The distributive law of multiplication over addition | Pre-Algebra | Khan Academy

Summary

This video explains the distributive property of multiplication over addition using a simple example: 4 × (8 + 3). It demonstrates both evaluating the expression directly and using the distributive law by breaking down the multiplication into separate terms: 4 × 8 + 4 × 3. The video emphasizes how the distributive property works by visualizing the multiplication as repeated addition and offers clear steps to simplify the expression to 44. The explanation is designed to make the distributive property easier to understand and apply in different mathematical contexts.

Takeaways

• 😀 The distributive property allows you to multiply a number by each addend in a sum before adding the results.
• 😀 The given expression starts as 4 times (8 + 3), which is an example of using the distributive law.
• 😀 The first instinct might be to evaluate what's inside the parentheses first, but using the distributive property requires multiplying the outside number by both terms inside the parentheses.
• 😀 The expression 4 times (8 + 3) can be rewritten as 4 times 8 + 4 times 3 using the distributive property.
• 😀 The distributive property is helpful because it distributes multiplication over addition, ensuring you multiply both terms within the parentheses.
• 😀 Simplifying the expression 4 times 8 + 4 times 3 gives the result of 44.
• 😀 By visualizing 8 plus 3 as objects (circles), you can better understand how multiplying the sum by 4 is like duplicating the entire sum multiple times.
• 😀 When you duplicate the sum (8 + 3) four times, it shows that you're multiplying both 8 and 3 by 4.
• 😀 Using the distributive property helps to break down the problem into smaller parts that are easier to solve.
• 😀 Visualizing multiplication helps make the distributive property easier to understand, as seen in the example where the sum is duplicated four times.
• 😀 The distributive property works because multiplying 4 by each term inside the parentheses (8 and 3) gives the same result as evaluating the sum first and then multiplying.

Q & A

What is the distributive property of multiplication over addition?

The distributive property of multiplication over addition states that when you multiply a number by a sum, you can distribute the multiplication to each term inside the parentheses. This means that a(b + c) = a*b + a*c.

How is the distributive property applied in the expression 4 × (8 + 3)?

In the expression 4 × (8 + 3), the distributive property is applied by multiplying 4 by both 8 and 3 separately. This results in 4 × 8 + 4 × 3, which simplifies to 32 + 12.

What is the result when you first simplify the parentheses in 4 × (8 + 3)?

When you first simplify the parentheses in 4 × (8 + 3), you calculate 8 + 3, which equals 11. Then, you multiply 4 by 11, resulting in 44.

Why is it important to use the distributive property in this case instead of simplifying the parentheses first?

Using the distributive property emphasizes how multiplication can be distributed across addition. It also helps to understand the relationship between the numbers and their multiplication, which is especially useful when working with more complex expressions.

How does the distributive property work visually in this example with circles?

Visually, the distributive property can be represented by imagining 8 circles and 3 circles. When multiplied by 4, the total set of circles is repeated four times, thus multiplying both 8 and 3 by 4 separately, showing that 4 × (8 + 3) equals 4 × 8 + 4 × 3.

What happens if you apply the distributive property correctly to 4 × (8 + 3)?

Applying the distributive property correctly to 4 × (8 + 3) gives the expression 4 × 8 + 4 × 3, which simplifies to 32 + 12, resulting in 44.

What mistake might someone make when first solving 4 × (8 + 3)?

A common mistake would be to first evaluate the sum inside the parentheses (8 + 3) and then multiply by 4, which would give 44, but this does not demonstrate the distributive property of multiplication.

How do we know that distributing 4 over (8 + 3) is the correct application of the distributive property?

Distributing 4 over (8 + 3) is correct because it reflects how multiplication works across addition. The distributive property allows you to split the multiplication into two parts (4 × 8 and 4 × 3) and then combine the results.

Why does the distributive property help when solving more complex mathematical expressions?

The distributive property helps by breaking down complex expressions into simpler, more manageable parts. This method allows you to work with smaller, easier numbers and simplify the calculations step by step.

What is the final simplified result of applying the distributive property to 4 × (8 + 3)?

The final simplified result of applying the distributive property to 4 × (8 + 3) is 32 + 12, which equals 44.