Arrays: Exploring Rows and Columns - Second Grade Mathematics

Objects in the real world are often organized in neat, orderly grids: eggs in a carton, window panes in a frame, stamps on a sheet, or marching bands on a football field. In mathematics, an orderly grid of objects arranged in straight horizontal rows and straight vertical columns is called an array. Arrays are one of the most powerful visual tools in mathematics because they organize equal groups into a rectangular structure, making counting, repeated addition, and area easy to understand. In this chapter, you will master the difference between rows and columns, learn how to name arrays, and explore how turning an array reveals the commutative property.

Rows Go Across, Columns Go Up and Down

To understand arrays, you must master the two directions that form any grid.

The Anatomy of an Array:

Row 1  ----->   [ * ] [ * ] [ * ] [ * ]   (Horizontal: left to right)
Row 2  ----->   [ * ] [ * ] [ * ] [ * ]
Row 3  ----->   [ * ] [ * ] [ * ] [ * ]
                  |     |     |     |
                  v     v     v     v
                 Col 1 Col 2 Col 3 Col 4  (Vertical: up and down)

Rows Are Horizontal (Side to Side)

Think of rowing a boat across a lake or the horizontal rows of seats in a movie theater. Rows go straight across from left to right. In the array above, there are 3 horizontal rows.

Columns Are Vertical (Up and Down)

Think of tall Greek or Roman marble columns that hold up the roof of a building. Columns stand straight up and down. In the array above, there are 4 vertical columns.

Naming an Array: Rows by Columns

Mathematicians always name an array by stating the number of rows first, followed by the number of columns: "Rows by Columns."

Naming Convention:
    Number of ROWS  x  Number of COLUMNS

The Array Above:
Has 3 Rows.
Has 4 Columns.
Array Name: "3 by 4 array" (written as 3 x 4).

The "RC" Memory Device

A helpful trick to remember the order is to think of a "Remote Control" car or "RC": Rows come first (R), Columns come second (C). If an array has 2 rows and 5 columns, it is a 2 by 5 array. If an array has 5 rows and 2 columns, it is a 5 by 2 array.

The Commutative Property: Turning an Array

What happens when you physically turn or rotate an array by a quarter turn?

Original Array (3 by 2):
*  *   <--- Row 1 (2)
*  *   <--- Row 2 (2)
*  *   <--- Row 3 (2)
Equation: 2 + 2 + 2 = 6 (3 rows of 2)

Rotate 90 Degrees (2 by 3):
*  *  *   <--- Row 1 (3)
*  *  *   <--- Row 2 (3)
Equation: 3 + 3 = 6 (2 rows of 3)

The Same Total, A New Perspective!

Look closely at both arrays: The 3 by 2 array has 6 stars. The 2 by 3 array has 6 stars. Rotating the array changed the rows into columns and the columns into rows, but NOT A SINGLE STAR WAS ADDED OR REMOVED! This visual proof demonstrates the Commutative Property of Multiplication: 3 x 2 = 2 x 3 = 6. Changing the orientation of the factors does not change the total product!

Practice Exercises

Exercise 1: Describe the difference between a row and a column in your own words, providing a real-world example of each.

Exercise 2: Look at an array with 4 horizontal rows, where each row contains 5 dots. State the array's name in "rows by columns" format, and calculate the total number of dots.

Exercise 3: Draw or describe a 2 by 6 array of stars. Then describe what the array looks like when rotated 90 degrees, including its new name.

Exercise 4: Explain how rotating a rectangular array proves that 4 x 3 and 3 x 4 have the exact same total.

Solutions and Explanations

Solution to Exercise 1: A row is a horizontal line of items running side to side from left to right, like rows of corn planted in a farm field. A column is a vertical line of items running straight up and down, like tall stone pillars supporting the porch of a building.

Solution to Exercise 2: The array has 4 rows and 5 columns, so its name is a "4 by 5 array" (written 4 x 5). To find the total number of dots, add 5 four times: 5 + 5 + 5 + 5 = 20 dots.

Solution to Exercise 3: A 2 by 6 array consists of 2 horizontal rows with 6 stars in each row (total 12 stars). When rotated 90 degrees, it stands tall with 6 horizontal rows and 2 columns. Its new name is a "6 by 2 array".

Solution to Exercise 4: A 4 by 3 array contains 4 rows of 3 items (total 12). When you rotate the grid 90 degrees, the 4 rows become 4 columns, and the 3 columns become 3 rows, creating a 3 by 4 array. Because no items were added or taken away during the turn, the total quantity of 12 items remains identical, proving that 4 x 3 = 3 x 4 = 12.