Dan Meyer has posted a number of “What Can You Do With This?” activities on his blog. (Activities is probably too prescriptive a word; they’re more in the nature of prompts for student thinking, noticing, and wondering.) One of the first was the image below, which he made by superimposing frames from a video camera. I thought of it again for two reasons: I wanted to show the students in my Secondary Math Methods course a prompt that can elicit student curiosity and questioning, and I also wanted to investigate how such a prompt might be enhanced by tools created in Web Sketchpad.
I recognize that the decision to incorporate specific tools encourages some lines of student thinking while discouraging others, but that doesn’t worry me much, as I can imagine using this image in developing the idea that projectile motion is parabolic, using it again a bit later to see if we can detect the effects of friction on the recorded motion, and revisiting it yet again in calculus to approximate the instantaneous rate of change in the ball’s position. To preserve some open-endedness, the first page’s text makes an effort to draw students’ attention away from the tools and toward the image. It’s not hard to elicit ten or a dozen things that students notice, and a similar number of things they wonder. By allowing no more than a single noticing or wondering from each student, and not giving away your hand by approving or disapproving of any of them, you can quickly have contributions from more than half the class just in launching this lesson.
In case the images on the tool icons haven’t already given away my agenda in this use of the picture, I’ll make it explicit now: I envision using this as a function-fitting activity in conjunction with studying the standard and factored forms of the quadratic function. I might use it in advance, asking students to note how each of the parameters changes the shape of the graph, and explaining that we’ll investigate these function forms in detail in the next few lessons. Or I might use it as review, giving students an incentive to revisit what they know about these two forms.
The picture (by leaving the ball’s path literally in the air) and the student wonderings provide both a real-world context and useful motivation for seeing where the ball goes after the images end. To find out, students will need to fit a function to the picture.
Press the button at the lower right corner of the picture to go to page 2, start with the topmost tool, and see how well you can fit a graph defined in that way to the ball’s flight. (When you click a Web Sketchpad tool, the complete construction appears, with the tool’s given objects glowing. To use the tool, drag each glowing object to specify how it connects with your existing sketch. You can drag it on top of an existing object to connect it to that object, or you can drag it to empty space to make it independent of the already-existing objects.) If at any time you want to undo one or more tools you’ve used, just press the left arrow above the tool icons.
Pages 3 and 4 are identical to page 2, allowing you to try one tool on each page and compare your results by flipping pages. I recommend to students using only one tool on each page, as otherwise the pages can become cluttered and confusing.
As you work to fit each of the three function forms to the ball’s trajectory, you may notice that you can change the position and scale of the coordinate system, which is itself an important lesson for students: they are permitted to choose a coordinate system that makes their work easier.
One last thought: In an introduction to the standard and factored forms of the quadratic function, the third tool serves as a teaser, giving students a taste of some fascinating future mathematics. But later on this same sketch would be useful to introduce students to a unit on fitting polynomial functions to data.
Another last thought: If you would like to use this with your class and want to have a fourth tool that uses the vertex form, let me know via the comments.