## Always Sometimes Never - Section 3: Direct Instruction

# Introduction to Polynomials

Lesson 1 of 16

## Objective: SWBAT classify polynomials and perform addition, subtraction and multiplication with polynomials.

## Big Idea: Remember linear and quadratic expressions from Algebra 1? These are both part of a larger set called polynomial expressions. Operations with polynomial expressions are a lot like operations with integers.

*80 minutes*

#### Warm-up

*20 min*

Warm-up- Polynomial Operations is designed to refresh students' algebra 1 skills with polynomial operations. It is always the case that some students do not remember how to perform polynomial addition, subtraction and multiplication, so I ask students to work together to remind each other of the procedures.

The warm-up may take a little longer than 20 minutes, depending on how rusty students' Algebra 1 skills are. When they have finished the warm up, I project the answers on the board and ask if they have questions. Volunteers are solicited to perform an operation at the board at the request of a student, explaining their work as they write it out [MP3].

#### Resources

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#### Vocabulary Reinforcement

*10 min*

There are many vocabulary terms unique to our study of vocabulary, and success in this unit requires students to use these terms according to their precise definition [MP6]. For this reason, I take time to teach the vocabulary explicitly early in this lesson.

I give each student a list of 36 polynomial vocabulary words that will be used in this unit. I instruct them to work in pairs to determine the words they know and can use in a sentence related to polynomials.

Students will use this word list later in the unit to make vocabulary posters, but now they simply discuss the vocabulary to activate their prior knowledge of polynomials. I find that providing a word list in a vocabulary-heavy lesson results in better participation in the lesson.

#### Resources

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#### Direct Instruction

*30 min*

This discussion and note-taking session is designed to review students' knowledge of polynomials and introduce students to the closed nature of polynomial addition, subtraction and multiplication. I use questioning and Cold Calling during this discussion and make sure that all students remain engaged.

First, I outline how polynomials are classified and run through some examples of putting polynomial expressions into standard form and then classifying the polynomial. As I introduce the vocabulary related to polynomial classification, I refer students to the word list they have from their partner discussion. The example polynomials that we used to practice classification stay on the board for the next part of the discussion.

To practice vocabulary and introduce the closure property, I then use quick polls to determine if students are able to determine whether statements about polynomial operations are "always," "sometimes," or "never" true. I talk briefly about raising the level of discussion with this type of question in the video, Always Sometimes Never. Some of the statements I use are included below.

- When you add a linear binomial to a linear binomial, you get another linear binomial
- When you add a quadratic trinomial to a linear binomial, you get a cubic trinomial
- When you multiply one polynomial by another polynomial you get a polynomial

We discuss the quick poll results and I ask students to use examples to support or disprove their statements [MP3]. If they have difficulty thinking of polynomial examples, I point out those on the board from the "classifying" part of the discussion.

From here we generalize the concept of closure to polynomials as a set [MP8]. Through questioning, I lead students to the idea that operations with polynomials follow the same set rules as operations with integers, which addresses CCSS APR.A.1.

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#### Introduction to Factoring

*20 min*

As a closing activity, I introduce the next day's topic: rewriting an expression by factoring. Factoring is a major topic in the polynomial unit because it is a great example of how writing a mathematical expression in a different, but equivalent way can reveal important properties [MP7].

I remind students of the definition of a greatest common factor (GCF). We do a few examples of finding the GCF with numbers and then move into some examples that include variables. Finally, I show students how to factor a polynomial expression by factoring out the GCF.

Tomorrow's warm up will be a game of BINGO with greatest common factors, and the evening's homework is to create their GCF BINGO card by doing some GCF factoring.

#### Resources

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- LESSON 1: Introduction to Polynomials
- LESSON 2: Seeing Structure in Expressions - Factoring GCF's and Quadratics
- LESSON 3: Connecting Polynomials to Sequences
- LESSON 4: Connecting Polynomials to Geometric Series
- LESSON 5: Quadratic Functions: Standard and Intercept Forms
- LESSON 6: Quadratic Functions: Vertex Form
- LESSON 7: Flexibility with Quadratic Functions
- LESSON 8: Connecting Quadratic Functions and Quadratic Equations
- LESSON 9: Solving Quadratic Equations
- LESSON 10: Quadratic Performance Task
- LESSON 11: Quadratic Modeling (DAY 1)
- LESSON 12: Quadratic Modeling (DAY 2)
- LESSON 13: Quadratic Modeling (DAY 3)
- LESSON 14: Quadratic Modeling (DAY 4)
- LESSON 15: Review Workshop: Polynomial Functions and Expressions
- LESSON 16: Unit Assessment: Polynomial Functions and Expressions