In math, an array refers to a set of numbers or objects that will follow a specific pattern. An array is an orderly arrangement (often in rows, columns or a matrix) that is most commonly used as a visual tool for demonstrating multiplication and division.

There are many everyday examples of arrays that help with understanding the utility of these tools for quick data analysis and simple multiplication or division of large groups of objects. Consider a box of chocolates or a crate of oranges that have an arrangement of 12 across and 8 down rather than count each one, a person could multiply 12 x 8 to determine the boxes each contain 96 chocolates or oranges.

Examples such as these aid in young students' understanding of how multiplication and division work on a practical level, which is why arrays are most helpful when teaching young learners to multiply and divide shares of real objects like fruits or candies. These visual tools allow students to grasp how observing patterns of "fast adding" can help them count larger quantities of these items or divide larger quantities of items equally amongst their peers.

## Describing Arrays in Multiplication

When using arrays to explain multiplication, teachers often refer to the arrays by the factors being multiplied. For example, an array of 36 apples arranged in six columns of six rows of apples would be described as a 6 by 6 array.

These arrays help students, primarily in third through fifth grades, understand the computation process by breaking the factors into tangible pieces and describing the concept that multiplication relies on such patterns to aid in quickly adding large sums multiple times.

In the six by six array, for instance, students are able to understand that if each column represents a group of six apples and there are six rows of these groups, they will have 36 apples in total, which can quickly be determined not by individually counting the apples or by adding 6 + 6 + 6 + 6 + 6 + 6 but by simply multiplying the number of items in each group by the number of groups represented in the array.

## Describing Arrays in Division

In division, arrays can also be used as a handy tool to visually describe how large groups of objects can be divided equally into smaller groups. Using the above example of 36 apples, teachers can ask students to divide the large sum into equal-sized groups to form an array as a guide to the division of apples.

If asked to divide the apples equally between 12 students, for example, the class would produce a 12 by 3 array, demonstrating that each student would receive three apples if the 36 were divided equally among the 12 individuals. Conversely, if students were asked to divide the apples between three people, they would produce a 3 by 12 array, which demonstrates the Commutative Property of Multiplication that the order of factors in multiplication does not affect the product of multiplying these factors.

Understanding this core concept of the interplay between multiplication and division will help students form a fundamental understanding of mathematics as a whole, allowing for quicker and more complex computations as they continue into algebra and later applied mathematics in geometry and statistics.