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I sometimes encounter students who ask questions like 'Why are we learning this if it won't be on the exam?' If there is time to spare I like to teach interesting applications or natural extensions of the material we have covered.
How do you respond to such students?
This is really two different questions: (1) applications of the standard material, and (2) additional material that extends the standard material.
For applications, my answer, addressed to my students, would be something like this. We have just learned math topic X, and I've presented a couple of its applications, A and B. These applications will help you to understand X better, will help you to understand why we would care about X in the first place, and may come up in later coursework in your major. You ask whether A and B will be on the exam. Applications on the exam could include A, B and C, where C is some application you've never seen. If you actually understand what the material means, and have seen a couple of applications, then you should also be able to figure out new applications for yourself. If you can't do this, then you don't understand the material.
For additional material, it really depends on what your motivation was. For example, if you're teaching first-year calculus but you want to introduce some vector calculus, the answer is probably that you just shouldn't. If the motivation is that it's super fun, then present it that way. Say, literally, that that's why. In general, make sure you can give a clear answer to the question and then preemptively answer it when you introduce the material.
Well, for me it would be a big deception if that happened to me, provided that the students didn't actually have reasons to complain. If so, avoid teaching this "extra material":
So, if it is not any of those cases, or similar, then yes, the situation must change. I assume we're talking about lazy / unconcious students who just want to pass, and they don't really care about learning or not.
So here my advice. In short, it is: "okay, you won't need it for my exam, but next year you'll be using this. The teacher won't have time to explain this, and (s)he will go very fast through this, so you better understand it now so that you can follow he class".
But it is important that you must transmit how much one suffers when one gets lost in class. The suffering of being in your seat and ask yourself "what am I doing here?" And you can't leave. That's stressing and agonizing. And not only you have a bad time in class. Then you don't understand anything. Since you don't udnerstand it, lessons go by and you keep at the start point, so you can lose a whole year. Make them imagine and "live" the situation. That's what I'd do.
EDIT: I didn't have much time and I left the answer like this. Now, re-reading it looks quite terrible . Of course, you should also do the opposite: transmit how satisfactoy is to learn something new, which helps people make sense of everything. Connecting concepts is always good. Those times when "suddenly everything makes sense" are enormously satisfying. You must transmit all this as well.
If you're going to teach it, than include it, or a credible threat of it (meaning sometimes on the exam).
Furthermore, if the class is not nailing the regular topics (high percentage of strong grades) than perhaps you shouldn't do extra topics but drill more on the ones that aren't strong yet. Remember drill is more effective than lecture.
It's not about how much you can "get through" but about how well you teach the majority of the students the topics they need to know. It's like sports. Fundamentals rule.
First, any time students ask "Is this on the test?", it makes me reach for a vomit bag. But my reaction is irrelevant compared to the student experience.
Most students [consider themselves] busy and [in their minds] you've already decided that the material is not important enough to assess for grading. This incentive of grading is, for many students, the one true signal of what is important enough work on. Once students know something isn't graded, they will turn their attention to friends, family, health, other classes, jobs, money, relationships, housing, etc. and other areas of life that they actually perceive as important.
Analogous situations:
You call customer service and are told by an automated recording for 2 straight hours, that your call is very important. How much do you care about such claims of importance relative to the fact that the company doesn't hire (incentivize) enough people to work in customer service?
Your boss says: "From now on, everyone in the company spends 10am-12pm every Monday, Wednesday, and Friday working on project X, under my direction. You're not going to get paid for those hours but I expect you to show up cheerfully and work as hard as you can anyway. Why? Because we don't just work for pay! We work for many other reasons, such as fun, learning, and natural interest. Come on, guys, be intrinsically motivated! Like me! Project X is cool! As a bonus, some of you might find the experience with project X useful someday. [Edit: Besides, I'm not sure how to measure your productivity at all for X, so there's another reason for me not to pay you for your labor.] Anyone who doesn't want to put in those hours is just lazy and instrumentally motivated and that's a big problem."
