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Transcript
4 Practical Geometry (Constructions) Objectives: Construct a line parallel to a given line Construct a triangle given three measurements to identify it uniquely Construct a quadrilateral given five measurements to identify it uniquely Before We Begin: Terms Properties Parallel lines and transversal Triangles Congruency Quadrilaterals Angle properties of parallel lines and transversal Congruency criteria for triangles Lesson Plan: (Chapter 10: NCERT Textbook for Class VII, Chapter 4: NCERT Textbook for Class VIII) Content Introduction to Practical Geometry Teacher's Activity Student's Activity Briefly review the terms and properties associated with parallel lines, triangles and congruency by providing clues and inviting the children to respond with appropriate terms and properties Explain how drawing parallel lines, triangles and quadrilaterals would involve using these properties to construct scaled models of the geometrical concepts, using a protractor, compass and ruler Revisit the various properties of parallel lines (transversal and angles), triangles and congruency Assignments Drawing a line parallel to a given line Drawing a triangle given three properties of the triangle Through the construction of a line parallel to a given line, point out the various components that go into geometrical construction: Rough diagram Understanding the properties required to construct the figure Writing clear and concise steps, using geometrical language (points, lines, angles, arcs, intersections, etc.) Constructing the fair diagram using a well-sharpened pencil Explain how a parallel line is constructed using one of the properties of transversal (e.g. interior alternate angles are equal) Encourage the students to try constructing a parallel line using another property (e.g. corresponding angles are equal) Understand how a property of parallel lines, with a transversal, can be used to construct a line parallel to a given line Understand the various components of a geometrical construction and neatly construct a line parallel to a given line Review the various congruency criteria for triangles: SSS, ASA, SAS and RHS Point out that if we are to construct a triangle that exactly matches a given triangle, then the congruency criteria need to match. Deduce that the minimum criteria required to construct a triangle thus need to match one of the congruency criteria Explain, with clear, written steps, the construction of a triangle, given 3 parameters that match one of the congruency criteria Identify which congruency criteria the given triangle parameters match Construct a triangle using given parameters, with a rough diagram, fair diagram and clear steps Exercise questions from Class VII textbook, Practical Geometry (Chapter 10) Drawing a quadrilateral given five properties of the quadrilateral Just as a triangle requires a minimum of 3 parameters to construct, invite students to identify the minimum parameters required to construct a quadrilateral (Hint: break up the quadrilateral into two triangles) Building on the congruency criteria covered for triangles, draw a rough diagram for the given quadrilateral to identify if it can be constructed. Construct a quadrilateral with the given parameters Understand how a minimum of 5 parameters are required to construct a quadrilateral, by dividing it into two triangles Using a rough diagram, understand how to construct the given quadrilateral Construct a quadrilateral and write steps to explain the construction Textbook examples Select questions from Ex 4.1, 4.2, 4.3, 4.4 Drawing a quadrilateral given less than 5 parameters Discuss special cases where the known properties of quadrilaterals can be used to derive the minimum parameters required to construct the quadrilateral Squares Rectangles Parallelograms (in general) Trapeziums Rhombus Kites Understand properties of the following special quadrilaterals: Squares Rectangles Parallelograms (in general) Trapeziums Rhombus Kites Use the known properties to construct a quadrilateral even if less than 5 parameters are given (e.g. one parameter is sufficient to construct a square: side of the square) Examples for each shape Ex 4.5 Introduce exceptions to the 3 parameter/5 parameter rules for triangles and quadrilaterals by asking children to construct figures that are not practically possible (e.g. triangle given three angles, Understand that the minimum parameter rule (3 for triangles and 5 for quadrilaterals) may not be sufficient in some cases Draw a rough diagram using the parameters given and analyse to see if Feasibility of drawing the given triangle/quadrilateral Analyse constructions given the following parameters for quadrilaterals: Four angles and a side quadrilateral given 3 sides and 2 nonincluded angles, etc.) Deduce that 3 parameters are the minimum number of parameters required to construct a triangle but these may not be sufficient in all cases. Similarly, deduce that 5 parameters are the minimum number of parameters required for quadrilaterals but these may not be sufficient in all cases. Point out the importance of drawing a rough diagram and analysing if a particular triangle or quadrilateral can be constructed with the given parameters Activity the parameters given are sufficient to construct the figure Explain with reason why a given set of parameters is not sufficient to construct a triangle or quadrilateral 3 sides, one included angle and one nonincluded angle 2 sides, a diagonal, two non-included angles Details below Self-Assessment and Test Why do we need to learn to construct geometrical shapes? We use various properties of parallel lines, triangles and quadrilaterals to construct these geometrical shapes. This process is a practical reinforcement of various theoretical concepts. e.g. Understanding why three parameters are required to prove congruency of two triangles. Construction is fun!! Later on, this develops into advanced graphics techniques that have varied applications in architecture, engineering, animations, etc. Activity: Children can choose to do one or more of the following: Construction of geometrical designs using ruler and compass only (worksheet provided with various complex design patterns) Scaled 2-D model of Poorna Scaled 3-D model of Poorna Self-Assessment: Topic Understanding of the topic Working with simple cases High comfort-level and confidence with the topic Drawing a line parallel to a given line I know the properties used to construct parallel lines. I can construct a line parallel to the given line using the property and steps demonstrated in class I can construct a line parallel to the given line using other properties of parallel lines (e.g. other angle properties of transversal, properties of parallelograms, etc.) Drawing a triangle given three properties of the triangle I understand that the parameters required to construct a triangle are the same as the criteria required to prove congruency of triangles (SSS, ASA, SAS, RHS) I can draw a rough diagram to indicate what parameters are given I can construct a triangle with the given parameters I can write the steps required to construct the triangle I can identify if a triangle can be constructed using the given parameters I can explain why a triangle cannot be constructed using the given parameters (e.g. angle sum property is not satisfied, three angles are not sufficient to construct a triangle, etc.) I can draw a rough diagram to indicate what parameters are given I can construct a quadrilateral with the given parameters I can write the steps required to construct the quadrilateral I can identify if a quadrilateral can be constructed using the given parameters I can explain why a quadrilateral cannot be constructed using the given parameters (e.g. which vertices cannot be located if four angles and one side are given) I can use properties of special quadrilaterals (square, rectangle, parallelogram, trapezium, rhombus, kite) to construct them even if fewer than 5 parameters are given Drawing a quadrilateral given five properties of the quadrilateral I understand that a minimum of 5 parameters are required to construct a quadrilateral I can divide a quadrilateral into two triangles and use properties of triangles to understand how to construct a quadrilateral Uses of geometrical constructions I understand the practical uses of geometrical constructions and can cite examples I can use a ruler, compass and protractor to draw a scaled version of simple 2-D objects e.g. geometrical patterns like below I can use a ruler and compass to draw a scaled version of complex 2-D objects e.g. I can use a ruler, compass and protractor to draw a scaled 2-D model of a simple 3-D shape (e.g. a house) Additional Comments: (Please write a few lines about your experience with the topic, whether you need to put in additional time or require teacher’s help, etc. Be as specific as possible)