My goal today, as usual, is to move from the concrete/real-world to the abstract/mathematical. In prior lessons, students developed three-column data tables based on the profit maximization problems and then used these tables to generate quadratic functions. The limitation in these cases was that all of the functions had an x-intercept of 0 because with a cost of $0 the profit would also be $0. Today, we will use four-column data tables to extend the same idea to a parabola that does not have 0 as an x-intercept.
My students will really want hints and instruction about the first data table—they ask, “What are we supposed to do?” I find that the open-endedness of this task actually makes students more willing to get started, because I answer all the questions this way:
When they say something like, “Why does this data table have 4 columns?” I say, “That is a great observation when you compare this to last week’s problems. What do you think is going on here?”
In a typical class, students come up with a lot of different ways of looking at these tables. Eventually, I tell them to treat the first column as the x-column and to find rules for each of the other columns in terms of this column. When students enter this data into the computer, they will find that it generates two lines and a parabola, which is shown on the graph on the second page. (Note: this is not the same data, but the same type of data). It is interesting to talk with the class about why the data looks like this way. Leaving the conversation loosely structured and informal allows students to stay engaged without worrying about whether their answers are right.
Teaching Note: Throughout this warm-up I try to encourage students who want more confirmation to find a way to check their answers using the online graphing calculator.
The Making Connections task is pretty quick, but I like to take the time to highlight some of the key connections between the data table and the graph. These include:
With some prompting, students should be able to generate these statements, or at least explain them, so it is worth taking the time to facilitate this by asking students to come up with ideas and then share some of those ideas with the class.
I introduce today's Exit Ticket by writing it on the board. I ask them to write 1 or 2 sentences using the key words listed. I ask students to write their sentences on a whiteboard. I find that this makes it easier for them to take risks and try out new words.
I ask students to share with a partner to make sure that their sentences make sense. Then, I read them as students leave. Strangely, they often end up writing sentences that really clearly summarize the big idea of the lesson, even with a more open-ended prompt.