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Distributive Property
Of Multiplication

To “distribute” means to divide one whole amongst many parts. The distributive property of multiplication is based on this concept and states that,

"Multiplying a number to a sum of two or more addends yields the same result as obtained by multiplying the number to each addend individually and then adding the products."

Let us consider an example to understand what this means. We have to carry out multiplication for the following expression:

4(2 + 3)

The ‘order of operations’ rule of mathematics tempts us to solve this expression using the following steps:

Step 1: Perform addition inside the parenthesis to get 4 x 5

Step 2: Perform multiplication to get the result = 20

The distributive property changes the approach. We first distribute the number being multiplied, over the addends. The new steps followed are:

Step 1: Split the parenthesis to distribute 4. We get (4 x 2) + (4 x 3)

Step 2: Perform multiplication inside the parenthesis to get 8 + 12

Step 3: Add the products to get the result = 20

We notice that the result is the same in both cases and the distributive property indeed holds true. Visually, we can represent this as:

Demonstrating the distributive property of multiplication

It is interesting to note that the distributive property of multiplication is valid not just for addition, but for subtraction as well. For example, let’s say we have to solve the expression:

3(5 - 2)

We can use the normal approach to calculate it as:

3(5 - 2) = 3 x 3 = 9

Using distributive property, we can calculate it as:

3(5 - 2) = (3 x 5) – (3 x 2) = 15 – 6 = 9

Once again, we can see that the result is the same in both cases and the distributive property holds as expected. In fact, the property is valid even when we have a combination of addition and subtraction inside the parenthesis, as shown below.

When dealing with a combination of addition and subtraction inside the parenthesis, we must be extra careful with the + and - signs. At this point, we can modify the statement of the distributive property of multiplication to now state that,

“Multiplying a number to a sum or difference of two or more numbers (called operands) yields the same result as obtained by multiplying the number to each operand individually and then adding or subtracting the products (depending on the operation).”

Why do we use the distributive property of multiplication?

From the examples above, it seems as if using the distributive property makes multiplication lengthy because we are using three steps to do what could have been in two steps. However, using the distributive property actually makes calculations easier in some cases. For example, let us consider the expression:

6(57)

Most children would find this multiplication complicated, but if we use the distributive property of multiplication over addition, we can simplify this expression as:

6(50 + 7) = 300 + 42

= 342

This expression is easier to solve because now we are multiplying 6 with a multiple of 10 and with a single-digit number. There is still some scope to simply the multiplication further. If we use the distributive property of multiplication over subtraction instead of addition, we will get:

6(60 - 3) = 360 – 18

= 342

Clearly, the aim is to carry out multiplication with multiples of 10 and with numbers as small as possible.This is obviously simpler than multiplying with a complicated number having two or more digits. Let us now consider an example with a three-digit number inside the parenthesis:

3(597) = 3(500 + 90 + 7)

= 1500 + 270 + 21

= 1791

In this example, it would be better to use subtraction. Can you guess why?

3(597) = 3(600 - 3)

= 1800 – 9

= 1791

By using subtraction, we reduce the number of times we have to multiply – we are doing it twice instead of thrice. Also, the number 3 is smaller than 9 and 7, which makes calculations fast.

In both the above examples, since the three-digit number is a multiple of hundred, we are essentially multiplying 3 with a one-digit number only – either 5 or 6 and simply appending two zeroes in the end. We have effectively reduced a three-digit multiplication with 397 into simple one-digit multiplication.

However, using direct subtraction always is also not a good idea. If we were to evaluate an expression such as 5(287) it would be better to use a mixture of addition and subtraction. Here’s why:

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  • Direct subtraction would give us 5(300 – 13) where 13 is a two-digit number.
  • So, we go one step ahead and split 13.
  • Wecaneitherwrite13as10+3or20–7.Theformerisabetteroptionbecause3<7.

Therefore,

5(286) = 5(300 - (10 + 3))

= 5(300 – 10 – 3)

= 1500 – 50 -15

= 1435

At this point you’re probably wondering – are there any rules to determine which approach to follow in which case? The answer is, no. Figuring out if and how you need to split the term in the parenthesis is something that one learns only with practice. Personal preferences matter too – some children might find two-digit multiplication easier and more fun than three-digit addition or subtraction.

As a parent or teacher, you must teach the concept of distributivity to children but don’t pressure them to always use it. Give them the freedom to choose the approach that they feel most comfortable with.

How does distributivity work with division?

Given that division is the inverse of multiplication, we cannot expect distributivity to work exactly the same in both. In multiplication, we have the freedom to split either of the two numbers being multiplied. For example, in 124 x 35, (100 + 20 + 4) x 35 as well as 124 x (30 + 5) will give the same answer when we distribute.

However, if we were to evaluate 124 ÷ 35, then (100 + 20 + 4) ÷ 35 will not give the same answer as 124 ÷ (30 + 5), as demonstrated below.

Distributivity as a building block for variables and unknowns

In addition to simplifying calculations, one big reason behind teaching the distributive property to children is that it serves as a preparation for dealing with variables like ‘x’ and ‘y’. When we have unlike terms inside the parenthesis, we often use distributivity to solve equations. For example, if we have to solve the equation 5(3 + x) = 17, we distribute 5 over the terms inside the parenthesis to get:

5(3 + x) = 17

or 15 + 5x = 17

or 5x = 2

or x = 5/2

How can you get children to practice the distributive property?

Now that you’re equipped with examples, tips, and tricks for distributivity, it’s time to help children apply the concept. The next time you go to the grocery store, ask your child to total the bill for you before you get to the billing counter. Teachers can share math worksheets with their students, based on the distributive property.

Give opportunities to children to apply the concept wherever possible!

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