I think the fundamental idea is quite easy but easily obscured. The argument can be given as a fully general mathematically rigorous proof. Since understanding the shutdown decision amounts to understanding a universally true mathematical result, specific examples can only help in illustrating the argument (doubtful here) and as a correctness check for the proof. What one needs to focus then is the logical structure of the argument, because the math is very, very easy.
Students need to be able to multiply, divide, add, subtract, understand what it means for one number to be larger than another number, and know the definitions of the various cost curves involved. For the last part, it is important to remind students of the difference between cost curves and costs- the same way we drill them to know the difference between demand curves and quantities demanded. What students do not need is knowing that the marginal cost curve intersects the average variable cost curve at the minimum of the latter. Indeed, student need not know what marginal costs are and the argument works even when the cost curves are not differentiable and marginal costs are undefined.
So here is the argument:
Shutting down means producing a quantity of 0 instead of producing a positive quantity. If a firm cares only about profits, it will shut down when shutting down provides a higher profit than not. Let C be the total cost curve. The total cost of producing a quantity of 0 is the fixed cost, C(0). The variable cost of producing a quantity is the total cost of producing that quantity minus the fixed cost, VC(Q)=C(Q)-C(0). Profit is revenue minus cost and, since we are in a competitive setting, revenue is simply price times quantity, PQ, and profit PQ-C(Q). So shutting down is the right decision if the profit of producing 0 is higher than the profit of producing a positive quantity. The difference between producing some positive quantity Q and producing 0 is
Dividing by Q (which we can, since Q is positive), we see this difference is negative if AVC(Q)>P, so the firm should shut down in that case. That the minimum average cost is larger than P means that the average cost is larger than P at all quantities. So shutting down is better than producing any other quantity.
Realistically, not all students at all places will be able to understand this argument. But students who do not understand this argument, don't understand why it is optimal to shut down when the minimum of the AVC curve is larger than P. Students might be able to correctly tell you that shutting down is optimal when given an MC curve, an AVC curve, and a price P by equating the first two curves, solving for Q, plugging into the AVC curve and getting a value larger than P. But they will not understand why it is optimal to shut down when the minimum of the AVC curve is larger than P.