## Using a Multiplication Table: Overview

The groundwork for multiplication was laid earlier for students when they used objects, number lines, and arrays to multiply. See Using Arrays to Show Multiplication Concepts. Students are now ready to use their understanding of multiplication to explore number patterns. A multiplication table is an excellent tool for discovering patterns.

Look at the multiplication table below. The numbers along the left side and the top are factors. The numbers inside are products.

Use the multiplication table below to locate the product of 9 and 7. Find the row for 9 and the column for 7. The number in the square where the row and column meet is the product.

9 x 7 = 63

The Commutative Property of Multiplication states that changing the order of the factors does not change the product. Use the multiplication table below to locate the product of 7 and 9.

7 x 9 = 63

Following is one of the more dramatic multiplication patterns that you can help students discover by using a multiplication table. The products of two factors that are the same form a pattern. The table below shows the product of 6 and 6.

6 x 6 = 36

Note that a square is formed. The product of any number multiplied by itself is called a **square number**.
Since 6 x 6 = 36, 36 is a square number.

The multiplication table below shows all the square numbers to 100. Notice that the squares form a diagonal across the multiplication table. Notice also that the products on one side of the diagonal mirror the products on the other side.

Students can explore square numbers as well as other patterns by using multiplication tables. Following are some of the other patterns your students will discover.

The products in the row and column for some factors are double the products in the row and column for other factors. Explore with students how the relationships between the factors in the rows and columns affect the relationship between the products of those factors. For example, since 8 is the double of 4, the products in the row for 8 are double the products in the row for 4.

The products in the rows and columns for 1, 3, 5, 7, and 9 alternate between even and odd numbers.

The sum of the digits in each product for the multiples of 9 is 9.

Students should be given a lot of practice using multiplication tables. They will use them again when they divide. See Multiplication Tables and Fact Families.

Once students understand how to multiply two factors, they are ready to proceed to multiplying three factors. While this may sound challenging to students at first, they will discover that by grouping factors together, it is really quite easy.

Parentheses are grouping symbols that tell which operations to perform first. The factors 1 and 3 are grouped together in the multiplication sentence below.

(1 x 3) x 5 = 15

3 x 5 = 15

This time, the factors 3 and 5 are grouped together.

1 x (3 x 5) = 15

1 x 15 = 15

You see that changing the grouping of factors does not change the product. This is called the Associative Property of Multiplication. Students should grasp the Associative Property of Multiplication easily because it works in much the same way as the Associative Property of Addition, which they studied earlier.