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 triangles are congruent or similar
Identify similar triangles and describe their properties
Apply tests for congruent triangles
Use simple deductive reasoning in numerical and non-numerical problems
Verify the properties of special quadrilaterals using congruent triangles
Construct proofs of geometrical relationships involving congruent or similar triangles / Mathematics K-10
›MA5.111MG describes and applies the properties of similar figures and scale drawings
›MA5.214MG calculates the angle sum of any polygon and uses minimum conditions to prove triangles are congruent or similar
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
CEO will provide guidance in this area
Example:
- The value of sacramentality celebrates the presence of God in every facet of creation.
- The Christian message is ultimately one of hope
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,ontological 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
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
- Context if you prefer the unit overview and context to be separate
- School focus for learning – eg blooms taxonomy, solo taxonomy, contemporary learning, habits of mind,BLP (building learning power)
- Any specific social and emotional learning which could be embedded into the unit eg enhanced group work
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
Content / Teaching, learning and assessment / Resources
Critical Question 1: What should students know and be able to do?
List outcomes and indicators
This is another opportunity to be explicit about the specific Catholic perspective(s) that students should more fully know and be able to apply, as a result of their engagement in this unit / Critical Question 3: How will we structure learning experiences to ensure students learn?
The learning is planned with identified results and appropriate evidence of understanding in mind. What will be taught (curriculum), and how should it be taught 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
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.
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
Students:
Use theenlargementtransformation to explain similarity (ACMMG220)
- describe two figures as similar if an enlargement of one iscongruent to the other
find examples of similar figures embedded in designs from many cultures and historical periods (Reasoning)
explain why any two equilateral triangles, or any two squares, are similar, and explain when they are congruent (Communicating, Reasoning)
investigate whether any two rectangles, or any two isosceles triangles, are similar (Problem Solving)
- match the sides andangles of similar figures
- name the vertices in matching order when using the symbol ||| in a similarity statement
- use the enlargement transformation and measurement to determine that thesize of matching angles and theratio of matching sides are preserved in similar figures
Solve problems using ratio and scale factors in similar figures (ACMMG221)
- choose an appropriate scale in order to enlarge or reduce a diagram
- construct scale drawings
- interpret and use scales in photographs, plans and drawings found in the media and in other key learning areas
- determine the scale factor for pairs of similarpolygons and circles
- apply the scale factor to find unknown sides in similar triangles
- calculate unknown sides in a pair of similar triangles using aproportion statement
- apply the scale factor to find unknown lengths in similar figures in a variety of practical situations
Given a pair of congruent shapes in different orientations students list the pairs of corresponding angles. They then write a congruence statement with the corresponding angles listed in the same order.
Each student constructs a triangle given three sides, two sides and the included angle, two angles and one side or the hypotenuse and one other side in a right angled triangle. They then write a set of instructions for their partner to construct a triangle congruent to their own
Students identify congruent figures in tessellations and art work (eg works by Escher, Vasarely and Mondrian) and find examples of congruent figures embedded in designs from other cultures and historical periods. They determine whether the design or pattern was created using a translation, reflection, rotation or a combination of transformations.
Groups of students identify similar figures in designs, architecture and artwork or landscaping in European formal gardens. They present their findings to the class.
Students explain why a pair of congruent figures in different orientations has the same area.
Students choose a pair of congruent figures from a series of shapes and write the correct congruence statement about the shapes.
Students are given circles of differing radii. In groups they match the circles which are congruent and deduce.
Which measurement condition is sufficient for congruence in circles? Each group reports their findings to the class.Students investigate the circumference and area of congruent circles and write a journal entry outlining their findings
Given a pair of similar shapes in different orientations students redraw the second figure so that the matching angles are in the same position as the first figure. They list the pairs of corresponding angles and then write a similarity statement with the matching angles listed in the same order.
In pairs students investigate the angle size and side lengths of a pair of similar 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 similar figures.
FINESSE
Facts .... General nature
Information ..... Given
Next.....
Equation (or formula)
Simplify/substitute
Solve
/
Geometry Vocabulary
Basic Congruency Figures
Basic Congruency Description
Another Description for Congruency
Congruency GeoBoard Exericise
Similarity Khan Academy Videos ALL GREAT!
Similarity Basic Description
Similar Figures and Triangle Questions
Great Visual of Similarity
Measuring Scale Factor
Trigonometry: similar triangles
Congruent triangles
See what happens to the image shape when you move the original shape around the graph. Experiment by moving the dilation centre as well as using different shapes from the boxes on the left of the screen. Compare the coordinates of the original and image shapes
Scale IT!!!
Great Interactive ~ Explore changes in area when a shape is acted on by a scale factor. Examine changes in volume when an object is acted on by a scale factor. Analyse the changes using whole numbers
Exploring Ratio and Proportion
Stage 5.2 - Properties of Geometrical Figures
Students:
Formulate proofs involvingcongruent triangles andangle properties (ACMMG243)
- write formal proofs of thecongruence of triangles, preserving matching order of vertices
- apply congruent triangle results to prove properties of isosceles and equilateral triangles:
conversely, if two angles of a triangle are equal, then the sides opposite those angles are equal
if the three sides of a triangle are equal, then each interior angle is 60°
- use the congruence of triangles to prove properties of the special quadrilaterals, such as:
the diagonals of a parallelogram bisect each other
the diagonals of arectangle are equal
Use theenlargementtransformations to explainsimilarity and to develop the conditions for triangles to be similar (ACMMG220)
- investigate the minimum conditions needed, and establish the four tests, for two triangles to be similar:
if two sides of a triangle are proportional to two sides of another triangle, and the included angles are equal, then the two triangles are similar
if two angles of a triangle are equal to two angles of another triangle, then the two triangles are similar
if the hypotenuse and a second side of aright-angled triangle are proportional to the hypotenuse and a second side of another right-angled triangle, then the two triangles are similar
explain why the remaining (third) angles must also be equal if two angles of a triangle are equal to two angles of another triangle (Communicating, Reasoning)
- determine whether two triangles are similar using an appropriate test
- apply geometrical facts, properties and relationships to find the sizes of unknown sides and angles of plane shapes in diagrams, providing appropriate reasons