The Distributive Property - Sixth Grade Mathematics

The distributive property connects multiplication and addition, providing the algebraic mechanism to eliminate parentheses by distributing a multiplier across every term inside a grouped sum or difference.

The Algebraic Law of Distribution

The distributive property states that multiplying a single factor by a sum or difference gives the exact same result as multiplying the factor by each term individually and then adding or subtracting the products.

The Distributive Property Formulas:
                      a * (b + c) = (a * b) + (a * c)
                      a * (b - c) = (a * b) - (a * c)

The Geometric Area Model of Distribution

The distributive property is visualized by calculating the area of a large rectangle partitioned into two smaller adjacent rooms.

Area Model for 4(x + 3):
              x              3
       +--------------+-------------+
     4 |   4 * x = 4x |  4 * 3 = 12 |
       +--------------+-------------+
Total Enclosed Area = 4x + 12.

Whether you calculate the total area as height times total width [4 * (x + 3)], or as the sum of the two interior rooms [4x + 12], the total area is identical. Therefore, 4(x + 3) = 4x + 12.

Expanding Algebraic Expressions Step-by-Step

When applying the distributive property, the outside factor must be multiplied by EVERY term inside the parentheses.

+-------------------------------------------------------------------------+
|                  DISTRIBUTING ALGEBRAIC EXPRESSIONS                     |
|                                                                         |
|   Step 1: Multiply the outside factor by the FIRST term inside.         |
|   Step 2: Multiply the outside factor by the SECOND term inside.        |
|   Step 3: Preserve the appropriate operation sign (+ or -).             |
|   Step 4: Combine any remaining like terms.                             |
+-------------------------------------------------------------------------+

Worked Examples:
Example A: Expand 5(3x + 7)
- Distribute 5 to 3x: 5 * 3x = 15x.
- Distribute 5 to 7:  5 * 7 = 35.
- Combine: 15x + 35.

Example B: Expand 6(2y - 5)
- Distribute 6 to 2y: 6 * 2y = 12y.
- Distribute 6 to 5:  6 * 5 = 30.
- Combine with minus sign: 12y - 30.

The Incomplete Distribution Mistake

The most common error is multiplying the outside factor only by the first term (writing 5(3x + 7) = 15x + 7). Remember that parentheses act like a package: the multiplier on the outside must be delivered to every resident inside the package.

Distributing with Fractional and Negative Multipliers

The distributive property applies to all rational numbers: Distributing a Fraction: 1/2 * (4x + 10) = (1/2 * 4x) + (1/2 * 10) = 2x + 5. Distributing a Negative Factor: Reverses all signs inside parentheses: -3(2x - 4) = (-3 * 2x) - (-3 * 4) = -6x - (-12) = -6x + 12.

Exercises

Exercise 1. Apply the distributive property to expand each expression: (a) 4(2x + 9); (b) 7(3m - 4); (c) 8(5 + 2y); (d) 1/3 * (6k - 15).

Exercise 2. Expand and simplify by combining like terms: 3(4x + 5) + 2x.

Exercise 3. Use an area model sketch or description to show why 5(2x + 3) equals 10x + 15.

Solutions

Solution to Exercise 1. For (a) 4(2x + 9): 4 * 2x + 4 * 9 = 8x + 36. For (b) 7(3m - 4): 7 * 3m - 7 * 4 = 21m - 28. For (c) 8(5 + 2y): 8 * 5 + 8 * 2y = 40 + 16y. For (d) 1/3 * (6k - 15): (1/3 * 6k) - (1/3 * 15) = 2k - 5.

Solution to Exercise 2. First, distribute 3 to the terms inside the parentheses: 3 * 4x + 3 * 5 = 12x + 15. The expression is now: 12x + 15 + 2x. Reorder and combine like terms: (12x + 2x) + 15 = 14x + 15.

Solution to Exercise 3. Draw a rectangle with a height of 5 units. Partition its width into two sections: the first section has a length of 2x, and the second section has a length of 3 units. The area of the first section is height times width: 5 * 2x = 10x. The area of the second section is 5 * 3 = 15. The total area of the entire rectangle is the sum of both sections: 10x + 15.