Understanding Area Models in Mathematics
An area model is a visual tool in mathematics that helps break down numbers to solve multiplication and division problems. It connects the concept of finding a rectangle’s area (A = l x w) to solving a multiplication or division problem. To use an area model for multiplication, the factors are the rectangle’s dimensions (length and width), and the product, or area, is the result. To use an area model for division, the total area and one dimension are known, and the other dimension is the unknown to be found.
Area models are particularly helpful for understanding the distributive property, as they allow students to see how large numbers can be decomposed into smaller, more manageable parts. This approach builds a strong foundation for conceptual understanding, helping students eventually transition to the more abstract standard algorithm.
Connecting Area Models to Multiplication and Division
Area Models and Multiplication
When used for multiplication, an area model breaks down larger numbers into smaller parts (using expanded form) to simplify calculations.
Example: Multiply 23 ×15
1. Draw a rectangle that represents a rectangle that has a length of 23 units and a width of 15 units.
2. Break apart the side lengths based on place value. The length of 23 will be broken up into 20 + 3 and 15 into 10 + 5.
3. Label the sides and draw lines to break apart the length and the width of the rectangle.
4. Now you have a rectangle that has been broken up into four smaller parts. Now it’s time to calculate the area of those smaller parts.
5. Multiply the dimensions of each smaller part of the rectangle to find the partial areas or partial products.
- 20 x 10 = 200
- 20 x 5 = 100
- 3 x 10 = 30
- 3 x 5 = 15
6. The partial areas, or partial products, are 200, 100, 30, and 15
7. Finally, add the partial areas to find the total area of the rectangle:
200 + 100 + 30 + 15 = 345
The total area, or product of 23 and 15 is 345 (23 x 15 = 345).
Area Models and Division
In division, an area model is used to find an unknown side length. The dividend represents the total area, and the divisor represents the known side length. The quotient is the missing side length.
This works because if A = l x w then, A ÷ w = l
Example: Divide 96 ÷ 4
1. Draw a rectangle with a width of 4 units and label it. The length of the rectangle is unknown.
2. The total area is 96. Break apart the total area into smaller partial areas that are easy to divide by 4. For example, you might break apart 96 as 40 + 40 + 16.
3. Since 4 x 10 = 40, the first and second partial lengths are 10 each.
4. Since 4 x 4 = 16, the last partial length is 4.
5. Now add the partial lengths to determine the total length, or the quotient.
6. 10 + 10 + 4 = 24
So 4 x 24 = 96, therefore, 96 ÷ 4 = 24.
If a student isn’t sure how to break apart the total area to make it easily divisible, they can “build up” to the area by gradually adding lengths. For example:
A student might start with a side length of 10, creating a partial area of 40. This leaves 56 of the area still unaccounted for. They can then add another side length of 10, covering an additional 40 of the area. Now only 16 of the area remains. The student knows that a side length of 4 will cover the remaining area. Adding the partial side lengths gives 10 + 10 + 4 = 14
Key Ideas for Teaching Area Model
Using Visual and Hands-On Tools for Area Models
Introduce area models with concrete manipulatives, such as square tiles and base ten blocks, as well as grid paper to help students see the connection between numbers and areas.
Multiplication example:
Have students use square tiles or base-ten blocks to build an array/area model. For example, to represent 23 × 15, the array would be 2 ten-rods and 3 unit cubes long, and 1 ten-rod and 5 unit cubes wide. The array is then “filled in” with ten-rods and unit cubes to complete the rectangle.
Once the array is complete, students can count by tens and ones to find the total area. Encourage students to break the array into smaller, more manageable sections, similar to how they would approach solving with an area model.
Tip: When using manipulatives to explore the area model, keep the factors relatively small. We recommend using factors less than 30 to ensure the building process is manageable for students.
Division example:
Use graph paper to represent the total area as a grid. For example, to model and solve 84 ÷ 6, students can start by drawing a rectangle with a width of 6. They can then gradually extend the length of the rectangle until the total area contains 84 squares. Encourage students to break 84 into smaller, more manageable parts as they solve for the missing dimension (quotient).
These hands-on tools bridge the gap between concrete and abstract understanding.
Connecting Area Models to Real-World Contexts
Present word problems that involve real-world scenarios that highlight the practicality of area models in solving problems. For example:
- A farmer wants to plant 24 rows of 12 plants each. How many plants will there be in total? Use an area model to break the problem into smaller parts.
- A rectangle has an area of 72 square feet, and one side is 8 feet long. What is the length of the other side?
Encouraging Flexibility in Decomposition
Highlight when students use multiple ways to decompose numbers when using the area model. For example:
When using an area model to solve 84 ÷ 3 one student might break apart the area of 84 as 30 + 30 + 12 + 12 while another student might decompose it as 60 + 24. Discuss their results as a class to highlight that the total can be broken down in different ways, but the outcome remains the same.
The Importance of Attending to Precision
When students draw an area model, especially for solving a multiplication problem, it’s important that they think of the factors as the dimensions and create a rectangle that accurately represents those dimensions. They should consider what a rectangle with those dimensions would look like. For example, is one side much longer than the other, or are the sides closer in length?
A 24 × 85 rectangle would be long and narrow:
While a 39 × 42 rectangle would be more square-like:
Students should also place the decomposing lines accurately. For instance, if a side length of 37 is broken into 30 and 7, the line should not be drawn in the middle but instead reflect that 30 is much longer than 7.
Accuracy is very important because a well-drawn area model can double as a self-checking tool. For example, if a student uses an area model to solve 37 × 42 and calculates the area of the 30 × 40 section as 120, but that section is clearly the largest in the model, they might notice the error and recognize the result as unreasonable. This visual connection strengthens both understanding and precision.
Common Misconceptions About Area Models
Misconception: Area Models Are Only for Multiplication
Some students may think area models only apply to multiplication. Reinforce their use for division by solving problems where the total area and one dimension are given, and the missing dimension is the quotient.
Misconception: Knowing How to Use the Area Model is Unnecessary, Especially if the Standard Algorithm is Known
Area models are powerful visual tools that deepen understanding for all learners by illustrating the structure of numbers and the distributive property. Following the steps of the standard algorithm doesn’t guarantee a student understands the underlying concepts. Use area models to show how they connect to the standard algorithms for multiplication and division, helping students see what is happening with the numbers conceptually.
Misconception: The Area is Always the “Answer”
When students use area models for multiplication, they learn that the area represents the product, which is generally the answer they’re looking for. However, in division, the structure changes: the area represents the dividend, and the quotient is found along the side of the rectangle as a dimension. This shift can be confusing if students have extensive experience with area models for multiplication before encountering division.
Address this by introducing area models for both multiplication and division together, rather than treating them as separate topics. This helps students understand that the area model is flexible and that what it represents depends on what you’re trying to find.
Additionally, before setting up the area model for division, have students write the related multiplication equation. For example, if solving 84 ÷ 6, write: 6 × ___ = 84. This clarifies that they’re finding the missing factor (the side length), not the product (the area).
Addressing these misconceptions and using thoughtful instructional strategies helps students build a deep understanding of how area models work across operations. When combined with hands-on tools and real-world connections, area models become a powerful visual strategy for developing place value understanding, computational fluency, and algebraic thinking.
