solve division problems with remainders using the area model

You are using an outdated browser and it's not supported. Please upgrade your browser to improve your experience.

LOGIN FOR PROGRAM PARTICIPANTS
PROGRAM SUPPORT

Solve Division Problems With Remainders Using Arrays And Area Models

Objective: Understand and solve division problems with a remainder using the array and area models.

In Lesson 15, students deepen their understanding of division by solving problems with remainders using both arrays and the area model.

There may be cases when our downloadable resources contain hyperlinks to other websites. These hyperlinks lead to websites published or operated by third parties. UnboundEd and EngageNY are not responsible for the content, availability, or privacy policies of these websites.

Grade 4 Mathematics Module 3, Topic E, Lesson 15

Prerequisites

Related Guides and Multimedia

Our professional learning resources include teaching guides, videos, and podcasts that build educators' knowledge of content related to the standards and their application in the classroom.

solve division problems with remainders using the area model

Long Division Alternative: The Area or Box Method

Long division is often considered one of the most challenging topics to teach. Luckily, there are strategies that we can teach to make multi-digit division easier to understand and perform.

The Box Method , also referred to as the Area Model , is one of these strategies. It is a mental math based approach that will enhance number sense understanding. Students solve the equation by subtracting multiples until they get down to 0, or as close to 0 as possible.

If you plan on teaching the partial quotients strategy in your classroom (which I highly recommend) the Box Method is a great way to get started. It uses the same steps as partial quotients , but is organized a bit differently.

Let’s learn how to perform the Box Method/Area Model for long division!

Wait! Are you looking for the Area Model for multiplication rather than division? Find it HERE.

Below, I have included both a video tutorial and step-by-step instructions.

VIDEO TUTORIAL

AREA MODEL/BOX METHOD FOR LONG DIVISION: STEP-BY-STEP INSTRUCTIONS

Suppose that we want to solve the equation 324÷2.

Step 1: First we draw a box. We write the dividend inside the box, and the divisor on the left side.

Step 2: We want to figure out how many groups of 2 can be made from 324. We will do this in parts to make it easier. We could start by making 100 groups of 2, since we know that we have at least this many groups. So we multiply 100×2 to make 200, and then take that 200 away from 324. Now we have 124 left.

Step 3: We make another box and carry the 124 over to it. Now let’s take away another easy multiply of 2. How about 50 groups of 2? We know that we can take out another 50 groups of 2 from 124. 50×2=100, so we take 100 from 124. Now we have 24 left.

Step 4: We make another box and carry the 24 over to it. We know that 12 groups of 2 makes 24, so let’s write a 12 on top and take away 24 from the 24. Now we end up with 0, so we know that we are finished our equation.

Step 5: Now we add the “parts” from the top of the boxes to find our quotient. 100+50+12=162, so we know that 324÷2=162.

ONE MORE EXAMPLE (WITH A REMAINDER)

Let’s take a look at one more example. In this example, we will solve 453÷4.

The Area Model for Multiplication

The area model is fantastic for multiplication as well! If you’d like to read about how to teach this in a concrete way, here’s a post you may be interested in.

BOX STRATEGY/AREA MODEL TASK CARDS

These task cards give students the opportunity to practice the box method/area model for long division in a variety of different ways. Students will calculate quotients, solve division problems, figure out missing dividends and divisors, think about how to efficiently solve an equation using the box method, and more. See the Box Method Task Cards HERE or the Big Bundle of Long Division Task Cards HERE .

THE LONG DIVISION STATION

The Long Division Station is a self-paced, student-centered math station for long division. Students gradually learn a variety of strategies for long division, the box method being one of them. One of the greatest advantages to this Math Station is that is allows you to target every student and their unique abilities so that everyone is appropriately challenged. See The Long Division Station HERE.

OR SEE ALL RESOURCES

Pingback: Partial Quotients: an alternative for traditional long division - Shelley Gray

Pingback: The Grid Method for Long Division - Shelley Gray

Pingback: Effective Alternatives for Long Division - Shelley Gray

Very well taught, understandable. Thanks. Substitute gets it.

That is wonderful! My son uses the box method for multiplication , but becomes visually frustrated and angry when he was figuring out long division today. I have never seen this method before in my life. Totally makes sense. Thank you

love it so easy to work with

Pingback: Primary, Junior Maths – Qaawyrdprimary

Free Fast Finisher Activities

Master the Multiplication Facts!

Make math relevant!

$solve division problems with remainders using the area model$

A fun math game!

The Best Way to Teach Multiplication Facts

Get in touch with us

Are you sure you want to logout?

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.

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.

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.

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:

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.

Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem? Right Angle Triangles A triangle with a ninety-degree […]

Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

How to Find the Area of Rectangle?

How to Solve Right Triangles?

Learn maths

Division using the Area Model

These lessons, with videos, examples, and solutions help Grade 4 students learn how to solve division problems without remainders using the area model.

Common Core Standards: 4.NBT.6, 4.OA.3

Topic E: Division of Tens and Ones with Successive Remainders

New York State Common Core Math Grade 4, Module 3, Lesson 20 Grade 4, Module 3, Lesson 20 Worksheets

The following figure shows an example of division using the area model. Scroll down the page for more examples and solutions.

NYS Math Grade 4, Module 3, Lesson 20 Concept Development Problem 1: Decompose 48 ÷ 4 from whole to part. Problem 2: Decompose 96 ÷ 4 from whole to part.

NYS Math Grade 4, Module 3, Lesson 20 Problem Set

NYS Math Module 3 Grade 4 Lesson 20 Homework

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Division using Models

Embedded in this unit of printable division worksheets are varied strategies for 3rd grade, 4th grade, and 5th grade kids to learn the basics of division with a range of hands-on tasks, such as cut-and-paste sharing activity, drawing objects for equal sharing, division by grouping objects, dividing by distributing objects in rows and columns of the array model, number line, bar model and area model. Gain a free access to some of these tasks.

» Division by Grouping Models

» Division Array Models

» Division using a Number Line

Division by Sharing | Cut-and-Paste Activity

Spin a story around the illustrations to introduce the concept of division. Snip the pictures, divide them into two equal parts and glue them in the columns and learn division by equal sharing in the process.

Division by Sharing | Drawing Activity

Blend fun and learning with this alternative pdf exercise for grade 3 kids. Read the scenario carefully, share the objects evenly, draw the correct number of specified objects in each group to complete the worksheets.

Pictorial models to divide numbers is an interesting strategy and most helpful in solving word problems. Represent the division equation as a rectangular bar and divide the bar into equal parts to find the quotient.

Representing Division in 3 Models | Activity

Recapitulate three division methods with this set of interesting 3-in-1 activity worksheet pdfs. Solve for the quotient by representing each division equation as a grouping model, an array model and on a number line.

Area Models Worksheets | Without Remainder – Level 1

Visually represent the division equation as a rectangle with these level 1 area models. Figure out the quotient by dividing the given 3-digit numbers inside the rectangle by the single-digit divisors. Repeat the process until you get a 0.

Area Models Worksheets | Without Remainder – Level 2

With double-digit divisors in place, these printable area models worksheets help 4th grade kids relate divisor and quotient to the width and length of the rectangle and the dividend to the area of the rectangle. Go on dividing!

Area Models Worksheets | With Remainder – Level 1

Direct students of grade 4 and grade 5 to solve the division equation using the area or box method by repeatedly subtracting the multiples until they can't be subtracted anymore and add up the partial quotients and find the remainder.

Area Models Worksheets | With Remainder – Level 2

These level 2 area models pdf worksheets present the area and the width of a rectangle and help kids identify the length by performing division. They continue dividing the area by width until they get a remainder < divisor and write down the quotient obtained as the length.

Division by Grouping Model Worksheets

Familiarize 3rd grade kids with the grouping strategy of division using this set of printable worksheets. Group objects, answer questions based on the model, complete the division statements, figure out left overs and much more.

(27 Worksheets)

Division Array Model Worksheets

Work out the quotient by observing arrays of objects distributed across rows and columns, answer questions, decipher arrays to write a division sentence, solve word problems and draw arrays to bolster division skills.

(16 Worksheets)

Division using a Number Line Worksheets

Interpret the division sentences on the number lines by drawing hops, read the number line and write the division equation, MCQs and many such activities, help develop division skills with ease.

(40 Worksheets)

Related Worksheets

» Division Tables and Charts

» Divisibility Rules

» Basic Division

» In and Out Boxes for Division

» Multiplication and Division Fact Family

Become a Member

Membership Information

What's New?

Printing Help

Testimonial

Members have exclusive facilities to download an individual worksheet, or an entire level.

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Unit 5: Lesson 4

Division with area models

Divide by 1-digit numbers with area models

Helping with Math

One of the first things children learn when they start learning how to multiply numbers is to make patterns with objects in an array. The children chart the manipulatives to discover they have a length and a width. Counting all the manipulatives, they find a total. Through this early experience, students begin to build a skill that will continue to develop all the way through high school algebra .

Common Core and school curriculums began emphasizing non-standard algorithms over the traditional methods. The purpose of the area model is the mathematical development of students. The purpose of developing methods is to gain a lasting understanding of the mechanics of math rather than finding an answer to a quick math problem. While the standard algorithm is often the most efficient way to solve a problem, it hides the reasoning behind the math from students learning to do more complicated math at a younger and younger age. Yes, the area model looks very different from the math many of us did as children, but the mechanics remain the same.

Area models and arrays are based on a simple concept: the length of a rectangle times the width equals the total area. Area models that students typically use are simple physical arrays.

The basic model lays the groundwork for learning that will continue throughout high school. How can this model be used to help young students better understand? The visual difference between what adding looks like compared to multiplication is the most important use of this model. It makes it more clear how different 7 + 3 is from 7 x 3. Students will need to know this distinction when studying the order of operations. When students have mastered multiplication facts , they move on to two-digit multiplication . It is here that the model takes the turn that many adults begin to be uncomfortable with math!

What is an Area Model?

In mathematics, an area model is a model or rectangular diagram used to solve multiplication and division problems, where the factors or the quotient and division determine the length and width of the rectangle.

By using number bonds, we can break one large area of the rectangle into several smaller boxes, making calculation easier. Next, we get the area of the whole, thereby determining the product or quotient.

You can multiply two 2-digit numbers with the area model by following the steps below:

Write the multiplicands in expanded form as tens and ones.

For example, 35 as 30 and 5 and 27 as 20 and 7.

Draw a 2 × 2 grid, or simply a box with 2 rows and 2 columns.

On the top of the grid (box), write the term for one of the multiplicands. One on the top of each cell.

Write the terms of the other multiplicand on the left side of the grid. One on the side of each cell.

In the first cell, write the product of the number and the tens. In the second and third cells, write the product of the tens and ones. In the fourth cell, write the product of the ones.

In order to determine the final product, add up all the partial products.

Here, for example, the area model has been used to multiply 27 and 35.

27 x 35 = ?

27 x 35 = (20 + 7) x (30 + 5)

Here we see how to find the product of a 3-digit number by a 2-digit number using the area model.

374 x 43 = ?

374 = 300 + 70 + 4

43 = 40 + 3

Hence, 374 x 43 = (300 + 70 + 4) x (40 + 3)

As mentioned earlier, we can use the area model for the division. Let us divide 825 by 5.

825 ÷ 5 = ?

825 ÷ 5 = 100 + 60 + 5 = 165

Another example to help you understand better

630 ÷ 18 = ?

630 ÷ 18 = 30 + 5 = 35

Some Interesting facts about the area model

Benefits using the Area Model

The “open-ended” approach provides entry points for students to begin solving larger division problems and multiplying large numbers.

Students will relate more by using and illustrating the “boxes” for the area/rectangular model. The rectangles represent an actual box or group of something and are symbolic.

In order to improve students’ understanding of the model and performance, you should encourage them to solve the problem “in a certain way”.

Usage of Area Models

As we introduce multiplication in primary , arrays and area models assist not only in supporting the development of proportional reasoning, but also teach us how to develop strategies that build number flexibility and the automaticity of math facts.

Arrays and area models can be used to illustrate many big ideas in mathematics including, but not limited to:

The term “array” may not be familiar to many people. Fortunately, the definition is fairly straightforward:

Mathematics defines an array as a group of objects ordered in rows and columns.

Despite their simplicity, they are extremely helpful when teaching multiplication and building conceptual understanding for more abstract ideas that require fluency with procedures.

Area Model of Multiplication

The area model of multiplication is often regarded as the most conceptually understandable multiplication strategy for young learners. In addition to the fact that this strategy builds upon the learner’s spatial reasoning (more than half of all young children follow this as a preferred learning technique), it also enables learners to isolate the partial products of multiplication problems, an essential step in learning the strategy for multiplication.

At the start of their maths journey, students are taught to:

Understanding the area model as a computational tool requires understanding the following:

Students should keep this in mind before starting the area model.

Consider 3 x 2, or “3 groups of 2”:

“3 groups of 2”

This may very well seem like a simple and or even unnecessary representation, but having a visual representation of the multiplication of the two numbers will always yield a rectangular array is an important concept. The unit not only shows the connections between Number Sense and Numeration and Measurement but also implicitly provides students with insight into how more abstract mathematics works later in grades 9 and 10.

To be able to skip count more fluently, students should continue to practice counting and quantity, which includes unitizing. The use of arrays allows students to continue developing their ability to unitize and work with composing and decomposing numbers.

All models, though different, deal with numeric length and width. It is possible to simplify algebraic expressions by using area models which do not require numeric values. A manipulative called algebra tiles are usually used to build algebraic area models.

Once students get a grasp of the basics, they can progress to more advanced use of the model that serves as a great precursor to encourage students to begin creating their own algorithms and gradually, connect to the standard algorithm.

Multiplying Polynomials using an Area Model

Multiplying polynomials is really about using the distributive process, but very often using it multiple times over each polynomial gets multiplied by everything else it turns out to be a bit tricky.

As you know, a rectangle’s area is equal to its width times its length. Let us illustrate this with an example.

Say we have this rectangle here and we state the top is 7 and the side is 6; the area would equal 7×6 or 42.

The same calculation would apply if we break this rectangle into four pieces, instead of seven we will break it into four and three, which is seven, and on the side here will make it one and five, which equals 42.

This is what we mean. 4×1 gives me this little area, 4×1 is 4, 3×1 is 3, 4×5 will be 20, 3×5 is 15, we divided this into four separate pieces and when we add them up, we will get 42. 4+3 is 7, 20 is 27, and 15 more is 42.

Therefore, we can use this idea of broken rectangles to multiply binomials. Here’s how.

We could do the product x+3 times 2x+1 by writing its rectangle. We are going to call this x+3, 2x+1 is going to be the side lengths, so now when we multiply each of these four things and add them together we will get the same answer.

So x times 2x is 2x squared, 3 times 2x is 6x, x times 1 is x, 3 times 1 is 3, so when we add all those together we will get 2x squared plus, 6x plus x is 7x plus 3 3. That is the answer for this product. This is the same answer you would have gotten using FOIL.

This method makes sure that each term gets multiplied by every other term. The x gets multiplied by 2x and 1 and the 3 gets multiplied by 2x and 1.

This area model is also really really useful when you get into big polynomials,

Area Model of Division

In division problems, the rectangular diagram or model is used to solve the factors or quotients and the divisor determines the rectangle’s length and width.

Reference: https://cdn-skill.splashmath.com/panel-uploads/GlossaryTerm/79184219049645d2a9a7815d4a1ea1d4/1562556661_shaded-part-area-model-multiplication.png

Let us take the example of a rectangle, where we find the area by multiplying the width and length. So if the length is 32 units and the width of 23 units we determine the area 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.

The rectangle can be further divided into smaller rectangles by measuring the length of each smaller rectangle again and again. In order to get the desired length, add these lengths together.

Consider a smaller rectangle of a height of 15 units and a length of 20 units as a starting point. This means the rectangle’s area is 300 square units, and the rest of the rectangle is 555 – 300 or 255 square units.

Now we have a sub-division area of 255 square units. Since 15 x 10 = 150, a new rectangle can be drawn with a height of 15 units and a length of 10 units.

Finally, 15 x 7 = 105 is found. As a result, the rectangle has a height of 15 units and a length of 7 units, totaling 105 square units.

The rectangle’s length is therefore 20 + 10 + 7 units, or 37 units.

To conclude, 555 ÷ 15 = 37.

Area Model Division With Remainders

As already discussed, arrays are objects that are grouped into columns and rows. The columns are vertical and the rows are horizontal. The dividend is the number being divided, and the divisor is how many numbers are in each group. When you divide something, it does not always divide evenly, and some numbers are leftover. These numbers are remainders. When splitting with arrays, you can see the remainder, which will help you visualize the math.

Commonly used Terms

28 ÷ 4 = 7, the number 7 is the quotient.

Question 1: Find the area using an area model.

Reference: https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A4ed6c5718265eaf4718dbb8c5e1121b6467f800946d1bf3ea7636e77%2BIMAGE_THUMB_POSTCARD_TINY%2BIMAGE_THUMB_POSTCARD_TINY.1

First, factor in and substitute the given dimensions into the formula for length and width.

In this case A=(16.2 mm)(2.3 mm)

Then, use the area model representation to find the answer. Change 16.2 and 2.3 to whole numbers and break up the numbers according to place value.

16.2 → 162 → 100 + 60 + 22.3 → 23 → 20 + 3

Next, find the areas of the smaller rectangles.

Add up the areas.

2,000 + 1,200 + 300 + 180 + 40 + 6=3726

Finally, include the decimal point into the sum. Calculate the number of decimal places in 16.2 and 2.3. There are a total of two decimal places. Move the decimal point two places to the left.

A = 37.26 mm2

The area is 37.26 square millimeters.

Question 2: Find the product using an area model.

In the first step, represent 1.5 horizontally and 2.5 vertically on the same area model.

Change 1.5 and 2.5 to quantities of hundredths.

1.5=150 hundredths

2.5=250 hundredths

Complete the rectangle by filling in the area.

Then, add up the number of units in each section.

The product of 1.5 times 2.5 is 3.75.

Reference: https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A1147299d881d24ab3de4ec995ff8504b528637a2c6e18ca37af7a56b%2BIMAGE_THUMB_POSTCARD_TINY%2BIMAGE_THUMB_POSTCARD_TINY.1

The area model helps students develop a rich understanding of multiplication and division through a variety of problem contexts and methods that elicit multiplicative thinking.

In truth, daily life presents us with various contexts that are multiplicative in nature and teachers must use appropriate strategies and models that resonate with children’s intuitions as they engage in these concepts.

Recommended Worksheets

Area of Other Quadrilaterals (Province Themed) Math Worksheets Multiplication and Division Problem Solving (Halloween Themed) Math Worksheets Multiplication of Polynomials (Universe Themed) Worksheets

Link/Reference Us

We spend a lot of time researching and compiling the information on this site. If you find this useful in your research, please use the tool below to properly link to or reference Helping with Math as the source. We appreciate your support!

<a href="https://helpingwithmath.com/area-model/">Area Model</a>

"Area Model". Helping with Math . Accessed on March 6, 2023. https://helpingwithmath.com/area-model/.

"Area Model". Helping with Math , https://helpingwithmath.com/area-model/. Accessed 6 March, 2023.

Additional Geometry Theory:

Latest worksheets.

The worksheets below are the mostly recently added to the site.

Probability of Compound Events (Black Friday Sale Themed) Math Worksheets

Sample Space (Spring Festival Golden Week Themed) Math Worksheets

Trapezoid (World Tourism Day) Math Worksheets

Collinear Points (Construction Themed) Math Worksheets

Estimating Difference (Be Eco-Friendly Themed) Math Worksheets

Factoring Perfect Square Trinomials (Earth Day Themed) Math Worksheets

Additive Inverse (Valentine’s Day Themed) Math Worksheets

Imperial System of Measurement (National Baking Week Themed) Math Worksheets

Estimating Product (Climate Change Themed) Math Worksheets

Multiplying By 11 (Word Games Themed) Math Worksheets

$SplashLearn$

Area Model Multiplication – Definition With Examples

What is area, area model of fractions, division without remainder, solved examples, practice problems, frequently asked questions.

To understand the area model in math, we must understand the meaning of area first. The space occupied by a flat shape or surface of an object is known as area. The area of the rectangle given below is the shaded part. Area of a rectangle is the base of the area model of solving multiplication and division problems.

Area of a rectangle $= \text{Length} \times \text{Width}$

$Area of a rectangle$

If the rectangle has a length equal to 25 units and width 18 units, then this area can be calculated by finding the product $25 \times 18$. In other words, when you consider the product $25 \times 18$, geometrically, we can interpret it as the area of a rectangle of length 25 units and width 18 units.

Related Games

A fraction area model represents a whole shape that is split into equal parts.

Suppose we are given a model representation of a fraction. To identify the fraction, we use the following steps:

$Area model for fractions$

Related Worksheets

$Add Fractions using Area Models Worksheet$

What Is an Area Model of Multiplication and Division?

In mathematics, an area model is a rectangular diagram that is used to multiply and divide two numbers or expressions, in which the factors or the quotient and divisor define the length and width of the rectangle. We can break down one large area of the rectangle into several smaller boxes, using number bonds, to make the calculation easier. Then we add to get the area of the whole, which is the product or quotient.

As mentioned earlier, Area of a shape is the space occupied by the shape . The area of the given rectangle is the shaded part.

$Area Model Multiplication – Definition With Examples$

If the rectangle has a length equal to 25 units and width 18 units, then this area can be calculated by finding the product $25 \times 18$. In other words, when you consider the product $25 \times 18$, geometrically we can interpret it as the area of a rectangle of length 25 units and width 18 units.

Further, we can divide this rectangle into smaller rectangles for easier calculations, and their areas can be added to get the total area of the rectangle. We can do this by breaking the lengths of the sides into smaller numbers using expanded forms. For example, 25 can be written as $20 + 5$. Similarly, the width 18 can be written as $10 + 8$. Using these, we can divide the original rectangle into 4 smaller rectangles as shown.

$Area Model Multiplication – Definition With Examples$

Now, it is easier to calculate the areas of these smaller rectangles and their sum gives the total area or the product $25 \times 18$.

The rectangle I has a length of 20 units and width of 10 units making an area of 200 square units. Similarly, the rectangle II has an area equal to 10 times 5 or 50 square units. Likewise, we can calculate the areas of rectangles III and IV as 40 sq. units and 160 sq. units respectively.

$Area Model Multiplication – Definition With Examples$

Now, the total area of the rectangle of sides 25 units and 18 units is calculated by adding these partial sums.

$200 + 50 + 40 + 160 = 450$ sq. units

One can check using the standard algorithm to multiply two 2-digit numbers that 25 times 18 is 450.

In a rectangular auditorium, there are 36 rows of chairs with 29 chairs in each row.

$Area Model Multiplication – Definition With Examples$

To find the total number of chairs we need to find the product 36 × 29.

Step 1 : Write the multiplicand and the multiplier using expanded forms.

$36 = 30 + 6$

$29 = 20 + 9$

$Area Model Multiplication – Definition With Examples$

Step 2 : Find the areas of the smaller rectangles.

$Area Model Multiplication – Definition With Examples$

Step 3 : Add the partial sums to get the total area.

$Area Model Multiplication – Definition With Examples$

Thus, there are 1044 chairs in the auditorium.

Area Model of Multiplication of Whole Numbers

Let’s understand the steps for the area model of multiplication with respect to different cases.

Multiplication of Two-digit Number by One-digit Number

Example: $65 \times 7$

Step 1: Write the multiplicands in expanded form.

In this example, $65 = 60 + 5$

Step 2: Draw a $2 \times 1$ grid or 1 row and 2 column box.

$2 digit × 1 digit multiplication$

Step 3: On the top of the grid, write “the tens and ones part of the multiplicand we expanded earlier. On the left side we write the other expanded multiplicand. In this example:

$Area Model Multiplication – Definition With Examples$

Step 4: In the first cell, write the product of the number 7 and the tens part of the number 65. In the second cell, write the product of the number 7 and ones part of 65. In this example:

$multiplying 2 digit × 1 digit using area model$

Step 5: Add the partial products i.e. numbers in each of the cells. In this example:

$420 + 35 = 455$

So, we can say that $65 \times 7 = 455$

Multiplication of Two-Digit Number by Two-Digit Number

Example: $52 \times 79$

Step 1: Write the multiplicand and multiplier, i.e., 52 and 79 in expanded form.

In this example, $52 = 50 + 2 and 79 = 70 + 9$

Step 2: Draw a grid of size 2 by 2.

$2-digit by 2-digit multiplication$

Step 3: On the top of the grid, write the expanded terms for one of the multiplicands as shown in the previous example. Mention the same for the other multiplicand on the left side.

$Area Model Multiplication – Definition With Examples$

Step 4: In the first cell, put down the product of the tens of both the numbers. In the second and third cells, put the product of the tens and ones of the numbers accordingly. In the fourth cell, put the product of the ones of both the numbers.

$multiplying 2 digit × 2 digit using area model$

Step 5: Add the partial products. In this example:

$3500 + 140 + 450 + 18 = 4108$

So, we can say that $52 \times 79 = 4108$

Multiplication of Three-Digit Number by Two-Digit Number

In the multiplication of a 3-digit number by a 2-digit number, we make a grid of $2 \times 3$.

For example: Multiply $248 \times 81$

$248 = 200 + 40 + 8$ and $81 = 80 + 1$

$2 digit × 3 digit multiplication using area model$

Adding the partial products, we get $16000 + 3200 + 640 + 200 + 40 + 8 = 20088$

So, the product $248 \times 81 = 20088$

Area Model of Multiplying the Fractions

Proper Fractions

We can also multiply two proper fractions using an area model. Suppose we have to multiply $\frac{3}{4}$ and $\frac{1}{2}$, we will follow the steps given below:

Step 1: Draw a rectangle and mark length on one side and breadth on the other side. In this case, mark $\frac{3}{4}$ as length and $\frac{1}{2}$ as breadth as $\frac{3}{4} \gt \frac{1}{2}$.

$Fraction multiplication$

Step 2: Look at the length. Divide the length into the parts as much as the denominator and shade the parts same as the numerator. In this case, divide the length into 4 equal parts and shade 3 out of them.

$Area Model Multiplication – Definition With Examples$

Step 3: Look at the breadth. Divide the breadth into the parts as much as the denominator and shade the parts the same as the numerator. In this case, divide the breadth into 2 equal parts and shade 1 out of them.

$Multiplying fractions using area model$

Step 4: See the common shaded parts. In this case, there are three shaded parts, so the numerator of the product is 3.

Step 5: See the total number of equal parts. In this case, the total number of equal parts are 8. So, the denominator of the product is 8.

Step 6: Write the product fraction as $\frac{Numerator}{Denominator}$. In this case, the product $= \frac{3}{8}$.

Area Model for Multiplying the Decimals

Suppose we have to multiply 3.6 and 2.2.

Step 1: Break the decimal into a whole number and decimal portion.

Here, $3.6 = 3 + 0.6$ and $2.2 = 2 = 0.2$

Step 2: Make a $2 \times 2$ grid

$Area model for multiplication of decimals$

Step 3: On the top of the grid, write one of the multiplicands, and on the left side of the grid, write the other one, as mentioned in the image.

$Multiplying decimals using area model$

Step 4: Multiply the whole part of one multiplicand with both the parts of the second multiplicand and fill the first row. Here, we are first multiplying 2 and 3 and then, 2 and 0.6.

$Multiplication of 2 decimal numbers using area model$

Step 5: Multiply the decimal part of the same multiplicand with both the parts of the second multiplicand and fill the second row. Here, we are multiplying 0.2 and 3 and then, 0.2 and 0.6.

$Multiplication of decimals using area model$

Step 6: Add the numbers in all 4 cells to find the answer.

So, $3.6 \times 2.2 = 6 + 1.2 + 0.6 + 0.12 = 7.92$

Area Model of Division of Whole Numbers

Area of a rectangle with the length of 48 units and a width of 21 units can be measured by multiplying 48 by 21. Similarly, a division problem like $352 \div 22$ can be represented geometrically as the missing dimension of a rectangle with an area of 352 square units and a side of 22 units. The division is considered to be complete when the difference we get at the end is less than the divisor. The difference we are left with at the end is known as the remainder.

Suppose we have to divide 432 by 16.

Step 1: Write the dividend inside the box and the divisor outside the box.

In this example:

$Division using area model$

Step 2: We will find a number divisible by 16 and less than or equal to 43. We get 32. We will add 0 after it. After eliminating 320 from 432, we get 112. And we will write quotient on the top, i.e., 20.

$Dividing numbers using area model$

Step 3: We move to the next box. Now we find a number divisible by 16 and less than equal to 112. We get $16 \times 7 = 112$.

$Using area model to divide numbers$

We get the answer: $432 \div 16 = 20 + 7 = 27$

Another example of area model of division is given below:

Area Model Multiplication – Definition With Examples

1. What fraction is given in the figure?

$Fraction representation using area model$

Solution: The denominator is 8. The numerator is 4.

So, the fraction is $\frac{4}{8}=\frac{1}{2}$.

2. Calculate the product of 48 and 6 by area model.

Alt tag: Product using area model

$48 \times 6 = 240 + 48 = 288$

3. Find the product of 39 and 42 using the area model.

$2 × 2 product using area model$

$39 \times 42 = 1200 + 360 + 60 + 18 = 1638$

4. Divide 5607 by 5 using the area model and write the quotient and remainder.

Solution:

$Division using area model$

Quotient $= 1000 + 100 + 20 = 1120$ and remainder $= 2$

5. Multiply $\frac{3}{5}$ and $\frac{2}{3}$ using an area model.

Solution:

$Multiplying 2 fractions using area model$

Total mutual shaded parts $= 6$

Total equal parts $= 15$

Fraction $= \frac{6}{15} = \frac{2}{5}$

Area Model Multiplication - Definition With Examples

Attend this quiz & Test your knowledge.

Which of the following model is not the correct representation of the fraction $\frac{3}{5}$?

$Area Model Multiplication – Definition With Examples$

What will be the value of the blank in the given model for division to find $4870 \div 79$ ?

$Area Model Multiplication – Definition With Examples$

What will be the missing number when we multiply 2.3 and 4.7?

$Area Model Multiplication – Definition With Examples$

What will be the missing multiplicands in the following model of multiplication?

$Area Model Multiplication – Definition With Examples$

What would be the resultant fraction when two fractions are multiplied using the given model?

$Area Model Multiplication – Definition With Examples$

What is the difference between area model and set model of fractions?

In the area model, we divide the whole rectangle—all of its area in equal parts, each with the same area. In the set model, the focus is on the number of objects rather than the area.

What operations can be done on fractions using the area model?

We can add, subtract, and multiply two fractions using the area model.

What is the use of the area model of multiplication?

The area model helps the students to understand how math works. The most important use of this model is to visually differentiate between addition and multiplication and demonstrate how distributive property uses both addition and multiplication.

Can we use an area model to multiply polynomial expressions?

Yes, we can use the area model to multiply polynomial expressions. Let’s look at the example given below. Here we are multiplying $(x + 2)$ and $(x + 4)$.

$Polynomial multiplication using area model$

What are partial products in an area model of multiplication?

Partial products are the resultant numbers when we multiply each digit of a number to each digit of another number, where every digit also maintains their place value. In the area model of multiplication, the numbers in all the cells, which when added, gives the final product, are known as partial products.

In the given example, when we multiply 19 and 27, we get 200, 180, 70 and 63 as the partial products.

Eureka math lesson 20 solve division problems without remainders using the area model

Students focus on interpreting the remainder within division problems both in by solving problems with remainders using both arrays and the area model.

Solve homework

Solve mathematic tasks

Lesson 20 Homework 4 3

Determine math equation

Math is the study of numbers, space, and structure.

Clarify mathematic equations

To solve a mathematical problem, you need to first understand what the problem is asking. Once you understand the question, you can then use your knowledge of mathematics to solve it.

Provide multiple methods

There are many ways to improve your writing skills, but one of the most effective is to practice writing regularly.

Paco solved a division problem by drawing an area model. EUREKA. MATH. Lesson 20: Solve division problems without remainders using the area model.

Recurring customers

Grade 4 Module 3: Homework Lesson 20

Solve division problems with remainders using the area model. Date: 8/28/13. 3.E.97. 2013 Common Core, Inc. Some rights reserved. commoncore.org.

Average satisfaction rating 4.8/5

Based on the average satisfaction rating of 4.8/5, it can be said that the customers are highly satisfied with the product.

Math is a way of solving problems by using numbers and equations.

Solving word questions

Word questions can be tricky, but there are some helpful tips you can follow to solve them.

Improve your theoretical performance

If you want to improve your theoretical performance, you need to put in the work.

Lesson 21 Homework 43

Lesson 20: Solve division problems without remainders using the area model. This work is derived from Eureka Math and licensed by Great Minds.

Get Homework

There's no need to dread homework, with a little organization and focus it can be a breeze.

Free time to spend with your family and friends

I love spending time with my family and friends, especially when we can do something fun together.

Determine mathematic questions

Math is a way of determining the relationships between numbers, shapes, and other mathematical objects.

Deal with math question

Math is all about solving equations and finding the right answer.

Get calculation support online

Looking for a little help with your math homework? Check out our online calculation assistance tool!

Solve equation

Area model division calculator

Math can be a challenging subject for many students. But there is help available in the form of Area model division calculator.

Get the Most useful Homework explanation

Solve mathematic question

Long Division Calculator with Remainders

Area model for division

Better than just an app

We are here to help you with all of your homework needs!

We are more than just an application, we are a community.

Deal with math equations

Solving math equations can be tricky, but with a little practice, anyone can do it!

Division calculator with remainder (/)

How to solve a division problem using an area model

Sal uses area models to divide 268/2 and 856/8. Sort by: Use your math skills while using the other things you,ve got. It's pretend and problem.

Area Model for Division

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

Division Using Area Models

Area model division.

Visualize solutions to multi-digit division problems by modeling them with virtual manipulatives. This interactive exercise focuses on calculating quotients

Looking for detailed, step-by-step answers? Look no further – our experts are here to help.

Get help from our expert homework writers!

No need to be a math genius, our online calculator can do the work for you.

What Is Area Model Division? Definition, Examples, Facts

Division Using an Area Model (Up to 4

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

Deal with mathematic question

Mathematics is a field of study that deals with numbers, shapes, and patterns.

Clarify mathematic problem

There's nothing more frustrating than being stuck on a math problem. But don't worry, there are ways to clarify the problem and find the solution.

Solve math questions

Solving math equations can be challenging, but it's also a great way to improve your problem-solving skills.

Explain mathematic tasks

Mathematics is the study of numbers, shapes, and patterns.

Interactive resources you can assign in your digital classroom from TPT.

Easel Activities

Easel Assessments

Unlock access to 4 million resources — at no cost to you — with a school-funded subscription..

division using area model step by step

All Formats

Resource types, all resource types, results for division using area model step by step.

Division Powerpoint ( Using long division, area model, and partial quotient)

Partial Quotients Step-By-Step Guide for Students

$Tech About Math$

4th Grade Division Strategy Deluxe Bundle Distance Learning

$Akins Math Shop$

Division Practice using Area Model - VISUAL Math Strategy [5.NBT.B6]

Differentiated Nearpod Dividing Fractions Using an Area Model for Math Centers

Internet Activities

Also included in: Webinar Bundle 2 Upper Grade - Special Offer - Distance Learning Differentiation

Anchor Chart Planogram Vol. 3 - Multiplication and Division

Division 4th Grade: Area Models and Partial Quotients: Templates 4.NBT.B.6

Math Doodle - Dividing with Partial Quotients - So EASY to Use! PPT Included!

$Science and Math Doodles$

Math Doodle - Area and Perimeter - So EASY to Use! PPT Included

4th Grade Multiplication & Division TEKS Tests - STAAR Review - Digital & Print

$Marvel Math$

Also included in: 4th Grade Multiplication and Division Unit - Word Problems, Practice, Games TEKS

$2 Step Word Problems | 3rd Grade Math Games$

2 Step Word Problems | 3rd Grade Math Games

Also included in: 3rd Grade Math Games Bundle | Math Task Cards

Math Doodle - Multiplying 2 Digit Numbers with Area Models - So EASY to Use!

Math Doodle - 2 Step Multiplication Using Algorithms - So EASY to Use!

Math Doodle - Dividing Using the Standard Algorithm - So EASY to Use!

Arrays, Area Models and the Commutative Property of Multiplication

Also included in: 3rd Grade Math TEKS Year Long Bundle - 3rd Grade Math Units

Math Doodle - Dividing with Models - So EASY to Use! PPT Included!

Area Model Division Lesson Powerpoint

5th Grade Multiplying Dividing Decimals & Word Problems Math Notes

Also included in: 5th Grade Year Long Math MEGA Growing Bundle

Math Doodle - Dividing with Remainders - So EASY to Use! PPT Included!

Area Model Division Practice

Multiplying and Dividing Fractions Activities

$From Math to Music$

Multiplication and Division Skills Anchor Charts - Bundle (3rd Grade)

Math Doodle - All About Area - So EASY to Use! PPT Included

Division of Whole Numbers Differentiated Interactive Notes

Also included in: ALL Whole Number Operations Differentiated Notes BUNDLE

TPT empowers educators to teach at their best.

Keep in Touch!

Are you getting the free resources, updates, and special offers we send out every week in our teacher newsletter?

Calculator Soup ®

Online Calculators

Long Division Calculator with Remainders

Calculator Use

Divide two numbers, a dividend and a divisor, and find the answer as a quotient with a remainder. Learn how to solve long division with remainders, or practice your own long division problems and use this calculator to check your answers.

Long division with remainders is one of two methods of doing long division by hand. It is somewhat easier than solving a division problem by finding a quotient answer with a decimal. If you need to do long division with decimals use our Long Division with Decimals Calculator .

What Are the Parts of Division

For the division sentence 487 ÷ 32 = 15 R 7

solve a long division problem with parts of division: dividend, divisor, quotient, remainder

How to do Long Division With Remainders

From the example above let's divide 487 by 32 showing the work.

set up the problem long division 487 divided by 32

Division Worksheets

Division worksheets for grade 3 through grade 6.

Our free division worksheets start with practicing simple division facts (e.g. 10 ÷2 = 5) and progress to long division with divisors up to 99. Exercises with and without remainders and with missing divisors or dividends are included.

Choose your grade / topic:

Grade 3 division worksheets, grade 4 mental division worksheets, grade 4 long division worksheets, grade 5 division worksheets, grade 6 division worksheets.

Division facts: drills and practice

Long division: drills and practice

Division flashcards

Topics include:

solve division problems with remainders using the area model

IMAGES

VIDEO

COMMENTS