Multiplying a Whole Number by a Fraction: Key Steps and Reasoning

Converting a Whole Number to a Fraction

• Whenever you need to multiply a whole number by a fraction, the first step is to rewrite the whole number as a fraction.
  • Technique: Place the whole number over $1$.
  • Example: The whole number $3$ becomes $\frac{3}{1}$.
  • Purpose: This gives the expression both a numerator and a denominator, allowing us to use the standard fraction‐multiplication rules.
  • Key insight: Writing $3$ as $\frac{3}{1}$ does NOT change its value; both represent “three wholes.”

Preserving the Rest of the Problem

• After converting the whole number, bring down all remaining factors unchanged.
  • In the transcript’s implicit example: we have $3 \times \frac{1}{2}$.
  • Rewriting the whole number yields $\frac{3}{1} \times \frac{1}{2}$.

Multiplying Fractions – “Straight Across” Rule

• With both factors now in fraction form, you can multiply numerators together and denominators together.
  • Using the example: $\frac{3}{1} \times \frac{1}{2} = \frac{3\times1}{1\times2} = \frac{3}{2}$.
  • Result interpretation: $\frac{3}{2}$ represents one and a half when expressed as a mixed number.

Conceptual Takeaways

• Equivalence Principle: Changing the form (whole → fraction) does not alter the underlying value.
• Operational Readiness: Fractions must share the same mathematical structure (numerator/denominator) before applying fraction operations.
• General Method: For any whole number $n$ and fraction $\frac{a}{b}$:
  $n \times \frac{a}{b} = \frac{n}{1} \times \frac{a}{b} = \frac{n\,a}{b}$
• This technique underpins not just multiplication but also division of fractions by whole numbers.