Students continue the thinking started in the Warm-up and explore what values of will make a factored expression equal 0. In this activity, students are using the structure of the equations to reason about the solutions (MP7), so technology is not an appropriate tool.

Identify students using different strategies to find values of . For example, some students may write down short equations to solve, such as , while others may skip writing that step and go directly to writing down the solution.

If students go from to , from to , or , consider asking:

• “Can you explain how you found the values of .”
• “How could substituting your answer for in the original equation help you check your work?”

Display the polynomial  for all to see, and ask "How could you figure out what values of make ?" (We could use the factors to identify the zeros of the function. We could graph  and see where the curve crosses the -axis.)

Then display a graph of for all to see, and check against their answer. The visual provided by the graph is helpful for understanding the relationship between zeros and factors that will be used throughout the unit and beyond. If needed, remind students that a zero of a function is an input that results in an output of 0. 

Select 2–3 previously identified students to share how they identified values of , starting with those who wrote short equations and continuing with those who worked out the solutions mentally. If not brought up in students' explanations, explicitly call out the common error that a factor of means that is a solution.

An important takeaway from this activity is that it takes only one of the factors equaling 0 at a specific input for the entire expression to equal 0. For example, if , then either or (or both) are 0. If needed, have a brief discussion about why this is true of 0 but not other numbers (for example, if , that doesn't guarantee that either or is 10).

In this partner activity, students take turns matching the equation of a polynomial written in factored form with either a graph or a description of a graph. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

All the polynomials in the activity are 3rd degree with similar factors, so students must pay close attention to signs as they match cards. In this activity, students use the structure of the equations to reason about the matches (MP7), so graphing technology is not an appropriate tool.

If students are unsure how to match cards 7 and 8, consider asking: 

• “Can you explain how you matched your graphs?”
• “How could using the factors to identify the zeros of the function, help you eliminate all possible matches but one?”

Once all groups have completed the Card Sort, 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?”

The purpose of this discussion is to help build student fluency identifying the horizontal intercepts of a graph of a polynomial from the equation of the polynomial written in factored form. Select 2–3 groups to share one of their sets of cards and how they matched the equation with a graph or description of a graph. Discuss as many different sets of cards as the time allows, making sure students connect features of the equation with features of the graph or description.