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MA4-18MG Angle Relationships Summary of substrands Duration 2 weeks MA3-16MG measures and constructs angles, and applies angle relationships to find unknown angles. Stage 3 - Angles 1 Estimate, measure and compare angles using degrees (ACMMG112) Construct angles using a protractor (ACMMG112) Stage 3 - Angles 2 Investigate, with and without the use of digital technologies, angles on a straight line, angles at a point, and vertically opposite angles; use the results to find unknown angles (ACMMG141) Unit overview The development of knowledge and understanding of angle relationships, including the associated terminology, notation and conventions, is of fundamental importance in developing an appropriate level of knowledge, skills and understanding in geometry. Angle relationships and their application play an integral role in students learning to analyse geometry problems and developing geometric and deductive reasoning skills, as well as problem-solving skills. Angle relationships are key to the geometry that is important in the work of architects, engineers, designers, builders, physicists, land surveyors, etc, Outcomes MA41WM communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols MA42WM applies appropriate mathematical techniques to solve problems MA43WM recognises and explains mathematical relationships using reasoning MA418MG identifies and uses angle relationships, including those related to transversals on sets of parallel lines Big Ideas/Guiding Questions How do you know when two lines are parallel? Students can measure the angle of various ramps, such as wheelchair access ramps and skateboard ramps. For example, if given that a wheelchair ramp must have a slope of 1 in 8, they could draw this and discover what angle it has to be. Could use the idea of the billiards table or a soccer goal to discover the as well as the geometry that is common and important in everyday situations, such as in nature, sports, buildings, astronomy,art, etc. angle that the ball needs to be hit or kicked on. Key words angle, obtuse, corresponding, reflex, co interior, degree, adjacent, diagonal, parallel, revolution, complementary, straight, supplementary, right-angled, perpendicular, vertically, opposite, protractor, transversal, equilateral, acute, alternate, interior, exterior, interval, complement, supplement, Students in Stage 4 should write geometrical reasons without the use of abbreviations to assist them in learning new terminology, and in understanding and retaining geometrical concepts, eg 'When a transversal cuts parallel lines, the cointerior angles formed are supplementary'. In Stage 4, students should use full sentences to describe the properties of plane shapes, eg 'The diagonals of a parallelogram bisect each other'. Students may not realise that in this context, the word 'the' implies 'all' and so this should be made explicit. Using the full name of the quadrilateral when describing its properties should assist students in remembering the geometrical properties of each particular shape. Students in Stage 4 should write geometrical reasons without the use of abbreviations to assist them in learning new terminology, and in understanding and retaining geometrical concepts. Catholic Perspectives School Free Design God has created the universe. A place of beauty and wonder. Since the beginning of time, man has pondered about the universe and the stars. It is the use of geometry that has helped develop a deeper understanding of our world. The importance of Euclidean Geometry is one of historical and practical use for the study of mathematics in today’s society. Euclidean Geometry is one of the oldest branches of mathematics, developed by Euclid in 300BC, and serves as a basis of modern mathematics that governs our world. (www.muchmoremath.com/geometry) Assessment Overview Generally, teachers should design specific assessment tasks that can be drawn from a variety of the following sources of information and assessment strategies: • student responses to questions, including open ended questions, • student explanation and demonstration to others, • questions posed by students, • samples of student work, • student produced overviews or summaries of topics, • investigations or projects, • students oral and written report • practical tasks and assignments, • short quizzes • pen and paper tests, including multiple choice, short answer questions and questions requiring longer responses, including interdependent questions ( where one answer depends on the answer obtained in the preceding part) • open book tests • comprehension and interpretation exercise • student produced worked samples, • teacher/student discussion or interviews • observation of students during learning activities including the student’s correct use of terminology • observation of a student participating in a group activity References can be made to the relevant end of chapter review or screening tests found in textbooks or other resource areas Content Teaching, learning and assessment Use the language, notation and conventions of geometry Open discussion on ‘what is an angle’ ‘where can you find angles’, ‘how do you measure angles’ define, label and name points, lines and intervals using capital letters label the vertex and arms of an angle with capital letters Interactive protractor on Smart board Look for angles in the environment or in pictures from magazines and newspapers. Group activity: Student to construct and label the vertex and arms of a number of angles for the other student to measure and name the angle. FREEFALL WORKSHEETS GEOGEBRA label and name angles using or Resources notation use the common conventions to indicate right angles and equal angles on diagrams Practical activity: Give a page of different size angles. Students to estimate the size of each angle then measure the angle recording their findings in a table. Could also include columns to name the angle and identify the type of angle. MATHLETICS ACTIVITIES: Ruler and Protractors Recognise the geometrical properties of angles at a point use the terms 'complementary' and 'supplementary' for angles adding to 90° and 180°, respectively, and the associated terms 'complement' and Some students may find the use of the terms 'complementary' and 'supplementary' (adjectives) and 'complement' and 'supplement' (nouns) difficult. Teachers should model the use of these terms in sentences, both verbally and in written form, eg, '50° and 40° are complementary angles', 'The complement of 50° is 40°'. Useful websites for worksheets, interactive games, maths charts, flashcards http://resources.woodlands-junior.kent.sch.uk/maths/units120.html http://www.topmarks.co.uk/Interactive.aspx?cat=22 http://www.helpingwithmath.com/ 'supplement' use the term 'adjacent angles' to describe a pair of angles with a common arm and a common vertex identify and name right angles, straight angles, angles of complete revolution and vertically opposite angles embedded in diagrams recognise that adjacent angles can form right angles, straight angles and angles of revolution (Communicating, http://www.sheppardsoftware.com/math.htm Students should be aware that complementary and supplementary angles may or may not be adjacent. Students are to be encouraged to give reasons when finding unknown angles. For some students formal setting out could be introduced. For example, ABQ = 70º (corresponding angles, AC PR) www.au.ixl.com Reasoning) Sector graphs could be used to demonstrate angles at a point. Identify corresponding, alternate and co-interior angles when two straight lines are crossed by a transversal (ACMMG163) identify and name perpendicular lines using the symbol for 'is perpendicular to' ( The other two results then follow using vertically opposite angles and angles on a straight line. Alternatively, the equality of the alternate angles can be seen by rotation about the midpoint of the transversal use the common conventions to indicate parallel lines on diagrams identify and name pairs of parallel lines using the symbol for 'is parallel to' ), eg Students could explore the results about angles associated with parallel lines cut by a transversal by starting with corresponding angles – move one vertex and all four angles to the other vertex by a translation. , eg define and identify 'transversals', including transversals of parallel http://www.mathopenref.com/parallel.html www.mathsisfun.com/geometry/parallel-lines www.mathplanet.com/education/geometry Using a diagram similar to the one below, students could be asked to find the values of each pronumeral and give reason, given one of them. http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/parallellinesrev1.shtml lines identify, name and measure alternate angle pairs, corresponding angle pairs and co-interior angle pairs for two lines cut by a Draw a pair of parallel lines, students can rule their own transversal and then measure each Rulers, protractor angle. transversal use dynamic geometry software to investigate angle relationships formed by parallel lines and a transversal (Problem Solving, Reasoning) recognise the equal and supplementary angles formed when a pair of parallel lines is cut by a 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 transversal Investigate conditions for two lines to be parallel (ACMMG164) use angle properties to identify parallel lines explain why two lines are either parallel or not parallel, giving a reason (Communicating, Reasoning) 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 http://www.mathopenref.com/parallel.html apply geometrical facts, properties and relationships to solve numerical problems such as finding unknown sides and angles in diagrams Eratosthenes’ calculation of the circumference of the earth used parallel line results. Solve simple numerical problems using reasoning (ACMMG164) Find the value of each variable and give reason. find the sizes of unknown angles embedded in diagrams using angle relationships, including angles at a point and angles associated with parallel lines, giving reasons explain how the size of an unknown angle was calculated (Communicating, Reasoning) Emphasise to the students when answering questions on angles, remember the following points: Look out for parallel lines (squares, rectangles, parallelograms and marked parallel lines) and see which of the above rules apply. Look out for lines of equal length. Remember that an isosceles triangle has two sides of equal length and that the angles which are directly opposite these sides are also equal. Diagrams are unlikely to be drawn to scale, so do not be tempted to measure! Do not assume things which may turn out to be untrue (for example, an angle is not necessarily a right angle just because it looks as though it should be!). It is likely that you will not be able to find the required angle immediately. Do not worry - just write as much information as possible onto the diagram. It should eventually lead you to the answer. If not, you might still get some marks for your working! Registration Evaluation Class: __________________________ Start Date: _______________________ Finish Date: ______________________ Teacher’s Signature: _______________________ Sample questions Highlight the response that best describes your view to the following statements and provide comments in the spaces provided. 1. The set text/s (if relevant) were suitable for the student needs and interests: STRONGLY AGREE 2. AGREE UNSURE STRONGLY DISAGREE There were sufficient and suitable resources to teach the unit: STRONGLY AGREE AGREE UNSURE STRONGLY DISAGREE 3. There was sufficient time to teach the set content: STRONGLY AGREE 4. AGREE UNSURE STRONGLY DISAGREE Assess the degree to which syllabus outcomes have been demonstrated by students in this unit: ........................................................................................................................................................................................................................................................................................................................................... ...................... ........................................................................................................................................................................................................................................................................................................................................... ...................... ........................................................................................................................................................................................................................................................................................................................................... ....................... 5. Evaluate the degree to which the diverse needs of learners have been addressed in this unit: ........................................................................................................................................................................................................................................................................................................................................... ...................... ............................................................................................................................................................................................................................................................................................................................................. .................... ........................................................................................................................................................................................................................................................................................................................................... ...................... 6. Comment on the effectiveness of pedagogical practices employed in this unit: ........................................................................................................................................................................................................................................................................................................................................... ...................... ........................................................................................................................................................................................................................................................................................................................................... ...................... ........................................................................................................................................................................................................................................................................................................................................... ...................... 7. Assessment was meaningful and appropriate to reflect student learning and achievement: ........................................................................................................................................................................................................................................................................................................................................... ...................... ........................................................................................................................................................................................................................................................................................................................................... ...................... ........................................................................................................................................................................................................................................................................................................................................... ...................... 8. Suggested program adjustments / other comments: ........................................................................................................................................................................................................................................................................................................................................... ...................... ........................................................................................................................................................................................................................................................................................................................................... ...................... ........................................................................................................................................................................................................................................................................................................................................... ......................