How Does Area Model Division Simplify Long Division?
An area model is a rectangular diagram or model used in mathematics to solve multiplication and division problems, in which the factors or quotient and divisor determine the rectangle length and width.
To make the calculation simpler, we can divide one wide region of the rectangle into several smaller boxes using number bonds. After that, we add to get the total field, which is the result or quotient.
Follow these steps to multiply two 2-digit numbers using the area model:
In expanded form, write the multiplicands as tens and ones.
For example, 27 is the same as 20 and 7, and 35 is the same as 30 and 5.
Create a 2 x 2 grid or a box of two rows and two columns.
On the top of the grid, write the words of one of the multiplicands (box). Each cell has one on the top.
Write the words of the other multiplicand on the left side of the grid. Each cell has one on each side.
In the first cell, write the product of the number on the tens. Then, in the second and third cells, write the product of the tens and ones. In the fourth cell, write the product of the ones.
To get the final product, combine all of the partial products.
Long division is often regarded as one of the most difficult topics to teach. Fortunately, we can teach techniques for the long division area model to make multi-digit division easier to understand and execute.
One of these methods is the Box Method, also known as the Area Model. It's a method focused on mental math that will help you understand numbers better. Students solve the equation by subtracting multiples until they reach zero, or as near as possible to zero.
Benefits of Teaching With the Area Model/Rectangular Array Model For Division or Rectangular Array Division
When taught the “open-ended” way, the Area Model for Division offers entry points for all students to begin solving larger division problems, regardless of their multiplication fact knowledge.
Students will relate division to "taking away" from what we have to build as many "equal classes" as possible by using and illustrating the "boxes" for the area/rectangular division model. (The rectangles are a symbolic representation of an actual box or group of something in my teaching.)
Students can double-check their work by using the very same division form, but starting with a different number.
Students should be encouraged to solve the problem “in a certain way” in order to improve their comprehension of the model and performance. (When working on example problems in small or large groups, I ask students who have "solved" the division problem if they can solve it in a different way using the same method.)
What is a Rectangular Model For Division?
An area model is a rectangular diagram or model used in mathematics to solve multiplication and division problems, in which the factors or quotient and divisor determine the rectangle's length and width.
What is the Area Model Division?
The area of a shape is the amount of space it takes up.
We can measure the area of a rectangle with a length of 32 units and a width of 23 units by multiplying 32 by 23. In other words, the product 32 x 23 can be represented geometrically as the area of a rectangle with a length of 32 units and a width of 23 units.
Similarly, a division problem like 555 ÷ 15 can be represented geometrically as the missing dimension of a rectangle with an area of 555 sq. units and a length of 15 units on one side.
The rectangle can be subdivided into smaller rectangles by measuring the length of each smaller rectangle again and again. To get the desired length, add these lengths together.
Consider a smaller rectangle with a height of 15 units and a length of 20 units as a starting point. As a result, the rectangle's area is 300 square units, and the rest of the rectangle is 555 – 300 or 255 square units.
We now have a sub-division area of 255 square units. Since 15 x 10 = 150, a new rectangle with a height of 15 units and a length of 10 units can be drawn.
Finally, 15 x 7 = 105 is found. As a result, the shaded rectangle above has a height of 15 units and a length of 7 units, totaling 105 square units.
As a result, the rectangle's length is 20 + 10 + 7 units, or 37 units. As a result, 555 ÷ 15 = 37.
Area Model Division With Remainders
Arrays are objects that are grouped into columns and rows. The columns are vertical and the rows are horizontal. The number being divided is the dividend, and the divisor indicates how many numbers are in each group. When you divide something, it doesn't always divide evenly, and some numbers are leftover. These numbers that are leftover are remainders. You can see the remainder when splitting with arrays, which will help you visualize the math.
Solved Example For Division Using Area Model Division Steps
Question 1 - Solve: 453 ÷ 4 Using Area Model Division Steps
Solution -
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We began by writing our dividend on the inside of the box and our divisor on the left.
First, we eliminated 100 four-person parties. This brought the total to 400. We got 53 when we subtracted 400 from 453.
We moved the 53 to the next box, then took out 10 more groups of four to make 40. We subtracted the 40 from the 53, leaving us with 13.
We moved the 13 to the next box, then removed three groups of four to make 12. We subtracted the 12 from the 13 and were left with one.
We can't take any more groups of four down, so we're down to one. We get 113 R1 by adding 100 + 10 + 3 + remainder 1 to get our final quotient.
FAQs on Area Model Division: Step-by-Step Guide for Students
1. What is the area model of division and how does it work?
The area model of division is a visual method that uses a rectangular diagram to solve division problems. In this model, the total area of the rectangle represents the dividend, and the length of one side represents the divisor. The goal is to find the length of the other side, which will be the quotient. It works by breaking the dividend into smaller, friendlier numbers and finding partial quotients for each part.
2. What are the main steps for solving a problem using the area model for division?
To solve a division problem using the area model, you can follow these general steps:
- Step 1: Draw a large rectangle. Write the divisor outside, along one of the sides (e.g., the width).
- Step 2: Break the dividend into numbers that are easy to divide by the divisor (e.g., using place value or friendly multiples).
- Step 3: Start with the largest part of your dividend. Place it inside the rectangle and divide it by the divisor to find the first partial quotient. Write this partial quotient above that section of the rectangle.
- Step 4: Subtract the amount you just divided from the total dividend to see what remains.
- Step 5: Continue the process with the remaining amount, creating new sections in the rectangle until the remainder is zero or less than the divisor.
- Step 6: Add all the partial quotients together to get the final answer.
3. How can you solve a 3-digit by 1-digit division like 168 ÷ 7 using the area model?
To solve 168 ÷ 7, you draw a rectangle with 7 on the side. First, break 168 into friendly numbers divisible by 7, like 140 and 28. In the first part of the box, use 140. Since 140 ÷ 7 = 20, you write 20 on top. Now, you have 168 - 140 = 28 left. In the second part of the box, use 28. Since 28 ÷ 7 = 4, you write 4 on top. Finally, add the partial quotients: 20 + 4 = 24. So, 168 ÷ 7 = 24.
4. How are remainders represented in the area model of division?
In the area model, a remainder is the final amount left over inside the box that is too small to be divided by the divisor. After you have subtracted the last possible multiple of the divisor from the dividend, the number that remains is the remainder. For example, in 152 ÷ 10, after accounting for 150 (10 x 15), you are left with 2. This leftover '2' cannot form a group of 10 and is therefore the remainder.
5. How is the area model for division different from the traditional long division method?
The main difference lies in their approach. The area model is a visual and conceptual method that emphasizes place value and number sense by breaking the dividend into flexible parts. In contrast, traditional long division is a more abstract and rigid algorithm that follows the sequence 'Divide, Multiply, Subtract, Bring Down'. The area model helps students understand *why* division works, while long division is a more procedural shortcut to find the answer.
6. Why is learning the area model important if I already know long division?
Learning the area model is important because it strengthens your number sense and deepens your understanding of division. It's not just about getting the answer; it's about seeing the relationship between multiplication and division visually. This method is particularly helpful for visual learners and makes it easier to work with larger numbers by breaking them into manageable chunks, which can reduce calculation errors and build confidence in mental maths.
7. Does the area model for division have any other names?
Yes, the area model for division is often referred to as the "box method" or "partial quotients method". All these names describe the same strategy of using a rectangular box to break down a division problem into smaller, simpler parts and then adding the partial answers to find the total quotient.
