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Polyhedra, Platonic Solids, and Euler’s FormulaAuthor(s): John QuintanillaDate/Time Lesson to be Taught: June 11, 2012Technology Lesson: YesNoCourse Description: Name: Summer Mathematical Enrichment Class for Girls 2012Grade Level: Mostly 3rd gradersHonors or Regular: HonorsLesson Source: JQObjectives:SWBAT build polyhedra from maps of polyhedra. SWBAT learn the definitions for types of polyhedra and parts of polyhedra.SWBAT accurately count the number of vertices, faces, and edges of the Platonic solids.SWBAT will learn and apply Euler’s formula. Texas Essential Knowledge and Skills:§111.23. Mathematics, Grade 7. (b)??Knowledge and skills(8) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world. The student is expected to: (B) make a net (two-dimensional model) of the surface area of a three-dimensional figureMaterials List and Advanced Preparations:Polyhedral maps of five Platonic solids and a truncated icosahedron (soccer ball). These can be obtained from or by Googling “polyhedral net”Worksheet for compiling resultsPaper for writing vocabulary wordsDice (an example of a Platonic solid)Post-AssessmentsScissorsTapePaperPencilAccommodations for Learners with Special Needs (ELL, Special Ed, 504, GT, learning styles, etc.): None provided below, though this could be added.5EsENGAGEMENT 1Time: 10 MinutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsOn a piece of paper, draw a figure with three sides that are exactly the same.Now draw a figure with four sides that are exactly the same.Those were too easy. Now draw a figure that has eight sides that are exactly the same.The figures we just did were easy because they were draw on a piece of paper. Let’s now start thinking about figures in three dimensions.Figures in three dimensions are called solids.Can you think of a solid so that all of its sides are the same?That’s right, a cube! Where have you seen cubes before?Can you think of any other solids which have all of its sides the same?What is the name of the shape you just drew?What is the name of the shape you just drew?What is the name of the shape you just drew?Have you ever seen any of these shapes while driving on a highway?If I gave you enough time, do you think it’s possible to draw a figure that has 12 sides that are exactly the same? 100 sides?What does “two dimensions” mean? What does “three dimensions” mean?Students draw equilateral triangles.“Triangle!” “Equilateral triangle!”Students draw squares.“Square!”Students struggle to draw a regular octagon.“Octagon!” [maybe]“Road signs!” “Stop sign!” “Yield sign!”“That’d be hard!”“Yes!”Two: “It’s flat”Three: “It’s not flat” Students write solid as a vocabulary word.“That’s hard!” [This could take some time before someone comes up with the correct answer.]“A cube!” “Ice!”“Dice!”[Probably silence. Someone may think of a pyramid.]Evaluation/Decision Point AssessmentStudent OutcomesOnce the students have confirmed their knowledge of regular polygons, we are ready to move into the next stage. Students have demonstrated geometrical and linguistic knowledge of regular polygons.EXPLORATION 1Time: 45 MinutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and Misconceptions It turns out that there are only five solids that have all sides the same. These are called the Platonic solids.[Writing on board]There is a tetrahedron, which has 4 sides. Each side is called a face.There is a cube, which has 6 faces.There is an octahedron, which has 8 faces.There is a dodecahedron, which has 12 faces. “Dodec” means 12 in Greek.And there is an icosahedron, which has 20 faces.What we’re going to do is build each of these by cutting and folding some paper. [Hands out polyhedral nets]Let’s start with the tetrahedron, and then build our way to the more complicated solids. Cut along the outline of each shape, and then fold on the solid lines. Use the tape to make the solid. You’ll notice that some of the regions are actually little flaps. You’ll use those to help tape the solid together.If you finish early, I’m also giving you the cutout for a truncated isocahedron. This is a semi-regular Platonic solid since it has both pentagons and hexagons.[Teacher helps students as they struggle to tape the solids, stopping every few minutes to display solids to the class.]Does anyone know what tetra means? Can you think of a word based on tetra?Does anyone know what octa means? Can you think of a word based on octa?Have you even seen one of these before?Think about sports.Does anyone here play soccer?Students write Platonic solids on their vocabulary sheets.[As the lesson continues, students write the name of each Platonic solid and the number of sides on their vocabulary sheets.]Tetris! Each shape in Tetris has four squares!Possible student question: “Does the cube have a long name too?” [Answer: hexahedron]Eight!Octagon!Octopus![Stunned silence]Oooooh. A soccer ball![Students start building the solids. Difficulties really start arising with the octahedron.]Evaluation/Decision Point AssessmentStudent OutcomesConstruction of the five Platonic solids.The students have constructed at least four of the five solids in the time allotted. EXPLANATION 1Time: 15 MinutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsLet’s think about shapes in two dimensions again. Good. Let’s now return to the Platonic solids.Good. Except that the each side of a polyhedron is called a face.Good. Each lines --- a sides of each face --- is called an edge.Good. Each corner is called a vertex.Ha, ha, I tricked you. Actually, if you have more than one vertex, they’re called vertices.On a piece of paper, what is a shape with three sides called?What are the parts of a triangle called?What parts do you see of a solid?If there’s more than one face, what are they called?What other parts of the solid do you see?If there’s more than one edge, what are they called?There’s one more part of the solid. What is it?If there’s more than one vertex, what are they called?So how many parts of a polyhedron are there?And how many parts of a triangle were there?Does this make sense?A triangle!Sides! Corners!Sides.[Students write face on their vocabulary sheet.]Faces!Lines![Students write edge on their vocabulary sheet.]Edges!Corners![Students write vertex on their vocabulary sheet.]Vertexes![Students write vertices on their vocabulary sheet.]3: faces, edges, vertices.2: sides and corners (or edges and vertices).No.Yes! Solids have 3 dimensions, but triangles only have 2 dimensions.Evaluation/Decision Point AssessmentStudent OutcomesTeacher names the attributes of polyhedra.Students are able to name the attributes of polyhedra. EVALUATION 1Time: 10 MinutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsPost-Assessment 1. ENGAGEMENT 2Time: 5 minutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsLet’s now count the number of faces, edges and vertices for each solids. Let’s begin with the tetrahedron. Please take out and hold the tetrahedron that you built. I’m going to pass out a chart. [Passes out charts]Next to the tetrahedron, please write the word tetrahedron. Then please write down the number of vertices, faces, and edges in the chart.How many faces does the tetrahedron have?How many edges does the tetrahedron have?How many vertices does the tetrahedron have?[Students identify, find, and hold their tetrahedra.]464Students enter tetrahedron 4, 4, 6 in the chart on the first line of the chart.Evaluation/Decision Point AssessmentStudent OutcomesReinforce definitions of face, edge, vertex.Students can correctly identify and count the number of faces, edges and vertices for a tetrahedron. EXPLORATION 2Time: 10 minutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsPlease write those names on the chart.OK, I’m going to turn you loose. Work in groups to count the number of faces, edges and vertices for the other Platonic solids. If you were able to build the truncated icosahedron (soccer ball), then count the parts for that too.What were the names of the other Platonic solids?Cube, octahedron, dodecahedron, icosahedron.[Students enter the names.][Students work in groups to count the parts of the other solids. They will probably begin to have difficulty with the dodecahedron.]Evaluation/Decision Point AssessmentStudent OutcomesReinforce definitions of face, edge, vertex.Students correctly count the number of faces, edges and vertices for a cube and an octahedron. Some students are able to correctly count for the others.EXPLANATION 2Time: 20 minutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsLet’s now review what you’ve found.Wow, lots of different answers for that one.Yes, it is. So let’s take another look at the easier Platonic solids to see if we can come up with a better way of counting the parts of the dodecahedron and the icosahedron.That’s right. So we’ve just come up with a different way of counting the number of edges: 4 triangles * 3 edges per triangle / 2 = 6.Let’s now think about the number of vertices.Excellent. Let’s now do the same thing for the cube.Excellent. Now I’m going to turn you loose. Do the exact same thing to recount the number of edges and vertices for the other three Platonic solids.If you finish early, you may want to think about how you could use this same idea to count the vertices and edges for the truncated icosahedron.[Teacher circles room, answering questions as necessary. Answers:Octa: F = 8, E = 12, V = 6Dodec: F = 12, E = 30, V = 20Icosa: F = 20, E = 30, V = 12]What did you find for the tetrahedron?What did you find for the cube?What did you find for the octahedron?What did you find for the dodecahedron?Why do you think you all got answers that are a little bit different?First off, how many triangles are in a tetrahedron?And how many edges does each triangle have?Four triangles, and three edges per triangle. So how many total edges are there?But wait… you told me before that the tetrahedron had only 6 edges. What happened?How many triangles are in a tetrahedron?And how many vertices does each triangle have?Four triangles, and three vertices per triangle. So how many total vertices are there?That’s right, it won’t be twelve. Last time, we saw that each edge belonged to two different triangles. How about the vertices?Good. There are four triangles, three vertices per triangle, but each vertex is triple-counted. So how many total vertices are there?How many squares are in a cube?And how many edges does each square have?And each edge belongs to how many squares?Good: six squares and four edges per square, but each edge belongs to two squares. So how many edges are there?And does that match what you found earlier?Good. Now tell me about the vertices.And does that match what you found earlier?4 vertices, 4 faces, 6 edges.8 vertices, 6 faces, 12 edges.6 vertices, 8 faces, 12 edges.[Answers vary.]It’s hard to count so many vertices and edges!Four!Three!Four times three, so twelve![After some thought]Oooh, each edge is counted twice!Four!Three!Well, it’s not going to be twelve.[After some thought]Oooh, each vertex belongs to three trianges!4 * 3 / 3 = 4 vertices.Six!Four!Two!6 * 4 / 2 = 12Yes!Six squares.Four vertices per square.Each vertex belongs to three squares.6 * 4 / 3 = 8Yes![Students work on the remaining three solids.]Evaluation/Decision Point AssessmentStudent OutcomesMove to the next phase once the students correctly compute the edges and vertices for the other Platonic solids. Students will understand how to use basic combinatorics to count the number of edges and vertices of regular Platonic solids.ELABORATION 2Time: 15 minutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsAll right, we’ve collected all of these numbers. Good observation. The edge has 6 faces and 8 vertices, but the octahedron has 6 vertices and 8 faces. Both have 12 edges.That’s a good question. This actually isn’t a coincidence, and there’s a whole branch of mathematics called graph theory that can explain why that happened.Excellent. This is called Euler’s [pronounced oilers] formula. This is named for Leonhard Euler, who was a famous Swiss mathematician who lived in the 1700s.It turns out this formula is true for any polyhedron, not just the five that we studied today. Does anyone recognize any patterns with these numbers?Does anybody see another pattern?This one’s a little more subtle. Let me give you a hint. For each solid, add the number of faces and the number of vertices.Now do you see a pattern?Great observation. How would you write this as a formula?[Stunned silence for a minute.]I’m not sure if this is right, but I see the same numbers repeated for the cube and the octahedron.Oooh. The dodecahedron has 12 faces and 20 vertices, but the icosahedron has 12 vertices and 20 faces. Both have 30 edges.Why did that happen?[Stunned silence.][Students add.]Oooh. The sum is two more than the number of edges!Number of vertices + number of faces = number of edges + 2Number of vertices + number of faces – number of edges = 2[Students write Euler’s formula on their vocabulary sheet.] Evaluation/Decision Point AssessmentStudent OutcomesStudents have inductively arrived at Euler’s formula.Students will understand Euler’s formula. EVALUATION 2Time: 10 minutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsPost-Assessment 2. ................
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