Example Of An Area Model

Understanding the Area Model: A Comprehensive Guide with Examples

The area model is a powerful visual tool used in mathematics, particularly in elementary and middle school, to illustrate fundamental concepts like multiplication, factoring, and area calculations. It provides a concrete representation of abstract mathematical ideas, making them easier to grasp for students of all learning styles. This comprehensive guide will explore the area model in detail, providing various examples and explaining its applications across different mathematical contexts. We'll delve into its use in multiplication, factoring polynomials, and even its connection to other mathematical concepts like distributive property.

What is the Area Model?

At its core, the area model leverages the concept of area – the space enclosed within a two-dimensional shape. We represent numbers or algebraic expressions as the dimensions of a rectangle, and the area of that rectangle visually represents the product or result of the operation. For example, to multiply 12 by 15, we can represent 12 as (10 + 2) and 15 as (10 + 5). These become the dimensions of a rectangle that we then divide into smaller rectangles representing partial products. The sum of the areas of these smaller rectangles gives us the final product. This method is particularly effective in demonstrating the distributive property of multiplication.

Area Model for Multiplication: Simple Examples

Let's start with straightforward examples to illustrate the basic mechanics of the area model in multiplication.

Example 1: Multiplying 12 x 15

1. Represent the numbers: We represent 12 as (10 + 2) and 15 as (10 + 5).

2. Draw the rectangle: Draw a rectangle with dimensions (10 + 2) and (10 + 5).

3. Divide the rectangle: Divide the rectangle into four smaller rectangles.

4. Calculate the area of each smaller rectangle:

   • 10 x 10 = 100
   • 10 x 2 = 20
   • 5 x 10 = 50
   • 5 x 2 = 10

5. Add the areas: 100 + 20 + 50 + 10 = 180. Therefore, 12 x 15 = 180.

This visually demonstrates the distributive property: 12 x 15 = (10 + 2)(10 + 5) = 10(10 + 5) + 2(10 + 5) = 100 + 50 + 20 + 10 = 180.

Example 2: Multiplying 23 x 14

1. Represent the numbers: Represent 23 as (20 + 3) and 14 as (10 + 4).

2. Draw and divide the rectangle: Draw a rectangle and divide it into four smaller rectangles.

3. Calculate the area of each smaller rectangle:

   • 20 x 10 = 200
   • 20 x 4 = 80
   • 3 x 10 = 30
   • 3 x 4 = 12

4. Add the areas: 200 + 80 + 30 + 12 = 322. Therefore, 23 x 14 = 322.

Area Model for Multiplication: Larger Numbers and Variables

The area model's strength lies in its scalability. It can easily handle larger numbers and even algebraic expressions involving variables.

Example 3: Multiplying 345 x 27

1. Represent the numbers: Represent 345 as (300 + 40 + 5) and 27 as (20 + 7).

2. Draw and divide the rectangle: Draw a rectangle and divide it into six smaller rectangles.

3. Calculate the areas:

   • 300 x 20 = 6000
   • 300 x 7 = 2100
   • 40 x 20 = 800
   • 40 x 7 = 280
   • 5 x 20 = 100
   • 5 x 7 = 35

4. Add the areas: 6000 + 2100 + 800 + 280 + 100 + 35 = 9315. Therefore, 345 x 27 = 9315.

Example 4: Multiplying Binomials (Algebra)

Let's consider multiplying two binomials: (x + 3)(x + 2).

1. Represent the expressions: The dimensions of the rectangle are (x + 3) and (x + 2).

2. Draw and divide: Draw a rectangle and divide it into four smaller rectangles.

3. Calculate the areas:

   • x * x = x²
   • x * 3 = 3x
   • x * 2 = 2x
   • 3 * 2 = 6

4. Combine like terms: x² + 3x + 2x + 6 = x² + 5x + 6. Therefore, (x + 3)(x + 2) = x² + 5x + 6. This visually demonstrates the FOIL method (First, Outer, Inner, Last) often used in algebra.

Area Model for Factoring Polynomials

The area model also works in reverse to help factor polynomials. Given a polynomial, we can use the area model to find its binomial factors.

Example 5: Factoring x² + 5x + 6

1. Draw a rectangle: Draw a rectangle and label its area as x² + 5x + 6.

2. Place the terms: Place the x² term in the top-left corner.

3. Find factors: We need to find two numbers that add up to 5 (the coefficient of x) and multiply to 6 (the constant term). These numbers are 3 and 2.

4. Fill the rectangle: Place 3x and 2x in the remaining two rectangles. The remaining corner will be filled with 6.

5. Determine the dimensions: The dimensions of the rectangle are (x + 3) and (x + 2). Therefore, x² + 5x + 6 = (x + 3)(x + 2).

Example 6: Factoring 2x² + 7x + 3

1. Draw a rectangle: Draw a rectangle with area 2x² + 7x + 3.

2. Place the leading term: Place 2x² in the top-left corner.

3. Find factors: We need two numbers that add to 7 and multiply to 6 (2 x 3). These numbers are 6 and 1.

4. Strategically fill the rectangle: Place 6x and x in the remaining rectangles. This necessitates placing a 3 and an x along one side, and a 2x and a 1 along the other.

5. Determine the dimensions: The dimensions of the rectangle are (2x + 1) and (x + 3). Therefore, 2x² + 7x + 3 = (2x + 1)(x + 3).

Area Model and the Distributive Property

The area model is intrinsically linked to the distributive property of multiplication. The distributive property states that a(b + c) = ab + ac. The area model visually represents this by breaking down the larger rectangle into smaller, easily calculable rectangles whose areas represent the partial products (ab and ac), the sum of which is the total product. This makes understanding the distributive property more intuitive and less abstract.

Beyond Basic Multiplication: Applications in Advanced Mathematics

While frequently used in elementary and middle school, the underlying principles of the area model extend to more advanced mathematical concepts. It can be used in:

• Higher-degree polynomials: The area model can be extended to multiply and factor polynomials of higher degrees, albeit with more complex diagrams.

• Matrices: The area model's principles are analogous to matrix multiplication, where the elements of the resulting matrix are calculated as the dot products of the rows and columns of the original matrices.

• Geometric problems: The area model can be used to solve geometric problems involving area calculations and relationships between dimensions.

Frequently Asked Questions (FAQ)

Q: Is the area model only for multiplication?

A: No, while primarily used for multiplication, the area model is also valuable for factoring polynomials, visually demonstrating the distributive property, and solving various geometric problems related to area.

Q: Can the area model handle negative numbers?

A: Yes, but it requires careful consideration of signs. Negative dimensions would be represented as negative lengths, affecting the signs of the partial products.

Q: Is there a limit to the size of the numbers the area model can handle?

A: Technically, there's no limit, but practically, the complexity of the diagram increases with larger numbers, making it less manageable. For very large numbers, traditional multiplication methods might be more efficient.

Q: How does the area model compare to other multiplication methods?

A: The area model offers a visual and intuitive approach, making it particularly helpful for students who struggle with abstract concepts. It excels in illustrating the distributive property and providing a concrete understanding of multiplication. While potentially slower for large numbers, its pedagogical value is significant.

Conclusion

The area model is a versatile and insightful tool for understanding multiplication, factoring, and the distributive property. Its visual nature makes complex mathematical concepts more accessible and understandable, fostering a deeper comprehension of fundamental mathematical principles. From simple multiplication of whole numbers to factoring complex polynomials, the area model provides a powerful and intuitive pathway to mastering these essential mathematical skills. Its adaptability and clear visual representation make it a valuable tool throughout the mathematics curriculum, benefiting students at various levels of mathematical understanding. By using this model, students develop a stronger foundation in algebra and prepare themselves for more advanced mathematical concepts.