This lesson builds on our work in yesterday's lesson. When students enter the room, I animate the Geogebra applet to attract their attention:
I ask the students, "what do you notice?" I tell the class to write down their observations with respect to how the purple line is moving between red and pink lines. Here are some images that reflect what they are looking at:
As the students record observations, and I will ask questions to focus their attention on the movement of the purple line. For example, I might say, "Is the y-intercept changing? What does this mean in the context of Mario collecting coins?" I expect that my students will make observations such as:
I try to continually return my students attention to the context of the problem. I like to ask them questions like,"What do the individual points represent?" I want them to be aware that each point tells you how much time has passed, and, how many coins he has at that point. I also ask if the points should be connected on the graph. This may be a point of contention. One could argue that we shouldn't connect points since Mario only collects individual coins and does not collect fractions of coins. I also like to ask if there are y-intercept values (or points on the graph) that can't apply to this problem. For example, "Does it make sense for the y-intercept to be negative?"
At regular intervals, I point out that this is "all part of the modeling process." Ultimately, since we are using an average rate to describe his coin collecting, we are modeling rather than precisely representing his coin collection.
Now, to launch this Follow-up Investigation, I show them an updated version of Mario's coin adventure:
I might not show the whole clip, since I plan to ask them a simple question:
What is different?
I want my students to notice that I have added milliseconds to our time measurement. The goal is to attempt to get a more precise measurement of his collection rate. I ask them to make predictions about what will happen with our data when we increase the accuracy of our measurement, typically they say it will increase the accuracy of our equation and prediction.
Today, they are in for a surprise! As quantum mechanics has taught us, zooming in on a more precise level can lead to more ambiguity and uncertainty.
Today's investigation begins with a data set with time measured in millisecond. I provide a list of data points and I ask the students to analyze it any way they can (a table, a graph, an equation, etc) in preparation for creating a model. Here is the data set:
My goal is for the students to read the table wither up/down or left/right and consider the changing slope in some way. As they explore the data I circulate and help students take notice of something quite surprising: Mario is slowing down.
Design Note: I spent a very long time creating this lesson, because I kept searching for time stamps that would lead to the correct prediction for the total number of coins. This was based on the assumption that Mario was moving at a constant rate (as he appears to be). However, the table clearly shows that he is slowing as he runs along.
Since this exploration is relatively open ended, I prepared some challenge problems in case students finish early. Here is the the spice rack I created for them: mario-spicerack.ppt
To summarize this lesson, we talk about the graph of Mario's journey and use either Desmos or Geogebra to highlight just how close to linear this data is:
As I display the graph and table given out in class and we share different ways of analyzing the data. This is mostly a review of slope and unit rate. As we talk, I am also only introducing the concept of line of best fit in a very informal way: It is a line that represents the data we have. I say, "If you have to draw a line for this data, where would it go?" I like to show how close his rate was to 4 coins per second (the linear regression gives a line with slope 3.9). I also like to talk about the y-intercept of this line (about 1.1). Since we have drawn this line to "fit" our data, the y-intercept no longer directly connects to the context of the problem. It has "moved" from the context and moved on the graph to better represent the data. I find teh y-intercept to be vital to further investigations with lines of best fit.
We might also talk about the Spice Rack problems. If we don't get to them today, we can work on them throughout the week.