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**Array|Definition & Meaning**

**Definition**

A **structured arrangement** of different items or things following a **definite pattern** or flow is known as an **array**. The things and items could be arithmetic **numbers, any object, or pictures** while the definite pattern could be **columns or rows**. The basic aim of arrays is to demonstrate the concept of multiplication and division and assist in counting the items.

By using arrays, **multiplication and division** relationships can be broken down fascinatingly**.** Because multiplication and division are **opposites** of each other, arrays illustrate this concept of **inverse operations**. An array can be used to **quickly analyze data** and easily multiply or divide groups of objects. Many examples of arrays are found in everyday life that illustrates the utility of these tools.

**Condition For Pattern Arrangement in an Array**

Consider the figure shown above, the arrow starting from **top to bottom** shows the **column arrangement** while the arrow starting from **left to right** shows the **row arrangement** of the array. Anything like numbers or objects arranged in a such manner makes an array.

**Rectangular Array**

When the **number of rows** and number of** columns** is **equal** in the count then it forms a r**ectangular array**. Consider the above figure when we count the number of rows and columns, we are getting 4, so there are **4 rows and 4 columns** forming a rectangular array.

Consider the above figure that is **not an array** because there is **no row or column** in the figure, so we can’t say this is an array.

This figure has 4 rows and 3 columns, so this is a **4 by 3 array**.

Considering the above example, we want to see if the above figures are **rectangular arrays or not. **

**In figure A**, there are **three triangles** in the first row and the second and third row has **two triangles**, so this is **not an array** because the array has not the same number of items in every row.

**In figure B** there are **two rows and four columns**, although **it is an array** it is **not satisfying** the condition for a **rectangular array** i.e., an equal number of rows and columns.

**In figure C** the first row has **four circles,** the second row has **two circles**, the third row has **three circles**, and the fourth row has **five circles** so there is no equal number of items in each row, so it is **neither an array nor a rectangular array**.

In figure D the number of rows and columns is three, so it is a rectangular array.

**Demonstrating Multiplication using Array**

**Multiplication** is commonly explained with arrays by referring to the **factors** being multiplied. An array of **sixteen oranges** arranged in **four columns of four rows** would be referred to as a **4 by 4 array**. Using the four-by-four array, we can easily understand that they will have 16 oranges in total if each **column represents** a group of **four oranges** and if there are **four rows** of these groups. You can easily calculate this number by **multiplying the number of items** in each group by the total number of groups in the array rather than counting each orange individually or **adding 4 + 4 + 4 + 4.**

Consider the above figure the concept of multiplication can be easily understood by the array as they act as a **visual aid** which makes it easier to understand. There are **two rows** and there are **four objects in each row** so we will simply multiply **2 by 4 which gives 8.**

**Demonstrating Division using Array**

Suppose there are **16 oranges**, and we are asked to **divide** these **oranges** among **two people** what we do will simply make an array of **2 by 8** as shown in the figure below.

**Demonstrating Inverse using Array**

Suppose there are **9 circles**, and they are **represented** as a **3 by 9 array** we know that **3 multiplied by 3** equal **9** and we also know its inverse that **9 divided by 3** equals** 3**. In simple words **division** is **undoing multiplication** and **multiplication** is **undoing division** so there is an **inverse relation** between them that could be easily understood by the figure shown below.

**Commutative Property of Multiplication**

Suppose we have **12 triangles**, and they are represented in two forms of arrays.

The first form is that there are **4 rows** and there are **3 objects in each row** so multiplying gives a total of **12 triangles count**.

The second form is that there are **3 rows and there are 4 objects** in each row so multiplying gives **12 triangles count**.

Thus, this **proved** the **commutative property of multiplication** whether the order or row or columns are interchanged the **final product** remains the **same**. The demonstration is shown in the figure below.

Developing a fundamental understanding of these arrays as a whole will help us understand the** interplay** between **multiplication and division**, enabling us to perform more **complex calculations** as we advance into **algebra** and then **advance mathematics**.

**Visual Examples**

Consider the following figures, using the concept of array studied above **write** the **number** of **rows, columns,** and the **total number of objects** along with the array size/order for each figure.

Figure 10 – Array Worksheet

### Solution

**In Figure A**

Number of rows = **3**

Number of columns = **3**

Total number of objects =** 9**

Array size/order = **3×3**

**In Figure B**

Number of rows = **3**

Number of columns = **2**

Total number of objects = **6**

Array size/order = **3×2**

**In Figure C**

Number of rows = **4**

Number of columns = **4**

Total number of objects = **16**

Array size/order = **4×4**

**In Figure D**

Number of rows = **3**

Number of columns = **6**

Total number of objects = **18**

Array size/order = **3×6**

Arrays are **most useful** when teaching students how to **multiply and divide** large shares of real objects, such as apples or oranges, since they help them understand how multiplication and division work practically. Students can use these **visual representations** to practice counting larger quantities of these items or dividing evenly among their peers by observing patterns of **quick addition**.

*All mathematical drawings and images were created with GeoGebra.*