## Distributing and Factoring Using Area

- Lesson

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.

Area for Distributive Property Activity Sheet

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: -2*a*+ 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**

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

**Extensions**

- Students can use this method to factor a monomial from expressions that have more than two terms, such as: 3
*x*^{3}+ 6*x*^{2}- 12*x*. - 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.

### Common Core State Standards – Mathematics

Grade 6, Geometry

- CCSS.Math.Content.6.G.A.1

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Grade 6, The Number System

- CCSS.Math.Content.6.NS.B.4

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).