##
* *Reflection: Staircase of Complexity
What if There is No Common Factor? - Section 4: Close

Today's ticket out worked well to set the stage for future thinking about factoring completely. Eventually, I will give students a "rule" that they should always look for the greatest common factor first. However, the concept is that when you are done factoring (however you start) all of your factors should be prime. This ticket out allows students to see that even after they factored once, their factors were still not prime and so one part of the expression could be factored again.

See What if...Student Work 1: This student realized that the expression was not factored completely but did not say to factor using the greatest common factor. For now, this is okay, if students realize that the resulting factors (2(2x-4)) still have a common factor. They will eventually get to the correct factorization of 4(x-2).

See What if...Student Work 2: This student was able to determine the greatest common factor and also showed a good understanding of how to find the original expression.

*Staircase of Complexity: Setting the Stage*

# What if There is No Common Factor?

Lesson 12 of 18

## Objective: SWBAT factor a polynomial expression made up of four terms with no GCF.

#### Launch

*5 min*

My plan is to use the think pair share strategy to meet my goals for this launch. I begin class by letting students think about the question posed on the factoring_polynomials_grouping slide independently.

As students begin to put down some ideas, I will make my way around the room to observe. I am looking for students who are making a connection to the idea that the factors must multiply to obtain the original expression.

Once students have had a couple of minutes to think independently, I let them pair up and discuss their ideas with a partner. Listen for students that are making sense of the problem and ask one or two pairs to share their reasoning with the class (MP3).

**Teaching Note**: I also discuss this opening in the beginning of the direct instruction portion of this lesson.

*expand content*

#### Direct Instruction

*10 min*

I prepared this video to introduce the factoring_polynomials_grouping_direct worksheet. I plan to use a gradual release as we progress through this series of questions. In the first example, I will do the entire problem with students. In the second example, I will help students with the procedure of factoring out the gcf from the first row and then let them finish factoring. In the third example, I hope to be able to allow my students to work independently.

*expand content*

#### Practice

*20 min*

For practice today, I will ask students to work with their partner on this practice activity from kutasoftware.com: factoring_polynomials_grouping_practice.

As they work on the worksheet, I will encourage my students to use an area model to structure the expression and then determine two binomial factors. Depending on the amount of time available, I will choose which practice problems to assign to students.

Today, I plan to have my students choose any eight problems and factor them into two binomials using the area model. I plan to post the answer key so that as students are working they can independently check their work. I also encourage students to check their work by multiplying the binomial factors to ensure their factors make sense (MP3).

**Teaching Note**: Some of the expressions require students to factor completely. This means that one or both of their binomial factors will have a greatest common factor that can be factored out. Some students will notice this but others may not. Don't be too concerned if students are not factoring completely as this will be addressed both in the closing of this lesson and in future lessons.

*expand content*

#### Close

*5 min*

To close today's lesson, I will ask students to complete factoring_polynomials_grouping_close as a Ticket out the Door. I will ask my students to record their response on a half sheet of paper.

In question Number 1, students will look at the structure (MP7) of the binomials factors to determine if they are factored completely (aka, are they both prime factors). I want my students to notice that 4x and 8 have a common factor of 4. In the second part of the question, I am trying to assess students' understanding of the connection between factors and the original expression. Again, I hope students realize that to find the original expression they need to multiply the factors.

I hope that by asking students to think in both directions (multiplying and factoring), I will help them to see the connection and gain deeper conceptual understanding of polynomial expressions.

*expand content*

James - I have been teaching factoring by grouping for 20 years, but this method is great! I can't wait to teach it this way. Thanks for all your good ideas.

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- LESSON 1: Adding and Subtracting Monomials
- LESSON 2: Adding and Subtracting Polynomials
- LESSON 3: More with Adding and Subtracting Polynomials
- LESSON 4: Polynomial Puzzles 1: Adding and Subtracting Polynomials
- LESSON 5: Multiply and Divide Monomials-Jigsaw day 1 of 2
- LESSON 6: Multiply and Divide Monomials-Jigsaw Day 2 of 2
- LESSON 7: Multiplying Higher Degree Polynomials
- LESSON 8: Multiplying Polynomials Investigation
- LESSON 9: Polynomial Vocabulary
- LESSON 10: Polynomial Puzzles 2: Distributive Property
- LESSON 11: Factoring Using a Common Factor
- LESSON 12: What if There is No Common Factor?
- LESSON 13: Factoring Trinomials
- LESSON 14: More with Factoring Trinomials
- LESSON 15: Polynomial Puzzles 3: Multiplying and Factoring Polynomials
- LESSON 16: Seeing Structure in Factoring the Difference of Squares
- LESSON 17: Factoring Completely
- LESSON 18: More with Factoring Completely