4-Column Quadratic Data Tables

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:

• “Well, we haven’t learned this yet, so just see what you can figure out.”

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:

• The left-hand column represents the x-coordinates for each function, while the other columns represent the y-coordinates.
• The middle two columns both increase at a constant rate, which is why the graph shows two linear functions.
• The fourth column is the product of the two middle columns, so it shows a parabola.
• The x-intercepts of the lines correspond to the x-intercepts of the parabola because if one of the factors is 0 the product will be 0 as well.

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.

Teaching Notes:

• Note that there are two problems here, one in which both the lines have a slope of 1 and one in which they do not. I ask most students to focus first on the problem in which the lines have the same slope and the extension is for students who tackle this problem easily. Either way, they are thinking about the same big ideas. 
• Students who work on the next level will realize eventually that the coefficient a in each form of the parabolic function rule is the product of the slopes of the two lines. It is interesting to discuss why this is true.

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.