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Distributive Property Calculator
Demonstrate and verify the distributive property a(b+c) = ab + ac with numbers. Expand expressions, see step-by-step verification, area model visual, and comparison table for multiple examples.
Distributive Property Calculator
Leave blank for two terms
Left Side: a(b + c)⧉
91.0000
7.00 × (10.00 + 3.00) = 7.00 × 13.00
Right Side: ab + ac⧉
91.0000
70.00 + 21.00
Verified Equal?⧉
✓ Yes
Both sides match — distributive property confirmed
Product ab⧉
70.0000
7.00 × 10.00
Product ac⧉
21.0000
7.00 × 3.00
Inner Sum (b+c)⧉
13.0000
What a multiplies in the left side
Area Model Visualization
Blue = ab, Yellow = ac. Total area = ab + ac = a(b + c)
Magnitude Comparison
a(b+c) = 91.00
ab = 70.00
ac = 21.00
Verification Table — Multiple Examples
| a | b | c | a(b+c) | ab+ac | Equal? |
|---|---|---|---|---|---|
| 2 | 5 | 3 | 16.00 | 16.00 | ✓ |
| 3 | 7 | 4 | 33.00 | 33.00 | ✓ |
| 5 | 10 | 2 | 60.00 | 60.00 | ✓ |
| 4 | 8 | 6 | 56.00 | 56.00 | ✓ |
| 7 | 9 | 1 | 70.00 | 70.00 | ✓ |
| 6 | 11 | 5 | 96.00 | 96.00 | ✓ |
| 10 | 3 | 7 | 100.00 | 100.00 | ✓ |
| -2 | 6 | 4 | -20.00 | -20.00 | ✓ |
| 8 | 12 | 3 | 120.00 | 120.00 | ✓ |
| 0.5 | 20 | 8 | 14.00 | 14.00 | ✓ |
Planning notes, formulas, and examples
About the Distributive Property Calculator
The distributive property is one of the most fundamental rules in algebra: a × (b + c) = a × b + a × c. It allows you to "distribute" multiplication over addition (or subtraction). This property is the basis for expanding expressions, factoring, mental math shortcuts, and the FOIL method. For example, 7 × 13 can be computed as 7 × (10 + 3) = 70 + 21 = 91. This calculator lets you enter values for a, b, and c to verify the property numerically. It shows both sides of the equation, the step-by-step expansion, and an area model visualization where a(b + c) is shown as a rectangle of height a and width (b + c), split into two subrectangles of areas ab and ac. You can also toggle subtraction mode to verify a(b − c) = ab − ac, and explore multiple examples with presets. A comparison table computes both sides for a range of values so you can see the property holds universally.
When This Page Helps
The distributive property is so fundamental that students must internalize it before tackling polynomial algebra, factoring, and mental math shortcuts. This calculator verifies the identity numerically for any values you enter, shows the step-by-step expansion, draws an area model for geometric intuition, and demonstrates with a 10-row comparison table that the property holds universally — including with negatives, decimals, and three-term expressions.
How to Use the Inputs
- Enter the multiplier a, first term b, and second term c.
- Select Addition (+) or Subtraction (−) for the operation inside parentheses.
- Optionally enter a third term d for the extended property a(b + c + d).
- Click a preset like "7(10+3)" or "−2(6+5)" to load a common example.
- Compare the Left Side a(b+c) and Right Side ab + ac to verify they match.
- Examine the area model diagram showing the rectangle split into sub-areas.
- Review the 10-row comparison table verifying the property across multiple values.
Formula used
a(b + c) = ab + ac
a(b − c) = ab − ac
Left side: a × (b + c)
Right side: (a × b) + (a × c)Example Calculation
Result: 7(10 + 3) = 7×10 + 7×3 = 70 + 21 = 91
Both sides equal 91, verifying that 7(10 + 3) = 7×10 + 7×3. The area model shows a 7×13 rectangle split into 7×10 and 7×3.
Tips & Best Practices
- Check that all inputs use the same scale and assumptions before trusting the result.
- Compare the answer with the worked example or a rough estimate to catch entry mistakes.
The Distributive Property as a Mental Math Strategy
The distributive property is the secret behind many mental math tricks. To compute 7 × 13 mentally, rewrite it as 7 × (10 + 3) = 70 + 21 = 91. To compute 8 × 99, write 8 × (100 − 1) = 800 − 8 = 792. These decompositions work because multiplication distributes over addition and subtraction. Practicing with this calculator builds the intuition to spot useful decompositions quickly.
Area Model and Geometric Interpretation
The area model provides a visual proof: a rectangle with height a and width (b + c) can be split into two rectangles with areas ab and ac. The total area is a(b + c) = ab + ac. This geometric interpretation extends to algebra tiles for polynomial multiplication: (x + 2)(x + 3) is a rectangle of area x² + 5x + 6, split into four sub-rectangles (x², 2x, 3x, 6). The area model is the foundation of the FOIL method and polynomial long multiplication.
Extending to Three or More Terms
The distributive property generalizes to any number of terms: a(b + c + d) = ab + ac + ad. This is used in polynomial expansion, matrix-vector multiplication (distributing a scalar over a sum of vectors), and sigma notation in calculus. The extended distributive property is also the basis for expanding products of sums like (a + b)(c + d) = ac + ad + bc + bd, which itself follows from applying distribution twice.
Sources & Methodology
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Frequently Asked Questions
It states that multiplication distributes over addition: a(b + c) = ab + ac. This means you can break a product into simpler parts.
Yes: a(b − c) = ab − ac. The distributive property applies to both addition and subtraction.
It is the foundation for expanding expressions, factoring polynomials, mental math, and the FOIL method for multiplying binomials.
A visual representation where a(b + c) is a rectangle of height a and width b + c, split into two rectangles with areas ab and ac.
Absolutely. For example, 3(5 + (−2)) = 3(3) = 9, and 3×5 + 3×(−2) = 15 − 6 = 9.
It extends to any number of terms: a(b + c + d) = ab + ac + ad. This is used in polynomial multiplication.
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