So another question we asked students was this one…
The times and positions of an object are given in the table below. What’s a good estimate for the speed of the object at the instant t=2?
| t(s) | x(m) |
| 0 | 2 |
| 1 | 4 |
| 2 | 10 |
| 3 | 20 |
| 4 | 34 |
| 5 | 52 |
Here’s what I’m interested in knowing…
First
If you had to write a multiple-choice version of this question, what options would you give students, and why? In other words, what approach or thinking underlies each choice? Are any of your choices degenerate?– meaning that the answer could point to very different thinking/approaches?
Second
What’s something you would want to ask a student as follow up (either generally or to particular answers)?
Teach. Brian. Teach. 
My students would approach this by calculating
, using the double interval as we practiced often. I’d probably prefer that they first graph it and think about why the double interval represents a better estimate than say 
. I don’t really love giving multiple choice options like this, but I think one distractor I would definitely put in would be 5m/s, to see if students can avoid x/t as a solution. In fact, I’d probably want to ask a follow up question as to why it isn’t x/t—and maybe even make the argument that since 5m/s is pretty close to 7m/s, shouldn’t this be good enough? Is there a case where x/t would produce a result that is way off of what the instantaneous velocity should be?
Yeah I’m not a big fan of making this question MC, necessarily, but I thought it’s an interesting exercise to force me to think about what approaches / what answers. And yes, simply taking x/t is biggest thing we saw. In our initial version, we had initial position be zero. So we are wondering how many students will now say 4 m/s.. finding the average velocity, vs. getting 5m/s.
I like your follow up question about asking why x/t is wrong. I also think, asking students if their estimate is likely to be an over or under estimate is a good question, especially if they look at the interval preceding or following t=2 seconds.
Another approach, instead of the surrounding interval method, would be to average the average velocities before and after… in this case, it would give the same answer, but it wouldn’t always.
I’m also curious what, if anything, changes if we start the clock running not at t = 0.
oops. That last comment was from me. I was logged into the wrong wordpress account.
Follow up:
Students who chose 8 m/s for this question scored in the 76-99 range on final exam
Students who chose 5 m/s for this question scored in the 53-79 range on final exam
These two answers account for 60% of students, with other 40% scattering among 4m/s, 6m/s, and 10m/s.
Smallish sample size but, it’s still a statistically significant difference (P = .005).