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 → Number System ↓


Fractions


In mathematics, a fraction represents a part of a whole or a division of a quantity into equal parts. It is a way of breaking numbers into smaller, more manageable pieces to understand them better. Understanding fractions is important because they are widely used in everyday life, from cooking recipes to measuring distances and even in financial calculations.

What is fraction?

A fraction has two numbers: the numerator and the denominator. The numerator shows how many parts we have and the denominator shows how many equal parts the whole is divided into.

Fraction = Numerator / Denominator

For example, in the fraction 3/4, the number 3 is the numerator, and 4 is the denominator. This fraction means that we have 3 parts out of a total of 4 equal parts.

Visualization of fractions

Example: half (1/2)

Consider the fraction 1/2. It means one part of two equal parts. In the above rectangle, the entire shape is divided into two equal parts, and one part is shaded.

Example: Quarter (1/4)

Consider the fraction 1/4. It means one part out of four equal parts. Here, the figure is divided into 4 equal parts out of which 1 part is shaded.

Types of fractions

There are several types of fractions:

  • Proper fraction: A fraction in which the numerator is smaller than the denominator. For example, 3/4.
  • Improper fraction: A fraction in which the numerator is greater than or equal to the denominator. For example, 5/3.
  • Mixed number: A combination of a whole number and a fraction. For example, 1 1/2
  • Equivalent fractions: Different fractions that represent the same value. For example, 1/2 is equivalent to 2/4 and 4/8.

Understanding proper and improper fractions

Proper fractions are easy to understand; they are always smaller than 1. For example, in 3/4, three parts of four equal parts combine to form a fraction that is smaller than the perfect one.

Improper fractions are fractions that are greater than or equal to 1. For example, 5/3 means that we have more than a perfect because 5 is greater than 3.

Visualizing improper fractions

Consider the improper fraction 5/3. You can see that we have taken a whole (3/3) and added 2 more pieces of 1/3 to get to 5/3.

Converting between mixed numbers and improper fractions

It is often necessary to convert between mixed numbers and improper fractions to simplify calculations and make them easier to understand.

Converting a mixed number to an improper fraction

Step 1: Multiply the whole number by the denominator. Step 2: Add the numerator to the result from Step 1. Step 3: Write the result from Step 2 over the original denominator. Example: Convert 2 2/3 to an improper fraction. Step 1: 2 x 3 = 6 Step 2: 6 + 2 = 8 Step 3: The improper fraction is 8/3.

Converting an improper fraction to a mixed number

Step 1: Divide the numerator by the denominator. Step 2: Write the quotient as the whole number. Step 3: Write the remainder over the original denominator. Example: Convert 11/4 to a mixed number. Step 1: 11 ÷ 4 = 2 (quotient) with a remainder of 3. Step 2: The mixed number is 2 3/4.

Equivalent fractions

Equivalent fractions are fractions that represent the same part of a whole. They may look different, but they have the same value.

Example of equivalent fractions

Consider the fraction 1/2:
  • 1/2
  • 2/4
  • 4/8
These fractions are equivalent because they all represent half of a whole.

Adding and subtracting fractions

To add and subtract fractions, the fractions must have the same denominator. If they don't, we must first convert them into equivalent fractions with the same denominator.

Example: Adding fractions with the same denominators

Add 2/8 and 3/8: Step 1: The denominators are the same. Step 2: Add the numerators: 2 + 3 = 5. Step 3: Write the result over the common denominator: 5/8.

Example: Adding fractions with different denominators

Add 1/2 and 1/3: Step 1: Find the Least Common Denominator (LCD) of 2 and 3, which is 6. Step 2: Convert each fraction to an equivalent fraction with denominator 6. 1/2 = 3/6 (multiplied both numerator and denominator by 3) 1/3 = 2/6 (multiplied both numerator and denominator by 2) Step 3: Add the equivalent fractions: 3/6 + 2/6 = 5/6.

Example: Subtracting fractions

Subtract 3/4 from 7/8: Step 1: Find the Least Common Denominator (LCD) of 4 and 8, which is 8. Step 2: Convert 3/4 to an equivalent fraction with denominator 8, which is 6/8. Step 3: Subtract the numerators: 7 - 6 = 1. Step 4: Write the result over the common denominator: 1/8.

Multiplying and dividing fractions

Multiplying and dividing fractions is a little different from addition and subtraction.

Example: Multiplication of fractions

Multiply 2/3 by 3/4: Step 1: Multiply the numerators: 2 x 3 = 6. Step 2: Multiply the denominators: 3 x 4 = 12. Step 3: Write the result as a fraction: 6/12, which simplifies to 1/2.

Example: Division of fractions

Dividing fractions involves multiplying by the reciprocal of the divisor.

Divide 3/5 by 6/7: Step 1: Write the reciprocal of 6/7, which is 7/6. Step 2: Multiply 3/5 by 7/6. Step 3: Multiply the numerators: 3 x 7 = 21. Step 4: Multiply the denominators: 5 x 6 = 30. Step 5: Write the result as a fraction: 21/30, which simplifies to 7/10.

Simplifying fractions

It is always a good practice to simplify fractions to their simplest form by dividing both their numerator and denominator by their Greatest Common Divisor (GCD).

Example: Simplifying a fraction

Simplify 18/24: Step 1: Find the GCD of 18 and 24, which is 6. Step 2: Divide both by 6. 18 ÷ 6 = 3 24 ÷ 6 = 4 Step 3: The simplified fraction is 3/4.

Conclusion

Fractions are a fundamental part of math that helps us understand how to divide and work with quantities less than a whole. Mastering fractions involves practicing with different types of fractions, understanding how to convert between different forms, and being able to perform operations such as addition, subtraction, multiplication, and division. Understanding fractions enhances number sense and is essential in our everyday lives. Continue to practice with different examples, visualize them when possible, and remember to simplify your fractions for clear and understandable results.


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