Math Background

Distributive Property of Multiplication: Overview

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.

base-ten blocks

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.

base-ten blocks

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.

    2 2  
    5 6 7
  2, 2 6 8
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.

    9 0 8
  6, 3 5 6
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  =  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.

base-ten blocks

The array can be rearranged in the following way.

base-ten blocks

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!

Houghton Mifflin Math Grade 4