Experience First

In this rich task, students must reason about what it means for a polynomial to have a certain factor and how this relates to the x-intercept and the remainder when dividing by that factor.

To prepare for this lesson, it is important to anticipate student responses and strategies. This will help you ask more purposeful questions as you monitor groups. We use Peg Smith and Margaret Stein’s 5 Practices for Orchestrating Productive Discussions as a framework for facilitating this lesson. Think about which correct and incorrect strategies students might use to solve this problem and what kinds of assessing and advancing questions you can ask to uncover student thinking.

When you introduce the task, emphasize that you are excited to see all the different ways that students are thinking about the problem. Express curiosity about students’ strategies. Give students at least 5 minutes of individual think time before allowing them time to work in groups. This is important so that students can make sense of the problems for themselves and contribute to their group instead of just immediately taking up the ideas of others.

When you are in the monitoring phase of the lesson, be looking for students that use the remainder theorem and the division method. If other methods are used, consider whether sharing them will help students make progress towards today’s learning goals. As you monitor groups, be thinking about what student ideas you want to surface during the whole class discussion. Decide which students you would like to explain their work in the debrief and in what sequence. You may choose to have a student present the division method first, whether or not they were able to complete the problem (working with the unknown coefficient is tricky in figuring out what term goes in each box), to highlight the possibility of an easier, more efficient strategy.

If no group uses the remainder theorem method, present the idea as having come from a student in a different hour or previous year.

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Formalize Later

The whole class discussion is the key component in consolidating students’ understanding. Have students revoice each other’s ideas to encourage good listening and to make sure all students have access to the ideas presented. Make sure the class works to understand a method before they critique it. This could sound something like: “Without saying if you agree or disagree, can someone restate Lal’s rationale for evaluating the function at x=3?”

When multiple solutions strategies are presented, have students compare and contrast them. Have groups decide which of the presented strategies they think is best and why.

By the end of the discussion, all students should be able to verbalize why when (x+3) is a factor of a polynomial, that x=-3 is a zero of the function and why c has to be chosen in such a way that dividing the polynomial by (x+3) gives no remainder.