Distributive Property Explained for a 12-year-old: Step-by-Step Examples, Visuals, and Practice

Learn the distributive property for a 12-year-old with easy explanations, area model visuals, step-by-step examples, and practice problems with answers. Master a(b + c) = ab + ac quickly.

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What is the distributive property?

The distributive property is a rule that helps you multiply a number by a group of numbers added or subtracted together. In algebra, it looks like this:

a(b + c) = ab + ac

That means if you multiply a by the sum of b and c, you get the same result as multiplying a by b and a by c, then adding the two products.

Simple number example

Calculate 3 × (4 + 5).

  1. Add inside the parentheses first: 4 + 5 = 9, so 3 × (4 + 5) = 3 × 9 = 27.
  2. Now use the distributive property: 3 × (4 + 5) = 3×4 + 3×5 = 12 + 15 = 27.

Both ways give 27, so it works!

Why it works (area model)

Imagine a rectangle that is 3 units tall and 9 units wide. Its area is 3 × 9 = 27. Now split the width into two parts: 4 and 5. The area becomes two smaller rectangles:

  • Left rectangle area: 3 × 4 = 12
  • Right rectangle area: 3 × 5 = 15

The total area is 12 + 15 = 27, which matches 3 × (4 + 5). The distributive property is just adding areas of parts.

Using subtraction

The distributive property works with subtraction too:

a(b - c) = ab - ac

Example: 2 × (10 - 3) = 2×10 - 2×3 = 20 - 6 = 14.

How to use it step-by-step

  1. Look inside the parentheses and find each term being added or subtracted.
  2. Multiply the number outside the parentheses by each term inside.
  3. Add or subtract the results, following the sign between the terms.

Practice examples with steps

1) 4 × (6 + 2)

Step a: 4×6 = 24. Step b: 4×2 = 8. Add: 24 + 8 = 32.

2) 5 × (7 - 3)

Step a: 5×7 = 35. Step b: 5×3 = 15. Subtract: 35 - 15 = 20.

3) (2 + 3) × 4 — multiplication works either side

Distribute 4: 2×4 + 3×4 = 8 + 12 = 20.

Using variables

If you have letters, it looks the same. For example:

3(x + 2) = 3x + 6

Distribute 3 to each term: 3×x = 3x and 3×2 = 6.

Fraction example

1/2 × (6 + 4) = 1/2×6 + 1/2×4 = 3 + 2 = 5.

Common mistakes to avoid

  • Forgetting to multiply every term inside the parentheses.
  • Mistakes with negative signs: remember to multiply the sign too (for example, 2×(5 - 8) = 2×5 - 2×8).
  • Mixing up the order: the outside number must be applied to each term separately.

Practice problems

  1. 3 × (5 + 4)
  2. 6 × (2 + 7)
  3. 4 × (9 - 3)
  4. 2(x + 5) — write it out as an expression
  5. 1/3 × (9 + 6)
  6. Factor: 12 + 15 by finding a common factor and using distributive property backwards

Answers

  1. 3 × (5 + 4) = 3×5 + 3×4 = 15 + 12 = 27
  2. 6 × (2 + 7) = 6×2 + 6×7 = 12 + 42 = 54
  3. 4 × (9 - 3) = 4×9 - 4×3 = 36 - 12 = 24
  4. 2(x + 5) = 2x + 10
  5. 1/3 × (9 + 6) = 1/3×9 + 1/3×6 = 3 + 2 = 5
  6. 12 + 15 = 3×4 + 3×5 = 3(4 + 5) = 3×9 = 27

Quick tip to remember

Think "distribute the outside number to every term inside the parentheses." If you do that, you will be using the distributive property correctly.

If you want, I can give you more practice problems, make a worksheet, or show animated pictures of the area model. Which would you like next?

Ask a followup question

Previously Asked Questions 1

#1
why would you ever use the distributive property if the other way to find the answer is faster?
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Great question — you’re right that sometimes just adding inside the parentheses first is faster. But the distributive property is a super useful tool for many situations. Here are the main reasons you’d use it, with simple examples a 12‑year‑old can try in their head:

1) Makes mental math easier

  • Example: 17 × 6. Instead of doing long multiplication, split 17 into 10 + 7 and use the distributive property: 6×(10+7) = 6×10 + 6×7 = 60 + 42 = 102. That’s quick in your head.
  • Another trick: 19 × 5 = (20 − 1) × 5 = 20×5 − 1×5 = 100 − 5 = 95.

2) Works well with awkward numbers (like fractions)

  • Example: (1/2) × (6 + 4) = (1/2)×6 + (1/2)×4 = 3 + 2 = 5. Sometimes splitting first helps you compute without a calculator.

3) You need it in algebra (variables)

  • If you have 3(x + 4) = 18, you must distribute to get 3x + 12 = 18 so you can solve for x. When letters are involved, you can’t “add inside” like normal numbers.

4) Helps simplify and factor expressions (backwards distributing)

  • If you see 12 + 15, you can factor out 3: 12 + 15 = 3(4 + 5) = 3 × 9 = 27. Factoring is useful when simplifying expressions or solving equations.

5) Helps avoid mistakes with negatives and signs

  • Distributing makes you multiply signs correctly. For example 2(5 − 8) = 2×5 − 2×8 = 10 − 16 = −6.

6) Visual and problem-solving uses (area model)

  • The distributive property explains why splitting a rectangle into parts and adding their areas works. This helps in geometry and real‑world problems.

When not to use it: if the numbers are tiny and adding first is quicker (like 3×(4+5)), just add then multiply. The distributive property is another tool in your toolbox — use it when it makes the work faster or simpler.

Want a few quick mental‑math problems to practice using distributive tricks? I can give 5 you can try in your head.

Asked 12/3/2025