Download Chapter 15: Reasoning in geometry Stage 4 – Year 8 Unit length: 2

Chapter 15: Reasoning
in geometry
Stage 4 – Year 8
Strand:
Measurement and Geometry
Rationale:
This topic develops deductive reasoning, and encourages clear setting
out of proofs where reasons accompany each step of an answer.
Geometry, as well as having practical value, can be used to develop
logical thinking and careful setting out of reasoning.
Properties of Geometrical Figures
A student:
• classifies, describes and uses the properties of triangles and
quadrilaterals, and determines congruent triangles to find unknown
side lengths and angles (MA4-17MG)
Angle Relationships
A student:
• identifies and uses angle relationships, including those related to
transversals on sets of parallel lines (MA4-18MG)
Outcomes:
Content
statements:
Resources:
• Recognise the geometrical properties of angles at a point
(NSW Stage 4)
• Identify corresponding, alternate and co-interior angles when two
straight lines are crossed by a transversal (ACMMG163)
• Investigate conditions for two lines to be parallel (ACMMG164)
• Solve simple numerical problems using reasoning (ACMMG164)
• Demonstrate that the angle sum of a triangle is 180° and use this to
find the angle sum of a quadrilateral (ACMMG166)
• Use the properties of special triangles and quadrilaterals to solve
simple numerical problems with appropriate reasoning (NSW Stage 4)
• Foundation worksheets
• Challenge worksheets
• Drag-and-drop activities
• GeoGebra activities
Unit length: 2.5 weeks
Angle Relationships; Properties
of Geometrical Figures 1
Substrand:
Teacher:
Dates taught:
________ to ________
Teacher reflection
Strengths
Weaknesses
• Interactive lessons
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0052 4
Literacy:
adjacent
alternate
angle
co-interior
complement
complementary
condition
corresponding
equilateral
exterior
isosceles
opposite
parallel
quadrilateral
reason
supplementary
supplement
transversal
triangle
vertically
Other
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0052 4
Student Book / eBook
15:01 Adjacent angles
15:02 Angles at a point
and vertically
opposite angles
15:03 Angles associated
with parallel lines
Content dot points
• use the terms 'complementary'
and 'supplementary' for angles
adding to 90° and 180°,
respectively, and the
associated terms 'complement'
and 'supplement’
• use the term 'adjacent angles'
to describe a pair of angles
with a common arm and a
common vertex
• identify and name right angles,
straight angles, angles of
complete revolution and
vertically opposite angles
embedded in diagrams
• use the common conventions
to indicate parallel lines on
diagrams
• identify and name pairs of
parallel lines using the symbol
for 'is parallel to'
• define and identify
'transversals', including
transversals of parallel lines
• identify, name and measure
alternate angle pairs,
corresponding angle pairs and
co-interior angle pairs for two
lines cut by a transversal
• recognise the equal and
supplementary angles formed
when a pair of parallel lines is
cut by a transversal
• use angle properties to identify
parallel lines
Register
Technology
Resources and suggestions
Familiarise students with angle names,
complementary angles and
supplementary angles by doing
Prep quiz 15:01 on p. 437.
Encourage students to explain their
reasons verbally and in a logical written
format.
Drag-and-drop
Finding the size of angles 1
Foundation worksheet 15:02 – Angle
relationships
Use the photos on pp. 444 and 454 to
stimulate conversation on the use of
angles in construction and design.
GeoGebra
Adjacent angles
Angles at a point
Vertically opposite angles
Using one angle to find three
more
Angles and parallel lines 1
Angles and parallel lines 2
Alternate, co-interior and
corresponding angles
Drag-and-drop
Finding the size of angles 2
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0052 4
Read through the blue box on p. 446 to
illustrate the links between angles in
parallel lines.
Fun spot 15:03 (p. 450)
Use the GeoGebra activities previewed
on p. 450.
15:04 Angle sum of a
triangle
15:05 Angle sum of a
quadrilateral
15:06 Isosceles and
equilateral
triangles
15:07 More involved
numerical
problems
Review
• justify informally that the
interior angle sum of 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 the angle sum
of a quadrilateral is 360°
• use the angle sum results for
triangles and quadrilaterals to
determine unknown angles in
triangles and quadrilaterals,
giving reasons
• find unknown sides and angles
embedded in diagrams, using
the properties of special
triangles and quadrilaterals,
giving reasons
• find the sizes of unknown
angles embedded in diagrams
using angle relationships,
including angles at a point and
angles associated with parallel
lines, giving reasons
Drag-and-drop
Finding the size of angles 3
Use parallel lines or three angles
placed on a straight line to show that
the angle sum of a triangle is 180o.
GeoGebra
Angle sum of a quadrilateral
The parallelogram family
Foundation worksheet 15:05 –
Triangles and quadrilaterals
Interactive lessons
Angles in simple polygons
Interior and exterior angles
in polygons
GeoGebra
Investigating triangle
properties
Isosceles and equilateral
triangles
Interactive lesson
Angles in triangles
GeoGebra
Geometric reasoning:
parallel lines
Geometric reasoning:
parallel lines and the angle
sum of a triangle
Geometric reasoning:
triangles and quadrilaterals
Interactive lesson
Problems involving
properties of angles and
parallel lines
Drag-and-drop
Maths terms 15
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0052 4
Use four angles placed at a point to
show that the angle sum of a
quadrilateral is 360o.
Use the GeoGebra activities previewed
on p. 457.
Use the two GeoGebra activities
previewed on p. 459.
Foundation worksheet 15:07 –
Finding the angles
Challenge worksheet 15:07 – The
logical prisoner
Use the GeoGebra activities previewed
on p. 462.
Challenge 15:07 – Can you explain
why? (p. 463)
Fun spot 15:07 (p. 463)
Maths terms 15 (p. 464)
Diagnostic test 15 Use the right-hand
column to assist in remediation when
errors occur. (p. 465)
Assignment 15A Exam-style
questions for revision (p. 467)
Assignment 15B Working
mathematically problems (p. 468)
Assignment 15C Use this cumulative
revision to review previous
topics. (p 469)
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0052 4