Fractions Rules and Bow Tie Shortcut

Overview

The lesson explains two main rules for working with fractions: multiplying/dividing and adding/subtracting. It introduces a bow tie shortcut for adding/subtracting without finding the lowest common denominator.

The Two Rules for Fractions

• Rule 1: Multiplication and division of fractions are straightforward procedures.
• Rule 2: Addition and subtraction require common denominators; use LCD or the bow tie shortcut.
• Adding and subtracting follow the same procedure; multiplying and dividing are closely related.

Key Concepts and Vocabulary

• Numerator: Top number of a fraction.
• Denominator: Bottom number of a fraction.
• Lowest Common Denominator (LCD): Smallest common multiple of denominators.
• Reciprocal: Fraction flipped upside down; a/b becomes b/a.
• Mixed Number: Whole number with a fraction (e.g., 2 1/5).
• Improper Fraction: Numerator ≥ denominator (e.g., 11/5).

Multiplying Fractions (Rule 1)

• Multiply numerators together; multiply denominators together.
• Example: 2/3 × 1/5 = (2×1)/(3×5) = 2/15.
• For mixed numbers, convert to improper fractions first, then multiply.

Dividing Fractions (Rule 1)

• Convert division to multiplication by the reciprocal of the right-hand fraction.
• Procedure: a/b ÷ c/d = a/b × d/c.
• Example: 2/3 ÷ 1/5 = 2/3 × 5/1 = 10/3.
• For mixed numbers, convert both to improper fractions before dividing.

Converting Mixed Numbers to Improper Fractions

• Formula: a b/c = (a×c + b)/c.
• Example: 2 1/5 = (2×5 + 1)/5 = 11/5.
• Example: 1 1/3 = (1×3 + 1)/3 = 4/3.

Adding and Subtracting Fractions (Rule 2)

• If denominators match: keep the denominator; add or subtract numerators.
• Example: 2/7 + 1/7 = (2+1)/7 = 3/7.
• If denominators differ: use LCD method or the bow tie method.

LCD Method (Standard Approach)

• Find the LCD of denominators.
• Rewrite each fraction with the LCD denominator using equivalent fractions.
• Add or subtract numerators; keep the LCD denominator.
• Example: 2/5 + 1/3
  • LCD = 15; 2/5 = 6/15; 1/3 = 5/15
  • 6/15 + 5/15 = 11/15.

Bow Tie Method (Shortcut for Add/Subtract)

• Works for all addition/subtraction fraction problems; may require reducing at the end.
• Pattern:
  • Numerator = (bottom-right × top-left) [use sign of the operator] (bottom-left × top-right).
  • Denominator = (left denominator × right denominator).
• Always start cross-multiplication from bottom-right to top-left, then bottom-left to top-right.

Bow Tie Examples

• Example (addition): 2/5 + 1/3
  • Numerator: (3×2) + (5×1) = 6 + 5 = 11
  • Denominator: 5×3 = 15
  • Result: 11/15.
• Example (subtraction): 4/9 − 1/2
  • Numerator: (2×4) − (9×1) = 8 − 9 = −1
  • Denominator: 9×2 = 18
  • Result: −1/18.
• Mixed numbers: Convert to improper fractions, then apply bow tie.
  • 3 1/3 + 2 3/5 → 10/3 + 13/5
  • Numerator: (5×10) + (3×13) = 50 + 39 = 89
  • Denominator: 3×5 = 15
  • Result: 89/15.

Structured Procedures and Examples

Key Terms & Definitions

• Fraction: Number expressed as a/b with b ≠ 0.
• Equivalent Fractions: Different forms representing the same value.
• Reduce/Simplify: Write a fraction in lowest terms by dividing numerator and denominator by GCF.
• Cross-Multiplication (Bow Tie): Crisscross products used in addition/subtraction to form the new numerator.

Action Items / Next Steps

• Practice Rule 1: Multiply and divide several proper, improper, and mixed number fractions.
• Practice Rule 2: Use both LCD and bow tie methods; compare results and simplify answers.
• Memorize the bow tie starting order: bottom-right to top-left first, then bottom-left to top-right.
• Review finding LCDs and reducing fractions to lowest terms.