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The content for this section is sub-divided
into the following areas:
Integers
Top
| Content |
Resources |
| use their
previous understanding of integers
and place value to deal with arbitrarily
large positive numbers and round
them to a given power of 10 |
|
| understand and use positive numbers, both
as
positions and translations on a number line order integers |
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| use the concepts and vocabulary of
factor (divisor),
multiple and common factor, highest
common factor, least
common multiple, prime
number and prime factor decomposition |
W
primes, factors and multiplies
A
Who wants to be a millionaire
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Powers and roots
Top
| Content |
Resources |
| use
the terms square, positive square root, negative
square root, cube and
cube root
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| use index notation for squares, cubes and
powers
of 10 |
W
introducing powers of 10
W
powers of 10
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| use
index laws for multiplication
and division of integer powers |
N
W indices |
| use standard
index form, expressed in conventional
notation and on a calculator
display |
W
W standard form |
Fractions
Top
| Content |
Resources |
| understand
equivalent fractions, |
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| simplifying a fraction by cancelling all
common
factors |
W
fractions |
| order fractions by rewriting them with a
common
denominator |
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Decimals
Top
| Content |
Resources |
| use
decimal notation and recognise that each
terminating decimal is a fraction |
W
money |
| recognise that recurring decimals are exact
fractions, and
that some exact fractions
are recurring decimals |
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| order
decimals |
W
decimals |
Percentages
Top
| Content |
Resources |
| understand
that ‘percentage’ means ‘number
of parts per 100’ and use this to compare
proportions |
O
W
W
O
percentage poster
W
W
mental percentages
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| interpret percentage as the operator ‘so
many
hundredths of’ |
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| use percentage in real-life situations |
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Ratio
Top
| Content |
Resources |
| use
ratio notation, including reduction to its
simplest form and its various links to fraction
notation |
O
equivalent ratios
W
fractions and ratio
W
ratio
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| maps and scale drawings, paper sizes and
gears |
W
W
map scales
W
ratio
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Number operations and the relationships between
them
Top
| Content |
Resources |
| add,
subtract, multiply and divide integers and
then any number |
W
add and subtract negative numbers
W
negative numbers
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| multiply or divide any number by powers
of
10, and any positive number by a number
between 0 and 1 |
W
powers of 10 |
| find the prime factor decomposition of
positive
integers |
|
| understand ‘reciprocal’ as
multiplicative inverse, knowing that any non-zero
number multiplied by its reciprocal
is 1 (and that zero has no reciprocal, because division by zero is not
defined) |
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| multiply and divide by a negative
number |
W
negative numbers |
| use index
laws to simplify and calculate the
value of numerical expressions involving
multiplication and division of integer
powers |
|
| fractional
and negative powers |
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| use inverse operations |
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| understanding
that
the inverse operation of raising a positive number to power n is raising the result of this
operation to power 1/n |
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| use brackets and the hierarchy of operations |
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| calculate a given fraction of a given
quantity,
expressing the answer as a fraction |
|
| express a given number as a fraction of
another |
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| add and subtract fractions by writing them
with
a common denominator |
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| perform short division to convert a simple
fraction
to a decimal |
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| distinguish between fractions with
denominators
that have only prime factors
of 2 and 5 (which are represented
by terminating decimals), and other
fractions (which are represented
by
recurring decimals)
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| convert a recurring decimal to a
fraction |
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| understand and use unit fractions as
multiplicative
inverses |
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| multiply and divide a fraction by an
integer,
and multiply a fraction by a unit fraction
and by a general fraction |
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| convert simple fractions of a whole to
percentages
of the whole and vice versa |
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| then understand the multiplicative
nature of percentages as operators calculate an original amount when given the transformed amount after
a percentage
change |
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| reverse percentage problems |
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| divide a quantity in a given ratio |
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Mental methods
Top
| Content |
Resources |
| recall
all positive integer complements to 100 |
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| recall all multiplication facts to
and
use them to derive quickly the
corresponding division facts |
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| recall the cubes of 2, 3, 4, 5 and 10 |
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| the facts that n^0=
1 and n^-1 = 1/n for
positive integers n, the
corresponding rule for negative numbers, n^1/2
and n^1/3 for any positive
number n |
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| recall the
fraction-to-decimal conversion of familiar
simple fractions |
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| round to the nearest integer and to one
significant figure |
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| round to a given number
of significant figures |
O
W
rounding off
W
rounding and accuracy
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| estimate answers to problems involving
decimals |
W
estimating |
| develop a range of strategies for mental
calculation |
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| derive unknown facts from those they
know |
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| convert between ordinary and standard
index form representations, converting to
standard index form to make sensible estimates
for calculations involving multiplication
and/or division |
W
W
standard form |
| add and subtract mentally numbers with
up
to two decimal places |
W
adding decimals |
| multiply and divide numbers with no
more than one decimal digit, using the commutative, associative, and distributive laws and factorisation where
possible, or place
value adjustments |
W
Decimal fun multiplication |
Written methods
Top
| Content |
Resources |
| use
standard column procedures for addition
and subtraction of integers and decimals |
|
| use standard column procedures for
multiplication
of integers and decimals, understanding
where to position the decimal point
by considering what happens if
they multiply equivalent fractions |
W
long multiplication
W
decimal multiplication
W
decimal division
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| use efficient methods to calculate with
fractions,
including cancelling common factors
before carrying out the calculation, recognising
that, in many cases, only a fraction
can express the exact answer |
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| solve simple percentage problems,
including increase and decrease |
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| reverse percentages |
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| represent
repeated proportional change using
a multiplier raised to a power compound
interest |
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| calculate an
unknown quantity from quantities
that vary in direct or inverse
proportion |
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| calculate with
standard index form |
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| use surds
and in
exact calculations, without
a calculator |
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| solve word problems about ratio and
proportion,
including using informal strategies
and the unitary method of solution |
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| rationalise
a denominator |
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Calculator methods
Top
| Content |
Resources |
| use
calculators effectively: know how to enter
complex calculations and use function
keys for reciprocals, squares and powers |
W
W
W Percentages |
| use an extended range of function keys,
including trigonometrical and
statistical functions relevant
across this programme
of study |
|
| use calculators,
or written methods, to calculate
the upper and lower bounds of calculations, particularly when working with measurements |
|
| enter a range of calculations, including
those
involving measures, time
calculations in which fractions of an hour must
be entered as fractions or as decimals |
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| understand the calculator display,
interpreting
it correctly, and knowing not to
round during the intermediate steps of a calculation |
|
|
use standard index form display and how to enter numbers in standard
index form
|
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| use
calculators for reverse percentage calculations
by doing an appropriate division. |
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| use calculators to explore exponential
growth and decay, using a multiplier and
the power key. |
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Solving numerical problems
Top
| Content |
Resources |
| draw
on their knowledge of operations and inverse operations (including powers
and roots), |
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| draw on their knowledge of the operations
and
the relationships between them, and of simple
integer powers and their corresponding
roots, to solve problems involving
ratio and proportion, repeated
proportional change, fractions, percentages and reverse
percentages, inverse
proportion, surds, measures and
conversion between measures, and compound measures defined within a particular
situation,
a
range of measures including
speed, metric units, and
conversion between metric and common
imperial units, set in a variety of contexts |
|
| 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 |
W
number 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. |
|
Use of symbols
Top
| Content |
Resources |
| distinguish
the different roles played by letter
symbols in algebra, using
the correct notational
conventions for multiplying or
dividing by a given number, and knowing that letter
symbols represent definite unknown numbers
in equations, defined quantities or
variables in formulae, general, unspecified
and independent numbers in identities
and in functions they define new expressions
or quantities by referring to known
quantities |
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| understand that the transformation of
algebraic
expressions obeys and generalised
the rules of arithmetic |
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| expand the product of two linear
expressions |
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|
manipulate algebraic expressions by
collecting like terms, by multiplying a single
term over a bracket, and by taking out
single common term factors, taking out common factors, factorising quadratic
expressions
including the difference of two
squares and cancelling common factors
in rational expressions
|
W
simplifying expressions
W
factorising
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| know the
meaning of and use the words ‘equation’, ‘formula’, ‘identity’ and
‘expression’ |
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Index notation
Top
| Content |
Resources |
| use
index notation for simple integer powers
and simple instances of index laws |
|
| substitute positive and negative numbers
into
expressions |
|
Equations
Top
| Content |
Resources |
| set
up simple equations |
W
forming equations
W
W
making equations
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| solve simple equations by using inverse
operations or by transforming both sides
in the same way |
W
W
solving equations
H
(Answer grid)
Number grid
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Linear equations
Top
| Content |
Resources |
| solve
linear equations, with integer or
fractional coefficients, in which the unknown appears
on either side or on both sides of the
equation |
W
W
solving equations
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| solve linear equations that require
prior simplification of brackets,
including those that have
negative signs occurring anywhere in
the equation, and those with a negative solution |
W
solving equations
W
equation match
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Formulae
Top
| Content |
Resources |
| use
formulae from mathematics and other subjects
expressed initially in words and then
using letters and symbols |
W
number machines |
| formulae for the area of a triangle, the area
enclosed
by a circle, |
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| substitute numbers into a formula |
W
W
4 in a line
W
W W
substitution
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| change the subject of a formula,
including cases where the subject
occurs twice, or where a power of the
subject appears |
W
W
W changing the subject |
| generate a formula |
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Direct and inverse proportion
Top
| Content |
Resources |
| set
up and use equations to solve word and other problems involving direct proportion or inverse proportion and
relate
algebraic solutions to graphical representation of
the equations |
W
proportion |
Simultaneous linear equations
Top
| Content |
Resources |
| find
the exact solution of two simultaneous
equations in two unknowns by
eliminating a variable, and
interpret the equations as lines and their
common solution as the point of intersection |
W
simultaneous equations |
| solve simple
linear inequalities in one variable, and represent the solution set on
a number line |
N
W
inequalities |
| solve several linear inequalities in two
variables
and find the solution set |
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Quadratic equations
Top
| Content |
Resources |
| solve
quadratic equations
by completing
the square and using the quadratic
formula |
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Simultaneous linear and quadratic equations
Top
| Content |
Resources |
| 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 or
non-linear in the other |
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Numerical methods
Top
| Content |
Resources |
| use
systematic trial and improvement to find approximate solutions of
equations where there is no
simple analytical method
of solving them. |
W
trial and improvement |
Sequences
Top
| Content |
Resources |
| generate
common integer sequences (including
sequences of odd or even integers, squared integers, powers of 2, powers
of 10, triangular numbers) |
|
| generate terms of a sequence using term-to-term and position-to-term
definitions of the
sequence |
W
number patterns
O
W
B
number sequences
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| use linear expressions to describe the nth term of an
arithmetic sequence, justifying its form by reference to
the activity or context from
which it was generated |
O
W
W sequences and formulae
W
number patterns
O&W
Sequences based on patterns
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Graphs of linear functions
Top
| Content |
Resources |
| use the
conventions for coordinates in the plane |
|
| plot points in all four quadrants |
O
W W co-ordinate grids
W
co-ordinate fred
W
co-ordinate pictures
W
orc co-ordinates
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| recognise (when values are given for
m and
c) that equations of the form y = mx + c correspond to straight-line graphs
in the coordinate plane |
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| plot graphs of functions in which
y is given
explicitly in terms of x, or implicitly |
W
understanding equations |
| find the gradient of lines given by
equations of the form y = mx + c (when values
are given for m and
c) |
|
| understand that the form
y = mx + c represents a straight line and that m
is the gradient of the line, and c is the value
of the y-intercept explore
the gradients of parallel lines |
N
W straight
line graphs |
| construct linear functions from real-life
problems
and plot their corresponding graphs |
|
| discuss and interpret graphs arising from
real
situations |
O
travel graphs |
Interpret graphical information
Top
| Content |
Resources |
| interpret
information presented in a range of
linear and non-linear graphs |
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| graphs describing trends, |
|
| conversion graphs, |
|
| distance-time graphs, |
O
travel graphs
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| graphs of height or weight
against age, |
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| graphs of quantities that vary
against time, such as employment |
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Quadratic functions
Top
| Content |
Resources |
| generate
points and plot graphs of simple
quadratic functions, then more general quadratic functions find
approximate solutions of a quadratic
equation from the graph of the
corresponding quadratic function |
W
quadratic graphs |
| find the intersection points of the
graphs of a linear and quadratic function, knowing that these are the approximate solutions of the
corresponding
simultaneous equations representing
the linear and quadratic functions |
W
W
W
graphical solutions of equations |
Other functions
Top
| Content |
Resources |
| plot
graphs of: simple
cubic functions, the exponential
function for
integer values of x and simple positive values of k, the
circular functions y =
sin x and y =
cos x, using
a spreadsheet or graph plotter
as well as pencil and paper |
|
| recognise the characteristic shapes of all
these
functions |
|
Transformation of functions
Top
| Content |
Resources |
| apply
to the graph of y = f(x) the transformations y = f(x) + a, y = f(ax), y = f(x
+ a), y = af(x)
for linear, quadratic, sine and cosine functions f(x) |
W
functions |
Loci
Top
| Content |
Resources |
| construct
the graphs of simple loci including
the circle for
a circle of radius r centred at the origin |
|
| find graphically the intersection points
of a given straight line with this
circle and know that this
corresponds to solving the two
simultaneous equations representing
the line and the circle. |
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