The start of my teaching career coincided with the mass introduction into math classrooms of handheld graphing calculators. I have learned and explored so much with these technologies that I cannot imagine teaching without these deeply inspiring tools. I first encountered Computer Algebra Systems (CAS) in 1999 when one of my AP Calculus students showed up with a newly-released TI-89. Since then, CAS have inspired, supported, and revolutionized my students’ thinking and my teaching even further. The following problem beautifully combines the powers of these technologies.

The standard equation form of a quadratic function is y = ax² + bx + c, where a, b, and c are real constants. Students often (usually?) learn through direct instruction that these graphs are parabolas and that a controls the parabolas’ vertical direction and makes them taller or shorter (although many textbooks describe this dimension as width). Parameter c moves the parabola vertically without changing its shape. Student exploration using sliders on graphing calculators can deliver these insights far more quickly and with deeper comprehension than direct instruction ever could.

Sliders now exist on many handheld graphers. In just ten seconds on the free online Desmos® calculator, I typed y = ax² + c and added sliders to create a student lab ready for exploration. You can play with the sliders on my creation at https://www.desmos.com/calculator/e31lbwcotb.

Imagine yourself a student seeing parabolic transformations for the first time and play with the sliders to change the graph. With some directed questions about when the parabola looks tall, moves up, points down, and so on, your students would be able to figure out and explain the transformations, gaining better understanding than any memorized textbook table could give.

Before seeing this approach, I had never considered the effect of the bx term on the parabola. Surely parabolas can do more than stretch and slide up and down. The linear term must have some effect, but it is not at all what you might expect. Make a prediction about how the graph of a standard form parabola changes when b varies. Using https://www.desmos.com/calculator/0fwuwn2ni0, play with the b slider and follow the path of the vertex. Are you surprised? Can you write an equation in terms of a and b for the vertex’s path?

Working in groups, my students are usually successful experimenting until they find the vertex’s path: y = ax² + c. You can enter this generic equation in Desmos and vary the sliders to confirm your hypothesis. A tighter proof uses algebra. My TI-Nspire CAS output is shown below. 

An equivalent result comes from the free online WolframAlpha® (http://www.wolframalpha.com/input/?i=solve+y%3Dax%5E2%2Bbx%2Bc+and+y%3D-ax%5E2%2Bc+for+%7Bx%2Cy%7D). Both CAS give two solutions: the y-intercept (0, c) and the generic location of the vertex (–b/(2a), (4ac – b²)/(4a)). That’s quite a find for students just encountering parabolic graphs for the first time!

Just as dynamic geometry software creates an exploratory environment for geometry classes, CAS and sliders on graphing software create an inspiring sandbox for algebraic thinking and discovery.

ADDITIONAL READING

Anita Schuloff’s “More on a Quadratic Function” (Reader Reflections, Mathematics Teacher Aug. 2014, vol. 108, no.1, p. 7) describes the experience of her precalculus students using The Geometer’s Sketchpad®. For further analysis using GeoGebra, see “Technology-Enhanced Discovery” by Chris Harrow and Lillian Chin (MT May 2014, vol. 107, no. 9, pp. 660–65).

CHRIS HARROW, casmusings@gmail.com, a National Board Certified Teacher, is the mathematics chair at the Hawken School in Cleveland, Ohio. He blogs and presents nationally on the educational uses of technology and on Computer Algebra Systems (CAS) in precollegiate mathematics. He is also the recipient of a Presidential Award for Excellence in Mathematics and Science Teaching.