## Developing The Question: Bike Dots

Let’s look at an example of developing the question versus rushing to the answer.

First, a video I made with the help of some workshop friends at Eanes ISD. They provided the video. I provided the tracking dots.

To develop the question you could do several things your textbook likely won’t. You could pause the video before the bicycle fades in and ask your students, “What do you think these points represent? Where are we?”

Once they see the bike you could then ask them to rank the dots from fastest to slowest.

It will likely be uncontroversial that A is the fastest. B and C are a bit of a mystery, though, loudly asking the question, “What do we mean by ‘fast’ anyway?” And D is a wild card.

I’m not looking for students to correctly invent the concepts of angular and linear velocity. They’ll likely need our help! I just need them to spend some time looking at the deep structure in these contrasting cases. That’ll prepare them for whatever explanation of linear versus angular velocity follows. The controversy will generate interest in that explanation.

Compare that to “rushing to the answer”:

How are you supposed to have a productive conversation about angular velocity without a) seeing motion or b) experiencing conflict?

See, we originally came up with these two different definitions of velocity (linear and angular) in order to resolve a conflict. We’ve lost that conflict in these textbook excerpts. They fail to develop the question and instead rush straight to the answer.

BTW. Would you do us all a favor? Show that video to your students and ask them to fill out this survey.

This is a series about “developing the question” in math class.

Featured Comment:

Bob Lochel, with a great activity that helps students feel the difference between angular and linear velocity:

I keep telling myself that I would love to try this activity with 50 kids on the football field, or even have kids consider the speed needed to make it happen.

Without some physical activity, some sense of the motion and what it is that is actually changing, then the problems become nothing more than plug and chug experiences.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Bob Lochel

August 26, 2014 - 10:43 am -

Dan, one of my favorite activities from my trig days are “trig whips”, an activity my colleague now uses all the time with all levels. The lessons, resources, and some video are on my blog: http://mathcoachblog.com/2012/10/02/experiencing-linear-and-angular-velocity/

I keep telling myself that I would love to try this activity with 50 kids on the football field, or even have kids consider the speed needed to make it happen.

Without some physical activity, some sense of the motion and what it is that is actually changing, then the problems become nothing more than plug and chug experiences.

2. #### Dan Meyer

August 26, 2014 - 1:48 pm -

@Bob, love that activity. If I didn’t feature it for Great Classroom Action, that’s a huge oversight. Pushed it to the top post.

3. #### ash

August 26, 2014 - 2:51 pm -

Great video.
To take a different aproach, what about the concept of understanting gear ratios?
Judging by the low cadence in the video, I’m guessing the rider is in 53 tooth chainring on the front and possibly an 18 tooth sprocket at the back. 53:18 or 2.94 – ie dot A (the wheel) rotates 2.94 times for every rotation of dots B or C (the crank).
Given the rear wheel is 700c, the diameter including a standard 23mm tire, ~ 27″. we can calculate distance the bike would travel for the ratio of 53:18.
Intriguing is a discussion about why we have gears and the effectiveness of using this gear at different speeds and on different inclines/descents.

4. #### Howard Phillips

August 26, 2014 - 2:52 pm -

There is a serious confusion between “speed” and “velocity”. Speed is a scalar, a simple number, and velocity is a vector quantity, with magnitude and direction. The two textbook samples just ignore this. I wonder whether any poor student would get that the average velocity of my house calculated between 9am yesterday and 9am today is zero. (It hasn’t gone anywhere!!).
Angular velocity appears to be used in place of angular speed or speed of rotation. That is, of course, rotation about the axis of rotation.
I like your bike example, especially as point D has a speed (constant) and an angular speed (not constant). This should cause some big discussion.

There are other horrors in the book excerpts. Does anybody want to express the angular speed of the earth in rads/sec, I always thought it was 1, in sensible units.

As for examples, a pre quartz clock is a minefield. I think they still make them in China!

5. #### travis

August 26, 2014 - 6:35 pm -

Is there a way to make the ‘splash shot’ [the first frame] be bike-less?

August 27, 2014 - 4:16 am -

These are the folks in my department! I am teaching linear and angular velocity today and was going to show this video. I’ll give the kids the link to your questions and have you answer if you provide me with the student feedback.

I think I’ll also develop a google form of my own just like yours.

7. #### James Key

August 27, 2014 - 8:11 am -

Love the video and the whole series. Perhaps a small improvement (in the spirit of “developing the question”) would be to show the bike with no dots, and ask *the students* what we could pay attention to. Maybe you lead with, “We’re going to be learning about things that move. Some things move in a straight line, and other things move in a circular pattern. Other kinds of motion are possible, but these two in particular are especially easy for us to study. With that in mind, watch this short video clip, and be prepared to tell me — what things are moving in a circular pattern? Anything moving in a straight line?”

The intent here is for students to identify the dots *themselves,* and then you can be like, “Here are some dots that I thought would be neat to watch for our lesson today.”