Factoring Using the Area Model
There are methods to make factoring less mysterious, and more fun, for students.
By Wendy Denzel
As a high school math teacher, I am not surprised when students struggle with factoring. It is one of the “dirty words” in algebra classes around the nation. Many students have been trying to factor for at least four years with little success. No wonder they are not thrilled about taking another math class in which they are required to do this task!
Solving the Mystery of Factoring
Factoring is often taught as an abstract algorithm. All too often, teachers tell their students, “You just do this and it works.” This type of explanation only serves to make math concepts even more "mysterious" and confusing for students. It doesn’t have to be! As I began preparing my factoring unit, I realized that I didn't want factoring to be mysterious any longer. I knew I needed a different approach, so I went searching for one.
Finding the Area of a Rectangle
Most students are comfortable with the idea of finding the area of a rectangle. They can picture the rectangle, they can count the square units, they know the procedure, and they have done this before and been successful. Factoring using the area model allows students to relate the concept of finding the area of a rectangle by multiplying length times width and then reversing the process; if given an area they can find the dimensions of the rectangle. It builds on students’ prior knowledge. It makes the concept concrete; they can picture the factoring process. Also, factoring using the area model provides a way for students to organize their thoughts.
I usually start the lesson by finding the area of a rectangle. I draw a rectangle on the board that has the dimensions of x + 2 units across the top and bottom, by x+ 5 units down the sides. Using the classic formula, area= length x width, students will quickly calculate the area of the rectangle by finding the area of each of the four parts and adding the parts to find the total area of the rectangle. We can then reverse the process to find the dimensions of another rectangle given the area of the parts.
Factoring Using the Area of a Rectangle
First, I draw a rectangle on the board and put the number 126 inside of it (the figure should be 14 by 9). In order to find the length and width of the sides, students must find the prime factors of 126. Since 126 is an even number, it is divisible by 2, 126 = 2 x 63, which when broken down is 2 x 9 x 7. Thus, the prime factorization of 126 is 2 x 3 x 3 x 7. Have students look closely at the rectangle. The top and bottom are about a third longer than the two sides. Ask students what numbers would make sense along the top and bottom, and the two sides? Remember, the length x the width must equal 126 when multiplied together. Tell your students that they must use a combination of the prime factors of 126 to find the dimensions. After some discussion, the students should realize that if they multiply 3 x 3 together, the answer of 9 would be appropriate along the sides. If they multiply 2 x 7 together, the answer of 14 would be appropriate along the top and the bottom – which are the longer sides of the rectangle. Sure enough, when multiplied together, 14 x 9 equals 126! Viola! They have found the dimensions of a rectangle by finding the prime factors of the area, and using their skills of spatial reasoning to assign the length and the width.
Factoring Made Fun
At first, most students do not even understand that they are factoring. It becomes almost a game and students ask, “Why didn’t someone show me this before?” They are excited to realize that factoring can be fun! Additionally, I often have my students use their whiteboards at their desks for this lesson. Students enjoy the ease of changing dimensions by simply erasing a answer if the dimensions they tried didn’t work. For more factoring lessons and worksheets see below.
Factoring Lessons and Worksheets
Students arrange tiles into rectangles of different dimensions, but the same area to help draw the connections between area and factoring. Students also investigate prime factorization and factor trees.
Students work in groups to discover strategies, rules, and procedures for factoring using the Punnett Square method (area model) for factoring trinomials.
Students work in groups using algebra tiles or algebra blocks and processed through stations set-up in the classroom. In these stations, students first show the multiplication of two binomials using the algebra tiles/blocks. Next, students remove the blocks leaving the rectangular array. Finally, students work to discover the original rectangular array.