Download MA4-17MG Properties of geometrical figures 1

MA4-17MG Properties of Geometrical Figures 1
Summary of Sub Strands
Duration
MA3-15MG manipulates, classifies and draws two-dimensional shape, including equilateral, isosceles and scalene triangles, and describes
their properties.
MA3-16MG measures and constructs angles, and applies angle relationships to find unknown angles.
Unit overview
Outcomes
Big Ideas/Guiding Questions
Students look at the relationship and
properties of geometrical figures and
learn to analyse geometrical
problems.
›
How important are angles in identifying a plane shape?
Geometry is common and important
in everyday situations, including in
nature, sports, buildings, astronomy,
art, etc
›
MA42WM applies appropriate
mathematical techniques to solve
problems
›
MA43WM recognises and explains
mathematical relationships using
reasoning
›
MA417MG classifies, describes and
uses the properties of triangles and
quadrilaterals, and determines
congruent triangles to find unknown
side lengths and angles
1
MA41WM communicates and
connects mathematical ideas using
appropriate terminology, diagrams and
symbols
What is ‘Euclidean Geometry.’
2 weeks
›
Key words
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angle, obtuse, corresponding, vertex, reflex, co interior, degree, adjacent, diagonal, parallel, revolution,
complementary, triangle, straight, supplementary, scalene, right-angled, perpendicular, vertically,
opposite, isosceles, protractor, transversal, equilateral, acute, alternate, quadrilateral, convex quadrilateral,
non-convex quadrilateral, line symmetry, rotational symmetry, axis, axes, square, rectangle, kite, rhombus,
parallelogram, trapezium, 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 co-interior 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.
2
This syllabus uses the phrase 'line(s) of symmetry', although 'axis/axes of symmetry' may also be used.
'Scalene' is derived from the Greek word skalenos, meaning 'uneven'. 'Isosceles' is derived from the Greek
words isos, meaning 'equals', and skelos, meaning 'leg'. 'Equilateral' is derived from the Latin
words aequus, meaning 'equal', andlatus, meaning 'side'. 'Equiangular' is derived from aequus and another
Latin word, angulus, meaning 'corner'.
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)
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• 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
Classify triangles according to their side and angle
properties and describe quadrilaterals (ACMMG165)
Students are taught to label and name both triangles and quadrilaterals. They
are taught to use common conventions for properties of triangles and
quadrilaterals and how to classify and identify different triangles.
Useful websites for worksheets, interactive
games, maths charts, flashcards
▪ label and name triangles (eg triangle
) and quadrilaterals (eg
and on diagrams
or
) in text
▪ use the common conventions to mark
equal intervals on diagrams
▪ recognise and classify types of triangles on the
basis of their properties
recognise that a given triangle may belong
to more than one class (Reasoning)
explain why the longest side of a triangle is
always opposite the largest angle
(Reasoning)
Do a quiz on what triangle am I?
For the key inquiry question, Identify the types of triangles that are used in
ramps. Which ones are the most commonly used and why?
Through investigation, discover that the longest side of a triangle is opposite
the largest angle and the shortest side is opposite the smallest angle.
Show that it is impossible to construct a triangle where the sum of the length of
two sides is smaller than the length of the third side by attempting it.
Sketch triangles by hand or using computer software
http://www.topmarks.co.uk/Interactive.a
spx?cat=22
http://www.helpingwithmath.com/
http://www.sheppardsoftware.com/math.htm
Investigative questioning.
Can you construct a triangle with side lengths of 5cm, 9cm 16cm?
explain why the sum of the lengths of two
sides of a triangle must be greater than the
length of the third side (Communicating,
Reasoning)
Can you construct a triangle with angles 2 obtuse angles?
sketch and label triangles from a worded or
verbal description (Communicating)
Can you construct an isosceles right angled triangle?
4
http://resources.woodlandsjunior.kent.sch.uk/maths/units120.html
Can you construct a triangle with 3 acute angles?
Can you construct a triangle with angles of 50o, 100o and 40o?
www.au.ixl.com
Can you construct triangle ABC with 〱 A 130o, 〱 B 50o and 〱 C 20o?
▪ distinguish between convex and non-convex
quadrilaterals
▪ investigate the properties of special
quadrilaterals (parallelograms, rectangles,
rhombuses, squares, trapeziums and kites),
including whether:
What could you say about the interval AC and AB?
Define the difference between convex and non-convex quadrilaterals. This can
be shown visually and/or with the formal definition.
Using a variety of methods, such as paper folding, measurement and dynamic
geometry software, discover the properties of the special quadrilaterals.
− the opposite sides are parallel
− the opposite sides are equal
Match up a given quadrilateral with its description.
− the adjacent sides are perpendicular
− the opposite angles are equal
− the diagonals are equal
Do a wanted or missing poster on different quadrilaterals with identifiable
features. Some may do one on a type of triangle. Eg.. Missing Eddie Equilateral
or Susie Square.
− the diagonals bisect each other
− the diagonals bisect each other at right
angles
− the diagonals bisect the angles of the
quadrilateral
use techniques such as paper folding or
measurement, or dynamic geometry
software, to investigate the properties of
quadrilaterals (Problem Solving, Reasoning)
sketch and label quadrilaterals from a
worded or verbal description
(Communicating)
▪ classify special quadrilaterals on the basis of
their properties
▪ describe a quadrilateral in sufficient detail for it
to be sketched (Communicating)
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Adjustment
Some students will need assistance with drawing various shapes and may
benefit from computer software. This may also assist in identifying shapes in
different transformations as well as showing composite shapes.
Identify line and rotational symmetries (ACMMG181)
▪ investigate and determine lines (axes) of
symmetry and the order of rotational symmetry
of polygons, including the special
quadrilaterals
determine if particular triangles and
quadrilaterals have line and/or rotational
symmetry (Problem Solving)
▪ investigate the line and rotational symmetries
of circles and of diagrams involving circles,
such as a sector or a circle with a marked chord
or tangent
▪ identify line and rotational symmetries in
pictures and diagrams, eg artistic and cultural
designs
Students could be asked “When is it possible to draw a line which would divide
a shape into two parts, so that when folded along that line the parts would
coincide exactly?”
Students explore the line symmetry of polygons by:
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cutting out shape templates and folding to determine axes of symmetry
using mirrors to check answers
exploring the school environment for examples of line symmetry and
sketching their findings
Shape templates
MIRA Mirrors
Lines of symmetry of plane shapes:
http://www.mathsisfun.com/geometry/symmetryline-plane-shapes.html
Adjustment
●
some students may require assistance when exploring the line
symmetry of shapes other than squares, rectangles and triangles.
Students explore rotational symmetry in the following activities:
●
copy a figure such as the one shown onto a piece of cardboard and cut
it out.
Place a pencil through the centre of the cardboard and rotate the figure 90˚
in a clockwise direction. The figure should look exactly as it did in its
initial position. The figure has rotational symmetry as it can be turned
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www.resources.woodlandsjunior.kent.sch.uk/maths
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less than 360˚ to match the original figure. The figure can be turned
four times before it returns to its initial position and with each turn it
appears identical. This means that the order of rotational symmetry is
four.
Using tracing paper to sketch polygons, including circles and diagrams
involving circles to explore the order of rotational symmetry.
Finding examples of rotational symmetry in the home or school
environment eg. Car tyres, flowers, windmills
Students investigate the occurrence of line and rotational symmetry in art and
design through:
●
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Demonstrate that the angle sum of a triangle is 180°
and use this to find the angle sum of a quadrilateral
(ACMMG166)
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a triangle is 180°, and that any exterior angle
equals the sum of the two interior opposite
angles
●
▪ use the angle sum of a triangle to establish that
quadrilaterals to determine unknown angles in
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Tracing paper
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Magazines, newspapers, internet access
Smartboard resources: ‘Symmetry’ and
‘Axial Symmetry’ folders.
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Blank paper
Scissors
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Use the interactive protractor on the
smartboard to model this.
www.mathsisfun.com/triangle.html
www.slideshare.net
www.mathwarehouse.com/geometry
A collection of images from media sources including magazines,
newspapers and the internet
Investigating the symmetry of cultural landmarks both locally and
internationally eg Taj Mahal, Sydney Harbour Bridge, Eifel tower.
Students draw any triangle on paper and label the interior angles a, b
and c.
The triangle is cut out and the vertices are torn off so that the
a, b and c are easily seen.
Students place the letters together so that the vertices are touching and
discuss their findings.
Alternatively, students could draw different triangles and measure the three
interior angles to establish the rule.
Students can also discover the exterior angle rule by drawing a diagram and
measuring angles or cutting out the two interior angles and placing them on the
exterior angle to show they are equivalent.
the angle sum of a quadrilateral is 360°
▪ use the angle sum results for triangles and
Cardboard
Scissors
Students discover the angle sum of a triangle by practical means:
▪ justify informally that the interior angle sum of
use dynamic geometry software or other
methods to investigate the interior angle
sum of a triangle, and the relationship
between any exterior angle and the sum of
the two interior opposite angles (Reasoning)
●
●
Adjustment
For extension, students could research the work of the ancient Greek
●
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triangles and quadrilaterals, giving reasons
mathematician Euclid who discovered that the angles in a triangle added up to
180˚.
Students explore the angle sum of a quadrilateral by:
●
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Drawing any quadrilateral and dividing it into two triangles noting that
the interior angle sum of a quadrilateral is equal to the angle sum of
two triangles.
Students could also establish the rule by a similar method to the
triangle earlier by drawing any quadrilateral and labeling the angles a,
b, c and d.
Adjustments
Students could investigate the interior angle sum of further polygons by
dividing the shape into triangles.
Teachers explain, with appropriate examples, how to write simple proofs to
find missing angles in triangles and quadrilaterals.
8
●
●
Blank paper,
Scissors
Use the properties of special triangles and
quadrilaterals to solve simple numerical problems
with appropriate reasoning
▪ find unknown sides and angles embedded in
diagrams, using the properties of special
triangles and quadrilaterals, giving reasons
recognise special types of triangles and
quadrilaterals embedded in composite
figures or drawn in various orientations
(Reasoning)
9
Teachers explain, with a variety of examples, how to find missing sides and
angles in special triangles and quadrilaterals.
Adjustment
Students could explore the use of linear equations to solve problems like the
one below.
Registration
Evaluation
Class:
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Start Date:
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Finish Date:
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Teacher’s
Signature:
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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
3. There was sufficient time to teach the set content:
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UNSURE
STRONGLY DISAGREE
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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