Arrays and Area Models - Third Grade Mathematics

When items are neatly organized into straight rows and columns, counting them becomes fast and orderly. In this chapter, you will explore arrays and area models—two of the clearest visual representations of multiplication in all of mathematics. Arrays connect visual patterns to numerical equations and lay the essential groundwork for understanding area in geometry.

What Is an Array?

An array is an orderly arrangement of objects aligned into horizontal rows and vertical columns with no gaps and no overlaps.

Row:    Runs horizontally from left to right ( <-----> )
Column: Runs vertically from top to bottom   (   ^   )
                                             (   |   )
                                             (   v   )

Look at this array of stars:

Column 1   Column 2   Column 3   Column 4   Column 5
   *          *          *          *          *      <- Row 1
   *          *          *          *          *      <- Row 2
   *          *          *          *          *      <- Row 3

Count the rows: there are 3 rows going across. Count the columns: there are 5 columns going up and down. Every row has exactly 5 stars. This is a 3 by 5 array, written as: 3 x 5 = 15 stars!

Reading and Writing Array Equations

Always state the number of rows first, followed by the number of columns:

Rows  x  Columns  =  Total Objects

Example Array:
# # # #
# # # #
# # # #
# # # #
# # # #

- Number of rows (horizontal): 5
- Number of items in each row (columns): 4
- Equation: 5 x 4 = 20

The Commutative Property Revealed by Arrays

One of the greatest beauties of an array is how easily it demonstrates the Commutative Property of Multiplication: changing the order of the factors does not change the product.

Look at what happens when you rotate an array by 90 degrees (a quarter turn):

Original Array (2 rows of 4):
o  o  o  o
o  o  o  o
Equation: 2 x 4 = 8

Rotated Array (4 rows of 2):
o  o
o  o
o  o
o  o
Equation: 4 x 2 = 8

Both arrays contain the exact same 8 dots! Therefore: 2 x 4 = 4 x 2 = 8. When you memorize your multiplication facts, every array you learn gives you two facts for the price of one!

Transitioning to Area Models

An area model replaces individual dots with a continuous geometric rectangle divided into unit squares (a grid).

       <--------- 6 units wide --------->
      +-----+-----+-----+-----+-----+-----+
      |  1  |  2  |  3  |  4  |  5  |  6  |  ^
      +-----+-----+-----+-----+-----+-----+  |
      |  7  |  8  |  9  | 10  | 11  | 12  |  3 units high
      +-----+-----+-----+-----+-----+-----+  |
      | 13  | 14  | 15  | 16  | 17  | 18  |  v
      +-----+-----+-----+-----+-----+-----+

In this area model:
- The height (number of rows) is 3 units.

- The width (number of columns) is 6 units.

- The total number of square units inside is 3 x 6 = 18 square units.
Area models bridge the gap between counting individual objects and measuring geometric shapes.

Breaking Apart Arrays (The Distributive Property Preview)

You can slice a large array into two smaller arrays to solve tricky facts.

Suppose you want to solve 7 x 6, but you are not sure of your 7 facts: Slice the 7 rows into 5 rows and 2 rows!

5 rows of 6: 5 x 6 = 30
2 rows of 6: 2 x 6 = 12
Combine them: 30 + 12 = 42!
Therefore: 7 x 6 = 42.

Breaking apart arrays makes complex multiplication manageable.

Chapter Practice Exercises

  1. Write the multiplication equation (Rows x Columns = Total) for each array: a. x x x x x x x x x x x x b. o o o o o o o o o o o o
  2. Draw or describe an array with 4 rows and 6 columns. What is the total number of items?
  3. Show how rotating an array shows that 3 x 5 = 5 x 3.
  4. A classroom has student desks arranged in 5 rows with 6 desks in each row. How many desks are in the classroom?
  5. A gardener planted tomato plants in an array of 4 rows with 8 plants in each row. Write the multiplication sentence and find the total number of tomato plants.

Solutions and Step-by-Step Answers

  1. Array equations: a. 4 rows and 3 columns -> 4 x 3 = 12. b. 2 rows and 6 columns -> 2 x 6 = 12.
  2. An array with 4 rows and 6 columns has 4 horizontal lines of 6 items each. Equation: 4 x 6 = 24 items.
  3. Original array has 3 rows and 5 columns (3 x 5 = 15). Turning it sideways gives 5 rows and 3 columns (5 x 3 = 15). The total number of objects (15) remains identical, proving 3 x 5 = 5 x 3.
  4. 5 rows of 6 desks: 5 x 6 = 30 desks in the classroom.
  5. 4 rows of 8 tomato plants: 4 x 8 = 32 tomato plants total.