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GCSE - Shape, Space and Measures

 

 

Foundation only

Foundation and Higher

Higher only

 

 

 

  

 

1. ANGLES AND PROPERTIES OF POLYGONS
2. PERIMETER, AREA AND VOLUME
3. REFLECTIONS AND ROTATIONS
4. PROPERTIES OF CIRCLES
5. TRANSLATION AND ENLARGEMENT

6. MEASURES

7. CONSTRUCTION AND LOCI

8. VECTORS

9. SIMILARITY AND CONGRUENCE

10. PYTHAGORAS' THEOREM

11. TRIGONOMETRY

12. 3D SOLIDS

 

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 right-angled 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 ½ 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 parallelogram-based 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 semi-circles

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 m² and cm² 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 2-D 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 2-D and 3-D 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 2-D 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 semi-circle 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 2-D 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 A-A*

 

 

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 2-D 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, square-based pyramids and other 3-D 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 2-D 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 2-D, then 3-D problems

Use Pythagoras' theorem to find any side of a right-angled 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 right-angled triangles, and use these to solve problems, including those involving bearings then use these relationships in 3-D 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 ½ ab sin C

Use the trigonometric formula for the area of a triangle

A

 

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

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

 

calculate the area of a triangle using ½ 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 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, 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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