Resource 4: Examples of symmetry in art and fabricsTeacher resource for planning or adapting to use with pupilsExamples of symmetry in Islamic artExamples of symmetry in Kenyan fabric patternsThe Kenyan flag is also symmetrical. Can your class find any other symmetrical flags?Resource 5: PolygonsPupil useYou will need to use the following 2D polygons:PentagonOctagonSquareSeptagonTriangleHexagonResource 6: Recording symmetryPupil use

Object

Lines of symmetry

Order of rotational symmetry

AcknowledgementsGrateful acknowledgement is made to the following sources:OtherResource 1 : Examples of symmetry found in nature: Original source: http://creative.gettyimages (Accessed 2008)Resource 2 : Examples of symmetry in Kenyan masks : Original sources: http://www.globalcraftsb2b.com/catalog/index.php?cPath=46 (Accessed 2008)http://www.soulyafrica.com/images_products/kenyan_mask_thumb.jpg (Accessed 2008)http://www.back2africa.com/products.php?pid=5 (Accessed 2008)Resource 4 : Examples of symmetry in art and fabrics : Original sources:Islamic arthttp://www.islamicarchitecture.org/art/images/ (Accessed 2008)http://blog.vcu.edu/arts/images/ (Accessed 2008)http://home.earthlink.net/ (Accessed 2008)http://artfiles.art.com/images/ (Accessed 2008)Kenyan fabric patternshttp://www.fotosearch.com/SIX010/ken-010/ (Accessed 2008)http://www.imaniworkshops.org/handbags.html (Accessed 2008)Every effort has been made to contact copyright holders. If any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangements at the first opportunity.Section 5: Teaching transformationsKey Focus Question: How can you develop confident mental modelling in geometry?Keywords: congruence; translation; reflection; transformation; multigrade; differentiation; practicalLearning OutcomesBy the end of this section, you will have:introduced pupils to transformation, congruence, translation and reflection;used cut-out shapes as a means to develop the mental transformation of geometric shapes;considered the challenges of differentiating this work for older and younger pupils, and tried some different approaches.IntroductionIn our daily lives we see many examples of shapes that have been modified (changed) or transformed.This section will help you develop your own subject knowledge about geometry and transformation, as well as your skills in developing your pupils’ understanding. Most of the resources in this section, therefore, are to support your subject knowledge as a teacher of mathematics.1. Using practical workIn geometry, ‘transformation’ means altering some geometric property of a shape, (such as rotating it or moving its position on the page) while keeping other properties of the shape the same (we say the shapes are ‘congruent’).An excellent way for pupils to model transformation is by using physical objects or looking at shapes in everyday life and how they are transformed e.g. in fabric patterns. While pupils are doing this, encourage them to talk with you and each other about what they are doing. Talking about how they are trying to manipulate the objects will improve their understanding of geometry and the language associated with it.Case Study 1: Planning a lesson in geometry with a colleagueMrs Ogola, a teacher in a primary school in Masindi, Uganda, was discussing her experience in teaching geometry to her pupils with a senior associate, Mrs Mwanga. She complained that pupils do not like this topic. Her pupils complained that geometry is very abstract, requiring much imagination. Apart from that, it bears little or no relation to real life. Therefore, she herself was not always enthusiastic about teaching it.Mrs Mwanga admitted to similar experiences, but encouraged her to try using a practical investigative approach and to encourage her pupils to talk about what they were doing. Together they planned a lesson in which pupils would carry out step-by-step activities using samples of fabrics with patterns that contain translations and variations of shapes (see Resource 1: Some Tanzanian fabric patterns). This can lead to pupils discovering the concepts to be learned themselves.Mrs Mwanga and Mrs Ogola both taught the lesson to their classes and then met afterwards to discuss how it went. Mrs Ogola was surprised at the level of her pupils’ thinking and how much they wanted to talk about what they were doing. Mrs Mwanga agreed that allowing pupils to talk about their work not only excited them, but also gave them confidence in their ability to do mathematics.Activity 1: Investigating congruent shapes To complete this activity, you will need a piece of cardboard and a pencil and ruler for each pair or small group of pupils, and several pairs of scissors.Ask your pupils to draw three different straight-sided shapes on their card and then cut out their shapes. They should number each of their cardboard shapes 1, 2 or 3. Next, on a separate piece of paper, ask your pupils to draw around each shape; then move the shapes any way they like without overlapping what they have already drawn, and draw round them again. Repeat this until the page is full of shapes, then label inside each outline with a letter (e.g. a, b, c…). (The finished work should be similar to Resource 2: Examples of congruent shapes.)Ask pupils to swap their work with another group. Can they find the outlines that were made with the same shape? (Younger children might need to use the cardboard shapes to help them.) Ask them to write down what they think is the answer – e.g. shape 1, outlines a, b, d, g.Using the cut-out shape, can they show you what has to happen in moving from one outline to another? Can they describe this in their own words?Early finishers can colour in their work, using one colour for outlines from the same shape. You could display these on the classroom walls, headed ‘Congruent shapes’.2. Differentiating workOne of the simplest transformations is translation. To translate a shape, we simply move its position on the page, up or down, left or right, but do not change the shape in any other way (see Resource 3: Translation).Because translating a shape is simple, even very young pupils can grasp the idea, especially if they have physical shapes to manipulate. For older pupils, the activity can be made more challenging by using x-y coordinates and calculation, rather than manipulating physical shapes.Case Study 2 and Activity 2 look at translation and how to differentiate tasks according to age and stage.Case Study 2: Extending understanding of translationMrs Kiboa teaches a multigrade class in which she has a group of older children who are doing well at mathematics. Feeling their current work had not been stretching them enough, Mrs Kiboa took an opportunity to let them enjoy a real challenge. (For more information on teaching multigrade classes, see Key Resource: Working with multigrade classes.)Mrs Kiboa had already introduced x-y coordinates to the whole class. One day, while most of the class were working on a triangle translation activity using cut-out shapes, Mrs Kiboa gave these four pupils extra support (see Resource 4: Translating and reflecting triangles).Drawing a triangle with labelled x-y axes on some grid paper, she asked the pupils what the coordinates of the three corners (vertices) were – they answered easily, and wrote their answers down. Next, she asked them, ‘What would happen if I were to move the shape six spaces to the right? What would the new x-y coordinates be?’ When they had answered correctly, she went on: ‘And if I moved the shape 3 spaces down?’ Mrs Kiboa went on in this fashion until she felt the pupils clearly understood what was happening.Next, she said to them, ‘Now, each of you set one another a problem – give coordinates for a triangle, and a translation to apply to the triangle. Write this down, then draw the triangle you have been set, calculate the translated coordinates, and draw the new position. If you do this correctly, you may then try shapes other than triangles to test each other with.’The pupils enjoyed the respect of their teacher, as well as the opportunity to work more freely and to challenge each other mathematically.Activity 2: Investigating translations practicallyMake sure pupils understand how to give x-y coordinates, through whole-class teaching. To differentiate the task for older or younger pupils, see the notes on differentiation in Resource 4.Ask pupils to draw and cut out a triangle, square and rectangle from a piece of squared paper: emphasise that each corner (or vertex) of their shapes should be at one of the ‘crosses’ on their grid paper by drawing an example on the board. No side should be more than 10 squares long.On a second piece of grid paper, ask pupils to draw and label x-y axes at least 20 squares long (see Resource 4).Putting one of their cut-out shapes on the paper so that its corners are on the ‘crosses’ of the grid, they should mark the vertices (a, b, c & d as appropriate), then draw the shape and write down the coordinates of each vertex.Ask them to move their shape to a new position (keeping it the same way up) and repeat this process.Ask your pupils: ‘What happens to the x coordinates between the two positions? Does the same thing happen to each coordinate? What happens to the y coordinates?’What parts of this activity caused difficulty for your pupils? How will you support them next time?3. A practical approach to ‘reflection’Translation is relatively simple, because it affects the coordinates of all vertices in the same way (for example, all x coordinates will increase or decrease by the same amount).Reflection is more mathematically complex, because you must treat each coordinate separately and in relation to another item – the location of the mirror line. Reflection therefore requires pupils to hold quite a number of different ideas in their minds at the same time (see Resource 4).Think about what familiar examples of reflection you might be able to use to help your pupils with this topic – perhaps some work you may have done on symmetry or patterns and designs in art using local traditional ideas. Consider how pupils could use cut-out shapes as they develop the ability to manipulate such shapes mentally.In addition, this part suggests you continue to encourage pupils to discuss their thinking – an important key in unlocking their understanding of mathematics.Case Study 3: Using group work to help think about reflectionsMrs Nkony, an experienced teacher in a primary school in Kilimanjaro, has taught the basics of reflection to her class. She now decides to help them discuss their activity and their findings.She knows that discussion is not merely answering short, closed questions, so she decides to set up a structure to help discussion among her pupils. She arranges them into pairs. They are asked to look at each other’s work, and make three observations about reflection that they will report back. For each observation, they must both be happy that they have found a way to describe or explain it as clearly as they can. When both members of the pair are in agreement that they have three clear observations, they are to put their hands up.Mrs Nkony then puts the pairs together to make fours, asking each pair to explain their observations to the other. She then asks the fours to decide on the three best or most interesting observations to feed back to the class.She realises that she could use this way of working in lessons other than mathematics. To find out what your pupils know and can do see Key Resource: Assessing learning.Key Activity: Thinking about reflectionsYour pupils could reuse the shapes they cut out of grid paper for Activity 2, or make some more if necessary.On a second piece of grid paper, ask pupils to draw and label x-y axes at least 20 squares long (see Resource 4).Putting one of their cut-out shapes on the paper so that its corners are on the ‘crosses’ of the grid, they should mark the vertices (a, b, c & d as appropriate) then draw the shape and write down the coordinates of each vertex (corner).Ask pupils to draw a vertical or horizontal mirror line on their grid. They should then draw the reflection of the shape on the other side of the mirror line (remind pupils that they may use the cut-out shape if it helps them) and write down the coordinates of the reflection.Challenge your pupils to work out the reflection coordinates without using the cut-out shape. Ask them to explain how they did it. Practise using lots of shapes so that pupils become confident.How well did you introduce and explain this work? How do you know this?Resource 1: Some Tanzanian fabric patternsTeacher resource for planning or adapting to use with pupilsResource 2: Examples of congruent shapesTeacher resource for planning or adapting to use with pupilsIf two shapes are congruent, they are identical in both shape and size.QuestionWhich of the following shapes are congruent?Answers:A and GD and IE and JC and HRemember that shapes can be congruent even if one of them has been rotated (as in A and G) or reflected (as in C and H).Resource 3: TranslationBackground information / subject knowledge for teacherYou will find a useful resource on the website below.http://www.bbc.co.uk/schools/ks3bitesize/maths/shape_and_space/transformations_1_2. shtmlThis website provides background information on translations, transformations and reflections as well as interactive activities over four pages that you can do to explore the concepts and ideas involved.If we translate a shape, we move it up or down or from side to side, but we do not change its appearance in any other way.When we translate a shape, each of the vertices (corners) must be moved in exactly the same way.Which of the following shapes are translations of triangle A?Answer: D and E are translations of triangle A.Resource 4: Translating and reflecting trianglesBackground information / subject knowledge for teacherx-y coordinates always give the ‘x’ (horizontal axis) value before the ‘y’ (vertical axis) value.So, in the illustration, the x-y coordinates for abc:a = 4, 8b = 4, 2c = 2, 2The translation to a_{1}b_{1}c_{1} increases the value of x by 12, and y by 9. So:a_{1} = 16, 17b_{1} = 16, 11c_{1} = 14, 11DifferentiationThis can be made simple, by moving a cut-out shape around the grid, drawing around it and recording the new coordinates.This can be made more challenging by giving coordinates for a shape and asking pupils to draw the shape. Then say how a translation affects the x-y values, and ask them to work out the new coordinates and redraw the position of the shape.In the illustration, the x-y coordinates for abc are:a = 4, 8b = 4, 2c = 2, 3Reflecting abc in a vertical ‘mirror line’ (x=8) gives an image (a_{1}b_{1}c_{1}) at new coordinates:a_{1} = 12, 8b_{1} = 12, 2c_{1} = 14, 3NoteThe object and its image are always at the same perpendicular distance (distance measured at right angles) form the mirror line, e.g. if ‘a’ is 4 squares from the mirror line, ‘a_{1}’ must also be 4 squares from the mirror line.Compare the x-y coordinates of abc and a_{1}b_{1}c_{1} and observe that a vertical mirror line leaves the y coordinates unchanged.Similarly, a horizontal mirror line would leave the x coordinates unchanged.Original Source: http://www.bbc.co.uk/schools (Accessed 2008)AcknowledgementsGrateful acknowledgement is made to the following sources:OtherResource 2 : Examples of congruent shapes:Original Source: http://www.bbc.co.uk/schools/ (Accessed 2008)Resource 3 : Translation :http://www.bbc.co.uk/schools/ks3bitesize/maths/shape_and_space/transformations_1_2. shtml (Accessed 2008)Resource 4 : Translating and reflecting triangles :Original Source: http://www.bbc.co.uk/schools (Accessed 2008)Every effort has been made to contact copyright holders. If any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangements at the first opportunity.