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Tangrams: making shapes Make a tangram by drawing an 8 x 8 square and cutting out the 7 shapes as shown below. Tangrams: making shapes Use all 7 shapes each time; they can be rotated and flipped over if needed. Individual pieces must not overlap each other. Try to make the following: • A rectangle • A large isosceles triangle • A parallelogram • An isosceles trapezium • An irregular pentagon …with a line of symmetry • An irregular hexagon …with 2 lines of symmetry Tangrams: Square Make a tangram by drawing an 8 x 8 square and cutting out the 7 shapes as shown below. Tangrams: Square Use all 7 shapes each time; they can be rotated and flipped over if needed. Individual pieces must not overlap each other. How many different ways can the large square be made? Tangrams: area Make a tangram by drawing an 8 x 8 square and cutting out the 7 shapes as shown below. Find the area of each of the individual shapes. Tangrams: fractions Look at the tangram below. What fraction of the original square is each of the individual shapes? Tangrams: perimeter Make a tangram by drawing an 8 x 8 square and cutting out the 7 shapes as shown below. Find the perimeter of each of the individual shapes. Tangrams: perimeter 2 Make a tangram by drawing an 8 x 8 square and cutting out the 7 shapes as shown below. Tangrams: perimeter 2 Use all 7 shapes each time; they can be rotated and flipped over if needed. Individual pieces must not overlap each other. Make several different shapes and find their perimeters. What is the largest perimeter you can make? What is the smallest perimeter you can make? Tessellations A tessellation is a regular pattern made of tiles placed so that it can continue in all directions. Every bit of the pattern should be repeated. Anyone looking at the pattern should be able to see exactly how it will continue. Some tessellations are created using just one shape, others use two or more shapes Which of the patterns on the following slides are tessellations? Quadrilateral Tessellations Which quadrilaterals tessellate? Try using one of the following as a tile: • Square • Rectangle • Parallelogram • Rhombus • Trapezium • Kite • ‘Irregular’ quadrilateral Regular Tessellations A regular tessellation is one in which only one regular polygon is used. Tiles must all be the same size and have to be placed edge to edge. How many regular tessellations are there? Prove that there can be no others. Semi-regular tessellations A semi-regular tessellation is one in which more than one regular shape is used. How many different ones can you find? Given that the largest number of sides of a polygon in a semi-regular tessellation is 12, can you prove that no others are possible? Teacher notes The idea of this month’s classroom resource is to take a simple starting point and use it in several ways to address different aspects of the curriculum. It is not expected that this is a sequence of lessons as different activities will be most suitable for different groups of pupils. Additionally, many of these activities would make for good sources of display work, just in case anyone’s thinking about school ‘open days’… Tangrams: overview This ancient puzzle can be used in a variety of ways, traditionally: • To recreate given shapes • To create new shapes However, additional activities and questions are: • Area: Given that the side length of the original tangram is 8 units, find the area of each of the smaller shapes making up the tangram • How many different ways can all 7 shapes be used to make the square. Rotations and reflections not permitted. • What fraction of the original shape are each of the 7 pieces? • If the original tangram has side length 8 units, what are the side lengths (and/or perimeters) of each of the 7 pieces? • Make shapes using all 7 pieces and find their perimeters. What is the smallest perimeter that can be made using all 7 shapes? What’s the maximum perimeter that can be made using all 7 shapes Tangrams This ancient puzzle can be used in a variety of ways, traditionally: • To recreate given shapes • To create new shapes The usual rule is that pieces cannot overlap, sometimes an additional rule is given that all adjacent shapes must meet at an edge/ partial edge. This activity is accessible for almost all pupils Some answers, although there are probably several possibilities for each: Tangrams Area: Given that the side length of the original tangram is 8 units, find the area of each of the smaller shapes making up the tangram. Suitable for most KS3 pupils, can be solved by: • Counting squares • Reasoning - using the smaller square and deducing that each of the smaller triangles is half the area and then using combinations of these to physically recreate the larger shapes • Finding what fraction of the whole each piece represents • Using formulae Tangrams How many different ways can all 7 shapes be used to make the square. Suitable for most KS3 pupils Just one arrangement, but there are several rotations and reflections that can be created. Ask pupils to make the square in as many different ways as they can and use their responses to initiate discussions about ‘same and different’ and transformations as an introductory activity for reflections and rotations. Tangrams What fraction of the original shape are each of the 7 pieces? Tangrams: perimeters If the original tangram has side length 8 units, what are the side lengths (and/or perimeters) of each of the 7 pieces? • • • • • • • A B C D E F G Tangrams: perimeters 2 Make shapes using all 7 pieces and find their perimeters. What is the smallest perimeter that can be made using all 7 shapes? What’s the maximum perimeter that can be made using all 7 shapes? Mathematics required: surds Encourage pupils to leave answers in surd form for addition purposes, although they may have to convert to decimals to compare some. Pupils can experiment with shapes of their own or be given the shapes on slides 12 & 13 to begin with. They might also be given a limitation of only using certain shapes i.e triangles and quadrilaterals. Tangrams: perimeters 2 Long thin shapes will have a larger perimeter. A square might be expected to have the minimum perimeter, but shapes such as the hexagon shown actually have smaller ones. This could lead to a discussion about the relationship between area and perimeter. For a fixed area, the closer a shape is to being circular, the smaller its perimeter will be. Tangrams: perimeters 2 A selection of answers are shown Tessellations: overview Tessellations make for engaging activities, which are accessible to most pupils. In Key Stage 3 the activities might begin with creating tessellations However, they also provide opportunities for utilising dynamic geometry software and also for reasoning and proving. Activities: Is it or isn’t it a tessellation Quadrilateral tessellations Regular tessellations Semi-regular tessellations Tessellations: is it or isn’t it? • • • • • 1 – yes 2 – yes 3 – no: not a regular pattern 4 – yes 5 – no: this is a pattern, but it’s not repeated Tessellations: Which quadrilaterals tessellate? All quadrilaterals tessellate This can be explored by pupils cutting out a template and drawing round it to create a tessellation. Another way to demonstrate this is to use Dynamic Geometry Software, using the Geogebra file ‘Quadrilateral Tessellation’. (free Geogebra software required). The quadrilateral in the top left hand corner of the page is the driver. Move the vertices of this shape and all others will change with it, maintaining a tessellation. Hence it can be demonstrated that all quadrilaterals tessellate. Tessellations: Regular tessellations Moving into reasonably simple proof, it can be shown that there are only 3 possible regular tessellations. It would be helpful for pupils to be given time to think about how they could prove this and perhaps have a class discussion rather than telling them how to prove it. Tiles are fitted edge to edge and hence meet at points. Since the angle sum must be 360°, the interior angle of the regular polygon must be a factor of 360. There are (at least) two ways to approach this. • Find all the factors of 360 and work out which ones are interior angles of regular polygons or • For a tessellation there must be 3 shapes meeting at a point - 2 wouldn’t be a point. Therefore the largest angle it could be would be 120° (hexagon) and the smallest regular polygon is a triangle 60° . This means that only 4 shapes need to be checked to determine if their interior angles are factors of 360° Tessellations: Semi-regular tessellations There are 8 semi-regular tessellations, although there is a 9th if a mirror image is permitted. Proof by exhaustion can be used to prove that there are no others, but the entire proof would be daunting. Providing the information that the largest number of sides for any regular polygon in a semi-regular tessellation is 12 makes the problem more accessible. Finding a logical and systematic way to identify combinations of interior angles of regular polygons which have a sum of 360° allows another proof by exhaustion.