1.4 Multiply Whole Numbers

Use Multiplication Notation

Suppose you were asked to count all these pennies shown in Figure 1.11.

Figure 1.11

Would you count the pennies individually? Or would you count the number of pennies in each row and add that number $3$ times.

Multiplication is a way to represent repeated addition. So instead of adding $8$ three times, we could write a multiplication expression.

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.

There are several symbols that represent multiplication. These include the symbol $\times$ as well as the dot, $·$, and parentheses $(\text{).}$

Model Multiplication of Whole Numbers

There are many ways to model multiplication. Unlike in the previous sections where we used $\text{base-10}$ blocks, here we will use counters to help us understand the meaning of multiplication. A counter is any object that can be used for counting. We will use round blue counters.

Example 1.40

Model: $3\times 8.$

Solution

To model the product $3\times 8,$ we’ll start with a row of $8$ counters.

The other factor is $3,$ so we’ll make $3$ rows of $8$ counters.

Now we can count the result. There are $24$ counters in all.

$3\times 8=24$

If you look at the counters sideways, you’ll see that we could have also made $8$ rows of $3$ counters. The product would have been the same. We’ll get back to this idea later.

Multiply Whole Numbers

In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section.

Table 1.4 shows the multiplication facts. Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.

What happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.

What happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and $1$ is called the multiplicative identity.

Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that $8+9=17$ is the same as $9+8=17.$

Is this also true for multiplication? Let’s look at a few pairs of factors.

When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.

To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.

We start by multiplying $3$ by $7.$

We write the $1$ in the ones place of the product. We carry the $2$ tens by writing $2$ above the tens place.

Then we multiply the $3$ by the $2,$ and add the $2$ above the tens place to the product. So $3\times 2=6,$ and $6+2=8.$ Write the $8$ in the tens place of the product.

The product is $81.$

When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.

When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.

Example 1.46

Multiply: $62(87).$

Solution

Write the numbers so each place lines up vertically.
Start by multiplying 7 by 62. Multiply 7 by the digit in the ones place of 62. $7\cdot 2=14.$ Write the 4 in the ones place of the product and carry the 1 to the tens place.
Multiply 7 by the digit in the tens place of 62. $7\cdot 6=42.$ Add the 1 ten we carried. $42+1=43$. Write the 3 in the tens place of the product and the 4 in the hundreds place.
The first partial product is 434.
Now, write a 0 under the 4 in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of 87 by 62. Multiply 8 by the digit in the ones place of 62. $8\cdot 2=16.$ Write the 6 in the next place of the product, which is the tens place. Carry the 1 to the tens place.
Multiply 8 by 6, the digit in the tens place of 62, then add the 1 ten we carried to get 49. Write the 9 in the hundreds place of the product and the 4 in the thousands place.
The second partial product is 4960. Add the partial products.

The product is $\mathrm{5,394}.$

Example 1.48

Multiply: $(354)(438).$

Solution

There are three digits in the factors so there will be $3$ partial products. We do not have to write the $0$ as a placeholder as long as we write each partial product in the correct place.

Example 1.49

Multiply: $(896)201.$

Solution

There should be $3$ partial products. The second partial product will be the result of multiplying $896$ by $0.$

Notice that the second partial product of all zeros doesn’t really affect the result. We can place a zero as a placeholder in the tens place and then proceed directly to multiplying by the $2$ in the hundreds place, as shown.

Multiply by $10,$ but insert only one zero as a placeholder in the tens place. Multiply by $200,$ putting the $2$ from the $12.$ $2·6=12$ in the hundreds place.

$$\begin{array}{}\\ \\ \hfill 896\\ \hfill \underset{\text{\_\_\_\_\_}}{\times 201}\\ \hfill 896\\ \hfill \underset{\text{\_\_\_\_\_\_\_\_\_\_}}{17920}\\ \hfill \mathrm{180,096}\end{array}$$

When there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example:

Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words. Now we’ll reverse the process and translate word phrases into math notation. Some of the words that indicate multiplication are given in Table 1.5.

Multiply Whole Numbers in Applications

We will use the same strategy we used previously to solve applications of multiplication. First, we need to determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify to get the answer. Finally, we write a sentence to answer the question.

Example 1.52

Humberto bought $4$ sheets of stamps. Each sheet had $20$ stamps. How many stamps did Humberto buy?

Solution

We are asked to find the total number of stamps.

Write a phrase for the total.	the product of 4 and 20
Translate to math notation.	$4\cdot 20$
Multiply.
Write a sentence to answer the question.	Humberto bought 80 stamps.

If we want to know the size of a wall that needs to be painted or a floor that needs to be carpeted, we will need to find its area. The area is a measure of the amount of surface that is covered by the shape. Area is measured in square units. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm.) on a side. A square inch is a square that is one inch on each side, and so on.

For a rectangular figure, the area is the product of the length and the width. Figure 1.12 shows a rectangular rug with a length of $2$ feet and a width of $3$ feet. Each square is $1$ foot wide by $1$ foot long, or $1$ square foot. The rug is made of $6$ squares. The area of the rug is $6$ square feet.

Figure 1.12 The area of a rectangle is the product of its length and its width, or $6$ square feet.