Multiplication from Equal Groups to Arrays

From equal groups to arrays

The first introduction to multiplication for students is as solving problems about equal groups. So, using 3 × 4 as an example, at first students interpret it as 3 groups of 4 and they represent it with concrete materials (3 groups of 4 objects) or with a drawing as shown below.

Initially students may find 3 × 4 by counting the dots one by one. When more confident with skip counting, they will count by fours. The critical understanding is that 3 × 4 = 4 + 4 + 4 (ie that multiplication can be solved by repeated addition).

The next representation for multiplication is as an array. Students need to be able to represent 3 × 4 not as 3 separate groups, but as 3 rows of 4 objects as shown below.

Learning to see the 3 groups of 4 in the array (and also to see 4 groups of 3) is an important step in students' understanding of multiplication. Recognising an array as an instance of multiplication opens the way to new mathematical ideas, including, for example, the area of a rectangle.

Calculating multiplication facts with arrays

As with other aspects of calculation, an important goal is to move students towards strategies that are more efficient than counting by ones. The visual pattern of arrays encourages students to use the more efficient strategy of skip counting. For example, an array showing 4 × 8 could be drawn as an array of 4 rows of 8 dots:

Students might count the dots by columns as 4, 8, 12, 16, 20, 24, 28, 32 or the rows as 8, 16, 24, 32. Some students might count the first row by ones then skip count the rest.

Once proficient using arrays, students may put out just one row of counters or draw just one row of dots and count these multiple times. To find the total of 6 × 8 a student might draw 8 dots

and then count by 8 six times: 8, 16, 24, 32, 40, 48.

Eventually students no longer need the partial model of counters or drawings and will skip count or recall the multiplication facts to solve multiplication or division problems.