##
* *Reflection: Discourse and Questioning
Factoring GCF and Grouping - Section 2: Factoring with GCF Algebraically and Graphically

A key shift in the Common Core is providing students more opportunities to evaluate and discuss mathematics. One method I use is to give students several solutions to a problem and have them do a think-pair-share on which solution is correct. In this lesson, I was able to implement this with an additional practice problem for factoring with GCF.

This particular problem has a common Misunderstanding that plagues students which is factoring out an entire term from a polynomial. So often students will just leave it off rather than writing a one in its place. I had students look at these two solutions and then discuss their opinion with their partner. The resulting class discussion brought out several ways to look at this problem and I felt like students were more aware of this issue than they would have been if I told them about it or even made them practice several examples.

*Discourse and Questioning: Questioning in the Common Core*

# Factoring GCF and Grouping

Lesson 6 of 15

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

#### Warm up and Homework Review

*10 min*

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.

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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 x^{2} +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) = x^{2} +4x. Some students may get something like x(x+4x) or even 4(x^{2}+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 x^{2} +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)= x^{2} +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.

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#### Factoring by Grouping

*17 min*

The next task is a factor by grouping area problem. I give the students the area of a rectangle as 2y^{3} + 2y^{2} + 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.

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#### Exit Ticket

*3 min*

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.

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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.

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Laws of Exponents
- LESSON 2: Sorting Polynomial Equations and Identities
- LESSON 3: Operations with Polynomials Day 1 of 2
- LESSON 4: Operations with Polynomials Day 2 of 2
- LESSON 5: Products of Polynomial Functions
- LESSON 6: Factoring GCF and Grouping
- LESSON 7: Factoring Trinomials
- LESSON 8: Special Factoring Situations
- LESSON 9: Polynomial Quiz and Factoring Puzzle
- LESSON 10: Factoring Completely
- LESSON 11: Modeling Equations and Functions with Factoring
- LESSON 12: Polynomial Inequalities
- LESSON 13: Polynomial Review Day 1
- LESSON 14: Polynomial Review Day 2
- LESSON 15: Polynomial Test