Multiplying Fractions & Mixed Numbers - Sixth Grade Mathematics

Multiplying fractions allows us to calculate fractional parts of quantities, determine geometric areas, and scale values accurately in real-world contexts.

Conceptual Meaning: Finding a Fraction of a Fraction

In whole number arithmetic, multiplication often represents repeated addition (for example, 3 * 4 means 3 groups of 4). When multiplying fractions, the word "of" frequently translates into multiplication. Finding 1/2 of 1/3 means partitioning one-third into two equal pieces and selecting one of them.

Visual Area Model for 1/2 * 2/3:
Divide unit square vertically into thirds: [ 1/3 | 1/3 | 1/3 ]
Shade 2 vertical thirds.
Now divide horizontally in half:
+-------------+-------------+-------------+
|  SHADED *   |  SHADED *   |    empty    |  <- Top Half
+-------------+-------------+-------------+
|    empty    |    empty    |    empty    |  <- Bottom Half
+-------------+-------------+-------------+
The entire square now contains 6 equal cells (2 * 3 = 6).
The doubly-shaded region contains 2 cells (1 * 2 = 2).
Result: 1/2 * 2/3 = 2/6 = 1/3.

The Standard Multiplication Algorithm

Unlike addition and subtraction, multiplying fractions does NOT require a common denominator. The algorithm operates directly across numerators and denominators.

               (a / b) * (c / d) = (a * c) / (b * d)

Rule: Multiply numerators together to find the new numerator.
      Multiply denominators together to find the new denominator.
      Simplify the product to lowest terms.

The Cross-Simplification Technique

Multiplying large numerators and denominators before simplifying can produce unwieldy numbers that are difficult to reduce. Cross-simplifying (also known as cross-canceling) divides any numerator and any denominator by their common factor BEFORE multiplying.

Evaluate: (9/16) * (8/15)
Without cross-simplifying: (9 * 8) / (16 * 15) = 72 / 240 (Difficult to reduce)

With cross-simplifying:
- Pair 9 (numerator) and 15 (denominator): Divide both by 3 --> 9 becomes 3, 15 becomes 5.
- Pair 8 (numerator) and 16 (denominator): Divide both by 8 --> 8 becomes 1, 16 becomes 2.
Expression simplifies to: (3 / 2) * (1 / 5) = (3 * 1) / (2 * 5) = 3/10.

Multiplying Whole Numbers and Mixed Numbers

To multiply whole numbers or mixed numbers, all factors must first be converted into improper fraction form.

+-------------------------------------------------------------------------+
|                  MULTIPLYING FRACTION COMBINATIONS                      |
+-------------------+-----------------------+-----------------------------+
| Combination       | Transformation Step   | Computational Example       |
+-------------------+-----------------------+-----------------------------+
| Whole Number *    | Express whole number  | 5 * (2/7) = (5/1) * (2/7)   |
| Fraction          | over 1:  W = W / 1    | = (5 * 2) / (1 * 7) = 10/7  |
+-------------------+-----------------------+-----------------------------+
| Mixed Number *    | Convert mixed number  | 2 and 1/3 * (3/4)           |
| Fraction          | to improper fraction  | = (7/3) * (3/4) = 7/4       |
|                   | first!                | = 1 and 3/4                 |
+-------------------+-----------------------+-----------------------------+
| Mixed Number *    | Convert BOTH mixed    | 1 and 1/2 * 2 and 2/5       |
| Mixed Number      | numbers to improper   | = (3/2) * (12/5)            |
|                   | fractions first!      | = (3 * 6) / (1 * 5) = 18/5  |
+-------------------+-----------------------+-----------------------------+

Multiplication as Scaling: Resizing Without Calculating

Understanding how multiplication affects magnitude is crucial for mathematical reasoning: Multiplying a number by a fraction strictly less than 1 (such as 3/4) results in a product smaller than the original number (for example, 20 * 3/4 = 15, and 15 < 20). Multiplying a number by a fraction equal to 1 (such as 4/4) leaves the value unchanged. Multiplying a number by a fraction greater than 1 (such as 5/3) results in a product greater than the original number.

Exercises

Exercise 1. Calculate the product and simplify using cross-simplification: (14/25) * (15/28).

Exercise 2. Calculate the product of the whole number and fraction: 12 * (5/8). Express the final result as a mixed number in simplest form.

Exercise 3. Calculate the area of a rectangular garden plot measuring 3 and 1/2 meters in length and 2 and 2/3 meters in width.

Solutions

Solution to Exercise 1. For (14/25) * (15/28): pair 14 and 28; divide both by 14, so 14 becomes 1 and 28 becomes 2. Pair 15 and 25; divide both by 5, so 15 becomes 3 and 25 becomes 5. The expression becomes (1/5) * (3/2). Multiply across numerators and denominators: (1 * 3) / (5 * 2) = 3/10.

Solution to Exercise 2. Write 12 as the fraction 12/1: (12/1) * (5/8). Cross-simplify 12 and 8 by dividing both by their common factor 4: 12 becomes 3, and 8 becomes 2. Multiply across: (3 * 5) / (1 * 2) = 15/2. Convert the improper fraction to a mixed number: 15 / 2 = 7 with a remainder of 1, yielding 7 and 1/2.

Solution to Exercise 3. To find the area of a rectangle, multiply length by width: Area = (3 and 1/2) * (2 and 2/3). Convert both mixed numbers to improper fractions: 3 and 1/2 = 7/2, and 2 and 2/3 = 8/3. The expression is (7/2) * (8/3). Cross-simplify 2 and 8 by dividing both by 2: 2 becomes 1, and 8 becomes 4. Multiply across: (7 * 4) / (1 * 3) = 28/3. Convert 28/3 to a mixed number: 28 / 3 = 9 with a remainder of 1. The area of the garden is 9 and 1/3 square meters.