Maths - Geometry

Maths - Geometry

1. Geometry

   • 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
     • Recall and use properties of angles
       • angles at a point
       • angles at a point on a straight line, including right angles
       • perpendicular lines
       • vertically opposite angles
   • Understand and use the angle properties of parallel and intersecting lines, triangles and quadrilaterals
     • Distinguish between scalene, isosceles, equilateral, and right-angled triangles
     • Understand and use the angle properties of triangles
     • Use the angle sum of a triangle is 180°
     • Understand and use the angle properties of intersecting lines
     • Understand and use the angle properties of parallel lines
     • Mark parallel lines on a diagram
     • Use the properties of corresponding and alternate angles
     • Understand and use the angle properties of quadrilaterals
     • Give reasons for angle calculations
     • Explain why the angle sum of a quadrilateral is 360°
     • Understand a proof that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices
     • Use the size/angle properties of isosceles and equilateral triangles
     • Understand the proof that the angle sum of a triangle is 180°
     • Recall and use these properties of angles in more complex problems
   • Calculate and use the sums of the interior and exterior angles of polygons
     • Use the angle sums of irregular polygons
     • Calculate and use the angles of regular polygons
     • Use the sum of the interior angles of an n-sided polygon
     • Use the sum of the exterior angles of any polygon is 36
     • Use geometric language appropriately and recognise and name pentagons, hexagons, heptagons, octagons and decagons
     • Use the sum of the interior angle and the exterior angle is 180o
     • Find the size of each interior angle or the size of each exterior angle or the number of sides of a regular polygon
     • Understand tessellations of regular and irregular polygons
     • Tessellate combinations of polygons
     • Explain why some shapes tessellate when other shapes do not
     • Calculate and use the sums of the interior angles of polygons
   • Recall the properties and definitions of special types of quadrilateral, including square, rectangle, parallelogram, trapezium, kite and rhombus
     • Recall the properties and definitions of special types of quadrilateral, including symmetry properties
     • List the properties of each, or identify (name) a given shape
     • Classify quadrilaterals by their geometric properties
   • Recognise reflection and rotation symmetry of 2-D shapes
     • Recognise reflection symmetry of 2-D shapes
     • Identify and draw lines of symmetry on a shape
     • Recognise rotation symmetry of 2-D shapes
     • Identify the order of rotational symmetry of a 2-D shape
     • Draw or complete diagrams with a given number of lines of symmetry
     • State the line symmetry as a simple algebraic equation
     • Draw or complete diagrams with a given order of rotational symmetry
   • Understand congruence and similarity
     • Recognise that all corresponding angles in similar figures are equal in size when the lengths of sides are not
     • Understand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles using formal arguments, and to verify standard ruler and a pair of compasses constructions
     • Understand similarity of triangles and of other plane figures, and use this to make geometric inferences
     • Complete a formal geometric proof of similarity of two given triangles
   • Use Pythagoras’ theorem in 2-D and 3-D
     • Understand, recall and use Pythagoras’ theorem in 2-D, then in 3-D problems
     • Understand the language of planes, and recognise the diagonals of a cuboid
     • Calculate the length of a diagonal of a cuboid
   • Distinguish between centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
     • Recall the definition of a circle and identify (name) and draw the parts of a circle
     • Understand related terms of a circle
     • Draw a circle given the radius or diameter
   • Use 2-D representations of 3-D shapes
     • Use 2-D representations of 3-D shapes
     • Use isometric grids
     • Draw nets and show how they fold to make a 3-D solid
     • Understand and draw front and side elevations and plans of shapes made from simple solids
     • Given the front and side elevations and the plan of a solid, draw a sketch of the 3-D solid
   • Describe and transform 2-D shapes using single or combined rotations, reflections, translations, or enlargements by a positive, fractional or negative scale factor and distinguish properties that are preserved under particular transformations
     • Describe and transform 2-D shapes using single rotations
     • Understand that rotations are specified by a centre and an (anticlockwise) angle
     • Find the centre of rotation
     • Rotate a shape about the origin, or any other point
     • Describe and transform 2-D shapes using single reflection
     • Understand that reflections are specified by a mirror line
     • Identify the equation of a line of symmetry
     • Describe and transform 2-D shapes using single translations
     • Understand that translations are specified by a distance and direction (using a vector)
     • Translate a given shape by the vector ( 2 -3)
     • Describe and transform 2-D shapes using enlargements by a positive and a negative or fractional scale factor
     • Understand that an enlargement is specified by a centre and a scale factor
     • Enlarge shapes using (0, 0) as the centre of enlargement
     • Enlarge shapes using centre other than (0, 0)
     • Find the centre of enlargement
     • Describe and transform 2-D shapes using combined rotations, reflections, translations, or enlargements
     • Distinguish properties that are preserved under particular transformations
     • Recognise that enlargements preserve angle but not length
     • Use congruence to show that translations, rotations and reflections preserve length and angle, so that any figure is congruent to its image under any of these transformations
     • Understand that distances and angles are preserved under rotations, reflections and translations so that any shape is congruent to its image
     • Describe a transformation
   • Use straight edge and a pair of compasses to carry out constructions
     • Use straight edge and a pair of compasses to do standard constructions
     • Construct a triangle
     • Construct an equilateral triangle
     • Understand, from the experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA triangles are not
     • Construct the perpendicular bisector of a given line
     • Construct the perpendicular from a point to a line
     • Construct the perpendicular from a point on a line
     • Construct the bisector of a given angle
     • Construct angles of 60o , 90o , 30o , 45o
     • Draw parallel lines
     • Draw circles and arcs to a given radius
     • Construct a regular hexagon inside a circle
     • Construct diagrams of everyday 2-D situations involving rectangles, triangles, perpendicular and parallel lines
     • Draw and construct diagrams from given information
   • Construct loci
     • Find and describe regions satisfying a combination of loci
     • Construct:
       • a region bounded by a circle and an intersecting line
       • given distance from a point and a given distance from a line
       • regions which may be defined by ‘nearer to’ or ‘greater than’
       • equal distances from two points or two line segments
     • All loci restricted to two dimensions only
   • Calculate perimeters and areas of shapes made from triangles, rectangles and other shapes
     • Measure shapes to find perimeter or area
     • Find the perimeter of rectangles and triangles
     • Calculate perimeter and area of compound shapes made from triangles, rectangles and other shapes
     • Recall and use the formulae for the area of a triangle, rectangle and a parallelogram
     • Calculate areas of shapes made from triangles and rectangles
     • Calculate perimeters of compound shapes made from triangles and rectangles
     • Find the area of a trapezium
     • Find the area of a parallelogram
     • Find the surface area of simple shapes (prisms) using the formulae for triangles and rectangles, and other shapes
   • Find circumferences and areas of circles
     • Find circumferences of circles and areas enclosed by circles
     • Recall and use the formulae for the circumference of a circle and the area enclosed by a circle
     • Use   3.142 or use the  button on a calculator
     • Find the perimeters and areas of semicircles and quarter circles
     • Calculate the lengths of arcs and the areas of sectors of circles
     • Answers in terms of  may be required
     • Find the surface area of a cylinder
   • Calculate volumes of right prisms and shapes made from cubes and cuboids
     • Calculate volumes of right prisms, including the triangular prism, and shapes made from cubes and cuboids
     • Recall and use the formula for the volume of a cuboid
     • Find the volume of a cylinder
     • Use volume to solve problems
   • Use the trigonometric ratios and the sine and cosine rules to solve 2-D and 3-D problems
     • Use the trigonometric ratios to solve 2-D and 3-D problems
     • Find angles of elevation and angles of depression
     • Find the angle between a line and a plane (but not the angle between two planes or between two skew lines)
     • Use the sine and cosine rules to solve 2-D and 3-D problems
     • Understand, recall and use trigonometric relationships in right-angled triangles, and use these to solve problems in 2-D and in 3-D configurations
   • Understand and construct geometrical proofs using circle theorems
     • Understand and use the fact 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
     • Find missing angles on diagrams
     • Give reasons for angle calculations involving the use of tangent theorems
     • 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 in a semicircle is a right angle
       • angles in the same segment are equal
       • opposite angles of a cyclic quadrilateral sum to 180o
       • alternate segment theorem
       • the perpendicular from the centre of a circle to a chord bisect the chord
   • Calculate the area of a triangle using 2 1 ab sin C
     • Calculate the area of a triangle given the length of two sides and the included angle
   • Solve mensuration problems involving more complex shapes and solids
     • Solve problems involving more complex shapes and solids, including segments of circles and frustums of cones
     • Find the surface area and volumes of compound solids constructed from cubes, cuboids, cones, pyramids, spheres, hemispheres, cylinders
     • Solve problems including examples of solids in everyday use
     • Find the area of a segment of a circle given the radius and length of the chord
   • Use vectors to solve problems
     • Understand and use vector notation
     • Calculate, and represent graphically, the sum of two vectors, the difference of two vectors and a scalar multiple of a vector
     • Calculate the resultant of two vectors
     • Solve geometrical problems in 2-D using vector methods
     • Apply vector methods for simple geometrical proofs