Math Background

Multiplication Tables and Fact Families: Overview

The multiplication table is a great tool for exploring number patterns and relationships. Students used a multiplication table earlier to learn multiplication facts. Multiplication tables are also useful for showing division concepts and fact families.

Look at the multiplication table below.

The numbers along the left side and top are factors. The numbers inside are products.

multiplication table

To use the multiplication table to find the product of 3 and 9, locate 3 in the first column and then find 9 in the top row. Follow the 3 row to where it meets the 9 column. The number in the square where the column and row meet is the product. See the shaded area in the table below.

3 x 9 = 27

multiplication table

Another way to find the product of 3 and 9 on the multiplication table is to locate 9 in the first column and then 3 in the top row. See the shaded area in the table below.

3 x 9 = 27

multiplication table

A multiplication table can also be used to find missing factors in multiplication and division sentences. Finding a missing factor in multiplication is similar to finding a quotient in division.

Five times n equals twenty; twenty divided by five equals n.

Use the multiplication table below to find the missing factor in 5 x n = 20.

Locate 5 in the first column and move across the row to 20. The number in the square at the top of the column is the missing factor.

5 x 4 = 20

multiplication table

To find the quotient in 20 ÷ 5 = n, follow the same steps as for 5 x 4 = 20 above. The number in the square at the top of the column is the quotient.

20 ÷ 5 = 4

Here is another way to show the same related number sentences on a multiplication table.

To find the missing factor in 5 x n = 20, locate 5 in the top row and move down the column to 20. Then move left from 20 to the end of the row. The number in the square at the end of the row is the missing factor.

5 x 4 = 20

multiplication table

To find the quotient in 20 ÷ 5 = n, follow the same steps as for 5 x 4 = 20 above. The number in the square at the left end of the row is the quotient.

20 ÷ 5 = 4

Making the connection between missing factors in multiplication sentences and quotients in division will help students better understand the relationship between the two operations. See Regrouping to Multiply and Divide. Provide students with many opportunities to practice these topics.

A multiplication table can also be used to reinforce students' understanding of other math concepts, such as the Commutative Property of Multiplication and inverse operations. Look at the multiplication table below.

multiplication table

The table shows 3 x 6 = 18. It also shows 6 x 3 = 18, because the Commutative Property of Multiplication states that changing the order of the factors does not change the product.

The inverse, or opposite, of multiplication is division. So the table also shows 18 ÷ 3 = 6 and 18 ÷ 6 = 3.

These four number sentences each use the same three numbers: 3, 6, and 18. Related number sentences that use the same numbers are called a fact family.

Some fact families have only two related number sentences. The multiplication table below shows the fact family for 7 and 49.

multiplication table

The fact family for 7 and 49 is 7 x 7 = 49 and 49 ÷ 7 = 7. There is only one multiplication sentence in this fact family because the factors are the same number. There is only one division sentence because the divisor and quotient are the same number.

As students gain experience using a multiplication table, encourage them to look for more number relationships and patterns, including doubles facts, skip counting by 5s and 10s, doubles and triples, and so on.


Houghton Mifflin Math Grade 3