Heights of Falling Objects

The purpose of the Warm-Up is partly for students to review key skills and partly to preview new material. I explicitly tell students to skip problems that they have fully mastered, and throughout the year I coach them to do this. Some students will spend lots of time on problems that have become routine for them, and they no longer need this practice, so they need to be coached to skip these problems. Other students skip problems that they don’t know how to do, so it is important to circulate and ask them to explain problems to you. Sometimes I ask students to do part of one problem just to make sure that they know how to do it. Ultimately, the goal is for students to assess their understanding of key problems and concepts each and every day and to use the warm-up as a tool to help them do this. Constantly facilitate these conversations with students by asking them how they are choosing which problems to work on.

The idea of today’s warm-up is that students will need to understand problem (2) and (3) in order to apply these skills in a new real-world situation dealing with falling objects. The first problem is tricky, however, and may take them some time. Encourage them to really think about how to set up a function for the first problem and have a stack of organizers available for students who like to use them (MP1).

Ask students to make sure that they understand problems (2) and (3) and explain to them that they will need both of these skills to be successful in today’s lesson. Give them lots of time to talk to their partners about these problems, and have some reference posters available showing how to determine whether a data table matches a quadratic function and also showing how to find the vertex of a the function.

When it seems like most students have had time to review these two key skills, ask them to find a new partner (or assign them a new partner if you want to rearrange the room a bit) and talk about problem 4. There is a lot going on in this problem, and it is great to take the time to ask students what the graph and the data table are really showing. What are x and y in this situation? What is happening to the height of the object as it is falling? Why does this happen? Discuss the context of the situation in some depth so that students can make more sense of the numbers and computations later (MP2). Students may or may not be able to find a function to fit this data.  It’s fine if they don’t, because that will be the focus of the day’s lesson.

The first question on this Exit Ticket may seem trivial, but producing precise explanation requires deeper understanding of many concepts (MP6). The goal is for students to be able to say that the function is not linear because the height does not change by the same amount over each time interval. Why doesn’t the height change linearly? This is because the object accelerates as it falls. (Students may have some prior knowledge of this from physics, but even if they don’t, this should fit with their intuition.)

The second question really gets at the idea of the vertex. The vertex (or maximum height) changes if we throw the object upwards rather than just letting it fall. How does this show up in the function rule? Ask students to explain their answers to this question using multiple representations (MP3).

The third question is asking students to make some connections between the real-world problem about falling objects and the more abstract problems about parabolas and vertices and different forms of functions (MP2). Though it may seem obvious, asking students to articulate this helps them develop better metacognition and enables them to think more broadly about the day’s lesson.