Lesson 4: Multiply Whole Numbers

New Concept

Multiplication is a fast way to handle repeated addition. We'll explore its symbols (e.g., $3 \times 8$), key properties, and the standard method for multiplying multi-digit whole numbers to solve real-world problems, such as finding an area.

What’s next

Now that you have the basics, you'll move on to interactive examples, master the multiplication steps through a series of practice cards, and apply your new skills in challenge problems.

Property

Multiplication is a way to represent repeated addition. We call each number being multiplied a factor and the result the product. We read $3 \times 8$ as three times eight, and the result as the product of three and eight.

Operation Symbols for Multiplication
To describe multiplication, we can use symbols and words.

Examples

• The expression $5 \times 7$ is read as "five times seven," and the result is "the product of five and seven."
• The expression $11 \cdot 4$ is read as "eleven times four," and the result is "the product of eleven and four."
• The expression $9(12)$ is read as "nine times twelve," and the result is "the product of nine and twelve."

Property

The product of any number and 0 is 0.

Examples

• To calculate $37 \cdot 0$, remember that the product of any number and zero is 0. So, $37 \cdot 0 = 0$.
• To find $(125)0$, multiplying by zero results in zero. The answer is $0$.
• The product $0 \times 8,912$ is also 0, because the order of factors does not change the outcome when multiplying by zero.

Explanation

Think of it this way: if you have zero groups of any number, you have nothing. Similarly, if you have many groups of zero, you still have nothing. That's why any number multiplied by zero is always zero.

Property

The product of any number and 1 is the number. 1 is called the multiplicative identity.

Examples

• To calculate $1 \cdot 63$, remember that the product of any number and one is the number itself. So, $1 \cdot 63 = 63$.
• To find $(250)1$, multiplying by one does not change the value. The result is $250$.
• The product $1 \times 7,654$ is simply $7,654$ because 1 is the multiplicative identity.

Explanation

Multiplying a number by 1 means you have exactly one group of that number, so its value (or identity) doesn't change. It's like the number is looking in a mirror; it stays the same.

Property

Changing the order of the factors does not change their product.

Examples

• We know that $6 \cdot 8 = 48$. Because of the Commutative Property, we also know that $8 \cdot 6 = 48$.
• To find the product of 15 and 3, you can calculate $15 \cdot 3 = 45$ or $3 \cdot 15 = 45$. Both give the same result.
• A floor plan shows a room that is 10 feet wide and 12 feet long. Its area is $10 \times 12 = 120$ square feet, which is the same as $12 \times 10 = 120$.

Explanation

Just like with addition, you can swap the numbers you're multiplying and still get the same answer. For example, having 3 rows with 5 apples each is the same total amount as having 5 rows with 3 apples each.

Property

To multiply two whole numbers to find the product:

1. Write the numbers so each place value lines up vertically.
2. Multiply the digits in each place value. Work from right to left, starting with the ones place in the bottom number. Multiply the ones digit of the bottom number by each digit in the top number. If a product in a place value is more than 9, carry to the next place value. Write the partial products, lining up the digits. Repeat for the tens place, hundreds place, and so on, using zeros as placeholders.
3. Add the partial products.

Examples

• To multiply $58 \times 4$: First, calculate $4 \times 8 = 32$. Write down the 2 and carry the 3. Then, calculate $4 \times 5 = 20$, and add the carried 3 to get 23. The product is $232$.
• To multiply $76 \times 32$: The first partial product is $2 \times 76 = 152$. The second partial product is $30 \times 76 = 2280$. Add them: $152 + 2280 = 2432$.
• To multiply $408 \times 7$: First, $7 \times 8 = 56$ (write 6, carry 5). Then, $7 \times 0 = 0$, plus the carried 5 is 5. Finally, $7 \times 4 = 28$. The product is $2,856$.

Explanation

When multiplying large numbers, we break the problem down. We multiply the top number by each digit of the bottom number one at a time. These smaller results, called partial products, are then added to get the final answer.