Lesson 3

Introducing Polynomials

3.1: Which One Doesn’t Belong: What are Polynomials? (5 minutes)

Warm-up

This warm-up prompts students to compare four expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Launch

Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular item does not belong, and together, find at least one reason each item doesn’t belong.

Student Facing

Which one doesn’t belong?

A: \(4 - x^2 + x^3 - 4x\)

B: \(2x^4 + x^2 - 5.7x + 2\)

C: \(x^2 + 7x - x^{\frac13} + 2\)

D: \(x^5 + 8.36 x^3 - 2.4 x^2 + 0.32x\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

The purpose of this activity is to more formally name features of polynomials, building on student language. If not used by students, remind them of vocabulary to describe different polynomial features, such as coefficient, leading coefficient, and term. Tell students that a polynomial function can always be written as a sum of terms, each of which is a constant times a whole number power of \(x\). Introduce students to degree, the largest exponent on a variable in a polynomial, as one way polynomials are classified. It may be helpful to show students that expression A, a 3rd degree polynomial, can be written as \((x - 1)(x - 2)(x + 2)\), so even if an expression for a polynomial is written in factored form, the distributive property can always be used to rewrite it to more clearly identify features like the degree. More about how to do this will be addressed in future lessons.

3.2: Card Sort: Equations and Graphs (15 minutes)

Activity

A sorting task gives students opportunities to analyze representations and structures closely and make connections (MP7). There is no need to formalize these connections at this time, since the activity is meant as an introduction to the mathematical work ahead and an opportunity for students to develop some basic intuition for what polynomials can look like.

Students should not use graphing technology to identify matches, and should focus on the structure of the expressions, such as the constant term or the value of the expression at specific values of \(x\), in order to identify which expressions and graphs belong to the same polynomial function.

Launch

Arrange students in groups of 2. Distribute 1 set of pre-cut slips to each group.

Conversing: MLR8 Discussion Supports. In their groups of 2, students should take turns explaining their reasoning for how they matched equivalent polynomial functions. Display the following sentence frames for all to see: “These cards represent the same function because. . . .” and “I noticed _____ so I . . . .” Invite the listener to press for additional details by referring to the structure of the expressions, such as the constant term, the highest degree, or the value of the expression at specific values of \(x\). This will help students justify how features of the equations and graphs of polynomials can be used to identify equivalent polynomial functions.
Design Principle(s): Support sense-making; Maximize meta-awareness

Student Facing

Your teacher will give you a set of cards. Group them into pairs that represent the same polynomial function. Be prepared to explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are unsure how to begin, remind them that they can strategically eliminate graphs by, for example, testing specific \(x\)-values (such as 0) to see if the output of the function appears to match the \(y\)-value of the graph.

Activity Synthesis

Once all groups have completed the matching, discuss the following:

  • “Which matches were tricky? Explain why.”
  • “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

Conclude the discussion by asking 2–3 students to describe Graph g and Graph e. Tell students that Graph g is a quadratic (a 2nd degree polynomial), which has either one minimum or one maximum. Graph e, however, is cubic (a 3rd degree polynomial), and the graphs of 3rd degree polynomials and higher can change directions several times. In the case of Graph e, there is a peak at about \(x=\text-3 \) and a valley at \(x=0\). The \(y\)-values at those two points are called the relative maximum and relative minimum of the graph, respectively. The relative maximum of the graph is about \(y=18\) and the relative minimum is \(y=0\). The word relative is used because while these are maximums and minimums relative to surrounding outputs, there are other outputs that have greater or lesser values. Graph d is an example of a polynomial function with both a relative minimum at about \(x=\text-2.5\) and an actual minimum, sometimes referred to as an absolute minimum, at about \(x=1\).

Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students share their reasoning about their matches, scribe their thinking on a visible display. Annotate the graphs to identify key characteristics that come from student discourse, and also to label the graphs while presenting the new vocabulary terms of relative minimum and maximum.
Supports accessibility for: Visual-spatial processing; Conceptual processing

3.3: Let’s Make Some Curves (15 minutes)

Activity

The goal of this activity is for students to graph polynomial equations to develop their understanding of the different features the graph representing a polynomial can have and to begin to connect the structure of the expression to the shape of the graph. This activity is meant to be an informal study in which students experiment with different coefficients and degrees, preparing students for the work ahead and building fluency graphing and changing graphing windows as appropriate for the technology. For students who are more comfortable graphing, ask them to come up with new characteristics that they can challenge one another to find equations for.

Launch

Provide access to devices that can run Desmos or other graphing technology. 

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students of understandings they bring from earlier coursework relating to the form of linear and quadratic equations. Invite students to create a list of what they already know about the connection between an equation and the appearance of its graph, and to add to the list as they make predictions and test using graphing technology.
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

Use graphing technology to write equations for polynomial functions whose graphs have the characteristics listed when plotted on the coordinate plane.

  1. A 1st degree polynomial function whose graph intercepts the vertical axis at 8.
  2. A 2nd degree polynomial function whose graph has only positive \(y\)-values.
  3. A 2nd degree polynomial function whose graph contains the point \((0,\text-9)\).
  4. A 3rd degree polynomial function whose graph crosses the horizontal axis more than once.
  5. A 4th degree or higher polynomial function whose graph never crosses the horizontal axis.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

For each of the following letters, find the equation for a polynomial function whose graph resembles the given letter: U, N, M, W.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may not be sure how to begin graphing polynomial functions. It may be helpful to display empty frames, such as \(y=ax^2+bx+c\) or \(y=\text{?}x^2+\text{?}x+\text{?}\) for the 2nd degree polynomials, to help students get started.

Activity Synthesis

Select 1–2 students per question to share their equations. While students may have used some variety of “guess and check,” encourage students to describe any strategies they identified for changing equations to meet specific criteria. For example, a student may have noticed the difference between the overall shape of graphs representing polynomials with even degree versus odd degree and used that knowledge to figure out an equation for the last question. If possible, display all student graphs for each question on one set of axes for all to see.

Conclude the discussion by asking students what features they did not see when graphing polynomials. It may be helpful to remind students of what graphs representing some other types of functions look like that they have seen in the past, such as piecewise functions or arithmetic and geometric sequences. Possible observations include things like how polynomials, unlike some piecewise functions, have no corners or gaps.

Representing, Conversing: MLR7 Compare and Connect. As students share their equations for polynomial functions, call students’ attention to how specific features of the polynomial function, such as degree, intercepts, and specific points, are represented in the equation and graph. Consider asking, “How is the degree represented in both the equation and graph?” or “How is the \(y\)-intercept represented in both the equation and graph?” Listen for and amplify the language students use to connect a specific feature of an equation to a specific feature of its corresponding graph. This will help students use precise mathematical language to make connections between the multiple representations of a polynomial function.
Design Principle(s): Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Invite students to write down any questions they still have about polynomial functions on slips of paper. Collect the slips after some quiet work time, then display the questions one at a time for all to see and consider. When possible, invite students to answer questions or provide arguments for why a polynomial could (or could not) have a specific feature. Some possible questions:

  • “Could the graph of a polynomial have a loop?” (A loop means there is more than 1 output for a given input, which isn’t possible since polynomials are functions.)
  • “How many terms could an expression for a polynomial have?” (Any number of terms is possible.)
  • “Does \(y=3\) count as a polynomial? What degree would it have?” (This is a polynomial and we can write this as \(y=3x^0\) to see that it is a 0-degree polynomial.)

3.4: Cool-down - Identifying Features (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

A polynomial function of \(x\) is a function given by a sum of terms, each of which is a constant times a whole number power of \(x\). Polynomials are often classified by the term with the highest exponent on the independent variable. For example, a quadratic function, like \(g(t)= 10+ 96t -16t^2\), is considered a 2nd-degree polynomial because the highest exponent on \(t\) is 2. Similarly, a linear function like \(f(x)=3x-10\) is considered a 1st-degree polynomial. Earlier, we considered the function \(V(x)=(11-2x)(8.5-2x)(x)\), which gives the volume, in cubic inches, of a box made by removing the squares of side length \(x\), in inches, from each corner of a rectangle of paper and then folding up the 4 sides. This is an example of a 3rd-degree polynomial, which is easier to see if we use the distributive property to rewrite the equation as \(V(x)=4x^3 - 39x^2 + 93.5x\).

Graphs of polynomials have a variety of appearances. Here are three graphs of different polynomials with degree 1, 3, and 6, respectively:

Graph of first degree polynomial on coordinate plane with no grid, origin O. Horizontal axis from negative 10 to 8, by 2's. Vertical axis from negative 30 to 20, by 10's.  The function is decreasing and linear.
 
Graph of third degree polynomial.
Graph of sixth degree polynomial.

Since graphs of polynomials can curve up and down multiple times, they can have points that are higher or lower than the rest of the points around them. These points are relative maximums and relative minimums. In the second graph, there is a relative maximum at about \((\text-3,18)\) and a relative minimum at \((2, 0)\). The word relative is used because while these are maximums and minimums relative to surrounding points, there are other points that are higher or lower.

In future lessons, we’ll explore connections between equations and graphs of polynomials and learn more about how the degree of a polynomial affects the shape of the graph.