Transformation and Symmetry EME497 Assignment 2 By Tamara Clinch
Transformation and Symmetry One of the three fundamental strands of shape and space in ANational Statement on Mathematics for Australian Schools (Australian Education Council, 1991) Dealing with the mathematical equivalent of changes of position, orientation, size and shape and symmetries in shapes and arrangements (Booker et. al, 2004)
Why are shape and space important? • Visual and spatial skills are essential in modern life and are used for a wide variety of practical tasks. (Booker et. al, 2004) • Concepts of space and shape are vital in describing and navigating our surroundings and designing solutions to many real world problems • E.g. architecture, civil and mechanical engineering, landscape and playground design, theatre set design, town planning, artists and craftspeople, fashion designers, publishers and advertising, packaging and transport….can you name some more?
Tasmanian Curriculum Opportunities to Learn from Mathematics/Numeracy K-10 Syllabus (Tasmanian Department of Education, 2008)
Tasmanian Curriculum Assessment Guidelines from Mathematics/Numeracy K-10 Syllabus (Tasmanian Department of Education, 2008)
Transformation The process by which an original shape is transformed into a new one, can include one or more of: Translation (slide) • Resizing (dilation, contraction, compression, enlargement) • Rotation (turn) Reflection (flip) (Pierce, 2006)
Symmetry Symmetrical shapes look the same when they are folded in half or when they are rotated Bilateral or line symmetry (reflection) Rotational symmetry (Pierce, 2006)
Classroom Activities A variety of activities are outlined that can be adapted to different ability levels within the same classroom. Lesson ideas which incorporate these and other activities are outlined in the next section. Activities build on understanding about space and shape as they become more complex. Concept Key • shown on each activity • highlight the main concepts addressed by this activity • the various activities also emphasise literacy , visual art, thinking and working mathematically, other maths concepts, ICT and more, however the concept key is limited to the subject area at hand. Bilateral Symmetry Rotation Pattern Making Rotational Symmetry Translation Reflection
Basic Paper Folding (Willis et. al., 2005, p.104, 108) • There are a variety of paper folding and paint printing (e.g. butterfly prints) or paper folding and cutting activities that can be conducted in a classroom • It is important to ask children to think about what they want to achieve before they paint or cut and what they think their painted shapes or cuts will look like after they unfold the paper so that they begin to visualise the results of their work. • For later stages, they can ‘design’ their work on a template before actually doing the painting or cutting to see if their cuts resulted in the shapes they wanted. (Mom Unplugged, 2008)
More Paper Folding Snowflakes (Willis et. al., 2005, p.108, 112) or use online instructions (Libbrecht, 1999) • Start with four folds and progress to more folds to create more complex patterns • Talk about which edges of the folded shape make which parts of the finished snowflake. This online snowflake (Popular Front, 2002) is great for enabling students to figure it out themselves. • Ask students to design their snowflake by drawing their intended cuts on a template of the folded shape. Ask if the snowflake turned out how they thought it would. • Display the designs with the snowflakes and, if possible, display the paper that was cut away (better for simpler versions) and encourage students to match cut away parts with snowflake and justify their choice.
Broken Windows(Willis et. al., 2005, p. 107) • Show drawings of broken windows and provide a choice of different-shaped pieces to fit in the hole. Explain that filling the hole with the right piece of glass can repair the window. • Encourage students to predict which piece will be required to fix each window and explain why. • Ask: How would you move that piece so it would fit in the hole? • Invite students to check they chose the correct piece by placing it over the hole.
Patterns (Willis et. al., 2005, p. 106, 121) • Have students visualise the effect of reflecting figures. • Invite them to draw half of a square (cat, boat) on card so that the edge of the card forms the mid-line of the drawing. Have them cut the half figure out, predict and draw what the whole figure will look like. • Encourage them to check by reflecting and tracing to make the whole figure. Repeat this to create a border or sequence of objects for a picture. • Which figures are reflections of the original shapes? How can you tell? • Have students construct a pattern on a tile measuring 9 cm by 9 cm by marking dots 3 cm apart on each edge and then joining the dots in some way. Then have them make 5 more tiles exactly the same. • Invite students to arrange them in a 2 x 3 array to make a pattern. • Ask: Are the squares a translation, rotation or reflection of the original?
More Patterns(Willis et. al., 2005, p. 109, 124) • Have students fold and cut paper to create different templates for stencilling and decide which designs will be more suitable for borders or corners than others. • Invite students to test their pattern or arrangement (perhaps using grid paper) before completing the final product. • Ask: What would your design look like if you put the stencil together in different ways (turning it sideways, turning it over)? How can you move the shape to make it fit into the corners better? • Encourage students to predict and draw other designs. • Invite students to select a pattern from a collection of border patterns, trace the pattern unit or motif and record in words how that unit is transformed. • Ask them to swap with a partner and use the tracing and the description to re-create and draw the border. Encourage partners to compare re-creations with the original borders and explain any differences. Ask: Was there enough information? What other information would be required to re-create the original design? Was it necessary to describe the distance between figures? Did you need to be more specific about the distance between the figures or the amount of turn or both?
Turning Things (Willis et. al., 2005, p. 145) • Ask students to list the things they have turned during the day (e.g. door handle, tap). Then have them draw a diagram of one of the things they have turned, mark its point of rotation and make a template. • Invite them to place the template over the original object and turn it around the point of symmetry. • Ask: In how many places in a full turn can this object be matched? How do you know? Draw out that for some things there is an infinite number of matching positions of symmetry and for others there is not.
3D Symmetry • Students use “Wedgits” or other 3D blocks to build their own creations that have at least one line of symmetry. • Students then designate the angle/direction from which a digital photo is taken to record their creation. • When photos are printed, students draw the line/s of symmetry on the photos for display. (Clinch, 2008)
Extension/Assessment Tasks Venn Diagram (Willis et. al., 2005, p. 147) Have a large Venn diagram drawn on the floor and labelled ‘translation’, ‘reflection’ and ‘rotation’. Ask students to sort a range of symmetrical objects and patterns according to the type of symmetry. Ask: Where would you place a pattern or a figure that has translation and reflection symmetry, but not rotational symmetry? Where would you place a kite (patterned plate, glass, plain t-shirt)? Design Brief (Willis et. al., 2005, p. 146) Have students produce specified symmetrical designs from a given figure. For example, ask them to design a reflecting floral border no wider than 3 cm around a piece of A4 paper. Invite them to choose from using folded paper, circular and square grid paper, or computer graphics. Encourage students to say what movements they used to produce their particular design. World Art (Willis et. al., 2005, p. 148) Invite students to research and collect examples of art and designs from around the world that exhibit reflection symmetry (e.g. Aboriginal rock art, Maori carvings, Italian sculpture, Escher designs). Encourage them to examine differences between how the figure is reflected over the mirror line. Ask: Are they reflected vertically or horizontally? Is the mirror line on an angle? Is the figure next to or apart from the mirror line? Have students make a display with written descriptions about the symmetry used in each.
Flexible Lesson Ideas Increasing Difficulty • Paper Folding & Painting • Butterfly/ inkblot paintings • Folded paintings of recognisable objects or patterns • Specifically designed folded painting of more than one object, a complex pattern or a scene Paper Folding & Cutting Cutting simple shapes on the fold line Cutting more complex shapes and predicting the outcome Designing then cutting a continuous frieze Cutting four pointed snowflakes Designing then cutting six pointed snow flakes
Flexible Lesson Ideas Increasing Difficulty • Tangrams & Pattern Blocks • Identify regular polygons shown in different transformations • Perform transformations (e.g. rotate triangle and discuss how it has changed) • ‘Broken Windows’ activity • Create simple figures then create same figure in different position. Describe the • transformations of shapes to make second figure. • As above, but communicate transformations to another student who can’t see the completed figure – see if they can create the same figure.
Flexible Lesson Ideas Increasing Difficulty Application of Knowledge 3D Symmetry activity Identify examples of various transformations and symmetry on regular polygons ‘Turning Things’ activity on rotational symmetry Show images of symmetry and transformation in patterns and objects and ask class members to identify elements they can see (see photo of ‘Whole Class Symmetry Activity) Complete a transformations or symmetry search in school yard, home or community and describe real world examples ‘Venn Diagram’ activity
Flexible Lesson Ideas Increasing Difficulty Patterns Identify symmetry and transformations in simple patterns Create simple patterns using a single type of transformation (e.g. 'Patterns’ reflection border pattern activity) Identify multiple transformations and/or symmetries in complex patterns (e.g. 'Patterns’ tile making activity) ‘World Art’ activity Create more complex patterns with multiple transformations and/or symmetries (‘More Patterns’ border pattern stencils activities) Describe the transformations and/or symmetries in a pattern using appropriate vocabulary so that another student can re-create the pattern ‘Design Brief’ activity
Symmetry DisplayShowing Grade 4 Snowflakes with their designs and 3D Symmetry (Clinch, 2008)
Whole Class Symmetry Activity Grade 4 students show examples of symmetry they can find in images of complex patterns displayed on interactive whiteboard. (Clinch, 2008)
Online Activities • Use the links already demonstrated to explore concepts of transformation and symmetry with online activities • Explore symmetry by building a variety of quilt blocks at this Teachers’ Lab site(Annenberg Media, 1997). • The ‘really hard’ level of this flash game, ‘The Shape Lab’, is all about symmetry (BBC, 2008). • This online pattern block applet (Archytech, 1998) is very useful for a variety of tasks and this site (Moore, 1994)has some great suggestions of lesson plans and activities for pattern blocks. • Symmetry Artist (Pierce, 2006)is a great online tool which creates symmetrical artwork.
Summary • Visual and spatial skills are essential in modern life and are used for a wide variety of practical tasks. (Booker et. al, 2004) • Transformation and symmetry are important concepts of shape and space and classroom activities that develop these concepts also develop visual and spatial skills of students. • This often neglected area of maths can lay the foundation for understanding complex geometry in later years and can also link maths to other curriculum areas and make it more accessible to students with a variety of learning styles.
Questions? • Any questions? • Any other ideas or suggestions that you have found worked well in your classrooms? • Resources available in hard copy or online at http://trclinch.edublogs.org
References Annenberg Media. (1997) Space and Shape in Geometry: Quilts on Teacher’s Lab website accessed on 10 Oct 08 at http://www.learner.org/teacherslab/math/geometry/shape/quilts/index.html Archytech (1998) Pattern Block Applet accessed on 10 Oct 2008 at http://www.arcytech.org/java/patterns/patterns_j.shtml Australian Education Council. (1991) A National Statement on Mathematics for Australian Schools. Curriculum Corporation. Carlton, VIC. BBC (2008) Shape Lab game on BBC Schools Bitesize Numeracy website accessed on 10 Oct 2008 at http://www.bbc.co.uk/schools/ks1bitesize/numeracy/shapes/index.shtml Booker, G., Bond, D., Sparrow, L. & Swan, P. (2004) Teaching Primary Mathematics. Pearson Education. Sydney, NSW. Clinch, T.R. (2008) Photographs of in class activities and student work for 3D Symmetry and Snowflakes taken at St. Michael’s Collegiate School , Hobart, during professional experience with grade four. Mom Unplugged. (2008) Unplug Your Kids. Weblog accessed on 10 Oct 08 at http://www.unplugyourkids.com/2008/06/01/paint-weekly-unplugged-project/ Moore, V. (1994) Primary Math Activities on Drexell University Math Forum website accessed on 10 Oct 08 at http://mathforum.org/varnelle/kgeo.html Libbrecht, K. (1999) Activities for Kidson SnowCrystals.com viewed on 10 Oct 08 at http://www.its.caltech.edu/~atomic/snowcrystals/kids/kids.htm Pierce,R. (2006) Translation and Symmetry on MathsIsFun.com accessed on 10 October at http://www.mathsisfun.com/geometry/index.html Popular Front. (2002) Create Your Own Snowflake online game viewed on 10 Oct 08 at http://www.popularfront.com/snowdays/ Tasmanian Department of Education. (2008) Tasmanian Curriculum: Mathematics –numeracy K – 10 syllabus and support materials. Hobart, TAS. Willis S., Devlin, W., Jacob, L., Powell, B., Tomazos, D., Treacy, K., Hardman, C., and Sherrard, P. (2005) First Steps in Mathematics: Space. Rigby. Melbourne, VIC.