Download Teaching Strategies

Course(s): Year 10 (Stage 5.1)
Unit of Work: Review of Space & Geometry
KLA
RE
ENG
MA

SCI
Indicative Hours: 15 h
HSIE PDHPE
Term: 3
CA
TAS
LOTE
Weeks: 7 to 10
Unit Description:
This topic gets covered in Stage 4. Its inclusion at this juncture is to review basic concepts and practise essential skills in preparation for the School
Certificate Examination.
The study of solid shapes imparts basic understanding of their dimensions that is an essential requirement in the Measurement strand. Also
students of Design & Technology and Art may apply skills developed in this topic, when they prepare diagrams, sketches and/or drawings for the
making of models or products; and students of Science will find this topic useful when drawing and building models of crystals.
The study of angles is the introduction to Deductive Geometry where students are required to support statements with reasons. A sound
understanding of this basic topic facilitates a better application of geometrical concepts to solve problems and theorems at a later stage.
The sequential build-up of geometrical concepts requires students to study triangles and basic quadrilaterals and utilise geometrical instruments to
construct these shapes according to stipulated parameters. The properties of special quadrilaterals are important in Measurement. For example, the
perpendicularity of the diagonals of a rhombus and a kite allow a rectangle of twice the size to be constructed around them, leading to formulae for
finding area. At this Stage, the treatment of triangles and quadrilaterals is still informal, with students consolidating their understandings of different
triangles and quadrilaterals and being able to identify them from their properties.
Similarity is linked with ratio in the Number strand and with map work in Geography. Similar and congruent figures are embedded in a variety of
designs (eg tapa cloth, Aboriginal designs, Indonesian ikat designs, Islamic designs, designs used in ancient Egypt and Persia, window lattice,
woven mats and baskets).
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Syllabus outcomes for each course:
SGS 4.1: Describes and sketches three-dimensional solids including polyhedra, and classifies them in terms of their properties.
SGS 4.2: Identifies and names angles formed by the intersection of straight lines, including those related to transversals on sets of parallel lines,
and makes use of the relationships between them.
SGS 4.3: Classifies, constructs, and determines the properties of triangles and quadrilaterals.
SGS 4.4: Identifies congruent and similar two-dimensional figures stating the relevant conditions.
Resources:
Mathematics Syllabus Years 7 – 10, 2003
New Century Mathematics Year 10 (Stage 5.1) – Chapter 6
CD-ROM included with textbook
Hotmaths website
Phase 1:
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Properties of Solids
 Describing solids in terms of their
geometric properties
number of faces
shape of faces
number and type of congruent faces
number of vertices
number of edges
convex or non-convex
 Identifying any pairs of parallel flat faces
of a solid
 Determining if two straight edges of a
solid are intersecting, parallel or skew
 Determining if a solid has a uniform
cross-section
 Classifying solids on the basis of their
properties
A polyhedron is a solid whose faces
are all flat.
A prism has a uniform polygonal
cross-section.
A cylinder has a uniform circular
cross-section.
A pyramid has a polygonal base and
one further vertex (the apex).
A cone has a circular base and an
apex.
All points on the surface of a sphere
are a fixed distance from its centre.
 Identifying right prisms and cylinders and
oblique prisms and cylinders
 Identifying right pyramids and cones and
oblique pyramids and cones
 Sketching on isometric grid paper
Teaching Strategies
Registration
 Demonstrate the describe solids in terms of their geometric
properties number of faces, shape of faces number and
type of congruent faces number of vertices number of
edges convex or non-convex
 Identify the pairs of parallel flat faces of a solid.
 Demonstrate if two straight edges of a solid are
intersecting, parallel or skew
 Explain the meaning of uniform cross- section
 Classify solids on the basis of their properties.
 Explain the meaning of Polyhedron.
 Explain the different properties.
Interpret and make models from isometric drawings
Recognise solids with uniform and non-uniform crosssections
Analyse three-dimensional structures in the environment to
explain why they may be particular shapes eg buildings,
packaging
Visualise and name a common solid given its net
Recognise whether a diagram is a net of a solid
Identify parallel, perpendicular and skew lines in the
environment
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Assessment
Strategies: what and
how?
 Teacher in-class
observation
 Questioning
 Participation
 Written assessment
task
shapes built with cubes
 Representing three-dimensional objects
in two dimensions from different views
 Confirming,
for
various
convex
polyhedra, Euler’s formula
F+V=E+2
relating the number of faces (F), the
number of vertices (V) and the number
of edges (E)
 Exploring the history of Platonic solids
and how to make them
 Making models of polyhedra
Phase 2:
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Angles at a Point
 Labelling and naming points, lines and
intervals using capital letters
 Labelling the vertex and arms of an
angle with capital letters
 Labelling and naming angles using A
and XYZ notation
 Using the common conventions to
indicate right angles and equal angles on
diagrams
 Identifying and naming adjacent angles
(two angles with a common vertex and a
common arm), vertically opposite angles,
straight angles and angles of complete
revolution, embedded in a diagram
 Using the words ‘complementary’ and
‘supplementary’ for angles adding to 90º
and 180º respectively, and the terms
‘complement’ and ‘supplement’
 Establishing and using the equality of
vertically opposite angles
Teaching Strategies
 Explain and demonstrate the labeling and naming of points,
lines and intervals
 Demonstrate and give examples of labeling vertices and
arms of angles with capital letters
 Demonstrate the different ways of labeling angles
 Demonstrate the common conventions indicating right
angles and equal angles on diagrams
 Demonstrate and explain how to identify and name
adjacent, vertically opposite angles, straight angles and
angles of complete revolution
 Explain the use of the words ‘complementary’ and
‘supplementary’ when describing angles
 Demonstrate the use of equality of vertically opposite
angles.
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
Phase 3:
Angles Associated with Transversals
 Identifying and naming a pair of parallel
Teaching Strategies
 Revision of parallel lines.
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lines and a transversal
 Using common symbols for ‘is parallel to’
( ) and ‘is perpendicular to’ ()
 Using the common conventions to
indicate parallel lines on diagrams
 Identifying, naming and measuring the
alternate angle pairs, the corresponding
angle pairs and the co-interior angle
pairs for two lines cut by a transversal
 Recognising the equal and
supplementary angles formed when a
pair of parallel lines are cut by a
transversal
 Using angle properties to identify parallel
lines
 Using angle relationships to find
unknown angles in diagrams
 Identify parallel and perpendicular lines in the class
environment e.g., opposite sides of the white board are
parallel but the adjacent sides are perpendicular and so on.
Explain the common conventions.
 Use diagrams to explain the different types of angles in
parallel lines.(corresponding, alternate and co-interior
angles)
 Revision of complementary and supplementary angles.
 Demonstrate and explain how to find the value of the
unknown angle by using the properties of two-dimensional
shapes and parallel lines results.
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
Construct a pair of perpendicular lines using a ruler and a
protractor, a ruler and a set square, or a ruler and a pair of
compasses
 Use dynamic geometry software to investigate angle
relationships
Phase 4:
Notation
Teaching Strategies
 Labelling and naming triangles (eg ABC)  Demonstrate the ability to label triangles and quadrilaterals.
and quadrilaterals (eg ABCD) in text and
 Demonstrate the common conventions to mark equal
on diagrams
intervals on diagrams
 Using the common conventions to mark
equal intervals on diagrams
 Demonstrate how to classify triangles on the basis of their
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Triangles
 Recognising and classifying types of
triangles on the basis of their properties
(acute-angled triangles, right-angled
triangles,
obtuse-angled
triangles,
scalene triangles, isosceles triangles and
equilateral triangles)
 Constructing various types of triangles
using geometrical instruments, given
different information
e.g., the lengths of all sides, two sides
and the included angle, and two angles
and one side
 Justifying informally by paper folding or
cutting, and testing by measuring, that
the interior angle sum of a triangle is
180º, and that any exterior angle equals
the sum of the two interior opposite
angles
 Using a parallel line construction, to
prove that the interior angle sum of a
triangle is 180º
 Proving,
using
a
parallel
line
construction, that any exterior angle of a
triangle is equal to the sum of the two
interior opposite angles
properties (acute-angled triangles, right angled triangles,
obtuse-angled triangles, scalene triangles, isosceles
triangles and equilateral triangles)
 Demonstrate how to construct triangles using geometrical
instruments, given different information eg the lengths of all
sides, two sides and the included angle, and two angles
and one side
 Students verify by paper folding or cutting, and testing by
measuring, that the interior angle sum of a triangle is 180º,
and that any exterior angle equals the sum of the two
interior opposite angles
 Sketch and label triangles from a given verbal description
 Describe a sketch in sufficient detail for it to be drawn
 Recognise that a given triangle may belong to more than
one class
 Recognise that the longest side of a triangle is always
opposite the largest angle
 Recognise and explain why two sides of a triangle must
together be longer than the third side
 Recognise special types of triangles and quadrilaterals
embedded in composite figures or drawn in various
orientations
 Determine if particular triangles and quadrilaterals have line
and/or rotational symmetry
 Apply geometrical facts, properties and relationships to
solve numerical problems such as finding unknown sides
and angles in diagrams
 Justify their solutions to problems by giving reasons using
their own words
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Phase 5:
Quadrilaterals
 Distinguishing between convex and nonconvex quadrilaterals (the diagonals of a
convex quadrilateral lie inside the figure)
 Establishing that the angle sum of a
quadrilateral is 360º
 Constructing
various
types
of
quadrilaterals
 Investigating the properties of special
quadrilaterals
(trapeziums,
kites,
parallelograms, rectangles, squares and
rhombuses) by using symmetry, paper
folding, measurement and/or applying
geometrical reasoning Properties to be
considered include :
opposite sides parallel
opposite sides equal
adjacent sides perpendicular
opposite angles equal
diagonals equal in length
diagonals bisect each other
diagonals bisect each other at right
angles
diagonals bisect the angles of the
quadrilateral
 Investigating the line symmetries and the
order of rotational symmetry of the
special quadrilaterals
 Classifying special quadrilaterals on the
basis of their properties
Teaching Strategies
 Sketch and label quadrilaterals from a given verbal
description
 Bisect an angle by applying geometrical properties e.g.,
constructing a rhombus
 Bisect an interval by applying geometrical properties e.g.,
constructing a rhombus
 Draw a perpendicular to a line from a point on the line by
applying geometrical properties e.g., constructing an
isosceles triangle
 Draw a perpendicular to a line from a point off the line by
applying geometrical properties e.g., constructing a
rhombus
 Use ruler and compasses to construct angles of 60º and
120º by applying geometrical properties e.g., constructing
an equilateral triangle
 Explain with the help of examples the difference between
convex and non-convex quadrilaterals.
 Explain the sum of the angles of a quadrilateral.
 Explain the different properties of quadrilaterals.
Phase 6:
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Circles
Teaching Strategies
 Identifying and naming parts of the circle  Explain that a circle consists of all points that are a given
distance from the centre and how this relates to the use of
and related lines, including arc, tangent
a pair of compasses
and chord
 Investigating the line symmetries and the  Use dynamic geometry software to investigate the
properties of geometrical figures
rotational symmetry of circles and of
diagrams involving circles, such as a
sector and a circle with a chord or
tangent
Phase 7:
Congruence
 Identifying
congruent
figures
by
superimposing
them
through
a
combination of rotations, reflections and
translations
 Matching sides and angles of two
congruent polygons
 Naming the vertices in matching order
º in a
when using the symbol
congruence statement
 Drawing
congruent
figures
using
geometrical instruments
 Determining the condition for two circles
to be congruent (equal radii)
Teaching Strategies
 Explain the meaning
superimposing objects.
of
the
word
congruent
by
 Give class examples of congruent shapes by using
diagrams, drawing found in media, design work etc.
 Also write the matching sides and vertices.
 Demonstrate and explain the four tests of congruency by
means of diagrams.
 Students can cut out different shapes and test congruency
by placing them on top of each other.
 Explain the difference between congruence and similarity.
Recognise congruent figures in tessellations, art and design
work
Phase 8:
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Similarity
 Using the term ‘similar’ for any two
figures that have the same shape but not
necessarily the same size
 Matching the sides and angles of similar
figures
 Naming the vertices in matching order
when using the symbol lll in a similarity
statement
 Determining that shape, angle size and
the ratio of matching sides are preserved
in similar figures
 Determining the scale factor for a pair of
similar polygons
 Determining the scale factor for a pair of
circles
 Calculating dimensions of similar figures
using the enlargement or reduction factor
 Choosing an appropriate scale in order
to enlarge or reduce a diagram
 Constructing scale drawings
 Drawing similar figures using geometrical
instruments
Teaching Strategies
 Demonstrate and explain the use of similar figures in
finding lengths in the environment where it is impractical to
measure directly eg heights of trees, buildings.
 Using scale factor demonstrate how to draw similar figures
(by enlarging or reducing the diagram)
 Explain similarity by enlarging cartoons and pictures and
hence demonstrate how to find the scale factor.
 Demonstrate and explain by means of examples how to
match the sides and angles of similar figures. Use
enlargement or reduction factor to find the dimensions of
similar figures.
 Explain how to construct similar figures using geometrical
instruments.
Interpret and use scales in photographs, plans and drawings
found in the media and/or other learning areas
Enlarge diagrams such as cartoons and pictures
Apply similarity to finding lengths in the environment where it
is impractical to measure directly eg heights of trees,
buildings
Apply geometrical facts, properties and relationships to solve
problems such as finding unknown sides and angles in
diagrams
Justify their solutions to problems by giving reasons using
their own words
Recognise that area, length of matching sides and angle
sizes are preserved in congruent figures
Recognise that shape, angle size and the ratio of matching
sides are preserved in similar figures
Recognise that similar and congruent figures are used in
specific designs, architecture and art work eg works by
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Escher, Vasarely and Mondrian; or landscaping in
European formal gardens
Find examples of similar and congruent figures embedded in
designs from many cultures and historical periods
 Use dynamic geometry software to investigate the
properties of geometrical figures
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REGISTRATION & TEACHER EVALUATION
Date Unit of Work was Started: ________________
Date Unit of Work was Completed: ________________
It is confirmed that the outcomes ticked () above have been completed.
Supporting Students with Special Needs
Students with Special Needs
in Class
Targeted Outcomes for these
Students
Accommodations/
Adaptations
(if different to rest of class)
(as necessary)
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Special Provisions Organised
(if applicable)
Comments
Teaching Strategies and Applications,
noting any additional ones
Use of resources
Need for extra resources
Student response (Students’ attitude to
the Unit)
Appropriateness and Achievement of
Outcomes
Timing and placement of topic
Depth of coverage
Alterations made to cater for range of
abilities in the class
Further points for improvement
Name of Teacher: ____________________________
Date: ___________________
Signature: __________________________________
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