Division with Area Model: Definition with Examples

“Today, we will discuss division with area model. We will see how to solve the area model with division. Also, we will review the division area model with the remainder. So, let us begin!”

“Area model with division is a handy trick. It is helpful to divide large numbers. This model will make the work easier. Want to know how? We are here to explain!”

First, we will have a look at what is the area model?

What is the Area Model?

The area model is a mathematical concept. The area model is a rectangular model or diagram. It is useful for problems of multiplication and division. Notably, the area model is also called the Box model. 

In this, we break one large rectangular area into some smaller boxes. We do this using the number bonds. It helps to make the calculation easier. Then we will add the found values. After this, we will get the area of the entire rectangle. It will be the final result. We get the product on multiplication or a quotient when using the area model for division.

Derivation of the Area Model

This model is named the area model because it is derived from the concept of finding the area of a rectangle. 

Area of a rectangle = length × breadth (l × b). 

Concerning this model, the quotient and divisor factors. The quotient and division ascertain the length and width here.

Division with Area Model

The Division with Area Model area model with division is very helpful in solving division problems. Long division is considered one of the most complex topics to learn. Notably, the area model has great usability here. Students can apply the long division with the area model. Division of large figures is very easy with it. This method is simple to understand and apply.

Division With Area Model focuses on mental math. With this method, we can better understand the numbers. In this, we solve the problem based on division by subtracting multiples. We continue this process until we get a zero. Either we will get a zero or a remainder digitnumbers.

Now, let us have a look at the merits of using the area model with division.

Merits of Using the Area Model (Rectangular Model) for Division

Here are some merits of division with the area model.

• The Area Model with division provides entry points for every student to start solving large division problems. For this, we should use this method in an open-ended way. It disregards their knowledge of multiplication.

• The students can easily correlate division to taking away from what we have. It is to create as many equal sections as possible. We use and represent sections or boxes for the area division model. (The rectangles can be assumed as symbols of an actual box or a rectangular object.)

• In this model, students can double-check their solutions. We use the same division form for this. But, we start with another number. It brings surety about accuracy.

• If the teacher encourages, students should try solving the division problem differently. It will help to enhance their understanding of the model. This will, in turn, enhance their performance. While solving examples, students can solve them differently. Only the way of finding the solution will be different. The method will remain the same.

Now, let us see how to solve division problems with the area model.

How to Solve Problems of Division With Area Model?

Here is an explanation for solving the area model with division.

The area of a rectangle or any shape is the amount of space.

Let us say:

We can calculate the area of a rectangle using the formula (l × b). So, consider a rectangle with a length of 12 units. It has a breadth of 8 units. We can find its area by multiplying 12 by 8. In other words, we can geometrically represent the product as –

12 × 8 is the area of a rectangle with a length of 12 units and a breadth of 8 units.

Similarly, we will now take a division problem. 

Let us solve 570 ÷ 15

We can represent 570 ÷ 15 geometrically. Here, 570 cm is the area of the entire rectangle. The measurement of one side is 15 cm. Now, we have to find the missing dimension of the rectangle with an area of 570 cm sq., having one side of 15 cm.

Here, we will divide the rectangle into smaller rectangles. Then, we will measure the length of each smaller rectangle again and again. We will continue this until we get a 0. To get the missing length, we will add all lengths together.

Things will get more clear on solving practically.

Step 1: Consider a large rectangle with a breadth of 15 cm. We will begin with its first section. It has a length of 25 units as a starting point. 

On solving, the area of this section of the rectangle is 375 cm. The rest of the rectangle is 195 cm (570 cm – 375 cm = 195 cm). 

Step 2: Now, we have the next section of the area of 195 cm. Since 15 x 10 = 150, the new rectangle will have a length of 10 cm. The rectangle will have a 15 cm breadth (as before).

On solving, the area of this section of the rectangle is 150 cm. The rest of the rectangle is 45 cm (195 cm – 150 cm = 45 cm).

Step 3: We will get the next section of the area 45 cm. Since 15 x 3 = 45, the new rectangle will have a length of 3 cm. The rectangle will have a 15 cm breadth (as before).

Step 4: Finally, 15 x 3 = 45 is found. As a result, the last section or rectangle will have a breadth of 15 units and a length of 3 cm. 

570-375	195-150	45-45
195	45	0

So, the length of the rectangle is 25 + 10 + 3 units = 38 cm. 

Hence, 570 ÷ 15 = 38.

Let us take another example.

Divide 4956byh 4.

• We will start with the dividend, i.e., 4956.

• First, we will write the product of 4 × 1200. We will get 156 by subtracting 4800 from 4956.

• We will move 156 to the next box/rectangle. We will get 36 when we subtract 120 from 156. 

• Then, we will write the product of 4 × 9, i.e., 32. Here, we will get the remainder as 0.

4956-4800	156-120	36-36
156	36	0

This was how to solve division with an area model when there is no remainder. Now, we will see how to solve division problems with remainders using the area model.

Division Area Model With Remainder

When we divide a number, it does not always divide completely. Some numbers are left at the end. These leftover numbers are remainders. In the area model with division, we perform division by splitting it into small rectangular sections. The number being divided here is the dividend. The divisor is common for all the sections (rectangles). When solving for each section, we will get the remainder. The area method helps to visualize the math easily.

Now, we will see an example showing how to solve division problems with remainders using the area model.

Let us solve 653 ÷ 5.

Step 1: We will start by writing the dividend, i.e., 653, in the first box. Our divisor (5) will be outside on the left.

Step 2: First, we will write the product of 5 × 100. In total, it will be 500. We will get 153 by subtracting 500 from 653.

Step 3: We will move 153 to the next box/rectangle. Then, we will write the product of 5 × 30. We will get 3 when we subtract 150 from 153. 

There remain only three.

5	100	30
653-500	153-150
153	3

We cannot take any more sections. We will get 130+R3 by adding 100 + 30 + remainder 3 to reach our final quotient.

This was how to solve area model divisions with remainders.

Here is another solved example.

Let us solve 5663 ÷ 4.

Step 1: We will start by writing the dividend, i.e., 4663, in the first box. Our divisor (4) will be outside on the left.

Step 2: First, we will write the product of 4 × 1000. We will get 653 by subtracting 4000 from 4663.

Step 3: We will move 663 to the next box/rectangle. Then, we will write the product of 4 × 150. We will get 63 when we subtract 600 from 663. 

Step 4: We will move 63 to the next box. Then, we will write the product of 4 × 15. We will get threebyn subtracting 60 from 63. 

4	1000	150	15
4663-4000	663-600	63-60
663	63	3

So, we will get 1165+R3 by adding 1000 + 150 + 65 + remainder 3 to reach our final quotient.

At a glance

An area model with division is a rectangular model or diagram. It is a mathematical concept. It helps to solve division problems. In this, the quotient and divisor determine the factors. We discussed well-explained solved examples above.

Now, the students know:

• how to perform division with area model
• how to solvethe division area model with the remainder

Students can solve any division problem with this rectangle or Box model.

Hope this article proves to be helpful for you.

Frequently Asked Questions

1. What is an example of division using the area model?

Ans. An example of division using the area model is finding the area of a rectangle that is 4 inches long and 3 inches wide. To find the area, you first need to know that one inch equals 2.54 centimeters. You can then multiply 4 by 2.54 to get 12.36 centimeters (cm). Next, you divide 12.36 by 3 to get 4.04 cm2, which is the area of a rectangle in square centimeters.

2. How do you solve 42 ÷ 3 using an area model?

Ans. Show a number bond to represent Maria’s area model. Start with the total and then show how the total is split into two parts. From the two parts, represent the total length using the distributive property and then solve. Solve 42 ÷ 3 using an area model. Draw a number bond and use the distributive property to solve for the unknown length.

3. How to solve 60 ÷4 using an area model?

Ans. To solve 60 ÷ 4 using an area model, you need to divide the large number into four equal parts. To do that, you’ll take a rectangle and divide it into four equal parts. You can then calculate the area of each part and multiply them together to get the answer: 10 x 10 = 100.

4. How do you solve a division problem with an area model?

Ans. To solve a division problem with an area model, first draw a number line. Then, divide the number line into equal parts, each representing one of your groups. Using the original measurement and dividing it by the group you’re working with, you can determine how many groups that original measurement is supposed to be divided into.

5. How do you solve Alfonso’s area model?

Ans. One way to solve Alfonso’s area model is to first find the lengths of the sides that make up each square. Then, you can use simple square root formulas to find the area of each square. Finally, you can add up the areas of all four squares.