Dividing Using Area Models

An area model turns a division problem into a picture of a rectangle. Remember that for any rectangle, area equals length times width. If you already know the area of a rectangle and one of its sides, you can figure out the missing side by dividing. That is exactly what a division problem asks you to do: the dividend is the area, the divisor is one known side, and the quotient is the missing side you are trying to find.

Instead of trying to divide a big number all at once, the area model lets you break the dividend into smaller, friendlier pieces (usually multiples of ten, one hundred, or another easy number), divide each piece separately, and then add the results together.

To divide using an area model, start by drawing one long rectangle. Label the height with the divisor, since that side length is already known. The area of the whole rectangle is the dividend. Your job is to find the total width, because the width represents the quotient.

Rather than guessing the whole width at once, split the rectangle into two or three smaller rectangles whose areas are easier chunks of the dividend to work with. Each smaller rectangle still has the same height (the divisor), but a smaller, friendlier area.

1. Write the divisor as the height of a rectangle.
2. Break the dividend into friendly chunks that add up to the original dividend, choosing chunks that divide evenly (or almost evenly) by the divisor.
3. Draw one smaller rectangle for each chunk, using the chunk as its area and the divisor as its height.
4. Divide each chunk by the divisor to find the width of that smaller rectangle. This width is a partial quotient.
5. Add all the partial quotients together to get the final quotient.

To divide \( 96 \div 8 \), think of a rectangle with height 8 and area 96. Splitting 96 into friendly chunks gives \( 80 + 16 \), since both chunks divide evenly by 8.

Each smaller rectangle gives a partial quotient: \( 80 \div 8 = 10 \) and \( 16 \div 8 = 2 \). Adding the widths together gives the full quotient, \( 10 + 2 = 12 \), so \( 96 \div 8 = 12 \).

Larger dividends often need three friendly chunks instead of two. For \( 468 \div 4 \), split 468 into \( 400 + 60 + 8 \), since each of those numbers divides evenly by 4.

Adding the partial quotients gives \( 100 + 15 + 2 = 117 \), so \( 468 \div 4 = 117 \). The area model keeps every piece manageable, even though the original dividend has three digits.

The trick to this strategy is picking chunks that are both easy to divide and add back up to the original dividend. Multiples of ten and one hundred are usually the friendliest choice, which is why it helps to be comfortable dividing multiples of 10 before working through an area model. If a chunk still feels too big, split it into an even smaller multiple of ten and try again.

An area model is really a visual version of dividing using place value: both methods break the dividend into place value parts and divide each part before combining the results. Once you are confident splitting numbers into friendly chunks and rectangles, the same thinking carries directly into dividing multi-digit numbers, where the chunks come from the digits themselves rather than being chosen freely.

• Forgetting that the chunks must add up exactly to the original dividend.
• Choosing a chunk that does not divide evenly, which makes the partial quotient hard to find.
• Adding the partial areas instead of the partial quotients when finding the final answer.
• Mixing up which side of the rectangle is the divisor and which is the missing quotient.

Related lessons

• Math 5