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- 1. From Flatland to Spaceland MATHEMATICS. 2 ESO by Rafael Cabezuelo Vivo nd May 2011
- 2. IDENTIFICACIÓN DEL MATERIAL AICLE TÍTULO FROM FLATLAND TO SPACELAND NIVEL LINGÜÍSTICO SEGÚN MCER A2.1 IDIOMA INGLÉS ÁREA/ MATERIA MATEMÁTICAS NÚCLEO TEMÁTICO GEOMETRÍA La unidad pretende pasar de las figuras planas a los GUÍON TEMÁTICO (no cuerpos tridimensionales, haciendo un recorrido por las principales características de éstos. Se utiliza lo aprendido más de 100 palabras) para describir objetos de uso cotidiano. Se incluye una autoevaluación. FORMATO PDF CORRESPONDENCIA 2º ESO CURRICULAR AUTORÍA RAFAEL CABEZUELO VIVO TEMPORALIZACIÓN APROXIMADA 6 SESIONES más una tarea final y una autoevaluación de contenidos y destrezas. COMPETENCIAS BÁSICAS Lingüística: Mediante la lectura comprensiva de textos, la resolución de problemas y el uso de descripciones. Matemática: Con la interpretación y descripción de la realidad a través del pensamiento matemático, aplicando los conocimientos a objetos y realidades cercanas. Interacción con el mundo: A través de las estructuras geométricas, desarrollando la visión espacial y relacionando las formas estudiadas con objetos cotidianos. Aprender a aprender: Por medio de la perseverancia, la sistematización, la autonomía y la habilidad para comunicar con eficacia los resultados del propio trabajo. OBSERVACIONES Los contenidos de las sesiones pueden exceder de una hora de clase real, especialmente cuando se llevan a cabo algún ‘role play’, trabajo grupal y/o manual. Las dificultades matemáticas pueden también suponer una necesidad de adaptación de la duración de cada actividad. Las actividades de postarea, al final de cada sesión o grupo de sesiones podían utilizarse todas como actividades finales, junto a la ficha de autoevaluación. Además, cada sesión puede utilizarse de forma independiente. Secuencia AICLE 2o ESO 2 From Flatland to Spaceland
- 3. TABLA DE PROGRAMACIÓN AICLE OBJETIVOS DE ETAPA CONTENIDOS DE CURSO TEMA O SUBTEMA MODELOS DISCURSIVOS TAREAS • Identificar las formas y relaciones espaciales que se presentan en la vida cotidiana, analizar las propiedades y relaciones geométricas implicadas y ser sensible a la belleza que generan al tiempo que estimulan la creatividad y la imaginación. • Integrar los conocimientos matemáticos en el conjunto de saberes que se van adquiriendo desde las distintas áreas de modo que puedan emplearse de forma creativa, analítica y crítica. • Poliedros y cuerpos de revolución. Desarrollos planos y elementos característicos. Clasificación atendiendo a distintos criterios. Utilización de propiedades, regularidades y relaciones para resolver problemas del mundo físico. • Volúmenes de cuerpos geométricos. Estimación y cálculo de longitudes, superficies y volúmenes. • Utilización de procedimientos para analizar los poliedros u obtener otros. • • • • Geometría Figuras planas Cuerpos geométricos Poliedros • • • • Cuerpos de revolución Áreas Volúmenes Historia de la Matemática • Comparar objetos reales con las figuras estudiadas • Analizar y describir objetos de uso cotidiano • Explicar procesos de cálculo • Análisis de las figuras planas como tarea previa de comprensión de los cuerpos sólidos. • Construcción y análisis de los cuerpos platónicos. • Cálculo de superficie y volumen de los principales poliedros y cuerpos de revolución (prismas, pirámides, cilindros y conos). • Descripción y análisis geométrico de un objeto cotidiano. FUNCIONES: Hacer descripciones, explicar cálculos. CONTENIDOS LINGÜÍSTICOS CRITERIOS DE EVALUACIÓN Secuencia AICLE 2o ESO ESTRUCTURAS: I think that…, The object/building is like a..., From my point of view… On one hand… on the other hand..., I (don't) agree with you, etc. a times/plus/minus/divided by b, one third (fractions), is equal to, etc. LÉXICO: apex, apothem, cone, cross section, cube, cubic, cylinder, dimension, dodecahedron, dodecahedron, edge, face, figure, flat, flatland, height, hexagon, icosahedron, irregular, n-sided, oblique, octahedron, pentagon, perimeter, pi (π), plane, platonic solids, polygon, polyhedron, prism, pyramid, radius, rectangle, regular, right, shape, side, slant length, solid, space, square, square root, squared, straight, surface-area, tetrahedron, triangle, vertex, volume, etc. • • • • • Relacionar a Escher con la geometría. Conocer y distinguir los cuerpos platónicos. Conocer y distinguir los principales poliedros y cuerpos de revolución. Calcular la fórmula de Euler Calcular el área y el volumen de prismas, pirámides, conos y cilindros. 3 From Flatland to Spaceland
- 4. Image 1 Image 2 What can you tell me about these images? Secuencia AICLE 2o ESO 4 From Flatland to Spaceland
- 5. Session 1 Pretask 1. Word cloud. Look at the words and the images above. First, listen and repeat the words. Then, fill in the gaps in the texts below. Gaps with the same letter must be filled with the same word. Image 1 shows a a)_____________ that can be drawn on a b)_____________ surface called a c)_____________ (it is like on an endless piece of paper). Our world has three d)_____________, but there are only two d)_____________ on a plane that are length and height, or x and y. A e)_____________ is a 2-dimensional shape made of f)_____________lines. Image 1 shows a e)_____________ named g)_____________, do you know more of them? h)_____________, i)_____________ and _____________. Image 2 shows a j)_____________, which is a three-dimensional k)_____________. k)_____________ Geometry is the geometry of threedimensional l)_____________, the kind of k)_____________ we live in. It is called three-dimensional, or 3-D because there are three dimensions: width, depth and height or x, y and z. A m)_____________ is a j)_____________that has 12 n)_____________ (from Greek -dodeca- meaning 12). Each face has 5 o)_____________, and is actually a pentagon. When we say m)_____________ we often mean regular m)_____________ (in other words all n)_____________ are the same size and shape), but it doesn't have to be. If you have more than one m)_____________ they are called p)_____________. It is one of the five platonic k)_____________. More k)_____________ are q)_____________, r)_____________, etc. Secuencia AICLE 2o ESO 5 From Flatland to Spaceland
- 6. Task: Living in Flatland 2. Listening. You are going to listen a text from the book "Flatland: A Romance of Many Dimensions". It is an 1884 science fiction novel by the English schoolmaster Edwin Abbott Abbott. Listen carefully and underline the words that you hear: flatland space figures triangles heptagons sheep surface readers squares paper sinking world rising countrymen lower pentagons curved lines edges my universe border 3. Now answer these questions about the text and Flatland: What kind of figures can you find in Flatland? Name as many of these figures as you can remember. It doesn't matter if they don't appear in the text: These are the characteristics of certain plane figures. Which figures are we talking about? They are 2-dimensional shapes. They are made of straight lines. The shape is "closed" (all the lines connect up) 4. Make questions using the information you can find in this website: http://www.mathsisfun.com/geometry/polygons.html Secuencia AICLE 2o ESO 6 From Flatland to Spaceland
- 7. Statements Questions Polygon comes from Greek. PolyWhere does the word polygon come from? or means "many" and -gon means "angle". What is the origin of the word polygon? It is an Icosagon. All sides has the same lenght and all angles are also equal. It is a polygon with, at least, one internal angle greater than 180º. Another name is Tetragon. 5. From the same book now read this text: "Place a penny on the middle of one of your tables in Space; and leaning over it, look down upon it. It will appear a circle. But now, drawing back to the edge of the table, gradually lower your eye (thus bringing yourself more and more into the condition of the inhabitants of Flatland), and you will find the penny becoming more and more oval to your view ; and at last when you have placed your eye exactly on the edge of the table (so that you are, as it were, actually a Flatland citizen) the penny will then have ceased to appear oval at all, and will have become, so far as you can see, a straight line." What is the writer describing? The editor of the book tells you to make three drawings as an illustration to make the text clear. Use the empty boxes provided next to the text. Secuencia AICLE 2o ESO 7 From Flatland to Spaceland
- 8. What I Learned • • • • • 6. True or False. The plural of polyhedron is polyhedrons. Flatland is a two-dimensioned country. There are lots of dodecahedra living in Flatland. Figures in Flatland can rise above or sink below the surface. A polygon is a closed plane figure made of stright lines. T/F T/F T/F T/F T/F 7. Classify these plane figures in the category they straight best (it can be more than one). Use X to select the category: Figure Name Polygons Not a Polygon Regular Irregular Concave Convex Triangle 8. Colour the regular polygons Secuencia AICLE 2o ESO 8 From Flatland to Spaceland
- 9. Session 2 Pretask 1. Vocabulary activation. Listen and repeat. Then match pictures and words: 2 1 5 4 Secuencia AICLE 2o ESO 9 3 6 From Flatland to Spaceland
- 10. 2. Answer these questions in groups of four: • • • • What does the building in the second picture look like? How many sides do you think the dice has? What is the main difference between the third and the sixth picture? Do you think that solid geometry is important in our lifes? Why? I think that… The object/building is like a... From my point of view… On one hand… on the other hand... I agree with you / I don’t with you Because... Task: Moving to Spaceland 3. Video You are going to see a video about how solids can be seen in a place like Flatland. The narrator is Maurits Cornelis Escher (1898-1972), most commonly known as M. C. Escher. He was a fascinating artist whose compositions are worldwide famous. Escher became fascinated by the mosaics and symmetries in Alhambra, when he visited it in 1922. Watch the video in this link: http://www.dimensions-math.org/Dim_E.htm (you may need to download the video before watching) Now read these questions about the video before watching it again. It is the time to resolve any doubt you can have, so ask your teacher anything you don't understand. • Complete the following table with the names, number of faces, vertices and edges for each polyhedron. Icosahedron Octahedron Dodecahedron Tetrahedron Secuencia AICLE 2o ESO Hexahedron 10 Cube From Flatland to Spaceland
- 11. Picture • Name Faces Vertices Edges Can you identify the polyhedron by its plane section? Write them here in order of appearance (from 5'10" to 6'20"). Secuencia AICLE 2o ESO 11 From Flatland to Spaceland
- 12. • In the video there is a second and colorful method to explain polyhedra to our flat friends, the lizards. Underline the correct answer. Stereographic projection Solid inflation Face colouring • Can you identify the polyhedron by it's plane proyection? Write them here in order of appearance (from 10'20" to 12'20"). • The Greek philosophers attributed a magical importance to these 5 solids, associating one of the fundamental elements from which the world is formed to each of them. What is the name for these figures? Underline the correct answer. Fantastic Solids Platonic Solids Plutonic Solids 4. Do it yourself Now it's your time. The best way to understand _________ ← Fill in from the last question solids is to build them. For this you have to follow the instructions given in this website (use models with tabs): http://www.mathsisfun.com/geometry/model-constructiontips.html Some games (role-playing games) use these solids as dice. To make them you have to write a number on each face, but you have to follow this simple rule: Opposite faces must always add up to the same value!! (use a regular cubic dice to check it) 5. Counting Faces, Vertices and Edges. If you count the number of faces (the flat surfaces), vertices (corner points), and edges of a polyhedron, you can discover an interesting thing: The number of faces plus the number of vertices minus the number of edges equals 2 . This can be written neatly as a little equation: Euler's Formula Secuencia AICLE 2o ESO F + V - E = 2 12 From Flatland to Spaceland
- 13. It is known as the "Euler's Formula", and is very useful to make sure you have counted correctly! Now it is your turn! Check Euler's Formula for the platonic solids, and now you can use your models to count everything!! Name Image Faces Vertices Edges F + V - E Dodecahedron Tetrahedron Icosahedron Hexahedron Octahedron What I Learned 6. Fill in the blanks. Use the words given to complete this summary. In this lesson on three-dimensional solids, you've seen a lot of _____________. But there are five special _____________, known collectively as the _____________, that are different from all the others. What makes the _____________ special? Well, two things, actually. 1. They are the only polyhedra whose _____________ are all exactly the same. Every _____________ is identical to every other _____________. For instance, a cube is a Platonic solid because all six of its _____________ are congruent _____________. Secuencia AICLE 2o ESO 13 From Flatland to Spaceland
- 14. 2. The same number of _____________ meet at each _____________. Every _____________ has the same number of adjacent _____________ as every other _____________. For example, three equilateral triangles meet at each _____________ of a _____________. No other _____________ satisfy both of these conditions. Consider a pentagonal prism. It satisfies the second condition because three _____________ meet at each _____________, but it violates the first condition because the _____________ are not identical; some are _____________and some are _____________. Secuencia AICLE 2o ESO 14 From Flatland to Spaceland
- 15. Session 3 Pretask In the following sessions (3 to 6) you are going to work with 3D solids. 1. Vocabulary activation. Listen and repeat. Then write the appropiate words under each picture (one word can be written under more than one picture): Secuencia AICLE 2o ESO 15 From Flatland to Spaceland
- 16. 2. Video: You are going to see a video about some properties of solids. http://www.brightstorm.com/math/geometry/volume/3-d-solid-properties Now you have to say if the following statements are true or false. You can review the video at home if you need it. • • • • • A vertex is a line where three or more faces intersect. Where two faces intersect you create an edge. Bases are polyhedra. The bases are lateral faces. Right and Oblique prisms have the same volume and surface area. (With the same bases and height) T/F T/F T/F T/F T/F Task: Square goes upward Flatland 3. Reading: Adapted from Flatland (p73-74.) In section sixteen a stranger named Sphere tries to reveal to the main character, Square, the mysteries of Spaceland. Sphere. Tell me, Mr. Mathematician ; if a Point moves Northward, and leaves a luminous tail, what name would you give to the tail? Square. A straight Line with two extremities. Sphere. Now the line moves parallel to itself, East and West, so that every point in it leaves behind it the tail of a Straight Line. What name will you give to this Figure? Square. A Square, with four sides and four angles. Sphere. Now open your imagination a little, and imagine a Square in Flatland, moving parallel to itself upward. Square. What? Northward? Sphere. No, not Northward ; upward ; out of Flatland altogether. I mean that every Point in you (because you are a Square), in your inside, passes upwards through Space. Each Point describes a Straight Line of its own. I was now impatient and under a strong temptation to launch my visitor into Space, or out of Flatland, anywhere, so that I could get rid of him. Instead I replied: Square. And what is the nature of this Figure? I hope you can describe it in the language of Flatland. Sphere. Oh, certainly. It is all plain and simple, but you must not speak of the result as being a Figure, but as a Solid. But I will describe it to you. - We began with a single Point, which of course being itself a Point has only one terminal Point. Secuencia AICLE 2o ESO - One Point produces a Line with two terminal Points. - One Line produces a Square with four terminal Points. Now you can answer to your own question: I, 2, 4, are evidently in Geometrical Progression. What is the next number? Square. Eight. Sphere. Exactly. The Square produces something which we call a Cube with eight terminal Points. Now are you convinced? Square. And has this Creature sides, as well as angles or what you call "terminal Points"? Sphere. Of course; and we call them faces. Square. And how many faces or sides will I generate by the motion of my inside in an "upward" direction, and whom you call a Cube? Sphere. How can you ask? And you are a mathematician! The side of anything is always, if I may so say, one Dimension behind the thing. Consequently, as there is no Dimension behind a Point, a Point has 0 sides ; a Line, has 2 sides (for the Points of a Line may be called by courtesy, its sides) ; a Square has 4 sides ; 0, 2, 4 ; what kind of Progression do you call that? What is the next number? Square. Arithmetical. Six. Sphere. Exactly. Then you see you have answered your own question. The Cube which you will generate will be bounded by six sides. You see it all now, eh? 16 From Flatland to Spaceland
- 17. 4. Text attack! Write the following questions in chronological order, then match them with the correct answers. Find the odd answer: • • • • • • • • • Where does the square need to move to create a cube? Why is Square impatient? Sphere sum up the whole process again, write it in the correct order: How is the square obtained? How can Square know the number of faces in a cube? How can Square know the number of vertices in a cube? What does Square understand when Sphere tells him that a square shoud move upwards to construct a cube? Why? What does Square want to do with Sphere? What would be the name of the cube faces in Flatland? • • • • • • • • • • Secuencia AICLE 2o ESO Here is how the figures are formed: Point → Line → Square → Cube. Square understand northward instead of upward, beacuse he lives in Flatland and it is difficult for him to undertstan the three-dimensional space. The square needs to move upwards to create a cube. Becouse he can't understand the three-dimensional space very well. He wants to launch Sphere into space or out of Flatland. Square launches Sphere into space, out of Flatland. Because they are in arithmetical progression, each number is two more than the previous one. So 0 for a point, 2 for a line, 4 for a square, ... and 6 for a cube. The name would be sides. Because they are in geometrical progression, each number is double of the previous one. So 1 for a point, 2 for a line, 4 for a square, ... and 8 for a cube. The line has to move parallel to itself, East and West. 17 From Flatland to Spaceland
- 18. 1. How is the square obtained? The line has to move parallel to itself, East and West. 2. 3. 4. 5. 6. 7. 8. 9. This is the odd answer: Can you write a title for this chapter of the story? Secuencia AICLE 2o ESO 18 From Flatland to Spaceland
- 19. What solid do you obtain if the square moves upward a length greater than (or less than) the size of the square? 5. Your turn! Now you have to make your own story. You are going to work in groups of four. The teacher will dictate the beginning of a text, in which you will have to describe how a pentagon from Flatland can be transformed into a pyramid. Then, when the teacher claps one member of the group will continue the text by writing another short paragraph about the transformation. Each time the teacher claps, you will pass the paper to a new group member who will write the next section of text. Continue like this until the circle is complete. The member of the group who wrote the first paragraph will also write the last one. When you finish, choose a spokesperson to read your text out loud to the rest of the class. Teacher’s dictation Sphere. Imagine, my friend, that you are now a pentagon in Flatland... Student's 1 text: Student's 2 text: Student's 3 text: Student's 4 text: Student's 1 text -last paragraph-: Secuencia AICLE 2o ESO 19 From Flatland to Spaceland
- 20. Session 4 Task: Prisms Surface-Area & Volume 1. Reading: Adapted from www.mathsisfun.com If you make the cross section of a cube you will get a square, and the cross section of this building is a triangle ... de pt h A cross section is the shape you get when cutting straight across an object. height These are the three dimensions that a solid has: width, depth and height width A solid that has the exact same polygon as its cross section all along its length is called a Prism The bases of the prism will be also the same polygon. According to the cross section or the base of a prism it can be named... Triangular prism Square prism Rectangular prism Pentagonal prism and so on... A square prim that has edges of equal length can be called a cube (or hexahedron) and each face will be a square. Do you remember the platonic solids? So a cube is just a special type of square prism, and a square prism is just a special type of rectangular prism, and They are all cuboids! If the cross section of a prism is a regular polygon (Equilateral Triangle, square, regular pentagon, regular hexagon, etc), then you have a Regular Prism. Secuencia AICLE 2o ESO 20 From Flatland to Spaceland
- 21. 2. Complete the following table with actual prismatic objects: Figure Base Poligon Cross section Name Regular 3. Calculate. Surface-area of a prism. The surface area of a prism is the sum of the area of all its faces. As the bases are polygons you will need to remember how to calculate their area. It is measured in squared units (f.i. m2, ft2) Secuencia AICLE 2o ESO 21 From Flatland to Spaceland
- 22. You will need to calculate the area of one base (Ab). s = 5 in (side) a = 3,44 in (apothem) H = 10 in (height) p⋅a 5⋅s⋅a = = 2 2 p stands for perimeter Ab = = 5⋅5⋅3.44 86 = = 43 in² 2 2 Then you will have to calculate the area of one face (Af), which will allways be a cuadrilateral. A f =s⋅H =5⋅10= 50 in² There are two bases and five lateral faces (in this example), so the total surface area will be: Atotal =2⋅Ab+5⋅A f =2⋅43+5⋅50=86+250= 336 in² Now calculate the following prisms' surface areas: Prism Base area Face area Surface area Regular prism Base side: 5 in Height: 12 in Base sides: 3 x 10 in Height: 17 in 4. Calculate. Volume of a prism. The Volume of a prism is simply the area of one of its bases times the height of the prism. It doesn't matter if it is a right or an oblique prism. It is measured in cubic units (f.i. cm 3, in3). For the previous example we can calculate the volume in this way: Secuencia AICLE 2o ESO 22 From Flatland to Spaceland
- 23. We already know the area of the base. Do you remember it? Ab = p⋅a = 2 Can it be that easy? I think you can do it on your own: V =Ab⋅H = ⋅ in² = in³ Complete the following table: Prism Base area Height Volume Regular prism Base side: 3 in Apothem: 2 in Height: 24 in Regular prism Base side: 4 in Apothem: 3,5 in Height: 11 in 5. Search. Bullring in Montoro has a prismatic shape. Can you tell me how many sides the base has? What is the name for this n-sided polygon? Secuencia AICLE 2o ESO 23 From Flatland to Spaceland
- 24. Session 5 Task: Pyramids Surface-Area & Volume 1. Video: You are going to see a video about regular pyramids surface-area. http://www.brightstorm.com/math/geometry/area/surface-area-of-pyramids Now you have to select the correct answer to the following questions. You can review the video at home if you need it. Then make groups of four to construct the questions and the answers. . h? e side t lengt e slan th of th triangular g is th the len f the What ngth is the height o nt le sla - The ngth is eight. f the h slant le o - The length is the s. th face What is nt leng the firs The sla t calcu - T he m lations an calc that th Wha - Th e m ulate th e man t sha an calc makes e surfa pes - Th e m ? u c - The do th an calc late the surf e of the trian e fac a ce o f ulate th gle firs - The faces o the len es o t. e surfa f f the gt ce o f t - The faces o the pyra pyra he pen h first. f the mi d face mid tagon h p s of first. have the p yramid h ave a tri ? angu ave yram ap lar id ha ve a entago shape. na squa re sh l shape . ape. gth. e slant len e-area? the surfac base and th faces areas. the e calculate e adding-up How can h base and th pentagon areas. ce-area by e surfa ing-up the a by add alculate th se and the - He can c ulate the surface-are by adding-up the ba a alc - He can c ulate the surface-are alc - He can c 2. Calculate: Calculate the surface area of these pyramids. To calculate the slant length you need to remember the Pythagorean Theorem. If you cut the pyramid by the apothem of the base and the apex (the top point) you can see a right triangle: Secuencia AICLE 2o ESO 24 From Flatland to Spaceland
- 25. Imagine a regular square pyramid. Let's calculate its surface area: Base = 10 in Height = 12 in Cutting the pyramid we have an isosceles triangle, that can be divided into two right triangles. Get focus on ABC, which is a right triangle with a 90º angle at C. AC = One leg of the right triangle = Heigth of the pyramid = 12 in CB = Other leg of the right triangle = one half of the base of the pyramid = 5 in AB = Hypotenuse of the right triangle = Slant length of the pyramid = Unknown AB 2= AC 2+CB 2 ; AB=√ AC 2+CB 2=√ 122+5 2=√ 144+25=√ 169= 13 in ← Slant length Area of the base: As it is a square its surface is: Area of one face: It is the area of a triangle: Ab =base⋅base=10⋅10= 100 in Af = 2 base⋅height 10⋅13 = = 65 in 2 2 2 There are one base and four lateral faces, so the total surface-area will be: Atotal =Ab +4⋅A f =100+4⋅65=100+260= 360 in 2 Now calculate the following pyramids' surface areas: Pyramid Base area Slant length Face area Surface-Area Tetrahedron (regular pyramid) Base side: 3 in Regular pyramid Base side: 3.2 in Height: 6 in Secuencia AICLE 2o ESO 25 From Flatland to Spaceland
- 26. 3. Calculate: Calculate the voume of these pyramids. The volume of a pyramid is very easy, you just have to calculate one third of the base area times the height of the pyramid. Let's make the previous example: We already know the area of the base. Do you remember it? Ab =base⋅base= 1 1 Can it be that easy? V = ⋅Ab⋅H = ⋅ 3 3 ⋅ = in 2 in³ Complete the following table: Pyramid Base area Height Volume Regular pyramid Base side: 4.5 in Apothem: 3.1 in Height: 12 in Regular pyramid Base side: 10 in Apothem: 8.66 in Height: 16 in Secuencia AICLE 2o ESO 26 From Flatland to Spaceland
- 27. Session 6 Task: Smooth down the Prism and the Pyramid 1. Reading: When the base of a prism changes from a polygon to a circle then you get a __________. Doing the same with a pyramid what you get is a _____________. The volume can be calculated in a similar way, but the surface-area is slightly different. Here you can see the formulas: Cylinder Cone 2 Bases: Ab =π⋅r 2 1 Side: A s=2⋅π⋅r⋅h Surface area Base: Ab =π⋅r 2 Side: A s=π⋅r⋅s=π⋅r⋅√ h 2+r 2 TOTAL: Atotal =2⋅Ab+ As=2⋅π⋅r⋅ r+h) ( Volume TOTAL: Atotal =Ab +As =π⋅r⋅(r +s) 1 1 V = ⋅Ab⋅h= ⋅π⋅r 2⋅h 3 3 2 V =Ab⋅h=π⋅r ⋅h 2. It's your turn! A sheet cylinder: Look at a sheet of paper, how many cylinders can be made using its dimensions? Secuencia AICLE 2o ESO 27 From Flatland to Spaceland
- 28. Calculate the surface area and the volume of the obtained cylinders, and then compare the values. The volume of a bucket Can you tink of the shape of a bucket? What is this shape like? D You will need the diameter or radius of both bases: D = 2.30 dm d Secuencia AICLE 2o ESO d = 1.80 dm height = 4.22 dm Think the best way to calculate the volume (1 dm 3 = 1 l). You could need Thales' theorem to find one missing data. 28 From Flatland to Spaceland
- 29. What I Learned From session 3 to 6 you have learned a lot of think about polyhedra and non-polyhedra 3d solids. Let's see what you can remember. 3. Word cloud Match the images with the words: Secuencia AICLE 2o ESO 29 From Flatland to Spaceland
- 30. 4. Do it yourself Most of our buildings have a polyhedra, cylinder or cone shape. Can you analyze one of them. Look at this church. You will have to calculate its surface-area and its volume. Notice that is full of Polyhedra. Use convenient units for all dimensions and calculus. Cut the different parts of the church and paste them on the next table: 30 ft 40 ft 20 ft 17,3 ft 50 ft 120 ft 50 ft 100 ft Secuencia AICLE 2o ESO 30 From Flatland to Spaceland
- 31. Tower: Tower roof: Type: Type: Base polygon: Base polygon: Side: Side: Apothem: Apothem: Surface-area: Height: Volume: Height: Surface-area: Roof: Type: Base polygon: Side: Volume: Height: Surface-area: Secuencia AICLE 2o ESO 31 Volume: From Flatland to Spaceland
- 32. Building: Type: Base polygon: Side: Height: Surface-area: Volume: Now complete the table: Surface-area Volume Building Roof Tower Tower Roof WHOLE CHURCH Secuencia AICLE 2o ESO 32 From Flatland to Spaceland
- 33. HOMEWORK. FINAL SUMMARY Final Task: Geometry around 1. Description Find a cool geometrical object at home. You have to create a presentation with the image of the object. You will have to explain later a few things about it to the rest of your classmates. • • • • • • • • What kind of geometrical object does it look like? Which are its faces, bases, edges, vertices or apex? Does it check Euler's Formula? Is it simple (just one shape) or complex (like the church)? What are its dimensions (you can sketch the object)? What are its surface-area and volume? How did you make the calculations? What is it for? Use your own pictures and text to make an eye-catching presentation. Remember not to overload the slides, it is better to use single phrases and good pictures to show what you want to say. Think about the student at the end of the class, and use large texts and adecuate colours. Please, don't forget to bring your object with you!! Secuencia AICLE 2o ESO 33 From Flatland to Spaceland
- 34. ASSESSMENT WORKSHEET. NAME: DATE: Your task is reflecting on what you have learned. Read the following statements about skills and knowledge you have learned during the project. Please, circle one of these options: YES NO NOT YET. Self-assessment chart I CAN I KNOW Organize vocabulary into categories Take notes from a listening or a video Get valuable information from different sources Describe images and pictures Summarize the main ideas from a text Participate in a role-play Understand plane figures Calculate Euler's formula with a polyhedron Calculate surface-area and volume of a solid Analyse and describe geometrically an object YES YES YES YES YES YES YES YES YES YES NO NO NO NO NO NO NO NO NO NO NOT YET NOT YET NOT YET NOT YET NOT YET NOT YET NOT YET NOT YET NOT YET NOT YET What is a platonic solid and their caractheristics Escher was very interested in geometry The concept of volume and surface-area What are the main polyhedra and how to identify them The difference between the main polyhedra and non-polyhedra The difference between the main polygons and non-polygons YES NO NOT YET YES NO NOT YET YES NO NOT YET YES NO NOT YET YES NO NOT YET YES NO NOT YET Feedback CONTENTS DEVELOPED SKILLS SUGGESTIONS FOR IMPROVEMENT Secuencia AICLE 2o ESO 34 From Flatland to Spaceland

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