Writing Equations from Rectangular Arrays - Second Grade Mathematics

A rectangular array is a treasure chest of mathematical relationships: depending on whether you look horizontally across its rows or vertically down its columns, the exact same array can generate two distinct repeated addition equations! Being able to write equations that represent both row-by-row counting and column-by-column counting is a vital second-grade skill that cements your understanding of equal groups. In this chapter, you will master writing row equations, writing column equations, and calculating the total number of items in any rectangular array up to 5 rows and 5 columns.

The Dual Perspectives of an Array

Consider a 3 by 4 rectangular array of circles:

The 3 by 4 Array:
(O)  (O)  (O)  (O)
(O)  (O)  (O)  (O)
(O)  (O)  (O)  (O)

Perspective 1: Counting by Rows (Horizontal Strips)

Look at the array as horizontal rows: Row 1 has 4 circles. Row 2 has 4 circles. Row 3 has 4 circles. There are 3 rows of 4 circles. Row Equation: 4 + 4 + 4 = 12

Perspective 2: Counting by Columns (Vertical Strips)

Now, shift your eyes and look at the exact same array as vertical columns: Column 1 has 3 circles. Column 2 has 3 circles. Column 3 has 3 circles. Column 4 has 3 circles. There are 4 columns of 3 circles. Column Equation: 3 + 3 + 3 + 3 = 12

Comparing Row and Column Equations

Let us place the two equations side by side:

Row Equation:      4 + 4 + 4 = 12        (Three 4s added together)
Column Equation:   3 + 3 + 3 + 3 = 12    (Four 3s added together)

Both Equations Equal Exactly 12!

The Structure of the Addends

Notice how the equations mirror the dimensions of the array: In the row equation, the addend is 4 (the row length), and it is repeated 3 times (the number of rows). In the column equation, the addend is 3 (the column height), and it is repeated 4 times (the number of columns). Writing both equations proves that you can decompose the total rectangle either horizontally or vertically without changing the sum.

Arrays up to 5 by 5: Essential Foundations

In second grade, you focus on arrays with up to 5 rows and up to 5 columns.

Common 2nd Grade Arrays:
2 by 3:  Row: 3 + 3 = 6            Col: 2 + 2 + 2 = 6
3 by 3:  Row: 3 + 3 + 3 = 9        Col: 3 + 3 + 3 = 9 (Square Array!)
4 by 2:  Row: 2 + 2 + 2 + 2 = 8    Col: 4 + 4 = 8
5 by 4:  Row: 4 + 4 + 4 + 4 + 4=20 Col: 5 + 5 + 5 + 5 = 20

The Special Case of Square Arrays

Notice the 3 by 3 array: it has 3 rows and 3 columns. Row Equation: 3 + 3 + 3 = 9 Column Equation: 3 + 3 + 3 = 9 When the number of rows equals the number of columns, the array forms a perfect square, and the row equation and column equation are completely identical!

Practice Exercises

Exercise 1: Look at an array that has 3 rows with 5 squares in each row. Write the repeated addition equation for the rows, write the repeated addition equation for the columns, and state the total.

Exercise 2: An array generates the row equation 2 + 2 + 2 + 2 = 8. How many rows does this array have, and how many items are in each row? Write the matching column equation.

Exercise 3: Draw or describe a 4 by 3 array of dots. Write both the row equation and the column equation that describe your drawing.

Exercise 4: Explain why a 4 by 4 array produces the exact same repeated addition sentence whether you count by rows or by columns.

Solutions and Explanations

Solution to Exercise 1: For an array with 3 rows of 5 squares: The row equation adds 5 three times: 5 + 5 + 5 = 15. The column equation adds 3 five times: 3 + 3 + 3 + 3 + 3 = 15. The total number of squares is 15.

Solution to Exercise 2: In the row equation 2 + 2 + 2 + 2 = 8, the number 2 is repeated 4 times. This means the array has 4 rows with 2 items in each row (a 4 by 2 array). The matching column equation looks at the 2 columns of 4 items: 4 + 4 = 8.

Solution to Exercise 3: A 4 by 3 array has 4 horizontal rows with 3 dots in each row. The row equation is 3 + 3 + 3 + 3 = 12. The column equation looks at the 3 vertical columns of 4 dots: 4 + 4 + 4 = 12.

Solution to Exercise 4: A 4 by 4 array is a square array where the number of rows (4) equals the number of columns (4). Counting by rows adds 4 four times (4 + 4 + 4 + 4 = 16), and counting by columns also adds 4 four times (4 + 4 + 4 + 4 = 16). Because the dimensions are identical, both perspectives generate the exact same repeated addition sentence.