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## 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.

## Prerequisites

## Related Guides and Multimedia

## Long Division Alternative: The Area or Box Method

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.

## ONE MORE EXAMPLE (WITH A REMAINDER)

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

- First we wrote our dividend inside the box, and our divisor on the left side.
- We took out 100 groups of 4 first. This made 400. We subtracted 400 from 453 and were left with 53.
- We carried the 53 to the next box, and then took out another 10 groups of 4 to make 40. We took the 40 away from the 53 and were left with 13.
- We carried the 13 over to the next box, and then took out 3 groups of 4 to make 12. We took the 12 away from the 13 and were left with 1.
- We cannot take any more groups of 4 out, so our remainder is 1. To find our final quotient, we add 100+10+3+remainder 1 to make 113 R1.

## The Area Model for Multiplication

BOX STRATEGY/AREA MODEL TASK CARDS

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## Division with Area Model: Definition with Examples

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

## What is the Area Model?

## Derivation of the Area Model

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

## Division with Area Model

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.

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.

Things will get more clear on solving practically.

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

- 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.

## Division Area Model With Remainder

This was how to solve area model divisions with remainders.

Here is another solved example.

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

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

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.

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

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

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

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

## Related topics

## Composite Figures – Area and Volume

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

## Ways to Simplify Algebraic Expressions

## Other topics

## How to Find the Area of Rectangle?

## How to Solve Right Triangles?

## Learn maths

## Division 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

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

- Alfonso solved a division problem by drawing an area model. a. Look at the area model. What division problem did Alfonso solve? b. Show a number bond to represent Alfonso’s area model. Start with the total and then show how the total is split into two parts. Below the two parts, represent the total length using the distributive property and then solve.
- Solve 45 ÷ 3 using an area model. Draw a number bond and use the distributive property to solve for the unknown length.
- Solve 60 ÷ 4 using an area model. Draw a number bond to show how you partitioned the area, and represent the division with a written method.
- Solve 72 ÷ 4 using an area model. Explain, using words, pictures, or numbers, the connection of the distributive property to the area model.
- Solve 72 ÷ 6 using an area model and the standard algorithm.

NYS Math Module 3 Grade 4 Lesson 20 Homework

- Maria solved the following division problem by drawing an area model. a. Look at the area model. What division problem did Maria solve? b. 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. Below 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.
- Solve 96 ÷ 6 using an area model and the standard algorithm.

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## Division using Models

» Division using a Number Line

Division by Sharing | Cut-and-Paste Activity

Division by Sharing | Drawing Activity

Representing Division in 3 Models | Activity

Area Models Worksheets | Without Remainder – Level 1

Area Models Worksheets | Without Remainder – Level 2

Area Models Worksheets | With Remainder – Level 1

Area Models Worksheets | With Remainder – Level 2

Division by Grouping Model Worksheets

Division Array Model Worksheets

Division using a Number Line Worksheets

» In and Out Boxes for Division

» Multiplication and Division Fact Family

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## Unit 5: Lesson 4

## Divide by 1-digit numbers with area models

## What is an Area Model?

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.

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.

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

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.

Another example to help you understand better

## Some Interesting facts about the area model

- The area model is also known as the box model.
- Finding the area of a rectangle is the basis of the area model of solving multiplication and division problems . Area of a rectangle = Length × Width.
- The area model of multiplication uses the distributive law of addition.
- Expanded forms can be used to multiply numbers with more than 2 digits.
- The division is incomplete if the difference is greater than the divisor. In the case where the difference is less than the divisor, then it is the remainder.
- Is it possible to start the process with another length of a rectangle? Yes, for example, in 555 ÷ 15, if we take into consideration the sub-rectangle of length 5 units then the area becomes 15 ⨉ 5 or 75 sq. units and the rest of the area becomes 555 − 75 = 480 sq. units. In a similar manner, the procedure can be continued.

## Benefits using the Area Model

## Usage of Area Models

- Multiplication
- Distributive Property with Whole Numbers
- Factoring (Common, Simple/Complex Trinomials)
- Completing the Square
- Multiplying a Binomial by Monomial
- Multiplying a Binomial by Binomial (aka FOIL)
- Finding Area with Whole Number Dimensions
- Perfect Squares & Square Roots

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

## Area Model of Multiplication

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

- Use a variety of tools and strategies to relate multiplication of one-digit numbers and division by one-digit divisors to real-life situations (e.g., place objects in equal groups, write repeated addition or subtraction sentences, use arrays, etc);
- Apply a variety of mental strategies to multiply 6 x 6 and divide to 36 ÷ 6, (e.g., skip counting doubles, doubles plus another set);
- Determine, through investigation, the properties of zero and one in multiplication (i.e., any number when multiplied by zero equals zero; any number when multiplied by 1 equals the original number) (Example problem: Using tiles create arrays that represent 3 x 3, 3 x 2, 3 x 1, and 3 x 0. Explain what you think will happen when you multiply any number by 1, and when you multiply any number by 0.);
- Use the area model to visualize products of two numbers;
- Consider using the area model to understand both the decomposition of numbers and the distributive property of multiplication,

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

- numbers can be decomposed into the sum of smaller parts; and
- the distributive property can be used to break down one large multiplication problem into several smaller ones.

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

Consider 3 x 2, or “3 groups of 2”:

## Multiplying Polynomials using an Area Model

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

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

## Area Model of Division

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

## Area Model Division With Remainders

- Distributive property : A property of multiplication that can be used to simplify problems, for example, 6 fours = 5 fours + 1 four or 6 × 4 = (5 × 4) + (1 × 4).
- Long division : A method of solving division problems; also known as the standard algorithm for division.
- Quotient : An answer that is obtained by taking one number and dividing it by another. For example, in

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

- Remainder : The number that is left over when a whole number is divided by a whole number, for example, 25 ÷ 6 = 4 with a remainder of 1.
- Standard algorithm: The standard steps that are followed to solve a specific type of problem. For example, the process of long division is a standard algorithm.

Question 1: Find the area using an area model.

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)

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

Next, find the areas of the smaller rectangles.

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

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.

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.

- https://in.ixl.com/maths/lessons/area-model-multiplication
- https://www.matific.com/in/en-in/home/maths-activities/episode/multiplication-algorithms-area-model-for-multiplication/
- https://www.splashlearn.com/math-vocabulary/multiplication/area-model-multiplication
- https://www.ck12.org/book/ck-12-middle-school-math-concepts-grade-6/section/4.8/

## Recommended Worksheets

## Link/Reference Us

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

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

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## Area Model Multiplication – Definition With Examples

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

## Related Games

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

- Step 1: The whole shape is divided into equal parts. We count these numbers. We write this number as the denominator. In this example, there are 3 equal parts, so the denominator is 3.
- Step 2: Count the number of shaded shapes given in the figure. We write this number as your numerator. There are 2 shaded parts. So, the numerator is 2.
- Step 3: We write it in the form of $\frac{Numerator}{Denominator}$. In this example, we get $\frac{2}{3}$.

## Related Worksheets

## What Is an Area Model of Multiplication and Division?

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

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.

Step 2 : Find the areas of the smaller rectangles.

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

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

In this example, $65 = 60 + 5$

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

## Multiplication of Two-Digit Number by Two-Digit Number

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

$3500 + 140 + 450 + 18 = 4108$

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 $81 = 80 + 1$

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

## 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

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

Suppose we have to divide 432 by 16.

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

Another example of area model of division is given below:

- The area model is also known as the box model.
- This model of multiplication uses the distributive law of multiplication.
- Expanded forms are used to multiply numbers with more than 1 digit.
- The area model of solving multiplication and division problems is derived from the concept of finding the area of a rectangle. Area of a rectangle $= \text{Length} \times \text{Width}$

1. What fraction is given in the figure?

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.

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

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

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

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

Total mutual shaded parts $= 6$

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 $\frac{3}{5}$?

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

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

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

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

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

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?

Can we use an area model to multiply polynomial expressions?

What are partial products in an area model of multiplication?

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## What Are the Parts of Division

For the division sentence 487 ÷ 32 = 15 R 7

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## How to do Long Division With Remainders

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

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Description Objective: Solve division problems with remainders using the area model. Lesson 21 focuses on division problems with remainders. Quotients and remainders are independent of each other but must both be included to give a complete response.

Create division equations with area models Divide by 1-digit numbers with area models Math > 4th grade > Division > Division with area models © 2023 Khan Academy Terms of use Privacy Policy Cookie Notice Division with area models CCSS.Math: 4.NBT.B.6 Google Classroom About Transcript Sal uses area models to divide 268÷2 and 856÷8. Sort by:

Solve the following problems using the area model. Support the area model with long division or the distributive property. 5. 49 ÷ 3 6. 56 ÷ 4 7. 58 ÷ 4 8. 66 ÷ 5 9. 79 ÷ 3 10. Seventy-three students are divided into groups of 6 students each. How many groups of 6 students are there? How many students will not be in a group of 6?

With this division strategy, students divide by breaking the dividend into its expanded form. Then, students use familiar multiplication facts to divide. It is suggested that this would be the...

Description 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. Downloads There may be cases when our downloadable resources contain hyperlinks to other websites.

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.

Solve Division Problems with Remainders Using an Area Model - YouTube AboutPressCopyrightContact usCreatorsAdvertiseDevelopersTermsPrivacyPolicy & SafetyHow YouTube worksTest new...

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.

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. Show Video Lesson NYS Math Grade 4, Module 3, Lesson 20 Problem Set

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

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.

White's Workshop. 4.8. (102) $4.00. PDF. Learning the area model is an important step in mastering division. Each worksheet has a model that shows students how to do the area model with and without remainders. This product is designed to support your 4th graders with 2 digits ÷ 1 digit, 3 digits ÷ 1 digit, and 4 digits ÷ 1 digit worksheets.

Division with area models. Division with area models . ... Hannah split up the area of a rectangle to help her solve the equation 348 ÷ 5 =? 348 \div 5=? 3 4 8 ÷ 5 =? 348, divided by, 5, equals, question mark. She splits the rectangle into parts that are easy to divide by 5 5 5 5 and a remainder. Complete the equations. a = a= a = a, equals

Finding the area of a rectangle is the basis of the area model of solving multiplication and division problems. Area of a rectangle = Length × Width. The area model of multiplication uses the distributive law of addition. Expanded forms can be used to multiply numbers with more than 2 digits.

Step 1: Write the multiplicand and the multiplier using expanded forms. 36 = 30 + 6 29 = 20 + 9 Step 2: Find the areas of the smaller rectangles. Step 3: Add the partial sums to get the total area. Thus, there are 1044 chairs in the auditorium. Area Model of Multiplication of Whole Numbers

Students focus on interpreting the remainder within division problems both in by solving problems with remainders using both arrays and the area model. Data Protection Data protection is an important issue that should be taken into consideration when handling personal information.

Area model division calculator - Learn how to solve long division with remainders, or practice your own long division problems and use this calculator to check. Math Index ... Once you know what the problem is, you can solve it using the given information.

Lesson 20: Solve division problems without remainders using the area model. 291 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015-Great Minds. eureka math.org This file derived from G4 -M3-TE-1.3.-06.2015 This work is licensed under a Creative Commons Attribution NonCommercial ShareAlike 3.0 Unported License.

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 ... 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.

Division using an area model is a fantastic, VISUAL strategy for students to use to learn and understand the concept of division. This 8-page workbook/packet includes 14 division problems all using the area model strategies. 5 problems do NOT have remainders, while the other 9 DO include remainders.

Math Video: Example. The video shows 7 carrots being divided amongst 3 friends; the leftover carrot is the remainder. This is shown first pictorially and then by an equation or division sentence. The key concepts are "equal groups" and "remainders". Practice worksheets for this type of question can be found here:

Put the 5 on top of the division bar, to the right of the 1. Multiply 5 by 32 and write the answer under 167. 5 * 32 = 160. Draw a line and subtract 160 from 167. 167 - 160 = 7. Since 7 is less than 32 your long division is done. You have your answer: The quotient is 15 and the remainder is 7.

Grade 5 division worksheets. Divide 3 or 4-digit numbers by 1-digit numbers mentally. Division with remainder 1-100, 1-1,000. Dividing by whole tens or hundreds, with remainders. Long division with 1-digit divisors, no remainders. Long division with 1-digit divisors, with remainders.

Division with Remainders - Sample Math Practice Problems The math problems below can be generated by MathScore.com, a math practice program for schools and individual families. References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program. In the main program, all problems are automatically ...