Learning Ladders Math Geometry, Shape and Space

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1 Learning Ladders Math Geometry, Shape and Space Step Year Geometry, Shape & Space DP Pending Regular Track (Benchmarked against IGCSE Extended Section Topic#32) Cosine Rules can find sides and angles from word problems Area of triangle using sine ( can work out the missing side using algebra) Solve complex trig problems set in three dimensions (Benchmarked against IGCSE Extended Section Topic#32) Cosine Rules can find sides and angles only from given triangles. Area of triangle using sine Solve simple trig problems set in three dimensions( finding sides and angles) (Benchmarked against IGCSE Extended Section Topic#32) Advanced Track (Benchmarked against IGCSE Additional Mathematics Sections Topic 10) Draw graphs such as y=asin(b(x+c))+d; y=acos(b(x+c))+d; y=atan(b(x+c))+d where a, b, c and d are positive integers Use identities ; ; ; ; in more challenging situations. Double angle identities for Sin2x and Cos2x (in more challenging situations) Solve simple trig equations using the above identities Vectors (Benchmarked against IGCSE Additional Mathematics Topic# 13 Vectors) Finding Vector Equation of Straight Lines and converting to Cartesian form and vice versa. (Benchmarked against IGCSE Additional Mathematics # Topic 10 ) Understand amplitude and periodicity and be able to model given data with trig graphs Draw graphs such as y=asin(b(x+c))+d; y=acos(b(x+c))+d; where a, b, c and d are positive integers Use identities ; ; ; ; in very simple situations. Double angle identities for Sin2x and Cos2x ( in simple situations) Solve simple trig equations using the above identities (Benchmarked against IGCSE Additional Mathematics Topic # 10 )

2 Can convert word problem of bearings to diagrams Work with Sine rule ( ambiguous case) Cosine Rules - can only find the sides Area of triangle using sine ( when all information is given) Solve simple trig problems set in three dimensions (can find sides but not angles) Know the six trig functions of angles(sine, cosine, tangent, secant, cosecant, cotangent) Understand amplitude and periodicity in very simple situations. Draw graphs of sinx, cosx and tanx. Use identities ; ; Solve simple trig equations using the above identities Prove simple identities Vectors (Benchmarked against IGCSE Additional Mathematics Topic # 13 Vectors) Geometry- Symmetry and Mensuration (Benchmarked against IGCSE Extended Section #31 and #28 Symmetry) Recognize and use rotational and line symmetry (including order of rotational symmetry) in two dimensions and properties of triangles, quadrilaterals and circles to arrive at conclusions. Volume and Surface Area of Prisms, Cylinders and Cones( compound shapes and reverse problems) Solve problems involving the arc length and sector area of a circle and reverse. Angles (Benchmarked against IGCSE Extended Section#29 Angle Properties) Work with angles in polygons Can apply skilfully his knowledge of angles in circles such as angles in a semi-circle; angle between tangent and radius, angle at the centre of circle is twice the angle at the circumference to find the necessary angles; angles in the same segment and cyclic quadrilaterals (Benchmarked against IGCSE Core Section #32) Solve word problems using bearing, Solve more challenging 3 D problems involving right triangles Use vectors in the form ( and ai + bj Know and use position vectors and unit vectors Find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars Scalar Product (Back tracked from DP) Vector Equation of Lines (Back tracked from DP) (Benchmarked against IGCSE Extended Section Topic#32) Solve word problems using bearing Cosine Rules can find sides and angles from word problems Area of triangle using sine ( can work out the missing side using algebra) Further (Benchmarked against IGCSE Additional Mathematics Circular Measure Topic # 9) Matrices Know and use the unit circle to solve problems. Work with positive and negative angles Know the sin, cos and tan values of 0, 30,45,60,90 and use these values to solve problems. Trigonometric identities ( use and apply in complex situations) (Removed because not in IBDP)

3 Find angles between planes Locus Not done because not in IBDP Vectors (Benchmarked against IGCSE Extended Mathematics Topic # 35 Vectors) Position vectors Geometry- Symmetry and Mensuration (Benchmarked against IGCSE Extended Section # 31 and #28 Symmetry) Recognize rotational and line symmetry (including order of rotational symmetry) in two dimensions and properties of triangles, quadrilaterals and circles directly related to their symmetries. Volume and Surface Area of Prisms, Cylinders and Cones ( compound shapes) Solve problems involving the arc length and sector area of a circle Angles (Benchmarked against IGCSE Extended Section#29 Angle Properties) Work with angles in circles such as angles in a semi-circle; angle between tangent and radius, angle at the centre of circle is twice the angle at the circumference, Angles in the same segment are equal; cyclic quadrilaterals (Benchmarked against IGCSE Core Section #32) Locus Solve simple word problems using bearings Solve simple 3 D problems involving right triangles. Find the angle between a line and a plane. Not done because not in IBDP Vectors (Benchmarked against IGCSE Extended Mathematics Topic # 35 Vectors) Use the sum and difference of two vectors to express given vectors in (Benchmarked against IGCSE Mathematics Extended Section #32) Solve simple word problems using bearings Cosine Rules can find sides and angles only from given triangles. Area of triangle using sine ( in-direct situations) Further (Benchmarked against IGCSE Additional Mathematics Circular Measure Topic # 9) Know and use the unit circle to some extent. Work with angles between 0 x 360 Know the sin, cos and tan values of 0,30,45,60,90 Be able to convert between radians and degrees for all angles. Trigonometric identities use and apply in simple situations Matrices (Removed because not in IBDP)

4 terms of two coplanar vectors Geometry- Symmetry and Mensuration (Benchmarked against IGCSE Extended Section # 31 and #28 Symmetry) Recognize line symmetry (including order of rotational symmetry) in two dimensions and properties of triangles, quadrilaterals and circles directly related to their symmetries. Build on knowledge of Volume and Surface Area of Prisms, Cylinders and Cones Solve problems involving the arc length and sector area of a circle for semi-circle, quarter circle. Angles (Benchmarked against IGCSE Extended Section#29 Angle Properties) Work with angles in polygons. Work with angles in circles such as angles in a semi-circle; angle at the centre of circle is twice the angle at the circumference. (Benchmarked against IGCSE Core Section #32) Solve simple word problems involving right triangle using sin, cos and tan Work with angles of elevation and depression using sin, cos and tan Can convert word problem of bearings to diagrams Locus Not done because not in IBDP Vectors (Benchmarked against IGCSE Extended Mathematics Topic # 35 Vectors) (Benchmarked against IGCSE Extended Section #32) Can convert word problem of bearings to diagrams Work with Sine rule ( ambiguous case) Cosine Rules - can only find the sides Area of triangle using sine ( when all information is given) Further (Benchmarked against IGCSE Additional Mathematics Circular Measure Topic # 9) Know what is a unit circle Work with angles greater than 90 o Be able to convert between radians and degrees for small angles. Matrices (Removed because not in IBDP) 21 9 Geometry: measurements 9Gs7 9Mt1 Use vectors in the form ( and ai + bj Find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars Represent vectors by directed line segments; Use the sum and difference of two vectors to express given vectors in terms of two coplanar vectors-in straight forward simple situations Know and use Pythagoras theorem to solve two-dimensional problems involving right-angled triangles Solve problems involving average speed

5 IGCSE Topic 32 Introduction to Sin(e), Cos(ine) and Tan(gent) in 2D and 3D shapes IGCSE Topic 26 Similarity and Congruence-Proofs Geometry: this part of Geometry not covered, because not in IBDP 9Gs3 Draw 3D shapes on isometric paper 9Gs4 Analyse 3D shapes through plans and elevations 9Gs5 Identify reflection symmetry in 3D shapes 9Gp9 Find by reasoning the locus of a point that moves at a given distance from a fixed point, or at a given distance from a fixed straight line 20 9 Geometry: angles 9Gs2 Solve problems using properties of angles, of parallel and intersecting lines, and of triangles, other polygons and circles, justifying inferences and explaining reasoning with diagrams and text Geometry: measurements 9Gp7 Use bearings (angles measured clockwise from the north) to solve problems involving distance and direction 9Ml1 Solve problems involving measurements in a variety of contexts 9Mt2 Use compound measures to make comparisons in real-life contexts, e.g. travel graphs and value for money 9Ma4 Calculate lengths, surface areas and volumes in right-angled prisms and cylinders. IGCSE Topic 31 Surface area and Volume of Prisms, Cylinder, Spheres, Cones and Pyramids IGCSE Topic 26 Similarity and Congruence-Simple Proofs 19 9 Geometry: measurements 9Ma2 Know that land area is measured in hectares (ha), and that 1 hectare = m²; convert between hectares and square metres 9Gp1 Tessellate triangles and quadrilaterals and relate to angle sums and half-turn rotations; know which regular polygons tessellate, and explain why others will not IGCSE Topic 31 Surface area and Volume of Prisms and Cylinders IGCSE Topic 32 Find area and perimeter of Sectors Use converse of Pythagoras theorem. Geometry: angles 9Gs1 Calculate the interior or exterior angle of any regular polygon; prove and use the formula for the sum of the interior angles of any polygon; prove that the sum of the exterior angles of any polygon is 360

6 IGCSE Topic 29 Angles in circles-angles in a semicircle, angles in the same segment, angles in a cyclic quadrilateral, angle between radius and tangent, angle at the centre Geometry: construction IGCSE Topic 27 Construct quadrilaterals with a compass. 9Gs6 Use a straight edge and compasses to: construct the perpendicular from a point to a line and the perpendicular from a point on a line inscribe squares, equilateral triangles, and regular hexagons and octagons by constructing equal divisions of a circle 18 8 Geometry: measurements 8Gp1 Find the midpoint of the line segment AB, given the coordinates of points A and B 17 8 Geometry: measurements 8Gs3 Know that the longest side of a right-angled triangle is called the hypotenuse 8Gs7 Draw simple nets of solids, e.g. cuboid, regular tetrahedron, square-based pyramid, triangular prism 8Ml1 Choose suitable units of measurement to estimate, measure, calculate and solve problems in a range of contexts, including units of mass, length, area, volume or capacity 8Ml2 mile 8Ma2 Derive and use formulae for the area of a triangle, parallelogram and trapezium; calculate areas of compound 2D shapes, and lengths, surface areas and volumes of cuboids 8Ma3 Use simple nets of solids to work out their surface areas 9Ma1 Convert between metric units of area, e.g. mm² and cm², cm² and m² and volume, e.g. mm³ and cm³, cm³ and m³; know and use the relationship 1 cm³ = 1 ml 9Gs7 Know and use Pythagoras theorem to solve two-dimensional problems involving right-angled triangles 8Gp4 Interpret and make simple scale drawings 9Gp8 Make and use scale drawings and interpret maps Geometry: angles 8Gs1 Know that if two 2D shapes are congruent, corresponding sides and angles are equal 8Gs2 Classify quadrilaterals according to their properties, including diagonal properties 8Gs4 Identify alternate angles and corresponding angles (and co-interior angles, IGCSE) 8Gs5 Understand a proof that: the angle sum of a triangle is 180 and that of a quadrilateral is 360 ;

7 the exterior angle of a triangle is equal to the sum of the two interior opposite angles 8Gs6 Solve geometrical problems using properties of angles, of parallel and intersecting lines, and of triangles and special quadrilaterals, explaining reasoning with diagrams and text Geometry: construction IGCSE Topic 27 Construct angles of 30, 60, 120 and 150 with a compass 8Gs9 Use a straight edge and compasses to construct: the midpoint and perpendicular bisector of a line segment; the bisector of an angle 8Gs10 Use a ruler and compasses to construct circles and arcs a triangle, given three sides (SSS) a triangle, given a right angle, hypotenuse and one side (RHS) 16 8 Geometry: measurements 8Mt1 Draw and interpret graphs in real life contexts involving more than one component, e.g. travel graphs with more than one person 9Ma3 Solve problems involving the circumference and area of circles, including by using the π key of a calculator. IGCSE Topic 31 Find the area of a kite and rhombus IGCSE Topic 31 Find Volume of prisms with a cross-sectional shape of parallelogram, trapezium, kite, circle, rhombus using formula V= Area of Cross Sectional x h Understand the link between capacity of liquids and capacity of solids and convert units within the metric system for capacity and volume Geometry: angles IGCSE Topic 29 Recognize if two (or more) angles are adjacent complementary or supplementary (IGCSE) Geometry: transformations 9Gp5 Recognise that translations, rotations and reflections preserve length and angle, and map objects on to congruent images, and that enlargements preserve angle but not length 9Gp6 Know what is needed to give a precise description of a reflection, rotation, translation or enlargement 15 7 Geometry: measurements 7Ma1 Know the abbreviations for and relationships between square metres (m²), square centimetres (cm²), square millimetres (mm²) 7Ma4 Calculate the surface area of cubes and cuboids from their nets Geometry: angles

8 7Gs7 Solve simple geometrical problems by using side and angle properties to identify equal lengths or calculate unknown angles, and explain reasoning 14 7 Geometry: measurements 7Mt1 Draw and interpret graphs in real life contexts involving more than one stage, e.g. travel graphs 7Ma3 Derive and use the formula for the volume of a cuboid; calculate volumes of cuboids 8Ma1 Know the definition of a circle and the names of its parts; know and use formulae for the circumference and area of a circle Geometry: angles 7Gs3 Name and identify side, angle and symmetry properties of special quadrilaterals and triangles, and regular polygons with 5, 6 and 8 sides 7Gs5 Start to recognise the angular connections between parallel lines, perpendicular lines and transversals 7Gs6 Calculate the sum of angles at a point, on a straight line and in a triangle, and prove that vertically opposite angles are equal; derive and use the property that the angle sum of a quadrilateral is 360 Geometry: transformations 8Gp2 Transform 2D shapes by rotation, reflection and translation, and simple combinations of these 8Gp3 Understand and use the language and notation associated with enlargement; enlarge 2D shapes, given a centre of enlargement and a positive integer scale factor 8Gs8 Identify all the symmetries of 2D shapes 9Gp2 Use the coordinate grid to solve problems involving translations, rotations, reflections and enlargements 9Gp3 Transform 2D shapes by combinations of rotations, reflections and translations; describe the transformation that maps an object onto its image 9Gp4 Enlarge 2D shapes, given a centre and positive integer scale factor; identify the scale factor of an enlargement as the ratio of the lengths of any two corresponding line segments Geometry: construction IGCSE Topic 27 Construct angle of 30, 60 and 90 degrees with the compass. IGCSE Topic 27 Construct a perpendicular and parallel line using a compass. IGCSE Topic 27 Construct a perpendicular bisector of a line using a compass. IGCSE Topic 27 Bisect an angle with the compass. 7Gs10 Use a ruler, set square and protractor to: (covered in Steps 16 and 17) measure and draw straight lines to the nearest millimetre; measure and draw acute, obtuse and reflex angles to the nearest degree; draw parallel and perpendicular lines;

9 construct a triangle given two sides and the included angle (SAS) or two angles and the included side (ASA); construct squares and rectangles; construct regular polygons, given a side and the internal angle) 13 7 Geometry: measurements 7Ml1 Choose suitable units of measurement to estimate, measure, calculate and solve problems in everyday contexts 7Ml2 Know abbreviations for and relationships between metric units; convert between kilometres (km), metres (m), centimetres (cm), millimetres (mm) kilometres (km), metres (m), centimetres (cm), millimetres (mm); tonnes (t), kilograms (kg) and grams (g); litres (l) and millilitres (ml) 7Ml3 Read the scales on a range of analogue and digital measuring instruments 7Mt2 Know the relationships between units of time; understand and use the 12-hour and 24-hour clock systems; interpret timetables; calculate time intervals 7Ma2 Derive and use formulae for the area and perimeter of a rectangle; calculate the perimeter and area of compound shapes made from rectangles Geometry: angles 7Gs1 Identify, describe, visualize and draw 2D shapes in different orientations 7Gs2 Use the notation and labelling conventions for points, lines, angles and shapes 7Gs4 Estimate the size of acute, obtuse and reflex angles to the nearest 10 7Gs8 Recognise and describe common solids and some of their properties, e.g. the number of faces, edges and vertices IGCSE Topic 27 Recognize if two (or more) angles are adjacent complementary or supplementary (IGCSE) Geometry: transformations 7Gs9 Recognise line and rotation symmetry in 2D shapes and patterns; draw lines of symmetry and complete patterns with two lines of symmetry; identify the order of rotation symmetry 7Gp1 Read and plot coordinates of points determined by geometric information in all four quadrants 7Gp2 Transform 2D points and shapes by: reflection in a given line; rotation about a given point; translation Know that shapes remain congruent after these transformations

10 12 6 6Gs3 Identify and describe properties of quadrilaterals (including the parallelogram, rhombus and trapezium), and classify using parallel sides, equal sides, equal angles 6Gs4 6Gs6 6Gp1 Recognise and make 2D representations of 3D shapes including nets Check that the sum of the angles in a triangle is 180, for example, by measuring or paper folding; calculate angles in a triangle or around a point Read and plot co-ordinates in all four quadrants Gs1 Classify different polygons and understand whether a 2D shape is a polygon or not 6Gs2 6Gs5 Visualise and describe the properties of 3D shapes, e.g. faces, edges and vertices Estimate, recognise and draw acute and obtuse angles and use a protractor to measure to the nearest degree Gs2 Recognise reflective and rotational symmetry in regular polygons 5Gs3 5Gs4 Create patterns with two lines of symmetry, e.g. on a pegboard or squared paper Visualise 3D shapes from 2D drawings and nets, e.g. different nets of an open or closed cube 5Gs6 Understand and use angle measure in degrees; measure angles to the nearest 5 ; identify, describe and estimate the size of angles and classify them as acute, right or obtuse 5Gs7 5Gp2 5Gp3 Calculate angles in a straight line Predict where a polygon will be after reflection where the mirror line is parallel to one of the sides, including where the line is oblique Understand translation as movement along a straight line, identify where polygons will be after a translation and give instructions for translating shapes 9 5 5Gs1 Identify and describe properties of triangles and classify as isosceles, equilateral or scalene 5Gs5 Recognise perpendicular and parallel lines in 2D shapes, drawings and the environment 5Gs6 Understand and use angle measure in degrees; measure angles to the nearest 10 degrees; identify, describe and estimate the size of angles and classify them as acute, right or obtuse 5Gp1 Read and plot co-ordinates in the first quadrant 8 4 4Gs1 Identify, describe, visualise, draw and make a wider range of 2D and 3D shapes including a range of quadrilaterals, the heptagon and tetrahedron; use pin boards to create a range of polygons. Use spotty paper to record results. 4Gs2 Classify polygons (including a range of quadrilaterals) using criteria such as the number of right angles, whether or not they are regular and their symmetrical properties 4Gs3 4Gs4 4Gs5 Identify and sketch lines of symmetry in 2D shapes and patterns Visualise 3D objects from 2D nets and drawings and make nets of common solids Find examples of shapes and symmetry in the environment and in art

11 7 4 4Gs5 Find examples of shapes and symmetry in the environment and in art 4Gp1 Describe and identify the position of a square on a grid of squares where rows and columns are numbered and/or lettered 4Gp2 Know that angles are measured in degrees and that one whole turn is 360 or four right angles; compare and order angles less than 180 4Gp3 Devise the directions to give to follow a given path 6 3 3Gs1 Identify, describe and draw regular and irregular 2D shapes including pentagons, hexagons, octagons and semi-circles 3Gs3 Identify, describe and make 3D shapes including pyramids and prisms; investigate which nets will make a cube 3Gs5 Draw and complete 2D shapes with reflective symmetry and draw reflections of shapes (mirror line along one side) 3Gs7 Identify 2D and 3D shapes, lines of symmetry and right angles in the environment 3Gs8 Identify right angles in 2D shapes 3Gp3 3Gp4 Use a set square to draw right angles Compare angles with a right angle and recognise that a straight line is equivalent to two right angles 5 3 3Gs4 Classify 3D shapes according to the number and shape of faces, number of vertices and edges) 3Gs6 Relate 2D shapes and 3D solids to drawings of them 3Gs7 Identify 2D and 3D shapes, lines of symmetry and right angles in the environment 3Gp1 3Gp2 Use the language of position, direction and movement, including clockwise and anti-clockwise Find and describe the position of a square on a grid of squares where the rows and columns are labelled 4 2 2Gs1 Sort, name, describe, visualise and draw 2D shapes (e.g. squares, rectangles, circles, regular and irregular pentagons and hexagons) referring to their properties; recognise common 2D shapes in different positions and orientations 2Gs2 shapes 2Gs3 2Gs4 2Gp1 2Gp2 2Gp3 Sort, name, describe and make 3D shapes (e.g. cubes, cuboids, cones, cylinders, spheres and pyramids) referring to their properties; recognise 2D drawings of 3D Identify reflective symmetry in patterns and 2D shapes; draw lines of symmetry Find examples of 2D and 3D shape and symmetry in the environment Follow and give instructions involving position, direction and movement Recognise whole, half and quarter turns, both clockwise and anti-clockwise Recognise that a right angle is a quarter turn 3 2 2Gs1 Sort, name, describe, visualise and draw 2D shapes (e.g. squares, rectangles, circles, regular and irregular pentagons and hexagons) referring to their properties;

12 2Gs2 2Gs4 2Gp1 Sort, name, describe and make 3D shapes (e.g. cubes, cuboids, cones, cylinders, spheres and pyramids) referring to their properties; Find examples of 2D and 3D shape and symmetry in the environment Follow instructions involving position, direction and movement 2 1 1Gs 2 Name and sort common 3D shapes (e.g. cube, cuboid, cylinder, cone and sphere) using features such as number of faces, flat or curved faces. Use them to make patterns and models 1Gs 3 Recognise basic line symmetry 1 Gp1 Use everyday language of direction and distance to describe movement of objects 1 1 1Gs 1Name and sort common 2D shapes (e.g. circles, squares, rectangles and triangles) using features such as number of sides, curved or straight. Use them to make patterns and models. 1 Gp1 Begin to use everyday language of direction and distance to describe movement of objects P KG Uses mathematical vocabulary to describe shape pictures ( This triangle has 3 sides and 3 corners ) Match some shapes by recognizing similarities and orientation Identifies and names 3D and 2D shapes Understands and uses positional language appropriately (under, over, behind, in front of, inside, outside, next to)