GCSE  Algebra

Foundation
only

Foundation and Higher

Higher only




USE
OF SYMBOLS
Specification

Learning
objectives 
Grade

Resources

distinguish
in meaning between the words
'equation', 'formula',
'identity' and 'expression'.
manipulate algebraic
expressions by collecting like
terms, by multiplying a single term
over a bracket, and by taking out
common factors

Simplify expressions
with one variable such as a + 2a + 3a 
F 
W
simplifying blockbusters
W factorising
W simplifying collect
a joke
W
simplifying
F
Expand over 1 bracket

Simplify expressions
with more than one variable 
E 
Multiply out
expressions with brackets such as 3(x+2) 
D 
Factorise expressions
over one bracket 
D 
Expand harder
expressions involving indices or more
than one bracket such as 3(x+2)5(2x1) 
C 
expand the product
of two linear expressions

Be able to expand
expressions like (x+2)(x4) 
C

W
quadratics simplifying and expanding
F
Expand 2 sets of brackets

SEQUENCES
Specification

Learning
objectives 
Grade

Resources

generate terms
of a sequence using termtoterm
and positiontoterm definitions
of the sequence
.

Continue a sequence of
numbers or diagrams 
G 
P W number patterns
W W
number sequences
W Flash cards

Write the terms of a
simple sequence 
G 
Find a particular term
in a sequence of positive numbers 
F 
Write the term to term
rule of a sequence involving positive
numbers 
F 
Find a
particular term in a sequence of
negative numbers or fractions 
E 
Write
the term to term rule of a sequence
involving negative numbers or
fractions 
E 
generate
common integer sequences
(including sequences of odd or
even integers, squared integers,
powers of 2, powers of 10,
triangular numbers)

Write the terms of a
sequence or a series of diagrams given
the nth term 
D 
P Sequence envelope

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

Describe the nth term
of a sequence or a series of diagrams 
C 
W
W
sequences and formulae
W W
number patterns
W nth term
W Sequences based on patterns
W
nth term

COORDINATES
AND GRAPHS OF LINEAR FUNCTIONS
Specification

Learning
objectives 
Grade

Resources

Understand that one
coordinate identifies a point on a
number line, two coordinates identify
a point in a plane, and three
coordinates identify a point in space,
using the terms 1D, 2D, and 3D.
use
the conventions for coordinates in
the plane
plot
points in all four quadrants
and use axes and coordinates
to specify points in all four
quadrants
locate
points with given coordinates

Use
coordinates in the first quadrant 
G 
P P
W
W coordinate grids
W coordinate
fred P
(answers)
W coordinate
pictures
W orc coordinates
W W
coordinate bingo
W 3D coordinates

Use
coordinates in all four quadrants 
F 
Draw lines
such as x=3 and y=x+2 
E 
Draw lines
such as y=2x+3 
D 
Solve
problems involving graphs, such as
finding where the line y=x+2 crosses the
line y=1 
D 
Use and
understand coordinates in 3D 
C 
find the coordinates of
points identified by geometrical
information

Solve geometrical
problems using coordinates such as
finding the 4th corner of a
parallelogram given the other 3

D 

find the coordinates
of the midpoint of the line
segment AB, given A and B. 
Find the midpoint of a
line segment

C 

recognise (when
values are given for m and c) that
equations of the form y = mx + c
correspond to straightline graphs
in the coordinate plane

Recognise
the equations of straight line graphs
such as y=4x+2 
C 
W W
W
straight line graphs

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 yintercept

Find the
gradients of straight line graphs 
C 
Find the
gradient and yintercept from the
equation of a straight line 
B 
investigate the
gradients of parallel lines

Explore
the gradients of parallel straightline
graphs 
B 
explore the
gradients of parallel lines
and
lines perpendicular to each
other

Explore
the gradients of perpendicular
straightline graphs 
A 
EQUATIONS
Specification

Learning
objectives 
Grade

Resources

set up simple
equations
solve
simple equations by using inverse
operations or by transforming both
sides in the same way
solve
linear equations, with
integer or fractional
coefficients,
in which the unknown appears on
either side or on both sides of
the equation
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

Solve
equations such as 3x=12 or x+5=9 
F

W forming equations
W W
making equations
W word equations
W X
W
solving equations
W X
(Answer grid)
Number grid
W
W
X X
solving equations
W
rearranging using given
operations
W
balancing
X
linear equations
W
Collect a letter
W
simple equations
P
domino loops
W
solving equations
W
equation match
W manipulating
equations
W solving
equations

Solve
equations suc as 3x1=9 or x/2 = 7 
E

Solve
equations such as 3x4=5 or 2(5x+1)=28 
D 
Solve
equations such as 3x12=2(x5) and
equations involving fractions or simple
algebraic fractions 
C 
Find a
solution to a problem by forming an
equation and solving it 
C 
Solve
equations with more than one algebraic
fraction 
B 
TRIAL AND
IMPROVEMENT
Specification

Learning
objectives 
Grade

Resources

use systematic trial and
improvement to find approximate
solutions of equations where there is
no simple analytical method of solving
them.

Form and solve
nolinear equations using trial and
improvement methods 
C

P
Starter
W W
W
F
trial and improvement
P
P
Examples

REALLIFE
GRAPHS
Specification

Learning
objectives 
Grade

Resources

interpret
information presented in a range of
linear and nonlinear graphs
construct linear
functions from reallife problems
and plot their corresponding
graphs
discuss and
interpret graphs arising from real
situations

Plot points of a conversion
graph and read off positive values 
F 
W
P
Conversion graphs without
questions
W conversion graphs
questions
W
drawing conversion graphs

Read from a conversion graph
for negative values 
E 
Interpret distancetime graphs 
E 
P
travel graphs without questions
W gradients and
area under graph
P P
W
animated travel graphs

Calculate simple average speeds
from distancetime graphs 
D 
Calculate complex
average speeds from distancetime
graphs 
C 
Interpret velocitytime graphs 
B 
Discuss and interpret graphs
modelling real situations 
B 
FORMULAE
Specification

Learning
objectives 
Grade

Resources

use formulae
from mathematics and other
subjects expressed
initially in words and then using
letters and symbols

Use a formula written in words 
G 
X number machines

Use a simple formula expressed
in symbols 
F 
derive/generate a formula

Write an expression from a
problem 
F 
substitute numbers
into a formula
substitute
positive and negative numbers into
expressions with indices.

Substitute positive numbers
into a simple formula 
F 
W W
4 in a line
W W
W
substitution
W
substitution

Substitute negative
numbers into a simple formula 
E 
Substitute negative
numbers into a complicated formula 
D 
change the subject
of a formula, including cases where the
subject occurs twice, or where a
power of the subject appears

Rearrange linear formulae such
as p=3q+5 
C 
W W
W
changing the subject 
Rearrange formulae that include
brackets, fractions and square roots 
B 
Rearrange formulae where the
variable appears twice 
A 
QUADRATIC FUNCTIONS
Specification

Learning
objectives 
Grade

Resources

factorise quadratic
expressions including the difference
of two squares and cancel common
factors in rational expressions
solve
quadratic equations by
factorisation, completing the square
and using the quadratic formula

Solve quadratic
equations such as x²8x+15=0 by
factorisation 
B 
W
F F Factorise expressions
F
F
Factorise and solve
P
P
W
F
Factorisation Formula
F
F
Complete the square

Solve
quadratic equations such as x²2x1=0
by using the quadratic formula

A

Solve
equations with algebraic fractions by
converting them to a quadratic equation 
A* 
Complete
the square 
A* 
Use
completing the square to solve equations
and find maximum and minimum values 
A* 
INEQUALITIES AND
REGIONS
Specification

Learning
objectives 
Grade

Resources

solve simple linear
inequalities in one variable, and
represent the solution set on a
number line

Solve
simple inequalities such as 3x<9 and
represent solutions on a number line 
C 
W W
inequalities

Solve
linear inequalities such as 4x3<10
or 4x<2x+7 and represent solutions
on a number line 
C 
Solve
linear inequalities such as
x+13>5x3 and represent solutions
on a number line 
B 
solve several linear
inequalities in two variables and
find the solution set

Solve a
set of linear inequalities in two
variables the solution as a region of a
graph 
B 

SIMULTANEOUS
EQUATIONS
Specification

Learning
objectives 
Grade

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

Solve a pair of
simultaneous equations in two unknowns,
such as 2x+y=5 and 3x2y=4 
B 
P
P
P
Starter Intructions Rules
W W
W
W
P
P
simultaneous equations
W spot the mistake

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

Know that simultaneous
equations can be represented by lines on
a graph and that the point of
intersection is the solution 
B 
P
W
W
W
graphical solutions of equations
W Graphical method

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 x² + y² = r²

Solve a pair of
simultaneous equations where one is
linear and one is nonlinear such as
y=3x2 and y=x² 
A 
P
Linear and Quadratic

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. 
Solve a
pair of simultaneous equations where
one is linear and one is nonlinear
such as x+5y=13 and x²+y²=13 
A* 
F
Quadratics 
Solve a
pair of simultaneous equations
graphically, such as y=x1 and
x²+y²=16 
A 
Solve a
pair of simultaneous equations
graphically, such as y=2x1 and
x²+y²=27 
A* 
OTHER
FUNCTIONS
Specification

Learning
objectives 
Grade

Resources

plot graphs of
functions in which y is given
explicitly in terms of x, or
implicitly; no table or axes
given
generate points and
plot graphs of simple quadratic
functions, then more general
quadratic functions and find
approximate solutions of a
quadratic equation from the graph
of the corresponding quadratic
function

Draw
graphs of simple quadratic functions
such as y=3x² and y=x²+4 
D 
W understanding
equations
W quadratic graphs
W
W
W
Graphical solutions

Draw
graphs of harder quadratic functions
such as y=x²2x+1 
C 
Find the
points of intersection of quadratic
graphs with lines 
C 
Use graphs
to find approximate solutions to
quadratic equations 
C 
plot
graphs of simple cubic functions,
the reciprocal function y=1/x
(with x not zero), the exponential
function y=k^{x} for
integer values of k and positive
values of x.
recognise
the characteristic shapes of all
the above functions

Complete
tables for, and draw graphs of cubic and
reciprocal functions 
B 
W
investigating graphs

Use graphs
of cubic and reciprocal functions to
solve equations 
B 
Solve
cubic equations by drawing appropriate
lines on graphs 
A* 
Plot and
sketch graphs of exponential functions 
A* 
Recognise
the shapes of graphs of functions 
A* 
TRANSFORMING
FUNCTIONS
Specification

Learning
objectives 
Grade

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, sin and
cos functions f(x)

Transform the
graph
of y = f(x), using the transformations
y = f(x)
+ a, y = f(ax), y = f(x + a), y =
af(x) for linear, quadratic, sin and
cos functions f(x)

A*

P
transformations of graphs
W
functions

ALGEBRAIC
PROOFS
Specification

Learning
objectives 
Grade

Resources

show step by step deduction in
problem solving 
Decide with a reason
whether a statement is true or false 
ED 

recognise the importance of
counterexamples 
Identify a counter
example 
D 
derive proofs using short chains
of deductive reasoning 
derive
proofs using short chains of deductive
reasoning 
C 
understand the difference between
a practical demonstration and a proof 
understand
the difference between a practical
demonstration and a proof 
C 
ALGEBRAIC MANIPULATION
Specification

Learning objectives

Grade

Resources

know the meaning of and use
the words equation, formula, identity
and expression
manipulate algebraic
expressions by collecting like
terms, by multiplying a single term
over a bracket, and by taking out
common factors

Be
able to collect like terms

E

W
blockbusters activity

multiply out simple expressions
with brackets 
D 

Factorise
simple expressions requiring one
bracket.

D


factorise quadratic
expressions including the difference
of two squares and cancel common
factors in rational expressions

expand
and simplify harder expressions

C

W
expanding & simplifying collect a
joke
W
expanding and simplifying
F
factorise where a=1
W
factorising including quadratics
F
factorise where a is greater than 1

Factorise
quadratic expressions where the
coefficient of x² is 1

B

Factorise
quadratic expressions where the
coefficient of x² is greater than 1

A

Simplify
rational expressions

B

Simplify
harder rational expressions

A*

expand the
product of two linear expressions

Expand
and simplify quadratic expressions where
the coefficient of x² is 1

B

W
W
expanding and simplifying
expressions including
two brackets

Expand
and simplify quadratic expressions where
the coefficient of x² is greater than 1

A

NONLINEAR
GRAPHS
AND GRAPHICAL SOLUTION OF EQUATIONS
Specification

Learning objectives

Grade

Resources

generate points and plot
graphs of simple quadratic functions,
then more general quadratic functions
and find approximate solutions of a
quadratic equation from the graph of
the corresponding quadratic function

Draw
graphs of simple quadratic functions
such as y=3x² and y=x²+4

D

P
explaining plotting curved graphs
W
quadratic graphs

Draw
graphs of harder quadratic functions
such as y=x²2x+1

C

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

Find
the points of intersection of quadratic
graphs with lines

C

P
explaining intersection points for
solving equations
W
W
graphical solutions of equations
W
using a graph plotter and deciding
which graphs to plot.

Use
graphs to find the approximate solutions
of quadratic equations

C

Use
points of intersection of a quadratic
graph such as y=x²2x4 with lines such
as y=2x+1 to solve equations like
x²2x4=2x+1 and simplify this to
x²4x5

A

Solve
equations using two graphs where the
equation needs manipulating to match the
graphs

A

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