1,491
edits
Changes
From Karnataka Open Educational Resources
→Resource Title and description
<!-- This portal was created using subst:box portal skeleton -->
<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;">
''[https://karnatakaeducation.org.in/KOER/index.php/%e0%b2%a4%e0%b3%8d%e0%b2%b0%e0%b2%bf%e0%b2%ad%e0%b3%81%e0%b2%9c%e0%b2%97%e0%b2%b3%e0%b3%81 ಕನ್ನಡ]''</div>
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
===Concept Map ===
===Concept Map===
{{#drawio:mmTriangles|interactive}}
===Learning Objectives===
===Learning Objectives===
==== Concept #: Formation of a triangle, elements of a triangle and its measures ====
==== Concept #: Formation of a triangle, elements of a triangle and its measures ====
The triangle is the basic geometrical figure that allows us to best study geometrical shapes. A quadrilateral can be partitioned into two triangles, a pentagon into three triangles, a hexagon into four triangles, and so on.These partitions allow us to study the characteristics of these figures. And so it is with Euclidean geometry—the triangle is one of the very basic parts on which most other figures depend. Here we will be investigating triangles and related its properties
The triangle is the basic geometrical figure that allows us to best study geometrical shapes. A quadrilateral can be partitioned into two triangles, a pentagon into three triangles, a hexagon into four triangles, and so on.These partitions allow us to study the characteristics of these figures. And so it is with Euclidean geometry—the triangle is one of the very basic element on which most other figures depend. Here we will be investigating triangles and related its properties
===== Activities # =====
===== Activities # =====
====== [[Interior and exterior angles in triangle]] ======
====== [[Interior and exterior angles in triangle]] ======
Interior angles are angles that are formed with in the closed figure by the adjacent sides.An exterior angle is an angle formed by a side and the extension of an adjacent side. Exterior angles form linear pairs with the interior angles.
Interior angles are angles that are formed with in the closed figure by the adjacent sides. An exterior angle is an angle formed by a side and the extension of an adjacent side. Exterior angles form linear pairs with the interior angles.
[[Category:Triangles]]
[[Category:Triangles]]
[[Category:Class 8]]
[[Category:Class 10]]
==== Concept #: Types of triangles based on sides and angles ====
==== Concept #: Types of triangles based on sides and angles ====
Variations in elements that make a triangle results in distinct triangles. Recognizing these variations helps in interpreting changes that are possible with in a triangle.
Variations in elements that make a triangle results in distinct triangles. Recognizing these variations helps in interpreting changes that are possible with in a triangle.
'''Video resource:''' Explanation of Types of Triangles by NCERT.
{{Youtube|SXYZd536Vao
}}
===== Activities # =====
===== Activities # =====
====== [[Angle sum property]] ======
====== [[Angle sum property]] ======
Interior angles of a triangle are in relation and also determine the type of angles that can forms a triangle. This also helps in determining an unknown angle measurement.
Interior angles of a triangle are in relation and also determine the type of angles that can forms a triangle. This also helps in determining an unknown angle measurement.
'''Video resource:''' Classroom activity for angle sum property of Triangle by NCERT
{{Youtube|BRDAXvQlzt0
}}
====== [[Angle sum property of a Triangle]] ======
====== [[Relation between interior and exterior angles in triangle|Exterior angle theorem]] ======
====== [[Relation between interior and exterior angles in triangle|Exterior angle theorem]] ======
Interior angle and the corresponding angle form a linear pair. This exterior angle in relation to the remote interior angles and their dependencies are deducted with the theorem.
Interior angle and the corresponding angle form a linear pair. This exterior angle in relation to the remote interior angles and their dependencies are deducted with the theorem.
====== [[Exterior angle property of a Triangle]] ======
==== Concept #: Construction of triangles ====
==== Concept #: Construction of triangles ====
These points of concurrencies, orthocenter, centroid, and circumcenter of any triangle are collinear that is they lie on the same straight line called the Euler line.
These points of concurrencies, orthocenter, centroid, and circumcenter of any triangle are collinear that is they lie on the same straight line called the Euler line.
=====Activities #=====
======Activities #======
======[[Exploring concurrent lines from given surroundings]]======
======[[Exploring concurrent lines from given surroundings]]======
Interactive activity to introduce concurrent lines using examples from our surroundings.
Interactive activity to introduce concurrent lines using examples from our surroundings.
=====Concept #: Concurrency of medians in triangles.=====
=====Concept #: Concurrency of medians in triangles.=====
Median of a triangle is a line segment from a vertex to the midpoint of the opposite side. A triangle has three medians. Each median divides the triangle into two smaller triangles of equal area. The medians of a triangle are concurrent and the point of concurrence is called the centroid. The centroid is always inside the triangle. The centroid is exactly two-thirds the way along each median. i.e the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.
Median of a triangle is a line segment from a vertex to the midpoint of the opposite side. A triangle has three medians. Each median divides the triangle into two smaller triangles of equal area. The medians of a triangle are concurrent and the point of concurrence is called the centroid. The centroid is always inside the triangle. The centroid is exactly two-thirds the way along each median. i.e the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.
[[Image:KOER%20Triangles_html_m404a4c0b.gif|link=]]
[[Image:KOER%20Triangles_html_m404a4c0b.gif|link=]]
=====Activities #=====
======Activities #======
======[[Marking centroid of a triangle|Marking centroid of the triangle]]======
======[[Marking centroid of a triangle|Marking centroid of the triangle]]======
This is a hands on activity to explore concurrent lines formed in a triangle when vertices are joined to the midpoints of the opposite side.
This is a hands on activity to explore concurrent lines formed in a triangle when vertices are joined to the midpoints of the opposite side.
======[[Medians and centroid of a triangle]]======
======[[Medians and centroid of a triangle]]======
The centroid of a triangle is where the three medians intersect. This activity will show you how to find the centroid and you’ll explore several geometric relationships related to centroid and medians.
The centroid of a triangle is where the three medians intersect. This activity will show you how to find the centroid and you’ll explore several geometric relationships related to centroid and medians.
=====Concept #: Concurrency of altitudes in triangles=====
=====Concept #: Concurrency of altitudes in triangles=====
The distance between a vertex of a triangle and the opposite side is called the altitude of the triangle. Altitude also refers to the length of the segment. Altitudes can be used to compute the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. A triangle has 3 altitudes. The intersecting point of 3 altitudes of a triangle is known as orthocentre of the triangle. This point may be inside, outside, or on the triangle. If the triangle is obtuse, it will be outside. If the triangle is acute, the orthocentre is inside the triangle. The orthocenter on a right triangle would be directly on the 90° vertex. From Greek: orthos - "straight, true, correct, regular" The point where the three altitudes of a triangle intersect. One of a triangle's points of concurrency.
The distance between a vertex of a triangle and the opposite side is called the altitude of the triangle. Altitude also refers to the length of the segment. Altitudes can be used to compute the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. A triangle has 3 altitudes. The intersecting point of 3 altitudes of a triangle is known as orthocentre of the triangle. This point may be inside, outside, or on the triangle. If the triangle is obtuse, it will be outside. If the triangle is acute, the orthocentre is inside the triangle. The orthocenter on a right triangle would be directly on the 90° vertex. From Greek: orthos - "straight, true, correct, regular" The point where the three altitudes of a triangle intersect. One of a triangle's points of concurrency.
=====Activities #=====
======Activities #======
======[[Altitudes and orthocenter of a triangle]]======
======[[Altitudes and orthocenter of a triangle]]======
An altitude of a triangle is a line segment that is drawn from the vertex to the opposite side and is perpendicular to the side. A triangle can have three altitudes. Point of intersection of these lines for different types of triangles is explored.
An altitude of a triangle is a line segment that is drawn from the vertex to the opposite side and is perpendicular to the side. A triangle can have three altitudes. Point of intersection of these lines for different types of triangles is explored.
One consequence of the Perpendicular Bisector Theorem is that the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
One consequence of the Perpendicular Bisector Theorem is that the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
=====Activities #=====
======Activities #======
[[Perpendicular bisectors and circumcenter of a triangle|'''Perpendicular bisectors and circumcenter of a triangle''']]
[[Perpendicular bisectors and circumcenter of a triangle|'''Perpendicular bisectors and circumcenter of a triangle''']]
If a point is on the bisector of an angle, then it is equidistant from the two arms of the angle.
If a point is on the bisector of an angle, then it is equidistant from the two arms of the angle.
=====Activities #=====
======Activities #======
======[[Angular bisectors and incenter of a triangle]]======
======[[Angular bisectors and incenter of a triangle]]======
The intersecting point of three lines which are the bisectors of three angles of a triangle that is the incenter and it's properties are examined.
The intersecting point of three lines which are the bisectors of three angles of a triangle that is the incenter and it's properties are examined.
Two objects are similar if they have the same shape, but not necessarily the same size. This means that we can obtain one figure from the other through a process of expansion or contraction, possibly followed by translation, rotation or reflection. If the objects also have the same size, they are congruent. Two triangles are said to be congruent to one another only if their corresponding sides and angles are equal to one another
Two objects are similar if they have the same shape, but not necessarily the same size. This means that we can obtain one figure from the other through a process of expansion or contraction, possibly followed by translation, rotation or reflection. If the objects also have the same size, they are congruent. Two triangles are said to be congruent to one another only if their corresponding sides and angles are equal to one another
==== Solved problems/ key questions (earlier was hints for problems).[edit | edit source] ====
==== Concept #: [https://karnatakaeducation.org.in/KOER/en/index.php/Basic_Proportionality_Theorem Basic Proportionality Theorem] ====
The concept of Thales theorem has been introduced in similar triangles. If the given two triangles are similar to each other then,
* Corresponding angles of both the triangles are equal
* Corresponding sides of both the triangles are in proportion to each other
=== Projects (can include math lab/ science lab/ language lab)[edit | edit source] ===
==== Solved problems/ key questions (earlier was hints for problems) ====
===Additional Resources===
==== Resource Title and description ====
1.[http://www.mathopenref.com/tocs/triangletoc.html Triangles]- This resource contain all information related to triangle like definition, types of triangle, perimeter, congruency etc.,
2. [http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRIA Triangles]- This helps to know the aplication of geometry in our daily life, it contain videos and interactives.
3. [http://www.mathopenref.com/tocs/congruencetoc.html Congruence]
4. [http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.SIM Similarity and Congruence]
5. The below video gives information about Angle Sum Property by Gireesh K S and Suchetha S S
{{#widget:YouTube|id=GfP8M5GwcdQ}}
====OER====
* Web resources:
** [https://en.wikipedia.org/wiki/Triangle Wikipedia, the free encyclopedia] : The website gives a comprehensive information on triangles from basics to in depth understanding of the topic.
** [https://jsuniltutorial.weebly.com/ix-triangles.html cbsemathstudy.blogspot.com] : The website contains worksheets that can be downloaded. Worksheets for other chapters can also be searched for which are listed based on class.
* Books and journals
* Textbooks
** NCERT Textbooks – [http://ncert.nic.in/textbook/textbook.htm?iemh1=7-15 Class 9]
* Syllabus documents
====Non-OER====
* Web resources:
** [https://www.brighthubeducation.com/middle-school-math-lessons/39674-triangle-properties-and-angles/ Bright hub education] : The website describes a lesson plan for introducing triangles and lists classroom problems at the end of the lesson that can be solved for better understanding.
** [http://www.cpalms.org/Public/PreviewResourceLesson/Preview/40261 CPALMS] : The website contains lesson plan, activities and worksheets associated with triangles.
** [https://www.urbanpro.com/cbse-class-9-maths-construction UrbanPro] : This website gives downloadable worksheets with problems on triangle construction.
** [https://schools.aglasem.com/59755 AglaSem Schools] : This website lists important questions for math constructions.
** [http://www.nios.ac.in/media/documents/SecMathcour/Eng/Chapter-12.pdf National Institute of Open Schooling]: This website has good reference notes on concurrency of lines in a triangle for both students and teachers.
** [http://www.mathopenref.com/ Math Open Reference] : This website gives activities that can be tried and manipulated online for topics on geometry.
* Books and journals
* Textbooks:
** Karnataka Govt Text book – Class 8 : [http://ktbs.kar.nic.in/New/website%20textbooks/class8/8th-english-maths-1.pdf Part 1] , [http://ktbs.kar.nic.in/New/website%20textbooks/class8/8th-kannada-maths-2.pdf Part 2]
* Syllabus documents (CBSE, ICSE, IGCSE etc)
=== Projects (can include math lab/ science lab/ language lab) ===
'''''Laboratory Manuals''' - Mathematics : [https://ncert.nic.in/pdf/publication/sciencelaboratorymanuals/classIXtoX/mathematics/lelm402.pdf Click here] to refer activity 15,16,18 and 20'' which explains the properties of Triangle.
=== Assessments - question banks, formative assessment activities and summative assessment activities ===
=== Assessments - question banks, formative assessment activities and summative assessment activities ===
# [[:File:Introducction to Triangles.pdf|Introduction to triangles]]
# [[:File:Types of Triangle by sides.pdf|Types of Triangle by sides]]
# [[:File:Properties of Triangles.pdf|Properties of Triangle]]
[[Category:Class 9]]
[[Category:Class 8]]
