Grade 6
  1. Number System  1.1. Whole Numbers  1.1.1. Place Value and Face Value  1.1.2. Comparing and Ordering Numbers  1.1.3. Rounding Numbers  1.1.4. Properties of Whole Numbers (Associative, Commutative, Distributive)  1.2. Integers  1.2.1. Introduction to Integers  1.2.2. Addition and Subtraction of Integers  1.2.3. Multiplication and Division of Integers  1.2.4. Properties of Integer Operations  1.3. Fractions  1.3.1. Types of Fractions  1.3.2. Equivalent Fractions  1.3.3. Simplification of Fractions  1.3.4. Addition and Subtraction of Fractions  1.3.5. Multiplication and Division of Fractions  1.3.6. Comparing and Ordering Fractions  1.4. Decimals  1.4.1. Understanding Decimals  1.4.2. Conversion between Fractions and Decimals  1.4.3. Addition and Subtraction of Decimals  1.4.4. Multiplication and Division of Decimals  1.4.5. Comparing and Ordering Decimals
  2. Algebra  2.1. Basics of Algebra  2.1.1. Variables and Constants  2.1.2. Expressions and Equations  2.1.3. Algebraic Terms and Coefficients  2.2. Solving Simple Equations  2.2.1. Balancing Equations  2.2.2. Solving One-Step Equations  2.2.3. Solving Two-Step Equations  2.3. Patterns and Sequences  2.3.1. Identifying Patterns  2.3.2. Arithmetic Sequences
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Equivalent Ratios  3.1.3. Simplifying Ratios  3.2. Proportion  3.2.1. Introduction to Proportion  3.2.2. Direct and Inverse Proportions  3.3. Percentages  3.3.1. Converting Ratios to Percentages  3.3.2. Percentage Increase and Decrease  3.3.3. Discount and Profit Calculations  3.3.4. Simple Interest Calculations in Percentages in Ratio and Proportion
  4. Geometry  4.1. Basic Geometric Shapes  4.1.1. Points, Lines, and Angles  4.1.2. Types of Angles  4.1.3. Intersecting and Parallel Lines  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Properties of Triangles  4.2.3. Sum of Angles in a Triangle  4.2.4. Congruence of Triangles  4.3. Quadrilaterals  4.3.1. Types of Quadrilaterals  4.3.2. Properties of Quadrilaterals  4.4. Circles  4.4.1. Radius and Diameter  4.4.2. Circumference and Area  4.4.3. Arcs and Chords  4.4.4. Sector and Segment
  5. Mensuration  5.1. Perimeter  5.1.1. Perimeter of Simple Shapes  5.2. Area  5.2.1. Area of Rectangles and Squares  5.2.2. Area of Triangles and Parallelograms  5.2.3. Area of Composite Figures  5.2.4. Area of Trapezium  5.3. Volume  5.3.1. Volume of Cubes and Cuboids  5.3.2. Volume of Cylinders  5.3.3. Volume of Cones  5.3.4. Surface Area of Simple Solids
  6. Data Handling  6.1. Introduction to Data  6.1.1. Types of Data  6.1.2. Organizing Data  6.1.3. Frequency Distribution  6.2. Graphs and Charts  6.2.1. Bar Graphs  6.2.2. Line Graphs  6.2.3. Pictographs  6.2.4. Pie Charts  6.3. Mean, Median, and Mode  6.3.1. Calculating Mean  6.3.2. Finding Median  6.3.3. Identifying Mode  6.3.4. Range of Data in Mean, Median, and Mode
  7. Probability  7.1. Basics of Probability  7.1.1. Understanding Probability  7.1.2. Probability of Simple Events  7.1.3. Experimental and Theoretical Probability
  8. Practical Geometry  8.1. Construction of Shapes  8.1.1. Constructing Triangles  8.1.2. Constructing Quadrilaterals  8.1.3. Constructing Circles and Arcs  8.2. Symmetry  8.2.1. Line of Symmetry  8.2.2. Rotational Symmetry  8.2.3. Translational Symmetry

Grade 6 ↓


Practical Geometry


Practical geometry is a fascinating element of mathematics that deals with the construction of different types of geometric shapes. In class 6 maths, practical geometry takes students to a dimension that transforms theoretical knowledge into physical structures. It allows us to explore shapes and structures that form the basis for more complex mathematical, architectural, and engineering concepts. In this lesson, we will cover the constructive aspect of geometry that aims to simplify understanding through practical application.

Basic tools and techniques

To get started with practical geometry, it is important to familiarize yourself with the basic tools and techniques used for construction. These simple tools help us create precise shapes and lines. The most commonly used tools are:

  • Ruler: Ruler is used to measure length and draw straight lines.
  • Compass: Compass is required for drawing circles and arcs.
  • Protractor: This instrument is used to measure and construct angles.
  • Set squares: These triangular tools are used to draw perpendicular and parallel lines.
  • Pencil: Pencil is a basic writing instrument used to mark points and draw shapes.

Creation of basic shapes

Drawing basic geometric shapes such as lines, angles and circles is the basis of practical geometry. Let's learn how each shape is constructed:

1. Construction of a line segment

A line segment is a part of a line that has two fixed end points. To construct a line segment of a specified length, for example, 5 cm, we perform the following steps:

  1. Place the ruler on a flat surface.
  2. Mark a point with a pencil at the zero (0) mark on the ruler.
  3. Mark another point at the 5 cm mark from this starting point.
  4. Connect these two points using the straight edge of the ruler.
    Diagram:
    Point A .------------------ Point B
                    5 cm

2. Construction of angle

An angle is formed by two rays meeting at a common endpoint, called the vertex. For example, to construct a 60 degree angle, follow these steps:

  1. Draw ray OA on the paper with a ruler.
  2. Place the protractor at the focal point O
  3. Make sure that the baseline of the protractor is aligned with the ray OA.
  4. Count up to 60 degrees on the protractor and mark a point B
  5. Remove the protractor and draw a ray OB through the marked point B
    Diagram:
    A ray (AO) - O (vertex) - B (point at 60 degrees)

3. Drawing a circle

A circle is a set of points that are equidistant from a given point called the center. To construct a circle with a radius of 4 cm:

  1. Place the compass pointer at the centre point O of the paper.
  2. Using the ruler, set the width of the compass to 4 cm.
  3. Rotate the compass 360 degrees to complete the circle.
    Diagram:
               , 
            ,   
          , 
         ( O )
          ,
            ,
               , 
    (radius 4 cm)

Advanced construction

Once we become familiar with these simple structures, we can move on to more complex shapes by combining basic structures.

4. Construction of perpendicular bisector

The perpendicular bisector is a line that divides another line into two equal parts and is perpendicular to it. Here is how to construct it on a line segment AB:

  1. Draw a line segment AB on the paper.
  2. Place the compass at a distance more than half the length of AB and draw an arc from points A and B
  3. The arcs intersect above and below the line. Mark these points P and Q
  4. Join P and Q, which will be the perpendicular bisector.
    Diagram:
    A -------|------- B
         PQ

5. Construction of angle bisector

An angle bisector divides an angle into two equal parts. To construct it on angle ABC:

  1. Construct angle ABC.
  2. Place the compass at point B and draw an arc that intersects both rays of the angle.
  3. From the points where the arc intersects the rays, draw two arcs inside the angle that intersects. Mark this as point D
  4. Draw ray BD. It bisects angle ABC.
    Diagram:
    B. A
      .D
      ,
       C

6. Construction of parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel. Here's how to build it with certain dimensions:

  1. Draw a side AB of the parallelogram with the specified length.
  2. At point A, use the protractor to construct an angle of your choice (eg. 60 degrees).
  3. Measure the required length of the side AD from this angle. Draw AD.
  4. Construct the same angle at point B Extend the direction of BC until it reaches the length of AD (parallel to AD).
  5. Connect points C and D to complete the parallelogram.
    Diagram:
    A - B
    ,
    DC (parallelogram)

7. Construction of a triangle

Triangles can be constructed when certain measurements are known. For example, if two sides and the angle between them are known, the triangle can be constructed as follows:

  1. Let us construct a triangle with sides AB = 5 cm, AC = 4 cm, and angle BAC = 60°.
  2. Draw a line AB of length 5 cm.
  3. At point A, use a protractor to mark a 60 degree angle.
  4. Measure 4 cm on the newly drawn ray and mark it as point C
  5. Form a triangle ABC by joining point C to point B
    Diagram:
             C  
           ,
          ,
      A--------B
      (60 degrees)

Application of practical geometry

Practical geometry is valuable in a variety of real-life applications beyond the classroom. Architects use geometric constructions in designing buildings, while engineers use these principles to create machinery components. Practical geometry serves as a foundation for many fields, making understanding and mastery essential in education.

Practice problems

Here are some simple problems you can try to expand your understanding of practical geometry:

  • Draw a rectangle of length 8 cm and width 5 cm.
  • Divide a line segment 10 cm long into four equal parts.
  • Construct a right-angled triangle whose one side measures 6 cm and the hypotenuse measures 8 cm.
  • From any point lying outside a given line, draw a line parallel to that line.

Conclusion

Practical geometry is more than just drawing shapes on paper. It is an essential skill that enhances spatial understanding and analytical skills. As you practice these constructions, visualizing shape and space becomes intuitive. These practical techniques serve as building blocks for more advanced geometric concepts as students progress in their mathematical education.


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Practical Geometry

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