This lesson applies previous knowledge of linear and quadratic functions to the concept of polynomial functions. Students will:
- classify polynomials based on their degree.
- write polynomials in standard form.
- examine the equation to determine how many roots a polynomial has.
- How are relationships expressed mathematically?
- How may data be arranged and portrayed to reveal the link between quantities?
- How are expressions, equations, and inequalities utilized to quantify, solve, model, and/or analyze mathematical problems?
- How can mathematics help us communicate more effectively?
- How may patterns be used to describe mathematical relationships?
- How can we utilize probability and data analysis to make predictions?
- How may detecting repetition or regularity help you solve problems more efficiently?
- How does the type of data effect the display method?
- How can mathematics help to measure, compare, depict, and model numbers?
- What factors determine whether a tool or method is appropriate for a specific task?
- How can we know whether a real-world scenario should be represented as a quadratic, polynomial, or exponential function?
- How would you explain the advantages of using multiple approaches to portray polynomial functions (tables, graphs, equations, and contextual situations)?
- Polynomial: An algebraic expression that contains one or more monomials.
- 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, \(y^3\), or 17.
- Standard Form: For a rational integral polynomial equation of degree n, \(a_0\)\(x^n\) + \(a_1\)\(x^{n-1}\) + … + \(a_n\) = 0.
- Trinomial: A polynomial with three terms.
- Coefficient: The numerical or constant multiplier of the variables in an algebraic term.
- index cards
- mini-whiteboards, whiteboard markers, and erasers/paper towel
- paper and markers if mini-whiteboards are not available
- poster graph paper
- copies of Lesson 1 Exit Ticket (M-A2-3-1_Lesson 1 Exit Ticket and KEY)
- copies of Polynomial Functions Graphic Organizer (M-A2-3-1_Polynomial Functions Graphic Organizer)
- Students will use the Think-Pair-Share activity to demonstrate their individual understanding of the relationship between the polynomial equation and its graph. They will also analyze each partner's representations.
- To complete the Lesson 1 Exit Ticket, students will classify, enumerate, determine power, and count the number of roots for four polynomial expressions.
Active Engagement, Modeling, and Explicit Instruction
W: This lesson teaches students to classify polynomials based on the variable's biggest exponent and express them using standard form. The lesson also demonstrates the relationship between the degree of the polynomial and the number of roots in the polynomial equation.
H: The initial problem set reinforces the necessity of distinguishing between like and unlike concepts. This is a technique that most students are familiar with and may use to identify increasingly difficult and complex expressions, giving them confidence when working with polynomial functions.
E: The Polynomial Functions Graphic Organizer helps students break down expressions into their parts. The site also helps them learn how to reconstruct individual terms to correctly classify expressions and create meaningful representations of polynomial equations.
R: In Activity 5 (Pairs), students use their understanding of the link between polynomial degrees, like and unlike terms, to classify polynomial expressions. They must also consider whether each pair accurately identified and classified the polynomial.
E: The Lesson 1 Exit Ticket assesses students' comprehension of the relationship between exponents, degree of polynomials, and like/unlike words. Students must demonstrate their understanding by naming the polynomial, counting the terms, expressing the degree, and noting the number of roots in the equation.
T: Group and partner work are utilized to foster collaboration among students. The emphasis should be on explaining mathematical ideas using vocabulary phrases that are specific to the subject. The class necessitates accurate note-taking skills to improve the learning experience and create a useful resource.
Logical, Sequential, and Visual Learners: Some students may benefit from seeing a step-by-step explanation of how to combine like terms in polynomial expressions. While writing or pointing to the terms, use their appropriate names.
O: This lesson lays the groundwork for future topics on polynomials and their application to real-world problems. It starts with an activity that engages students by activating prior information and demonstrating where they will take their knowledge. Students are exposed to vocabulary that they will need for the easy tasks that allow them to investigate polynomials. Students observe the relationships between polynomial equations and their graphs and then build their own connections. Students have time to evaluate the class's topics, receive timely feedback, and learn how to apply this new knowledge in the following lesson.
After this lesson, students will grasp the definition of a polynomial and how polynomial functions can be used to describe real-world situations. They will understand the connection between their prior understanding of linear and quadratic functions and polynomial functions. They will understand terms like degree, binomial, and trinomial. In Lesson 2, students will expand on their previous understanding by learning how to express real-world situations with polynomial functions. Students will be able to write polynomials in standard form and classify them based on degree and number of terms. They will be able to graph polynomials in factored form because they will understand the relationship between the number of roots and the degree of the polynomial.
To begin the lesson, engage students' prior knowledge. Post the following problems on the board:
1. 3x + 4x
2. 5x² − 3x²
3. 7x + 2x² − x + 8x²
"What does it mean to combine like terms?"
Write the following on the board:
"If we can't combine the terms, we should arrange them in an order that makes sense. Does anyone have any ideas on how to rearrange the terms?"
If students do not offer any suggestions, ask, "How are books organized in the library? How are words arranged in a dictionary? What does it mean to arrange a list in 'descending' order? Try sorting the terms on the board in descending order."
Students will likely come up with the following:
5\(x^4\) + 4 + 3x² − x − 2\(x^3\) (put the coefficients in descending order)
5\(x^4\) + 4 + 3x² − 2\(x^4\) − x (put the coefficients in descending order, ignoring signs)
5\(x^4\) - 2\(x^3\) + 3x² - x + 4 (correctly put the terms in descending order)
Inform the students who wrote the final equation that they have just written a polynomial in standard form.
Distribute the Polynomial Functions Graphic Organizer (see M-A2-3-1_Polynomial Functions Graphic Organizer in the Resources folder). The following is what students should include in their graphic organizers:
Polynomial: A sum of terms with variables raised to whole-number exponents.
Here, "quadratic" refers to something to the second power (the Latin: quadratum indicates square, as the area of a square).
Coefficient: The result of multiplying a number by a variable; the leading coefficient is the number that comes before the leading term.
Polynomials can be categorized according to the number of terms they contain.
Zero degree: constant (for example, 5).
Monomial: a polynomial with one term (for example, 4x).
Binomial: a polynomial with two terms (for example, 4x - 3).
Trinomial: a polynomial with three terms. (for example, 2x² + 4x − 3.)
A polynomial is any term that has four or more terms.
Polynomials can also be categorized according to their degree (the largest exponent).
First-degree: commonly known as "linear" (for example, 5x + 1)
Second-degree: also known as "quadratic" (example: 9x² - 5x + 3).
Third-degree, commonly known as "cubic" (example: 4\(x^3\) + 2x² − x + 8).
Fourth-degree: often called "quartic,"
(for example, 8\(x^4\) − 5\(x^3\) + 6x² + x − 3)
Fifth-degree, also called "quintic"
(for example, \(x^5\) + 2\(x^4\) – 5\(x^3\) – \(x^2\) + 8x + 4).
Polynomial equations with degrees greater than five cannot be solved using methods other than approximation.
Polynomials are typically written in standard form, with exponents descending order (largest to smallest).
Standard Form: 6\(x^4\) + 2\(x^3\) − x² + 3x − 5
Nonstandard form: 9x + 3x² − 4\(x^5\) + \(x^3\) + 2\(x^4\)
Activity 1 (Auditory): Defining Polynomials
Students are required to compose two polynomials in standard form on an index card and submit them individually. Read aloud one polynomial at a time, and have students answer the following questions:
"Is it a polynomial?" (Some students may have written one incorrectly.)
"What degree is the polynomial? Identify it as linear, quadratic, cubic, quartic, or none of the above."
"How many terms does the polynomial contain? Identify it as a monomial, binomial, trinomial, or none of the above."
Activity 2 (Auditory): Writing Polynomials in Standard Form
Speak out loud the polynomials that aren't in standard form using the index cards and the ones that weren't used in Activity 1.
"Use the standard form to write the polynomial."
"What is the degree of the polynomial?"
"How many terms does the polynomial contain? So, what is its name?"
"Pair up and check each other's work."
Activity 3: Think-Pair-Share
Post one line graph and one parabola graph on the board (examples: y = x - 2 and y = (x + 1)² - 4). "With your partner, write the equations for each graph. Apply what you've learned today to write them." Ask pairs to share their answers and explain how they came up with their equations.
"Can anyone detect a link between the graphs and his/her equations? What does the equation indicate about the graph?"
Remind students that plotting the graph without a graphing calculator means substituting values for x, marking the associated f(x) as an ordered pair, and then plotting enough points to complete the graph.
If students are struggling, remark, "Look at your graphic organizer. What did you learn today about polynomials?"
After a few minutes, have one partner from each pair stand up. Provide the partner with this instruction: "Look at where the graph crosses the x-axis and look at the degree of the polynomial." Students will return this information to their partners and attempt to make the connection. Ensure that the quadratic function given as an example has two x-intercepts.
After a few minutes, select a few volunteers to share their opinion of the connections. Some students may have noticed that the degree of the polynomial equals the number of roots (x-intercepts) in the graph.
Activity 4: Think-Pair-Share
Before completely describing the topic, draw a graph of a cubic function and a graph of a quadric function on the board. "Find the individual roots (x-intercepts) of the graphs. Once you've written them, talk with your partner about the roots and degrees of each graph. How are they connected?" If some pairs need extra learning, pair them with a partner who understands the concept. As a class, partners can discuss their conclusions about the degrees of polynomials and the number of roots in graphs.
Activity 5: Pairs
In pairs, one student supplies the degree and number of terms, and the other student constructs a polynomial in standard form with the same degree and number of terms. The first student evaluates the work and either agrees or explains why it is wrong. When they come to an agreement, they switch roles.
Activity 6: Whole Class
Write the terms listed below on paper and tape them to the board.
"Let's build four polynomials out of these 18 terms. That means we will not combine like terms. Put the polynomials into standard form. You will come up to the board one or two at a time to advance one term. Those seated at their desks should remain silent and not help the students who are at the board. If there is no negative sign in front of a phrase, use the plus sign. Once each term is in a polynomial, we'll check to see if it's in standard form. If it is, we shall classify the polynomial based on its degree (e.g., linear, quadratic, etc.) and the number of roots."
During this activity, you can count how many students correctly said their term and how many did not, to see how well the class knows.
This game may easily be duplicated and repeated with new sets of terms, and the individual sections can be customized to meet the needs of individual students or a specific class.
Use the Lesson 1 Exit Ticket (M-A2-3-1_Lesson 1 Exit Ticket and KEY) to quickly determine if students comprehend the topics. Have students complete the table on the Lesson 1 Exit Ticket. "The number of roots can be used to find the degree of polynomial equations. In the upcoming lesson, we will use roots to determine the factors of polynomials. "
Extension:
Provide students with the following exercises:
1. (3x − 1)(x² + 4x − 21)
2. (−2x + 4)(x − 2)
3. (x + 3)(\(x^3\) + 2x² − x + 4)
Answer the questions below for each exercise:
A. What are the degrees of each polynomial?
1. Linear, quadratic
2. Linear, linear
3. Linear, cubic
B. Multiply the two polynomials with area models to simplify your answer.
1. 3\(x^3\) + 11x² – 67x + 21
2. –2x² + 8x + 8
3. \(x^4\) + 5\(x^3\) + 5x² + x + 12
C. What is the degree of the polynomial?
1. Cubic
2. Quadratic
3. Quartic
D. What should be the number of real roots of your polynomial?
Cubic: 3; Quadratic: 2; Quartic: 4
