We should understand that our salah is a way of calling on Allah, and that we can call on Him anytime we want--and really, the more the better. Just making requests of Allah implies and fosters our belief and reliance on Him alone. Asking Him means we recognize His Hearing, His Knowledge and Power, while also acknowledging our powerlessness. So it seems that du'a is a fitting place to start.
We can call on Allah whenever we want, and know that He hears us.
And when My servants ask you, [O Muhammad], concerning Me - indeed I am near. I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me [by obedience] and believe in Me that they may be [rightly] guided.
Moreover, we have the benefit of Allah's response to our supplications. Firstly we might get the thing we are asking for. But if not, if Allah withholds it from us, then instead He may protect us from some other hardship we might have to face, or else it will become a reward for us on the Day of Judgment. So there's no reason not to make du'a--it's a win-win-win scenario.
But in case a person's du'a is not accepted, there may be a sin from which the person has not repented. In which case, the solution is to repent. Repentance is a topic on its own, but briefly this means the person should acknowledge the incorrect or sinful behavior in which he is or was engaged in, and correct it, ask forgiveness from Allah for it, and abstain from it in the future.
There is also an amazing story from the time of Moses and the Children of Israel. While they were wandering in the desert and suffering from a drought, Moses prayed to Allah for rain. But instead of rain it only got hotter and dryer. And then Allah revealed to Moses that one man among the Children of Israel had been sinning against Allah for some 40 years, so Moses was instructed to have that man leave the group before Allah would bring the rain.
So Moses went to the people and explained--he didn't know who the man was, but asked that the man, whoever he was, would leave. But imagine if you were that man? You knew your sins but nobody else around you knew. How hard would it be then to expose yourself? But yet if you didn't, then you and everyone else might die of thirst. And he recognized that all this time Allah had covered his sins. So he prayed, sincerely, to Allah for forgiveness and for Allah to continue to cover his sins. And then it started to rain.
But Moses asked Allah why the rain was coming even though the sinful man had not left. And Allah revealed that the same man had repented of his sins so Allah had allowed the rain for all of the people. Then Moses wanted to know who the man was, because of whom all of the Children of Israel had been first deprived of and then blessed with rain. But Allah refused, telling Moses that He had hid the man's sins for 40 years, so would He now expose them after repentance?
It's such a beautiful story and filled with loads of reminders for us today. That no matter how long we have sinned, Allah still accepts repentance. That our sins might not just be affecting us, and that if we find our prayers not being answered, perhaps we should try to correct ourselves and ask for forgiveness. And remember that we may call on Allah at any time for any need, and we should. It confirms and strengthens our belief in Allah and as an act of worship it can nurture our relationship with Him. And once we understand that, we can discuss salah with an appreciation of it as an act of submission and worship and so much more than a simple ritual.
There is much more that can be said about du'a--be patient for the result instead of hastily anticipating it, or declaring that Allah didn't answer; the best times to make du'a are after performing a good deed, like after salah or before breaking one's fast. The last third part of the night is also a great time to make du'a, as is while one is in salah during the prostration. Moreover, there are some etiquettes about making du'a in the first place which I think I will not include in this part of the class. Mostly, I don't have very much time, and I also don't want students to feel bogged down in "rules" when it comes to du'a. Is there something I'm omitting but I should include? Please let me know!