grade level/course: grade 6
TRANSCRIPT
Page 1 of 3 FabLab@WCCUSD 05/12/16
GradeLevel/Course:Grade6Lesson/UnitPlanName:PlatonicSolids—UsinggeometricnetstoexplorePlatonicsolidsanddiscoveringEuler’sformula.Rationale/LessonAbstract:Anactivitywherethestudentsusenetstoexploresurfacearea,andthetermsvertices,sidesandfaces.Timeframe:1–2classperiodsCommonCoreStandard(s):6.G.4:Representthree-dimensionalfiguresusingnetsmadeupofrectanglesandtriangles,andusethenetstofindthesurfaceareaofthesefigures.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
InstructionalResources/Materials:• Cubenets–includedattheendofthislessonplan.• HistoryInformation–includedattheendofthislessonplan.• Platonicsolidnetmasters–includedattheendofthislessonplan.• 3dprintedPlatonicsolidnets.Thestlfilesforthe3dprintedPlatonicsolidnetsare
locatedontheFabLabwebsite:http://www.wccusd.net/Page/5262• PlatonicSolidsPattern(i.e.,Euler’sFormula)worksheet–includedattheendofthis
lessonplan.Activity/Lesson:Day1:
1. Askstudentstodefinetheterms:faces,edges,andvertices:
Avertexisacorner.Avertex(plural:vertices)isapointwheretwoormorestraightlinesmeet.
Anedgejoinsonevertexwithanother. Anedgeisalinesegmentthatjoinstwovertices.
Afaceisanindividualsurface.Afaceisanyoftheindividualsurfacesofasolidobject.
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2. DefineGeometricNet:Ageometricnetisa2-dimensionalshapethatcanbefoldedtoforma3-dimensionalshapeorasolid.Oranetisapatternmadewhenthesurfaceofathree-dimensionalfigureislaidoutflatshowingeachfaceofthefigure.Asolidmayhavedifferentnets.Netsarehelpfulwhenweneedtofindthesurfaceareaofthesolids.
3. Reviewwithstudentswhatacubeisandaskthemhowmanyfaces,edges,andverticesdoesacubehave.
4. Handoutthecubenetsandaskstudentstocutthemout,foldalongthelinesandseeiftheycanmakeacubeoutofallofthenets.Note:thereareonly11netsforacubeshownbelow.
Thewebsitebelowisanactivitywherestudentsclickonanet,answeriftheythinkitcanmakeacube,andwatchtheanimationtodetermineiftheyarecorrect.Itisagreatextensiontothislesson.
https://illuminations.nctm.org/activity.aspx?id=3544
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Day2:
5. ExplaintostudentsthattodaytheywilllearnaboutPlatonicSolids.UsetheHistoryInformationpages–includedattheendofthislessonplan.
a. DefinewhatPlatonicsolidsareb. SharehistoryofPlatonicsolids
6. Nowthatstudentsunderstandthetermsfaces,edges,vertices,nets,andPlatonicsolids,handoutthePlatonicSolidsPatternsworksheet.Completethesecondrow,aboutacube,withstudents.
7. HandoutthePlatonicSolidsPatternsworksheetandmodels.Youcanusethe3dprintedPlatonicSolidsmodelsortheincludedPlatonicSolidNetmastersforthispartofthelesson.Usingthemodels,ingroups,studentscompletethePlatonicSolidsPatternsworksheet.
8. Usingthemodels(3dprintedorconstructedmodels),reviewtheanswerstothePlatonicSolidsPatternsworksheet.
9. Tellstudentsthatafamousmathematician,Euler,discoveredtheformulaF+V−E=2forPlatonicsolidsusingmodels–justliketheyhavedone.
Assessment:YoucanusethePlatonicSolidsPatternsworksheetastheassessmentofstudentunderstandingofthislesson.
Philip Gonsalves
HistoryInforma.onPages
Philip Gonsalves
In three-dimensional space, a Platonic solid is
made by using polygons (triangles and squares)
congruent regular polygonal faces with the same
number of faces m
eeting at each vertex. Only
five solids meet those criteria, and each is
named after its num
ber of faces.
What is a Platonic Solid?
Philip Gonsalves
The Platonic Solids — There are only 5 of them
.
Philip Gonsalves
The Greek philosopher Plato, w
ho was born around 430
B.C
., wrote about these five solids in a w
ork called Timaeus.
The Platonic solids themselves w
ere discovered by the early Pythagoreans, perhaps by 450 B
.C. There is evidence that
the Egyptians knew about at least three of the solids; their
work influenced the Pythagoreans.
Philip Gonsalves
ThePlatonicsolidswereknow
ntotheancientGreeks,andw
eredescribedbyPlato.Inthisw
ork,Platoequatedthetetrahedronwiththe"elem
ent"fire,thecubewithearth,
theicosahedronwithw
ater,theoctahedronwithair,andthedodecahedronw
iththestuffofw
hichtheconstella.onsandheavensw
eremade(Crom
well1997).
Philip Gonsalves
Plato mentioned these solids in w
riting, and it was he w
ho identified the solids w
ith the elements com
monly believed to
make up all m
atter in the universe.
Philip Gonsalves
Back in history, they didn’t have plastic and m
ade m
athematical m
odels out of rocks. Below
are samples
of the Platonic solids that were m
ade out of rocks.
Philip Gonsalves
Cube
Tetrahedron
Octahedron
Dodecahedron
Icosahedron
Name: Date: Period:
Platonic Solids Patterns Count and record the number of faces ( )F , vertices ( )V , and edges ( )E for each of the
Platonic Solids. Then calculate the values for F V E+ − . The Tetrahedron is already done.
Name Figure Faces ( )F
Vertices ( )V
Edges ( )E F V E+ −
Tetrahedron
4 4 6 4 4 6 2+ − =
Hexahedron (cube)
______ ______ ______ ________
Octahedron
______ ______ ______ ________
Dodecahedron
______ ______ ______ ________
Icosahedron
______ ______ ______ ________
Answer these questions on the back side of this paper.
1. What pattern did you observe in the last column? 2. Determine whether or not this relationship holds with an ordinary box (rectangular prism). 3. Explore this relationship with another polyhedron that has faces, vertices, and edges but is
not a rectangular prism or Platonic Solid.
Gensburg 2004
F + V– E