Wednesday’s Lesson — The Trickster Squirrel

by George Collison

The concept of function, which links a change in one variable to a corresponding change in another, is key to understanding and applying mathematical models to the real world. By studying and comparing functional relationships in different representations, students gain access to some of the deeper ideas of secondary mathematics, including slope as rate of change.

Getting beyond rise over run

The Trickster Squirrel activity focuses on the graphical definition of a function using the rate of change of distance over segments of time. Students shift attention from seeing the graph as a static image, or perhaps a trajectory, to seeing the two variables—time and distance—and rate of change as interconnected.

Democratizing big ideas

Jim Kaput of the University of Massachusetts, Dartmouth, relentlessly argued that what tradition calls pre-calculus and calculus represents a collection of ideas formed by historical accident. One need not wait for the formal symbolic machinery of calculus to approach the mathematics of change. The vast majority of our students are, by this approach, blocked from access to very important ideas for them and for our society. To ensure equity of access, a thread on functions and rates of change should be integrated into the curriculum from early grades.

Getting to know the Seeing Math™ Qualitative Grapher

The Seeing Math Qualitative Grapher, like Kaput’s SimCalc, explores change through graphically defined piecewise functions. Traditional symbolic form may be difficult for many students, but expressing functions graphically is quite easy. Adding linear or curved elements of the graph in the Q-Grapher changes the way a virtual object moves. Units are not specified on either axis. The user defines meaning for either axis or the value of the grid lines. Access the Qualitative Grapher on the CD or at our website. (Note: You need Java 1.3.1 or higher to run the interactive.)

Using the Q-Grapher

The grapher opens with two line segments on the graph. The x-axis shows time in seconds. The y-axis shows the height of the object (a box) at each second. All graphs are continuous.

• Click play (the right arrow icon) and watch the motion of the object on the left. As it goes up or down, a red line moves across the graph.
• To add a straight or curved line to the graph, click the desired icon along the bottom of the screen.
• To change the position or angle of any line, drag its dark square "handles" (n).
• To remove the most recently added line segment, click the large arrow on the bottom far right.
• Use the Change Object pull down list button on the bottom left to change the box to a dollar or a billiard ball.

Use these features to become familiar with the software. Try adding a linear or curved segment and pressing the Play button. What happens when you change the angle of a line segment? When you delete a segment?

The Trickster Squirrel

Trickster Squirrel is a mischievous creature that lives in a tree overhanging a spot where students like to play. Trickster has a toolbox of cunning gadgets like springs, parachutes, and pipes that he uses to alter expected paths of balls thrown his way.

Challenge 1: A ball is tossed up vertically. At the top of its path, out of the students’ sight, Trickster Squirrel grabs it and holds onto it for two minutes. Trickster throws the ball up higher; as it comes down, it hits a student on the head. Use the Q-Grapher to animate this motion. Explain your reasons behind selecting each segment of the graph.

Challenge 2: Students are intrigued by the unexpected behavior of the ball. They toss another. This time Trickster catches it, holds it for one minute, attaches a small parachute, and lets it go. The ball floats down slowly. A student catches it, holds on a bit, removes the parachute, then throws it up to a lower height so the squirrel cannot get it. The student catches it again.

Challenge 3: Make up your own Trickster Squirrel activity. Describe the event using speed, not distances, to determine the graph. How does the shape of these graphs compare to the actual trajectories of the ball? List similarities and differences.

Getting beyond distance/time

Challenge 4: Create a narrative about a relationship between two quantities and animate it using the Qualitative Grapher. Do not use time and distance. Consider a “nervousness meter” that measures heart and breathing rates and imagine a student at a school dance. The student is nervous about asking (or being asked) to dance. Make up a narrative of the student’s movement around the room, approaching or avoiding situations. Make a graph of the nervousness meter readings vs. time using the Q-Grapher. Explain for each segment the rate of change of values on the nervousness meter.

In the classroom

Students can use the Qualitative Grapher to explore dynamic relationships in linear and non-linear functions. Where are there maxima or minima? Where are rates changing the most rapidly, the least rapidly, or not at all? As students experience graphically defined functions in the Q-Grapher, they will be better equipped to interpret graphs and move more easily to generalized and more formal definitions of mathematical functions.

George Collison (george@concord.org) is Senior Curriculum Author for the Seeing Math project.