Imaginings in shape and space TYPE: Main OBJECTIVE(S): DESCRIPTION: OVERVIEW: EQUIPMENT: Begin to identify and use angle, side and symmetry properties of triangles and quadrilaterals; solve geometrical problems involving these properties, using step-by-step deduction and explaining reasoning with diagrams and text; use 2-D representations to visualise 3-D shapes and deduce some of their properties. 3 screens. 1 shows a square which is 'cut' to make other shapes (tangram). 2 shows 8 shapes and you are asked which could be views of a square. 3 shows a square with beads constrained to move on its edges where you have to make shapes. Tangrams, square shapes and shapes in a square. A search of Google (www.google.co.uk) using 'Tangram' will give many interesting sites, both in the UK and abroad. A balloon with a square drawn on it may be useful for screen 2. Geogebra is very useful for screen 3 where you can also use pinboards or geoboards. Commercial tangrams are available (magnetic ones can work well). Teacher notes include a photocopiable master of an 8 cm by 8 cm and 16cm by 16 cm tangram and another two masters give the silhouettes for the shapes here based on the 8 by 8 cm square. A final master provides 5 grids to draw shapes for the 'inside a square' activity. TYPE: OBJECTIVE(S): DESCRIPTION: OVERVIEW: EQUIPMENT: Plenary Begin to identify and use angle, side and symmetry properties of triangles and quadrilaterals; solve geometrical problems involving these properties, using step-by-step deduction and explaining reasoning with diagrams and text; use 2-D representations to visualise 3-D shapes and deduce some of their properties. 4 screens. 1 works through constructing Tangram pieces on a square grid. 2 works through constructing Tangram pieces by folding a square piece of paper. 3 allows pupils to construct polygons by moving 4 connected, variable points on a square frame. Tangrams, square shapes and shapes in a square. None specific. Imaginings in shape and space... 1 Main Whiteboard and Screen information... 2 Plenary Whiteboard and Screen information... 5 Tangrams 1... 9 Tangrams 2... 10 Tangrams 3... 11 Inside a square... 12 Spire Maths interactive files available in a flash format at: https://spiremaths.co.uk/ia/ Unfortunately they will not work on ipads or iphones. http://jamtecstoke.co.uk/ Page 1 of 12
Main Whiteboard and Screen information Screen 1: The tangram A square made of seven parts is shown on screen. After a short while the seven parts move to the edge one at a time to form a candle with flame. You are then shown the silhouette of a runner. You can drag and drop the pieces onto the runner to make it without overlap. Pieces are dragged by holding at the centre and rotated by dragging near the corner. There are 4 other shapes a cat, cat's head, house and figure three. The last of these can create a problem if pupils place the square at the top of the shape since the parallelogram then cannot be put there (and this is where it belongs) - this may not be a problem if pupils have cut out shapes since the parallelogram can then be turned over and will fit at the bottom of the shape. Key points: pupils enjoy making and using their own tangram (the instructions for folding and drawing are in the plenary); have pupils discuss this and try to visualise how to do it without any of the parts; some pupils will require the silhouettes visible to be able to complete these; some pupils will need the exact size silhouettes, some will manage with half size silhouettes while a few will be able to get by without either; one or two pupils may be able to visualise how to put the shapes together and do it very quickly. http://jamtecstoke.co.uk/ Page 2 of 12
Screen 2: Square shapes You are given 8 shapes: rectangle, ellipse, trapezium, kite, isosceles trapezium, circle, rhombus and square (45 degrees from the horizontal) and asked to state which could be a view of a square. When you have decided the shapes are moved to the side and it is suggested that you use a square with the data projector to attempt to make the given shapes. It is not easy to show any of these with the data projector, so practice first. For one or two of the shapes you may be able to draw a square on the screen with the pen and then use the 'keystone' features of your projector. Key points: do not move too quickly from the initial screen since you want pupils to think about the problem and how to try to demonstrate that any of these might be a view of a square; discussion is the key to this; some of your pupils may be aware of the distortion caused by an overhead projector when you project things well above the level of the projector lens; many pupils may be happy to exclude the ellipse and circle, but you could ask what the square would look like drawn on a balloon and then blown up (and then do it); all the rest are possible, but it does depend on the square (whether it is rigid or not); note that when you use anything square shaped with the projector you are probably technically using a square based prism. http://jamtecstoke.co.uk/ Page 3 of 12
Screen 3: Inside a square A 6 by 6 square is shown. On each side of the square is a bead (a moveable point) that is constrained to move along the one side of the square. Each bead starts at the midpoint of the side and each bead is joined to the two beads that are closest to it (so they are joined to make a square at the start). Each bead can be moved independently along its side. You are asked to make and describe the shapes you have made. Key points: have pupils discuss this and try to visualise what shapes can be made; some pupils will find this easy, others will not; you may wish to set the task asking which quadrilaterals can be made (and how); also which triangles can be made (once someone has suggested that this is possible); all quadrilaterals except the arrowhead can be made - though some pupils will have problems with another square (excluding the 6 by 6 and the start) and a rectangle; in all cases ask 'how do you know it is what you say it is' to help improve reasoning; a good individual activity on Geogebra; you can also use pinboards or geoboards to help with this; you may want pupils to record the work, but it is much less interesting if you just ask pupils to do it on paper and pupils are likely to gain less from it. http://jamtecstoke.co.uk/ Page 4 of 12
Plenary Whiteboard and Screen information Screen 1: Drawing a tangram Pupils are invited to work through the construction of a tangram on an 8 by 8 square grid. An animated construction, superimposed on the grid, is revealed by clicking '>' The animation can be stopped and started at any time by using '>' as a toggle. A line by line solution can be revealed by using '>>' and a complete solution by using '> '. Steps can be reversed in a similar manner. At each stage, brief notes explain what is happening in the animation. The use of colour highlights the work in process. At the end the 7 piece tangram is shown on the screen, all pieces coloured differently and without grid lines. Key points: pupils enjoy making and using their own tangram and should be encouraged to work through the process; accuracy and discussion of symmetry should be encouraged. http://jamtecstoke.co.uk/ Page 5 of 12
Screen 2: Folding a tangram Pupils are invited to work through the construction of a tangram by folding an 8 by 8 square. An animated construction, representing the various folds of the paper, is revealed by clicking '>' The animation can be stopped and started at any time by using '>' as a toggle. A line by line solution can be revealed by using '>>' and a complete solution by using '> '. Steps can be reversed in a similar manner. At each stage, brief notes explain what is happening in the animation and clarify the folding process. The use of colour highlights the work in process and the nature of the folds. At the end the 7 piece tangram is shown on the screen, all pieces coloured differently. Key points: pupils enjoy making and using their own tangram and should be encouraged to work through the process; accuracy and discussion of symmetry should be encouraged. http://jamtecstoke.co.uk/ Page 6 of 12
Screen 3: Inside a square A 6 by 6 square is shown. On each side of the square is a bead (a moveable point) that is constrained to move along the one side of the square. Each bead starts at the midpoint of the side and each bead is joined to the two beads that are closest to it (so they are joined to make a square at the start). Each bead can be moved independently along its side. You are asked to make and describe the shapes you have made. Key points: have pupils discuss this and try to visualise what shapes can be made; some pupils will find this easy, others will not; you may wish to set the task asking which quadrilaterals can be made (and how); also which triangles can be made (once someone has suggested that this is possible); all quadrilaterals except the arrowhead can be made - though some pupils will have problems with another square (excluding the 6 by 6 and the start) and a rectangle; in all cases ask 'how do you know it is what you say it is' to help improve reasoning; a good individual activity on either Geometer's SketchPad or Cabri Geometre; you can also use pinboards or geoboards to help with this; you may want pupils to record the work, but it is much less interesting if you just ask pupils to do it on paper and pupils are likely to gain less from it. http://jamtecstoke.co.uk/ Page 7 of 12
Screen 4: Vocabulary Vocabulary present: 2D, 3D, Axes, Circle, Concave, Construct, Convex, Cube, Cuboid, Cylinder, Delta, Distance, Draw, Edge, Equilateral, Face, Hemisphere, Irregular, Isosceles, Kite, Measure, Net, Parallelogram, Perpendicular, Polygon, Prism, Protractor (angle measurer), Pyramid, Quadrilateral, Rectangle, Regular, Rhombus, Right-angled, Ruler, Scalene, Set square, Sketch, Sphere, Square, Tetrahedron, Trapezium, Triangle, Vertex, Vertices. Spire Maths interactive files available in a flash format at: https://spiremaths.co.uk/ia/ Unfortunately they will not work on ipads or iphones. http://jamtecstoke.co.uk/ Page 8 of 12
Tangrams 1 http://jamtecstoke.co.uk/ Page 9 of 12 https://spiremaths.co.uk/ia/
Tangrams 2 http://jamtecstoke.co.uk/ Page 10 of 12 https://spiremaths.co.uk/ia/
Tangrams 3 http://jamtecstoke.co.uk/ Page 11 of 12 https://spiremaths.co.uk/ia/
Inside a square Shape name Shape properties Shape name Shape properties Shape name Shape properties Shape name Shape name Shape properties Shape properties http://jamtecstoke.co.uk/ Page 12 of 12 https://spiremaths.co.uk/ia/