Difference between revisions of "Similarity and Congruence"

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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
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[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
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|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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|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://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; "|
 
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
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[http://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]
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[http://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]
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[http://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]
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[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
 
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
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[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
 
|}
 
|}
 
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
 
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
  
 
= Concept Map =
 
= Concept Map =
−
<mm>[[5a._Similar_and_Congruent_triangles.mm|Flash]]</mm>
 
 
__FORCETOC__
 
__FORCETOC__
 +
[[File:similar_ and _congruent triangles.mm|flash]]
 +
 
= Textbook =
 
= Textbook =
 
To add textbook links, please follow these instructions to:  
 
To add textbook links, please follow these instructions to:  
Line 39: Line 40:
 
''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.''
  
−
===Activity No #1 ===
+
===Activity No #1 Identifying similar shapes, similar triangles===
 
{| style="height:10px; float:right; align:center;"
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
−
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
+
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
|}
−
 
+
* '''Estimated Time''' - 80 minutes (40 +40)
−
 
+
* '''Materials/ Resources needed''' - Blackboard, Geogebra files + projector, Calculator
−
 
−
 
 
−
 
−
 
 
−
 
 
−
 
−
==== Material and Resources Required ====
 
−
 
−
Blackboard
 
−
 
 
−
 
−
Geogebra files +
 
−
projector
 
−
 
 
−
 
−
Calculator
 
−
 
 
−
 
−
 
 
−
'''[[Pre-requisites/Instructions]] '''
 
−
 
 
−
 
−
* Planar figures and triangles
 
−
* Draw pairs of figures on the board [ both similar and dissimilar]; they can identify overlap of congruent figures
 
−
* Ask the children to identify
 
−
* If the children know the names of the theorem, ask them to explain- ask them what is SSS, AAA, ASA
 
−
* Show ratio and give the idea of proportionality
 
−
* Geogebra files. When I change the sides/ proportion, the triangles change in size. But the proportion remains the same, angle remains the same
 
−
* With calculator they verify proportion (this is very very useful for involving the whole class) they all can see the proportion remains constant though the size changes
 
−
* Show the arithmetic behind the proportion
 
−
 
−
==== Evaluation ====
 
−
 
−
[Activity evaluation -
 
−
What should the teacher watch for when you do the activity; based on
 
−
what they know change]
 
−
 
 
−
 
−
* Confusion between congruence and similarity
 
−
* When they give the theorem, if they cannot identify included side and angle
 
−
* When there is a wrong answer, identify what is the source of the confusion – sides, ratio and proportion
 
−
 
−
* Direct substitution
 
−
 
 
−
 
 
−
* '''Estimated Time'''
 
−
* '''Materials/ Resources needed'''
 
 
* '''Prerequisites/Instructions, if any'''
 
* '''Prerequisites/Instructions, if any'''
 
* '''Multimedia resources'''
 
* '''Multimedia resources'''
 +
[[Image:KOER%20Triangles_html_m3d25043b.jpg|300px|left]]<br><br>Suchetha . S. S  Asst. Teacher ( Mathematics ) GJC Thyamagondlu. Nelamangala Talluk Bangalore Rural District doing the activity below on a lesson on similar triangles using GeoGebra in the classroom<br><br><br><br><br><br><br><br>
 
* '''Website interactives/ links/ Geogebra Applets'''
 
* '''Website interactives/ links/ Geogebra Applets'''
 
* '''Process (How to do the activity)'''
 
* '''Process (How to do the activity)'''
 +
#Draw figures on the boards - planar figures and triangles
 +
#Draw pairs of figures on the board [ both similar and dissimilar]; they can identify overlap of congruent figures
 +
#Demonstrate using Geogebra files and discuss
 +
#As part of the procedure, the teacher can use the following points to evaluate the activity as well as evaluate student understanding
 +
*Confusion between congruence and similarity
 +
*When they give the theorem, if they cannot identify included side and angle
 +
*When there is a wrong answer, identify what is the source of the confusion – sides, ratio and proportion
 
* '''Developmental Questions (What discussion questions)'''
 
* '''Developmental Questions (What discussion questions)'''
−
* '''Evaluation (Questions for assessment of the child)'''
+
# Can you identify similar figures?
 +
# If the children know the names of the theorem, can they explain SSS, AAA, ASA?
 +
# What does a ratio of side mean? What is proportionality?
 +
# In the Geogebra files, when I change the sides/ proportion, the triangles change in size. But the proportion remains the same, angle remains the same
 +
#With calculator  verify the proportion (this is very very useful for involving the whole class) they all can see the proportion remains constant though the size changes
 +
#Show the arithmetic behind the proportion
 +
*'''Evaluation'''
 +
#Direct substitution in simple textbook problems
 +
#With a pencil and a ruler, how can you estimate the height of a pole
 
* '''Question Corner'''
 
* '''Question Corner'''
  
−
===Activity No # ===
+
===Activity No # 2 Review of the concepts of similarity and congruence===
 
{| style="height:10px; float:right; align:center;"
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
−
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
+
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
|}
−
* '''Estimated Time'''
+
* '''Estimated Time''' - 30 minutes
−
* '''Materials/ Resources needed'''
+
* '''Materials/ Resources needed''' -Projector, Computer
 
* '''Prerequisites/Instructions, if any'''
 
* '''Prerequisites/Instructions, if any'''
 
* '''Multimedia resources'''
 
* '''Multimedia resources'''
 +
{{#widget:YouTube|id=BI-rtfZVXy0}}
 
* '''Website interactives/ links/ Geogebra Applets'''
 
* '''Website interactives/ links/ Geogebra Applets'''
 
* '''Process (How to do the activity)'''
 
* '''Process (How to do the activity)'''
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* '''Evaluation (Questions for assessment of the child)'''
 
* '''Evaluation (Questions for assessment of the child)'''
 
* '''Question Corner'''
 
* '''Question Corner'''
−
 
  
 
==Concept #==
 
==Concept #==
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{| style="height:10px; float:right; align:center;"
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
−
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
+
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
|}
 
* '''Estimated Time'''
 
* '''Estimated Time'''
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{| style="height:10px; float:right; align:center;"
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
−
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
+
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
|}
 
* '''Estimated Time'''
 
* '''Estimated Time'''
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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
−
 
−
 
−
== Evaluation ==
 
−
 
−
== Self-Evaluation ==
 
−
 
−
== Further Explorations ==
 
−
 
−
== Enrichment Activities ==
 
−
 
−
= Pythagorean Theorem =
 
−
 
−
Pythagoras' Theorem was discovered by
 
−
Pythagoras, a Greek mathematician and philosopher who lived
 
−
between approximately 569 BC and 500 BC.
 
−
 
−
 
−
Pythagoras'
 
−
Theorem states that:
 
−
 
−
 
−
In
 
−
any right-angled triangle, the square of the hypotenuse is equal
 
−
to the sum of the squares of the other two sides. That is:
 
−
 
−
 
−
[[Image:KOER%20Triangles_html_m2f096af4.png]]
 
−
 
−
 
−
[[Image:KOER%20Triangles_html_4bd439df.png]]
 
−
 
−
 
−
 
−
 
−
'''Pythagoras' Theorem in
 
−
Three Dimensions'''
 
−
 
−
 
−
A
 
−
three-dimensional object can be described by three measurements -
 
−
length, width and height.
 
−
 
−
 
−
[[Image:KOER%20Triangles_html_679fe2f6.png]][[Image:KOER%20Triangles_html_m570261d2.png]]
 
−
 
−
 
−
We
 
−
can use Pythagoras' Theorem to find the length of the longest
 
−
straw that will fit inside
 
−
 
−
 
−
the
 
−
box or cylinder.
 
−
 
−
 
−
== Evaluation ==
 
−
 
−
== Self-Evaluation ==
 
−
 
−
== Further    Explorations ==
 
−
 
 
−
== Enrichment    Activities ==
 
−
 
−
= See    Also =
 
−
 
−
 
−
 
−
 
−
 
−
 
−
 
−
 
−
 
−
= Teachers    Corner =
 
−
 
−
Suchetha . S. S  Asst. Teacher ( Mathematics ) GJC Thyamagondlu. Nelamangala Talluk Bangalore Rural District doing a lesson on similar triangles using GeoGebra in the classroom
 
−
 
−
[[Image:KOER%20Triangles_html_m3d25043b.jpg|600px]]
 
−
 
−
 
−
== GeoGebra Contributions ==
 
−
# The GeoGebra file below to understand Similar Triangles <br>
 
−
## Similar Triangles Part 1 http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_1.html
 
−
## Download ggb file here http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_1.ggb
 
−
## Similar Triangles Part 2 http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_2.html
 
−
## Download ggb file here http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_2.ggb
 
−
## Similar Triangles Part 3 http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_3.html
 
−
## Download ggb file here http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_3.ggb
 
−
## See a video to understand this concept http://www.youtube.com/watch?v=BI-rtfZVXy0
 
−
 
−
# The GeoGebra file below verifies the Thales theorem <br>
 
−
## Thales Theorem http://www.karnatakaeducation.org.in/KOER/Maths/thales_theorem.html
 
−
## Download ggb file here http://www.karnatakaeducation.org.in/KOER/Maths/thales_theorem.ggb
 
−
## See a video that proves this theorem http://www.youtube.com/watch?v=Y-6yYsuGLoc
 
−
 
−
= Books =
 
−
 
−
 
−
 
−
 
−
 
−
= References =
 
−
== Enrichment    Activities ==
 
−
 
 
−
=== Activity 2 Similar Triangles ===
 
−
 
−
'''Learning Objective'''
 
−
 
−
 
−
To show similar planar
 
−
figures, discuss congruence and properties of congruent/ similar
 
−
triangles
 
−
 
−
 
−
 
−
 
−
 
−
==== Material and Resources Required ====
 
−
 
−
Blackboard
 
−
 
−
 
−
Geogebra files +
 
−
projector
 
−
 
−
 
−
Calculator
 
−
 
−
 
−
 
−
'''[[Pre-requisites/Instructions]] '''
 
−
 
−
 
−
* Planar figures and triangles
 
−
* Draw pairs of figures on the board [ both similar and dissimilar]; they can identify overlap of congruent figures
 
−
* Ask the children to identify
 
−
* If the children know the names of the theorem, ask them to explain- ask them what is SSS, AAA, ASA
 
−
* Show ratio and give the idea of proportionality
 
−
* Geogebra files. When I change the sides/ proportion, the triangles change in size. But the proportion remains the same, angle remains the same
 
−
* With calculator they verify proportion (this is very very useful for involving the whole class) they all can see the proportion remains constant though the size changes
 
−
* Show the arithmetic behind the proportion
 
−
 
−
==== Evaluation ====
 
−
 
−
[Activity evaluation -
 
−
What should the teacher watch for when you do the activity; based on
 
−
what they know change]
 
−
 
−
 
−
* Confusion between congruence and similarity
 
−
* When they give the theorem, if they cannot identify included side and angle
 
−
* When there is a wrong answer, identify what is the source of the confusion – sides, ratio and proportion
 
−
 
−
* Direct substitution
 

Latest revision as of 15:40, 17 May 2017

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

To add textbook links, please follow these instructions to: (Click to create the subpage)

Additional Information

Useful websites

Reference Books

Teaching Outlines

Concept #1 - Understanding similarity

Learning objectives

  1. To show similar planar figures, discuss congruence and properties of congruent/ similar triangles

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.

Activity No #1 Identifying similar shapes, similar triangles

  • Estimated Time - 80 minutes (40 +40)
  • Materials/ Resources needed - Blackboard, Geogebra files + projector, Calculator
  • Prerequisites/Instructions, if any
  • Multimedia resources
KOER Triangles html m3d25043b.jpg



Suchetha . S. S Asst. Teacher ( Mathematics ) GJC Thyamagondlu. Nelamangala Talluk Bangalore Rural District doing the activity below on a lesson on similar triangles using GeoGebra in the classroom







  • Website interactives/ links/ Geogebra Applets
  • Process (How to do the activity)
  1. Draw figures on the boards - planar figures and triangles
  2. Draw pairs of figures on the board [ both similar and dissimilar]; they can identify overlap of congruent figures
  3. Demonstrate using Geogebra files and discuss
  4. As part of the procedure, the teacher can use the following points to evaluate the activity as well as evaluate student understanding
  • Confusion between congruence and similarity
  • When they give the theorem, if they cannot identify included side and angle
  • When there is a wrong answer, identify what is the source of the confusion – sides, ratio and proportion
  • Developmental Questions (What discussion questions)
  1. Can you identify similar figures?
  2. If the children know the names of the theorem, can they explain SSS, AAA, ASA?
  3. What does a ratio of side mean? What is proportionality?
  4. In the Geogebra files, when I change the sides/ proportion, the triangles change in size. But the proportion remains the same, angle remains the same
  5. With calculator verify the proportion (this is very very useful for involving the whole class) they all can see the proportion remains constant though the size changes
  6. Show the arithmetic behind the proportion
  • Evaluation
  1. Direct substitution in simple textbook problems
  2. With a pencil and a ruler, how can you estimate the height of a pole
  • Question Corner

Activity No # 2 Review of the concepts of similarity and congruence

  • Estimated Time - 30 minutes
  • Materials/ Resources needed -Projector, Computer
  • Prerequisites/Instructions, if any
  • Multimedia resources

  • Website interactives/ links/ Geogebra Applets
  • Process (How to do the activity)
  • Developmental Questions (What discussion questions)
  • Evaluation (Questions for assessment of the child)
  • Question Corner

Concept #

Learning objectives

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.

Activity No #

  • Estimated Time
  • Materials/ Resources needed
  • Prerequisites/Instructions, if any
  • Multimedia resources
  • Website interactives/ links/ Geogebra Applets
  • Process (How to do the activity)
  • Developmental Questions (What discussion questions)
  • Evaluation (Questions for assessment of the child)
  • Question Corner


Activity No #

  • Estimated Time
  • Materials/ Resources needed
  • Prerequisites/Instructions, if any
  • Multimedia resources
  • Website interactives/ links/ Geogebra Applets
  • Process (How to do the activity)
  • Developmental Questions (What discussion questions)
  • Evaluation (Questions for assessment of the child)
  • Question Corner

Hints for difficult problems

Project Ideas

Math Fun

Usage

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

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