Lesson 6

This warm-up prompts students to compare four equations. 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. In particular, encourage students to be clear about any connections they make between the equation and what the corresponding graph must look like. 

Monitor for any students who use the distributive property to rewrite the first equation in standard form.

Arrange students in groups of 2–4. Display the equations 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.

During the discussion, ask students to explain the meaning of any terminology they use, such as constant term or coefficient. Also, press students on unsubstantiated claims.

If not brought up during the discussion, select previously identified students to share how they rewrote the first equation, including any organizing techniques they used, such as a diagram.

In an earlier activity, students graphed the function describing the relationship between the volume of a box made from a single piece of paper and the side length of the squares cut from the corners of the piece of paper. In this activity, students identify features of a similar polynomial function from the context and expression alone without a graph, so technology is not an appropriate tool during the activity. During the discussion, students use two equivalent forms of the expression to identify features of the graph, including intercepts.

Monitor for students using different strategies to rewrite the equation in standard form to share during the discussion.

Some students may not be sure how to identify the degree or leading term of the polynomial written in factored form. Encourage these students to try writing the expression in a different form, referring back to the discussion of the previous activity.

The goal of this discussion is for students to see some of the advantages and disadvantages of the two main forms polynomials are written in: factored form and standard form.

Begin the discussion by selecting 2–3 previously identified students to share their strategies for rewriting the equation in standard form. Depending on student familiarity with the distributive property, discussion of how to find the degree and leading term may need additional time and practice for students to explore different strategies and find one that works for them, such as using a diagram to stay organized while multiplying.

Display \(V(x)=(14-2x)(8.5-2x)(x)\) and \(V(x)=4x^3 - 45x^2 + 119x\) for all to see. Invite students to choose one and then use it to identify a feature of the graph of the polynomial until all major features (such as intercepts) have been identified. If possible, add these features to a set of scaled axes throughout the discussion and then sketch in the cubic curve of \(y=V(x)\) at the conclusion as a small preview of the work students will do in future lessons investigating the general shape of polynomials with different degrees.

This activity is optional practice that not all classes may need. If students struggle with multiplying in the previous activity, this activity may be useful.

This activity prompts students to notice the structure that relates expressions in factored form to their equivalent counterparts in standard form. When translating between these structures, it’s very helpful to have a way of staying organized. In this activity, students see one such way of staying organized: diagrams. 

When working with negative numbers, such as linear factors of the form \((x-n)\), it is helpful to think of subtracting \(n\) as adding \(\text-n\), and labeling the diagram accordingly. For example, for the last expression, \((x-1)(x-7)\), one side of the diagram should be labeled with \(x\) and -1, and the other labeled with \(x\) and -7.

Depending on students’ needs, not all of the following examples may be necessary. The first example is a completed diagram and the multiplication it represents. Ask students where they see multiplying in the diagram and how they could use it to find the result of the multiplication. The next diagram has not been filled in. Students should say which expressions are being multiplied (\((x-6)(x-2)\)) before filling in the diagram. The third diagram is the completed version.

Arrange students in groups of 2. Ask them to work on the first problem independently and then check answers with their partner.

Invite students to share their diagrams. If some students used methods other than diagrams to multiply their partner’s expression, invite them to share and explain their method.

In this activity, students graph two related cubic polynomials to investigate the effect on input-output pairs of a function when the function is multiplied by a constant. Students are also invited to use mathematical language such as relative minimums and maximums to describe with precision (MP6) as they say how the features of the graphs are similar and different. They use two equivalent forms, standard and factored, to find properties of the functions.

This activity also asks students to multiply three factors together to rewrite polynomials in standard form, increasing the level of difficulty from the previous activity. Monitor for students who complete the multiplication in different ways. For example, given three factors \(A\), \(B\), and \(C\), some students may multiply \((A \boldcdot B)(C)\), while others choose \((A)(B \boldcdot C)\). Students should understand that order of multiplication with polynomials, like that of integers, does not change the product.

This activity works best when each student has access to devices that can run the Desmos applet, because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, display the applet from the digital version of the activity for all to see during the synthesis. 

Some students may be unsure of the notation used to describe the graphing windows in the first problem. It may be helpful for them to make a sketch of the axes and mark off points before setting a window size on their graphing technology.

The purpose of this discussion is for students to understand that multiplying by a constant \(k\) causes all output values of a function to be \(k\) times farther from the horizontal axis, which in turn means \(k\) has no effect on the zeros of a function.

Begin the discussion by selecting 2–3 previously identified students to share how they rewrote \(f(x)\) and \(g(x)\) in standard form. Display different methods used for all to see and compare. If all students multiplied the factors in the same order, demonstrate how multiplying in a different order results in the same product.

Ask students, “How can you identify the constant term without multiplying out the entire expression?” (With an expression like \(\frac12 (x-2)(x+3)(x-7)\), the constant term is 21 and comes from \((\frac12)(\text-2)(3)(\text-7)\).)

If not brought up during the discussion, it is important to note that all three polynomials have the same degree and the same zeros, yet the three have distinct outputs for all other inputs. The idea that knowing the degree and zeros of a polynomial is insufficient for identifying a specific polynomial will be revisited in future lessons. If time allows, ask students to write an equation for a fourth function with the same degree and zeros as the first three.