area model — Definition, Formula & Examples
An area model is a rectangle drawn to represent multiplication or division, where the sides of the rectangle stand for the factors and the inside represents the product. You split the rectangle into smaller parts to make the calculation easier to see and solve.
An area model is a visual representation of arithmetic operations using the area of a rectangle. For multiplication, the length and width correspond to the two factors, and the total area equals the product. For division, the total area represents the dividend, one side represents the divisor, and the unknown side represents the quotient. Partial products or partial quotients are found by decomposing one or both factors into place-value parts.
How It Works
To use an area model for multiplication, draw a rectangle and label one side with the first factor and the other side with the second factor. Break each factor into its place-value parts — for example, split 24 into 20 and 4. This creates smaller rectangles inside the big one, and you multiply to find each partial product. Add the partial products together to get the final answer. For division, you start with the total area (the dividend) and one known side (the divisor), then figure out how wide each section needs to be to use up the dividend. The area model is different from an array, which uses rows and columns of individual dots or objects — the area model uses continuous rectangular regions instead.
Worked Example
Problem: Use an area model to multiply 23 × 15.
Decompose each factor: Break 23 into 20 + 3 and 15 into 10 + 5. Your rectangle will have four smaller sections.
Find each partial product: Multiply across each pair of parts to fill in the four sections of the rectangle.
Add the partial products: Combine all four sections to get the total area, which is the product.
Answer: 23 × 15 = 345
Another Example
Problem: Use an area model to divide 156 ÷ 12.
Set up the rectangle: Draw a rectangle with a total area of 156. Label one side 12 (the divisor). You need to find the other side (the quotient).
Fill friendly chunks: Start with an easy partial quotient. 12 × 10 = 120, so the first section has width 10 and area 120. Subtract to find what remains.
Fill the remaining area: Now handle the leftover 36. Since 12 × 3 = 36, the next section has width 3.
Add the widths: The total quotient is the sum of the section widths.
Answer: 156 ÷ 12 = 13
Why It Matters
Area models appear throughout 3rd- through 5th-grade math as a bridge between basic facts and the standard algorithm for multi-digit multiplication and long division. They help you see why the distributive property works, which becomes essential in algebra when you multiply expressions like . Understanding this visual strategy also builds a foundation for calculating area in geometry.
Common Mistakes
Mistake: Forgetting to multiply every pair of parts — for example, only computing 20 × 10 and 3 × 5 while skipping 20 × 5 and 3 × 10.
Correction: Every section of the rectangle must be filled. If you split the factors into two parts each, you get four partial products, not two. Check that the number of sections equals the number of parts on one side times the number on the other.
Mistake: Mixing up which numbers go on the sides versus inside the rectangle.
Correction: The factors (or divisor and quotient) are the side lengths. The numbers written inside the rectangle are the partial products (areas of each section), not the factors themselves.