The Art of Angles

1. Playing with Polygons

Suggested Learning Intentions

• To develop vocabulary about the angle and side properties of triangles and quadrilaterals
• To classify triangles according to their side and angle properties and describe quadrilaterals

Sample Success Criteria

• I can describe a polygon using its properties
• I can use side and angle properties of triangles and quadrilaterals to describe their differences and similarities
• I can identify polygons and non-polygons within a two-dimensional image
• I can explain and justify my thinking using manipulatives
• One computer per student
• Cardboard box
• Thick card, rulers, scissors
• Mondrian artworks: docx PDF
• Vocabulary flash cards: docx PDF
• Visualising shapes Images: docx PDF
• Polygon capture cards: docx PDF
• Frayer model template: pptx PDF

In this phase students will focus on developing the vocabulary required to distinguish one quadrilateral from another, and to identify the key characteristics of polygons.

Prior to beginning this stage, enable students by teaching vocabulary, and creating a word wall of key vocabulary for later reference.

Keep a note of student responses to the following task for formative assessment.

Provide each student with some thick card and ask them to draw at least two different polygonal shapes (regular and irregular shapes), and then cut them out. Place the shapes inside a cardboard box, with a covered slot large enough for students to place their hands inside.

Ask students to reach into the box to choose one shape and describe the shape they can feel to the other students, without removing it from the box or looking at the shape. The other students draw the shape, based on the description provided. Together students try to determine the name of the shape, based on the features described. The student then takes the shape out of the box to show the class, who compare the shape to their drawing and confirm its name. Repeat several times with different students describing the shapes.

Enable students who had incorrect responses in the drawing task by providing them with vocabulary flash cards (available in the Materials and Texts section above). These simple terms are matched to images to prompt students to recall their meanings.

1. Polygon art

Provide students with a copy of the Mondrian artworks (available in the Materials and Texts section). Piet Mondrian produced different styles of art during his career and is known generally for his use of line and shape. Facilitate a discussion about the shapes that students can see in the art. Provoke a discussion by asking what polygons can be seen in the work.

Question prompts include:

• What polygons can you see?
• How do you know that is a polygon?
• Can you explain why one of the shapes you can see isn’t a polygon

Ask students to justify why it is easy to find polygons in ‘Composition A’ and hard to find polygons in ‘Composition’ and ‘Mill in the evening’.

This conversation should result in a shared class understanding of the term polygon. You can find a comprehensive definition of polygons in the Glossary for the Mathematics Curriculum.

It is possible to complete this task using art from more contemporary artists. Students may enjoy the work of Tony Robbin, Kerrie Poliness, John Nixon or Bradd Westmoreland

To formalise a definition of the terminology, ask students to show what they have learnt using the Frayer model. A downloadable Frayer model template is available in the Materials and texts section above. Students use the Frayer model to define and state facts about polygons. Invite students to give examples and non-examples from the artworks. Encourage students to annotate their examples and non-examples with explanations or justifications.

This activity draws on ideas from NRICH: Stringy Quads.

Show students a clip of Bangarra Dance Theatre performing the ‘Ngarrindjeri: String Games’ portion of the ‘Unaipon’ production (0:41-1:26 of the link). Ask students to note the polygons they saw formed within the string.

To assist students in understanding the artistic performance you may like to present some of the information about David Unaipon’s life and the Ngarrindjeri people.

Students work in small groups with a piece of rope or string. Give students a set of properties and ask them to work as a team to make a polygon that fits the clue. You may consider clues such as:

• Can you make a four-sided shape with all different side lengths?
• Can you make a three-sided shape with two angles the same?
• Can you make a four-sided shape with two right angles?

Enable groups by displaying a wide range of polygons within the classroom.

To further explore the Bangarra Dance Theatre production consider using some of the teaching ideas or questions for consideration presented on the Bangarra website.

3. Visualising shapes

This activity draws upon ideas from NRICH: Shape Draw and Playing Mathematically: Visualising Shapes.

1. In pairs students will participate in a description and drawing game.
2. Arrange students so that pairs cannot see each other’s page.
3. Display a list of taboo words that cannot be used during this activity. For example:
triangle, isosceles triangle, right triangle, scalene triangle, square, rectangle, rhombus,
parallelogram, kite, trapezium, circle, oval, star.
4. Have one person from each pair describe the following shape to the other. Students should use only words, no gestures and definitely no words from the taboo list.
5. The second player should use the clues to draw what they think is being described. Clues may need to be repeated more than once.

A simple spoken description for a rectangle might be: "The shape has four sides. The shape has four right angles. The shape has two long sides and two short sides."

A simple polygon is included as the first shape to assist students in understanding the task. This will allow students to get quick feedback from each other about the style of the instructions that are helpful.

Once students achieve success with a simple polygon, provide a set of visualising shapes images (available in Materials and Texts) for students to describe to each other. Students should keep the shapes face down until they are required.

Enable students by including more images produced from single shapes or two shapes adjacent tp each other.

Extend students by overlapping polygons in more complicated ways.

4. Polygon capture

This activity draws upon ideas from William Carroll cited at Playing Mathematically: Polygon Capture.

In this game students will read formal mathematical language on the playing cards. Students will practice identifying the way the properties are shown in shapes by matching them to the polygon cards.

This game is played in pairs. Each pair needs a set of polygon capture cards (available in the Materials and texts section above). Cards are marked with a green P for polygon cards, blue A for angle cards and red S for side cards.

In pairs:

1. Arrange side and angle property cards in a pile each, face down.
2. Lay the polygon cards in an array, face up.
3. Choose a player to go first. This player should turn over one angle and one side card. The player keeps any polygons that match both the properties turned. The player should announce when they have finished searching the polygons.
4. The next player can capture any polygon that matches the properties turned that were missed by the previous player before beginning their turn. The new player then turns over new angle and side property cards and captures polygons that match both properties.
5. If the angle or side property cards run out, they should be shuffled and put back in play.
6. The game ends when only two polygon cards remain.
7. The winner is the person who captured the most polygons.

Students use paper or a digital collaboration tool to produce a class visual maths wall. Create a blank wall and share access with the whole class or provide slips of paper for students to illustrate. Encourage students to upload images and use text notes to label them with terms and definitions. Prompt students to identify two key images or definitions they used in each of the Get Started and Go deeper activities.

Enable students who are unsure of what to include by asking them to add the ideas they recorded on the Frayer model during the Polygon Art activity. Ask students to upload, print or draw an image of the art as either an example or non-example of a polygon. Support students to use the text notes to annotate their example with a justification.

You could prompt student thinking by adding these words to the wall before providing student access: polygon, quadrilateral, triangle, acute, obtuse, right angle, side.

This maths wall can be worked on collaboratively and if in digital format, moves easily between classrooms within the school or home. If your class always uses the same room, you may prefer to have a physical maths wall in the classroom.

Alternatively, students could produce an individual visual maths wall as a formative assessment. Provide a list of key terms that students should include on their wall to guide their thinking towards the most important vocabulary.

Bangarra Dance Theatre Australia, 2015. Unaipon. [Online]
Available at: https://www.bangarra.com.au/learning/resources/eresources/unaipon/
[Accessed 15 March 2022].

Bangarra Dance Theatre Australia, 2015. Unaipon: Creating Unaipon. [Online]
Available at: https://www.bangarra.com.au/learning/resources/eresources/unaipon/creating-unaipon/
[Accessed 15 March 2022].

Bangarra Dance Theatre Australia, 2015. Uniapon: Before Viewing. [Online]
Available at: https://www.bangarra.com.au/learning/resources/eresources/unaipon/before-viewing/
[Accessed 15 March 2022].

Desmos, n.d. Polygraph - Basic Quadrilaterals. [Online]
[Accessed 15 March 2022].

Mondrian, P., 2018. Composition, Piet Mondrian, 1916. [Online]
Available at: https://commons.wikimedia.org/wiki/File:Composition,_Piet_Mondrian,_1916.jpg
[Accessed 15 March 2022].

Mondrian, P., 2020. Composition A, Piet Mondrian, 1923. [Online]
Available at: https://www.wikiart.org/en/piet-mondrian/composition-a-1923
[Accessed 15 March 2022].

Mondrian, P., 2021. Mill in the evening, Piet Mondrian, 1905. [Online]
Available at: https://www.wikiart.org/en/piet-mondrian/mill-in-the-evening-1905
[Accessed 15 March 2022].

National Gallery of Victoria, n.d. Bradd Westmoreland. [Online]
Available at: https://www.ngv.vic.gov.au/explore/collection/artist/14056/
[Accessed 15 March 2022].

National Gallery of Victoria, n.d. John Nixon. [Online]
Available at: https://www.ngv.vic.gov.au/explore/collection/artist/1706/
[Accessed 15 March 2022].

National Gallery of Victoria, n.d. Kerrie Poliness. [Online]
Available at: https://www.ngv.vic.gov.au/explore/collection/artist/1747/
[Accessed 15 March 2022].

Note.ly, 2013. All your notes in one place. [Online]
Available at: https://note.ly/landing/index.php
[Accessed 15 March 2022].

Robbin, T., n.d. Development of Painting & Sculpture. [Online]
Available at: http://tonyrobbin.net/work.htm
[Accessed 15 March 202].

Trevaskis, J., 2016. Playing Mathetmatically. [Online]
Available at: https://webmaths.files.wordpress.com/2016/01/playing-mathematically-2016.pdf
[Accessed 15 March 2022].

University of Cambridge, n.d. NRICH: Shape Draw. [Online]
Available at: https://nrich.maths.org/10368
[Accessed 15 March 2022].

University of Cambridge, n.d. NRICH: Stringy Quads. [Online]
Available at: https://nrich.maths.org/2913
[Accessed 15 March 2022].