Houghton Mifflin Mathematics Teacher Support Grade 3 Grade 3
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Lesson Ideas

     Using Arrays to
     Show Multiplication

  Introducing the Concept
  Developing the Concept

What Is It?  

Using Arrays to Show Multiplication Concepts

Students can more readily develop an understanding of multiplication concepts if they see visual representations of the computation process. For example, they can picture students in a marching band arranged in equal rows or chairs set up in rows in an auditorium. These arrangements all have something in common; they are all in rows and columns. An arrangement of objects, pictures, or numbers in columns and rows is called an array. Arrays are useful representations of multiplication concepts.

This array has 4 rows and 3 columns. It can also be described as a 4 by 3 array.

This array has 5 rows and 4 columns. It is a 5 by 4 array.

Notice that the rows in each array are equal. Think of the rows as equal groups. Your students used equal groups to multiply in second grade. Look at this example.

When equal groups are arranged in equal rows, an array is formed.

When you show students the connection between equal groups and arrays, students can easily understand how to use arrays to multiply. They will use arrays again in Chapter 8 to divide.

Look at the multiplication sentence that describes the array below. The numbers in multiplication sentences have special names.

The numbers that are multiplied are called factors. The answer is called the product.

Now look at what happens to the factors and product in the multiplication sentence when the array is turned on its side.

The order of the factors changed, but the product stayed the same. When the order of the factors in any multiplication sentence changes, the product does not change. This is called the Commutative Property of Multiplication. Students should be familiar with the Commutative Property because it also applies to addition. They studied the Commutative Property of Addition in Chapter 3. See Addition and Subtraction.

Help students realize that by applying the Commutative Property, they know twice as many multiplication facts. For example, if they know 8 times 5 = 40, then they also know 5 times 8 = 40.


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