Difference between revisions of "Graphs And Polyhedra"

From Karnataka Open Educational Resources
Jump to navigation Jump to search
 
(96 intermediate revisions by 8 users not shown)
Line 1: Line 1:
 +
<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#ffffff; vertical-align:top; text-align:center; padding:5px;">
 +
''[http://karnatakaeducation.org.in/KOER/index.php/೧೦ನೇ_ತರಗತಿಯ_ನಕ್ಷೆ_ಮತ್ತು_ಬಹುಮುಖಘನಾಕೃತಿ ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ]''</div>
 +
 
<!-- This portal was created using subst:box portal skeleton  -->
 
<!-- This portal was created using subst:box portal skeleton  -->
 
<!--        BANNER ACROSS TOP OF PAGE        -->
 
<!--        BANNER ACROSS TOP OF PAGE        -->
 
{| id="mp-topbanner" style="width:100%;font-size:100%;border-collapse:separate;border-spacing:20px;"
 
{| id="mp-topbanner" style="width:100%;font-size:100%;border-collapse:separate;border-spacing:20px;"
 
|-
 
|-
−
|style="width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+
| 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:_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]
−
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+
| 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:_Pedagogy Teaching of Mathematics]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
−
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+
| 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/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
−
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+
| 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:_Topics Topics in School Mathematics]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
−
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+
| 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/Text_Books#Mathematics_-_Textbooks Textbooks]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
−
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
+
| 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/Maths:_Question_Papers Question Bank]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
 
|}
 
|}
Line 20: Line 23:
  
 
= 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]
 +
 
 +
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| 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 =
−
#history of Graph Theory
+
==Concept #1 Representation of a Graph==
−
#
+
===Learning objectives===
 +
#To define what is node.
 +
#to define what is arc
 +
#To define what is Region
 +
#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]]
  
−
==Concept #==
+
Activity #2
 +
[[Graphs_And_Polyhedra_Representation_of_a_Graph_activity_2| Graph Theory]]
 +
 
 +
==Concept #2 Types of Graphs==
 
===Learning objectives===
 
===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
+
Activity No #1<br>
−
Introduction to Graphs
 
  
 +
[[Graphs_And_Polyhedra_regular_polyhedrons_activity_1#Activity_-_Construction_of_Regular_Polyhedrons | Construction of regular polyhedrons]] <br>
  
−
{{#widget:YouTube|id=HmQR8Xy9DeM}}
+
Activity No #2
  
−
#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.''
 +
 
 +
===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 #==
+
==Concept # 4 Traversibility of a graph==
 
===Learning objectives===
 
===Learning objectives===
 +
#To Identify even order node
 +
#To Identify Odd order node
 +
#Condition for Traversibility
 +
#Condition for Non- Traversibility of 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
+
Activity No #1 [[Graphs_And_Polyhedra_concept4_activity1#Activity_-_Transversable_Networks| Transversable_Networks]]<br>
−
#Activity No #2
+
Activity No #2 [[Graphs_And_Polyhedra_concept4_activity1#Activity_-_Transversable_Networks| Eulers formula verification]]
 +
 
 +
==Concept # 5 Shapes of Polyhedrons==
 +
===Learning objectives===
 +
#Recognize regular and irregular polyhedron
 +
#Can write differences between regular and irregular polyhedron
 +
 
 +
 
 +
===Notes for teachers===
 +
''there can only be 5 platonic polyhedrons.''
 +
=Poly Hydrens=
 +
==Definition==
 +
 
 +
===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]]
 +
[https://www.mathsisfun.com/]
 +
 
 +
==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=
 
=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 =
 
= 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 =
Line 71: Line 175:
  
 
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]]

Latest revision as of 04:34, 5 November 2019

ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ

The Story of Mathematics

Philosophy of Mathematics

Teaching of Mathematics

Curriculum and Syllabus

Topics in School Mathematics

Textbooks

Question Bank

While creating a resource page, please click here for a resource creation checklist.

Concept Map

[maximize]


Textbook

  1. Karnataka text book for Class 10, Chapter 17 - Graphs And Polyhedra
  1. NCERT book on Graphs

Additional Information

More on Networks
Extending Graph Theory

Useful websites

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.

Developing stories: Understanding graphs

Other useful websites

  1. Wikipedia page for Graph Theory
  2. For More Informations on Platonic Solids
  3. For interactive Platonic Solids

Reference Books

Click here for DSERT 10 th Text book chapter Graph Theory
Introduction to Graph Theory, By Douglas B.West/

Teaching Outlines

Concept #1 Representation of a Graph

Learning objectives

  1. To define what is node.
  2. to define what is arc
  3. To define what is Region
  4. 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 Introduction to Graphs

Activity #2 Graph Theory

Concept #2 Types of Graphs

Learning objectives

  1. To identify Plane Graph
  2. To identify Non-Plane 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

Construction of regular polyhedrons

Activity No #2

Concept #3 Eulers formula for graph

Learning objectives

  1. Generalization of Euler's formula
  2. 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.

Activities

Activity No #1 Verification of Euler's Formula for Graphs
Activity No #2 Activity on verification of eulers formula

Concept # 4 Traversibility of a graph

Learning objectives

  1. To Identify even order node
  2. To Identify Odd order node
  3. Condition for Traversibility
  4. 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 Transversable_Networks
Activity No #2 Eulers formula verification

Concept # 5 Shapes of Polyhedrons

Learning objectives

  1. Recognize regular and irregular polyhedron
  2. Can write differences between regular and irregular polyhedron


Notes for teachers

there can only be 5 platonic polyhedrons.

Poly Hydrens

Definition

Activities

Activity No #1 Construction of regular octahedron and recognising th elements of Polyhedrons
Activity No #2 Polyhedra_Elements [1]

Concept # 6 Elements of Polyhedrons

Learning objectives

  1. Recognizes vertexes faces and edges of a polyhedron
  2. 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 Construction of regular octahedron and recognising th elements of Polyhedrons
Activity No #2 Polyhedra_Elements

Concept # 7 Euler's Formula for Polyhedrons

Learning objectives

  1. Can count number of vertexes faces and edges of a polyhedron
  2. 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 Activity on Eulers Theorem
Activity No #2 Work sheet on Verification of Eulers Formula for Ployhedrons

Assessment activities for CCE

Check your basic knowledge on Polyhedrons
| 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.


koning4.jpg


Image Courtesy : http://mathworld.wolfram.com/KoenigsbergBridgeProblem.html

For solution click here

Project Ideas

Math Fun

Usage

Create a new page and type {{subst:Math-Content}} to use this template

Retrieved from "https://karnatakaeducation.org.in/KOER/en/index.php?title=Graphs_And_Polyhedra&oldid=32326"