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Unit 5: Lesson 4
Division with area models, want to join the conversation.
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Area Model Division
What is the area model division, introducing area model division, benefits of using area model division, solved examples, practice problems, frequently asked questions.
When they first learn how to divide numbers in school, most people are taught the traditional method of finding a common denominator and then doing long division.
However, there’s an alternative that can make the problem of dividing fractions or other numbers much easier. It’s called the area model method. Let’s learn more about it!
The area of a shape is the space occupied by the shape .
If a rectangle has a length equal to 32 units and a width equal to 23 units, then we can find its area by calculating the product $32 \times 23$.
In other words, when we consider the product $32 \times 23$ geometrically, it can be interpreted as the area of a rectangle of length 32 units and width 23 units.
Similarly, we can geometrically interpret a division problem, say $555 \div 15$, as the missing dimension of a rectangle of area 555 square units and one side length 15 units.
- We can divide this rectangle into several small rectangles. Then, we can calculate the length of each small rectangle and add them together to find the length of the large rectangle.
- First, consider a small rectangle of width 15 units and length 20 units. The area of this rectangle is 300 square units. So the area of the rest of the rectangle is $555$ $–$ $300 = 255$ square units.
Now, we have an area of 255 square units left. Since $15 \times 10 = 150$, another rectangle of width 15 units and length 10 units can be created.
We are now left with an area of $255$ $–$ $150 = 105$ square units. As $15 \times 7 = 105$, the shaded rectangle has a width of 15 units and a length of 7 units.
Thus, the length of the large rectangle is $20 + 10 + 7 = 37$ units. Therefore, $555 \div 15 = 37$.
Let’s learn how to use area model division to divide 825 by 5.
Step 1: Let’s start by breaking 825 into 500, 300, and 25, which are fairly easy to divide by 5.
Step 2: Now, let’s divide these partial dividends by the divisor 5 to get partial quotients:
500 divided by 5 gives us 100,
300 divided by 5 gives us 60,
and lastly, 25 divided by 5 gives us 5.
Step 3: Finally, let’s add all partial quotients to get the final quotient:
100 + 60 + 5 gives us 165. Now, we have our final product, which is 165.
Area Model Division with a Remainder
Sometimes, you’ll want to divide numbers that don’t divide evenly from one another. In these cases, it can be helpful to use an area model.
Let’s look at some division area model examples.
Divide 443 by 4 using area model division.
Step 1: Let’s break the numbers into 400, 40, and 3.
Step 2: Dividing 400 by 4, we get 100,
40 divided by 4 gives us 10,
Since 3 cannot be divided by 4, it will be our remainder.
Step 3: Now, we need to add 100 and 10 together to get a quotient, which gives us 110.
So, our final product is quotient $= 110$, remainder $= 3$.
Area Model for Dividing Decimal Numbers
Dividing decimal numbers can be challenging, but it doesn’t have to be. When you know how to use a few tricks like reducing decimals or using an area model, dividing decimals becomes much easier. Let’s learn the steps through area model division problems.
Divide 225.5 by 5 using area model
Step 1: We can divide the decimal numbers exactly like the regular ones by removing the decimal and adding it in the final step.
So instead of dividing 225.5 by 5, we will divide 2255 by 5.
Step 2: Now, let’s break them into easily divisible numbers, which are: 2000, 200, 50, and 5.
Step 3: Now, let’s divide each by 5:
2000 divided by 5 is 400
200 divided by 5 is 40
50 divided by 5 is 10
5 divided by 5 is 1
Step 4: Now, let’s add them together: $400 + 40 + 10 + 1$, giving us 451.
Step 5: Finally, let’s add the decimal we removed earlier. So, 451 will become 45.1, which is our final quotient.
1. It makes it easier to divide fractions or decimal numbers by multiplying them instead of trying to use long division with both parts.
2. It is a good way to help students remember how subtraction or “take away” works.
3. It helps students solve long divisions easily and also helps them understand the process of dividing numbers.
1. Divide 728 by 14 using the area model.
Solution : Let’s start by breaking down 728 into 700 and 28.
Dividing 700 by 14 gives us 50, and 28 divided by 14 is 2.
Now, let’s add these two to get the quotient $(50 + 2)$, which gives us 52.
2. Divide 624 by 3 using the area model of division.
Solution : Let’s start by breaking down 624, which gives us 600 and 24.
600 divided by 3 gives us 200, and 24 divided by 3 gives us 8.
On adding the two, we get 208.
3. Divide 3213 by 4 using the area model of division.
Solution : Let’s start by breaking down 3214, which gives us 3200 and 14.
3200 divided by 4 gives us 800,
14 divided by 4 gives us 3, with a remainder of 2.
So, the final answer is 803, with a remainder of 2.
4. What is the quotient when you divide 4977 by 7 using area model?
Solution : First, let’s break down 4977 into 4900 and 77.
Dividing 4900 by 7, we get 700,
and 77 divided by 7 gives us 11
Adding these numbers $(700 + 11)$ gives us 711
So, the final answer is 711.
Attend this quiz & Test your knowledge.
What is the quotient when you divide 345 by 5 using the area model?
What is the answer when you divide 246 by 4 using the area model?
What is the quotient when you divide 29.7 by 3 using the area model?
What is the quotient when you divide 69.20 by 4 using the area model?
Can area model division be used for fractions?
Yes, you can efficiently use area model division to divide fractions.
Can the area model be used for multiplication?
Yes, you can use area models to solve a multiplication problem with large numbers. The multiplicands will be the length and breadth of a rectangle, and their product will be the area of the rectangle.
To find the area, you divide the rectangle into smaller segments. The sum of the areas of the smaller rectangles gives you the total area, which will be the product. This method is also called box multiplication.
What is the purpose of the area model division?
Area model division helps students visualize and make longer division problems easier to solve.
Why is the area model for division important?
Division using area models gives students another perspective on math equations, which is extremely valuable since not all kids (or adults) think in the same way.
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Division Using an Area Model
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- We'll learn how to do division with an area model.
- We'll also see that this strategy can help us divide 2, 3 and 4 digit numbers more efficiently!
- And we'll see that division using an area model can help us count fossils, plan a pizza party, and help our community.
To separate into equal groups; to share in a fair way, so everybody gets the same amount.
There are 30 French fries. Since I have 4 friends with me, each of us gets 6 fries.
To get close to 90, I could multiply 6 x 10 = 60. That leaves 30. Combining that 30 with the 6 from the ones column, I get 36, so now I want to get close to 36. I can multiply 6 x 6 = 36. The total, 10 + 6 = 16, is the answer.
Since 12×5=60, that means 12×50=600.
To find the answers to multiplication problems; to find the area of a rectangle.
The width is the dividend, 5.
Start with the first place value. 300 ÷ 5 = 60. Then divide 20 ÷ 5 = 4, so the answer is 60+4=64.
It’s mostly the same. The only change is that at the end, you divide 24 ÷ 5, so there is a remainder of 4.
Make a rectangle with a width of 6. Have three sections with areas 200, 20, and 6. Start in the first section. To get close to 200, 6×30=180. Combine the leftover 20 with the 20 already in the second section. To get close to 40, 6×6=36. Combine the leftover 4 with the 3 already in the third section. 6 goes into that section once, with a remainder of 1. The answer is 30+6+1 and the remainder, or 37 r 1.
The rectangle still has a width of 6, but now the sections have areas 2000, 200, and 30. Start in the first section. To get close to 2,000, 6×300=1,800. Combine the leftover 200 with the 200 already in the second section. Then, 6×60=360. Combine the leftover 40 with the 30 in the third section. Since 6×11=66, there is a remainder of 4. You don’t really need a fourth section with the 4 in it. The answer is 300+60+11=371 with a remainder of 4, or 371 r 4.
The process of partitioning into equal groups.
The result of division.
The amount being divided.
The number the dividend is being divided by.
Repeated addition of equal groups.
An operation that reverses the result of another operation. Multiplication and division are inverse operations.
Write a number as a sum of the value of each digit. 234 in expanded form is 200 + 30 + 4
A rectangle divided into sections to organize your calculations.
You can use an area model to solve division problems by representing the number being divided as the area of a rectangle and the known factor as one of the side lengths. The quotient is the length of the other side.
To better understand division using an area model…
LET’S BREAK IT DOWN!
There are 78 fossils in April's fossil kit. In the kit, there are equal amounts of 6 different kinds of fossils. Solve 78 ÷ 6 to find out how many of each kind there are. You can use an area model to solve division problems. First, let's think back to how to solve a multiplication problem with an area model. For example, to find 6 ✕ 24, imagine a rectangle with a length of 24 units and a width of 6 units. The area inside the rectangle represents the answer to the multiplication problem. The length, 24, can be split up into its tens (20) and its ones (4). Since 6 ✕ 20 = 120, and 6 ✕ 4 = 24, the total area inside the box is 120 + 24, or 144 square units. That means 6 ✕ 24 = 144. Now let's get back to our division problem. Turn the division problem into a multiplication problem by asking 6 times what number is 78. Now show the problem with an area model. Imagine a rectangle. The length of one side of the rectangle is 6. The length of the other side is unknown. The area is 78. Split the area into two sections by splitting 78 into two sections using place value. One section has an area of 70 and the other section has an area of 8. Then, think about what number you can multiply by 6 to get close to 70. Multiplying 6 by 11 results in 66. Write 11 above the first section. Then subtract 70 – 66 = 4 to find the leftover area. Move the leftover area to the next section and combine it with the 8 in that section for a total of 12. What number can you multiply by 6 to get 12? That's exactly 2. Write 2 above the second section. Since 11 + 2 = 13, that means 78 ÷ 6 = 13. There are 13 of each type of fossil. Try this one yourself: Use an area model to divide 95 by 4. Compare your strategy and your answer with another student.
Planning a Pizza Party
Marcos and April are helping to organize a graduation party for their school. Based on a survey of all the kids and their guests, they need to order 359 pizzas. They need to order an equal amount of 4 different types. April divides 359 ÷ 4 to find how many of each type to order. She draws a rectangle with a width of 4. The length of the other side is unknown. She splits the rectangle into three sections: 300, 50, and 9. April knows that 4 ✕ 50 = 200, so she writes 50 above the first section of the rectangle. That leaves 300 － 200 = 100, so she adds 100 to the next section along with the 50 to get 150. April knows she can get close to 150. Since 4 ✕ 30 = 120, she writes 30 above the second section of the rectangle. That leaves 150 － 120 = 30, which she adds to the 9 in the last section to get 39. April can get pretty close to 39 because 4 ✕ 9 = 36. She writes 9 above the third section of the rectangle. Then she subtracts 39 – 36 = 3. She can’t split 3 into 4 equal groups, so 3 is the remainder. To find the answer, add up the lengths of all 3 sections: 50 + 30 + 9 = 89 with a remainder of 3. Since they need a little more than 89 of each type of pizza, they should round up and order 90 of each type of pizza. Try this one yourself: Use an area model to divide 267 by 7. Compare your strategy and your answer with another student.
Filling Egg Cartons
Some chickens laid 372 eggs. Adesina is putting the eggs into cartons. Each carton holds 12 eggs. How many cartons will there be? To find the number of cartons needed, divide 372 ÷ 12. Set up an area model. The area model is a rectangle. One side has a length of 12 and the other side is unknown. The model has 3 sections broken up by place value: 300, 70, and 2. For the first section, 12 ✕ 20 = 240. Write 20 above this section. That leaves 300 － 240 = 60 left over. Add 60 to the 70 already in the second section, for a total of 130. You can get close to 130, because 12 ✕ 10 = 120. Write 10 above this section. That leaves 130 － 120 = 10 left over. Add the leftover area, 10, to the 2 already in the third section to get an area of 12. 12 ✕ 1 = 12, so write 1 above this section. Since 12 – 12 = 0, there is no remainder. Add up the length to find the answer: 20 + 10 + 1 = 31. There will be 31 full cartons of eggs. Try this one yourself: Use an area model to divide 378 by 14. Compare your strategy and your answer with another student.
Adesina wants to make ribbon pins for school spirit week. She buys a spool of ribbon that is 5,629 centimeters long. Each pin needs 25 centimeters of ribbon. How many pins can she make? To find the number of pins she can make, find 5,629 ÷ 25. Set up a rectangle with four sections. The width of the rectangle is 25. Label each section with: 5,000, 600, 20, and 9. Since 200 ✕ 25 = 5,000, write 200 above the first section. For the second section, multiply 20 ✕ 25 to get 500. Write 20 above the second section. Then combine the leftover area, 600 － 500 = 100, with the 20 in the third section. Then figure out how many times 25 goes into 120. Since 4 ✕ 25 = 100, write 4 above the third section. The leftover area is 120 – 100 = 20. Combine the leftover 20 with the 9 in the fourth section. Since 25 goes into 29 once, write a 1 above the fourth section. Subtract 29 – 25 = 4. Since you can’t divide 4 into 25 evenly, there is a remainder of 4. So, the total number of ribbon pins that can be made is 200 + 20 + 4 + 1 = 225. Adesina just won't use the last 4 centimeters of ribbon. Try this one yourself: Use an area model to divide 3,368 by 15. Compare your strategy and your answer with another student.
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Use an area model to divide 75 by 15.
Use an area model to divide 387 by 12.
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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.
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.
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.
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.
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.
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4th Grade GoMath 4.6 - Divide using the Area Model. Watch later. Share. Copy link. Info. Shopping. Tap to unmute.
With this division strategy, students divide by breaking the dividend into its expanded form. Then, students use familiar multiplication
In this video I explain the Area Model method of division. As high school math is my specialty, I should say I "attempt" to explain this
Sal uses area models to divide 268÷2 and 856÷8. Sort by: Top Voted. Questions ... if the question was 12 divided by 3 how can you make 12 my multiplication.
Benefits of Using Area Model Division ... 1. It makes it easier to divide fractions or decimal numbers by multiplying them instead of trying to use long division
You can use an area model to solve division problems by representing the number being divided as the area of a rectangle and the known factor as one of the side
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
Divide large numbers with this handy trick! Learn how to use area models to solve 3-digit and 4-digit division problems in this fun, free lesson!
Visualize solutions to multi-digit division problems by modeling them with virtual manipulatives. This interactive exercise focuses on calculating quotients
Students develop an understanding of remainders. They use different methods to solve division problems. You can expect to see homework that asks your child to