Lesson Plan

Classifying Polynomials

• Assessment Anchors
• Eligible Content
• Big Ideas
• Families of functions exhibit properties and behaviors that can be recognized across representations. Functions can be transformed, combined, and composed to create new functions in mathematical and real world situations.
• Mathematical functions are relationships that assign each member of one set (domain) to a unique member of another set (range), and the relationship is recognizable across representations.
• Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
• Patterns exhibit relationships that can be extended, described, and generalized.
• Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
• There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
• Concepts
• Algebraic properties, processes and representations
• Exponential functions and equations
• Polynomial functions and equations
• Competencies
• Extend algebraic properties and processes to quadratic, exponential, and polynomial expressions and equations and to matrices, and apply them to solve real world problems.
• Represent a polynomial function in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the solution of the associated polynomial equation to each representation.
• Represent a quadratic function in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the solution of the associated quadratic equation to each representation.
• Represent exponential functions in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the growth/decay rate of the associated exponential equation to each representation.

Objectives

[IS.3 - All Students]

This lesson connects previous experience and knowledge of linear and quadratic functions to the concept of polynomial functions. Students will:

• classify polynomials by their degree. [IS.4 - Struggling Learners]

• write polynomials in standard form. [IS.5 - All Students]

• determine the number of roots a polynomial has by looking at the equation. [IS.6 - Struggling Learners]

Essential Questions

How are relationships represented mathematically?
How can data be organized and represented to provide insight into the relationship between quantities?
How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?
How can mathematics support effective communication?
How can patterns be used to describe relationships in mathematical situations?
How can probability and data analysis be used to make predictions?
How can recognizing repetition or regularity assist in solving problems more efficiently?
How does the type of data influence the choice of display?
How is mathematics used to quantify, compare, represent, and model numbers?
What makes a tool and/or strategy appropriate for a given task?
• How can we determine if a real-world situation should be represented as a quadratic, polynomial, or exponential function?

• How do you explain the benefits of multiple methods of representing polynomial functions (tables, graphs, equations, and contextual situations)?

Vocabulary

• Polynomial: An algebraic expression that contains one or more monomials. [IS.1 - Struggling Learners]  [IS.2 - Struggling Learners]

• Binomial: A polynomial with two terms.

• Degree of Polynomial:The greatest exponent of the variables in the expression; for 7x2 + 5x + 8, the degree is 2.

• Monomial:A single term, such as x, y3, or 17.

• Standard Form: For a rational integral polynomial equation of degree n, a0xn + a1xn-1 + … + an = 0.

• Trinomial: A polynomial with three terms.

• Coefficient: The numerical or constant multiplier of the variables in an algebraic term.

Duration

120–180 minutes [IS.7 - All Students]

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

Formative Assessment

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• In the Think-Pair-Share activity, students will represent their individual knowledge of the relationship between the polynomial expression and its graph as a function. They will also evaluate the representations of their individual partner. [IS.11 - Struggling Learners]

• To complete the Lesson 1 Exit Ticket, students will classify, enumerate terms, identify power, and identify number of roots for four polynomial expressions.

Suggested Instructional Supports

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Active Engagement, Modeling, Explicit Instruction W: This lesson introduces students to classifying polynomials according to the greatest exponent of the variable and expressing them in standard form. The lesson also shows students the correspondence between the degree of the polynomial and the number of roots of the polynomial equation. H: The initial problem set helps to remind students of the importance of like terms and discriminating between like and unlike terms. This is a familiar and accessible skill for most students and using it to classify more difficult and complex expressions will give them confidence in working with polynomial functions. E: The Polynomial Functions Graphic Organizer gives students a tool with which they can dissect the component parts of an expression. The resource also supports their learning how to reconstruct the individual terms in order to appropriately classify expressions and equips them to make meaningful representations of polynomial equations. R: In Activity 5 (Pairs), students have to rethink the relationship between the degree of the polynomial, which terms are like and unlike, and apply that knowledge to classifying polynomial expressions. They must also engage in reflecting on whether or not each partner has correctly identified and classified the polynomial. E: The Lesson 1 Exit Ticket helps students evaluate their individual understanding of the relationship between exponent, degree of polynomial, and like/unlike terms. Students must represent their understanding by naming the polynomial, counting the terms, expressing the degree, and indicating the number of roots of the equation. T: Group and partner work is used throughout so that students can help each other. Emphasis should be placed on communicating mathematical ideas with the specific vocabulary words appropriate to the concepts. The lesson requires accurate note-taking skills to enhance the learning experience while creating a useful resource (notes). Logical/Sequential/Visual Learners: It may be helpful for some students to see a step-by-step listing of the process used when combining like terms in polynomial expressions. Speak the name of the terms appropriately while writing or pointing to them. O: This lesson is a building block for the lessons to come and is necessary for students to understand polynomials and how they are used to solve real-world situations. It begins with an activity that engages students because it activates prior knowledge and shows students where they will take their knowledge. Vocabulary is introduced that students need to know for the activities, which are simple and give students time to explore polynomials. Students see how polynomial equations are related to their graphs and then make connections on their own. Students have time to review the lesson’s concepts, receive timely feedback, and learn how they will use this new information in the next lesson.

 IS.1 - Struggling Learners Consider the following steps with regard to vocabulary for struggling learners: Use of a graphic organizer (e.g., Frayer Model, Verbal Visual Word Association, Concept Circles). Introduce new vocabulary using student friendly definitions. Review words with students. Provide opportunities for student s to apply the new/reviewed words. IS.2 - Struggling Learners Consider reviewing the difference between “expression” and “equation”.  Struggling learners may need to have a graphic organizer that contains both words and definitions w/examples of each. IS.3 - All Students (blank) IS.4 - Struggling Learners Consider modeling this to struggling learners. IS.5 - All Students Consider reviewing this term and providing a graphic organizer showing examples . IS.6 - Struggling Learners Consider modeling this to struggling learners. IS.7 - All Students Consider pre-teaching the concepts critical to the lesson.  Use formative assessments throughout the lesson to determine level of student understanding. Use follow-up reinforcement as necessary. IS.8 - Struggling Learners Consider providing struggling learners with written and/or pictorial examples of each of these.  Consider allowing struggling learners to use a graphing calculator for parts of the lesson. IS.9 - All Students Consider reviewing linear and quadratic functions and polynomial functions for struggling learners.  Provide examples on graphic organizers such as a concept map.  Review vocabulary terms; degree, binomial and trinomial and provide students with student friendly terms and examples. IS.10 - Struggling Learners Consider reviewing this with struggling learners and provide examples. IS.11 - Struggling Learners Consider using multiple forms of formative assessment for struggling learners.

Instructional Procedures

• View

After this lesson, students will know what a polynomial is and will understand that polynomial functions are another way to represent real-world situations. They will see the link between their previous knowledge of linear and quadratic functions and polynomial functions. They will know vocabulary such as degree, binomial, and trinomial. Students will build upon this knowledge in Lesson 2 and will learn to use polynomial functions to represent real-world situations. [IS.9 - All Students] Students will be able to write polynomials in standard form and classify polynomials by degree and number of terms. They will be able to graph polynomials in factored form because they will understand the connection between the number of roots and the degree of the polynomial.

To introduce the lesson, activate students’ prior knowledge. Place the following problems on the board.

1. 3x + 4x

2. 5x2 − 3x2

3. 7x + 2x2x + 8x2

What does it mean to combine like terms?”

Write the following on the board:

If we can’t combine the terms, then we should arrange them in an order that makes sense. Does anyone have an idea of how to rearrange the terms?”

If students don’t volunteer any ideas, ask, “How are books sorted in the library? How are words sorted in a dictionary? What does it mean to put a list in ‘descending’ order? [IS.10 - Struggling Learners] Try putting the terms that are on the board in descending order.”

Students will probably come up with the following:

5x4 + 4 + 3x2x − 2x3 (put the coefficients in descending order)

5x4 + 4 + 3x2 − 2x3x (put the coefficients in descending order, ignoring signs)

5x4 − 2x3 + 3x2x + 4 (correctly put the terms in descending order)

Tell the students who wrote the last expression that they have just written a polynomial in standard form.

Hand out the Polynomial Functions Graphic Organizer (see M-A2-3-1_Polynomial Functions Graphic Organizer in the Resources folder). Below is what students should add to their graphic organizers.

Polynomial: A sum of terms that has variables that are raised to whole-number exponents.

Quadratic, in this context, means to the second power (Latin: quadratum is square, as area of a square).

Coefficient: The number that is being multiplied by the variable; the number in front of the leading term is called the leading coefficient.

• Polynomials can be classified by the number of terms they have.

• Zero degree: constant (example: 5).

• Monomial: a polynomial that has one term (example: 4x)

• Binomial: a polynomial that has two terms (example: 4x − 3)

• Trinomial: a polynomial that has three terms (example: 2x2 + 4x − 3)

Any term with four or more terms is a polynomial.

• Polynomials can also be classified by their degree (the largest exponent)

• First-degree: also known as “linear” (example: 5x + 1)

• Second-degree: also known as “quadratic” (example: 9x2 − 5x + 3)

• Third-degree: also known as “cubic” (example: 4x3 +2x2x + 8)

• Fourth-degree: also known as “quartic”

(example: 8x4 − 5x3 + 6x2 + x − 3)

• Fifth-degree: also known as “quintic”

(example: x5 + 2x4 – 5x3x2 + 8x + 4).

Polynomial equations of degree greater than five are not solvable by methods other than approximations.

• Polynomials are usually written in standard form, where the exponents are in descending order (largest to smallest).

• Standard Form: 6x4 + 2x3x2 + 3x − 5

• Nonstandard form: 9x + 3x2 − 4x5 + x3 + 2x4

Activity 1 (Auditory): Defining Polynomials

Individually students should write two polynomials in standard form on an index card and hand it in. Read aloud one polynomial at a time and ask students to answer the following questions:

• Is it a polynomial?” (some students may not have written one correctly)

• What degree polynomial is it? Classify it as linear, quadratic, cubic, quartic, or none of the above.”

• How many terms does the polynomial have? Classify it as a monomial, binomial, trinomial, or none of the above.”

Activity 2 (Auditory): Writing Polynomials in Standard Form

Using the index cards and the polynomials that were not used in Activity 1, read aloud the polynomials not in standard form.

• Write the polynomial in standard form.”

• What is the polynomial’s degree?”

• How many terms does the polynomial have? So what is its name?”

• Pair up and check each other’s work.”

Activity 3: Think-Pair-Share

On large graph paper on the board, post one graph of a line and one graph of a parabola (examples: y = x − 2 and y = (x + 1)2 − 4). “With your partner, write the equations of each graph. Use what you have learned today to write them.” Ask pairs to share their answers and how they came up with their equations.

Does anyone see a link between the graphs and his/her equations? What does the equation say about the graph?”

Remind students that plotting the graph without a graphing calculator means substituting values for x and marking the corresponding f(x) as an ordered pair, and then continuing to plot a sufficient number of points to complete the graph.

If students struggle, say, “Look at your graphic organizer. What were some things you learned today about polynomials?”

After a few more minutes, have one partner from each pair come up. Tell the partners: “Look at where the graph crosses the x-axis and look at the degree of the polynomial.” Students will take this information back to their partners and try to make the connection. Make sure the quadratic function used as an example has two x-intercepts.

After a few minutes, choose a few volunteers to give their opinion of the connections. Some students are likely to have realized that the degree of the polynomial is the same as the number of roots (x-intercepts) of the graph.

Activity 4: Think-Pair-Share

Before thoroughly explaining the concept, put a graph of a cubic function and the graph of a quartic function on the board. “Individually, find the roots (x-intercepts) of the graphs. When you have written them, discuss the roots with your partner and the degrees of each graph. How are they linked?” If some pairs need opportunity for additional learning, place them with a pair that understands the concept. As a whole class, pairs can share their conclusions about polynomials’ degrees and the number of roots that the graphs have.

Activity 5: Pairs

In pairs, one student gives the degree and the number of terms and the second student writes a polynomial in standard form with that same degree and number of terms. The first student checks the work and either agrees or explains why it’s incorrect. Once they agree, they switch roles.

Activity 6: Whole Class

Write the following terms on sheets of paper and tape them to the board.

5x4     −8x5     −3x3     −x     6     4x2     −7x2     5x     6x4

2x3     10x3     x2     −3x     1     5     6x     −2x2     −3

Let’s make four polynomials with these 18 terms. That means we will not be combining like terms. Put the polynomials in standard form. You will be coming up to the board one or two at a time to move one term. Those who are sitting in their desks should remain quiet and not help the students who are at the board. If there is not a negative sign in front of a term, use the plus sign. Once every term is in a polynomial we will look at each one and check that it is in standard form. If it is, we will classify the polynomial by degree (including whether it’s linear, quadratic, etc.) and how many roots the polynomial should have.”

During this activity, you can tally how many students got their term correct and how many didn’t, just to gauge how well the class understands.

It’s easy to duplicate this activity and repeat it with different sets of terms and it may be useful to customize the individual elements to suit the needs of individual students or a particular class.

Use the Lesson 1 Exit Ticket (M-A2-3-1_Lesson 1 Exit Ticket.doc and M-A2-3-1_Lesson 1 Exit Ticket KEY.doc) for a quick way to evaluate whether students understand the concepts. Have students fill out the table on the Lesson 1 Exit Ticket. “The number of roots can be used to find the degree of polynomial equations, and in the next lesson we will be using roots to determine the factors of polynomials.”

Extension:

• Give students the following exercises:

1. (3x − 1)(x2 + 4x − 21)

2. (−2x + 4)(x − 2)

3. (x + 3)(x3 + 2x2x + 4)

Answer the following questions for each exercise:

A. What is the degree of each polynomial?

2. Linear, linear

3. Linear, cubic

B. Multiply the two polynomials using area models and simplify your answer (write in standard form).

1. 3x3 + 11x2 – 67x + 21

2. –2x2 + 8x + 8

3. x4 + 5x3 + 5x2 + x + 12

C. What is the degree of your polynomial?

1. Cubic