The cubic functions two roots quick-thinking question is meant to be an open-ended launch that allows all students to bring something to the table. At the most basic level, some students might just come up with one function that fits. Some of my students still hadn’t made the connection between x-intercepts and factors, so they used computers to guess-and-check to find an equation that passes through the given points. Other students were able to make the connection to the roots by choosing the right factors, but didn’t know how to make the function cubic without adding another root. Still other students were able to make a generalization about all the different functions that could fit these requirements.
I asked students to think for a few minutes, then talk with a partner. I wrote many different functions on the board, including some that didn’t fit the requirements in particular ways, such as:
I want students to discuss the Zero Product Property and the key definition of a cubic function.
It is okay if not all students understand all aspects of this discussion, because the main investigation of the day will focus on these same questions. The purpose of this quick discussion is to get the ideas out there so that throughout the rest of the day, we can refer back to these conversations while students tackle the investigation.
I made two versions of the assessment because I wasn’t sure which one would make students think more, and I ended up trying both of them out in different contexts. Roots and Cubic Functions V1 turned out to be a good starting point for everyone—I ended up slicing this up into strips.
I assigned each student a partner who had a similar level of understanding of the content. I got everyone started with the first problem—I asked them to show sketches of their graphs on graph paper and to come up with a generalization as soon as they could. This flexibility enabled students to spend as much or as little time as they wanted on each problem. Some students spent most of the class period finding different functions that fit the requirements of the first problem before they were able to make a generalization while other students were able to make generalizations without even finding particular functions.
I continued to distribute the problems on strips and each pair of students accomplished a different number of problems, which was fine with me as long as they were focused and on track throughout the class period. I cared more about the quality of their generalizations than I did about the number of problems they worked on.
For students who finished all the problems in the first document, I asked them to formulate a generalization about the possible number of roots of cubic functions. Roots of Cubic Functions V2 asked them to do this.