Fractions

Chapter at a Glance

This chapter covers the math of fractions, focusing on visual representations, equivalent fractions, and fractions greater than one. Using hands-on tools like fraction kits, grid shading, and division models, students learn how to identify, generate, and compare fractions. The chapter introduces comparing fractions by using $1$ and $\frac{1}{2}$ as benchmarks, and covers real-world problems like sharing parathas, slicing pizzas, and measuring the walking distance of ants.

Key Definitions & Terminology

• Fraction: A representation of part of a whole, written as a numerator (the number of parts selected) over a denominator (the total number of equal parts in the whole).
• Equivalent Fractions: Different fractions that represent the exact same portion of a whole (e.g., $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{3}{6}$).
• Benchmark Fraction: A common fraction (like $\frac{1}{2}$ or $1$) used as a reference point to compare other fractions.
• Mixed Number: A number made up of a whole number and a proper fraction (e.g., $2\frac{1}{2}$).
• Fractions Greater Than 1: Improper fractions where the numerator is larger than the denominator (e.g., $\frac{5}{2}$), representing more than a single whole.

Formulas, Rules & Properties

• Wholes Rule for Comparison:
  Fractions can only be compared directly if the wholes from which they are taken are of identical size.
• Generating Equivalent Fractions:
  Multiply or divide both the numerator and the denominator of a fraction by the same non-zero number:
  $$\frac{a}{b} = \frac{a \times n}{b \times n}$$
• Comparing Fractions with the Same Denominator:
  If denominators are equal, the fraction with the larger numerator is larger:
  $$\frac{x}{d} > \frac{y}{d} \quad \text{if } x > y$$
• Comparing Fractions with the Same Numerator:
  If numerators are equal, the fraction with the smaller denominator is larger (since the whole is divided into larger parts):
  $$\frac{n}{x} > \frac{n}{y} \quad \text{if } x < y$$
• Comparing with $\frac{1}{2}$:
  A fraction $\frac{a}{b}$ is:
• Larger than $\frac{1}{2}$ if $2a > b$.
• Smaller than $\frac{1}{2}$ if $2a < b$.
• Equal to $\frac{1}{2}$ if $2a = b$.

Core Concepts & Topics

• Fraction Kit Investigations:
• Making a whole using different fractional blocks (e.g. one $\frac{1}{2}$ piece and two $\frac{1}{4}$ pieces make a whole).
• Showing equivalencies: e.g. breaking a $\frac{1}{2}$ piece into two equal parts yields two $\frac{1}{4}$ pieces.
• Sameer's Grid Division Method:
• Drawing horizontal division lines across a shaded $\frac{1}{3}$ rectangle to show how it is equivalent to $\frac{2}{6}$, $\frac{3}{9}$, and $\frac{4}{12}$.
• Fractions on a Number Line:
• Dividing the distance between $0$ and $1$ into equal intervals (like halves or fourths) and mapping fractions greater than $1$.
• Food Share Calculations (Parathas & Pizzas):
• Finding the total amount of paratha eaten (e.g., $9$ pieces of $\frac{1}{4}$ paratha $= \frac{9}{4} = 2\frac{1}{4}$ parathas).
• Sharing pizza slices ($2$ pizzas cut into $3$ slices each). Transferring slices to calculate final shares.
• Benchmark Comparisons:
• Comparing fractions with $1$ (e.g., since $\frac{7}{8} < 1$ and $\frac{8}{6} > 1$, then $\frac{7}{8} < \frac{8}{6}$).
• Comparing fractions with $\frac{1}{2}$ (e.g., since $\frac{3}{6} = \frac{1}{2}$ and $\frac{5}{8} > \frac{1}{2}$, then $\frac{5}{8} > \frac{3}{6}$).

Worked Examples

• Fraction Kit Relations:
• Question: How many $\frac{1}{6}$ pieces make $\frac{1}{3}$?
  Solution: $2$ pieces (since $\frac{2}{6} = \frac{1}{3}$).
• Question: How many $\frac{1}{12}$ pieces make $\frac{1}{4}$?
  Solution: $3$ pieces (since $\frac{3}{12} = \frac{1}{4}$).
• Completing Equivalent Fractions:
• $\frac{2}{5} = \frac{4}{10} = \frac{\mathbf{6}}{15} = \frac{\mathbf{8}}{20} = \frac{20}{\mathbf{50}} = \frac{40}{\mathbf{100}}$
• $\frac{5}{9} = \frac{25}{\mathbf{45}}$ (numerator multiplied by 5, so denominator $9 \times 5 = 45$).
• Paratha Accumulation:
• Question: Baba ate $5$ pieces of $\frac{1}{2}$ paratha. How many parathas did he eat?
  Solution: $5 \times \frac{1}{2} = \frac{5}{2} = 2\frac{1}{2}$ parathas.
• Benchmark Comparison Exercises:
• Compare $\frac{2}{9}$ and $\frac{4}{7}$:
  • $\frac{2}{9} < \frac{1}{2}$ (since $2 \times 2 = 4 < 9$).
  • $\frac{4}{7} > \frac{1}{2}$ (since $4 \times 2 = 8 > 7$).
  • Therefore, $\frac{2}{9} < \frac{4}{7}$.
• Compare $\frac{11}{14}$ and $\frac{7}{20}$:
  • $\frac{11}{14} > \frac{1}{2}$ (since $11 \times 2 = 22 > 14$).
  • $\frac{7}{20} < \frac{1}{2}$ (since $7 \times 2 = 14 < 20$).
  • Therefore, $\frac{11}{14} > \frac{7}{20}$.
• Ant Walking Problem:
• Question: If the length of an ant is $\frac{1}{4}\text{ cm}$, what is the total length of $16$ such ants walking in a single line?
  Solution: $16 \times \frac{1}{4}\text{ cm} = \frac{16}{4}\text{ cm} = 4\text{ cm}$.
• Family Relation Riddle:
• Question: If the only brother of your father's sister had a child, what would be the child's relationship with you?
  Solution: Your father's sister is your aunt. Her only brother is your father. His child is you or your sibling (brother/sister).

Practical Activities & Experiments

• DIY Fraction Kit: Folding paper strips to create pieces for $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{6}$, $\frac{1}{8}$, and $\frac{1}{12}$, and testing different combinations to build a whole.
• Grid Shading: Using colored pencils on a grid sheet to shade fractions and identify equivalent sections.
• Number Line jump game: Drawing a large fraction number line on the floor and jumping to locate values like $\frac{5}{4}$ and $\frac{7}{2}$.
• Play-Dough Sharing: Cutting round clay discs into halves, thirds, and fourths to model paratha-sharing word problems.