Here are some more student responses to this question, which I posted before. Each of these has interesting nugget to ponder over.
If it only took 3 seconds for the ball to drop 45 meters, it would make sense for the ball to move 15 meters every second. However, an object cannot fall faster than 9.8m/s if it has simply been dropped.
Because 15m/s is the average speed. The truth is that while this object is being dropped it accelerates 9.8m/s*s every time so the object speed probably reached a little more than 15 m/s at some point.
It would be assumed this is correct because 45m divided by 3s gives you 15m/s. This is incorrect because acceleration is calculated in m/s/s and therefore every second the ball falls its it accelerates 9.8 m/s so by the time the ball hits the ground it is traveling 58.8 m/s, traveling 9.8m/s after the first second, 19.6m/s after the second second, and 58.8m/s at the end of the third second.
A person would think that because they simply would divided 3 seconds by 45 meters. The reason this wrong because the question is asking for average speed. The average speed is the total time distance travel divided the total time.
One might this that the speed is 15 m/s because velocity is equal to the distance divided by times and 45 divided by 3 is 15. I’m not sure as to why this isn’t true but my guess would be that since the ball is being dropped it is in free fall which would means the acceleration of the ball is constant. Also, another thought is that since the object is falling in a downward position the acceleration would be negative so this could also depict a negative speed. But, honestly, I’m not sure. Please explain more in class.
Teach. Brian. Teach. 
I’m super confused by the third student’s response. They seem to be totally understanding it, until all of a sudden they say the ball goes from 19.6 m/s to 58.8 m/s in one second. What thinking led to that?
That’s such a great question. I love that you are asking it.
I think they multiplied 19.6 by 3… Maybe they were thinking, so the speed gets up to 9.8, and then doubles to 19.8, and then it triples to 58.8.
Okay, so now I see it mathematically, but I’m still wondering what belief about physics led them to do that? Or, I guess I’m really wondering whether they actually understood what they said when they wrote “every second the ball falls its it accelerates 9.8 m/s”, or were they just parroting something they remember hearing you say as you talked about acceleration in class? Because it seems like if they did understand that statement, they would never make the mistake they did. Or maybe they do understand it, and the multiplying thing was just a silly mistake that I’m reading way too much into. I guess that’s the kind of thing you just need a more prolonged interaction with the student to diagnose.