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Course(s): Year 10 (Stage 5.1) Unit of Work: Review of Space & Geometry KLA RE ENG MA SCI Indicative Hours: 15 h HSIE PDHPE Term: 3 CA TAS LOTE Weeks: 7 to 10 Unit Description: This topic gets covered in Stage 4. Its inclusion at this juncture is to review basic concepts and practise essential skills in preparation for the School Certificate Examination. The study of solid shapes imparts basic understanding of their dimensions that is an essential requirement in the Measurement strand. Also students of Design & Technology and Art may apply skills developed in this topic, when they prepare diagrams, sketches and/or drawings for the making of models or products; and students of Science will find this topic useful when drawing and building models of crystals. The study of angles is the introduction to Deductive Geometry where students are required to support statements with reasons. A sound understanding of this basic topic facilitates a better application of geometrical concepts to solve problems and theorems at a later stage. The sequential build-up of geometrical concepts requires students to study triangles and basic quadrilaterals and utilise geometrical instruments to construct these shapes according to stipulated parameters. The properties of special quadrilaterals are important in Measurement. For example, the perpendicularity of the diagonals of a rhombus and a kite allow a rectangle of twice the size to be constructed around them, leading to formulae for finding area. At this Stage, the treatment of triangles and quadrilaterals is still informal, with students consolidating their understandings of different triangles and quadrilaterals and being able to identify them from their properties. Similarity is linked with ratio in the Number strand and with map work in Geography. Similar and congruent figures are embedded in a variety of designs (eg tapa cloth, Aboriginal designs, Indonesian ikat designs, Islamic designs, designs used in ancient Egypt and Persia, window lattice, woven mats and baskets). 1 Syllabus outcomes for each course: SGS 4.1: Describes and sketches three-dimensional solids including polyhedra, and classifies them in terms of their properties. SGS 4.2: Identifies and names angles formed by the intersection of straight lines, including those related to transversals on sets of parallel lines, and makes use of the relationships between them. SGS 4.3: Classifies, constructs, and determines the properties of triangles and quadrilaterals. SGS 4.4: Identifies congruent and similar two-dimensional figures stating the relevant conditions. Resources: Mathematics Syllabus Years 7 – 10, 2003 New Century Mathematics Year 10 (Stage 5.1) – Chapter 6 CD-ROM included with textbook Hotmaths website Phase 1: 2 Properties of Solids Describing solids in terms of their geometric properties number of faces shape of faces number and type of congruent faces number of vertices number of edges convex or non-convex Identifying any pairs of parallel flat faces of a solid Determining if two straight edges of a solid are intersecting, parallel or skew Determining if a solid has a uniform cross-section Classifying solids on the basis of their properties A polyhedron is a solid whose faces are all flat. A prism has a uniform polygonal cross-section. A cylinder has a uniform circular cross-section. A pyramid has a polygonal base and one further vertex (the apex). A cone has a circular base and an apex. All points on the surface of a sphere are a fixed distance from its centre. Identifying right prisms and cylinders and oblique prisms and cylinders Identifying right pyramids and cones and oblique pyramids and cones Sketching on isometric grid paper Teaching Strategies Registration Demonstrate the describe solids in terms of their geometric properties number of faces, shape of faces number and type of congruent faces number of vertices number of edges convex or non-convex Identify the pairs of parallel flat faces of a solid. Demonstrate if two straight edges of a solid are intersecting, parallel or skew Explain the meaning of uniform cross- section Classify solids on the basis of their properties. Explain the meaning of Polyhedron. Explain the different properties. Interpret and make models from isometric drawings Recognise solids with uniform and non-uniform crosssections Analyse three-dimensional structures in the environment to explain why they may be particular shapes eg buildings, packaging Visualise and name a common solid given its net Recognise whether a diagram is a net of a solid Identify parallel, perpendicular and skew lines in the environment 3 Assessment Strategies: what and how? Teacher in-class observation Questioning Participation Written assessment task shapes built with cubes Representing three-dimensional objects in two dimensions from different views Confirming, for various convex polyhedra, Euler’s formula F+V=E+2 relating the number of faces (F), the number of vertices (V) and the number of edges (E) Exploring the history of Platonic solids and how to make them Making models of polyhedra Phase 2: 4 Angles at a Point Labelling and naming points, lines and intervals using capital letters Labelling the vertex and arms of an angle with capital letters Labelling and naming angles using A and XYZ notation Using the common conventions to indicate right angles and equal angles on diagrams Identifying and naming adjacent angles (two angles with a common vertex and a common arm), vertically opposite angles, straight angles and angles of complete revolution, embedded in a diagram Using the words ‘complementary’ and ‘supplementary’ for angles adding to 90º and 180º respectively, and the terms ‘complement’ and ‘supplement’ Establishing and using the equality of vertically opposite angles Teaching Strategies Explain and demonstrate the labeling and naming of points, lines and intervals Demonstrate and give examples of labeling vertices and arms of angles with capital letters Demonstrate the different ways of labeling angles Demonstrate the common conventions indicating right angles and equal angles on diagrams Demonstrate and explain how to identify and name adjacent, vertically opposite angles, straight angles and angles of complete revolution Explain the use of the words ‘complementary’ and ‘supplementary’ when describing angles Demonstrate the use of equality of vertically opposite angles. Recognise and explain why adjacent angles adding to 90º form a right angle Recognise and explain why adjacent angles adding to 180º form a straight angle Recognise and explain why adjacent angles adding to 360º form a complete revolution Find the unknown angle in a diagram using angle results, giving reasons Phase 3: Angles Associated with Transversals Identifying and naming a pair of parallel Teaching Strategies Revision of parallel lines. 5 lines and a transversal Using common symbols for ‘is parallel to’ ( ) and ‘is perpendicular to’ () Using the common conventions to indicate parallel lines on diagrams Identifying, naming and measuring the alternate angle pairs, the corresponding angle pairs and the co-interior angle pairs for two lines cut by a transversal Recognising the equal and supplementary angles formed when a pair of parallel lines are cut by a transversal Using angle properties to identify parallel lines Using angle relationships to find unknown angles in diagrams Identify parallel and perpendicular lines in the class environment e.g., opposite sides of the white board are parallel but the adjacent sides are perpendicular and so on. Explain the common conventions. Use diagrams to explain the different types of angles in parallel lines.(corresponding, alternate and co-interior angles) Revision of complementary and supplementary angles. Demonstrate and explain how to find the value of the unknown angle by using the properties of two-dimensional shapes and parallel lines results. Apply angle results to construct a pair of parallel lines using a ruler and a protractor, a ruler and a set square, or a ruler and a pair of compasses Apply angle and parallel line results to determine properties of two-dimensional shapes such as the square, rectangle, parallelogram, rhombus and trapezium Identify parallel and perpendicular lines in the environment Construct a pair of perpendicular lines using a ruler and a protractor, a ruler and a set square, or a ruler and a pair of compasses Use dynamic geometry software to investigate angle relationships Phase 4: Notation Teaching Strategies Labelling and naming triangles (eg ABC) Demonstrate the ability to label triangles and quadrilaterals. and quadrilaterals (eg ABCD) in text and Demonstrate the common conventions to mark equal on diagrams intervals on diagrams Using the common conventions to mark equal intervals on diagrams Demonstrate how to classify triangles on the basis of their 6 Triangles Recognising and classifying types of triangles on the basis of their properties (acute-angled triangles, right-angled triangles, obtuse-angled triangles, scalene triangles, isosceles triangles and equilateral triangles) Constructing various types of triangles using geometrical instruments, given different information e.g., the lengths of all sides, two sides and the included angle, and two angles and one side Justifying informally by paper folding or cutting, and testing by measuring, that the interior angle sum of a triangle is 180º, and that any exterior angle equals the sum of the two interior opposite angles Using a parallel line construction, to prove that the interior angle sum of a triangle is 180º Proving, using a parallel line construction, that any exterior angle of a triangle is equal to the sum of the two interior opposite angles properties (acute-angled triangles, right angled triangles, obtuse-angled triangles, scalene triangles, isosceles triangles and equilateral triangles) Demonstrate how to construct triangles using geometrical instruments, given different information eg the lengths of all sides, two sides and the included angle, and two angles and one side Students verify by paper folding or cutting, and testing by measuring, that the interior angle sum of a triangle is 180º, and that any exterior angle equals the sum of the two interior opposite angles Sketch and label triangles from a given verbal description Describe a sketch in sufficient detail for it to be drawn Recognise that a given triangle may belong to more than one class Recognise that the longest side of a triangle is always opposite the largest angle Recognise and explain why two sides of a triangle must together be longer than the third side Recognise special types of triangles and quadrilaterals embedded in composite figures or drawn in various orientations Determine if particular triangles and quadrilaterals have line and/or rotational symmetry Apply geometrical facts, properties and relationships to solve numerical problems such as finding unknown sides and angles in diagrams Justify their solutions to problems by giving reasons using their own words 7 Phase 5: Quadrilaterals Distinguishing between convex and nonconvex quadrilaterals (the diagonals of a convex quadrilateral lie inside the figure) Establishing that the angle sum of a quadrilateral is 360º Constructing various types of quadrilaterals Investigating the properties of special quadrilaterals (trapeziums, kites, parallelograms, rectangles, squares and rhombuses) by using symmetry, paper folding, measurement and/or applying geometrical reasoning Properties to be considered include : opposite sides parallel opposite sides equal adjacent sides perpendicular opposite angles equal diagonals equal in length diagonals bisect each other diagonals bisect each other at right angles diagonals bisect the angles of the quadrilateral Investigating the line symmetries and the order of rotational symmetry of the special quadrilaterals Classifying special quadrilaterals on the basis of their properties Teaching Strategies Sketch and label quadrilaterals from a given verbal description Bisect an angle by applying geometrical properties e.g., constructing a rhombus Bisect an interval by applying geometrical properties e.g., constructing a rhombus Draw a perpendicular to a line from a point on the line by applying geometrical properties e.g., constructing an isosceles triangle Draw a perpendicular to a line from a point off the line by applying geometrical properties e.g., constructing a rhombus Use ruler and compasses to construct angles of 60º and 120º by applying geometrical properties e.g., constructing an equilateral triangle Explain with the help of examples the difference between convex and non-convex quadrilaterals. Explain the sum of the angles of a quadrilateral. Explain the different properties of quadrilaterals. Phase 6: 8 Circles Teaching Strategies Identifying and naming parts of the circle Explain that a circle consists of all points that are a given distance from the centre and how this relates to the use of and related lines, including arc, tangent a pair of compasses and chord Investigating the line symmetries and the Use dynamic geometry software to investigate the properties of geometrical figures rotational symmetry of circles and of diagrams involving circles, such as a sector and a circle with a chord or tangent Phase 7: Congruence Identifying congruent figures by superimposing them through a combination of rotations, reflections and translations Matching sides and angles of two congruent polygons Naming the vertices in matching order º in a when using the symbol congruence statement Drawing congruent figures using geometrical instruments Determining the condition for two circles to be congruent (equal radii) Teaching Strategies Explain the meaning superimposing objects. of the word congruent by Give class examples of congruent shapes by using diagrams, drawing found in media, design work etc. Also write the matching sides and vertices. Demonstrate and explain the four tests of congruency by means of diagrams. Students can cut out different shapes and test congruency by placing them on top of each other. Explain the difference between congruence and similarity. Recognise congruent figures in tessellations, art and design work Phase 8: 9 Similarity Using the term ‘similar’ for any two figures that have the same shape but not necessarily the same size Matching the sides and angles of similar figures Naming the vertices in matching order when using the symbol lll in a similarity statement Determining that shape, angle size and the ratio of matching sides are preserved in similar figures Determining the scale factor for a pair of similar polygons Determining the scale factor for a pair of circles Calculating dimensions of similar figures using the enlargement or reduction factor Choosing an appropriate scale in order to enlarge or reduce a diagram Constructing scale drawings Drawing similar figures using geometrical instruments Teaching Strategies Demonstrate and explain the use of similar figures in finding lengths in the environment where it is impractical to measure directly eg heights of trees, buildings. Using scale factor demonstrate how to draw similar figures (by enlarging or reducing the diagram) Explain similarity by enlarging cartoons and pictures and hence demonstrate how to find the scale factor. Demonstrate and explain by means of examples how to match the sides and angles of similar figures. Use enlargement or reduction factor to find the dimensions of similar figures. Explain how to construct similar figures using geometrical instruments. Interpret and use scales in photographs, plans and drawings found in the media and/or other learning areas Enlarge diagrams such as cartoons and pictures Apply similarity to finding lengths in the environment where it is impractical to measure directly eg heights of trees, buildings Apply geometrical facts, properties and relationships to solve problems such as finding unknown sides and angles in diagrams Justify their solutions to problems by giving reasons using their own words Recognise that area, length of matching sides and angle sizes are preserved in congruent figures Recognise that shape, angle size and the ratio of matching sides are preserved in similar figures Recognise that similar and congruent figures are used in specific designs, architecture and art work eg works by 1 0 Escher, Vasarely and Mondrian; or landscaping in European formal gardens Find examples of similar and congruent figures embedded in designs from many cultures and historical periods Use dynamic geometry software to investigate the properties of geometrical figures 1 1 REGISTRATION & TEACHER EVALUATION Date Unit of Work was Started: ________________ Date Unit of Work was Completed: ________________ It is confirmed that the outcomes ticked () above have been completed. Supporting Students with Special Needs Students with Special Needs in Class Targeted Outcomes for these Students Accommodations/ Adaptations (if different to rest of class) (as necessary) 1 2 Special Provisions Organised (if applicable) Comments Teaching Strategies and Applications, noting any additional ones Use of resources Need for extra resources Student response (Students’ attitude to the Unit) Appropriateness and Achievement of Outcomes Timing and placement of topic Depth of coverage Alterations made to cater for range of abilities in the class Further points for improvement Name of Teacher: ____________________________ Date: ___________________ Signature: __________________________________ 1 3 1 4