Teaching about LC filters
Your filter is an ideal resonant circuit - practically non-existent! Components, especially inductors are lossy and signal sources also have something, typically a series resistance. Finally there's a load in the output. It also takes something. The resulted circuit no more has a pole (=zero denominator) at the resonant frequency. If there's low enough resistance load resistor, this circuit can be a 2nd order low pass filter.

This filter pulls the signal source output on the knees at the resonant frequency if there's some resistance in the source and the output is unloaded. It's extremely important to understand it qualitatively - the LC series impedance is lowest at the resonant frequency and the resistance of the signal source makes with it a frequency dependent voltage divider. You can calculate what's left at the output, but you can see it also by simulating. 

You should do some simulations to see the effect of the various resistances. Even this site has a good enough simulator. 

Calculations with equations really are complex, but doable. Advanced filter designers  know some shortcuts how to use already calculated filter tables, how to transform them to other frequencies and how to change component values for different signal source and load resistances. The theory has been there already about 80 years. It's packed also in filter design software and printed to handbooks.

For teaching you maybe should show results and the methods qualitatively. Don't alienate people with heavy equation manipulations. More likely let them find something wanted with simulations. 

If  transfer function manipulation with complex numbers is the main subject, then simulation can be useful to reveal errors. 

You can calculate and plot transfer functions with Excel, but simulator can show the frequency response of a circuit without forcing one to write complex equations which are error prone.

Time domain analysis with simulators, Excel and formally with Laplace transforms can be interesting, but have you time for it? You know. My opinion is that you should show with simulations that too high resonant peak in the frequency response causes too much unwanted ringing when the signal has pulses.