The Art of Geometry - Mathematical Perspectives

The Art of Geometry Samantha Allsopp

Historical and Cultural Context Geometry is an incredibility old field of mathematics and in itself formed the foundations of other mathematical concepts and practices, including proofs and algebra.1 As with any mathematical field, Geometry is one touched by many hands and cultures,1-4 and its development most frequently attributed to the Greeks.5 Great mathematicians, such as Thales, Pythagoras, and Euclid, laid the foundations for modern geometry,5 which over time have been refined and built upon. Despite the fact that history paints Greek mathematicians as those contributing most to the development of Geometry, mathematicians in Egypt have long made use of geometry and geometrical formulae, calculating area, perimeter and volumes of numerous geometric shapes1-4 as evidenced by the Rhind Papyrus.2 The Rhind Papyrus, dating back to approximately 1650 BCE has been an invaluable development in understanding ancient Egyptian mathematics and as an example of the use and importance of geometry within this ancient civilisation.2,3 Right angle, isosceles and equilateral triangles are all specifically referenced and used in Egyptian diagrams and calculations.6 Additionally, patterns to find the area of regular geometric shapes were in consideration, as were methods of finding the area of composition shapes, for utilitarian purposes.2 This indicates that such knowledge and mathematical thinking has long been of importance to human civilisations. As a further note on the early development of Geometry, certain theories and understandings were in place long before they were ‘officially invented’. Remarkably, Babylonians made use of the Pythagorean Theorem well before Pythagoras had even developed it.2 It is also interesting to note that the importance and development of Geometry is not limited to that which is strictly mathematical. Geometry has been important in many sectors of life for thousands of years4 and its understanding has lead to some of the most magnificent structures of the world, including the Pyramids of Giza. Geometry has, in its thousands of years of existence, also held value in the realm of art, architecture, tradition and spirituality. Simply considering the use of Geometry in ancient temples, churches, art works and man-made structures, it is clear that Geometric shapes and symbols take on important cultural and religious value. Plato, for example, felt triangles were quite literally

a divine structure, theorizing that they not only made up the Platonic solids, which constitute the elements, but all matter, as a result of the purity and stability associated with such shapes.7 The world of art is rich with geometric inspiration, as evidenced by particular styles such as post modernism, cubism, art deco and geometric abstraction, the works of renowned artists, such as Escher and Picasso, as well as specific examples, such as Islamic art, Celtic designs, gothic architecture and stained glass. Geometric patterns have long been in use in art, particularly that of the beautifully tessellated Islamic art, in which such art patterns are linked to tradition, culture and spirituality.8 Islamic art and architecture has spanned a great length of time and distance and has an incredibly strong with Geometry and other mathematical elements.9 In looking at the evolution of Islamic art, Bonner8 has included a lovely array of art pieces which attest to the naturally geometric basis of Islamic art. In keeping with the Greek influence on Geometry, ancient Greek art also includes geometric influence, whereby an entire period of art was named the Geometric period, lasting around 200 years from 900 BCE.10 This art style is regarded as that last Mycenaean-Greek art form before foreign influence, stemming from a new period of relative peace and stability, which enabled artwork detail to develop into the quality of that seen during this period.10,11 Greek artwork during this time would cover pottery and other clay based or sculpted items, and use repeating lines and shapes that illustrate a new sense of style and technical skill in Greek culture.10,11

Teaching Resource Summary The following resource is a unit of learning (program of activities) pertaining to Geometry, specifically looking at the different types of triangles, the angles within triangles, and calculations of area involving triangles. This resource aims to show that geometry is everywhere and to provide students and teachers with the ability to approach mathematics in a creative way. It does this by presenting activities that teach and reinforce mathematical content in the context of art and creation, allowing students to practise mathematical skills of calculation, problem solving, measuring and using mathematical equipment (calculator, ruler and protractor) in a way that is more engaging. This resource may be used to introduce the different types of triangles, to reinforce the fact that geometry is not something that is only found within a maths class, and to provide activities for students to practise and to implement their learning without completing exercise after exercise. Application This unit of learning is primarily targeted at students in Year 7 and may be used as the following: ⋅

Introduction to geometry

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Consolidation of learning

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Motivation or engagement tools

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Homework or study

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Informal assessment

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Summative assessment

The activities planned in my resource also lend themselves to adaption for other year levels and topics. For example each activity can easily be amended with little effort to suit year 8 and 9 Geometry, by including quadrilateral shapes, and can also incorporate the topics of congruency, symmetry or volume, if so desired. I chose not to incorporate these aspects myself, simply because I did not feel my program needed to be any longer than it already is. The premise of my activity program can easily be applied to other areas of mathematics. Engaging students in the demonstration of their learning through creative ventures, and the production of artistic or communicative media, is an approach that could, and perhaps should, be applied more readily to mathematical content. My program contains numerous activities that are designed to flow on from one another; however, each activity would still be effective and add variation into normal mathematics classroom practices, if applied individually. Program Note 1: This program focuses only on how to use the given resources and does not specify student instruction or background teaching that may be required, as such things are completely dependent on the class and teacher and so cannot be prescribed. Note 2: the creation of art work and the demonstration of calculations and triangle types will need additional time. Setting aside multiple lessons or asking students to complete for homework is necessary.

Lesson 1

Tasks How many triangles can you see?

Requirements

Choose 3 problems from resource 1, where students must decide how many triangles there are in the given image.

Resource 1 Protractors Rulers

1. Students complete one problem individually. The problem is given on the board and students are to find as many triangles as possible. Ask the class to share how many triangles they identified. 2. Students complete 2 more problems from those given in resource 2, with a partner. Students then swap partners and explain to one another how they came to their answer. 3. Using the examples given, students are asked to measure the sides and angles of 3 different triangles from those identified from each problem. Students draw those triangles in their book, and are asked to define how many degrees are in a triangle. Students may work individually or in pairs. 4. Students are asked to colour code some of the triangles identified from their problems, defining them as either right angle, isosceles or equilateral. Providing multiple, smaller copies of individual problems may be advisable to allow the same problem to be used in multiple ways to demonstrate different triangles. Sorting triangles Students measure triangles, sort them and perform calculations.

2

1. Using resource 2, students cut out the information table from section one and glue into their books. 2. In pairs, students sort and glue the triangles into the sorting table, identifying each triangle based on their sides and angles. 3. Think, pair, share: Can a right angled triangle also be an equilateral triangle? Why? 4. Think, pair, share: Can an equilateral triangle have an obtuse angle? Why? 5. Think, pair, share: Can a triangle have more than one obtuse angle? Why? 6. Think, pair, share: Can a right angle triangle have an obtuse angle? Why? 7. In pairs, students calculate the area of each triangle using 2 different calculations (ie. use 2 different sides as the base of the triangle). Triangle Hunt In small groups students travel around the school in search of different triangles. Students record their findings in the table, resource 3. Initiate a class discussion of where triangles were found when students return to class.

Resource 2 with the sorting table printed on coloured paper Protractors Scissors and glue

Resource 3 Protractors Rulers

Random Grouping Students draw a triangle in their book, or on spare paper, determine the type of triangle they have drawn and the angle of each vertex. Students must find a different partner to usual, pairing themselves with someone who has a triangle with a similar area, type or angle. Students remain with partner for next activity.

Spare paper Protractors

Table Top Maths: Desk Triangles Introduction to composite area calculation. Students calculate area as well as measure and label triangles on an unorthodox medium. See ‘worked examples – table top maths’ for an example. 1. Students calculate the area of the top of their desk. 2. Using whiteboard markers and/or thin tape, students separate their desk into 2, 3, 4, 5, 6 and 7 triangles, measuring the angles of each vertex, labelling each triangle and recalculating the area of the desk each time by adding together the area of each triangle. 3. Students record their calculations in their book, including a brief drawing of how they have sectioned their table. 4. Class discussion: was your calculated area always the same? Were there are inconsistencies? Why?

3

Protractors Rulers Thin tape Whiteboard markers Desk space

Note: If worried about whiteboard markers staining the desk, place tape where you wish you draw the edges of the triangles, and then mark your lines on top of the tape strips. Summary Sheet Students create a summary page for the types of triangles and angles observed thus far. Students create triangles to cut out and glue to a summary sheet. See ‘worked examples – summary sheet’ for an example. Irregular Shapes Students create a straight lined, irregular shape for their partner, cutting the shape out from coloured paper. Their partner must break the given shape up into triangles and find the total area of the shape. Both students in the pair create their shape then calculate the area simultaneously. The shape may be returned to the original student of the pair, and they also calculate the area of the shape, using the back of the card. Students compare their answers and work together to iron out inconsistencies if there are any. Otherwise, students give their partner another irregular shape to calculate the area of. Students glue there shapes into their book and include a description for how the area of irregular shapes can be calculated. See ‘worked examples – irregular shapes’ for an example.

Coloured paper Rulers Protractors Scissors Glue Coloured paper Rulers Protractors Scissors Glue

Irregular Shapes Field Trip

4

In small groups, students are to find an irregular shape and attempt to discern its area using triangles. Students may use chalk or masking tape to break their finding into triangles. Once calculated students return to class. The class visits each group’s site, with each group showing how they obtained the area for their shape. Triangles In Art, Architecture, Culture And Modern Society Students are shown images of geometry in modern and historical mediums, using resource 4. Possibility for student discussion or research of the reason behind or history of geometry in art and culture. Students are then shown the following videos: ⋅ https://youtu.be/pg1NpMmPv48 ⋅ https://aeon.co/videos/discover-the-secret-geometry-ofbeauty-the-mathematics-of-form-and-pattern Triangle Page Segments Working in small groups or pairs, students break up an A4 page into triangles. The page must contain at least 1 type of every triangle. Thin tape or ruled lines are used to mark the boundaries of each triangle. The triangles are colour coded to demonstrate that at least 1 of each type has been included. See ‘worked examples – triangle page segments’ for an example. Geometric Art Students complete 2 pieces of geometric art. This activity may be used to consolidate learning, as a way of assessment or as a standalone activity. Students complete their art individually. See ‘worked examples – geometric art’ for examples of possible art types. Students should be encouraged to take inspiration from past or present art forms. 1. Irregular shape art Students create a modern art that is geometric in nature and comprised only of straight lines. Students must calculate the area taken up by their art using calculation. The geometric piece is to take up a solid section, but not encompass the entire page. Working out is to be completed on supporting documents (i.e. spare page or scanned copy of the art). 2. Types of triangles art Students are to create a piece of art that demonstrates at least one of each type of triangle (right angle isosceles and scalene, obtuse isosceles and scalene, acute equilateral, isosceles and scalene). The piece may take up the whole page. Proof that each triangle type is included needs to be evident on supporting documents (i.e. spare page or scanned copy of the art).

Measuring tape Chalk Masking tape Rulers

Resource 4 Projector Projector screen or whiteboard Internet Teacher Laptop

Plain or coloured A4 paper Rulers Thin tape

Ruler s Protractors Art supplies

Resource 1 Problem

Solution

Resource 2 Types of Triangles Cut out the table below and glue it into your book.

a a

3 identical sides 3 identical angles

a

a

a

2 identical sides 2 identical angles

c

no identical sides or angles

a b

Cut out the following triangles and sort into the table you have been given.

Name:

Date:

Sorting Triangles

Right Angle

Obtuse

Acute

Equilateral

Isosceles

Scalene

Resource 3 Triangle Hunt Look around the school and find as many of each triangle as you can. Complete the table as you go. Type of triangle

Where found

Number found

Right Angle

Acute

Obtuse

Scalene

Isosceles

Equilateral

Resource 4 A powerpoint presentation may be downloaded from the following links: PPT: https://drive.google.com/open?id=1B5UzjLLJaisBEnXyK-bzDteusbvpGAYE PDF: https://drive.google.com/open?id=1UYP38I-DXzBE7PU89vCyG333NeSknvcA

Worked Examples – Table Top Maths

Credit: Samantha Allsopp


Worked Examples – Summary Sheet


Worked Examples – Irregular Shapes


Worked Examples – Triangle Page Segments




Worked Examples – Geometric Art


Rationale Summary The activities contained in this learning program are age and ability appropriate for junior and middle secondary students, with all content in line with the Australian and West Australian Mathematics Curriculum.12 This activity is laced with historical relevance, drawing influence from Geometry in art, in the context of the current mathematics curriculum. This resource, as a result, is simply a platform for engaging students with the mathematical concepts in the discipline of Geometry and can easily be adapted to suit ones needs, mathematical focus or narrative. The creation of art using Geometry is intended to imitate historical and modern uses of Geometry, where numerous cultures and styles use Geometry as the basis for creative pieces. The activities are designed to incorporate collaborative learning, creativity and hands-on activities in an effort to implement effective pedagogical practices. Learning and Teaching Strategies For this unit of learning I have decided to focus on 3 main approaches: collaborative learning, implementing creativity and hands-on activities. By focusing on these key teaching and learning strategies, I believe this unit of learning can be effective, engaging and memorable for students, allowing them to have an increased sense of freedom and inventiveness. Collaborative learning is often deemed highly effective and a useful teaching tool, based on the ideas of social constructivism and the fact that students benefit academically, personally and developmentally from working with others.13-16 Mathematics as a subject regularly becomes a practice of memorisation and routine repetition, with students expected to commit formulae or processes to memory, simply so that they may complete their next set of questions. Unlike other subject areas, where students are more routinely exposed to alternative forms of learning, practise and communication, mathematics can often be approached solely as an individual, exercise after exercise regimen, creating an environment that is both monotonous and uninspiring. As a result, many students find mathematics boring, and call for the subject to be more fun and engaging.17,18 Creativity is highly valued in society, even within professional industries, and is something many individuals want to see more of in their education.17,19 Creativity is also seen as important in child development and thus education,14,20 forming one of the WA Curriculum General Capabilities.21 I believe incorporating creativity and into mathematics will increase student interest and satisfaction, ensuring they still practise the required skills and understandings, but enjoy themselves while doing so. Asking students to create something from their learning also features as a higher order thinking skill in Bloom’s Taxonomy, which is a favourable feature of quality learning.16,22 Using a creative medium to reinforce or teach content can be incredibly effective and should not be overlooked as a useful way to improve student outcomes and engagement.20 Integrating art into learning can support multiple learning and cognitive abilities, provide a safe and enjoyable learning environment and allow students to demonstrate or practise understanding of content in meaningful ways.23 Children can be immersed in learning intellectually, emotionally, physically through art, which can increase student engagement, the depth of discussion and participation and thus potentially increasing student outcomes.18,23 In competing against the idea that students often find maths boring, or

unenjoyable, using art to teach and reinforce maths can improve student perceptions of mathematics and help students to see the relevance of maths.18,24 Also, maths and visual art are inextricably linked, particularly in the field of geometry, with both fields experiential and illustrative in nature.25 These two fields can undoubtedly complement one another, as evidenced by the work of Escher and Da Vinci25 and incorporating art into maths can thus increase the relevancy of maths and introduce more authentic, experiential learning into the classroom. The use of hands-on activities within the classroom can increase student understanding and retention, by implementing a more concrete learning approach (which favours knowledge construction) and more student-centred, experiential learning, which is shown to have positive implications for student achievement and development.15,16 Curriculum Links Mathematics The content of this particular unit of learning relates to the Year 7 Western Australian Mathematics Curriculum, under the strand Measurement and Geometry.12 This unit of learning comprises of students creating and defining angles, triangles and utilising formulas to solve problems and find solutions, so is linked directly to the following curriculum codes: ACMMG159, ACMMG165 and ACMMG166. These activities can also be adapted to meet other Year 7 curriculum codes or Year 8 content (see ‘extension/enrichment’), as well as Year 9 Measurement and Geometry. One of the tasks included does relate to irregular or composite shapes, which is a feature of the year 9 curriculum (ACMMG216), however as I have focused solely on the use of triangles and straight lined shapes, I do not believe this will be unusually challenging for students. This unit of learning also corresponds with numerous General Capabilities:21 ⋅

Mathematical content: numeracy

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Summary page: literacy

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Collaborative nature: personal and social capability

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Historical and cultural perspective: ethical understanding and intercultural understanding

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Creative requirements and problem solving: critical and creative thinking

Links to Other Areas Geometry is linked with many other areas of education, industry and culture. One’s ability to calculate area, measure angles and utilise Geometric formulae, properties and reasoning is important in fields such as science, astronomy, agriculture, construction, architecture, design, engineering, technology, surveying, sporting and art. I have chosen to emphasise the role of geometry in art in as this is not only interesting and aesthetically pleasing, but also links to the history and culture surrounding the use and importance of Geometry. Understandings of Geometry, angles and Geometric formulae also corresponds to both junior and senior secondary physics (light, vectors, forces, inclined planes), chemistry (volume, surface area). The cultural, historical and artistic components of Geometry also link with the HASS curriculum (relating to the History strand, looking at the ancient world of Egypt, Greece, Rome, India or China)26 and Arts Curriculum (for example ACAVAM118, ACAVAM121, ACAVAR123, ACAVAR124).27 Considering the culture

and locale of Geometric art, can also lend itself to the consideration of Asian culture and art history, linking to the cross curriculum priority of Asia and Australia's engagement with Asia.28 Extension/Enrichment ⋅

Additional practice: ask students to calculate the perimeter and area of a set number of shapes in their geometric art.

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Historical enrichment and extension: students calculate the area of triangles using Heron’s formula.

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ICT integration: utilise online kaleidoscope or art software.

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Widening the focus: use to demonstrate symmetry (with the use of a Mira mirror).

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Enhancing problem solving: give students a set of right angle triangles, scalene triangles, equilateral triangles and isosceles triangles and ask students to come up with their own definition of each triangle type based on the pattern they observe within the groups (i.e. do not tell students what constitutes each triangle type until after all students have formed their own definition)

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Changing or extending the focus: ask students to incorporate tessellations or reflections in their geometric art (ACMMG181).

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Incorporation of further curriculum links: focus the activities on 3D shapes rather than 2D shapes (ACMMG160, ACMMG161) or incorporate angles and properties associated with parallel and transverse lines (ACMMG163, ACMMG164).

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Year 8: incorporate quadrilateral shapes or prisms and ask students to identify such shapes in their art, including calculations as appropriate (ACMMG196, ACMMG198, ACMMG202).

References 1. Stillwell, J. (2010). Mathematics and its history (3rd ed.). New York, NY: Springer. 2. Merzbach, U. C. & Boyer, C. (2011). A History of Mathematics (3rd ed.). Hoboken: Wiley. 3. Ostermann, A. & Wanner, G. (2012). Geometry by Its History. New York, NY: Springer. 4. Scriba, C. J., & Schreiber, P. (2015). 5000 years of geometry: Mathematics in history and culture. New York, NY: Springer. 5. Verstraelen, L. (2014). A concise mini history of geometry. Kragujevac Journal of Mathematics, 38(1), 5–21. http://dx.doi.org/10.5937/KgJMath1401005V 6. De Young, G. (2009). Diagrams in ancient Egyptian geometry: Survey and assessment. Historia Mathematica, 36(4), 321–373. http://dx.doi.org/10.1016/j.hm.2009.02.004 7. Plato. (1891). Timaeus. In B. Jowett (Ed.), Dialogues of Plato, translated into English with analyses and introductions by B. Jowett (3rd ed.). Retrieved from https://ebooks.adelaide.edu.au/p/plato/p71ti/timaeus.html 8. Bonner, J. (2017). Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction. New York, NY: Springer. 9. Kaplan, C. S., & Salesin, D. H. (2004). Islamic star patterns in absolute geometry. ACM Transactions on Graphics (TOG), 23(2), 97–119. http://dx.doi.org/10.1145/990002.990003 10. The Metropolitan Museum of Art. (2004). Heilbrunn Timeline of Art History: Geometric Art in Ancient Greece. Retrieved from http://www.metmuseum.org/toah/hd/grge/hd_grge.htm 11. Encyclopædia Britannica, inc. (2016). Geometric style. Retrieved from https://www.britannica.com/art/Geometric-style 12. School Curriculum and Standards Authority (SCSA). (2017a). Western Australian Curriculum: Mathematics v8.1. Retrieved from https://k10outline.scsa.wa.edu.au/home/p-10-curriculum/curriculumbrowser/mathematics-v8 13. Bernero, J. (2000). Motivating Students in Math Using Cooperative Learning (Dissertation/Theses). Retrieved from ERIC. (ED446999) 14. Churchill, R. et al., (2016). Teaching: Making a difference (3rd ed.). Milton, Queensland: John Wiley. 15. Duchesne, S. & McMaugh, A. (2016). Educational Psychology (5th ed.). Melbourne: Cengage Learning. 16. Venville, G., & Dawson, V. (2012). The Art of teaching science: for middle and secondary school (2nd ed.). Crows Nest, N.S.W.: Allen & Unwin.

17. Fielding-Wells, J., & Makar, K. (2008). Student (dis) engagement in mathematics. In Annual Conference of the Australian Association for Research in Education (AARE), Brisbane, Australia. Retrieved from https://s3.amazonaws.com/academia.edu.documents/7013497/mak08723.pdf?AWS AccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1507913311&Signature= bBK0PT3lKASaLavGb7R6Sv7zGiI%3D&response-contentdisposition=inline%3B%20filename%3DStudent_dis_engagement_in_mathematics .pdf 18. Granger, T. (2000). Math Is Art. Teaching Children Mathematics, 7(1), 10–13. Retrieved from ProQuest. 19. Adobe Systems Incorporated. (2012). Creativity and Education: Why it matters. Retrieved from http://www.adobe.com/aboutadobe/pressroom/pdfs/Adobe_Creativity_and_Educati on_Why_It_Matters_study.pdf 20. Miller, D. L. (2015). Cultivating creativity. English Journal, 104(6), 25–30. 21. School Curriculum and Standards Authority (SCSA). (2017b). General Capabilities in the Australian Curriculum. Retrieved from https://k10outline.scsa.wa.edu.au/home/p-10-curriculum/general-capabilitiesover/general-capabilities-overview/general-capabilities-in-the-australiancurriculum 22. Anderson, L.W., & Krathwohl, D.R. (Eds.). (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom’s Taxonomy of Educational Objectives. New York, NY: Longman. 23. Lynch, P. (2007). Making meaning many ways: An exploratory look at integrating the arts with classroom curriculum. Art Education, 60(4), 33–38. 24. Kurz, T. L, & Bartholomew, B. (2013). Conceptualizing Mathematics Using Narratives and Art. Mathematics Teaching in the Middle School, 18(9), 552–559. http://dx.doi.org/10.5951/mathteacmiddscho.18.9.0552 25. Bickley-Green, C. (1995). Math and Art Curriculum Integration: A Post-Modern Foundation. Studies in Art Education, 37(1), 6–18. http://dx.doi.org/10.2307/1320488 26. School Curriculum and Standards Authority (SCSA). (2017c). Humanities and Social Sciences. Retrieved from https://k10outline.scsa.wa.edu.au/home/teaching/curriculum-browser/humanitiesand-social-sciences 27. School Curriculum and Standards Authority (SCSA). (2017d). Visual Arts. Retrieved from https://k10outline.scsa.wa.edu.au/home/teaching/curriculum-browser/thearts/visual-arts2 28. School Curriculum and Standards Authority (SCSA). (2017be). Cross-Curriculum Priorities. Retrieved from https://k10outline.scsa.wa.edu.au/home/teaching/crosscurriculum-priorities2/cross-curriculum-priorities