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Line 23: − <mm>[[Graphs And Polyhedrons.mm|Flash]]</mm>+ + − + + + + − + + + − + + + + − + + − + − #Defining a Graph, node arc and Region − #Framing Euler's Formula for graphs − #Verifying Euler's Formula N + R = A + 2 for given Plane graphs − #Drawing graphs for given N,R and A − #Identifying the Traversible graphs − #Explaining and using the condition for Traversible graphs − #defining a Polyhedra − #Framing Euler's formula for Polyhedra − # verifying Euler's formula for the given Polyhedra − − − − '''Representation of a Graph''' Line 57:
Line 57: + + + + + + + + + + + + + + − #Activity No #1+ − =Introduction to Graphs=+ + − {{#widget:YouTube|id=HmQR8Xy9DeM}}+ + + + − #Activity No #2+ + − + + + + + + + + + + + + − {{#widget:YouTube|id=wmOWLkBlarY}}+ + + − + + + + + − + + + − #Activity No #1+ − + + + + − + + + + − = Hints for difficult problems = − Statement : The Königsberg bridge problem : if the seven bridges of the city of Königsberg (left figure; Kraitchik 1942), formerly in Germany but now known as Kaliningrad and part of Russia, over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began. This is equivalent to asking if the multigraph on four nodes and seven edges (right figure) has an Eulerian cycle. This problem was answered in the negative by Euler (1736), and represented the beginning of graph theory. − http://photonics.cusat.edu/images/koning4.jpg + + + + + + + − Image Courtesy : http://mathworld.wolfram.com/KoenigsbergBridgeProblem.html+ + + + + + + + + + + + − Interpretation :+ + − [[File:bridge .jpeg|400px]] + − For Konigsberg, let us represent land with red dots and bridges with black curves, or arcs: − Thus, in its stripped down version, the seven bridges problem looks like this:+ − The problem now becomes one of drawing this picture without retracing any line and without picking your pencil up off the paper. Consider this: all four of the vertices in the above picture have an odd number of arcs connected to them. Take one of these vertices, say one of the ones with three arcs connected to it. Say you're going along, trying to trace the above figure out without picking up your pencil. The first time you get to this vertex, you can leave by another arc. But the next time you arrive, you can't. So you'd better be through drawing the figure when you get there! Alternatively, you could start at that vertex, and then arrive and leave later. But then you can't come back. Thus every vertex with an odd number of arcs attached to it has to be either the beginning or the end of your pencil-path. So you can only have up to two 'odd' vertices! Thus it is impossible to draw the above picture in one pencil stroke without retracing.+ Line 116:
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added Category:Networks and Polyhedra using HotCat
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''[http://karnatakaeducation.org.in/KOER/index.php/೧೦ನೇ_ತರಗತಿಯ_ನಕ್ಷೆ_ಮತ್ತು_ಬಹುಮುಖಘನಾಕೃತಿ ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ]''</div>
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]
| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
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= Concept Map =
= Concept Map =
[[File:Graphs And Polyhedrons.mm|Flash]]
__FORCETOC__
__FORCETOC__
= Textbook =
= Textbook =
[http://www.ncert.nic.in/ncerts/textbook/textbook.htm?hemh1=15-16 NCERT book on Graphs]
#[http://ktbs.kar.nic.in/New/Textbooks/class-x/english/maths/class-x-english-maths-chapter17.pdf Karnataka text book for Class 10, Chapter 17 - Graphs And Polyhedra]
#[http://www.ncert.nic.in/ncerts/textbook/textbook.htm?hemh1=15-16 NCERT book on Graphs]
=Additional Information=
=Additional Information=
[http://www.mhhe.com/math/ltbmath/bennett_nelson/conceptual/netgraphs/graphs.htm| More on Networks]<br>[http://resources.esri.com/help/9.3/arcgisengine/dotnet/e084da94-d4f7-4da7-86ed-7df684ff2144.htm| Extending Graph Theory]
==Useful websites==
==Useful websites==
[http://en.wikipedia.org/wiki/Graph_theory Wikipedia page for Graph Theory]
The document linked below gives few ideas in using story telling as a tool for understanding, interpreting and constructing graphs. Suggestions on how to assist students in making connections between graphs and the real world have also been given here.
[http://www.tess-india.edu.in/sites/default/files/imported/57360/SM15_AIE_Final.pdf Developing stories: Understanding graphs]
[http://www.enchantedlearning.com/math/geometry/solids/ For More Informations on Platonic Solids]
Other useful websites
# [http://en.wikipedia.org/wiki/Graph_theory Wikipedia page for Graph Theory]
# [http://www.enchantedlearning.com/math/geometry/solids/ For More Informations on Platonic Solids]
# [http://www.mathsisfun.com/platonic_solids.html/ For interactive Platonic Solids]
==Reference Books==
==Reference Books==
http://dsert.kar.nic.in/textbooksonline/Text%20book/Kannada/class%20x/maths/Kannada-class%20x-maths-contents.pdf
[http://dsert.kar.nic.in/textbooksonline/Text%20book/Kannada/class%20x/maths/Kannada-class%20x-maths-contents.pdf| Click here for DSERT 10 th Text book chapter Graph Theory]<br>
[http://toihoctap.wordpress.com/2013/02/13/introduction-to-graph-theory-and-solution-manual-by-douglas-b-west| Introduction to Graph Theory, By Douglas B.West/]
= Teaching Outlines =
= Teaching Outlines =
==Concept #1 Representation of a Graph==
==Concept==
===Learning objectives===
===Learning objectives===
#To define what is node.
#To define what is node.
#To represent a Graph with node, Arc and Regions
#To represent a Graph with node, Arc and Regions
===Notes for teachers===
Here we should remember in any Graph a point which is not represented by letter cannot be considered as NODE
===Activities===
Activity #1
[[Graphs_And_Polyhedra_activities_Activity1| Introduction to Graphs]]
Activity #2
[[Graphs_And_Polyhedra_Representation_of_a_Graph_activity_2| Graph Theory]]
==Concept #2 Types of Graphs==
===Learning objectives===
#To identify Plane Graph
#To identify Non-Plane Graph
===Notes for teachers===
===Notes for teachers===
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
===Activities===
===Activities===
Activity No #1<br>
[[Graphs_And_Polyhedra_regular_polyhedrons_activity_1#Activity_-_Construction_of_Regular_Polyhedrons | Construction of regular polyhedrons]] <br>
Activity No #2
==Concept #3 Eulers formula for graph==
===Learning objectives===
#Generalization of Euler's formula
#Verification of Euler's formula for Networks
===Notes for teachers===
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
=Graph Theory=
===Activities===
Activity No #1
[[Graphs_And_Polyhedra_Concept_3_Eulers_formula_for_graph_activity_1#Activity_-_Verification_of_Euler.27s_Formula_for_Graphs|Verification of Euler's Formula for Graphs]]<br>
Activity No #2 [[Graphs_And_Polyhedra_Concept_traversibility#Multimedia_resources| Activity on verification of eulers formula]]
==Concept # 4 Traversibility of a graph==
===Learning objectives===
#To Identify even order node
#To Identify Odd order node
#Condition for Traversibility
#Condition for Non- Traversibility of Graph
===Notes for teachers===
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
===Activities===
Activity No #1 [[Graphs_And_Polyhedra_concept4_activity1#Activity_-_Transversable_Networks| Transversable_Networks]]<br>
Activity No #2 [[Graphs_And_Polyhedra_concept4_activity1#Activity_-_Transversable_Networks| Eulers formula verification]]
==Concept #==
==Concept # 5 Shapes of Polyhedrons==
===Learning objectives===
===Learning objectives===
#Recognize regular and irregular polyhedron
#Can write differences between regular and irregular polyhedron
===Notes for teachers===
===Notes for teachers===
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
''there can only be 5 platonic polyhedrons.''
=Poly Hydrens=
==Definition==
===Activities===
===Activities===
Activity No #1
#Activity No #2
[[Graphs_And_Polyhedra_Activities_6_Octahedron#Activity_-_Recognising_the_elements_through_the_construction_of_octahedron_in_origami|Construction of regular octahedron and recognising th elements of Polyhedrons]]<br>
Activity No #2
[[Graphs_And_Polyhedra_Concept_7_polyhedra_elements#Activity_-_Polyhedra_Elements| Polyhedra_Elements]]
[https://www.mathsisfun.com/]
=Assessment activities for CCE=
==Concept # 6 Elements of Polyhedrons==
===Learning objectives===
#Recognizes vertexes faces and edges of a polyhedron
#Can count number of vertexes faces and edges of a polyhedron
===Notes for teachers===
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
===Activities===
Activity No #1
[[Graphs_And_Polyhedra_Activities_6_Octahedron#Activity_-_Recognising_the_elements_through_the_construction_of_octahedron_in_origami|Construction of regular octahedron and recognising th elements of Polyhedrons]]<br>
Activity No #2
[[Graphs_And_Polyhedra_Concept_7_polyhedra_elements#Activity_-_Polyhedra_Elements| Polyhedra_Elements]]
==Concept # 7 Euler's Formula for Polyhedrons==
===Learning objectives===
#Can count number of vertexes faces and edges of a polyhedron
#Verifies Euler's formula for a given polyhedron
===Notes for teachers===
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
===Activities===
Activity No #1 [http://karnatakaeducation.org.in/KOER/en/index.php/Graphs_And_Polyhedra/concept7/activity1| Activity on Eulers Theorem] <br>
Activity No #2 [[:File:G1-eulerworksheet.pdf| Work sheet on Verification of Eulers Formula for Ployhedrons]]
=Assessment activities for CCE=
[http://wps.pearsoned.com.au/mfwa1-2/62/16069/4113811.cw/-/4113819/index.html| Check your basic knowledge on Polyhedrons]<br>[http://www.mathsisfun.com/geometry/platonic-solids-why-five.html | Why there are only 5 platonic solids?]
= Hints for difficult problems =
Statement : The Königsberg bridge problem : if the seven bridges of the city of Königsberg (left figure; Kraitchik 1942), formerly in Germany but now known as Kaliningrad and part of Russia, over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began.
http://photonics.cusat.edu/images/koning4.jpg
Image Courtesy : http://mathworld.wolfram.com/KoenigsbergBridgeProblem.html
For solution click [[Graphs_and_polyhedra_problems|'''here''']]
= Project Ideas =
= Project Ideas =
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template
[[Category:Networks and Polyhedra]]