Introduction to Multiplication

Introduction

In this lesson, you will learn the basics of multiplication.

These videos illustrate the lesson material below. Watching the videos is optional.

Multiplication Basics

Ancient mathematicians realized that adding the same number over and over would always result in the same answer. For example, two added together three times equals six, or $2 + 2 + 2 = 6$ . To simplify these equations, multiplication was developed, which is a new equation that represents repeated addition. When two is added 3 times, you can multiply 2 by 3 which is equal to 6: $2 \times 3 = 6$

Because these equations always lead to the same answer, they are known as multiplication facts. It is important to memorize these facts so you can more easily do arithmetic and better understand other principles, like fractions and algebra.

Ordering Multiplication Equations

In addition equations, you can rearrange the order of the numbers and get the same answer. For example, $2 + 7$ and $7 + 2$ are both equal to 9. Multiplication is the same way.

Example 1

Both $2 \times 3$ and $3 \times 2$ are equal to 6. The figures below help clarify this concept. In either equation, the number of blocks remains the same.

2 multiplied by 3 equals 6. This is represented by 3 rows each with 2 squares.

Figure 1

3 multiplied by 2 equals 6. This is represented by 2 rows each with 3 squares.

Figure 2

Example 2

The same principle applies for all multiplication facts. Figure 3 is a visual representation of 8 added together 5 times. This is another example of repeated addition.

8 plus 8 plus 8 plus 8 plus 8 equals 40. This is represented by 5 rows each with 8 squares.

Figure 3

Consider this another way: $8 \times 5$ .

8 multiplied by 5 equals 40. This is represented by 5 rows each with 8 squares.

Figure 4

You have the same set of blocks below, but instead of 8 added 5 times, you have 5 added 8 times. Notice it gives the same answer of 40.

5 plus 5 plus 5 plus 5 plus 5 plus 5 plus 5 plus 5 equals 40. This is represented by 8 rows each with 5 squares.

Figure 5

Remember: The rule of multiplication allows you to multiply repeated addition problems.

5 multiplied by 8 equals 40. This is represented by 8 rows each with 5 squares.

Figure 6

Figures 4 and 6 show that both $8 \times 5$ and $5 \times 8$ are equal to 40.

Multiplication Tables

Figure 7 is a multiplication table. You can use this multiplication chart to help you learn multiplication facts. Use it to make flashcards to help you memorize the facts. Memorizing multiplication facts will be helpful to you and make math easier as you continue through the course.

Figure 7

If you want to know what a certain equation equals, find the intersection between the two numbers that you are multiplying. For example, if you want to know what $8 \times 6$ equals, find the place on the multiplication table where 8 and 6 intersect, and you will see the answer is 48.

Multiplication by 0, 1, or 10

Multiplication equations with 0, 1, or 10 follow basic rules outlined below:

Multiplication of 0
Any number multiplied by 0 equals 0.

$3 \times 0 = 0$
$84 \times 0 = 0$
$1538976 \times 0 = 0$

Multiplication of 1
Any number multiplied by 1 remains the same number.

$5 \times 1 = 5$
$99 \times 1 = 99$
$53702 \times 1 = 53702$

Multiplication of 10
Any number multiplied by 10 is the same number but with a new zero at the end.

$5 \times 10 = 50$
$44 \times 10 = 440$
$19875 \times 10 = 198750$

In other words, multiplying by 10 moves all the numbers one place value to the left, or one place value higher.

Things to Remember

Multiplication is just repeated addition.
Memorizing multiplication facts will be beneficial now and in the future.
Any number multiplied by 0 equals 0.
Any number multiplied by 1 remains the same.
Any number multiplied by 10 is the same number with a 0 added to the end of the original number.

Practice Problems

$1 \times 4 = ?$ (
Video Solution

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Solution: 4
Details:

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$4 \times 3 = ?$ (
Solution

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Solution: 12
Details:
Start with 4 circles to represent the 4 in the problem.

Each circle is assigned 3 black dots to represent the 3 in the problem.
This grouping represents the multiplication of $4 \times 3$ .

Find the answer to the multiplication problem by counting the black dots only.

The answer is 12.

Another way to look at this is as rows and columns.
Here are three columns and four rows of circles.

In other words, three columns of four circles or four rows of three circles.
Either way you look at it, there are 12 total circles.

Therefore, $4 \times 3 = 12$ and $3 \times 4 = 12$ .

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$8 \times 4 = ?$ (
Video Solution

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Solution: 32
Details:

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$9 \times 8 = ?$ (
Solution

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Solution: 72
Details:
The circles represent the 9 in the multiplication problem.

Each circle now contains 8 images of a smiley face. These represent the 8 in the problem.

There are 9 groups of 8 smiley faces. Count the smiley faces only.

This is the same as 8 added together 9 times. (It is also the same as 9 added together 8 times.)
This represents the answer to $9 \times 8$ . The total count is 72.

The answer to $9 \times 8$ is 72.

Another way to look at this is as repeated addition.
Nine added together eight times is the same as $9 \times 8$ .

Similarly, 8 added together 9 times is $8 \times 9$ .
An exciting thing about multiplication is that it is commutative.

This means $9 \times 8 = 8 \times 9 = 72$ .

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$0 \times 65 = ?$ (
Solution

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Solution:
0

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$84 \times 1 = ?$ (
Solution

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Solution:
84

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$49 \times 10 = ?$ (
Solution

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Solution:
490

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