I begin this lesson by asking my students to pair-share what they think a "mathematical formula" is. While they're talking I walk around making observations about which conversations might be interesting to share with the class. I look for things like discussions of chemical formulas or familiar mathematical formulas like the Pythagorean. I call time after a few minutes or when the talk dies down (or turns to other topics!) and call on teams to share what they've talked about. (MP6) I allow anyone who wants to share do so, while I serve as scribe and summarize their comments on the board. When there are no more comments I ask my students to review the board and look for any common themes in what they shared. Usually this will include at least a couple of the more common math formulas like the Pythagorean, area of a rectangle, and area of a circle. I ask questions to move my students toward a consensus about what these have in common and how we can define "formula". For example, I might ask what the symbols in each of the formulas represent to help my students recognize that formulas often include symbols that stand for something real and measurable.
When we have a clear picture of what a formula is, I ask my students to consider where these formulas came from. This can be a very interesting discussion because few of my students think of themselves as mathematicians and view most mathematics as something done be people very different from themselves. I generally serve more as moderator than lecturer for this discussion because I've found it helpful for my students to articulate their beliefs about math so that they can more easily accept that they can actually derive formulas themselves!
As the discussion winds down, I tell the students they will have several challenges today.
You will need copies of the Data Challenge and measuring tools for this part of the lesson. I tell my students the first challenge will be to collect data with a partner. Because I want them to discover a fairly clear relationship I have a few sets of data to suggest that will give good results. I distribute the Data Challenge and tell my students they have ten minutes to collect their data. As the teams are working I walk around giving encouragement and assistance as needed. (MP1) When all the teams are done, I ask them to complete the questions on the handout and be ready to share in about five minutes.
I randomly select teams to present their results and the equations they developed to represent that data. I encourage discussion and critiquing by the remainder of the class to help the students presenting strengthen their abilities to explain and justify their work. It also helps the students critiquing learn to make focused and appropriate comments. (MP3)
When everyone has presented I congratulate my students on creating their own formulas for some basic ideas - the area of a rectangle using length and width and speed using time and distance. I explain that they won't have to collect data for the remaining formulas. Instead they will be showing how each formula can be derived from the given information. I tell them that they get to continue working in teams but that I expect every student to complete their own copy of the Deriving Formulas handout so that they'll have it as a reference. As they're working I again walk around offering encouragement and redirection as needed. (MP1) When everyone is done, I collect these paper so I can review how my students are doing with the idea of deriving. I explain why I choose to have my students derive formulas they are already familiar with in my Formula 1 video.
To close this lesson I ask my students to pair-share what they think the value they see in being able to derive formulas, then summarize their thoughts on notecard, including which formula they liked the best and why. These summaries give me some insight into how my students are viewing the process of deriving and whether they are putting any value on that process beyond the immediate answer to a question.