Introduction

An area model is a large rectangle containing two (or more) smaller rectangles. A sketch of a typical area model is shown below:

We complete this area model by finding the areas of the blue and pink rectangles and placing these values on our diagram:

  • Let's first look at the blue rectangle (the one on the left-hand side). Its length is 10, and its width is 4. Therefore, the area of this rectangle must be 10 \times 4 = {\color{blue}40}. Let's add this to our model:

  • Now, let's look at the pink rectangle (the one on the right-hand side). Its length is 6, and its width is 4. Therefore, the area of this rectangle must be 6 \times 4 = {\color{blue}24}. Let's add this to our model:

And we're done!

Note: Area models are not usually drawn to scale. In other words, the lengths shown on the model might not be in the correct proportion.

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Example: Completing an Area Model

From left to right, find the missing numbers in the area model below.

EXPLANATION

We need to look at the blue and pink rectangles:

  • Let's first look at the blue rectangle (the one on the left-hand side). Its length is 60, and its width is 4. Therefore, the area of this rectangle must be 60 \times 4 = {\color{blue}240}. Let's add this to our model:
  • Now, let's look at the pink rectangle (the one on the right-hand side). Its length is 3, and its width is 4. Therefore, the area of this rectangle must be 3 \times 4 = {\color{blue}12}. Let's add this to our model:

Therefore, from left to right, the missing numbers are 240 and 12.

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Practice: Completing an Area Model

Insert the missing number in the area model.

a
b
c
d
e
Practice: Completing an Area Model

Insert the missing numbers in the area model:

a
b
c
d
e
Using Area Models to Represent Multiplication Problems

Area models can be used to represent multiplication problems.

For example, let's return to the following area model:

Now, let's look at the big rectangle:

  • its length is 10+6=16

  • its width is 4

  • its area is {\color{blue}{40}}+{\color{blue}{24}}=64

Since the area of the big rectangle equals its length multiplied by its width, this model represents the following multiplication problem:

16 \times 4= 64

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Example: Interpreting an Area Model

The area model above can be used to represent the following multiplication problem:

32 \times \fbox{[math]\phantom{00}[/math]} = 192

What is the missing number?

EXPLANATION

Let's look at the big rectangle:

  • its length is 30+2=32

  • its width is 6

  • its area is 180+12=192

Hence, this area model can be used to represent the following multiplication problem:

32 \times \fbox{6}= 192

Therefore, the missing number is 6.

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Practice: Interpreting an Area Model

The area model above can be used to represent the following multiplication problem:

\[ 42 \times \fbox{$\phantom{00}$} = 210 \]

What is the missing number?

a
 $5$
b
 $6$
c
 $4$
d
 $8$
e
 $7$
Practice: Interpreting an Area Model

The area model above can be used to represent the following multiplication problem:

\[ \fbox{$\phantom{00}$} \times 7 = 126 \]

What is the missing number?

a
 $8$
b
 $10$
c
 $17$
d
 $18$
e
 $7$
Practice: Interpreting an Area Model

Complete the following multiplication problem represented by the area model above.

a
b
c
d
e
Example: Multiplying Numbers Using a Partially Filled Area Model

Using the area model below, find the value of 29 \times 6.

EXPLANATION

First, we compute the areas of the blue and pink rectangles. This gives the following picture:

Now, notice the following regarding the big rectangle:

  • its length is 20 + 9 = 29

  • its width is 6

  • its area is 120 + 54= 174

Therefore, this area model can be used to represent the following multiplication problem:

29 \times 6 = 174

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Practice: Multiplying Numbers Using a Partially Filled Area Model

Insert the missing numbers in the area model.

a
b
c
d
e
Practice: Multiplying Numbers Using a Partially Filled Area Model

Insert the missing numbers in the area model.

a
b
c
d
e
Multiplying One-Digit Numbers by Two-Digit Numbers Using Area Models

Let's solve the following multiplication problem using an area model:

{\color{blue}{16}} \times {\color{red}{8}}

The number 16 in expanded form is {\color{blue}{16}} = {\color{blue}10}+{\color{blue}6}. So, we place {\color{blue}10} and {\color{blue}6} above the left and right rectangles, and {\color{red}8} to the left of the big rectangle.

Next, we compute the areas of the blue and pink rectangles. This gives the following picture:

Now, notice the following regarding the big rectangle:

  • its length is 10 + 6 = 16

  • its width is 8

  • its area is 80 + 48 = 128

Therefore, this area model can be used to represent the following multiplication problem:

16 \times 8 = 128

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Example: Multiplying Two-Digit Numbers by One-Digit Numbers

Using the area model below, find the value of 53 \times 8.

EXPLANATION

First, we set up the model. The number 53 can be expressed in expanded form as {\color{blue}50}+{\color{blue}3}. So, we place {\color{blue}50} and {\color{blue}3} above the left and right rectangles, respectively, and {\color{red}8} to the left of the big rectangle.

Next, we compute the areas of the blue and pink rectangles. This gives the following picture:

Now, notice the following regarding the big rectangle:

  • its length is 50 + 3 = 53

  • its width is 8

  • its area is 400 + 24 = 424

Therefore, this area model can be used to represent the following multiplication problem:

53 \times 8 = 424

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Practice: Multiplying Two-Digit Numbers by One-Digit Numbers

The area model below represents the multiplication problem $23 \times 4.$ Insert the missing numbers.

a
b
c
d
e
Practice: Multiplying Two-Digit Numbers by One-Digit Numbers

The area model below represents the multiplication problem $42 \times 7.$ Insert the missing numbers.

a
b
c
d
e
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