PFC102R/102/2011SCHOOL OF EDUCATION Department of Teacher Education PFC102R TEACHING MATHEMATICS AND NATURAL SCIENCE TUTORIAL LETTER: PFC102R/102/2011 SEND THIS ASSIGNMENT IN AN ASSIGNMENT COVER MARKED AS ASSIGNMENT 02 Open Rubric 2 Dear Student Information on assignment 02 You must complete two sections, Section A and B for Spatial orientation (Geometry). The purpose for Section B is to grant the student an opportunity to discover how geometry should be taught at school. The objective for Section A is to upgrade the student’s own knowledge concerning Learning Area content. This assignment includes writing, calculating, drawing as well as the construction of “paper-and-stick” models. Take careful notice of the fact that the nets of the models need to be handed in with the rest of the assignment. A net is a twodimensional shape which can be folded to make a three-dimensional object (The Cambridge Mathematics Dictionary for schools 2009:84). This assignment should be seen as a “learn-and-do” manuscript for the student. Take notice that this assignment should not be seen as an example of how to teach the Learning Area to young learners. What is important, though, is the fact that the materials are presented in the correct order in which geometrical concepts should be learnt by learners. In other words, geometry teaching should start with threedimensional (3D) objects (Polyhedra) moving through the unfolding (nets) of these objects and ending in two-dimensional (2D) representations thereof (polygons). This assignment 02 will be returned to every student as soon as it has been assessed. Please spend enough time on this assignment. Think carefully about the aim and meaning of each component in the structure. You will understand that a large part of the assignment is devoted to convey to you the correct terminology and definitions. Please feel free to contact me if you have any queries. Further information can be find in your prescribed book as well as the internet. Acknowledgement: The majority of the drawings that are used in this assignment were taken from Serra, M. 1997. Discovering Geometry – An inductive approach. Key Curriculum Press: Berkeley, USA. Best of luck! Kind regards. Dr. A. M. Dicker AJH van der Walt Building Room 7-43 UNISA (Pretoria Campus) Tel.: (012) 429-4630 Fax: (012) 429-4900 Mail: [email protected] Departmental secretary: (012) 429-4583 Include examples of the work of the learners such as diagrams. 3. Remember that the van Hiele thought levels only apply to geometry. What can you deduce about the geometrical knowledge of your learners? 5. The group can be very small. Describe the thought levels as defined by Van Hiele through which the children progressed when doing this activity – see the prescribed book for these though levels. Your assignment will be assessed with the help of the following rubric: Statistics of learners correctly presented Activities selected correctly Teaching strategies correctly described Thought levels correctly indicated Geometrical knowledge of learners discussed Examples of learners’ work included YES YES YES YES YES YES NO NO NO NO NO NO . You should preferably do this activity with a group of children. It is important to include the following information about the children. Age group Number of children Number of boys and girls The time it took to complete each of the two activities For each activity do the following: 1. 4. Make a photocopy of the activity and stick it in your answer book 2. Discuss the teaching strategies for this activity.3 PFC102R/102 SECTION A ASSIGNMENT 02 Please note that no permission will be granted for late submission for this assignment due to the fact that it will not be possible to mark and return in time before the examination! Due date: 31 MAY 2011 You have to select two activities from the chapter containing Geometry in your textbook (any edition of the prescribed book). even 2 – 5 children will do. models etc. A polygon is a flat figure bounded by line segments. we really mean that the face is a triangular region. The word polyhedron means many faces. Triangle POY is one of the four triangular faces in the polyhedron below. is called a polyhedron. For example: Polyhedron (b) has 5 faces of which four are triangles and one is a square. we identify the face by naming the polygon that encloses it. Solids with flat surfaces In this assignment you will first learn about one important group of geometric solids called polyhedra. The flat polygonal surfaces of a polyhedron are called it faces. A solid formed by polygons that enclose a single region of space.4 SECTION B Polyhedra and Polygons 1. Name the other three faces ______________________________ ______________________________ ______________________________ . Later in this assignment you will study two types of polyhedrons: prisms and pyramids. When we say that the face of a polyhedron is a triangle. Although a face of a polyhedron includes the polygon and its interior region. The word polygon means many sides. Here are some examples of polyhedra: Note the faces of the polyhedra which are polygons. 1. then the polyhedron is called a regular polyhedron. ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ A point of intersection of three or more edges is called a vertex of the polyhedron. _____________________________ _____________________________ _____________________________ A polygon is classified by its number of sides. 4. Point P is one of the four vertices of the polyhedron. Name the other five. The prefix penta means “five” so a pentahedron is a polyhedron with five faces. The prefixes for polyhedrons are the same as they are for polygons – A polygon with four sides is called a tetragon. and each face is congruent to the other faces. The regular polyhedron shown underneath has 12 faces and is called a regular dodecahedron. 3.5 PFC102R/102 The line segment where two faces meet is called an edge. 2. Write down the other three. Line segment PO is one of six edges in the polyhedron. How would you classify a polyhedron with eight faces? ________________________________________________________________________ If each face of a polyhedron is enclosed by a regular polygon. a polyhedron is classified by its number of faces. and the faces meet at each vertex in exactly the same way. . An altitude of a prism is a perpendicular . The faces that are not the bases are called the lateral faces. __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ Prisms are classified by their bases. is called a right prism. 5. The lateral faces meet to form the lateral edges. A prism that is not a right prism is called an oblique prism. The two shaded faces of each prism are called the bases of the prism. For example a prism with triangular bases is a triangular prism. which is a special kind of polyhedron.6 Each solid shown below is a prism. Try to combine these statements into a definition for a prism. and prisms with hexagonal bases are hexagonal prisms. Triangular Prism Hexagonal Prism Rectangular Prism A prism whose lateral faces are perpendicular to the bases. The height of a prism is the length of an altitude. The perpendicular line segment from the vertex to the plane of the base is the altitude of the pyramid. The pyramids of Egypt are square pyramids because they have square bases.7 PFC102R/102 line segment from one base to the plane of the other. and the height is the length of the altitude. The common vertex of the lateral faces is the vertex of the pyramid. Pyramids Each solid shown below is a pyramid. . The faces that are not the bases are called the lateral faces. The shaded face of each pyramid is called the base of the pyramid. The lateral faces meet to form the lateral edges. Pyramids are also classified by their bases. The circle on the base of a hemisphere determines a great circle. __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________ Complete: A sphere is a ……………………………………………………………………………………… …………………………………………………………………………………………………………………………. The radius of the cylinder is the radius of the base. Like a prism. however. . Can you think of others? Write them down. a cylinder has two bases that are both parallel and congruent.8 2. A cylinder that is not right is oblique. A hemisphere is half a sphere. A solid with a curved surface can not be a polyhedron. 6.. The fixed distance is called the radius. The segment connecting the centers of the circles is called the axis of the cylinder. The height of a cylinder is the length of an altitude. compact discs (CD’s) plumbing pipes are shaped like cylinders.……………………………………………………………………………………………………………………….………………… . A sphere can be thought of as a three-dimensional circle. If the axis of a cylinder is perpendicular to the bases then the cylinder is a right cylinder. The equator is a great circle. Soup cans. or sphere. All the longitude lines on a globe of the earth are great circles. and the fixed point is the center. An altitude of a cylinder is a perpendicular segment from the plane of one of the bases to the plane of the other base. The most well known solid with a curved face that all sports fans known well is the ball. Solids with curved surfaces Polyhedrons are geometric solids with flat surfaces. Oranges are one example of spheres found in nature. ………………………………………………………………………………………………………. the bases of cylinders are circles as well as their interiors. Another solid with a curved surface is a cylinder. Instead of polygons. The other cones are oblique. Plato reasoned that because all objects are three-dimensional. both natural and human. obey the neat precise rules of arithmetic. must be in the solid shape of regular polyhedrons. a cone has a base and a vertex. The radius of a cone is the radius of the base. The vertex of a cone is a point not in the same plane as the base. Like a pyramid. If the line segment connecting the vertex of a cone with the center of its base is perpendicular to the base. Pythagoras and others and explained the nature of all things in his dialogue Timaeus. Pythagoras (572-497 b C) conjectured that numbers rule the universe. The height of a cone is the length of the altitude. The altitude of a cone is the perpendicular segment from the vertex to the plane of the base. In Plato’s view. Plato (429-347 b C) combined many of the ideas of Empedocles. All forces. fire and water – from Empedocles. The base of a cone is a circle and it’s interior. and these five geometric solids are commonly called the Platonic solids. then it is a right cone. fire. air. air and water. They were important to ancient Greek scholars who placed great emphasis on the study of science. which are explainable by mathematics. all things are composed of the five different atoms. atoms. The fifth atom makes up the . Funnels and ice cream cones are shaped like cones.9 PFC102R/102 A third type of solid with a curved surface is a cone. He took the idea for four of these atoms – earth. their smallest part. Empedocles (499-430 b C) studied nature and believed that all things are composed of four elements namely earth. There are only five regular polyhedrons. The Five Platonic Solids Regular polyhedrons have intrigued mathematicians for thousands of years. Greek philosophers saw the principles of mathematics and science as the guiding forces of the universe. THE MATHEMATICAL MODEL FOR FIRE IS CALLED A REGULAR TETRAHEDRON (4 FACES) THE MATHEMATICAL MODEL FOR WATER IS CALLED A REGULAR ICOSAHEDRON (20 FACES) THE MATHEMATICAL MODEL FOR AIR IS CALLED A REGULAR OCTAHEDRON (8 FACES) THE MATHEMATICAL MODEL FOR EARTH IS CALLED A REGULAR HEXAHEDRON (6 FACES) THE MATHEMATICAL MODEL FOR COSMOS IS CALLED A REGULAR DODECAHEDRON (12 FACES) 7. You will have to count the number of vertices (V). Plato assigned the shape of each of the five regular solids to each of the five atoms. 8. Use paper and make the nets of three of the above mentioned polyhedra and send them together with this assignment. Build each of the above mentioned polyhedra. 9. edges (E). Polyhedron Tetrahedron Icosahedron Octahedron Hexahedron Dodecahedron Vertices Faces Edges .10 cosmos: the stars and planets in the sky. before you can fill in the table below. and faces (F) of each polyhedron. The sides of the polyhedra should not be longer than 5 cm. Use toothpicks and modeling clay or jelly tots (sweets) to stick the models together at the vertices (sweets you can eat part of the model after you have finished!) You will have to make the models to answer the questions that follow. faces and edges in a polyhedron. use it to answer these questions. how many edges will it have? _____________________________________________________________________ 12. This formula was discovered by the Swiss mathematician – Leonard Euler (1707-1783). What is this formula? _____________________________________________________________________ Now that you have discovered the formula relating the number of vertices. If a solid has 6 faces. and is commonly known as Euler’s formula. F and E (or some combination of two or three of these operations) you can create a formula that will work for all polyhedrons. edges and f aces of a polyhedron. By adding or subtracting V. Look for patterns in the table to discover the relationships. If a solid has 7 faces and 12 edges. what are all the possible combinations of vertices and edges it can have? _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Try to draw the polyhedra to verify that the combinations of faces.11 PFC102R/102 10. how many vertices will it have? _____________________________________________________________________ 13. . There is a special relationship among the number of vertices. If a solid has 8 faces and 12 vertices. 11. vertices and edges are possible. What would each of the five Platonic solids look like when unfolded? Use the pictures on the previous page to help visualize the solids.12 Plato argued that fire atoms are in the shape of regular tetrahedrons because fire is the lightest atom and the tetrahedron has the least number of faces. because the cube is very stable. water. that Plato argued it must be in the shape of the atoms of the cosmos. Plato further reasoned that because fire. (The cut-open figures of solids are called the net of the solid. How many other patterns which agrees with the restrictions above. Therefore the water atom is the shape of the regular icosahedron. One face is missing. are possible for a regular hexahedron? Sketch them.) 15. like earth. The fifth and remaining regular polyhedron. and fire must have these shapes. It followed that because air is the second lightest of these three atoms. Complete the picture below to show what the regular tetrahedron would look like. In addition. . is so unlike the others. Complete the picture of what the regular hexahedron would look like if it were cut open along the lateral edges and three top edges. the dodecahedron. Two faces are missing. or regular hexahedrons. therefore. air and water react most often with one another. it must be in the shape of the octahedron because the octahedron has the second least number of faces. having pentagonal faces. then air. regular icosahedron. Because the faces of the regular octahedron. 14. they must be composed of atoms that are similar in shape. There is more than one possible way to unfold each polyhedron. Plato then reasoned that the earth atoms are in the shape of cubes. then unfolded. if it were cut open along the three lateral edges and unfolded into one piece. Note: There are 36 hexaminoes (figures build with six squares) of which 12 are nets of a cube but only six of them agree with the restriction in your question. it must be responsible for the sharp sting of fire. and regular tetrahedron are equilateral triangles. the regular tetrahedron has the sharpest points and. The regular octahedron is similar to the icosahedron but has only eight equilateral triangles as faces. Below is a picture of what the regular icosahedron would look like if it were cut along some edges and unfolded into one piece. The regular dodecahedron is made with 12 regular pentagons.13 PFC102R/102 16. They would resemble two flowers. Complete the picture below to show what the net of an octahedron would look like if it were cut along some edges and unfolded into one piece. 18. each having five pentagon-shaped petals around a centre pentagon. When folded back together. If half of the dodecahedron were cut along edges connecting the petals and then unfolded. or c ______________________________________________________________ HINT: TRACE. FOLD AND PASTE 17. Two faces are missing. what would it look like? Complete the pattern for half the dodecahedron. the five top triangles meet at one top point. The edge labelled X will line up with which edge: a b. Suppose you were to cut the dodecahedron into equal parts. . The following list of objects is given. The first one is done as an example.14 19. triangular prism triangular pyramid rectangular prism square prism octagonal prism heptagonal pyramid square pyramid pentagonal prism hexagonal prism hemisphere cylinder sphere cone 1 2 3 4 5 6 7 8 9 Die Tomb of Egyptian rulers Container for a scoop of ice-cream A box of breakfast cereal A round kitchen bowl Birthplace of a bee Stop sign Toblerone chocolate Children’s play tent A B C D E F G H I J K L square prism 10 Moon 11 Can of tuna fish 12 Book . Colour the nets below the cube so that it will make the cube in the picture once it is folded. The top half of the cube below is coloured red and the bottom half is coloured blue. (The nets in the answer are already partially coloured in red. Complete the table by linking each of the every day objects to the geometrical objects.) Use the two colours when you answer the question! 20. Examples of polygons Examples of figures that are not polygons Each line segment is called a side of the polygon. Each endpoint where the sides meet is called a vertex of the polygon. A convex polygon is a polygon in which no line segment connecting two vertices is outside the polygon. formed by connecting line segments endpoint to endpoint with each segment intersecting exactly two others. Convex polygons .15 PFC102R/102 Polygons A polygon is a closed geometric figure in a plane. A concave polygon is a polygon in which at least one line segment connecting two vertices is outside the polygon. If two sides share a common vertex. and if the four sides of quadrilateral CAMP are congruent to the four corresponding sides of quadrilateral SITE. If two vertices of a polygon are connected by a side. For example. . Two line segments or two angles are congruent if and only if they have the same measures. Polygons that are exactly the same size and shape are congruent polygons. you can use the ▲ symbol. then they are consecutive adjacent sides. For example. When the polygon is a triangle. For convex polygons this means two things: If the angles and the sides of one polygon are congruent to the corresponding angles and sides of another polygon. then they are consecutive or adjacent vertices. then the two polygons are congruent. For example. You can also call it pentagon DCBAE. if the four angles of quadrilateral CAMP are congruent to the four corresponding angles of quadrilateral SITE. the letters of the corresponding congruent angles should be written in an order that indicates the correspondences.16 Concave polygons When referring to a specific polygon.▲ ABC means triangle ABC. the pentagon underneath can be referred to as pentagon ABCDE. then quadrilateral SITE is congruent to quadrilateral CAMP. When you write the symbolic statement of congruence of the two figures. list in succession the capital letters representing consecutive vertices. The perimeter of a polygon is the sum of the lengths of its sides. it follows that their four pairs of corresponding angles and four pairs of corresponding sides are also congruent.17 PFC102R/102 The definition also means that if two polygons are congruent. NI and TP. and ENTIP are PN. if it is given that quadrilateral CAMP is congruent to quadrilateral SITE. TE. then their corresponding angles and sides are congruent. . For example. The polygon shown below has a perimeter of 37 cm. The figure below shows the diagonals of polygon ALRE are AR and LE. 18 In the figure below. The figures shown below are equiangular polygons. The figures shown below are not equiangular polygons. FL. The figures shown below are equilateral polygons. DE and CU are not diagonals. . (All sides are equal) The figures shown below are not equilateral polygons. 21.19 PFC102R/102 The figures shown below are regular polygons. (All sides and all angles are equal.) The figures shown below are not regular polygons. Name each of the polygons below: . An angle bisector in a triangle g. Quadrilateral b. A right angled triangle ABC h. The altitude of a triangle f. Make a figure next to each of the following to explain its meaning: a. A median of a triangle e. Parallelogram c.20 22. Isosceles triangle ABC d. An equilateral triangle ABC . An octagon Complete this assignment and place it with Section A in an assignment cover. .21 PFC102R/102 i. Complementary angles m. AB//CD k. Vertical opposite angles l. Supplementary angles n. A hexagon o. A line segment AB j.