The Distributive Property

Core Concepts

Learn how to break apart numbers using the distributive property to make multiplication much easier.

The Distributive Property

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What is the rule for the Distributive Property?

The rule is $a \times (b + c) = (a \times b) + (a \times c)$. This means you can multiply a number by each part of a sum and then add the results.

How do you break apart the number $47$ using place value to prepare for the distributive property?

You write it in expanded form: $40 + 7$.

Using the distributive property, rewrite the problem $6 \times 58$.

You would rewrite it as $6 \times (50 + 8)$, which then becomes $(6 \times 50) + (6 \times 8)$.

Solve $6 \times 58$ using the partial products from the distributive property: $(6 \times 50) + (6 \times 8)$.

$$ (6 \times 50) + (6 \times 8) = 300 + 48 = 348 $$

How do you break apart a three-digit number like $243$ for the distributive property?

You write it in expanded form based on place value: $200 + 40 + 3$.

Rewrite the problem $5 \times 243$ using the distributive property.

Rewrite it as $5 \times (200 + 40 + 3)$, which equals $(5 \times 200) + (5 \times 40) + (5 \times 3)$.

Find the partial products for $5 \times 243$ using the distributive property.

The partial products are:\ $5 \times 200 = 1000$\ $5 \times 40 = 200$\ $5 \times 3 = 15$

What is the final product of $5 \times 243$ after finding the partial products ($1000$, $200$, $15$)?

$$ 1000 + 200 + 15 = 1215 $$

Why is the distributive property a key concept for multi-digit multiplication?

It is the mathematical reason why methods like the area model and the standard algorithm work. It allows us to break a large, difficult problem into several smaller, easier problems.

Show how $9 \times 99$ can be solved easily with subtraction and the distributive property.

Think of $99$ as $(100 - 1)$. Then, $9 \times (100 - 1) = (9 \times 100) - (9 \times 1) = 900 - 9 = 891$.

The Distributive Property

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