# The Art of Angles

Mathematics / Visual Arts

Level 6Level 7Level 8

This learning sequence aims to develop student understanding of shapes and intersecting lines, including the properties of the angles and line segments. Students will explore how to justify their reasoning in solving problems. Students will have opportunities to use art as a medium to develop their understanding of using spatial reasoning and geometric properties to solve problems. The sequence provides opportunities for students to view shapes, angles, and lines in artworks.

 Big understandings Geometric thinking requires visualisation, interpretation, and construction. Geometric reasoning involves logical sequencing of connected statements. To convince others in mathematics we need to be able to justify our answers.

The sequence has been written by teachers for teachers. It has been designed to provide students with rich, engaging learning experiences that address the Victorian Curriculum. The sequence consists of four flexible stages, including suggested learning intentions.

Overview of stages

• 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

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• 3. Dancing Between the Lines

Suggested Learning Intentions

• To explore the properties of angles where a transversal crosses parallel lines
• To make connections between line and angle forms in artwork and line and angle properties in mathematics
• To use properties of lines and angles to discuss visual elements in artworks

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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

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• 4. Untangle Polygons and Angles

Suggested Learning Intentions

• To develop logical reasoning to solve problems involving lines, angles, and polygons
• To articulate mathematical reasoning through providing justifications of steps in problem solving
• To use visual representations to support problem solving

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• Prior knowledge

Before you commence this sequence, it is suggested that you ensure your students are familiar with measuring and drawing angles on a straight line, at a point and vertically opposite angles. Students should also be able to name triangles, and basic quadrilaterals, noting the number of sides and angles they have.

You can find support for building students’ understanding of these concepts in the following sections of the Mathematics Curriculum Companion: VCMMG171, VCMMG174, VCMMG202, VCMMG231.

Teaching strategies

The Mathematics Curriculum Companion provides teachers with content knowledge, suggested teaching and learning ideas as well as links to other resources. Resources are organised by Mathematics strands and sub-strands and incorporate the proficiencies: Understanding, Fluency, Problem Solving and Reasoning. The Companion is an additional resource that you could refer to when you are planning how you might use

The sequence highlights opportunities to apply the High Impact Teaching Strategies (HITS), which are a component of the Victorian Teaching and Learning Model

The sequence highlights the use of a variety of scaffolding practices to help support students in the learning process.

This sequence employs the following teaching strategies:

• Concrete manipulatives
• Differentiated teaching
• Student collaboration
• Questioning
• Feedback
• Multiple exposures
• Structured lessons
• Metacognition

Vocabulary

Students should be able to understand and use the following concepts and terms by the end of the learning sequence:

 Acute angle Polygon Alternate angle Quadrilateral Co-interior Reasoning Complementary Reflex angle Corresponding Right angle External angle Straight angle Internal angle Transversal Obtuse angle Triangle

You can find definitions of some of these terms in the Glossary for the Mathematics Curriculum.

It is recommended that the explicit teaching of vocabulary occur throughout this learning sequence. The Literacy in Mathematics section of the Literacy Teaching Toolkit provides several teaching strategies with  worked examples demonstrating how teachers can use literacy to support student understanding of mathematical language. A further set of strategies demonstrate how teachers can develop students' literacy skills to support their mathematical problem solving.

Assessment

Opportunities for formative and summative assessment are identified at different stages of the learning sequence. Look for the 'Assessment Opportunity' icon.

You may want to develop a rubric to assess students’ progress. A range of Formative Assessment resources are available from the Victorian Curriculum and Assessment Authority. This includes a Guide to Formative Assessment Rubrics, a series of modules to support you to develop your own formative assessment rubrics, and sample rubrics across six curriculum areas that demonstrate how you can put formative assessment rubrics into practice in the classroom.

In developing a rubric, you may wish to co-construct assessment criteria with your students. Each stage of the sequence provides sample success criteria for students working at Level 7.

The Victorian Curriculum and Assessment Authority has published annotated work samples that provide teachers with examples of student learning achievement at multiple levels for each strand of the Mathematics Curriculum.

Victorian Curriculum connections

Contents

Level 6

This sequence addresses content from the Victorian Curriculum in Mathematics, Visual Arts and Critical and Creative Thinking. It is primarily designed for Level 7, but also addresses the following content descriptions from Level 6:

 Content description Stage Mathematics Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles (VCMMG231) Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Investigate the effect of combinations of transformations on simple and composite shapes, including creating tessellations, with and without the use of digital technologies (VCMMG229) Assemble your Angles Visual Arts Explore visual arts practices as inspiration to create artworks that express different ideas and beliefs (VCAVAE029) Playing with Polygons Assemble your Angles  Dancing Between the Lines Untangle Polygons and Angles Identify and describe how ideas are expressed in artworks by comparing artworks from different contemporary, historical and cultural contexts, including artworks by Aboriginal and Torres Strait Islander peoples (VCAVAR032) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Critical and Creative Thinking Identify and form links and patterns from multiple information sources to generate non-routine ideas and possibilities (VCCCTQ023) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Investigate common reasoning errors including contradiction and inconsistency, and the influence of context (VCCCTR024) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Investigate thinking processes using visual models and language strategies (VCCCTM029) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum: Mathematics Level 6:

• Students solve problems using the properties of angles and investigate simple combinations of transformations in the plane, with and without the use of digital technology.

Level 7

This sequence addresses content from the Victorian Curriculum in Mathematics, Visual Arts and Critical and Creative Thinking. It is primarily designed for Level 7, and addresses the following content descriptions:

 Content description Stage Mathematics Describe translations, reflections in an axis, and rotations of multiples of 90° on the Cartesian plane using coordinates. Identify line and rotational symmetries (VCMMG261) Assemble your Angles Classify triangles according to their side and angle properties and describe quadrilaterals (VCMMG262) Playing with Polygons Untangle Polygons and Angles Demonstrate that the angle sum of a triangle is 180° and use this to find the angle sum of a quadrilateral (VCMMG263) Assemble your Angles Untangle Polygons and Angles Identify corresponding, alternate and co-interior angles when two straight lines are crossed by a transversal (VCMMG264) Dancing Between the Lines Untangle Polygons and Angles Investigate conditions for two lines to be parallel and solve simple numerical problems using reasoning (VCMMG265) Dancing Between the Lines Visual Arts Explore visual arts practices as inspiration to explore and develop themes, concepts or ideas in artworks (VCAVAE033) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Identify and connect specific features of visual artworks from different cultures, historical and contemporary times, including artworks by Aboriginal and Torres Strait Islander peoples (VCAVAR039) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Critical and Creative Thinking Synthesise information from multiple sources and use lateral thinking techniques to draw parallels between known and new solutions and ideas when creating original proposals and artefacts (VCCCTQ034) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Examine common reasoning errors including circular arguments and cause and effect fallacies (VCCCTR035) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum: Mathematics Level 7:

• Students classify triangles and quadrilaterals.
• Students name the types of angles formed by transversals crossing parallel lines and solve simple numerical problems involving these lines and angles.

Level 8

This sequence addresses content from the Victorian Curriculum in Mathematics, Visual Arts and Critical and Creative Thinking. It is primarily designed for Level 7, but also addresses the following content descriptions from Level 8:

 Content description Stage Mathematics Develop the conditions for congruence of triangles (VCMMG292) Untangle Polygons and Angles Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning (VCMMG293) Playing with Polygons Assemble your Angles Untangle Polygons and Angles Visual Arts Explore visual arts practices as inspiration to explore and develop themes, concepts or ideas in artworks (VCAVAE033) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Identify and connect specific features of visual artworks from different cultures, historical and contemporary times, including artworks by Aboriginal and Torres Strait Islander peoples (VCAVAR039) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Critical and Creative Thinking Synthesise information from multiple sources and use lateral thinking techniques to draw parallels between known and new solutions and ideas when creating original proposals and artefacts (VCCCTQ034) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Examine common reasoning errors including circular arguments and cause and effect fallacies (VCCCTR035) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Playing with Polygons Assemble your Angles Dancing Between the Lines Untangle Polygons and Angles

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum: Mathematics Level 8:

• Students identify conditions for the congruence of triangles and deduce the properties of quadrilaterals.
• Students use tools, including digital technology, to construct congruent shapes.

Learning Progressions

The Numeracy Learning Progressions support teachers to develop a comprehensive view of how numeracy develops over time. You can use the Numeracy Learning Progressions to:

• identify the numeracy capability of your students
• plan targeted teaching strategies, especially for students achieving above or below the age-equivalent expected level in the Victorian Curriculum: Mathematics
• provide targeted feedback to students about their learning within and across the progressions.

The sequence is related to the following progressions:

 Learning progression Understanding Geometric Properties Angles and lines Geometric properties

Click on the Learning Progression to access more detailed descriptions of student learning at each level.