Syllabus content Pedagogy Teaching ideas€¦ · two-dimensional shapes from descriptions of their features . Part 1 . Classify and determine properties of triangles and quadrilaterals

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 JULY 2014

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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,


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() {}

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KCARTWRIGHT5File AttachmentVertically opposite angles applet




geogebra.xml

KCARTWRIGHT5File AttachmentAdjacent angles applet



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





geogebra.xml

KCARTWRIGHT5File AttachmentAngles in a right angle applet




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


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�

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


settings.xml

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Jul 17-10:38 AM What do these shapes have in common?

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Jul 17-10:38 AM Do you know what this is called?

KCARTWRIGHT5File AttachmentAngles in shapes Notebook

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.


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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.


Stage 3 |The protractor | Lesson one

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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


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?


3

Steps


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

http://www.teachersofindia.org/en/users/azim-premji-foundationhttp://www.teachersofindia.org/en/activity/handmade-math-tools-protractor

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.

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http://www.teachersofindia.org

http://www.teachersofindia.org/en

http://www.teachersofindia.org/en/users/azim-premji-foundation

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:

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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

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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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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

http://www.schools.nsw.edu.au/learning/7-12assessments/naplan/teachstrategies/yr2013/index.php?id=numeracy/nn_spac/nn_spac_s3bi_13

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


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



KCARTWRIGHT5File AttachmentLuna Park angles BLM

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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.

http://www.aamt.edu.au/Webshop/Entire-catalogue/Developing-Mathematics-with-Pattern-Blocks

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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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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


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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Steps


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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Purpose

Outcomes

Key idea

Lesson concept

Materials

Teaching point

KCARTWRIGHT5File AttachmentPattern Blocks Lessonplan 1

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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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Steps


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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Steps


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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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


Naming practice

Zdena PethersFile AttachmentNaming Practices

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.


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

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 =


Reasoning in geometry

X + 61 + 29 + 90 + 61

X = 119

Page 2

Zdena PethersFile AttachmentReasoning in geometry

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Page 2


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Excerpt from: Mathematics Stage 4 – Angles (Centre for Learning Innovation) can be found on TaLe – secondary teachers – item code X00LB.

Reasoning and parallel lines


Reasoning and parallel lines

Page 2

Zdena PethersFile AttachmentReasoning and parallel lines

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


Investigation – Angles in quadrilaterals

Investigation – angles in quadrilaterals

Zdena PethersFile AttachmentInvestigation- angles in quadrilaterals

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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


Investigation – Quadrilaterals

Investigation – angles in quadrilaterals

Zdena PethersFile AttachmentInvestigation- quadrilaterals

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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, GeoGebra applets and teaching ideas Conferences

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Syllabus content Pedagogy Teaching ideas€¦ · two-dimensional shapes from descriptions of their features . Part 1 . Classify and determine properties of triangles and quadrilaterals