Free-fall Circle Counting: I can’t imagine ever not doing this

Today in class, we did a few circle counts for free-fall. [I’m pretty sure I stole this idea from Frank Noschese, who probably stole it from someone else Sadie Estrella, who blogs about it here. I can’t seem to the find the found the blog post where Frank wrote about it].

Of course, counting circles is more from elementary math classes, but it works for college physics really well. So with free fall, students in a circle “count off” the velocity of a thrown object each second, using 10 m/s/s change.

Easy ones might be like 30 m/s, 20 m/s, 10 m/s, 0 m/s, -10 m/s, -20 m/s, -30 m/s.

Medium ones like  25 m/s, 15 m/s, 5 m/s, -5 m/s, -15 m/s, – 25 m/s

Hard ones like:  18 m/s, 8 m/s, -2 m/s, – 12 m/s

I had student “circles” look more like motion diagrams, so a big “U” where the positive velocities head one way and the negative velocities head back down. I didn’t spend too much time on this, but we certainly could have. The activity was pretty enlightening for students, but what this activity actually did was provide us with fodder for later sense-making. Our conversations all day were always referencing back to the counting activity.

For example, one clicker question later in day was what does velocity vs. time graph look like for an arrow shot straight into the air and falling back down. Our discussion of the different answer choices was immensely helped by talking about it in relationship to the circle counting. “We keep counting down by -10, the count never turns around”.  Another clicker question about the acceleration at the top, circle counting really helped. “There’s never a time where you don’t count down by “10”… even if you are the person who says zero… you had to count down by ten, and the next person had to count down by 10″. No one is allowed to say the same number the previous person said. The final question it helped with was, “A ball is dropped from a height of 45m and takes 3s to hit the ground. What’s the instantaneous speed just before impact?” Tempting to say 15 m/s.

Anyway, if you are thinking of trying this. I highly encourage it. It forces participation in good way. Formative assessment is pretty easy. It provides lots of opportunities to “stop” and discuss issues that come up.

Notes:

1. I think it’s probably important for students to say the units and to enforce it (and maybe even say “moving at a velocity of – 15 m/s”). Later students will start incorrectly count free-fall distance by 10m (each second). This came up in our 45 m in 3 seconds clicker question, where at least one student arrived at answer of 15 m/s… by counting down from 45m three times to 15m.  We had to talk about how that would be constant velocity of – 10 m/s.  Forcing them to say units or the phrase won’t eliminate this, but it can’t hurt?
2. Plan ahead on what examples you want to give, what possible “stop and discuss” issues might come up. A good time to discuss is after mistakes, or even after any long pauses. Students tend to pause more when going from positive to negative, especially if it’s like 2 m/s to – 8 m/s.  In skip counting good questions come from, “How did you know to say __ without counting?” The discussion can be about strategy, but it can also be about keeping it connected to what’s physically happening.
3. If doing it again, I would definitely do a few with the whole class, but then maybe give them examples to work out in smaller groups, and then “present” their motion diagram count. The whole class is nice, because it models and you can have those “stop”, but small group might work well, too.

Further extensions of this activity I didn’t do, but would consider for next time could include:

— Skip counting, like only saying the velocity every so often (having students count time in between)… so a drop from rest, that only counts velocity every 3 seconds would be First person says “0 m/s”, Second person just says (1s), Third person says  (2s),  But next person has to say” -30 m/s”… etc,… so (4s), (5s), then “-60 m/s. “

— Counting in intervals other than one second… I think 1/10th of a second is pretty important.

— Counting with other accelerations.

–[Much later]  Adding in distances:  Like for an acceleration of 2 ft/s/s from rest, you would count “0 ft/s”, the next person would count “2 ft/s” and take 1 ft step (because average velocity was 1 ft/s for that s”… then the next person would say, “4 ft/s” but would take 3 ft step, etc.  Our tiles are 1ft, so this wouldn’t be hard. It’s like a live motion diagram.