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.

Invite each group to share one reason why a particular set of three go together. 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 of which three go together, attend to students’ explanations and ensure that 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," and to clarify their reasoning as needed. Consider asking:

• “How do you know ?”
• “What do you mean by ?”
• “Can you say that in another way?”

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 without a graph, so technology is not an appropriate tool. During the discussion, students use two equivalent forms of the expression to identify features of the graph, including intercepts.

Monitor for students who use these different strategies when determining the degree and leading term of the polynomial in the first question:

• Rewrite the equation in standard form. Some students may complete the multiplication in different ways. For example, given three factors , , and , some students may multiply , while others choose .
• Use diagrams to organize their work.
• Multiply  .

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 can be helpful to have a way to visually organize the work. Use this activity if students would benefit from additional practice using diagrams to organize their multiplication.

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

This activity asks students to multiply two factors together to rewrite polynomials in standard form, scaffolding the level of difficulty to prepare students for the upcoming activity. 

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

In the digital version of the activity, students use an applet to visualize the effect of multiplying a polynomial by a constant, . The applet allows students to experiment with different values of . 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, displaying the applet would be helpful during the Activity Synthesis.