Lesson 3

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.

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.

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.

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.

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

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.

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 \(x=1\).

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.

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

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.

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.