GCSE  Shape,
Space and Measures

Foundation
only

Foundation and Higher

Higher only




ANGLES
AND PROPERTIES OF POLYGONS
Specification

Learning objectives

Grade

Resources

distinguish between
acute, obtuse, reflex and right angles. 
Recognise acute, obtuse,
reflex and right angles 
F 
P protractors 
estimate the size of an
angle in degrees 
Estimate
angles and measure them accurately 
F 
understand and use angle measure
using the associated language 
measure and draw angles
to the nearest degree 
recall and use properties of
angles at a point, angles on a
straight line (including right
angles), perpendicular lines, and
opposite angles at a vertex
U se parallel lines, alternate
angles and corresponding angles

Use properties of angles
at a point and angles on a straight line 
F 
P W
W
W
angle facts
W
Mixed angle Facts
W
Parallel Lines

Understand the terms
'perpendicular' and 'parallel' 
F 
use angle properties of
equilateral, isosceles and rightangled
triangles

Use angle
properties of triangles and quadrilaterals 
C 
W
Triangles and angles
P
triangles and straight lines
P exterior angles
W
mixed angle facts
F
F
solve angles given as algebra
(answers)

explain why the angle sum of
any quadrilateral is 360 degrees

understand a proof that the
exterior angle of a triangle is equal to
the sum of the interior angles at the
other two vertices

prove that
the sum of angles in a triangle is 180° 
C 
understand the properties of
parallelograms and a proof that the
angle sum of a triangle is 180 degrees

calculate and use the sums of
the interior and exterior angles of
quadrilaterals, pentagons and hexagons.

Calculate exterior and
interior angles of regular polygons 
C 
calculate and use the angles of
regular polygons

Solve angle problems
involving regular polygons 
C 
use
bearings to specify direction


D 
W W
measuring bearings
F
Follow instructions
P
F Bearings treasure maps
P
bearings protractors

classify quadrilaterals by
their geometric properties
recall
the essential properties and definitions
of special types of
quadrilateral, including square,
rectangle, parallelogram, trapezium
and rhombus
distinguish between lines and line
segments

know the geometric
properties of special types of
quadrilateral. 
C 
W quadrilateral
properties
W shapes
P parallelogram or
trapezium?

PERIMETER,
AREA AND VOLUME
Specification

Learning
objectives 
Grade

Resources

find
areas and perimeters of rectangles,
recalling the formula, understanding
the connection to counting squares
and how it extends this approach

find the perimeter of
a shape by counting sides of squares 
G

W area and perimeter
investigation
W
counting squares
W
W
Area and perimeter of rectangles
P who wants to be a
millionaire

Find the area of a
shape by counting squares 
G

Estimate the area of
an irregular shape by counting squares 
G 
Work out the area and
perimeter of a simple rectangle such as
3m by 8m 
F 
Work out the area and
perimeter of a harder rectangle such as
3.6m by 7.2m 
E 
recall and use the
formulae for the area of a
parallelogram and a triangle
use their knowledge of
rectangles, parallelograms and
triangles to deduce formulae for the
area of a parallelogram, and a
triangle, from the formula for the
area of a rectangle

Find the
area of a triangle, parallelogram, kite
and trapezium 
D 
W
triangles for measuring area
W
W
parallelogram and
trapezium
W
area and volume worksheet
P
missing lengths

calculate perimeters
and areas of shapes made from
triangles and rectangles

Find the
area and perimeter of compound shapes 
D 
P rectangle and
triangle
W
W
W
Compound shapes
W
Perimeter of rectangles and
compound shapes
W Area, perimeter,
angles puzzle
P millionaire
cards
W
W Compound shapes 
area

calculate the area of a
triangle using ˝ ab sin C

Use the
trigonometrical formula for the area of
a triangle 
A


find circumferences of
circles and areas enclosed by
circles, recalling relevant formulae

Calculate
the circumference of a circle to an
appropriate degree of accuracy 
D 
P W
circles
X
Generates random questions
P circles  shaded
area
W circles test
W Circles
W Circumference

Calculate
the area of a circle to an appropriate
degree of accuracy 
D 
calculate
the perimeter of a semicircle 
C 
calculate
the area of a semicircle 
C 
find volumes of
cuboids, recalling the formula and
understanding the connection to
counting cubes and how it extends
this approach

Find the
volume of a solid by counting cubes and
stating units 
G 
P
P
volume by counting cubes

Find the
volume of a cube or cuboid 
E 
Find the
height of a cuboid, given volume, length
and breadth 
E 
find the surface area
of simple shapes using the area
formulae for triangles and
rectangles

Calculate
the surface area of prisms 
C 

calculate
volumes of right prisms and of
shapes made from cubes and cuboids
solve problems
involving surface areas and volumes
of prisms, cylinders,
pyramids,
cones
and spheres

Calculate
the surface area of cylinders 
C 
W cylinders
P
P
cone volume
P
cone surface area
W prisms
W P
prisms

Calculate
the volumes of triangular prisms and
parallelogrambased prisms 
C 
Find the
volume and surface area of pyramids,
cones and spheres 
A 
solve problems
involving more complex shapes and
solids, including segments of
circles and frustums of cones.

Find
the volume the top cone of a truncated
cone 
A 

Find the
volume of the frustum of a truncated
cone 
A* 
calculate the lengths
of arcs and the areas of sectors of
circles

Find
the length of a major arc of a circle 
A 
W
perimeter and area problems involving
semicircles
W
complex circle problems

Find the
area of a major arc of a circle 
A 
Find the
area of a segment of a circle 
A 
convert between area
measures, including square
centimetres and square metres, and
volume measures, including cubic
centimetres and cubic metres.

Change
between area measures such as m˛ and cm˛ 
D 
W
converting area and volume measures

Convert
between measures of volume 
C 
understand the
difference between formulae for
perimeter, area and volume by
considering dimensions

Use
dimensions to differentiate between
formulae for length, area and volume 
B 
W dimensions

REFLECTIONS
AND ROTATIONS
Specification

Learning
objectives 
Grade

Resources

recognise and
visualise rotations including
rotational symmetry of 2D shapes

Give the order of
rotation symmetry of a 2D shape

F 
P
rotational symmetry
P
example
P
W
rotations
W rotations
W
W X
transformations
W rotations and
reflections
P
P
P
P
P
P
Find centre of Rotation

measure the
angle of rotation using right
angles, simple fractions of a
turn or degrees.
understand that
rotations are specified by a
centre and an (anticlockwise)
angle
rotate a
shape about the origin
or any other
point.

Describe
fully rotations about the origin 
D

Describe fully rotations about any
point 
C

Rotate
shapes about any point 
C

Find the
centre of rotation and describe it fully 
C

recognise and
visualise reflections including
reflection symmetry of 2D and 3D
shapes.

Draw a line of symmetry
on a 2D shape

G 
Draw all the lines of
symmetry on a 2D shape

F 
understand
that reflections are specified by
a mirror line, (only using
a line parallel to an axis or
in a given mirror line such as y=x
or y=x)

Draw the
reflection of a shape in a mirror line 
G

P
example
P examples
W
W X
transformations
W P
P
reflections in a line
W rotations and
reflections
P reflections and
equations of lines

Draw the
line of reflection for 2 shapes 
F

Reflect
shapes in the axes of a graph 
E

Reflect
shapes in lines parallel to the axes,
such as x=2 and y=1 
D

Reflect
shapes in lines such as y=x and y=x 
C

Describe
fully reflections in a line 
D

Identify
reflection symmetry in 3D solids 
D

transform triangles
and other 2D shapes by rotation,
reflection and combinations of these
transformations, recognising that
these transformations preserve
length and angle, so that any figure
is congruent to its image under any
of these transformations
use congruence
to show that rotations and
reflections preserve length and
angle.
distinguish
properties that are preserved
under particular transformations

Name, draw
or complete 2D shapes from information
about their symmetry 
F


Combine
reflections and rotations 
C

PROPERTIES OF
CIRCLES
Specification

Learning
objectives 
Grade

Resources

recall the definition
of a circle and the meaning of
related terms, including centre,
radius, chord, diameter,
circumference, tangent and arc,
sector and segment

Name the parts
of a circle 
G

P
W words and formulae
W
diameter and radius
W Circumference and
diameter

understand that the
tangent at any point on a circle is
perpendicular to the radius at that
point
understand
and use the fact that tangents
from an external point are equal
in length
explain
why the perpendicular from the
centre to a chord bisects the
chord

Use the angle
and tangent/chord properties of a circle 
B


prove and use the facts
that the angle subtended by an arc
at the centre of a circle is twice
the angle subtended at any point on
the circumference, the angle
subtended at the circumference by a
semicircle is a right angle, that
angles in the same segment are equal
and that opposite angles of a cyclic
quadrilateral sum to 180 degrees

Prove
the angle and
tangent/chord properties of a circle 
A


prove and use the
alternate segment theorem

Use and prove
the alternate segment theorem 
A


TRANSLATION
AND ENLARGEMENT
Specification

Learning
objectives 
Grade

Resources

recognise
translations
understand that
translations are specified by a
distance and direction and
understand vector notation in this
context

Translate
a shape using a description 
D 
P
P
translations
W
translations
P
Summary of translations

Translate
a shape using a vector 
C 
understand
enlargements are specified
by a centre and positive scale
factor
identify the scale
factor of an enlargement as the
ratio of the lengths of any two
corresponding line segments and
apply this (to triangles only)
recognise,
visualise and construct
enlargements of objects (only
positive scale factors greater
than one, then positive scale
factors less than one)
Use f
ractional
and negative scale factors

Give a
scale factor of an enlarged shape 
F


Enlarge a
shape by a positive scale factor 
E

P
P
enlargements
P
P
shapes to enlarge
P
examples
W
enlargements
W
scale factors

Find the
measurements and dimensions of an
enlarged shape 
E

Enlarge a
shape by a positive scale factor from a
given centre 
D 
Enlarge a
shape by a fractional scale factor 
C 
Enlarge a
shape by a negative scale factor 
A 
transform
triangles and other 2D shapes by
rotation, reflection, translation
and combinations of these
transformations

Be able to
show how a shape will tessellate 
E 
P
W
tessellations

Transform
shapes by a combination of translation,
rotation and reflection 
C 
distinguish properties
that are preserved under
particular transformations
, recognise that
enlargements preserve angle but not
length, understand the
implications of enlargement for
perimeter,
understand
and use
the
implications of enlargement for
area and volume
,
understand
and use simple
examples of the relationship
between / the effect
of enlargement
on areas and volumes of
shapes and solids.
use congruence
to show that translations preserve
length and angle.
Distinguish
between formulae for perimeter, area
and volume by considering dimensions

Compare
the area and perimeter of an enlarged
shape with the original shape 
C 
P
enlargements and area

Distinguish
between formulae for perimeter, area and
volume by considering dimensions 
B 
Compare
areas and volumes of enlarged shapes
(see also, the section on similar
shapes) 
A 
MEASURES
Specification

Learning
objectives 
Grade

Resources

make
sensible estimates of a range of
measures in everyday settings

Decide which metric
unit to use for everyday measurements 
G 
P P
units of measure
W units of length
P
W litres and
millilitres
P
the weight is right

make
sensible estimates of a range of
measures in everyday settings

F 
interpret
scales on a range of measuring
instruments, including those for
time and mass

interpret
scales on a range of measuring
instruments, including those for
time and mass

F 
W measures
questions

know
rough metric equivalents of pounds,
feet, miles, pints and gallons
convert
measurements from one unit to
another
know
that measurements using real
numbers depend
on the choice of unit

know
rough metric equivalents of pounds,
feet, miles, pints and gallons

F 
P Who wants to be a
millionaire
W W
metric and imperial units
W metric units
W writing lengths
using decimals
W X
litres and millilitres
W
W
measurements revision

Convert
between metric and imperial units 
F 
understand and use
compound measures, including speed
and density

Solve simple speed
problems 
E 
P
P
understanding speed
W
W
Speed
W Speed, Distance,
Time
P
P
Millionaire cards

Solve more difficult
speed problems 
C 
Understand and use
compound measures such as speed and
density 
C 
recognise that
measurements given to the nearest
whole unit may be inaccurate by up
to one half in either direction

Recognise accuracy in
measurements given to the nearest whole
unit 
C 
P
minimum and maximum values
W measure is
approximate

use calculators, or
written methods, to calculate the
upper and lower bounds of
calculations, particularly when
working with measurements

Find the upper and
lower bounds of more difficult
calculations with quantities given to
various degrees of accuracy 
AA* 

CONSTRUCTION
AND LOCI
Specification

Learning
objectives 
Grade

Resources

measure and draw
lines to the nearest millimetre and
draw angles to the nearest
degree 
Measure a line accurately to the
nearest mm 
G


Measure or draw accurately an
angle to the nearest degree 
F 
draw
approximate
constructions of triangles
and other 2D shapes using a ruler and
protractor, given information about
their side lengths and angles
understand,
from their experience of constructing
them, that triangles satisfying SSS,
SAS, ASA and RHS are unique, but SSA
triangles are not

Draw a triangle given SSS, SAS,
SSA, RHS, or ASA 
E 
W accurate drawing
W Scale drawings

Understand constructions that
lead to a unique triangle and those which
do not 
D 
construct specified
cubes, regular
tetrahedra, squarebased pyramids
and other 3D shapes from given
information

Recognise the net of a simple
solid such as a cuboid 
G

P
3D shapes
P
drawing 3D shapes

Draw the net of a simple solid
such as a cuboid 
F

Construct and recognise the nets
of 3D solids 
D

use straight edge and
compasses to do standard
constructions, including an
equilateral triangle with a given
side , the midpoint and
perpendicular bisector of a line
segment, the perpendicular from a
point to a line, the perpendicular
from a point on a line, and the
bisector of an angle.

Draw a quadrilateral such as a
kite or a parallelogram with given
measurements 
D 
W loci
W locus and construction

Construct the perpendicular
bisector of a line 
C 
Construct the perpendicular from
a point to a line 
C 
Construct the perpendicular from
a point on a line 
C 
Construct angles of 60° and 90° 
C 
Construct the bisector of an
angle 
C 
find loci, both by
reasoning and by using ICT to produce
shapes and paths

Solve loci problems, such as
identifying points less than 3cm from a
point P. 
C 
P
W
P drawing loci

Construct accurately loci, such
as those points equidistant from two fixed
points 
C 
VECTORS
Specification

Learning
objectives 
Grade

Resources

calculate, and
represent graphically the sum of two
vectors, the difference of two
vectors and a scalar multiple of a
vector
understand and use the
commutative and associative
properties of vector addition
solve simple geometrical
problems in 2D using vector methods.

Add, subtract and multiply
vectors to solve vector geometry
problems 
A 
W vector problems
P
vector proofs
P
exam questions

Understand the relationship
between parallel and perpendicular
vectors 
A 
Solve more difficult vector
geometry problems 
A* 
calculate the resultant
of two vectors

Find the resultant of two
vectors 
A* 
SIMILARITY
AND CONGRUENCE
Specification

Learning
objectives 
Grade

Resources

understand congruence
understand
and use SSS, SAS, ASA and RHS
conditions to prove the congruence
of triangles using formal arguments,
and to verify standard ruler and
compass constructions

Match one side and one
angle of congruent triangles, given some
dimensions 
C 
W congruent shapes
P
congruence and similarity
F
(F)
Congruent Halves (answers)

Prove that two triangles
are congruent 
A 
Prove the construction
theorems 
A 
understand similarity of
triangles and of other plane figures,
and use this to make geometric
inferences

Match
sides and one angles of similar
triangles, given some dimensions 
B 
P
similar shapes
W W
similar shapes

Find the area of a 2D
shape, given the area of a similar shape
and the ratio 
A 
Find the volume of a 3D
shape, given the area of a similar solid
and the ratio 
A 
PYTHAGORAS' THEOREM
Specification

Learning
objectives 
Grade

Resources

understand, recall and
use Pythagoras’ theorem in 2D,
then 3D
problems

Use Pythagoras' theorem to find
any side of a rightangled triangle 
C

P
animation of Pythagoras' theorem
W notes
W
W
W
Pythagoras' theorem questions

Use Pythagoras' theorem to
find the height of an isosceles triangle 
C 
Use Pythagoras' theorem in
practical problems 
C 
Use Pythagoras' theorem in 3D
problems 
A 
find the length of line
segment AB, given the coordinates of
A and B. 
Find the distance between two
points from their coordinates 
B


TRIGONOMETRY
Specification

Learning
objectives 
Grade

Resources

understand, recall and
use trigonometrical relationships in
rightangled triangles, and use these
to solve problems, including those
involving bearings
then use these
relationships in 3D contexts,
including finding the angles
between a line and a plane (but
not the angle between two planes
or between two skew lines)

Use trigonometry to find a side
in a right angled triangle 
B

W
investigating trigonometry
W
W W
W
W
trigonometry

Use trigonometry to find an
angle in a right angled triangle 
B 
Use trigonometry to find sides
and angles in 3D 
A* 
Find an angle between a line and
a plane 
A* 
plot the graphs of
circular functions y=sinx and y=cosx.
draw, sketch and
describe the graphs of trigonometric
functions for angles of any size,
including transformations involving
scaling in either or both the x and y
directions

Sketch
and draw trigonometric graphs 
A 

Understand
the graphs of trigonometric functions for
graphs of any size 
A* 
calculate the area of a
triangle using ˝ ab sin C

Use the trigonometric
formula for the area of a triangle 
A


use the sine and cosine
rules to solve 2D and 3D problems

Use the sine and
cosine rule to find the missing side in
any triangle 
A 

calculate the area of a
triangle using ˝ ab sin C

Use the
formula for the area of a non right angled
triangle 
A 

3D
SOLIDS
Specification

Learning
objectives 
Grade

Resources

investigate /
explore the geometry of
cuboids (including cubes), and
shapes made from cuboids

Recognise and name 3D solids 
G 

Sketch 3D solids 
G 
Draw a cuboid on an isometric
grid and mark its dimensions 
F 
use 2D representations of
3D shapes and analyse 3D shapes
through 2D projections and
crosssections, including plan and
elevation.

Draw plans and elevations of 3D
solids 
D 
W solids and nets
P P
W
isometric grids
P P
W
W plans and elevations
P
4 in a row  plans and elevations

Recognise the net of a simple
solid such as a cuboid 
D 
Draw the net of a simple solid
such as a cuboid 
D 
Construct and recognise the nets
of 3D solids 
D 
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