e.g.
travel graphs and value for money), using ICT as appropriate |
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know and use the formula for the
area of a rectangle; calculate the perimeter and area of shapes made from
rectangles |
derive and use formulae for the
area of a triangle, parallelogram and trapezium; calculate areas of compound
shapes |
know and use the formulae for the
circumference and area of a circle |
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calculate the surface area of
cubes and cuboids |
know and use the formula for the
volume of a cuboid; calculate volumes and surface areas of cuboids and shapes
made from cuboids |
calculate the surface area and
volume of right prisms |
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5 Statistics |
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5.1
Specifying a problem, planning and collecting data |
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suggest possible answers, given a
question that can be addressed by statistical methods |
discuss a problem that can be
addressed by statistical methods and identify related questions to explore |
suggest a problem to explore using
statistical methods, frame questions and raise conjectures |
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decide which data would be
relevant to an enquiry and possible sources |
decide which data to collect to
answer a question, and the degree of accuracy needed; identify possible
sources; consider appropriate sample size |
discuss how different sets of data
relate to the problem; identify possible primary or secondary sources;
determine the sample size and most appropriate degree of accuracy |
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plan how to collect and organise
small sets of data from surveys and experiments: • design data collection sheets or
questionnaires to use in a simple survey |
plan how to collect the data;
construct frequency tables with equal class intervals for gathering
continuous data and two-way tables for recording discrete data |
design a survey or experiment to
capture the necessary data from one or more sources; design, trial and if
necessary refine data collection sheets; construct tables for gathering large
discrete and continuous sets of raw data, choosing suitable class intervals;
design and use two-way tables |
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collect small sets of data from
surveys and experiments, as planned |
collect data using a suitable
method (e.g. observation, controlled experiment, data logging using ICT) |
gather data from specified
secondary sources, including printed tables and lists, and ICT-based sources,
including the internet |
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5.2
Processing and representing data |
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calculate statistics for small
sets of discrete data: • find the mode, median and range,
and the modal class for grouped data |
calculate statistics for sets of
discrete and continuous data, including with a calculator and spreadsheet;
recognise when it is appropriate to use the range, mean, median and mode and,
for grouped data, the modal class |
calculate statistics and select
those most appropriate to the problem or which address the questions posed |
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construct, on paper and using ICT,
graphs and diagrams to represent data, including: • bar-line graphs |
construct graphical
representations, on paper and using ICT, and identify which are most useful
in the context of the problem. Include: • pie charts for categorical data |
select, construct and modify, on
paper and using ICT, suitable graphical representations to progress an
enquiry and identify key features present in the data. Include: • line graphs for time series |
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work through the entire handling
data cycle to explore relationships within bi-variate data, including
applications to global citizenship, e.g. how fair is our society? |
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5.3
Interpreting and discussing results |
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interpret diagrams and graphs
(including pie charts) and draw simple conclusions based on the shape of
graphs and simple statistics for a single distribution |
interpret tables, graphs and
diagrams for discrete and continuous data, relating summary statistics and
findings to the questions being explored |
interpret graphs and diagrams and
make inferences to support or cast doubt on initial conjectures; have a basic
understanding of correlation |
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compare two simple distributions
using the range and one of the mode, median or mean |
compare two distributions using
the range and one or more of the mode, median and mean |
compare two or more distributions
and make inferences, using the shape of the distributions and appropriate
statistics |
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write a short report of a
statistical enquiry, including appropriate diagrams, graphs and charts, using
ICT as appropriate; justify the choice of presentation |
write about and discuss the
results of a statistical enquiry using ICT as appropriate; justify the
methods used |
review interpretations and results
of a statistical enquiry on the basis of discussions; communicate these
interpretations and results using selected tables, graphs and diagrams |
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5.4
Probability |
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use vocabulary and ideas of
probability, drawing on experience |
interpret the results of an
experiment using the language of probability; appreciate that random
processes are unpredictable |
interpret results involving
uncertainty and prediction |
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understand and use the probability
scale from 0 to 1; find and justify probabilities based on equally likely
outcomes in simple contexts; identify all the possible mutually exclusive
outcomes of a single event |
know that if the probability of an
event occurring is p then the probability of it not occurring is
1 − p; use diagrams and tables to record in a
systematic way all possible mutually exclusive outcomes for single events and
for two successive events |
identify all the mutually
exclusive outcomes of an experiment; know that the sum of probabilities of
all mutually exclusive outcomes is 1 and use this when solving problems |
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estimate probabilities by
collecting data from a simple experiment and recording it in a frequency
table; compare experimental and theoretical probabilities in simple contexts |
compare estimated experimental
probabilities with theoretical probabilities, recognising that: • if an experiment is repeated the
outcome may, and usually will, be different |
compare experimental and
theoretical probabilities in a range of contexts; appreciate the difference
between mathematical explanation and experimental evidence |
