When introducing my high school students to Calculus, I begin by asking them if they would like to see why the volume of a cone is what it is; they invariably respond “yes”. This leads immediately into some diagrams and discussion on what they already know and information that is missing. I tell them up front that the sum of squares formula will be needed; this is derived through telescopic sums, a rich task in itself.
A diagram of a cone is then sectioned into cylindrical slabs and, through further discussion, it is decided that a reasonable estimate for volume of the entire cone could be determined if these slabs were very thin, there volumes calculated and then summed. The basic idea behind integration has now been considered and off we go.
By the end of day 2, students have seen how the discrete representation (Riemann sums) gives way to its continuous counterpart (integration) through use of the power rule. We work through several examples with uniform subintervals and then develop the power rule for integration using non-uniform partitions on f(x)=x^n. The FTC follows shortly thereafter; students are then given some time to practice finding area by setting up simple integrals and working through the process.
Simple differential equations soon follow, as shown below.
Rates of change (and related rates) can very easily be folded into the fabric of our work from thoughtful consideration of the topic shown above. This is a necessary outcome as the task of deriving integrals of logarithms, trig functions, etc. is too demanding at this level. Derivatives of these functions are, however, well within the grasp of high school students. Once these are derived, the inverse nature of integrals and derivatives can be exploited to continue on our little journey.
I realize that this approach isn’t for everyone but it works for me and my students seem to enjoy their work as well.
That is all for now. Thanks for reading.
samuelson mathxp 