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Welcome back! This is our third issue of The Mathematical
Bridge. This issue focuses on the Geometry section of the
Measurement and Geometry strand and looks more closely at the
progress of Two-Dimensional Space, Angles and Angle Relationships
across Stage 3 and 4 Mathematics. We see these connections as
important as the links between shapes and their angle properties
become foundational knowledge as students develop geometric
deductive and reasoning skills. The associated language, symbols,
notation and conventions are all essential in developing an
appropriate level of understanding.
We hope you find these resources useful and we welcome any
feedback and/or suggestions.
Katherin Cartwright, Mathematics Advisor K-6 and Zdena Pethers,
R/Numeracy Advisor 7-12
Getting the right angle In the new NSW mathematics K-10 syllabus
Angles is now its own substrand and appears in Stages 2 and 3. We
currently teach angles in primary as outcome b in Two- Dimensional
Space and angles are introduced earlier, in Stage 1.
Although it is now separate, the connections between angles and
shapes should still be strongly emphasised, particularly in Stage 3
where we begin to introduce properties of shapes, all shapes have
angle properties. This will lead into the Stage 4 substrands of
Angle Relationships and Properties of Geometric Figures. These
connecting concepts are also further developed as we link
measurement and geometry through finding area of shapes and volumes
of objects/ solids.
The first mention of angles in our new syllabus is actually in
Stage 2 in Two-Dimensional Space 1:
• recognises the vertices of two-dimensional shapes as the
verticesof angles that have the sides of ashape as their arm
• identify right angles in squares andrectangles
It will therefore be necessary to explore the concept of angles
as a measure of turn in the environment and in shapes prior to
expecting students to be able to recognise them as vertices. When
developing your scope and sequence of learning in Stage 2, teaching
Angles and Two-Dimensional Space together will provide students
with a deep understanding of the concept of describing shapes and
their features. This is a prior skill to seeing angles as
properties of shapes.
It is always best to start with the known then move to the
unknown. The environment provides students with a wide variety of
angles being used in the real world. They can explore, identify,
describe, investigate and draw angles from pictures and photographs
of familiar places.
Syllabus content Pedagogy Teaching ideas
PUBLIC SCHOOLS NSW – LEARNING AND LEADERSHIP DIRECTORATE ISSUE
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Starting with hands on tasks is also important, beginning with
concrete examples and activities then applying this knowledge to
abstract concepts is essential for all students at every stage of
learning from Early Stage 1 to Stage 4 and beyond. This enables
students to try a broader range of strategies to solve problems
when using geometric thinking and allows students to feel
comfortable in taking risks, trialling ideas, testing theories and
predictions.
Right Angle Challenge
From www.nrich.maths.org/2812
This activity requires students to manipulate two sticks to make
right angles.
Egyptian Rope
From www.nrich.maths.org/982
This activity allows students to use a knotted rope to make
various triangles. Students can then investigate angle features of
shapes.
Teachers can also link angles to time in the hands of a clock or
to themselves by looking at angles you can make with your body.
There is also much to be explored about angles in sport both with
playing fields and also looking at the amount of turn Olympic
divers, discus throwers, ice skaters and gymnasts make as part of
their sport.
A different angle…. As students work mathematically and think
like mathematicians, we encourage them to ask and pose questions
and prove their reasoning. There are a number of aspects and
concepts around angles and lines that would start robust discussion
in the classroom. Questions like: How do we know lines are
parallel? Can you prove your reasoning? How would you explain what
horizontal and vertical lines are?
Can students do this without the knowledge of right angles and
what perpendicular means?
http://www.mathsisfun.com/perpendicular-parallel.html
In my thinking, right angles and perpendicular lines are a bit
of a ‘chicken and the egg’ conversation. It is difficult to explain
one without referencing the other. This can be particular difficult
when students understanding of right angles (Stage 2 ) does not yet
refer to exact measuring of 90 degrees using protractors (Stage
3).
These kinds of justifications are not reflected in the
mathematics syllabus until Stage 4. However, to assist students in
developing sound knowledge and understanding of these concepts, it
may need to be discussed from as early as Stage 1.
GeoGebra has many applets that are interactive to show students
how angles move.
Vertically opposite angles GeoGebra Applet
Adjacent angles GeoGebra Applet
Angles at a point GeoGebra Applet
Angle relationships In Stage 4, there is stronger emphasis on
more formal understanding of angle relationships, including the
associated terminology, notation and conventions, as this is of
fundamental importance in developing an appropriate level of
knowledge, skills and understanding in geometry.
Angle relationships and their application play an integral role
in students learning to analyse geometry problems and developing
geometric and deductive reasoning, as well as problem-solving
skills. Angle relationships are key to the geometry that is
important in the work of architects, engineers,
PUBLIC SCHOOLS NSW – LEARNING AND LEADERSHIP DIRECTORATE ISSUE
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http://www.nrich.maths.org/2812http://www.nrich.maths.org/982http://www.mathsisfun.com/perpendicular-parallel.htmlhttp://www.mathsisfun.com/perpendicular-parallel.html
geogebra_thumbnail.png
geogebra_javascript.js
function ggbOnInit() {}
geogebra.xml
KCARTWRIGHT5File AttachmentVertically opposite angles applet
geogebra_thumbnail.png
geogebra_javascript.js
function ggbOnInit() {}
geogebra.xml
KCARTWRIGHT5File AttachmentAdjacent angles applet
geogebra_javascript.js
function ggbOnInit() {}
geogebra.xml
KCARTWRIGHT5File AttachmentAngles at a point applet
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designers, builders, physicists, land surveyors etc. as well as
the geometry that is common and important in everyday situations,
such as in nature, sports, buildings, astronomy, art, etc.
(Angle Relationships Stage 4 Mathematics K-10 Syllabus
online)
In Stage 4, students are expected to use the correct terms and
know their meanings. Students should use terms such as
complementary, supplementary, adjacent and vertically opposite as
they communicate their reasoning in solving problems involving
angles at a point.
Complementary angles add up to 90˚ GeoGebra Applet
Supplementary angles add up to 180˚ GeoGebra Applet
Students should also be proficient in using diagrams and symbols
when applying mathematical techniques and reasoning to solve
problems involving angle relationships. For example, using the
correct notation for right angles and equal angles in diagrams and
using capital letters when naming points and intervals.
Students are also expected to identify, understand and use angle
relationships related to transversals on sets of parallel lines and
use the terms alternate, corresponding and co-interior when
referring to these angles. They should be able to solve problems
involving angles related to parallel lines, and justify why two
lines are parallel.
Alternate angles between parallel lines are equal
Corresponding angles on parallel lines are equal
Co-interior angles between parallel lines are supplementary
Students are encouraged to investigate and develop some of these
relationships using their knowledge and skills about angle
properties from Stage 3 (e.g. vertically opposite, straight
angles).
Properties of Geometrical Figures Study of angle relationships
links smoothly to the investigation of properties of geometrical
figures. Angle sum of a triangle is 180 ˚ and angle sum of a
quadrilateral is 360˚, and students can use a variety of ways to
justify this. They build on their work in Stage 3 relating to the
side and angle properties of triangles and quadrilaterals, in a
more structured and formal way.
A triangle can be classified by its angle relationships as
right, scalene, equilateral or isosceles. Note the convention of
marking equal sides by identical markers.
A right-angled triangle has one angle of 90˚
A scalene triangle has no equal angles and no equal sides.
An equilateral triangle has all angles of 60˚ and all sides
are
equal.
An isosceles triangle has two equal angles opposite two
equal
sides.
It is important to expose students to different orientations of
the special triangles from those shown above. They need to be able
to identify them by their properties in ANY orientation.
Angle relationships are also important properties of
quadrilaterals and together with information about
60
60
60
PUBLIC SCHOOLS NSW – LEARNING AND LEADERSHIP DIRECTORATE ISSUE
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geogebra_thumbnail.png
geogebra_javascript.js
function ggbOnInit() {}
geogebra.xml
KCARTWRIGHT5File AttachmentAngles in a right angle applet
geogebra_thumbnail.png
geogebra_javascript.js
function ggbOnInit() {}
geogebra.xml
KCARTWRIGHT5File AttachmentSupplementary angles applet
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their sides, can be used to solve geometrical problems. Students
can explore properties of triangles and quadrilaterals through
investigations and find unknown angles by developing a series of
logical steps.
For example: Find the size of ∠ BCE.
Note the convention: parallel lines are shown by identical
arrows, and a right angle at E is indicated by a square.
Here is one way a student could solve this problem:
1) ∠ ABC = 70˚ since it is co-interiorwith ∠ DAB between
parallel linesAD and BC (co-interior angles aresupplementary – add
up to 180˚)
2) ∠ BCD = 110˚ since it is co-interior with ∠ ABC
betweenparallel lines AB and DC (co-interior angles
aresupplementary)
3) ∠ BCE = 70˚ since it issupplementary to ∠ BCD
(straightangle)
Another way to solve this problem could be:
1) ∠ ABC = 70˚ since it is co-interiorwith ∠ DAB between
parallel linesAD and BC (co-interior angles aresupplementary – add
up to 180˚)
2) ∠ BCE = 70˚ since it is alternatewith ∠ ABC between parallel
linesAB and DE (alternate angles areequal)
There are often several ways to solve a geometrical problem and
students should be encouraged to
present different solutions to each other, showing reasoning and
justifying their thinking. This deepens their conceptual
understanding of angle relationships.
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: e.g. 'When
a transversal cuts parallel lines, the co-interior angles formed
are supplementary'.
Transformations Students should also build on their work in
Stage 3 on transformations. Translation, rotation and reflection,
are called “congruence” transformations as the figures remain
identical and side and angle relationships remain unchanged. In
enlargements, lengths of sides change but angle relationships
remain identical.
http://www.youtube.com/watch?v=F1M1MncPq2c
Other interesting websites:
http://www.resources.det.nsw.edu.au/Resource/Access/f9a38f90-8e0d-492b-9a5d-4a46dd3f5c22/1
http://lrrpublic.cli.det.nsw.edu.au/lrrSecure/Sites/Web/geometer/
http://www.schools.nsw.edu.au/learning/7-
12assessments/naplan/teachstrategies/yr2013/index.php?id=numeracy/n
n_spac/nn_spac_s4b_13
Other Resources www.nrich.maths.org
Making Sixty
This activity asks students ti investigate and prove angle
properties related to triangles, retangles using knowledge of
congruent triangles.
Right angles
In this activity students explore creating triangles using
points around a circle.
110˚
E
PUBLIC SCHOOLS NSW – LEARNING AND LEADERSHIP DIRECTORATE ISSUE
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http://www.youtube.com/watch?v=F1M1MncPq2chttp://www.youtube.com/watch?v=F1M1MncPq2chttp://www.resources.det.nsw.edu.au/Resource/Access/f9a38f90-8e0d-492b-9a5d-4a46dd3f5c22/1http://www.resources.det.nsw.edu.au/Resource/Access/f9a38f90-8e0d-492b-9a5d-4a46dd3f5c22/1http://www.resources.det.nsw.edu.au/Resource/Access/f9a38f90-8e0d-492b-9a5d-4a46dd3f5c22/1http://lrrpublic.cli.det.nsw.edu.au/lrrSecure/Sites/Web/geometer/http://lrrpublic.cli.det.nsw.edu.au/lrrSecure/Sites/Web/geometer/http://www.schools.nsw.edu.au/learning/7-12assessments/naplan/teachstrategies/yr2013/index.php?id=numeracy/nn_spac/nn_spac_s4b_13http://www.schools.nsw.edu.au/learning/7-12assessments/naplan/teachstrategies/yr2013/index.php?id=numeracy/nn_spac/nn_spac_s4b_13http://www.schools.nsw.edu.au/learning/7-12assessments/naplan/teachstrategies/yr2013/index.php?id=numeracy/nn_spac/nn_spac_s4b_13http://www.schools.nsw.edu.au/learning/7-12assessments/naplan/teachstrategies/yr2013/index.php?id=numeracy/nn_spac/nn_spac_s4b_13http://www.schools.nsw.edu.au/learning/7-12assessments/naplan/teachstrategies/yr2013/index.php?id=numeracy/nn_spac/nn_spac_s4b_13http://www.nrich.maths.org/http://nrich.maths.org/6355http://nrich.maths.org/2847http://nrich.maths.org/6355�http://nrich.maths.org/2847�
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ISSUE 1 | FEBRUARY 2014
Continuum of learning Mathematics K-10 Measurement and Geometry
Strand
Stage 2 Stage 3 Stage 4 Two-Dimensional Space: A student
manipulates, identifies and sketches two-dimensional shapes,
including special quadrilaterals, and describes their features
MA2-15MG
Two- Dimensional Space: A student manipulates, classifies and
draws two-dimensional shapes, including equilateral, isosceles and
scalene triangles, and describes their properties MA3-15MG
Properties of Geometric 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
Part 1 Identify and describe shapes as ‘regular’ or
‘irregular’
Describe and compare features of shapes, including the special
quadrilaterals
Part 1 Identify, name and draw right-angled, equilateral,
isosceles and scalene triangles
Explore angle properties of the special quadrilaterals and
special triangles Classify and draw regular and irregular
two-dimensional shapes from descriptions of their features
Part 1 Classify and determine properties of triangles and
quadrilaterals
Identify line and rotational symmetries Determine the angle sums
of triangles and quadrilaterals
Use properties of shapes to find unknown sides and angles in
triangles and quadrilaterals giving a reason
Part 2 Identify congruent figures
Identify congruent triangles using the four tests
Note: the key ideas listed above for Two-dimensional Space are
only those that relate to angles, this is not a list of all key
ideas for Two-Dimensional Space Angles: A student identifies,
describes, compares and classifies angles MA2-16MG
Angles: A student measures and constructs angles, and applies
angle relationships to find unknown angles MG3-16MG
Angle Relationships: A student identifies and uses angle
relationships, including those related to transversals on sets of
parallel lines MA4-18MG
Part 1 Identify and describe angles as measures of turn
Compare angle sizes in everyday situations
Identify ‘perpendicular’ lines and ‘right angles’
Part 2 Draw and classify angles as acute, obtuse, straight,
reflex or a revolution
Part 1 Recognise the need for formal units to measure angles
Measure, compare and estimate angles in degrees (up to 360°)
Record angle measurements using the symbol for degrees (°)
Construct angles using a protractor (up to 360°)
Describe angle size in degrees for each angle classification
Part 2 Identify and name angle types formed by the intersection
of straight lines, including ‘angles on a straight line’, ‘angles
at a point’ and ‘vertically opposite angles’
Use known angle results to find unknown angles in diagrams
Use the language, notation and conventions of geometry
Apply the geometric properties of angles at a point to find
unknown angles with appropriate reasoning
Apply the properties of corresponding, alternate and co-interior
angles on parallel lines to find unknown angles with appropriate
reasoning
Determine and justify that particular lines are parallel
Solve simple numerical exercises based on geometrical
properties
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Stage 2 Teaching Ideas- Two-Dimensional Space and Angles These
lesson ideas are specifically for Stage 2 and form prior knowledge
that is required for students in Stage 3, as angles has its
foundation in their relation to Two-Dimensional shapes. You may
like to explore these concepts with your Stage 3 students to gain
knowledge of their current understandings. Strand: Measurement and
Space Substrand: Two-Dimensional Space 1 Outcomes: WM2-1WM uses
appropriate terminology to describe, and symbols to represent,
mathematical ideas WM2-3WM checks the accuracy of a statement and
explains the reasoning used MA2-15MG manipulates, identifies and
sketches two-dimensional shapes, including special quadrilaterals,
and describes their features Students: Compare and describe
features of two-dimensional shapes, including the special
quadrilaterals
recognise the vertices of two-dimensional shapes as the vertices
of angles that have the sides of the shape as their arms
identify right angles in squares and rectangles group
parallelograms, rectangles, rhombuses, squares, trapeziums and
kites using one or more
attributes, eg quadrilaterals with parallel sides and right
angles identify and describe two-dimensional shapes as either
'regular' or 'irregular', eg 'This shape is a
regular pentagon because it has five equal sides and five equal
angles' To be taught in conjunction with… Strand: Measurement and
Space Substrand: Angles 1 Outcomes: WM2-1WM uses appropriate
terminology to describe, and symbols to represent, mathematical
ideas MA2-16MG identifies, describes, compares and classifies
angles Students: Identify angles as measures of turn and compare
angle sizes in everyday situations (ACMMG064)
identify 'angles' with two arms in practical situations, eg the
angle between the arms of a clock identify the 'arms' and 'vertex'
of an angle
Activity 1: Exploring angles on shapes
Pose questions to students to gain understanding of their
knowledge of shapes and if they can identify angles in
two-dimensional shapes T: What do these shapes have in common? S:
All 2D shapes, all have straight lines/ sides, all regular shapes,
all ‘flat’, all have ‘corners’, all have vertices… T: When we look
at the vertices, do you know another name or way of describing
them? Note: If students do not say ‘angle’ or ‘right angle’ show
them the next image
Where else can you see them in the other shapes? See if the
students can also locate angles in the classroom, in pictures,
photos and on objects
PUBLIC SCHOOLS NSW – LEARNING AND LEADERSHIP DIRECTORATE ISSUE
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KCARTWRIGHT5File AttachmentAngles in shapes Notebook
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7 Activity 2: Sorting and grouping shapes Provide students with
an assortment of quadrilaterals (shapes cut out of paper) use both
regular and irregular shapes
Have students work in pairs to sort the shapes into piles and
share their reasons for sorting the shapes. Do students only sort
shapes according to ‘squares’ ‘rectangles’ etc or do students look
at other features like angles or length of sides? You may need to
prompt students to expand or explore other ways to sort the shapes
Could you sort the shapes into group of those with right angles and
those without? What about shapes that have parallel sides? What
about shapes with all sides equal? What do you notice about shapes
that have all sides equal? Specifically about their angles You
could pose the statement that ‘all shapes that have all sides of
equal length have all angles of equal size’ Allow the students to
investigate this and prove it to be true or false. Providing
reasons and discussing strategies and the processes they used to
explore the problem. Students may like to pose their own
investigations about angles and shapes. Activity 3: Angle Arms Many
students have the misconception that the length of the angle’s arm
influences the size of the angle. E.g. ‘the longer the arms of the
angle, the greater the angle size’ Draw a right angle on the board
or IWB Now draw another one (with longer arms) Ask the students:
Which angle is larger? How do you know? Why do you think that?
Where do we measure the angle? How could we check? What could we
use to check? (Students may suggest a square pattern block, a piece
of paper) Does it matter how long the arms are? Does this change
the size of the angle? Provide students with a square pattern block
(the orange one) or a square of Brennex paper to take around the
class and outside to find right angles where the arms are different
lengths or in different orientations. Many students also believe
there must be a ‘left’ angle if there is a right angle, for this
reason it is important to provide examples in different positions
and orientations, including on the diagonal. Also allow them to
find right angles in different locations.
PUBLIC SCHOOLS NSW – LEARNING AND LEADERSHIP DIRECTORATE ISSUE
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Stage 3 Teaching Ideas- Angles
Start with concrete and then move to abstract This lesson is
from the Teaching Space and Geometry K-6 CD, this resource is
currently being updated to align to the new mathematics K-10
syllabus outcomes and will be available online for NSW DEC teachers
in the future.
PUBLIC SCHOOLS NSW – LEARNING AND LEADERSHIP DIRECTORATE ISSUE
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Stage 3 |The protractor | Lesson one
1
Lesson one
Make a protractor
Using the properties of angles to construct a protractor
Purpose
Students need to apply their knowledge of the properties of
specific angles to make generalisations about the measurement of
angles.
Outcomes Describes and represents mathematical situations in a
variety of ways using mathematical terminology and some conventions
MA3-1WM
Selects and applies appropriate problem solving strategies,
including the use of digital technologies, in undertaking
investigations MA3-2WM
Gives a valid reason for supporting one possible solution over
another MA3-3WM
Measures and constructs angles, and applies angle relationships
to find unknown angles MA3-16MG
Key idea Construct angles using a protractor (up to 360º) Lesson
concept A straight angle is 180º.
Materials
a semi-circle of light cardboard for each student a large
protractor a class set of protractors pencils, rulers
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Stage 3 |The protractor | Lesson one
2
Teaching Point
Protractor: a tool for measuring angles.
Steps
Questions and discussion
Show the students a large protractor.
Organise the students into pairs and provide each student with a
semi-circle of cardboard.
Ask the pairs of students to discuss how they could use a
semi-circle of cardboard to construct a protractor without the use
of another protractor.
Have selected students share their strategies with the
class.
What is a protractor? When would you need to use a
protractor?
What are the features of a protractor? How do you use a
protractor? What are the units for measuring angles? What is the
symbol for degree?
Which angle would you locate first on the protractor? Why? What
angles could you immediately mark on your semi-circle? Where would
you mark these angles on your semi-circle? How could you accurately
determine where 90° would be on your protractor? How could you
determine an angle half this size? What size would that angle
be?
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Stage 3 |The protractor | Lesson one
3
Steps
Questions and discussion
Have the students construct their protractors and then join with
another pair to discuss, compare and modify them if they wish.
When their protractors are completed, have the students use a
classroom protractor to verify the accuracy of their markings.
Have the students use their protractors to measure and record
angles within the room.
What obtuse angle could you find using the same strategy? How
could you mark a 60° angle? How could you mark a 30° angle? What
obtuse angles could you mark out similarly? Which angles could you
estimate and mark?
KCARTWRIGHT5File AttachmentMake a protractor lessonplan
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Stage 3 Teaching Ideas- Angles Creating angle testers is one way
of allowing students check and justify their angle estimations
accurately without using a commercial protractor. Students in the
stages before Stage 3 may have had experience with using bendable
straws, a pipe cleaner in a straw, to look for angles smaller than
or larger than a right angle. With this angle tester, students can
find a greater variety of angles and can also explore the angle
properties of two-dimensional shapes, specifically triangles. This
lesson is by Azim Premji Foundation and can be found here
http://www.teachersofindia.org/en/activity/handmade-math-tools-protractor
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http://www.teachersofindia.org/en/users/azim-premji-foundationhttp://www.teachersofindia.org/en/activity/handmade-math-tools-protractor
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Published on Teachers of India
(http://www.teachersofindia.org)
Home > Handmade math tools: Protractor
Handmade math tools: Protractor
By Azim Premji Foundation | Jan 25, 2014
Through paper-folding many concepts in mathematics can be better
understood. For instance, by making creases in paperand making a
protractor, as Ashish Gupta from APF demonstrates, students can
discover the relationships between linesand angles.
Duration: 01 hours 00 minsIntroduction:
“Children often set mathematics aside as a cause for concern,
despite their limited exposure to it (Hoyles 1982)”
Mathematics is considered a boring subject by many children.
Teaching mathematics to children is thus a big challenge forthe
teachers. Through paper folding many concepts in mathematics can be
dealt with and this hands-on approach alsomakes it more
enjoyable.
A protractor is a geometric tool which is commonly used to
measure or draw any angle ranging 0 to 360 . Let us mix mathwith
paper-folding and construct our own protractor…
Objective:
Paper folding is a fascinating activity that leads to “active
mathematical experiences”. Making creases in a piece of paperand
forming straight lines is an interesting way of discovering and
demonstrating relationships between lines and angles.Before going
through the journey of making a handmade protractor using paper, we
need to understand some basicassumptions of paper folding–
Paper can be folded so that the crease formed is a straight
line.Paper can be folded so that the crease passes through one or
two given points.Paper can be folded so that a point can be made
coincident with another point on the same sheet.Paper can be folded
so that a point on the paper can be made coincident with a given
line on the same sheet and theresulting crease is made to pass
through a second given point. This, provided that the second point
is not in theinterior of a parabola that has the first point as
focus and the given line as directrix.Paper can be folded so that
straight lines on the same sheet can be made coincident.Line and
angles are said to be congruent when they can be made to coincide
by folding the paper.
('Mathematics through paper folding' by Alton T. Olson:
University of Alberta Edmonton, Alberta.)
Please note the limitations of a handmade protractor:
It can only be used for measuring angles that are multiples of
15 degrees.It is only a measuring device and cannot be used for
constructing angles.
Activity Steps:
Materials required: Sheets of paper and a pencil.
Step 1: Fold the square sheet of paper exactly in the middle to
form a crease which divides the sheet into two equalhalves.
1
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Step 2: Fold the top right corner of the sheet to a point on the
crease so that the point on the crease, the top right cornerand the
top left corner altogether form a triangle. As you can see, the
triangle thus obtained is a right angled triangle withthe other two
angles being 30 and 60 degrees.
Step 3: Now fold the lower right corner over the above triangle
to form another right angled triangle.
Step 4: In the last fold, take the lower left corner and fold it
in such a way that it meets the edge of the first right
angledtriangle as shown...
The Handmade Protractor is now ready to use!
Let’s identify the angles:
2
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Category: Classroom ResourcesSubject: MathematicsBoard: All
boardsGrade: Class 3-5
Class 6-8License: CC BY-NC-SA
Usage: Some angles can be directly measured using this
protractor while others can be measured by adding two angleslike
45� and 60�. Similarly you can unfold it and by adding two 60�
angles measure 120� etc.
Paper Folding is an interesting and 'active' way of learning
many mathematical concepts. The key concept in this wholeprocess of
making the protractor is dividing the straight angle (180�) into
three equal parts of each 60�.
References & Credits:
Mathematics through paper folding - Alton T. Olsen: University
of Alberta Edmonton. Paper protractor - Arvind Gupta Special
credits to Sir Jose Paul – learnings from his workshopsin 2006.
Source URL:
http://www.teachersofindia.org/en/activity/handmade-math-tools-protractor
3
https://www.youtube.com/watch?feature=player_detailpage&v=zFNBEZ9-8t8
http://www.teachersofindia.org/en/activity/handmade-math-tools-protractor
Handmade math tools: Protractor
Azim Premji FoundationFile AttachmentHandmade math tools
lessonplan
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10
Stage 3 Teaching Ideas- Angles This activity is one of many that
can be accessed via the NAPLAN Teaching Strategies website. There
are a
number of activities to the teaching of angles. Strand:
Measurement and Space Substrand: Angles 1 Outcomes: WM3-1WM
describes and represents mathematical situations in a variety of
ways using mathematical terminology and some conventions MA3-16MG
measures and constructs angles, and applies angle relationships to
find unknown angles Students:
identify that a right angle is 90°, a straight angle is 180° and
an angle of revolution is 360° identify and describe angle size in
degrees for each of the classifications acute, obtuse and
reflex
use the words 'between', 'greater than' and 'less than' to
describe angle size in degrees (Communicating)
Activity: Angle Card Matching
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http://www.schools.nsw.edu.au/learning/7-12assessments/naplan/teachstrategies/yr2013/index.php?id=numeracy/nn_spac/nn_spac_s3bi_13
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larger than a rightangle but smaller
than a straightangle?
right angle= 90 °
also called a full turn?
acute angle= 180 °
smaller than a rightangle?
obtuse angle= 360 °
larger than a straight
angle but smaller than a full turn?
straight angle< 90 °
straight like a line? reflex angle>180 ° and 90 ° and
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Stage 3 Teaching Ideas- Angles Strand: Measurement and Space
Substrand: Angles 2 Outcomes: WM3-1WM describes and represents
mathematical situations in a variety of ways using mathematical
terminology and some conventions MA3-16MG measures and constructs
angles, and applies angle relationships to find unknown angles
Students:
identify and name angle types formed by the intersection of
straight lines, including right angles, 'angles on a straight
line', 'angles at a point' that form an angle of revolution, and
'vertically opposite angles'
recognise right angles, angles on a straight line, and angles of
revolution embedded in diagrams (Reasoning) identify the vertex and
arms of angles formed by intersecting lines (Communicating)
recognise vertically opposite angles in different orientations and
embedded in diagrams
(Reasoning) Activity: Identifying angles (lesson idea by Nagla
Jebeile) This activity requires students to identify angles in the
environment in pictures. We have used a sample photo. You may find
it more useful to use images that the students take of the school,
school community or home environment. Choosing images that display
a variety of angles (including vertically opposite angles, adjacent
angles and angles at a point) is important as these are all new
angle types for students in Stage 3. The Ferris wheel is a great
image to explore for vertically opposite angles, adjacent angles
and angles at a point. Scissors are also a good example of
vertically opposite angles that provide a concrete example for
students. What types of angles can you see? Draw the different
types of angles you can find
Adjacent
Acute
Obtuse
Right
Vertically Opposite
Straight Line
Revolution
Reflex
http://upload.wikimedia.org/wikipedia/commons/9/9f/Luna_Park-Sydney-Australia.JPG
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JULY 2014
http://upload.wikimedia.org/wikipedia/commons/9/9f/Luna_Park-Sydney-Australia.JPG
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Task: What type of angles can you see? Draw the different types
of angles you can find
Adjacent
Acute
Obtuse
Right
Vertically Opposite
Straight Line
Revolution
Reflex
http://upload.wikimedia.org/wikipedia/commons/9/9f/Luna_Park-Sydney-Australia.JPG
http://upload.wikimedia.org/wikipedia/commons/9/9f/Luna_Park-Sydney-Australia.JPG
KCARTWRIGHT5File AttachmentLuna Park angles BLM
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12
Stage 3 Teaching Ideas- Angles in Two-Dimensional Space
Strand: Measurement and Space Substrand: Two-Dimensional Space 1
Outcomes: WM3-1WM describes and represents mathematical situations
in a variety of ways using mathematical terminology and some
conventions WM3-2WM selects and applies appropriate problem-solving
strategies, including the use of digital technologies, in
undertaking investigations WM3-3WM gives a valid reason for
supporting one possible solution over another MA3-15MG manipulates,
classifies and draws two-dimensional shapes, including equilateral,
isosceles and scalene triangles, and describes their properties
Students: Classify two-dimensional shapes and describe their
features
explore by measurement side and angle properties of equilateral,
isosceles and scalene triangles explore by measurement angle
properties of squares, rectangles, parallelograms and rhombuses
Activity: Exploring angles using pattern blocks There are number
of great activities that use pattern blocks to look angles and
angle relationships of two-dimensional space. Two lessons that
explore angles using patterns blocks come from our Teaching about
angles Stage 2 book. This book is currently being rewritten to
align the lessons to the new mathematics K-10 syllabus outcomes.
These two lessons attached are in draft form but can be used in the
classroom. We welcome any feedback about the success of these
lessons.
This book, Developing Mathematics with Pattern Blocks by Paul
Swan and Geoff The book can be purchased through AAMT for $40 for
members. It has a number of wonderful lessons that use pattern
block for angles, other special relationships and also for
Fractions.
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http://www.aamt.edu.au/Webshop/Entire-catalogue/Developing-Mathematics-with-Pattern-Blocks
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3
Lesson one
Pattern Blocks Creating, describing and drawing patterns using
pattern blocks
Purpose
Students need to identify angles as measures of turn, be able to
compare angle sizes and understand how manipulating the size of
angles within shapes changes the shape.
Outcomes
MA2-1WM uses appropriate terminology to describe, and symbols to
represent, mathematical ideas
MA2-3WM checks the accuracy of a statement and explains the
reasoning used MA2-15MG manipulates, classifies and sketches
two-dimensional shapes,
including special quadrilaterals, and describes their features
MA2-16MG identifies, describes, compares and classifies angles
Key idea
Identify and describe angles as measures of turn Compare angles
sizes in everyday situations
Lesson concept
Joining two angles will form larger angles, including angles on
a line (a straight angle), by creating a larger amount of turning
between the two outer arms.
Materials
• Pattern blocks: Each set of pattern blocks consists of orange
squares, green equilateral
triangles, yellow regular hexagons, red trapeziums with angles
of 60º and 120º, and two types of rhombus (a blue rhombus with
angles of 60º and 120º, and a white rhombus with angles of 30º and
150º).
• Pencils and paper • Digital camera • Pattern block picture
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3
Teaching point
• The most interesting patterns (mathematically speaking) are
those that are formed using only one or two different pattern
blocks. Students who have not used pattern blocks previously should
be encouraged to investigate and create patterns, without any
restrictions. If students are already familiar with pattern blocks,
they may be asked to use only one or two different blocks.
Steps
Questions and discussion
Organise the students into groups and distribute a large number
of pattern blocks to each group. Display a hexagon to the whole
class either on an electronic white board or by holding up a
cardboard hexagon.
Show the picture of the pattern blocks to the class. Ask a
student to indicate the hexagons.
Display a rhombus.
If the students name the shape a diamond or are unable to
correctly name it, tell them that it is a rhombus.
Display a trapezium.
What is the name of this shape? Why do you think it named a
hexagon? Find the blocks that have a hexagonal face amongst your
group’s pattern blocks.
What is the special name of this quadrilateral?
The origin of the word rhombus is from the Greek word for
something that spins. Why do you think that the early Greeks
related spinning to this shape? What do you notice about the sides
of a rhombus? Why isn’t a rhombus called a square?
Find the blocks that have a face that is a rhombus amongst your
group’s pattern blocks. What is the special name of this
quadrilateral? What do you notice about the sides of a
trapezium?
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3
Steps
Questions and discussion
Draw the students’ attention back to the picture of the pattern
blocks and have students indicate the trapeziums and then find the
blocks that have a face that is a trapezium amongst their pattern
blocks. Ask the students to compare the faces of their pattern
blocks.
Select students to describe a shape and have the rest of the
class hold up a pattern block that matches the description.
Instruct each group to use their pattern blocks to create their
own pattern where the blocks fit together without gaps or overlaps.
Have each group take a photo of their pattern.
Have the students record their observations. Focus the students’
attention back to the original pattern block photograph.
Have each group share the photograph of their pattern and their
recordings. Note: Group presentations may need to occur at
follow-up times rather than having all groups present at the
conclusion of this lesson. Time may be needed for each group to
prepare their presentation.
How are the angles on each shape similar or different? How are
the sides of the shapes similar of different? Why do the pattern
blocks that you have chosen fit together without leaving gaps? What
do you notice about the length of the sides? What do you notice
about the angles where you have joined shapes?
Do you notice anything about the angles in this picture that is
different from the angles in the pattern your group has
created?
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3
Purpose
Outcomes
Key idea
Lesson concept
Materials
Teaching point
KCARTWRIGHT5File AttachmentPattern Blocks Lessonplan 1
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3
Lesson two
Windmill patterns
Fitting pattern blocks around a point to compare the sizes of
the pattern block angles
Purpose
Students need to be able to compare angles and identify angles
as features of two-dimensional shapes
Outcomes
MA2-1WM uses appropriate terminology to describe, and symbols to
represent, mathematical ideas
MA2-2WM selects and uses appropriate mental or written
strategies, or technology to solve problems
MA2-3WM checks the accuracy of a statement and explains the
reasoning used MA2-16MG MA2-15MG
identifies, describes, compares and classifies angles
manipulates, identifies and sketches two-dimensional shapes,
including special quadrilaterals, and describes their features
Key idea
Compare angle sizes in everyday situations
Lesson concept
The size of an angle is measured by the amount of turning
between its two arms. The more turn, the larger the angle.
Materials
• Interactive Whiteboard • Pattern block • Paper and pencils •
Mini whiteboards
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3
Steps
Questions and discussion
Display using blocks or on an interactive whiteboard, blue
pattern blocks placed together around a point. Separate the blocks
and point to the pattern of lines made by the joins between the
blocks.
Select a student to draw the lines in the middle of the pattern.
Then remove the blocks to reveal the lines.
Select students to identify and describe the angles between two
of the lines. Do the students use the word acute, or less than a
right angle? (both are acceptable) Organise the students into pairs
and provide them with pattern blocks and mini whiteboards. Instruct
the students to repeat the process of joining the blocks around a
centre point for each of the other pattern block types. Then,
separating the blocks and drawing the lines between each block.
Have the students investigate the relationship between a pattern
block angle size and the number of blocks needed. Ask the students
to record the results in a table.
As a class, discuss the results.
How would you describe this pattern? Where is the centre point
of this pattern?
Can you point to an angle? How would you describe the size of
the angle?
Why were more blocks needed for some of the patterns? What can
you say about the angles
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3
Steps
Questions and discussion
Are students able to see that as the angle size increases (the
amount of turn), you need less blocks to make the pattern. Ask the
students to investigate the relationship between their pattern and
another block by placing a different shape block on top of their
line drawings and recording the results. Have the students explain
their investigations with other students. Then, select students to
share their results with the whole class.
when you compare the two different rhombuses? Compare the angles
on the square and the rhombus. What do you notice? Compare the
angles on the hexagon and the rhombus. What do you notice?
Purpose
Outcomes
Key idea
Lesson concept
Materials
KCARTWRIGHT5File AttachmentWindmill Patterns Lessonplan 2
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13
Stage 4 Teaching ideas – Angle relationships Strand: Measurement
and Geometry Substrand: Angle Relationships Outcomes: A student
MA4-18MG identifies and uses angle relationships, including those
related to transversals on sets of parallel lines MA4-1WM
communicates & connects mathematical ideas using appropriate
terminology, diagrams & symbols MA4-2WM applies mathematical
techniques to solve problems MA4-3WM recognises and explains
mathematical relationships using reasoning
In Stage 3, students investigate angle relationships in a more
informal way, finding different types of angles in their
environment, developing a conceptual understanding of what angles
are and their relationship to their world. In Stage 4, students are
expected to manipulate angles in a more abstract way, formally name
them and solve problems that involve angle relationships. Excerpt
from: Mathematics Stage 4 – Angles (Centre for Learning Innovation)
can be found on TaLe – secondary teachers – item code X00LB.
Naming practice
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Naming practice
Zdena PethersFile AttachmentNaming Practices
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14 Once students have had a chance to work through activities
like those below, a good way to differentiate learning for all
students would be to ask them to work in pairs of small groups, and
make up diagrams and similar problems for other students. This will
deepen their understanding of angle relationships and give all
students the opportunity to work at a level that is appropriate to
their ability.
Using Angle Relationships In each diagram, use the angle given,
to find the value of each pronumeral, giving your reasons. Do not
measure the angle using a protractor as the diagrams are not drawn
to scale.
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Using Angle Relationships
In each diagram, use the angle given, to find the value of each
pronumeral, giving your reasons. Do not measure the angle using a
protractor as the diagrams are not drawn to scale.
Zdena PethersFile AttachmentUsing Angle Relationships
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15 Excerpt from: Mathematics Stage 4 – Angles (Centre for
Learning Innovation) can be found on TaLe – secondary teachers –
item code X00LB.
Reasoning in geometry
X + 61 + 29 + 90
X =
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Reasoning in geometry
X + 61 + 29 + 90 + 61
X = 119
Page 2
Zdena PethersFile AttachmentReasoning in geometry
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16
Page 2
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17
Excerpt from: Mathematics Stage 4 – Angles (Centre for Learning
Innovation) can be found on TaLe – secondary teachers – item code
X00LB.
Reasoning and parallel lines
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Reasoning and parallel lines
Page 2
Zdena PethersFile AttachmentReasoning and parallel lines
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18 Excerpt from: Mathematics Stage 4 – Angles (Centre for
Learning Innovation) can be found on TaLe – secondary teachers –
item code X00LB.
Page 2
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Stage 4 Teaching ideas – Properties of geometrical figures
Strand: Measurement and Geometry Substrand: Properties of
Geometrical Figures Outcomes: A student: MA4-17MG classifies,
describes and uses the properties of triangles and quadrilaterals,
and determines congruent triangles to find unknown side lengths and
angles MA4-1WM communicates & connects mathematical ideas using
appropriate terminology, diagrams & symbols MA4-2WM applies
mathematical techniques to solve problems MA4-3WM recognises and
explains mathematical relationships using reasoning The following
investigation activities could be done in pairs or small groups
where students are encouraged to discuss, justify and give reasons
for their decisions.
Syllabus PLUS Series Recordings
Excerpt from: Mathematics Stage 4 – Properties of geometrical
figures (Centre for Learning Innovation) can be found on TaLe –
secondary teachers – item code X00L9.
Investigation – Angles in quadrilaterals
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Investigation – Angles in quadrilaterals
Investigation – angles in quadrilaterals
Zdena PethersFile AttachmentInvestigation- angles in
quadrilaterals
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20
Excerpt from: Mathematics Stage 4 – Properties of geometrical
figures (Centre for Learning Innovation) can be found on TaLe –
secondary teachers – item code X00L9.
Investigation – Quadrilaterals
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Investigation – Quadrilaterals
Investigation – angles in quadrilaterals
Zdena PethersFile AttachmentInvestigation- quadrilaterals
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21
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Syllabus PLUS Keep an eye out for the Syllabus PLUS Maths K-6
Series 4 in SchoolBiz Term 3, week 1. Flyer attached.
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Further information Learning and Leadership Directorate
Primary Mathematics Advisor
[email protected]
Secondary Mathematics AC Advisor
[email protected]
Secondary Mathematics Advisor
[email protected]
Level 3, 1 Oxford Street Sydney NSW 2000
9266 8091 Nagla Jebeile 9244 5459 Katherin Cartwright
© July 2014 NSW Department of Education and Communities
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Welcome back!Katherin Cartwright, Mathematics Advisor K-6 and
Zdena Pethers, R/Numeracy Advisor 7-12Getting the right angleA
different angle….Angle relationshipsProperties of Geometrical
FiguresTransformations Other interesting websites:Other
ResourcesContinuum of learning Mathematics K-10 Measurement and
Geometry StrandStage 2 Teaching Ideas- Two-Dimensional Space and
AnglesStage 3 Teaching Ideas- AnglesStage 3 Teaching Ideas-
AnglesStage 3 Teaching Ideas- Angles Stage 3 Teaching Ideas-
AnglesStage 3 Teaching Ideas- Angles in Two-Dimensional SpaceStage
4 Teaching ideas – Angle relationshipsStage 4 Teaching ideas –
Properties of geometrical figuresSyllabus PLUS Series
RecordingsSubscription link DEC Mathematics Curriculum
networkSyllabus PLUSResourcesScootle MANSWGeoGebra Institute,
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