The Art of Angles

Suggested Learning Intentions

• To investigate the way that angles in shapes relate to each other
• To manipulate angles to improve understanding of angle properties in polygons

Sample Success Criteria

• I can estimate the size of an angle when turning an object
• I can identify polygons and angles made when lines cross
• I can prove the sum of the internal angles in a triangle or quadrilateral
• I can use a range of manipulatives to explain and justify my solutions
• Uccello Artwork: docx PDF
• Angle Origami: pptx
• Angle Origami student handout: docx PDF (optional)
• Monuments: docx PDF (Area for further exploration)
• Origami paper
• Cardboard

Give small groups or pairs of students a piece of origami paper each. Ask students to make a 60-degree angle with their piece of paper without using any other tools.

Provide students with time for exploration and to attempt to complete the task.

After allowing time for exploration, facilitate a discussion about the different approaches students took to forming the angle. Ask students to consider what tools might make the task easier.

Show students the Angle-a-trons video.

Ask students to explain in their own words how Vi Hart was able to make a 180 degree angle-a-tron so confidently. Provide students with paper to create 180, 90, 45 and 60 degree angle-a-trons of their own. Observe students as they create their angle-a-trons, using a protractor to determine the level of accuracy as students make their folds. Encourage students to be accurate in their folding.

In pairs, students record the steps they took to create an angle-a-tron that requires at least two folds to create.  Ask each pair to share their instructions with another pair of students to test the quality of information provided. Each pair provides feedback for improvement in the quality of the instructions given.

Enable students who require further support by providing access to explicit instructions with diagrams. If required instructions are available for download from NRICH.

Extend students by challenging them to answer the question posed in the angle-a-tron video at 1:25 ‘how do you make an angle-a-tron that completes the circle?’

During the video Vi Hart produces artworks by tracing angle-a-trons repeatedly across the page. Invite students to create an artwork of their own using one or more angle-a-trons.

1. Polygon pictures on paper

Review the definitions of polygon and vertex before beginning this activity. Each student cuts out one polygon from a piece of thin cardboard. Ask students to mark one vertex of their polygon as the rotation point.

Students trace their polygon, then rotate it at the marked vertex and trace it again. Students should repeat rotation and tracing until the polygon rotates back to its starting position.

Students can choose any angle of rotation. By exploring different angles, students will discover that angles that divide evenly into 360 degrees will create patterns with symmetry.

• Would a different rotation angle make an interesting pattern?
• Are some rotation angles better than others for creating pattens?
• Would the pattern be more interesting with each rotation angle the same size or different?

Invite students to repeat the task using different conditions to explore the patterns that can be made.

2. Digital polygon pictures

Ask students to create an image using Geogebra. Prompt students to:

1. Draw a polygon: Choose the ‘polygon’ tool then click the position of each vertex.
2. Rotate around a point: Open ‘more’ tools and ‘rotate around a point’. Choose the object to rotate, a point to rotate around and the angle of the rotation.

Facilitate a discussion about the ease of use of paper and Geogebra as tools to create images. This activity is completed in two modes to provide students with multiple exposures to a similar task.

Provide students with images of Uccello’s artwork (available in the Materials and texts section above). Paolo Uccello installed this marble-tiled artwork on the floor of Basilica of St Mark in Venice in the 15th century. Uccello used symmetry and mathematics to create tiling that looks three dimensional. Students compare their polygon pictures with Uccello’s mosaic, and consider these questions:

• Which of the images created are artworks?
• When does mathematical drawing become art?

It is not expected that a class would come to an agreement about answering these questions.

3. Origami

It is recommended to do this task before the Dancing Between the Lines stage. The fold lines created in this task can help students visualise the terms 'parallel' and 'transversal'.

Present slides 12 to 15 of the Angle Origami slideshow to students (available in the Materials and texts section above). Before sharing the slideshow with students, delete slides 1-11, which provide teacher notes including the answer to the problem and considerations about how you might present the task to your class.

Provide students with a copy of the Angle Origami slideshow and allow them to progress through the steps.

Students will:

1. Fold a piece of origami paper
2. Identify the angles made when folding
3. Identify the shapes made when folding
4. Identify the internal angle sums of the shapes

Students will work through the activity at their own pace. Discuss that the comments made in green on the slideshow will assist them to keep the required records of their work for submission at the end of the task.

Enable students who require support by reading the steps for the origami to them, emphasising key words. Have students trace the fold lines using a ruler and pencil.

In addition to the extension tasks suggested within the slideshow, you could extend students by discussing ways to record their justifications about internal angle sums on slide 20. Show students how to follow simple mathematical conventions about arrangement of reasoning when providing a proof.

### Areas for further exploration

1. Hexaflexagons

The basic mathematics of folding equilateral triangles can be practised as a mathematical recreation break by exploring hexaflexagons. The ways that the video presents the mathematics of hexaflexagons, in exploring the pathways that can be taken in the flexing, is complicated and well beyond this level.

2. Orthogonal and isometric drawing

Provide students with images of monuments (available in Materials and Texts). Encourage students to look at these three-dimensional structures and consider how they might be drawn using two different drawing techniques.

• Isometric drawing is an example of a paraline technique. The object appears as three-dimensional with the third dimension shown as parallel lines receding on the page.
• Third angle orthogonal drawing shows each of the dimensions as a separate view. These views are drawn beside each other on the page. Top, front and side views are generally used.

Further detail on these techniques is provided in the technical drawing specifications and Design and Technologies glossary. Note that students build drawing skills throughout their school education as part of the Visual Communication and Design curriculum.

Tearing Paper

Ask students complete the following steps:

1. draw a triangle and a quadrilateral on their page
2. check that they have drawn polygons
3. mark the corners of the polygons with arcs to show their angles
4. cut out their shapes
5. tear two corners off their triangle
6. paste the remaining section of their triangle on their page
7. move the two torn triangle corners so they touch the third at the vertex to create a single angle
8. repeat steps 5 to 7 for the corners of the quadrilateral.

Between the steps of this task, prompt students to share their reasoning by inviting them to discuss what is different about the triangle and quadrilateral at each step.

Facilitate a discussion with students about what they notice when they place their corners together. What do they notice when the four angles of the quadrilateral sit beside each other? What do they notice when the three angles from the triangle sit beside each other?

To further extend student thinking about internal angles of quadrilaterals allow students time to explore Interactive Quadrilaterals. By using the digital tool students can quickly explore different quadrilaterals. Encourage students to toggle the buttons on and off to notice the available features in the interactive tool.

Ask students to expand their visual maths wall created during the Reflect and Consolidate phase of the Playing with Polygons stage. Brainstorm a list of key terms with students that should be included on their wall. Prompts include: angle, rotate, internal angle sum.

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