Multiplication and Division with Fractions

This chapter helps future elementary teachers understand the concepts and reasoning
behind multiplying and dividing fractions. It provides step-by-step methods, visual reasoning,
and teaching tips for use in the classroom.

1. Multiplying Fractions

1.1 The Rule: “Multiply Across”

For any two fractions:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

If one factor is a whole number \( n \), write it as \( \frac{n}{1} \).

Steps:

1. Multiply the numerators (top numbers).
2. Multiply the denominators (bottom numbers).
3. Simplify the result, if possible.

Example 1

$$\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$$

Example 2 (with whole number)

$$3 \times \frac{5}{8} = \frac{3}{1} \times \frac{5}{8} = \frac{15}{8} = 1\tfrac{7}{8}$$

1.2 Why It Works

Multiplying by a fraction means taking a part of a part.

• \(\frac{a}{b}\) means “take \( a \) parts out of \( b \).”
• \(\frac{c}{d}\) means “take \( c \) parts out of \( d \).”
• When you multiply, you find \(\frac{a}{b}\) of \(\frac{c}{d}\), giving \(\frac{a \times c}{b \times d}\).

Visual model (area/rectangle):

1. Start with a 1×1 square; divide into \( b \) vertical parts and shade \( a \) of them → \( \frac{a}{b} \).
2. Divide each piece into \( d \) horizontal parts; shade \( c \) of them → \( \frac{c}{d} \) of what remains.
3. The overlap represents \( \frac{a \times c}{b \times d} \) of the whole.

1.3 Simplifying by Cross-Cancellation

You can make multiplication easier by simplifying before multiplying.

Example

$$\frac{4}{9} \times \frac{3}{8}$$

• \(4\) and \(8\) share a factor of \(4\): \(4 \div 4 = 1\), \(8 \div 4 = 2\).

$$\frac{1}{9} \times \frac{3}{2} = \frac{3}{18} = \frac{1}{6}$$

1.4 Common Student Mistakes

• Adding denominators instead of multiplying:
  \(\frac{2}{3} \times \frac{1}{4} \neq \frac{3}{7}\).
• Forgetting to write whole numbers as fractions.
• Forgetting to simplify at the end.
• Ignoring sign rules (positives/negatives).
• Not estimating to check reasonableness.

Estimation check:
\(\frac{2}{3} \times \frac{4}{5} \approx 0.67 \times 0.8 = 0.53\), so the answer should be about one-half.

2. Dividing Fractions

2.1 The Rule: “Invert and Multiply”

To divide by a fraction, multiply by its reciprocal (flip it!).

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

Steps:

1. Keep the first fraction the same.
2. Flip (take the reciprocal of) the second fraction.
3. Change the division sign to multiplication.
4. Multiply and simplify.

Example 1

$$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\tfrac{7}{8}$$

Example 2

$$\frac{5}{6} \div 2 = \frac{5}{6} \times \frac{1}{2} = \frac{5}{12}$$

Example 3

$$2 \div \frac{3}{4} = \frac{2}{1} \times \frac{4}{3} = \frac{8}{3} = 2\tfrac{2}{3}$$

2.2 Why It Works

Division asks, “How many groups of this size fit into that total?” If

$$\frac{a}{b} \div \frac{c}{d} = x,$$

then

$$\frac{c}{d} \times x = \frac{a}{b}.$$

To solve for \( x \), multiply both sides by \( \frac{d}{c} \). That’s why “invert and multiply” works.

2.3 Common Pitfalls

• Forgetting to invert the second fraction.
• Dividing numerator-by-numerator and denominator-by-denominator (incorrect).
• Inverting the first fraction instead of the second.
• Assuming division always makes the result smaller (not true for fractions!).
• Dividing by zero — never allowed.

Quick check: If you divide by a small fraction (like \( \frac{1}{2} \)),
your result should get larger, not smaller.

3. Practice Problems

Try these problems on your own:

1. \( \frac{7}{12} \times \frac{3}{5} \)
2. \( 4 \times \frac{2}{9} \)
3. \( \frac{5}{8} \div \frac{3}{4} \)
4. \( \frac{9}{10} \div 2 \)
5. \( 3 \div \frac{7}{11} \)
6. \( \frac{14}{15} \times \frac{5}{7} \)

4. Teaching Tips for Future Educators

• Use concrete models: Fraction bars, area models, and circle models build understanding of “part of a part.”
• Connect to real-life stories: Example: “Sally has \( \frac{2}{3} \) of a pizza and gives \( \frac{1}{4} \) of it to Leo.
  How much of the pizza does Leo get?” → \( \frac{2}{3} \times \frac{1}{4} = \frac{1}{6} \).
• Encourage estimation: Predict whether the answer should be larger or smaller than 1 before calculating.
• Address misconceptions: Multiplying fractions doesn’t always make numbers bigger; dividing fractions doesn’t always make numbers smaller.
• Show every step: Write the reciprocal step clearly during division problems.
• Promote mathematical talk: Ask students to explain why “invert and multiply” works, not just how.