Lesson Objectives

- Learn how to label the parts of a multiplication problem as a factor or product
- Learn about the commutative property of multiplication
- Learn about the associative property of multiplication
- Learn about the identity property of 1
- Learn about the multiplication property of 0
- Learn about the distributive property of multiplication

## Multiplying Whole Numbers

Multiplying whole numbers is simply a shortcut for repeated addition. The multiplication operation can be accompanied by a variety of symbols. Most common in elementary school we use either symbol: “x” or “•” placed between the numbers being multiplied. Suppose we think about the following scenario: We have 4 rows with 3 boxes in each row. How can we find the total number of boxes? We could obviously count each box individually, but this is slow and not practical for larger examples. We could also use addition, we have 3 boxes in each row and we have 4 such rows. We could find the sum as:

3 + 3 + 3 + 3 = 12

Additionally, we could use a shortcut: 4 rows of 3 is the same as 4 times 3, which equals 12. The multiplication here is a shortcut for repeated addition. It may seem trivial with this example, but suppose we had 25 rows with 3 boxes in each row. This new scenario would make for a very long and tedious addition problem. We would have to add twenty-five occurrences of the number 3. Instead of using addition, we can simply use multiplication to get our answer:

25 x 3 = 75

We will first learn the single-digit multiplication facts via memorization of the times tables:

Times Table (1 - 9): Once we have memorized the multiplication table for all of the single-digit numbers, we can combine these facts with vertical multiplication to multiply numbers as large as we would like.

4 x 3 = 12

Example 1: Label the parts of the multiplication problem: 9 x 7 = 63 The 9 and 7 are being multiplied together, these are the factors. The 63 is the result of the multiplication, this is our product.

Example 2: Rewrite 6 x 5 using the commutative property of multiplication

To complete this problem, simply change the order of the factors. The 5 will be first or the leftmost number and the 6 will now come second.

6 x 5 = 5 x 6

In each case, the result is 30.

6 x 5 = 30

5 x 6 = 30

Changing the order of the factors (6,5) did not change the product (30).

Example 3: Rewrite (2 x 3) x 7 using the associative property of multiplication

To complete this problem, simply change the grouping of the factors. Notice how the parentheses are around the 2 x 3 portion of the problem. We will switch the grouping and place parentheses around 3 x 7.

(2 x 3) x 7 = 2 x (3 x 7)

In each case, the result is 42.

From the order of operations, we work inside of parentheses first.

(2 x 3) x 7 = 6 x 7 = 42

2 x (3 x 7) = 2 x 21 = 42

Changing the grouping did not change the product. In each case, the result of the multiplication is 42.

Example 4: Find the product of 635 x 1

635 x 1 = 635

Any number (635) multiplied by 1, remains unchanged (635).

Example 5: Find the product of 72,255 x 0

72,255 x 0 = 0

Any number (72,255) multiplied by 0, results in 0.

5(6 + 3)

This is the same as writing: 5 x (6 + 3)

How can we solve this problem? We will get to the order of operations soon, but for now, we need to know that when parentheses or grouping symbols are present, we must work inside first. If we find the sum of what’s inside of parentheses 6 and 3, we would get 9. We can replace the (6 + 3) with 9.

5(6 + 3) = 5 x 9

Now we can simply multiply and obtain our answer of 45.

5(6 + 3) = 5 x 9 = 45

The alternative way to solve this problem is with the use of the distributive property. We can distribute the 5 to each number inside of parentheses. This is another way of saying we will remove the parentheses and multiply the 5 by each number inside. 5(6 + 3) = 5 x 6 + 5 x 3

At this point, you may or may not know what operations to perform here. We have both addition and multiplication. We must always multiply before we add, so 5 x 6 and 5 x 3 are completed separately before addition occurs.

5(6 + 3) = 5 x 6 + 5 x 3 = 30 + 15 = 45

This property may seem like too much work. After all, we can simply add inside the parentheses first, and then multiply and get the same result. In many cases in Algebra, you will encounter situations where a variable is inside of parentheses and we can’t simply work inside. In these cases, we will rely on the distributive property to remove parentheses so we can continue to work through our problem.

Example 6: Find the product of 3(6 - 4) using the distributive property of multiplication

3(6 - 4) = 3 x 6 - 3 x 4 = 18 - 12 = 6

Multiplication is done before subtraction. Multiply 3 x 6 and 3 x 4 separately before subtraction occurs.

You can check this result by working inside of parentheses first and solving the problem.

3(6 - 4) = 3(2) = 6

Either way, the result is 6, again the distributive property may seem like more work for now, but we will rely on this property throughout our study of Algebra.

3 + 3 + 3 + 3 = 12

Additionally, we could use a shortcut: 4 rows of 3 is the same as 4 times 3, which equals 12. The multiplication here is a shortcut for repeated addition. It may seem trivial with this example, but suppose we had 25 rows with 3 boxes in each row. This new scenario would make for a very long and tedious addition problem. We would have to add twenty-five occurrences of the number 3. Instead of using addition, we can simply use multiplication to get our answer:

25 x 3 = 75

We will first learn the single-digit multiplication facts via memorization of the times tables:

Times Table (1 - 9): Once we have memorized the multiplication table for all of the single-digit numbers, we can combine these facts with vertical multiplication to multiply numbers as large as we would like.

### How do we Identify the Parts of a Multiplication Problem?

When working with a multiplication problem, the numbers being multiplied together are known as factors. The result of the multiplication is known as the product. In our first example, we looked at the multiplication problem:4 x 3 = 12

- 4 - Factor
- 3 - Factor
- 12 - Product

Example 1: Label the parts of the multiplication problem: 9 x 7 = 63 The 9 and 7 are being multiplied together, these are the factors. The 63 is the result of the multiplication, this is our product.

- 9 - Factor
- 7 - Factor
- 63 - Product

### Commutative Property of Multiplication

We learned in the properties of addition lesson, that we could add two or more numbers in any order and not change the sum. This property is known as the commutative property of addition. In multiplication, we have the same property. The commutative property of multiplication states that we can multiply two or more numbers in any order without changing the product.Example 2: Rewrite 6 x 5 using the commutative property of multiplication

To complete this problem, simply change the order of the factors. The 5 will be first or the leftmost number and the 6 will now come second.

6 x 5 = 5 x 6

In each case, the result is 30.

6 x 5 = 30

5 x 6 = 30

Changing the order of the factors (6,5) did not change the product (30).

### Associative Property of Multiplication

We learned in the properties of addition lesson, that we could group the addition of three or more numbers in any order and not change the sum. This property is known as the associative property of addition. In multiplication, we have the same property. The associative property of multiplication states that we can group the multiplication of three or more factors in any order and not change the product.Example 3: Rewrite (2 x 3) x 7 using the associative property of multiplication

To complete this problem, simply change the grouping of the factors. Notice how the parentheses are around the 2 x 3 portion of the problem. We will switch the grouping and place parentheses around 3 x 7.

(2 x 3) x 7 = 2 x (3 x 7)

In each case, the result is 42.

From the order of operations, we work inside of parentheses first.

(2 x 3) x 7 = 6 x 7 = 42

2 x (3 x 7) = 2 x 21 = 42

Changing the grouping did not change the product. In each case, the result of the multiplication is 42.

### Identity Property of 1

The identity property of 1 states that any number multiplied by 1 remains unchanged.Example 4: Find the product of 635 x 1

635 x 1 = 635

Any number (635) multiplied by 1, remains unchanged (635).

### Multiplication Property of 0

The multiplication property of 0 states that the product of zero and any number is 0.Example 5: Find the product of 72,255 x 0

72,255 x 0 = 0

Any number (72,255) multiplied by 0, results in 0.

### Distributive Property of Multiplication

The distributive property of multiplication states that multiplication is distributive over addition and subtraction. To understand this property, we must first know that a number placed outside of parentheses implies multiplication. If we saw a problem such as:5(6 + 3)

This is the same as writing: 5 x (6 + 3)

How can we solve this problem? We will get to the order of operations soon, but for now, we need to know that when parentheses or grouping symbols are present, we must work inside first. If we find the sum of what’s inside of parentheses 6 and 3, we would get 9. We can replace the (6 + 3) with 9.

5(6 + 3) = 5 x 9

Now we can simply multiply and obtain our answer of 45.

5(6 + 3) = 5 x 9 = 45

The alternative way to solve this problem is with the use of the distributive property. We can distribute the 5 to each number inside of parentheses. This is another way of saying we will remove the parentheses and multiply the 5 by each number inside. 5(6 + 3) = 5 x 6 + 5 x 3

At this point, you may or may not know what operations to perform here. We have both addition and multiplication. We must always multiply before we add, so 5 x 6 and 5 x 3 are completed separately before addition occurs.

5(6 + 3) = 5 x 6 + 5 x 3 = 30 + 15 = 45

This property may seem like too much work. After all, we can simply add inside the parentheses first, and then multiply and get the same result. In many cases in Algebra, you will encounter situations where a variable is inside of parentheses and we can’t simply work inside. In these cases, we will rely on the distributive property to remove parentheses so we can continue to work through our problem.

Example 6: Find the product of 3(6 - 4) using the distributive property of multiplication

3(6 - 4) = 3 x 6 - 3 x 4 = 18 - 12 = 6

Multiplication is done before subtraction. Multiply 3 x 6 and 3 x 4 separately before subtraction occurs.

You can check this result by working inside of parentheses first and solving the problem.

3(6 - 4) = 3(2) = 6

Either way, the result is 6, again the distributive property may seem like more work for now, but we will rely on this property throughout our study of Algebra.

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