What is Area Model?
An area model is a rectangular diagram used in mathematics to solve multiplication problems, in which the factors getting multiplied define the length and width of a rectangle. So the product is the area of the rectangle. Hence, it is known as the “Area model for multiplication”.
For example: 18 \( \times \) 11 can be represented as,
To make the calculation easier, we can divide one large portion of the rectangle into multiple smaller rectangles by breaking the factors into smaller numbers.
For instance, in 18 \( \times \) 11, 18 can be written as 10 + 8, then the equation becomes,
(10 + 8) \( \times \) 11
Now let’s apply distributive property of multiplication,
(10 \( \times \) 11) + (8 \( \times \) 11)
Here again 11 can be written as 10 + 1,
10 \( \times \) (10 + 1) + 8 \( \times \) (10 + 1)
Apply distributive property,
10 \( \times \) 10 + 10 \( \times \) 1 + 8 \( \times \) 10 + 8 \( \times \) 1
We can see here that 18 \( \times \) 11 can be modeled as 4 rectangles with areas 10 \( \times \) 10, 10 \( \times \) 1, 8 \( \times \) 10 and 8 \( \times \) 1. So the final product will be the sum of these areas,
= 100 + 10 + 80 + 8
= 198
Hence, 18 \( \times \) 11 = 198.
Splitting Factors and Partial Products
The factors of the multiplication equation should be split according to their place values.
For instance, in the above example, factors 18 and 11 are split according to tens and ones place values (18 = 10 + 8 or 1 tens and 8 ones and 11 = 10 + 1 or 1 tens and one).
In the above example, the area model has 4 rectangles. The areas of each of these rectangles are known as the partial products for the product 18 \( \times \) 11. The final product is the sum of the partial products.
The number of partial products in a multiplication equation depends on the number of digits in the factors. For example, if we multiply a two digit number with a single digit number then there will be 2 partial products. Similarly if we multiply 2 two digits numbers then there will be 4 partial products.
Here we will focus on multiplication of 2 digit numbers only and so the number of partial products will be 4.
Let’s take another example, 22 \( \times \) 16.
Write 22 as 20 + 2, that is, 2 tens and 2 ones,
= (20 + 2) \( \times \) 16
Apply distributive property,
= 20 \( \times \) 16 + 2 \( \times \) 16
Write 16 as 10 + 6 that is, 1 tens and 6 ones
= 20 \( \times \) (10 + 6) + 2 \( \times \) (10 + 6)
Apply distributive property,
= 20 \( \times \) 10 + 20 \( \times \) 6 + 2 \( \times \) 10 + 2 \( \times \) 6
= 200 + 120 + 20 + 12
So the multiplication equation can be modeled as,
The partial products or the areas of the 4 rectangles are 200, 120, 20 and 12.
The sum of these areas gives us the final product.
= 352
Solved Multiplication of Double Digit Examples Using Area & Models
Find the product of the following sets of numbers.
Example 1: Solve 15 \( \times \) 12 using an area model.
Solution:
Write 15 as 10 + 5 and 12 as 10 + 2.
= (10 + 5) \( \times \) (10 + 2)
Apply distributive property to get partial products,
= (10 \( \times \) 10) + (10 \( \times \) 2) + (5 \( \times \) 10) + (5 \( \times \) 2)
= (100) + (20) + (50) + (10) [Multiply]
= 180 [Add partial products]
Example 2 : Find the result of the given area model.
Solution:
14 \( \times \) 12
As per model write 14 as 10 + 4 and 12 as 10 + 2
= (10 + 4) \( \times \) (10 + 2)
Apply distributive property to get partial products,
= (10 \( \times \) 10) + (10 \( \times \) 2) + (4 \( \times \) 10) + (4 \( \times \) 2)
= (100) + (20) + (40) + (8) [Multiply]
= 168 [Add partial products]
Example 3: Solve 24 \( \times \) 22 using area model.
Solution:
Write 24 as 20 + 4 and 22 as 20 + 2
= (20 + 4) \( \times \) (20 + 2)
Apply distributive property to get partial products,
= (20 \( \times \) 20) + (20 \( \times \) 2) + (4 \( \times \) 20) + (4 \( \times \) 2)
= (400) + (40) + (80) + (8) [Multiply]
= 528 [Add partial products]
Example 4: Cynthia drew out a plan for the house of a couple and it looked like the picture shown below. Help her find the area of each of these sections and also the total area covered.
Solution:
Each section is in the shape of a rectangle. Hence the area will be,
10 \( \times \) 20 = 200 square units
10 \( \times \) 4 = 40 square units
5 \( \times \) 20 = 100 square units
5 \( \times \) 4 = 20 square units
The total area covered will the sum of the areas of each sections,
200 + 40 + 100 + 20 = 360 square units
Hence the area of each section is 200 square units, 40 square units, 100 square units and 20 square units. The total area is 360 square units.
Example 5: A rectangular plot of 45 m length and 19 m breadth is available to Alan. He needs to divide the area into four sections such that there is a large office space and 3 smaller spaces for a storage space, a pantry and a washroom. Find the area of each of these smaller sections of land and also find the complete area as well.
Solution:
Using partial products, the given area can be divided into smaller sections.
The total area of the land will be:
45 \( \times \) 19 square meters
To divide it into 4 sections such that there is 1 relatively large section for office space and 3 smaller sections for storage, pantry and washroom we will apply the area model to the product 45 x 19.
Write 45 as 40 + 5 and 19 as 10 + 9,
45 \( \times \) 19
= (40 + 5) \( \times \) (10 + 9)
Apply distributive property to get partial products,
= (40 \( \times \) 10) +(40 \( \times \) 9) + (5 \( \times \) 10) + (5 \( \times \) 9)
= (400) + (32) + (50) + (45) [Multiply]
Each of the partial products are the areas of the different sections. So, the given area can be divided into 400 m², 32 m²,50 m²and 45 m²
The total area will be,
400 + 360 + 50 + 45
= 855
Hence, the three smaller areas are 32 square meters, 50 square meters and 45 square meters. And the total area of the given stretch of land is 855 square meters.
When multiplying a 2 digit number by a 3 digit number using the area model, we will get 6 partial products.
If a number is broken down into 2 or more addends, then the sum of the product of each addend with another number is the same as multiplying both the numbers directly.