## Distributing and Factoring Using Area

• Lesson
6-8
1

In this lesson, expressions representing area of a rectangle are used to enhance understanding of the distributive property.  The concept of area of a rectangle can provide a visual tool for students to factor monomials from expressions.

Students often struggle with the distributive property when they are first introduced to the rule. The idea of multiplying everything by the common factor is often a battle that teachers have with low-achieving or young algebra-learners. By introducing area to students as a way to represent multiplication of terms, students have exposure to another tool for understanding and remembering why the distributive property is so.

Students can also write expressions as area to find common factors and for factoring. This process is essentially "undoing" the distributive property.

### Area Representation of the Distributive Property

The first activity sheet can be used to introduce the concept of the distributive property but this sheet is designed to acquaint students with the area representation for the distributive property, rather then develop the concept for first exposure to the rule.

The first section introduces students to the idea of writing the area of a rectangle as an expression of the length × width, even when one or more dimensions may be represented by a variable.

The next section acquaints students with the thought of writing the length of a segment consisting of two parts as a sum.

The key section is next, having students represent the area of each rectangle two ways to distribute the common factor among all parts of the expression in parentheses.

### Area Representation to Find a Common Factor

When students are presented with a figure such as the one below, they can be asked to determine the dimensions that yield the area expressed.

• The second page helps students recognize and factor out an integer common factor.
Help students who pause when the common factor is a negative number. Be sure that they change the sign of the second term. (Example: -2a + 10 = -2(a – 5) )
• The third page helps students recognize and factor out a variable common factor or an integer other than the leading coefficient.
Encourage students who pause when the common factor may not seem to follow a pattern. Students will have to consider both terms when deciding on a common factor.
• The fourth page helps students recognize and factor out a common factor that may be an integer, a variable or a product of both an integer and a variable to find the greatest common factor.
Students should recognize that when they find a common factor, it may not be the greatest common factor.

Assessment Options

1. Ask students to write about the relationship between the product and sum representation of the area model.
2. Ask students to write about what they think distributing has to do with the distributive property.

Extensions

1. Students can use this method to factor a monomial from expressions that have more than two terms, such as: 3x3 + 6x2 - 12x.
2. Students can use this method to multiply two binomials, such as: (x + 5)(x + 2).

Questions for Students

1. What is the area of a rectangle?

[Length × Width.]

2. What is a common factor?

[A number that divides into two or more numbers.]

3. Why is it important to factor out the greatest common factor?

[It is important to use the greatest common factor because it will create the most simplified factorization.]

Teacher Reflection

• What are some meaningful word problems that can be given to students that demonstrate how distributing and factoring are useful?
• How could you extend this lesson to help teach students how to multiply two binomials? Why (or why not) would you teach the procedural FOIL afterwards?
• Where did students struggle the most? What pedagogical techniques could you explore in helping those students?
• What are the benefits (and pitfalls) of working with a partner on this activity sheet?

### Learning Objectives

In this lesson, students will:

• Represent the area of a rectangle whose sides may be variables.
• See a new representation of the distributive property.
• Use area of rectangles as a tool to factor monomials from expressions.

### NCTM Standards and Expectations

• Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.
• Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations.