1.4 Multiply Whole Numbers

Multiplication of Whole Numbers

Correct multiplication notation

There are a few different ways to write multiplication, and you'll see all of them as you move through math courses.

• The multiplication symbol ($\times$) is the most familiar: $3 \times 4$ means 3 multiplied by 4.
• Parentheses work too: $3(4)$ means the same thing as $3 \times 4$.
• The dot operator ($\cdot$) shows up in algebra: $a \cdot b$ means $a$ multiplied by $b$. This avoids confusion with the variable $x$.

In a multiplication expression like $5 \times 6$, the two numbers being multiplied are called factors, and the answer is called the product. So the product of 5 and 6 is 30.

Visual models for multiplication

Visual models make it easier to see what multiplication actually represents.

• Equal groups: $3 \times 4$ means 3 groups with 4 objects in each group. If you picture 3 bags with 4 marbles each, you can count 12 marbles total. This connects multiplication to repeated addition: $4 + 4 + 4 = 12$.
• Arrays: $2 \times 3$ can be shown as 2 rows of 3 objects. Arrange 6 coins into 2 rows and 3 columns, and you've got a $2 \times 3$ array.

Arrays are especially useful for seeing the commutative property. A $2 \times 3$ array (2 rows, 3 columns) has the same number of objects as a $3 \times 2$ array (3 rows, 2 columns). Either way, you get 6. The order of the factors doesn't change the product.

Efficient multiplication techniques

Once you understand what multiplication means, the next step is getting faster at it.

• Memorize your multiplication tables up to $10 \times 10$. Quick recall of basic facts makes every other technique easier. Practice regularly so these stay sharp.
• Use the distributive property to break apart harder problems. For example:

$12 \times 15 = (10 + 2) \times 15 = (10 \times 15) + (2 \times 15) = 150 + 30 = 180$

You're splitting 12 into 10 and 2, multiplying each part by 15, then adding the results.

• Use the commutative property to pick the easier order. If $7 \times 13$ feels awkward, think of it as $13 \times 7$ instead. The answer is the same either way.
• Multiply by powers of 10 using a shortcut: just add zeros. $6 \times 10 = 60$ (add one zero). $6 \times 100 = 600$ (add two zeros). $6 \times 1000 = 6000$ (add three zeros).
• The standard multiplication algorithm is your go-to method for multi-digit numbers. Here's how it works for $24 \times 13$:

1. Multiply 24 by the ones digit of 13: $24 \times 3 = 72$
2. Multiply 24 by the tens digit of 13: $24 \times 1 = 24$, but since it's in the tens place, write it as 240
3. Add the partial products: $72 + 240 = 312$

Word problems to expressions

Turning a word problem into a math expression is a skill you'll use constantly. Here's how to approach it:

1. Spot the multiplication keywords. Words like "times," "product," "each," "per," "doubled," and "tripled" all signal multiplication.
2. Identify the quantities. Figure out what numbers you're working with. Sometimes they're given directly; sometimes you'll use a variable like $x$ for an unknown.
3. Write the expression. Combine the quantities with multiplication.

For example: "A bookshelf has 4 shelves, and each shelf holds 6 books. How many books total?" The word "each" tells you to multiply. The expression is $4 \times 6 = 24$ books.

Another example: "A baker makes 3 times as many muffins as cupcakes. If she makes $x$ cupcakes, how many muffins does she make?" The expression is $3 \times x$, or simply $3x$.

Real-world multiplication applications

Multiplication shows up whenever you're working with equal groups, repeated quantities, or rectangular measurements.

• Total items in equal groups: If there are 5 boxes with 12 cookies in each box, the total is $5 \times 12 = 60$ cookies.
• Area of rectangles: A room that's 8 feet long and 6 feet wide has an area of $8 \times 6 = 48$ square feet. You multiply length by width.
• Total cost of identical items: If a pen costs $2 and you buy 7 pens, the total cost is $7 \times 2 = 14$ dollars.

In each case, multiplication replaces the need to add the same number over and over.

Properties of multiplication

These four properties are rules that always hold true for multiplication. Knowing them helps you simplify problems and check your work.

• Commutative property: You can multiply factors in any order. $4 \times 7 = 7 \times 4 = 28$.
• Identity property: Any number multiplied by 1 stays the same. $9 \times 1 = 9$.
• Zero property: Any number multiplied by 0 equals 0. $15 \times 0 = 0$. No matter how large the number, multiplying by zero wipes it out.
• Associative property: When multiplying three or more numbers, grouping doesn't matter. $(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24$. This is handy when one grouping is easier to compute than another.