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Framework for developing schemes of work for the geometry curriculum for ages 14 -16 CURRICULUM CONTENT GRADES G - F GRADES E - D GRADES C - B TEACHING T OPIC G RADES A – A* OPPORTUNITIES INVESTIGATION AND ILLUSTRATION EUCLIDEAN P LANE GEOMETRY Distinguish Know and use angle, Construct hierarchical between, acute, side, diagonal and classification of 2 -D and 3-D right, obtuse and symmetry properties shapes. reflex angles. of quadrilaterals and classify quadrilaterals by their geometric properties. Know and use the angle sum of a triangle, the sum of angles at a point, and the sum of angles at a point on a line. Understand proofs Understand proofs that the that the angle sum exterior angle of a triangle is of a triangle is 180°. equal to the sum of the interior angles at the other two vertices. Know and use angle properties of equilateral, is osceles and right angled triangles. Understand why the angle sum of any quadrilateral is 360°. Know and use angle properties of intersecting and parallel lines and the consequent properties of parallelograms. Know and use the sum of the exterior angles of polygons. Calculate and use Know and use angle and the angles of regular symmetry properties of polygons. polygons. AND Angle sum of polygons by practical demonstration. USE OF ICT PROOF Exploring properties of quadrilaterals. Angle sum of a triangle by practical demonstration. Use formal arguments to verify standard ruler and compass constructions. D EDUCTION Angle sum of a triangle using parallel lines. Angle sum of polygons using triangles. Angle sum of a quadrilateral. Angle sum of a quadrilateral. Angle sums of polygons. Angle sums of polygons. CONTEXT AND APPLICATION Use of dynamic geometry to investigate and illustrate History of Euclidean geometry. Use of Logo to investigate polygons and stars. Use of dynamic geometry to investigate and illustrate. Fabric and graphic design. Tessellations. Use of dynamic geometry to Angle properties of investigate and Angle properties intersecting and parallel lines illustrate. of intersecting and and parallelograms. parallel lines and parallelograms. Use of dynamic geometry to investigate and illustrate. 1 CURRICULUM CONTENT GRADES G - F GRADES E - D GRADES C - B TEACHING T OPIC G RADES A – A* OPPORTUNITIES INVESTIGATION AND ILLUSTRATION Know and use the fact that the angle between a tangent and a radius is a right angle. Recognise congruence of shapes in different orientations. Understand and use tangent and Know and use the chord properties of circles. alternate segment theorem. Know and use the facts that the angle subtended by an arc at the Understand the proofs of centre of a circle is twice the the facts that the angle angle subtended at any point on subtended by an arc at the circumference, including the the centre of a circle is special case of a semicircle, and twice the angle that angles in the same segment subtended at any point on are equal. the circumference, that angles in the same Know and use angle properties segment are equal and of cyclic quadrilaterals. that opposite angles of a cyclic quadrilateral are equal. Understand the Use knowledge of similarity to Understand and use SSS, concept of similarity solve problems involving plane SAS, ASA and RHS and be able to shapes. conditions to prove the identify similar congruence of triangles. shapes. D EDUCTION AND PROOF Angle properties Angle properties relating to relating to circles. circles. Use of dynamic geometry to investigate and illustrate. Relationships between lengths in similar shapes. Use of dynamic geometry to investigate and illustrate. Understand various proofs of Use Pythagoras’s Pythagoras’s theorem, and use theorem to solve Pythagoras’s theorem to solve problems in 2D and 3D. problems in 2D. Pythagoras’s theorem. Various proofs of Pythagoras’s theorem. Pythagorean triples. Derivation of the formula for the distance between two points. Use trigonometry to solve 2D problems including bearings and angles of elevation and depression. Similar triangles leading to trigonometrical ratios. Use right angled triangle trigonometry to solve problems in 2D and 3D including finding the angle between a line and a plane. USE OF ICT Solve problems involving intersecting planes, pyramids etc. Use of a spreadsheet to assist with investigation. CONTEXT AND APPLICATION Tangent/chord theorem and the distance o f the horizon from different heights above the sea. History of the relationship between the sides of right angled triangles (Babylonians, Chinese and Greeks). Bearings. Using angles of elevation and depression to determine heights and distances. Astronomy. 2 CURRICULUM CONTENT GRADES G - F GRADES E - D GRADES C - B TEACHING T OPIC G RADES A – A* OPPORTUNITIES INVESTIGATION AND ILLUSTRATION Calculate the area of a triangle Use the sine and cosine using A = 1/2 absinC. rules to solve 2 -D and 3-D problems. Use distances to locate objects in 3D space. Investigate close packing of spheres, crystal lattices etc. GEOMETRY Find the coordinates Calculate the length of a line of the midpoint of segment in 2-D given the the line segment AB, coordinates of the end points. given points A and B. Understand the concept of gradient and use triangles to calculate gradient. Understand and use 3D coordinates. Investigation of the distance between two points on a coordinate grid. Investigation of gradients. Calculate the length of a line segment in 3D given the coordinates of the end points. Understand simple properties of points, lines and planes in 3D space. E.g. y = 4 is a plane, two planes generally intersect in a line. AND PROOF USE OF ICT Areas of triangles Proof of sine and cosine rules. using A = 1/2absinC Derivation of the formula for and A = 1/2bh. the area of a triangle. C OORDINATE Use coordinates Use coordinates in all in the first four quadrants. quadrant. D EDUCTION CONTEXT AND APPLICATION Calculation of the tetrahedral bond angle in the diamond crystal lattice. Use of a graph plotting program. Derivation of a general result Use of dynamic for the distance between two geometry and points on a coordinate grid. graph plotters , including graphic calculators to investigate and illustrate. CAD Navigation problems involving bearings. Locating positions on a map or grid. Link with algebra and y = mx +c. Gradients of roads and slopes. Locating positions on a map or grid. Air traffic control. Computer images in medicine and engineering. 3 CURRICULUM CONTENT GRADES G - F GRADES E - D GRADES C - B TEACHING T OPIC G RADES A – A* OPPORTUNITIES INVESTIGATION AND ILLUSTRATION 3D GEOMETRY Construct nets of Construct nets of cuboids. pyramids, prisms cones and cylinders Recognise nets from given of, prisms, information. pyramids cylinders and cones Understand and use Explore polyhedra whose faces 2D representations are regular polygons. of 3D objects including isometric Euler’s rule drawings of shapes F + V = E + 2. made from cuboids, simple sections, plans and elevations Carry out reflections of simple shapes in given mirror lines on Cartesian axes. Identify all lines of symmetry for 2D shapes. Reflect and describe reflections of simple shapes in a range of mirror lines on Cartesian axes and use computer packages to reflect shapes. D EDUCTION AND PROOF Use of equipment to make and handle models. USE OF ICT CAD. CONTEXT AND APPLICATION Design of packaging. Crystal structures. Cross curricular links with design technology. More difficult sections of solid shapes. Investigation. Euler’s rule. Existence of only 5 Platonic solids. Polyhedral forms in the natural world – pollen grains, viruses etc. Use rotations to prove Pythagoras’s theorem. Kaleidoscopes. Existence of only 5 Platonic solids linked to regular tessellations. S YMMETRY, TRANSFORMATIONS AND VECTORS Investigate two reflections being equivalent to a rotation. Symmetry in art. Islamic design & architecture. 4 CURRICULUM CONTENT GRADES G - F GRADES E - D GRADES C - B TEACHING T OPIC G RADES A – A* OPPORTUNITIES INVESTIGATION AND ILLUSTRATION Understand and use vertical and horizontal displacement for location and movement. Understand and use vector notation for translation. And use computer packages to translate shapes. Understand and use vectors in the context of translation including inverse translation, repeated translations and combinations of translations. Describe combinations of translations as a single vector. Calculate and represent graphically the sum and difference of two vectors and a scalar multiple of a vector. D EDUCTION AND PROOF USE OF ICT CONTEXT AND APPLICATION Investigate repeated translations and associated vector arithmetic. Transformations of graphs of algebraic and trigonometrical functions. Investigate enlargement of shapes. Combinations of rotations and enlargements to generate spirals. Calculate the resultant of two vectors. Translations and tesselations. Understand and use the commutative and associative properties of vector addition. Solve problems in 2D using vector methods. Enlarge shapes Enlarge shapes using Enlarge shapes using a centre of Enlarge 3D shapes. using any positivea centre of enlargement and any scale whole number enlargement and factor and describe scale factor. any positive whole enlargements. number scale factor . Solve simple problems involving enlargement of 3-D shapes. Identify order of Rotate a shape rotational about the origin symmetry. through multiples of 90°. Photographs. Desk top publishing. Rotate shapes using any centre of rotation and any specified angle and determine the centre and angle of rotation. 5 CURRICULUM CONTENT GRADES G - F GRADES E - D GRADES C - B TEACHING T OPIC G RADES A – A* OPPORTUNITIES INVESTIGATION AND ILLUSTRATION Use practical equipment to investigate simple tiling patterns. Demonstrate that any triangle will tessellate. Investigate polygons and combinations of polygons that will and will not tessellate. Determine which regular polygons will tessellate either singly (regular) or in combination with others (semi-regular). S YMMETRY, Devise instructions for a computer to generate and transform shapes. AND PROOF USE OF ICT CONTEXT AND APPLICATION Tessellations. Use of Logo and Pattern design. dynamic geometry packages. TRANSFORMATIONS AND VECTORS (CONTINUED ) Apply simple combinations of Apply more complex transformations and describe the combinations of results using a single transformations and transformation. describe the results using a single transformation. Distinguish properties that are preserved under particular transformations. D EDUCTION Investigate combinations of transformations. Stretch shapes using an invariant horizontal or vertical line and a scale factor. Estimate lengths. DRAWING , CONSTRUCTION AND LOCI 6 CURRICULUM CONTENT GRADES G - F GRADES E - D GRADES C - B TEACHING T OPIC G RADES A – A* OPPORTUNITIES INVESTIGATION AND ILLUSTRATION Choose and use appropriate instruments and units for measuring length. Use geometrical Understand through experience Construct the equipment to of constructing that triangles circumscribed and inscribe construct 2D shapes. satisfying SSS, SAS, ASA and RHS circles of triangles using are unique but SSA triangles are geometrical equipment or Use computer not. computer packages. packages to construct shapes. Find and sketch loci Construct standard loci using from practical geometrical equipment and examples. computer packages. Interpret and use scale drawing including maps and plans. D EDUCTION AND PROOF USE OF ICT Use of dynamic geometry for drawing loci. CONTEXT AND APPLICATION DIY retail. Graphic design. Use of Logo for construction of shapes. Construct Prove the validity of standard Carry out circumcircles, constructions using standard incircles, centroids, congruence. constructions exscribed circles. using dynamic geometry. Construct unknown centres of rotations. Construct polygons. Maps, plans and elevations. Templates for DIY. Garment making Use of CAD Scale diagrams, maps, blueprints and models Room planning Estimate the size of an angle in degrees. Use of software Snooker problems. packages on estimating angles. 7 CURRICULUM CONTENT GRADES G - F GRADES E - D GRADES C - B TEACHING T OPIC G RADES A – A* OPPORTUNITIES INVESTIGATION AND ILLUSTRATION D EDUCTION AND PROOF Use a 360°angle Understand, measure and use measurer to simple bearings. draw and measure angles. USE OF ICT CONTEXT AND APPLICATION Orienteering. MENSURATION Find perimeters of simple shapes Understand the Derive and use the concept of area formula for the area and find areas by of a rectangle. counting squares. Investigat e areas of rectangles. Derivation of the formula for the area of a rectangle. Understand the derivation of and use the formulae for the areas of parallelograms and triangles. Investigate the Derivation of the formulae for relationships the areas of parallelograms between areas of and triangles. triangles, rectangles and Prove that parallelograms with parallelograms. the same base and heights have the same area. Calculate the areas Solve problems involving lengths, of plane compound areas and volumes in plane shapes. shapes and right prisms. Investigate relationships between perimeters and areas and surface areas and volumes. Investigate volumes of parallelepipeds. Use of Practical spreadsheets for measurement of trial and real objects. improvement. Calculations from plans or blueprints. Fencing problems. Maximum box problems Calculation of the discharge of a river. 8 CURRICULUM CONTENT GRADES G - F GRADES E - D GRADES C - B TEACHING T OPIC G RADES A – A* OPPORTUNITIES INVESTIGATION AND ILLUSTRATION Know and use Calculate lengths of simple arcs formulae for finding and areas of simple sectors of circumferences and circles. areas of circles. Calculate the lengths of arcs and areas of sectors and segments of circles. Circumferencearea relationship for circles. D EDUCTION AND PROOF Prove the formula for the area of a circle by considering any polygon [A=0.5r x perimeter] circumscribing a circle and taking the limit as the polygon tends to a circle. USE OF ICT CONTEXT AND APPLICATION Latitude problems. Understand the concept of volume and find the volume of cuboids by counting cubes. Derive and use the formula for the volume of cuboids. Find the surface areas of prisms Solve problems involving including cylinders. volume and surface areas of pyramids, cones, spheres, composite shapes and frustums. Understand the relationships between lengths, areas and volumes of similar figures with whole number scale factors. Understand the relationships between lengths, areas and volumes of similar figures with any scale factors. Distinguish between formulae for length, area and volume by considering dimensions. 9 CURRICULUM CONTENT GRADES G - F GRADES E - D GRADES C - B TEACHING T OPIC G RADES A – A* OPPORTUNITIES INVESTIGATION AND ILLUSTRATION Use network diagrams to represent information on simple maps. Understand and use Understand and use Euler’s rule Represent solids as the terms arcs, R + N = A + 2. networks. nodes and regions as they relate to simple networks and traversability. NETWORKS Investigation of traversability of networks. D EDUCTION AND PROOF USE OF ICT CONTEXT AND APPLICATION Topological maps. Bridges of Konigsberg. Four Colour Theorem. 10