Factoring GCF and Grouping

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Objective

Students will be able to factor polynomials using the GCF and grouping.

Big Idea

Multiple methods for finding the factors of polynomials are addressed and analyzed in this lesson.

Warm up and Homework Review

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm Up- Factoring with GCF and Grouping which asks students to find the greatest common factor of two monomials.

I also use this time to correct and record any past Homework.

Factoring with GCF Algebraically and Graphically

20 minutes

This lesson is a scaffolding lesson to ensure that my students are well prepared to succeed with the Common Core objectives for Algebra 2.  Please watch my Video Narrative on why I am including this and my other factoring lessons.  

This lesson begins with a rectangle whose area is x2 +4x.  I ask the students to figure out the length and width of the rectangle (Math Practice 4).  They work on it with their partner and then we discuss it as a class. The goal is that the students come to the conclusion that x(x+4) = x2 +4x.  Some students may get something like x(x+4x) or even 4(x2+x).    A remedy for this issue is to have the students distribute what they got to make sure that it multiplies to the original expression.  It is also helpful to remind them that (x+4) means a single number that is 4 units larger than x.  We then talk about how x and x+4 are considered factors of x2 +4x, just like 4 and 5 are factors of 20, so we call this process of finding factors “factoring”. 

Factoring Graphically

Next, I pull up a graph of f(x)= x2 +4x as well as its two factors.  In the previous lesson, we took two functions and multiplied them to get a third function.  We also looked at some of the characteristic of each function and how those characteristics affect the product.  The students are going to take that knowledge and reverse it (Math Practice 7).  Again, I will let the students talk with their partner and then we will discuss as a class. 

The goal is that the students recognize the intersections on the x-axis are meaningful.  They have dealt with finding solutions graphically but haven’t specifically solved quadratic equations graphically.  This may take scaffolding if none of the students find the connections themselves.  Once the students understand that the intersections on the x-axis are the zeros of the quadratic and can be used to reproduce the function, I point out the beauty of using this method to factor or check factoring. 

We look at another graph and the students find the factors directly from the graph without the equation.  I then pull up the equation and the students check their work algebraically.  Now that they have a decent idea of how to find factor graphically, we look at the limitations of this method (Math Practice 5).  I pull up a graph of four separate functions with the same roots as the problem they just did as well as their equations.  I give the students a chance to discuss in partners why this proves to be a problem.  The key is that this is an acceptable method as long as students are aware of the limitations. 

Guided Practice

The students will factor a couple of problems algebraically.  This portion can be eliminated or extended depending on the needs of a given class.   If I need more problems, I usually just make them up off the top of my head.  A text book is another good resource for this.  

Factoring by Grouping

17 minutes

The next task is a factor by grouping area problem.  I give the students the area of a rectangle as 2y3 + 2y2 + 3y + 3 and ask them to find the length and the width.  The students will quickly find that taking out the greatest common factor is unhelpful here.  Many will probably feel a bit lost after a short period of time.  This is where I will model factor by grouping.  Please watch my video on using the box method for factor by grouping. 

I have a specific flow when teaching factoring that is very effective.  I start with GCF and then grouping.  I then use what they learned in grouping to teach factoring trinomials.  Finally, we put it all together.  I have had great success with this method and here is more detailed video (just under 8 minutes) of how this looks:

They will then try several practice problems to solidify this technique.  I do not insist that students use one particular method.  As long as they are successful, they can choose whatever method they want.   For students who dislike the box method, I show them the classic method for factor by grouping.  The reason I teach the box over the classic method is that negative terms tend not to be an issue with the box while they can be with the classic method as shown in this video.  

Exit Ticket

3 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

Today's Exit Ticket asks students to factor a polynomial using grouping.

Homework

The first two problems in this assignment asks students to find the factors of the graph of a quadratic polynomial. The remainder of the assignment has four problems for each factor with GCF and factor by grouping problems.  These skills are all used frequently in Algebra 2 content.  

This assignment was created using Kuta Software, a product I would highly recommend to any mathematics teacher.