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📖 Mathematics

Std 6
Curriculum Book Guide
1 2 3 4 5 6 7 8 9 10
Chapters
1 PATTERNS IN MATHEMATICS 2 Lines and Angles 3 Number Play 4 Data Handling and Presentation 5 Idli-Vada Game PRIME TIME 6 PERIMETER AND AREA 7 Fractions 8 Playing with Constructions 9 SYMMETRY 10 Integers (The Other Side of Zero)
7
Chapter 7
Skill: 60%

Fractions

Fractions

Chapter at a Glance

This chapter introduces fractions as equal shares of a whole. It details fractional units (unit fractions), proper and improper fractions, and mixed numbers. Students learn to represent and compare fractions on a number line, construct and use a fraction wall to identify equivalent fractions, simplify fractions to their lowest terms, and perform addition and subtraction. The chapter also outlines the history of fraction notation in ancient India and presents puzzles on writing numbers as the sum of distinct unit fractions (Egyptian fractions).

Key Definitions & Terminology

  • Fraction: A number representing equal parts of a whole, formed by dividing a whole unit or a group of units into equal shares.
  • Fractional Unit (or Unit Fraction): A fraction with $1$ in the numerator, representing one of the equal parts of a whole (e.g. $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$).
  • Numerator: The top number in a fraction, showing the number of fractional units being counted.
  • Denominator: The bottom number in a fraction, showing the total number of equal parts in the whole.
  • Proper Fraction: A fraction where the numerator is smaller than the denominator, representing a value less than 1.
  • Improper Fraction: A fraction where the numerator is greater than or equal to the denominator, representing a value greater than or equal to 1.
  • Mixed Fraction: A number representing a value greater than 1, written as a whole number part and a proper fraction part (e.g. $2 \frac{2}{3}$).
  • Equivalent Fractions: Fractions that represent the same value or position on the number line, despite having different numerators and denominators (e.g. $\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$).
  • Simplest Form (or Lowest Terms): A fraction whose numerator and denominator have no common factor other than 1.
  • Bhinna: The Sanskrit word for fraction, meaning "broken".

Formulas, Rules & Properties

  • Comparison Rules:
  • Same Denominator: The fraction with the larger numerator is greater (e.g., $\frac{5}{7} > \frac{4}{7}$).
  • Same Numerator: The fraction with the smaller denominator is greater, because dividing a whole into fewer parts makes each part larger (e.g., $\frac{1}{5} > \frac{1}{9}$ or $\frac{4}{7} > \frac{4}{8}$).
  • Mixed Fraction to Improper Fraction:
    $$W \frac{N}{D} = \frac{W \times D + N}{D}$$
  • Simplification to Lowest Terms:
    Divide both the numerator and the denominator of a fraction by their Highest Common Factor (HCF).
  • Brahmagupta's Addition & Subtraction Rules:
    To add or subtract fractions with different denominators, first convert them to equivalent fractions with a common denominator, then add or subtract the numerators and keep the common denominator:
    $$\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$$

Core Concepts & Topics

  • Fractions as Division:
    Sharing $U$ units equally among $P$ people gives each person a share of $\frac{U}{P}$ units. This is written as the division fact:
    $$U \div P = \frac{U}{P}$$
  • Visual Representation Tools:
  • Paper Strip Folding: Folding a paper strip in halves, quarters, and eighths to understand unit fraction spacing.
  • Fraction Wall: A visual diagram of horizontal rows representing fractions from a whole down to tenths. It shows how fractional segments align (e.g. three $\frac{1}{6}$ segments align exactly with one $\frac{1}{2}$ segment).
  • Number Line representation:
    Every fraction corresponds to a unique point on a number line. Between any two integers, there are infinitely many fractional numbers.
  • History of Fraction Notation:
  • Writing the numerator directly over the denominator (without a line) originated in India (e.g. in the Bakhshali Manuscript, c. 300 CE).
  • The horizontal fraction bar separating the two numbers was introduced in the 12th century by Moroccan mathematician Al-Hassar.
  • Brahmagupta (628 CE) codified the general rules for adding and subtracting fractions in his treatise Brahmasphutasiddhanta.
  • Ancient Indian texts (the Sulba-sutras and the Rig Veda) mention fractions: $\frac{3}{4}$ is referred to as tri-pada (comparable to teen paav in Hindi or mukkaal in Tamil).
  • Egyptian Fractions:
  • Ancient Egyptians and Babylonians wrote general fractions as the sum of distinct unit fractions (e.g. $\frac{19}{24} = \frac{1}{2} + \frac{1}{6} + \frac{1}{8}$).

Worked Examples

  • Guavas and Packets:
  • Guavas: 3 equal guavas weigh 1 kg $\rightarrow$ each weighs $\mathbf{\frac{1}{3}\text{ kg}}$.
  • Packets: 1 kg rice in 4 equal packets $\rightarrow$ each packet weighs $\mathbf{\frac{1}{4}\text{ kg}}$.
  • Sugarcane Juice: 4 friends share 3 glasses $\rightarrow$ each gets $\mathbf{\frac{3}{4}\text{ glass}}$.
  • Mixed and Improper Fraction Conversion:
  • $\frac{9}{2} = \mathbf{4 \frac{1}{2}}$
  • $\frac{47}{9} = \mathbf{5 \frac{2}{9}}$
  • $3 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \mathbf{\frac{13}{4}}$
  • $7 \frac{2}{3} = \frac{7 \times 3 + 2}{3} = \mathbf{\frac{23}{3}}$
  • Fraction Wall matching:
  • Question: How many pieces of length $\frac{1}{6}$ make a length of $\frac{1}{2}$? How many make a length of $\frac{1}{3}$?
    Solution: $\frac{3}{6} = \frac{1}{2}$ (takes 3 pieces); $\frac{2}{6} = \frac{1}{3}$ (takes 2 pieces).
  • Simplifying Fractions:
  • $\frac{16}{20} = \frac{16 \div 4}{20 \div 4} = \mathbf{\frac{4}{5}}$
  • $\frac{36}{60} = \frac{36 \div 12}{60 \div 12} = \mathbf{\frac{3}{5}}$
  • $\frac{126}{147} = \frac{126 \div 21}{147 \div 21} = \mathbf{\frac{6}{7}}$
  • Comparing and Ordering:
  • Compare $\frac{8}{3}$ and $\frac{5}{2}$: LCM $= 6$. $\frac{16}{6} > \frac{15}{6} \rightarrow \mathbf{\frac{8}{3} > \frac{5}{2}}$.
  • Ascending order of $\frac{7}{10}, \frac{11}{15}, \frac{2}{5}$: LCM $= 30$. $\frac{21}{30}, \frac{22}{30}, \frac{12}{30} \rightarrow \mathbf{\frac{2}{5} < \frac{7}{10} < \frac{11}{15}}$.
  • Fraction Addition and Subtraction:
  • Volume of paint: Rahim mixes $\frac{2}{3}$ litres of yellow paint with $\frac{3}{4}$ litres of blue paint.
    $$\text{Volume} = \frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12} = \mathbf{1 \frac{5}{12}\text{ litres}}$$
  • Jaya's walking distance: School is $\frac{7}{10}\text{ km}$ away. She rides an auto for $\frac{1}{2}\text{ km}$ and walks the rest.
    $$\text{Walk distance} = \frac{7}{10} - \frac{1}{2} = \frac{7}{10} - \frac{5}{10} = \frac{2}{10} = \mathbf{\frac{1}{5}\text{ km}}$$
  • Park running speed: Jeevika takes $\frac{10}{3}\text{ mins}$, Namit takes $\frac{13}{4}\text{ mins}$. Who is faster and by how much?
    Solution: Convert to common denominator 12: Jeevika $= \frac{40}{12}\text{ mins}$, Namit $= \frac{39}{12}\text{ mins}$. Namit takes less time, by $\frac{40}{12} - \frac{39}{12} = \mathbf{\frac{1}{12}\text{ minute}}$.
  • Egyptian Fraction puzzle:
  • Question: Find three distinct unit fractions that sum to 1. Find four distinct unit fractions that sum to 1.
    Solution:
    • Three fractions: $\mathbf{\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1}$.
    • Four fractions: $\mathbf{\frac{1}{2} + \frac{1}{3} + \frac{1}{8} + \frac{1}{24} = 1}$ (one of 6 solutions).

Practical Activities & Experiments

  • Paper Crease Fraction Strips: Folding a paper strip in half, then half again, and opening it to identify the crease intervals as $\frac{1}{4}, \frac{2}{4}, \frac{3}{4}$.
  • Fraction Wall assembly: Creating horizontal paper strips representing division scales from 1 unit down to tenths, aligning them to check for equivalent lengths.
  • Roti Sharing Demonstration: Drawing round circles to illustrate sharing 3 rotis among 4 children.
  • Number Line Plotting: Marking and plotting mixed numbers like $\frac{5}{2}$ and proper fractions like $\frac{3}{10}$ on a number line.
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