Area Model for Multiplication

What is an area model for multiplication?

An area model for multiplication, sometimes called the box method or grid method, represents a multiplication problem as the area of a rectangle. The sides of the rectangle correspond to the factors being multiplied.

By breaking the sides down into smaller sections based on place value, the large rectangle is divided into smaller, simpler boxes. The area model is a direct visual application of the distributive property of multiplication, which states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.

The factors are the numbers being multiplied, and the product is the final answer.

Partition factors by place value

To begin using the area model, write each factor in expanded form. Expanded form breaks a number into the value of its digits based on their place value positions.

For a two-digit number, partition it into tens and ones. For example, the number $47$ is partitioned into $40$ and $7$. For a three-digit number, partition it into hundreds, tens, and ones. The number $352$ becomes $300$, $50$, and $2$.

This partitioning is the key to the area model because it converts difficult multiplication facts into manageable multiples of ten, which are much easier to calculate mentally.

Build the rectangle

Draw a rectangle and divide it into a grid based on the number of digits in each partitioned factor.

If multiplying a two-digit number by a one-digit number, draw a rectangle divided into two columns and one row. Place the expanded form of the two-digit number along the top, writing the tens above the first column and the ones above the second column. Place the one-digit factor along the left side.

If multiplying a two-digit number by a two-digit number, draw a rectangle divided into two columns and two rows, creating a grid with four smaller boxes. Write the first expanded factor across the top and the second expanded factor down the left side.

Find and add partial products

The area model works by filling in the interior boxes. Each box represents a smaller multiplication problem.

Multiply the number at the top of a column by the number at the side of a row to find the area of that specific box. The answer written inside the box is called a partial product. Complete this step for every box in the grid.

Once all the partial products are calculated, add them together to find the final total product. Aligning the numbers correctly by place value is critical during this step. Sometimes, finding the total involves multiplication with regrouping, requiring careful addition of the columns.

Area model versus long multiplication

The area model and long multiplication both calculate the same final product, but they organize the mathematical steps differently.

The area model emphasizes the value of each digit, making it easier to see how $30 \times 20$ becomes $600$. Long multiplication is a more compact algorithm that requires less writing space but relies heavily on carrying digits correctly.

Worked examples

Example 1: Multiplying a two-digit number by a one-digit number

Question: Use an area model to calculate $6 \times 48$.

Method:

1. Partition $48$ into its expanded form: $40 + 8$.
2. Draw a rectangle with two columns. Write $40$ and $8$ along the top. Write $6$ along the side.
3. Multiply to find the partial products:

• $6 \times 40 = 240$
• $6 \times 8 = 48$
• Add the partial products together: $240 + 48$.

Answer: $288$.

Check: $6 \times 50$ is $300$. Because $48$ is $2$ less than $50$, subtract $6 \times 2$ from $300$. $300 - 12 = 288$. The answer is correct.

Example 2: Multiplying a two-digit number by a two-digit number

Question: Use an area model to calculate $36 \times 42$.

Method:

1. Partition both numbers into expanded form: $36$ becomes $30 + 6$, and $42$ becomes $40 + 2$.
2. Draw a $2 \times 2$ grid. Place $30$ and $6$ along the top. Place $40$ and $2$ down the side.
3. Multiply the top and side numbers for each box to find the four partial products:

• Top-left box: $40 \times 30 = 1{,}200$
• Top-right box: $40 \times 6 = 240$
• Bottom-left box: $2 \times 30 = 60$
• Bottom-right box: $2 \times 6 = 12$
• Add all four partial products together: $1{,}200 + 240 + 60 + 12$.

Answer: $1{,}512$.

Check: Use an alternate method. $36 \times 40 = 1{,}440$. $36 \times 2 = 72$. $1{,}440 + 72 = 1{,}512$. The answer is correct.

Common mistakes

When using the area model, errors usually happen during the calculation of partial products or the final addition step.

Forgetting placeholder zeros

A frequent mistake is dropping zeros when multiplying tens. For example, when calculating $30 \times 20$, a student might write $60$ instead of $600$. Always count the zeros in the factors to check the magnitude of the product.

Adding instead of multiplying

When filling the boxes, it is easy to accidentally add the factors instead of multiplying them. For instance, calculating $40 \times 6$ and writing $46$ rather than $240$. Keep the operation firmly in mind for every box.

Misaligning columns during addition

After correctly finding the partial products, a student might misalign the ones, tens, and hundreds columns when adding them up. Always write the final addition sum vertically and check that the digits line up strictly by place value.

Frequently asked questions

Can the area model be used for three-digit numbers?

Yes. To multiply a three-digit number by a two-digit number, partition the three-digit number into hundreds, tens, and ones. The resulting grid will have three columns and two rows, generating six partial products to add together.

Does the area model work with decimals?

Yes. Partition the decimal into its whole number and fractional parts, such as dividing $3.4$ into $3$ and $0.4$. Multiply the parts using the same box layout, taking care to place the decimal point correctly in each partial product.

Why is it called the box method?

It is called the box method because the required rectangle is divided into smaller square or rectangular boxes. Each box provides a dedicated space to write a partial product.