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|MA5.3-16MG : Properties of Geometrical Figures | Mathematics Stage 5 Year 9/10 |
|Summary of Sub Strands |Duration |
|S4 Properties of Geometrical Figures |5 weeks |
| |Detail: 5 weeks, … lessons per week (…hours) |
|Unit overview |Outcomes |Big Ideas/Guiding Questions |
|Develops and applies results for proving that |Mathematics K-10 | |
|triangles are congruent or similar |MA5.1-11MG describes and applies the properties of | |
|Identify similar triangles and describe their |similar figures and scale drawings | |
|properties |MA5.2-14MG calculates the angle sum of any polygon | |
|Apply tests for congruent triangles |and uses minimum conditions to prove triangles are | |
|Use simple deductive reasoning in numerical and|congruent or similar | |
|non-numerical problems |MA5.3-16MG proves triangles are similar, and uses | |
|Verify the properties of special quadrilaterals|formal geometric reasoning to establish properties | |
|using congruent triangles |of triangles and quadrilaterals | |
|Construct proofs of geometrical relationships | | |
|involving congruent or similar triangles | | |
| | |Key Words |
| | |Congruent, construction, enlargement, matching, proportion, reduction, reflection, rotation, scale, |
| | |scale factor, similar, similarity, transformation, translation, corresponding sides and angles |
|Catholic Perspectives |School Free Design |
|TALK TO MICHAEL FOR WHAT TO PUT HERE |This is a free design area for schools to add local additional areas. This could include: |
|CEO will provide guidance in this area |Context if you prefer the unit overview and context to be separate |
|Example: |School focus for learning – eg blooms taxonomy, solo taxonomy, contemporary learning, habits of mind,|
|The value of sacramentality celebrates the presence of God in every facet of creation. |BLP (building learning power) |
|The Christian message is ultimately one of hope |Any specific social and emotional learning which could be embedded into the unit eg enhanced group |
| |work |
|Mathematics, Reality, and God | |
|Paul A. Schweitzer | |
|DOI:10.1093/acprof:oso/9780199795307.003.0013 | |
|Simplicity and symmetry are the heart of beauty in mathematics. Beauty often motivates mathematicians| |
|and physicists. Einstein said that his theory of general relativity had to be true because it was so | |
|elegant. Archimedes was thrilled with his discovery that the ratio of the volume of a cylinder | |
|tightly enclosing the volume of a sphere is 3:2. Mathematics offers beauty without defects. Salvador | |
|Dali produced two religious paintings that have important mathematical components. Mathematics have | |
|very precise norms for proving theorems, but these generally don’t apply to ordinary life or other | |
|academic disciplines. Kurt Gödel brilliantly proved that a mathematical system could be proven either| |
|complete or consistent, but not both. This means mathematics is open to the transcendent, as must | |
|other disciplines be as well, since they are less precise than mathematics. Every type of rational | |
|discourse must be judged according to its own procedures and limitations. By developing n-space, the | |
|mind shows it is made in the image of God. It is helpful to compare theology with mathematics. Both | |
|subjects always have new problems to solve. It is now known that Gödel developed a proof for the | |
|existence of God based on the ontological argument. | |
|Keywords: beauty, golden,mean, Einstein, Dali, Gödel, completeness, consistency, theology, ontologi| |
|cal argument | |
| | |
|Below Connected Website | |
| | |
|Numeracy and the Catholic World View | |
|Numeracy operates within a variety of social contexts. From a Catholic perspective, numeracy must be | |
|imbued with a vision of the innate dignity of all students, as created in the image and likeness of a| |
|loving, generous and creating God. Teachers in Catholic schools have an obligation to not only teach | |
|their students the skills and knowledge to be numerate, but to teach from a Catholic perspective. | |
|Teachers are called to challenge their students to use the skills and knowledge they have acquired to| |
|bring about social change in the world. | |
| | |
|Below is from St Josephs Narrabeen | |
|[pic] | |
| | |
|Below is from Mount St Patrick College, Murwillumbah | |
|PRIMARY AIM | |
|The primary aim of the Department, as a whole, is to inculcate the skills, knowledge and attitudes as| |
|outlined in the syllabuses with a Catholic Perspective. | |
| | |
|GENERAL AIMS | |
|• To provide a structured and caring environment for the learning of Mathematics. | |
|• To develop the significance and relevance of Mathematics in everyday life. | |
|• To attempt to equip all student with the Mathematical skills and knowledge which will help | |
|them to cope with everyday life. | |
|• To make Mathematics meaningful and relevant to students. | |
|• To make Mathematics interesting and enjoyable. | |
|• To teach students to think clearly and logically. | |
|• To teach students good study habits. | |
|• To bring student to the realisation that they are not just learning Mathematics to pass | |
|examinations. | |
|• To develop staff professionally. | |
|• To foster the language of Mathematics as a form of communication. | |
|• To provide a sense of justice and equity in Mathematics regardless of racial origin or | |
|religion and to avoid stereotyping of roles for each sex. | |
| | |
|THE NATURE OF MATHEMATICS LEARNING | |
|Mathematics is learnt by individual students at different rates. | |
|It must be remembered that:- | |
|• students learn best when motivated | |
|• students learn Mathematics through interacting and reflecting. | |
|• students learn Mathematics through investigating. | |
|• students learn Mathematics through language. | |
|• students learn Mathematics as individuals in the context of cultural, intellectual, | |
|physical and social growth. | |
| | |
|CATHOLIC PERSPECTIVE | |
|'We are committed to the development of Catholic schools which demonstrate a special concern for, and| |
|understanding of, the uniqueness of each person.' | |
| | |
|Tick Points History | |
|CATHOLIC (GOSPEL) VALUES: | |
| GV1 Celebration | |
| GV2 Common Good | |
|GV3 Community | |
|GV4 Compassion | |
| GV5 Cultural Critique | |
| GV6 Faith | |
| GV7 Hope | |
|GV8 Human Rights | |
| GV9 Joy | |
|GV10 Justice | |
| GV11 Peace | |
| GV12 Reconciliation | |
| GV13 Sacredness of Life | |
|GV14 Stewardship of Creation | |
| GV15 Service | |
|GV16 Wisdom | |
| | |
|Tick Points Science | |
|Catholic Perspective Keywords | |
|Awe and Wonder | |
|Celebration | |
|Common Good | |
|Charity | |
|Commitment to community | |
|Community Conservation | |
|Compassion | |
|Courage | |
|Cultural Critique | |
|Dignity of each human person | |
|Endurance / perseverance | |
|Faith | |
|Family | |
|Forgiveness | |
|Global Solidarity and the Earth Community | |
|Hope | |
|Hospitality | |
|Human Rights Justice | |
|Joy | |
|Justice | |
|Love | |
|Multicultural Understanding | |
|Peace | |
|Reconciliation | |
|Reverence | |
|Sacredness of Life | |
|Service | |
|Sense of wonder | |
|Servant leadership | |
|Stewardship of Creation | |
|Structural Change | |
|Self Respect (Self Esteem) | |
|Truth | |
|Assessment Overview |
|Critical Question 2: How will we know that students have learned it? |
|As the syllabus outcomes form the focus of the unit, it is necessary to identify the specific evidence of learning to be gathered through teaching, learning and assessment activities that will |
|demonstrate knowledge, skills and understanding. The evidence of learning provides the basis for the selection of content and the planning of the learning experiences within the units. This evidence |
|will enable teachers to make judgements about student achievement in relation to the syllabus outcomes and identified content. |
|Include assessment for, as, of learning |
|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 |Resources |
|Critical Question 1: What should students know and be able to do? |Critical Question 3: How will we structure learning experiences to ensure students | |
|List outcomes and indicators |learn? | |
|This is another opportunity to be explicit about the specific |The learning is planned with identified results and appropriate evidence of | |
|Catholic perspective(s) that students should more fully know and be|understanding in mind. What will be taught (curriculum), and how should it be taught | |
|able to apply, as a result of their engagement in this unit |best (pedagogy), in light of the established goals? What sequence best suits the | |
| |desired results? How will we make learning engaging and effective, given the goals | |
| |and evidence required? | |
| | | |
| |Critical Question 4: How will we respond when students do not learn it or when they | |
| |already know it? | |
| |How do we ensure individual students who need additional time and support for | |
| |learning receive timely and effective intervention? | |
| |How do we differentiate the learning? | |
| |Changes to grouping/instruction | |
| |Different ways to deliver the content | |
| |Different ways for students to demonstrate the learning | |
| |Learning environment is considered | |
| |How will we make learning challenging when student know more than anticipated? | |
| |Adjustments | |
| |Teachers may make adjustments to teaching, learning and assessment practices for some| |
| |students with special education needs, so that they are able to demonstrate what they| |
| |know and can do in relation to syllabus outcomes/catholic perspectives and content. | |
| |The types of adjustments made will vary based on the needs of individual students and| |
| |occurs at the time of learning. | |
| |These may be: | |
| |Adjustments to the assessment process, e.g. additional time, rest breaks, quieter | |
| |conditions, or the use of a reader and /or scribe or specific technology | |
| |Adjustments to assessment activities, e.g. rephrasing questions or using simplified | |
| |language, fewer questions or alternative formats or questions | |
| |Alternative formats for responses, e.g. written point form or notes, scaffolded | |
| |structured responses, short objective questions or multimedia presentations. | |
| | | |
| |Student Reflection | |
| |Students reflect on the demands of the unit of work and the assessment activity. | |
| |They can record their findings about their own processes of learning by constructing | |
| |a PMI chart (plus, minus and interesting) to evaluate the topic | |
| |and the learning by addressing the following questions: | |
| |What did you get the most out of (P)? | |
| |What did you like the best (P)? | |
| |What did you think needed to be developed further (M)? | |
| |What was the most interesting thing you did or learnt (I)? | |
| |How has this unit developed your understanding of the subject? | |
| |What have you learnt about yourself as a learner? | |
| | | |
| | | |
|Stage 5.1 - Properties of Geometrical Figures | |[pic] |
|Students: | |Geometry Vocabulary |
|Use the enlargement transformation to explain similarity (ACMMG220)|Students are given a variety of polygons. By measuring side lengths and angles they |
|describe two figures as similar if an enlargement of one |match congruent figures and hence name matching sides and angles. |Basic Congruency Figures |
|is congruent to the other [pic] |[pic] |
|recognise that if two figures are similar, they have the same shape| |Basic Congruency Description |
|but are not necessarily the same size (Reasoning) |Given a pair of congruent shapes in different orientations students list the pairs of|
|find examples of similar figures embedded in designs from many |corresponding angles. They then write a congruence statement with the corresponding |1GL.html |
|cultures and historical periods (Reasoning) [pic] |angles listed in the same order. |Another Description for Congruency |
|explain why any two equilateral triangles, or any two squares, are | | |
|similar, and explain when they are congruent (Communicating, |Each student constructs a triangle given three sides, two sides and the included |Congruency GeoBoard Exericise |
|Reasoning) [pic] |angle, two angles and one side or the hypotenuse and one other side in a right angled|
|investigate whether any two rectangles, or any two isosceles |triangle. They then write a set of instructions for their partner to construct a |0/dahl1/CongruentShapesAC.html |
|triangles, are similar (Problem Solving) [pic] |triangle congruent to their own |Similarity Khan Academy Videos ALL GREAT! |
|match the sides and angles of similar figures [pic] | |
|name the vertices in matching order when using the symbol ||| in a |Students identify congruent figures in tessellations and art work (eg works by |ty |
|similarity statement [pic] |Escher, Vasarely and Mondrian) and find examples of congruent figures embedded in | |
|use the enlargement transformation and measurement to determine |designs from other cultures and historical periods. They determine whether the design|Similarity Basic Description |
|that the size of matching angles and the ratio of matching sides |or pattern was created using a translation, reflection, rotation or a combination of |
|are preserved in similar figures [pic] |transformations. |2GL.html |
|use dynamic geometry software to investigate the properties of |[pic] | |
|similar figures (Problem Solving) [pic] [pic] |Groups of students identify similar figures in designs, architecture and artwork or |Similar Figures and Triangle Questions |
|Solve problems using ratio and scale factors in similar figures |landscaping in European formal gardens. They present their findings to the class. |
|(ACMMG221) |Students explain why a pair of congruent figures in different orientations has the |s&topic_id=285&parent_name=Geometry+-%3E+Similar+|
|choose an appropriate scale in order to enlarge or reduce a diagram|same area. |Figures+and+Proportions |
|enlarge diagrams such as cartoons and pictures (Reasoning) |Students choose a pair of congruent figures from a series of shapes and write the |Great Visual of Similarity |
|construct scale drawings |correct congruence statement about the shapes. | |
|investigate different methods for producing scale drawings, | |Measuring Scale Factor |
|including the use of digital technologies (Communicating, Problem |Students are given circles of differing radii. In groups they match the circles which|
|Solving) [pic] [pic] |are congruent and deduce. |html |
|interpret and use scales in photographs, plans and drawings found |[pic] |Trigonometry: similar triangles |
|in the media and in other key learning areas [pic] |Which measurement condition is sufficient for congruence in circles? Each group |
|determine the scale factor for pairs of similar polygons and |reports their findings to the class.Students investigate the circumference and area |html |
|circles |of congruent circles and write a journal entry outlining their findings |Congruent triangles |
|apply the scale factor to find unknown sides in similar triangles | |
|calculate unknown sides in a pair of similar triangles using |Given a pair of similar shapes in different orientations students redraw the second |html |
|a proportion statement |figure so that the matching angles are in the same position as the first figure. They|See what happens to the image shape when you move|
|apply the scale factor to find unknown lengths in similar figures |list the pairs of corresponding angles and then write a similarity statement with the|the original shape around the graph. Experiment |
|in a variety of practical situations |matching angles listed in the same order. |by moving the dilation centre as well as using |
|apply the scale factor to find lengths in the environment where it |[pic] |different shapes from the boxes on the left of |
|is impractical to measure directly, eg heights of trees, buildings | |the screen. Compare the coordinates of the |
|(Problem Solving) [pic] |In pairs students investigate the angle size and side lengths of a pair of similar |original and image shapes |
| |figures. They find the ratio of corresponding sides. Pairs share their results and |
| |draw the conclusion that angle size and the ratio of matching sides are preserved in |.html |
| |similar figures. | |
| | |Scale IT!!! |
| |FINESSE |Great Interactive ~ Explore changes in area when |
| |Facts .... General nature |a shape is acted on by a scale factor. Examine |
| |Information ..... Given |changes in volume when an object is acted on by a|
| |Next..... |scale factor. Analyse the changes using whole |
| |Equation (or formula) |numbers |
| |Simplify/substitute |
| |Solve |.html |
| | |Exploring Ratio and Proportion |
| | |
| | |html |
| | | |
|Stage 5.2 - Properties of Geometrical Figures | |[pic] |
|Students: | |Congruency Khan Academy Videos ALL GREAT!!! |
|Formulate proofs involving congruent triangles and angle properties| |
|(ACMMG243) |Students apply geometrical facts and properties of similar figures to solve numerical|t-triangles |
|write formal proofs of the congruence of triangles, preserving |problems such as finding unknown sides and angles. |Congruency Proofs Examples |
|matching order of vertices [pic] [pic] |Students justify their solutions to problems by giving reasons in their own words. |
|apply congruent triangle results to prove properties of isosceles |[pic] |GP4/PracCongTri.htm |
|and equilateral triangles: [pic] |PROOF SET UP |Great Explanations and Question Downloads |
|if two sides of a triangle are equal in length, then the angles |Basic Geometry Pre Thought (translate/rotate/reflect and labelling of shapes sides |
|opposite the equal sides are equal |and angles) |uence |
|conversely, if two angles of a triangle are equal, then the sides |Congruency is SAME / EXACT / EQUAL |Geometry Vocabulary |
|opposite those angles are equal |Reasoning and Proofs - similar concepts to the solution guide to Science Practicals |
|if the three sides of a triangle are equal, then each interior |written set up |Basic Congruency Description |
|angle is 60° | |
|use the congruence of triangles to prove properties of the special |AIM. PROOF |1GL.html |
|quadrilaterals, such as: [pic] | |Congruent Triangle Description |
|the opposite angles of a parallelogram are equal |METHOD. REASONING |
|the diagonals of a parallelogram bisect each other | |gles.html |
|the diagonals of a rectangle are equal |CONCLUSION. THEREFORE |Interior Angle Sum Description |
|Use the enlargement transformations to explain similarity and to | | |
|develop the conditions for triangles to be similar (ACMMG220) |Students apply the four congruency tests to pairs of triangles to prove that they are|Game Interior Angles Matching |
|investigate the minimum conditions needed, and establish the four |congruent. Examples should include numerical and non-numerical problems. | |
|tests, for two triangles to be similar: [pic] [pic] | |Geometry In Architecture |
|if the three sides of a triangle are proportional to the three |For each statement made in the middle section a justified Reason with Knowledge from | |
|sides of another triangle, then the two triangles are similar |previous Geometry Outcomes MUST be given. |Congruency Questions |
|if two sides of a triangle are proportional to two sides of another|[pic] [pic] |
|triangle, and the included angles are equal, then the two triangles| |s&topic_id=283&parent_name=Geometry+-%3E+Congruen|
|are similar |Given the solution to a non-numerical proof showing that two triangles are congruent |t+Triangles |
|if two angles of a triangle are equal to two angles of another |students supply the reasons. |Another Description for Congruency |
|triangle, then the two triangles are similar |Students verify by paper folding that the altitude of an isosceles or equilateral | |
|if the hypotenuse and a second side of a right-angled triangle are |triangle divides it into two congruent triangles. They then apply the congruency |Congruency GeoBoard Exericise |
|proportional to the hypotenuse and a second side of another |tests to formally prove this and hence deduce the required properties. |
|right-angled triangle, then the two triangles are similar |Students answer open ended questions such as “when are two equilateral or isosceles |0/dahl1/CongruentShapesAC.html |
|explain why the remaining (third) angles must also be equal if two |triangles congruent?” |Basic Congruency Figures |
|angles of a triangle are equal to two angles of another triangle |Students prove that the diagonal of a parallelogram divides it into two congruent |
|(Communicating, Reasoning) [pic] |triangles. From this they establish with guidance, the properties of a parallelogram,| |
|determine whether two triangles are similar using an appropriate |such as opposite sides are equal, opposite angles are equal, and that the diagonals |Find out about congruent and non-congruent |
|test [pic] |bisect each other |triangles and the conditions required to make |
|Apply logical reasoning, including the use of congruence and |After establishing the results for a parallelogram groups within the class |them. Use line segments and angles to build two |
|similarity, to proofs and numerical exercises involving plane |investigate other special quadrilaterals. Each group studies a different |congruent triangles for three different |
|shapes (ACMMG244) |quadrilateral and applies the congruency results to establish its properties. Each |combinations of sides and angles. |
|apply geometrical facts, properties and relationships to find the |group member becomes an “expert” on their quadrilateral and then returns to a “home” |
|sizes of unknown sides and angles of plane shapes in diagrams, |group where all “experts” share their knowledge. Class results are then shared. |html |
|providing appropriate reasons [pic] |Students assess their peers by writing a problem involving one of the four congruency|Geometry Constructions |
|recognise that more than one method of solution is possible |tests in a numerical exercise to find an unknown side or angle and then trialing it |
|(Reasoning) [pic] |on a partner. |-clients/geometry-constructions/ |
|compare different solutions for the same problem to determine the |Through measurement and calculation they establish the elements preserved in similar |Angle Constructions YouTube |
|most efficient method (Communicating, Reasoning) [pic] |triangles, namely angle size and the ratio of corresponding sides. Students write up |
|apply the properties of congruent and similar triangles, justifying|their findings in their own words. |369932 |
|the results (Communicating, Reasoning) [pic] |[pic] | |
|apply simple deductive reasoning to prove results for plane shapes |Students answer questions such as “why are any two equilateral triangles similar?” or|This module is for study by an individual teacher|
|[pic] |“are any two isosceles triangles similar?” |or group of teachers. It: |
|define the exterior angle of a convex polygon [pic] |By measurement and calculation groups of students establish the tests sufficient to |• looks at approaches to developing pupils’ |
|establish that the sum of the exterior angles of any convex polygon|show that two triangles are similar, ie two angles are equal, sides are proportional |visualisation and geometrical reasoning |
|is 360º [pic] |or two sides are proportional and the included angle is equal. Group results are |skills; |
|use dynamic geometry software to investigate the constancy of the |shared and students then apply these tests to a variety of pairs of triangles to |• considers progression towards geometric proof. |
|exterior angle sum of polygons for different polygons (Reasoning) |determine whether they are similar. |
|[pic] [pic] |Students complete a cloze passage on the tests for similar triangles |stants/Mathematics%20Study%20Modules/5%20Geometri|
|apply the result for the interior angle sum of a triangle to find, |Students are given a triangle of known dimensions. They then calculate the side |cal%20Reasoning%201.pdf |
|by dissection, the interior angle sum of polygons with more than |lengths of triangles similar to the first after applying various enlargement and |Love this way of Proving Angle Sum Triangles |
|three sides |reduction factors. | |
|use dynamic geometry software to investigate the interior angle sum|Given a triangle of known dimensions students construct a triangle which is similar |Angle Sum of Any Polygon |
|of different polygons (Reasoning) [pic] [pic] |to it |
|express in algebraic terms the interior angle sum of a polygon with|By first writing statements about the ratio of corresponding sides in similar |D=L765 |
|n sides, eg [pic] (Communicating) [pic] |triangles students are then able to solve equations which enable them to find an |Algebraic Expressions of Angle Sum Polygons |
|apply interior and exterior angle sum results for polygons to find |unknown side. |
|the sizes of unknown angles [pic] |Use dynamic geometry software to investigate these proofs |1DP.html |
| |Exterior Angles of Any Convex Polygon |The geometer's warehouse is a web-based, |
| |[pic] |multimedia resource focusing on the geometry of |
| | |the Stage 4 and Stage 5 Mathematics syllabus. It |
| |Interior Angle Sum of Triangles |comprises 70 dynamic html worksheets, each |
| |[pic] |exploring a different outcome in Stage 4 and |
| |Numerical Problems of Polygon Angle Sums |Stage 5 geometry. Laptop friendly resource |
| |[pic] |
| |Students use shadow sticks to determine the heights of buildings or trees in the |es/Web/geometer/ |
| |playground |Method1,2,3,4 and 5 with Explanations of Interior|
| |[pic] |Angle Sum |
| |FINESSE |
| |Facts .... General nature |rior%20angle.htm |
| |Information ..... Given | |
| |Next..... |Interior Angles in Triangles ~ Find an active |
| |Equation (or formula) |triangle in a photograph. Work out its angles by |
| |Simplify/substitute |applying principles of opposite angles, |
| |Solve |complementary angles, supplementary angles and |
| |Evaluate |the sum of interior angles. Watch a video showing|
| | |how triangles are used in buildings and other |
| | |structures |
| | |
| |Decide whether figures are congruent and which transformations might have been |html |
| |performed to convert one figure to another, eg for the figures shown. |Construct triangles of various sizes to explore |
| |[pic] [pic] |the sum of all interior angles |
| |Experiment with minimum conditions under which triangles will be congruent. This |
| |could be approached by having students construct triangles with particular |html |
| |measurements for sides and angles, and testing whether they are congruent by |Exterior Angle Sum |
| |superposing | |
| |Investigate whether a triangle can be copied with the following techniques: |Compare the shadows cast by a small stick and a |
| |[pic] |larger object such as a column of a Greek |
| |using compasses and drawing the three sides |building. Check the angles at which the shadows |
| |using a protractor only and transferring the three angles |are cast and confirm whether the two objects |
| |using two sides and any angle using two angles and any one side |create similar triangles. Use equivalent ratios |
| |Investigate types of quadrilaterals and determine |to calculate the lengths of a triangle's sides: |
| |which must contain congruent triangles |the height of an object or the length of a |
| |which may contain congruent triangles |shadow. Explore how right-angled triangles can be|
| |which cannot contain congruent triangles |used in measurement, including special properties|
| |how many congruent triangles can be found in each |of the isosceles right-angled triangle. |
| |Develop a set of congruence tests for quadrilaterals |
| |In groups, write procedure texts for various triangle constructions. Give these to |html |
| |another group to firstly decide which they think will be congruent, then to construct| |
| |and check | |
| |[pic] | |
| |Explain geometrical constructions using congruence eg perpendicular from a point to a| |
| |line, bisector of an interval, bisector of an angle, perpendicular to a line from a | |
| |point on the line | |
| |[pic] | |
|Stage 5.3 - Properties of Geometrical Figures § |Each of the Resources given in next column are great lessons |Introduction to Geometry Proofs |
|Students: |Investigate who first proved that the angle in a semicircle is a right angle |
|Formulate proofs involving congruent triangles and angle properties|[pic] |on-to-geometry-proofs/ |
|(ACMMG243) |Prove by congruence that the angles opposite equal sides in an isosceles triangle are|Geometric Reasoning with Geogebra Java |
|construct and write geometrical arguments to prove a general |equal |
|geometrical result, giving reasons at each step of the argument, |Prove that if an angle bisector of a triangle is perpendicular to the opposite side, |etric_Reason/AS1.6_home.htm |
|eg prove that the angle in a semicircle is a right angle [pic] |then the triangle is isosceles |Geometry Proof Ideas for Dummies |
|Apply logical reasoning, including the use of congruence and |Solve problems like: Prove that the quadrilateral formed by the lines joining |
|similarity, to proofs and numerical exercises involving plane |midpoints of the sides of any quadrilateral will result in a parallelogram |egies-in-geometry.html |
|shapes (ACMMG244) |Solve problems using deductive reasoning |Review of Quadrilaterals UTube |
|write formal proofs of the similarity of triangles in the standard | | |
|four- or five-line format, preserving the matching order of |[pic] |Scaling Down Visual |
|vertices, identifying the scale factor when appropriate, and |[pic] |
|drawing relevant conclusions from this similarity [pic] [pic] |[pic] |html |
|prove that the interval joining the midpoints of two sides of a |[pic] |Scaling Up Visual |
|triangle is parallel to the third side and half its length, and the|[pic] |
|converse (Communicating, Problem Solving) [pic] |[pic][pic][pic][pic][pic][pic] |html |
|establish and apply for two similar figures with similarity ratio |Converse Proofs |Surface Area Scaling Up |
|[pic] the following: [pic] |[pic] |
|matching angles have the same size |[pic] |html |
|matching intervals are in the ratio [pic] |[pic] |Volume Scaling |
|matching areas are in the ratio [pic] |[pic] |
|matching volumes are in the ratio [pic] | |html |
|solve problems involving similarity ratios and areas and volumes |[pic] | |
|(Problem Solving) |Adjustments | |
|state a definition as the minimum amount of information needed to |Provide a scaffold – give part of the proof with gaps to be filled |Downloadable Text Ratio Proportion and Similarity|
|identify a particular figure [pic] [pic] |Work out what extra information is needed to answer |
|prove properties of isosceles and equilateral triangles and special| |ooks/Geometry/Amsco_Geometry_Textbook_Chapter_12.|
|quadrilaterals from the formal definitions of the shapes: [pic] | |pdf |
|[pic] | | |
|a scalene triangle is a triangle with no two sides equal in length | |Area and Volume of Similar Solids |
|an isosceles triangle is a triangle with two sides equal in length | |
|an equilateral triangle is a triangle with all sides equal in | |imilar-Solids/ |
|length | | |
|a trapezium is a quadrilateral with one pair of opposite sides | |Review of Quadrilaterals ~ Determine the features|
|parallel | |of a quadrilateral and identify it as a |
|a parallelogram is a quadrilateral with both pairs of opposite | |parallelogram, a rectangle, a rhombus, a |
|sides parallel | |trapezium, a square or a kite |
|a rhombus is a parallelogram with two adjacent sides equal in | |
|length | |html |
|a rectangle is a parallelogram with one angle a right angle | |Exploring Kites |
|a square is a rectangle with two adjacent sides equal | |
|use dynamic geometry software to investigate and test conjectures | |.html |
|about geometrical figures (Problem Solving, Reasoning) [pic] [pic] | | |
|prove and apply theorems and properties related to triangles and | |Great Proofs ~ examples |
|quadrilaterals: [pic] | |
|the sum of the interior angles of a triangle is 180º | |ams_and_rectangles.html |
|the exterior angle of a triangle is equal to the sum of the two | |Very Strange Shape Sorted Factory |
|interior opposite angles | |
|if two sides of a triangle are equal, then the angles opposite | |html |
|those sides are equal; conversely, if two angles of a triangle are | | |
|equal, then the sides opposite those angles are equal | | |
|each angle of an equilateral triangle is equal to 60º | |summarize the relationship between various kinds |
|the sum of the interior angles of a quadrilateral is 360º | |of quadrilaterals |
|the opposite angles of a parallelogram are equal | |
|the opposite sides of a parallelogram are equal | |ry/11-geometry-xmlplus |
|the diagonals of a parallelogram bisect each other | | |
|the diagonals of a rhombus bisect each other at right angles | |AWESOME Collection of MANY Proofs |
|the diagonals of a rhombus bisect the vertex angles through which | | |
|they pass | |UTube ~ How to write Proofs |
|the diagonals of a rectangle are equal | | |
|recognise that any result proven for a parallelogram would also | | |
|hold for a rectangle (Reasoning) [pic] | | |
|give reasons why a square is a rhombus, but a rhombus is not | |Full Proof Description and Worksheets Including |
|necessarily a square (Communicating, Reasoning) [pic] | |Tests |
|use a flow chart or other diagram to show the relationships between| |$20Triangles$|
|different quadrilaterals | |20Proofs.pdf |
|(Communicating) [pic] | | |
|prove and apply tests for quadrilaterals: [pic] | |Theorems and Properties with Converse Statements |
|if both pairs of opposite angles of a quadrilateral are equal, then| |
|the quadrilateral is a parallelogram | |GPB/theorems.htm |
|if both pairs of opposite sides of a quadrilateral are equal, then | | |
|the quadrilateral is a parallelogram | | |
|if all sides of a quadrilateral are equal, then the quadrilateral | |$20Proofs.pdf|
|is a rhombus | | |
|solve numerical and non-numerical problems in Euclidean geometry | | |
|based on known assumptions and proven theorems [pic] | |Maths is Power |
|state possible converses of known results, and examine whether or | | |
|not they are also true (Communicating, Reasoning) [pic] | | |
|Critical Question 1: What should students know and be able to do? |Critical Question 3: How will we structure learning experiences to ensure students | |
|List outcomes and indicators |learn? | |
|This is another opportunity to be explicit about the specific |The learning is planned with identified results and appropriate evidence of | |
|Catholic perspective(s) that students should more fully know and be|understanding in mind. What will be taught (curriculum), and how should it be taught | |
|able to apply, as a result of their engagement in this unit |best (pedagogy), in light of the established goals? What sequence best suits the | |
| |desired results? How will we make learning engaging and effective, given the goals | |
| |and evidence required? | |
| | | |
| |Critical Question 4: How will we respond when students do not learn it or when they | |
| |already know it? | |
| |How do we ensure individual students who need additional time and support for | |
| |learning receive timely and effective intervention? | |
| |How do we differentiate the learning? | |
| |Changes to grouping/instruction | |
| |Different ways to deliver the content | |
| |Different ways for students to demonstrate the learning | |
| |Learning environment is considered | |
| |How will we make learning challenging when student know more than anticipated? | |
| |Adjustments | |
| |Teachers may make adjustments to teaching, learning and assessment practices for some| |
| |students with special education needs, so that they are able to demonstrate what they| |
| |know and can do in relation to syllabus outcomes/catholic perspectives and content. | |
| |The types of adjustments made will vary based on the needs of individual students and| |
| |occurs at the time of learning. | |
| |These may be: | |
| |Adjustments to the assessment process, e.g. additional time, rest breaks, quieter | |
| |conditions, or the use of a reader and /or scribe or specific technology | |
| |Adjustments to assessment activities, e.g. rephrasing questions or using simplified | |
| |language, fewer questions or alternative formats or questions | |
| |Alternative formats for responses, e.g. written point form or notes, scaffolded | |
| |structured responses, short objective questions or multimedia presentations. | |
| | | |
| |Student Reflection | |
| |Students reflect on the demands of the unit of work and the assessment activity. | |
| |They can record their findings about their own processes of learning by constructing | |
| |a PMI chart (plus, minus and interesting) to evaluate the topic | |
| |and the learning by addressing the following questions: | |
| |What did you get the most out of (P)? | |
| |What did you like the best (P)? | |
| |What did you think needed to be developed further (M)? | |
| |What was the most interesting thing you did or learnt (I)? | |
| |How has this unit developed your understanding of the subject? | |
| |What have you learnt about yourself as a learner? | |
| | | |
| | | |
|Registration |Evaluation |
|Class: __________________________ |Teachers evaluate the extent to which the planning of the unit has remained focused on the syllabus outcomes. After the unit has been |
|Start Date: _______________________ |implemented, there should be opportunity for both teachers and students to reflect on and evaluate the degree to which students have |
|Finish Date: ______________________ |progressed as a result of their experiences, and what should be done next to assist them in their learning. |
|Teacher’s Signature: _______________________ |Evaluation reflects: |
| |The effectiveness of the program in meeting the diverse needs of students and identifies curriculum adjustments |
| |Level to which syllabus outcomes have been demonstrated by students |
| |The effectiveness of pedagogical practices employed |
| |Suggested program adjustments |
| |Elements of the school’s Contemporary Learning Framework |
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 |AGREE |UNSURE |STRONGLY DISAGREE |
2. 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 |AGREE |UNSURE |STRONGLY DISAGREE |
4. Assess the degree to which syllabus outcomes have been demonstrated by students in this unit:
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5. Evaluate the degree to which the diverse needs of learners have been addressed in this unit:
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6. Comment on the effectiveness of pedagogical practices employed in this unit:
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7. Assessment was meaningful and appropriate to reflect student learning and achievement:
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8. Suggested program adjustments / other comments:
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