Supporting planning for shape, space and measures in Key Stage 4: objectives and key indicators

Transcription

1 1 of 7 Supporting planning for shape, space and measures in Key Stage 4: objectives and key indicators This document provides objectives to support planning for shape, space and measures in Key Stage 4. The objectives in the first column are intended to help pupils progress from level 5 at the end of Key Stage 3 to grade C at the end of Key Stage 4. Those in the second column are intended to help pupils progress from level 6 to grade B, and those in the third column are intended to help pupils progress from level 7 to grade A/A*. Objectives that are highlighted in blue are additional to those that have been taken from the Framework for teaching mathematics: Years 7, 8 and 9. Objectives that are highlighted in bold are key indicators of the target grade. These can be useful in tracking pupils progress.

2 2 of 7 Using and applying shape, space and measures Problem solving Solve increasingly demanding problems and evaluate solutions; explore connections in mathematics across a range of contexts: number, algebra, shape, space and measures, and handling data; generate fuller solutions. Select the problem solving strategies to use in geometrical work; discuss the appropriateness of the selected strategies/approaches. Solve increasingly demanding problems and evaluate solutions; explore connections in mathematics across a range of contexts: number, algebra, shape, space and measures, and handling data; generate fuller solutions. Select and combine known facts and problem solving strategies to solve geometrical problems of increasing complexity. Solve increasingly demanding problems and evaluate solutions; explore connections in mathematics across a range of contexts: number, algebra, shape, space and measures, and handling data; generate fuller solutions. Analyse alternative approaches to geometrical problems and give reasons for pursuing or rejecting particular approaches. Communicating Communicate clearly through effective use of symbols and geometrical diagrams. Communicate clearly through effective use of symbols and geometrical diagrams. Use precise formal language and exact methods for analysing geometrical shapes and diagrams. Reasoning Use language and symbols effectively to present a convincing, reasoned argument. Appreciate the difference between mathematical explanation and experimental evidence. Show step-by-step deduction in solving geometrical problems. Explore connections in geometry; pose questions and investigate. Distinguish between practical demonstration and proof; know underlying assumptions, recognising their importance and limitations, and the effect of varying them. Show step-by-step deduction in solving more complex geometrical problems. Explore connections in geometry; pose questions and investigate. Understand the necessary and sufficient conditions under which generalisations, inferences and solutions to geometrical problems remain valid. Explore connections in geometry; pose questions and investigate. Pose extra constraints and investigate whether particular cases can be generalised further. Suggest extensions to problems; conjecture and generalise; identify exceptional cases or counter examples, explaining why.

3 3 of 7 Geometrical reasoning Distinguish between conventions, definitions and derived properties. Identify alternate and corresponding angles; understand a proof that: the sum of the angles of a triangle is 180º and of a quadrilateral is 360º the exterior angle of a triangle is equal to the sum of the two interior opposite angles. Know, and use, that if two 2-D shapes are similar, corresponding angles are equal and corresponding sides are in the same ratio. Understand from this that any two circles and any two squares are mathematically similar while in general any two rectangles 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 compass constructions. Classify quadrilaterals by their geometric properties. Explain how to find, calculate and use: the sums of the interior and exterior angles of quadrilaterals, pentagons and hexagons the interior and exterior angles of regular polygons. Solve problems using properties of angles, of parallel and intersecting lines, and of triangles and other polygons, justifying inferences and explaining reasoning with diagrams and text. Understand and apply Pythagoras theorem when solving problems in 2-D. Understand and apply Pythagoras theorem when solving problems in 2-D and to simple cases in 3-D. Understand and use Pythagoras theorem to solve 3-D problems.

4 4 of 7 Know the definition of a circle and the meaning of terms, including centre, radius, chord, diameter, circumference, tangent, arc, sector and segment. Understanding that inscribed regular polygons can be constructed by equal division of a circle; know that the tangent at any point on the circle is perpendicular to the radius at that point. Visualise and use 2-D representations of 3-D objects; analyse 3-D shapes through 2-D projections, including plans and elevations. Solve problems involving surface areas and volumes of right prisms. Solve problems involving the area and circumference of a circle. Understand and use trigonometrical relationships in right-angled triangles, and use these to solve problems, including those involving bearings. 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. 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 on a cyclic quadrilateral sum to 180º. Solve problems involving surface areas and volumes of cylinders. Solve problems involving lengths of circular arcs and areas of sectors. Use trigonometrical relationships in right-angled triangles to solve 3-D problems including finding the angles between a line and a plane. Calculate the area of a triangle using ½ ab sin C. Draw, sketch and describe the graphs of trigonometric functions for angles of any size, including transformations involving scalings in either or both of the x and y directions. Use the sine and cosine rules to solve 2-D and 3-D problems. Prove and use the alternate segment theorem. Solve problems involving surface areas and volumes of cylinders, pyramids, cones and spheres. Understand and use the formulae for the length of a circular arc and area and perimeter of a sector. Solve problems involving more complex shapes and solids, including segments of circles and frustums of cones.

5 5 of 7 Transformations and coordinates Transform 2-D shapes by combinations of translations, rotations and reflections, on paper and using ICT; know that translations, rotations and reflections preserve length and angle and map objects on to congruent images; identify reflection symmetry in 3-D shapes. Use congruence to show that translations, rotations and reflections preserve length and angle. Enlarge 2-D shapes, given a centre of enlargement and a whole-number scale factor, on paper and using ICT; extend to enlarging 2-D shapes, given a fractional scale factor; recognise the similarity of the resulting shapes. Use any point as the centre of rotation; measure the angle of rotation, using fractions of a turn or degrees; understand that translations are specified by a vector. Use positive fractional and negative scale factors to enlarge shapes. Identify the scale factor of an enlargement as the ratio of the lengths of any two corresponding line segments; recognise that enlargements preserve angle but not length, and understand the implications of enlargement for perimeter. Understand and use the effects of enlargement on perimeter. Understand and use the effects of enlargement on areas and volumes of shapes and solids. Use and interpret maps and scale drawings. Find the coordinates of the midpoint of the line segment AB, given the points A and B; calculate the length of AB. Understand and use vector notation to describe transformation of 2-D shapes by combinations of translations. Calculate and represent graphically the sum of two vectors. 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. Understand and use the commutative and associative properties of vector addition. Solve simple geometrical problems in 2-D using vectors.

6 6 of 7 Measures and construction Use units of measurement to calculate, estimate, measure and solve problems in a variety of contexts; convert between area measures (mm 3 to cm 3, cm 3 to m 3 and vice versa) and between volume measures (mm 3 to cm 3, cm 3 to m 3 and vice versa). Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half of the unit in either direction. Apply knowledge that measurements given to the nearest whole unit may be inaccurate by up to one half of the unit in either direction. Understand and use measures of speed (and other compound measures such as density or pressures) to solve problems. Understand the difference between formulae for perimeter, area and volume in simple contexts by considering dimensions. Understand the difference between formulae for perimeter, area and volume by considering dimensions. Construct specified cubes, regular tetrahedra, squarebased pyramids and other 3-D shapes. Use straight edge and compasses to construct: the mid-point and perpendicular bisector of a line segment the bisector of an angle the perpendicular from a point to a line the perpendicular from a point on a line. Use straight edge and compasses to construct a triangle, given RHS; use ICT to explore constructions of triangles and other 2-D shapes; know from experience of constructing them that triangles given SSS, SAS, ASA or RHS are unique, but that triangles given SSA or AAA are not. Deduce and use formulae for the area of a triangle, parallelogram and trapezium; calculate areas of compound shapes made from rectangles and triangles.

7 7 of 7 Know and use the formula for the volume of a cuboid; calculate volumes and surface areas of cuboids and shapes made from cuboids. Calculate lengths, areas and volumes in right prisms and cylinders. Know and use the formulae for the area and circumference of a circle. Find the locus of a point that moves according to a Extend to more complex rules involving loci and simple given rule, both by reasoning and using ICT. construction.