# How to multiply numbers quickly using the distributive property

Since multiplication is a shortcut for addition, it is important to show students illustrations of multiplication that imply addition. In 3 x 24, there are three groups of 24 added together. This model demonstrates the ease with which a student can find the product. At first, have your students count the tens blocks, and then have them count the ones blocks to enhance the concept of addition. They will count 6 tens and 12 ones. Then demonstrate how to regroup the 12 ones to make one ten block and 2 ones. A final count will yield 7 tens, 2 ones, or 72.

Although some exercises may not involve regrouping, have your students check them all to decide whether regrouping is needed. Some students may decide to add 3 groups of 24. This method involves regrouping the ones. Further regrouping resources are available in Grade 3, Regrouping to Multiply and Divide.

Once students become familiar with using the models for multiplying, have them record the multiplication alongside the model to show each individual step, including the regrouping, if needed. In this way, the base-ten blocks that are used to aid multiplication also lead to the development of the algorithm. The algorithm becomes dependent upon each action used with the blocks. With repeated practice in writing and developing the algorithm, the base-ten blocks can eventually be eliminated.

However, allow students the choice of working with the blocks to help students link to the symbolic algorithm. As a final check for the answer, have them estimate the product by rounding the factors. If the estimate and actual product are close, then students can tell that the answer is reasonable.

When multiplying three-digit numbers by one-digit numbers, the algorithm develops into a step-by-step process, which is dependent upon the number of digits in the greater factor.

Help students realize that multiplying with three-digit numbers results in a product of 3 or 4 digits. For example, in 4 x 567, students must first multiply ones, then tens, then hundreds.

Regrouping is needed only when there are more than 9 ones, more than 9 tens, or more than 9 hundreds.

 First, find 4 x 7 ones = 28 ones. Write 8 in the ones place of the product. Regroup 20 ones as 2 tens. Write 2 in the tens place. Second, find 4 x 6 tens = 24 tens. Add the regrouped 2 tens, which gives 26 tens.Write 6 in the tens place of the product. Regroup 20 tens as 2 hundreds. Write 2 in the hundreds place. Third, find 4 x 5 hundreds = 20 hundreds. Add the regrouped 2 hundreds. Write 2 in the hundreds place of the product. Regroup 20 hundreds as 2 thousands. Write 2 in the thousands place. Check the product by rounding 567 to 600. Then 4 x 600 = 2,400. Since 2,268 is close to 2,400, the product is reasonable.

Slowly guide students to multiply greater numbers. Help them realize that the algorithm developed through the use of the base-ten blocks can be extended to include numbers with more than three digits. It is always important to offer the base-ten blocks as further proof that the algorithm developed results in accurate products.

Factors with zeros, such as 306 or 670, need special attention. Frequently, students forget the need for zero as a placeholder.

Alert students to such situations and remind them that the product of any number and zero is zero.

Emphasize that these numbers are treated just as any other factor, so the multiplication algorithm, regrouping rules, and base-ten blocks will still work as methods for arriving at accurate products.

 First, find 7 x 8 ones. For the result 56, write 6 in the ones place. Regroup 50 ones as 5 tens. Second, find 7 x 0 tens, or 0. Add the 5 regrouped tens. Write 5 in the tens place. Third, find 7 x 9 hundreds, or 63 hundreds. Write 3 in the hundredsplace. Regroup 60 hundreds as 6 thousands. Check by rounding 908 to 900. Then find 7 x 900, or 6,300. Since 6,356 is close to 6,300, the product is reasonable.

Sometimes zeros in the factors represent multiples of 10, 100, or 1,000. To find 7 x 300, review the basic fact 7 x 3 = 21, then write as many zeros as are in the factors. Write 21, followed by two zeros, or 2,100 to record the product.

Start with one-digit factors to emphasize the use of the basic facts and the multiples of 10, 100, or 1,000. Review the use of the Associative Property to reacquaint students with the idea that the grouping of factors can be changed.

 4 x 600 = 4 x (6 x 100) Replace 600 with 6 x 100 = (4 x 6) x 100 Use the Associative Property to change the grouping of factors. = 24 x 100 Use the basic fact: 4 x 6 = 24. = 2, 400 Write the same number of zeros in the product as are in the factors.

Next, introduce two factors that are multiples of 10, 100, or 1,000. The use of patterns in developing work with multiples often allows students to visualize an idea and put it into words. Furthermore, this represents a good time to reacquaint students with the Commutative Property. The Commutative Property allows the order of factors to be changed without changing the product.

 4 x 3 = 12 Basic Fact 4 x 30 = 120 By the Associative Property, 4 x (3 x 10) = (4 x 10) = (4 x 3) x 10. 40 x 30 = 1,200 By the Commutative and Associative Properties, (4 x 10) x (3 x 10)becomes (4 x 3) x (10 x 10) or 12 × 100. 400 x 30 = 12,000 By the Commutative and Associative Properties, (4 x 100) x (3 x 10) becomes(4 x 3) x (100 x 10) or 12 x 1,000.

Patterns showing products of multiples generate student interest and often open lines of communication. Encourage students to explain patterns by writing them in a journal.

The multiplication operation has several mathematical properties. As demonstrated, the Commutative Property allows the order of factors to be changed without changing the product. We write the property as a x b = b x a.

This property allows us to multiply in an order that we may find more convenient. We can find 4 x 56 or 56 x 4 because the product does not change. The Associative Property allows factors to be grouped in different ways without changing the product.

We write the property as (a x b) x c = a x (b x c). Both properties are especially useful for mental math.

Notice that the Commutative and Associative Properties use the single operation of multiplication. On the other hand, the Distributive Property uses two operations: multiplication and addition. It is an essential key to success in algebra.

The Distributive Property allows you to “distribute” a number to each of the addends within parentheses. It offers another way of solving a problem. We write the property as a x (b + c) = (a x b) + (a x c).

Although powerful, the Distributive Property can be intricate. Here is an example. This array shows 3 x 6.

The array can be rearranged in the following way.

The array now shows 3 x (4 + 2). The factor 6 has been replaced by the sum of 4 and 2. You can see that there are now three groups of 4 and three groups of 2. This is written as (3 x 4) + (3 x 2). Mathematicians say that the number 3 has been “distributed” to 4 and to 2.

Notice that the total number of items always remains the same. Using this example, you can find the sum of 4 and 2, then multiply by 3. Or you can distribute the 3 and create two products that are then added.

 3 x (4 + 2) = 3 x 6 3 x (4 + 2) = (3 x 4) + (3 x 2) = 18 = 12 + 6 = 18

The true power of the Distributive Property becomes evident when you use mental math. You can astonish your students by finding products of larger numbers in your head. Then show them your “trick” by using the Distributive Property.

 4 x 109 = 4 x (100 + 9) 9 x 78 = 9 x (80 − 2) = (4 x 100) + (4 x 9) = (9 x 80) − (9 x 2) = 400 + 36 = 720 − 18 = 436 = 702

The Distributive Property can also be used “backwards.” If the same number is used as a factor more than once, you can apply the property to help you “condense” the problem.

For example, (32 x 4) + (32 x 6) = n can be solved by finding the two products, then adding. However, (32 x 4) + (32 x 6) = n can also be solved by realizing that 32 had been distributed to the numbers 4 and 6.

The property can simplify problems, making them easier for mental math.

 (32 x 4) + (32 x 6) = 128 + 192 (32 x 4) + (32 x 6) = 32 x (4 + 6) = 320 = 32 x 10 = 320

Help your students enjoy multiplication. Bring in advertisements and newspaper or magazine articles that show the essential nature of multiplication. Encourage open forums for students to share their ideas. Multiplication becomes a lifelong skill—one that is essential to almost all careers, professions, and trades. Become an integral part of that learning!

## Distributive Property of Multiplication – Elementary Mathematics

In today’s post, we will look at some examples of the distributive property. But first, we have to remember what this property consists of.

As you know, multiplication has different properties, among which we point out:

• Commutative Property
• Associative Property
• Neutral Element
• Distributive Property
• Well, the distributive property is that by which the multiplication of a number by a sum will give us the same as the sum of each of the sums multiplied by that number.
• For example:
• 3 x (4 + 5) = 3 x 4 + 3 x 5
• But we can also apply the distributive property in the other direction, then calling out a common factor, and thus:
• 2 x 6 + 2 x 9 = 2 x (6 + 9)
• Let’s look at two examples:
• Distributive: 8 x (13 – 1) = 8 x 13-8 x 1 = 8 x 13-8
• Remove common factor: 12 x 3 x 2 + 3 x 6 + 7 x 3 = 3 x (12 x 2 + 6 + 7)

### To understand this better, let’s see an example of Distributive Property in a Word Problem: Mary is preparing for her birthday party, at which she will distribute sweets to all her friends. To do this, she will put 5 strawberry, 4 lemon and 3 peppermint candies in each bag.  She has decided to give away 10 bags of candy.

How many candies are given away altogether?

To solve the problem, it is important that we know the number of candies of each kind in each bag, and the number of bags.

Therefore, we can solve this problem in two different ways:

• We find the total number of candies that she will put in each bag, and then multiply by the number of bags:

5 + 4 + 3 = 12 candies in each bag

12 x 10 = 120 candies in total

• We find the total number of candies of each flavor and then add:
1. 5 pieces of strawberry candies in 10 bags: 5 x 10 = 50 strawberry candies
2. 4 lemon candies in 10 bags: 4 x 10 = 40 lemon candies
3. 3 peppermint candies in 10 bags: 3 x 10 = 30 peppermint candies
4. We add all the candy: 50 + 40 + 30 = 120 candies in total
5. We see that the two paths have obtained the same result, so we can choose the path that we find easier.

What did you think about this post on examples of the distributive property? I hope it helped you to better understand the distributive property of multiplication! To keep learning, register and try Smartick for free.

• The Distributive Property of Multiplication
• Properties of Multiplication
• Learn the Different Properties of Multiplication
• Review the Different Properties of Multiplication
• Applying the Commutative Property of Addition and Multiplication in a Problem

## Decomposing the multiplicand: The distributive property of multiplication

 Example 1.    3 × 24 = 3 × 20  +  3 × 4 = 60  +  12 = 72.

We expanded 24 into 20 + 4, and then “distributed” 3 to each one.

Look:

 3 × 24 = 24 + 24 + 24 = 20 + 4  +  20 + 4  +  20 + 4 = 20 + 20 + 20  +  4 + 4 + 4 = 3 × 20  +  3 × 4.

The repeated addition of 24 is equal to the repeated addition of 20, plus the repeated addition of 4.

(For a general proof, see Appendix 4.)

Example 2.   Multiply
5 × 37  mentally.

Technique.  Distribute 5 to 30 + 7.  Say:

“150 plus 35 is 185.”

Multiply the numbers as you read them, from left to right.  The last number you say is the answer.

Example 3.   Multiply mentally  8 × 46.

• “320 + 48
= 368.”
• Example 4.   800 × 460
• Ignore the final 0's and multiply  8 × 46.  But we just saw
that
• 8 × 46 = 368.
• Therefore:
• 800 × 460 = 368,000.
• This is “368” with three 0's.

Example 5.   Multiply  6 × 7.30.  (Treat problems with decimal points as dollars and cents.)

Technique.  Expand 7.30 mentally into
7 + .30    Then

 6 × 7.30 = 42 + 1.80 (Lesson 9) = 43.80.

Example 6.   What is the price of five items that cost \$3.25 each?

Answer.  Since  4 × \$.25 = \$1.00, then  5 × \$.25 = \$1.25.  Say,

“5 × 3.25 = 15 + 1.25 = 16.25″

Example 7.   Multiply  2 × 438  mentally.

 Solution. 2 × 4 3 8 = 800 60 16 + + = 876.

The point is to say each partial sum.  Look at  2 × 438  and say,

“860 + 16 is 876.”

(Again, in 438, the 4 signifies 400, and the 3 signifies 30. Lesson 2.)

Example 8.   Multiply 4 × 709.

 Solution. 4 × 709 = 4 4 4 × 700  +   × 0  +   × 9 = 2800 + 0 + 36 = 2836.
• Note:   Any number times 0, or 0 times any number, is 0.
• Therefore, to calculate  4 × 709, simply ignore the 0 and say:
• “2800 + 36 is 2836.”

Example 9.   Multiply 8,000 × 4,310.

Technique.   Ignore the final 0's:

 8 × 4 3 1 = 3200 240 8 + + = 3440 + 8 = 3448.

Now replace the four 0's:

8,000 × 4,310 = 34,480,000

Example 10.   How much is 20% of \$68?

Solution.   10% of \$68 is \$6.80. (Lesson 4.)  Therefore, 20% is

2 × \$6.80 = \$12 + \$1.60 = \$13.60.

Example 11.   How many hours are there in one week?  How many minutes are there?

Solution.  There are 24 hours in one day, and
there are 7 days in a week.  Therefore,

1. 7 × 24 = 140 + 28 = 168 hours.
2. Now, in each hour there are 60 minutes. To multiply
3. 60 × 168,
4. ignore the 0 and multiply
 6 × 1 6 8 = 600 360 48 + + = 960 + 48 = 1008.
• Now replace the 0 we ignored:
• 10080.
• In one week, then, there are 10,080 minutes.
• At this point, please “turn” the page and do some Problems.
• or
• Continue on to Section 3:  Multiplying by rounding off
• Section 1 of this Lesson