Ideas beyond Number. Teacher s guide to Activity worksheets

Transcription

1 Ideas beyond Number Teacher s guide to Activity worksheets Intended outcomes: Students will: extend their knowledge of geometrical objects, both 2D and 3D develop their skills in geometrical reasoning and trigonometry solve problems using existing knowledge, skills and understanding find relationships make and justify generalisations have fun! Outline of unit: The unit focuses on regularity and symmetry in geometry. In particular, the emphasis is on regular polygons and regular polyhedra. There are opportunities for students to explore some of the solid geometry by experimenting with regular polygonal faces joined to make three dimensional vertices (leading to the discovery that there are only five Platonic solids and perhaps that there are thirteen Archimedean solids). It is hoped that experimentation will not be the only outcome, but rather that students will develop arguments which conclude that, for example, there can only possibly be five Platonics. It is this higher level of thinking that we are seeking to promote.

2 It is hoped that students will have opportunities to make these figures so that they can explore their properties. This is a rich place for young mathematicians to be using simple trigonometry to calculate distances and angles which can be difficult to visualise. If this making phase is to be valuable in its own right, it may be useful for students to explore the construction of regular polygons with a straight edge and a pair of compasses: which are possible and which are not? One possible route through the suggested activities might be: 1. A bit of revision: names, properties of regular 2D and 3D objects they know. 2. Polygons: Which regular polygons tessellate on their own (e.g. hexagons)? or in combination with others (e.g. octagons with squares)? What if they all have to have the same side length? Does that make any difference? Which regular polygons can we construct using a straight edge and a pair of compasses only? Can we calculate the measurements of our polygons angles, areas etc? Can we generalise it all, so that it becomes a spreadsheet activity? 3. Regular polyhedra: Which convex polyhedra can be made by using exactly one type and size of regular polygon as a face, with every vertex identical? What if we lift the restriction on only using one type of polygon but retain the constraint that all vertices should be identical? What about lifting the restriction that the polyhedra should be convex? Can any new ones be made? Can we calculate the measurements of these polyhedra angles between faces, heights, radius of circumscribing sphere, etc? Can we generalise about these measurements? (Is this too difficult?) What polyhedra fill space a sort of 3D tessellation? Cubes do (is that too obvious?) but are there any others, either alone or in combinations? Page 2 of 14

3 Activity sheet 1 Regular polygons and tesselation Which regular polygons tessellate? Square tiling is all around us, but are there any others? Questions 1. What is it about the angles and side lengths of these square tiles that guarantees the bottom left tile will fit exactly into the space? 2. What is it about regular octagons that prevents them tiling on their own, but does allow them to tessellate with another kind of regular polygon? (Which kind?) 3. Make sketches (or accurate drawings, if you prefer) to show how particular combinations of regular polygons tessellate. 4. Can you find a tessellation that involves regular decagons, either on their own (No! Why not?) or in combination with something else? A notation for the vertices of our tilings Look at the square tiling pattern at the top of the page. At every vertex, four squares meet. One way in which we might describe these vertices is (square, square, square, square). Page 3 of 14

4 A similar tiling of hexagons would have vertices described as (hexagon, hexagon, hexagon). 5. Describe each of your tilings using this system. Page 4 of 14

5 Activity sheet 2 Constructing polygons This part of the unit is concerned with regular polygons. In particular, we will be looking at how we can use our knowledge of the symmetrical properties of these polygons to construct them, using the traditional instruments of Euclidean geometry: a pair of compasses and a straight edge only NO PROTRACTORS! In order to tackle most of these, we will need to be able to easily and accurately find the mid-point of a given line segment; construct a perpendicular to a given line at a given point, and to bisect an angle. Discuss, experiment, investigate: What property of a rhombus (or a kite) can we use to locate the midpoint of a given line segment? What property of a rhombus (or a kite) can we use to bisect an angle? What angles can we construct? Can we use angle bisection to find any other useful angles? What regular polygons can be constructed, then, using just these construction techniques? Is it possible to construct a regular pentagon? Are there any polygons that can t be constructed, do you think? What are the angle properties of the polygons we can construct? What are the areas of the polygons we can construct, if the side length is one unit? Can we use a spreadsheet to generate these calculations efficiently? What is the radius of the circumcircle? The inscribed circle (this has every side of the polygon as a tangent to the circle)? Page 5 of 14

6 What angles can we construct? It is likely that students will have encountered the division of a circle into six equal arcs, thus generating angles which are integer multiples of. Can they explain why that works? Can we use angle bisection to find any other useful angles?,,, and multiples of around these are now available What regular polygons can be constructed then, using just these construction techniques? 3, 4, 6, 8 and 12-gons are probably as far as this is going to need to go. Is it possible to construct a regular pentagon? The answer is yes! It just needs more than the two techniques used above. See Activity sheet 3. Why might pentagons be useful to us? Once we ve constructed a regular pentagon, angle bisection gets us decagons and 15-gons. Not to mention their use in solids! It has been proven (not by us!) that 7-gons, 9-gons, 11-gons and 13-gons cannot possibly be constructed using compass and straight-edge techniques. However our list does now run to 3, 4, 5, 6, 8, 10, 12 and 15 sided regular polygons, and doubles of these. Page 6 of 14

7 Activity sheet 3 Pentagons A regular pentagon has sides which subtend an angle of at the centre of its circumscribing circle. We cannot use multiples of and angle bisection alone to create this angle. Tasks Consider this sequence of bisections: A. Construct and. Bisect the angle between them to find. B. Bisect the angle between and to find. Close, but no cigar. C. Bisect the angle between and to find. D. Bisect the angle between and to find. Agonising, isn t it? Only of a degree out and, to all intents and purposes this might do if we were just looking for something which looks okay but, hey, we re mathematicians! E. What would you do now? How close can you get? Use a spreadsheet or a calculator, if that would help. How many operations does it take before you think you are close enough? Page 7 of 14

8 Now consider the rather beautiful triangle below: Question 1. What type of triangle is it? How do you know? What does this triangle have to do with our quest to construct a regular pentagon? Task Now bisect angle, and let the bisector intersect at. Page 8 of 14

9 Question 2. What do you notice: about triangle? About triangle? There is a powerful relationship between the triangles and which we can use to help us work out the relationship between lengths of sides. If we could construct the distance we need, rather than worrying about bisecting angles all the time, we might be in luck. Let s imagine that our original triangle,, has side = 2 units. It would be helpful if we could know (and construct!) the length corresponding to. Let s call that distance. 3. What other distances on our diagram are now equal to? Can we express any other distances in terms of? 4. The ratio of base to equal side in our similar triangles must be the same, so = Which side should replace the question mark in the equation above? 5. Replace the names of the distances in this relationship with their lengths (where known) or the expressions in terms of. 6. How can this help you to find the exact value of? (For a hint, see the end of the worksheet.) Can you show that is equal to? 7. Now how can we construct that distance, when all we know for sure is that we have a line () of length 2? Well, we can find a distance of 1 unit by bisecting. Can we use distances of 1 and 2 units to find? Think about it. If we can find, can we find a distance of? I think so! Page 9 of 14

10 Task A. Now to the pentagon! You may well have already formulated your own ideas about how to use what we ve done to construct your regular pentagon. If so, great! Go ahead and do it. If not, take a look at the following. is a radius of length 2. Construct, a radius, perpendicular to. Construct point, the midpoint of. How long are, and? Bisect angle, to find on. Now construct, perpendicular to. is now one side of your pentagon. Set your compasses to as radius and step around the circle to find the other vertices. Question 8. How is this linked to what we looked at earlier? Where is the in this diagram? Page 10 of 14

11 Is there another way? Tasks What about: B. Start with a line of length 2, in the middle of a clean sheet of paper. C. Construct the perpendicular to at and mark a distance 2 along it, at. D. Construct the midpoint of to find, a distance of 1 from. E. Join. F. With centre and radius, draw an arc to intersect at. What is the length of? G. Construct the perpendicular bisector of. H. Now fix your compasses to the radius from your original line. With centre, draw an arc to intersect with the perpendicular bisector of at. I. is now the,, triangle that we were considering. J. If we can construct a angle, we can construct our regular pentagon! K. Finish the pentagon. Questions 9. In what ways are these two methods linked? 10. Are there more efficient methods? Did you find one of your own? Hint for finding the value of : Form the equation =. Rearrange it to form a quadratic equation: = Expand it, tidy it up. Then solve it exactly. Do not use a calculator. Page 11 of 14

12 Activity sheet 4 Regular Polyhedra When we were considering whether particular regular polygons tessellate or not, we will have seen that three regular pentagons, arranged as shown below, leave a small gap: Questions 1. What is the angle between the two free sides of the pentagons at the gap? 2. Now imagine joining the tiles with a flexible hinge, along the common sides and then bringing together the two free sides which form this gap. What do you see? If you re having difficulty visualising it, why not make a model? (You don t have to keep constructing these objects. You may be able to get your hands on some ready made tiles (ATM Mats, for example) or you may want to use a computer drawing package to do this for you. The picture above was created using nothing more sophisticated than Microsoft Word s Drawing Tools. I drew one regular pentagon Hint: Choose Basic Shapes, Pentagon and then Drag and draw it while holding Shift down to keep the shape regular and then simply copied and pasted it twice more. Very quick, very easy.) Page 12 of 14

13 This 3D vertex (could we call it?) might possibly be repeated to give a closed, convex regular polyhedron. Then again, it might not. Let s find out. What we are going to be looking at here is whether particular 3D vertices repeat to give us regular polyhedra. One regular polyhedron with which we are all familiar is the cube. All its vertices are that is, three squares meeting at one point. Questions 3. What would the angle of the gap be, if we had begun with three squares lying flat, as in our pentagon example, above? 4. Why can t we have a polyhedron with vertices? Or vertices? 5. Is there a polyhedron with vertices? What others can you find? Do you know the names for the polyhedra you find? One very important idea here is that the vertices should all be identical. The vertex (triangle, square, triangle, square) produces a particularly beautiful polyhedron called a cuboctahedron. The missing angle, if these were to be laid flat would be: = Now if we were to make a vertex, the same gap would appear in our attempted tiling. Questions 6. Does the vertex repeat to give the same polyhedron? (This may well be worth trying. Make enough tiles to build one. Start with a single vertex: triangle, triangle, square, square. Now add tiles to it, so that each vertex is the same. What happens?) Page 13 of 14

14 If we insist on all the vertices being identical, all the faces being regular polygons, and any polyhedra being convex, how many different polyhedra can be made? (This is not a trivial question although the answer has been known for a couple of thousand years, it took some powerful minds to get to it.) Of particular interest amongst this group of polyhedra are those that only use one kind of polygon the socalled Platonic Solids (the cube is one of these!). (Why is it impossible for there to be a Platonic solid with octagonal faces?) All of the other polyhedra use at least two types of face. The amazingly-named rhombicosidodecahedron (!) uses three different polygons. Its vertices are of the form. Task Join forces with someone and make it. (Can you predict how many of each face will be needed? How many vertices there will be? How many edges will the finished polyhedron have? I do mean predict, not guess! It might be helpful to begin by looking at much simpler models.) Question 7. Are there any relationships between the type of face, the number of faces, the number of vertices and the number of edges which are always true? Here s one to get you started. Cube: Vertex description: Number of faces: 6 Number of vertices: 8 Number of edges: 12 Page 14 of 14