## Multiplication by Two-Digit Numbers: Overview

Your students' ability to understand and use properties of operations and numbers will help them become better problem solvers. You will find that many students have already heard of the Distributive Property of multiplication, but they may not have yet made the connection to how it can aid them in understanding multiplication. (See Distributive Property of Multiplication.)

The Distributive Property of Multiplication is often explained by mathematicians as follows: Multiplying the sum of two or more addends by a number is the same as multiplying each addend by the number and then adding the products, or a x (b + c) = (a x b) + (a x c). Both of these explanations can be confusing to students. For example, they may not know that the Distributive Property gives them the opportunity to break apart a multiplication problem to make two or more simpler problems. A visual example may help explain the purpose of the Distributive Property.

An array is one way to show the product of two numbers.

Here is that same array broken apart into two smaller arrays, giving students a visual method of understanding the Distributive Property (see below).

You will want students to understand that breaking a multiplication problem into two smaller problems makes the multiplication easier, and the choice of which smaller problems to use can make the multiplication easier as well.

For example, 7 x 538 can be broken into many different smaller problems: 7 x (537 + 1), 7 x (521 + 11 + 6), and 7 x (520 + 10 + 8), to name a few. However, smaller problems that involve the multidigit numbers broken into expanded form are the easiest to compute. For ease in computing, using the Distributive Property, 7 x 538 can be broken into:
7 x (500 + 30 + 8) = (7 x 500) + (7 x 30) + (7 x 8) = 3,500 + 210 + 56 = 3,766.

You will want students to apply this knowledge to the vertical algorithm for the multiplication of multidigit numbers by one-digit numbers. Point out to students the similarities in the computations (note the color-coding).

Students are always happy to learn shortcuts to solving math problems. Therefore, your students will be eager to know about regrouping when multiplying.

For example:

Your students can then adapt these same concepts to multiplying with numbers that contain zeros.

Using the Distributive Property:
6 x 409 = 6 x (400 + 9) = (6 x 400) + (6 x 9) = 2,400 + 54 = 2,454.

Using the vertical algorithm with regrouping:

Show students that using a combination of properties of operations and numbers and multiplication with regrouping helps when multiplying any whole number by multiples of 10, 100, and 1,000.

Take 400 x 239, for example.

First, apply factoring and then the Associative Property of Multiplication.

400 x 239 = 239 x (100 x 4) = (4 x 239) x 100

Then apply multiplication with regrouping for
(4 x 239).

Thus, 400 x 239 = 100 x (239 x 4) or 100 x 956 = 95,600.

You may find that multiplying numbers by two-digit numbers is troublesome for your students. It may be helpful if they understand the connection between the Distributive Property and the multiplication of numbers by two-digit numbers.

For example:

Using the Distributive Property,

 72 x 538 = (70 + 2) x 538 = (70 x 538) + (2 x 538) = 37,660 + 1,076 = 38,736

Using the vertical algorithm with regrouping:

Point out to students the similarities in the above computations (note the color-coding).

As your students' ability to use properties of operations and numbers increases, you will notice that they can more easily determine which operation should be used to solve a particular word problem. Your students already know that there are clues to choosing the required operation. These clues are often in the form of words that indicate whether to add, subtract, multiply, divide, or to perform any combination of the four. However, not only will your students be able to identify these clues, they will also be more aware that often more than one operation is needed and that there is more than one way to solve a problem.