Geometry is a very important branch of mathematics. It is one of the most artistic parts of mathematics. By definition, geometry is a branch of mathematics that deals with shapes, lines, angles, graphs and sizes of a variety of things we see in everyday life. If not taught properly, students close the doors of a career in an extremely creative world. Hence, we need some best practices and strategies to teach this subject to students.
Types of Geometry
There are many subtopics under the main topic of geometry. They are extremely important and have a world of applications.
We will see some of them below.
1. Algebraic Geometry
Algebra is an important branch in mathematics. It involves solving linear and polynomial algebraic equations. This branch of geometry has applications in fields such as cryptography, String Theory, etc.
2. Discrete Geometry
This is another term for the basics of geometry that involve points, lines, circles, triangles, etc. It involves various topics like polyhedra, polytopes, etc. Real-life applications are packing of cells, organic chemistry, etc.
3. Differential Geometry
Differential geometry is a branch of geometry that deals with differentiation and integration, the soul of calculus.
4. Euclidean Geometry
This is the most commonly used branch of geometry whose application is seen every day in our lives. It is the study of solid and plane figures. It has real-life applications in computer science, architecture, crystallography, etc.
5. Convex Geometry
This is a more detailed version of Euclidean geometry that studies convex sets. It is used in optimization and functional analysis.
This branch of geometry deals with incidence structures that focus on lines and helps with mapping and has a world of applications in geography.
A very important aspect of Euclidean Geometry is plane geometry. This deals with the study of 2D shapes.
Two Dimensional Shapes (2D Shapes)
In layman terms, these are shapes that can be represented on a piece of paper. If rephrased mathematically it could be said that this is a plane shape. It is represented only on an X Y plane.
|Triangle||3||3||3||Pizza Slice, Pyramids of Giza, Road Signs|
|Square||4||4||4||Chessboard, Coasters, Keys on a keyboard|
|Rectangle||4||4||4||License Plates, Playing Cards, Indian Rupee Notes|
|Circle||0||0||0||Coins, Pizza, Wheels|
|Semi-Circle||2*||2*||2||Chinese Fans, Umbrella, Bowls|
|Parallelogram||4||4||4||The neck of a Guitar, United States Postal Service Symbol|
|Rhombus||4||4||4||Kites, Baseball Field, Jewellery design|
|Trapezium||4||4||4||Shape of handbags, Bridge Truss Supports, Popcorn Tins|
* Semi Circles have 1 straight side/edge and 1 curved side/edge.
Angles in Geometry
Yet another extremely important aspect of geometry is the study of angles. It has its uses in trigonometry, astrology and many more important fields of study.
According to the definition, an angle is made of two rays that meet at a common point and the angle is formed in this vertex. The main elements that make an angle are “lines” and “vertex”.
Note: Angles go counterclockwise from the initial line to the terminal line. Types of Angles
Angles have two major classifications:
1. Based on Magnitude
2. Based on Rotation
Types of Angles based on Magnitude
1. Acute Angle
An acute angle lies between 0° and 90°.
2. Obtuse Angle
The obtuse angle is the opposite of an acute angle. It is greater than 90° but less than 180°.
3. Right Angle
A right angle is another name for an angle that is formed by two perpendicular lines. Two perpendicular lines make an angle of 90o.
4. Straight Angle
A straight angle is an angle made by two lines overlapping each other at an angle of 180o. 5. Reflex Angle
This is a type of angle that is greater than 180° and less than 360°. Reflex angle along with the other acute or obtuse angle makes 360o. A Reflex angle is always larger than the other angle which together makes 360o.
Angle Types based on Rotation
1. Positive Angles
Positive angles are measured counterclockwise from the base. Positive angles are more common in geometry. They are primarily formed in the (+x,+y) plane.
2. Negative Angles
Negative angles are those angles which are measured clockwise from the base. They are formed in the (x, -y) plane.
NOTE: There are two more basic types of angles which are known as complementary and supplementary angles. Complementary angles sum up to, 90°, and Supplementary angles sum up to 180°.
For more information on angles, please see here.
There are hundreds of formulae involved in geometry since there are so many shapes involved. We have tried to sum up as many formulae as we could in the table below.
|Circle||1*||0||2 x π x r||Π x r2|
|Semi-Circle||2**||2||(Π x r) + 2 x r||0.5 x Π x r2|
|Triangle||3||3||a + b + c***||0.5 x b x h|
|Square||4||4||4 x a||a2|
|Rectangle Parallelogram||4||4||2 x (a + b)||a x b|
*Circle has one curved edge.
** Semi Circle has one curved edge and one straight edge
*** a, b, c, d are lengths of a quadrilateral
r: Radius of a circle
a, b, c, d: Lengths of Edges of a Shape
|Cube||12||6||8||6 x a2||a3|
|Cuboid||12||6||8||2(lw +lh+hw)||L x b x h|
|Prism||9||5||6||(p x h) + (2 x b)||B x h|
|Cone||1||2||1||Π x r (r + (h2+r2)0.5)||Π x r2 x (h/3)|
|Cylinder||2||3||0||(2 x Π x r x h) + ( 2 x Π x r2)||Π x r2 x h|
|Pyramid||8||5||5||(12 x p x l) + B||(l x w x h)/3|
r: Radius of a Circle
a: Length of a side of a cube
l, w, h: Length, Width, and Height of a Rectangle
B: Area of a Base
p: Perpendicular of a Prism/Pyramid
h: Height of a Prism/Pyramid
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Strategies to Teach Geometry in 2020
This is a common method to learn various types of shapes. It involves a cardboard piece, some thumbtacks, and strings. The goal is to form as many shapes as possible using the string and the thumbtacks in a uniform arrangement.
Puzzles are a good way to learn many concepts, not only geometry. It teaches planning, prediction, and gives children a good idea about the shapes.
3. Index Cards
Make index cards with all formulae involved in Geometry and help students revise from these. They are compact and make the subject seem less intimidating.
4.Pencils and Rulers
This is a game that involves students making many shapes with the stationery that is easily available. This is a fun game that can make the lesson exciting.
A series of questions are asked in a short period of time and tests children’s’ quick and perfect responses.
There are tools available online which students can use to sort shapes into appropriate slots. This gives them a sense of shapes, sizes, area, volume, and depth.
A scrapbook of all the possible shapes that the kids see around them every day. An entire book that shows off their observations found in a day.
This is a game where students are expected to find shapes all over their house. The sooner they find the correct shape, they win.
9.Dough, Dough, Dough your Shapes
Playdough is something that has been entertaining children for years. It helps kids in various ways. If shown how to use the right way, play dough can be highly educational.
10. Shape Sort
This is a game where children sort things around the house by their shapes. It helps kids to learn as well as have fun.
All human beings are a combination of shapes. Help kids figure this out and watch them draw the funniest combinations of shapes.
12. Spot the Shape
Animated movies are the best way to take this game forward. While watching the movie, suddenly pause it and ask kids to jolt down the various shapes they can observe.
Hope you find this article useful! I am sure the pointers listed about will help in an exciting journey for you and your kids/students.