The early feedback on Function Carnival has been quite kind. To recap, a student’s job is to graph the motion on three rides:

- The Cannon Man’s graph is piecewise quadratic and linear.
- The Bumper Car’s graph is piecewise linear, which has thrown a bunch of students.
- The Ferris Wheel’s graph is sinusoidal.

But we found ourselves wondering if there were other rides and other graphs and other great ideas we had missed. So we’re kicking this out to you in this week’s Curriculum Confab:

What would be a worthwhile ride to include in Function Carnival? What would you graph? Why is it important?

I’ll post some great responses shortly.

**Previously**:

In the last confab, we looked at a math problem inspired by Waukee Community School District’s decision to let their buses idle all night. Molly showed us how to make a good problem out of it, and a lousy problem also. Great confabbing, people.

## 44 Comments

## Chris Lusto

February 3, 2014 - 10:09 amThe Zipper would be cool as a follow-up to the Ferris Wheel. (a) It’s oblong instead of circular, so that’s a bit of an interesting curve ball from the usual stuff. (b) The individual cars also rotate, so you have to account for two motions at once. DESMOSIFY IT!

## Jason Dyer

February 3, 2014 - 10:33 amI’d go for a Drop Tower with velocity vs. time (starts out free fall, so positive and increasing, but eventually drops to 0).

## Jonathan Claydon

February 3, 2014 - 11:06 amOn the upper end of the scale might be tea cups (circular motion-ception) and a pendulum ride, like those pirate ships that take a while to build momentum before whirling you 360º.

## Michael P

February 3, 2014 - 11:48 amDouble Ferris wheel? Trig teachers would love ya for it.

## John Golden

February 3, 2014 - 11:56 amAll good ideas so far. I’d like to see a roller coaster for both height vs time and speed vs time. It’s another one that the coaster would strongly suggest a wrong answer for height vs time, but the speed is quite challenging to graph.

On Jason’s idea I’d love to see one of those drop towers that goes asymptotic to the ground at the end.

## Mike S

February 3, 2014 - 1:14 pmI’d second Jason above with the Drop Tower – but I’d like it to start on the ground. What was missing from the Carnival is something that stays still for a few seconds. It’s nice to see a good horizontal line as time passes, motionless.

## Jason Dyer

February 3, 2014 - 2:36 pm@Mike S: Agreed.

It’d be great if one of these was an actual video, Dan Meyer style. Any reasoning behind the flash animations?

## Megan Schmidt

February 3, 2014 - 3:40 pmI’m not sure what the programming would be like for this, but I think lots of excellent questions could be asked about the Mad Tea Party ride or its carnival equivalent.

http://en.wikipedia.org/wiki/File:Mad_Tea_Part_WDW.jpg

## Jim Hays

February 3, 2014 - 5:50 pmOh, so many options! The first few that I’ve thought of in no particular order:

1) Something with a jump discontinuity. The easiest way I see to do this is to add a velocity option for some of the existing rides. Unfortunately a macro-scale position graph cannot achieve this, unless we allow for telelportation (as in “Height Off Ground” in graphingstories.com).

Rationale: Students may be under the assumption that all graphs can be drawn by putting a pen down, moving it, then lifting it back up someplace else.

2) Parametric. The bumper cars offer a fantastic opportunity to be converted to a parametric equations exercise (as does the Ferris wheel, actually, but I prefer bumper cars).

Rationale: While the audience is different here, I know I initially struggled (as I believe did many other students), in getting an intuitive sense of what was going on witTh parametric. We need some good exploratory opportunities for students to develop an intuition here.

3) Step function. In addition to a jump continuity as explained above, a step function might be nice. My initial thought was someone waiting in line, with the y axis as the number of people ahead of them, but reflection reveals this is fraught with certain difficulties. Instead I suggest graphing the amount of ice cream on your cone: Get three scoops, one at a time, then lick the remainder until it is gone.

Rationale: Such a graph could help illustrate the difference between step graphs and say, linear functions.

4) I agree with John about a coaster, and I think this would be especially true if it goes through a loop.

Rationale: This is touched on by bumper cars already in that backwards motion does not mean the graph needs to go backwards, but unlike bumper cars, this one is self intersecting which doesn’t mean the result needs to be self-intersecting.

5) Something where you graph multiple curves representing distinct objects interacting with one another on the same graph (in different colors perhaps). Go-Karts might be a good option for this one.

Rationale: The graph would allow students to interpret things like “Who is in the lead?” and “What does it mean when the graphs intersect?”

Basically any of the kinds of graphs from graphingstories.com would make good entries.

## hunter patton

February 3, 2014 - 6:19 pmSea Dragon would be cool.(pendulum) Would the students recognize time continues to increase, or would they try to bring the function line back where it started?

http://www.beechbend.com/SeaDragonSelect.jpg

## Pam T

February 4, 2014 - 6:31 amHow about a water flume?

## Dan Meyer

February 4, 2014 - 6:39 am@

Chris, you thinking height v. time for the zipper?@

Jason, help me out with the Drop Tower. I’m seeing constant positive velocity as the carriage heads up to its drop height. Then negative velocity as it drops. I may have the wrong ride in mind, though@

Michael P&Megan Schmidt, I’m curious if something like The Scrambler gets the job done here.Mike S:We tried for that with Bumper Cars. No dice?

Jason Dyer:Hey, no one’s a bigger fan of Dan Meyer videos than me, let me tell you. But the recursive feedback is a bit more complicated to pull off in video. Extracting and reattaching a PNG layer is easier than extracting and reattaching a video layer.

@

Jim, thanks for the ideas and detailed rationales. Lots to think about here. The step function, in particular, is appealing but it’s unclear to me what we do if a student just scribbles away on the grapher.## Chuck

February 4, 2014 - 6:42 amMaybe a roller coaster if you are graphing speed over time (rather than height, so it doesn’t conform to the shape of the ride)

## misscalcul8

February 4, 2014 - 8:12 amAm I being too simple by saying elevator or escalator?

## Liz Padilla

February 4, 2014 - 9:05 amWhat about the tilt-a-whirl, since body motion is needed to have it twirl in a certain direction, to complete the circle and possible make it twirl faster? I hope I am not showing my age here.

## Jason Dyer

February 4, 2014 - 10:09 am@Dan: I was thinking of starting with the drop ride at top. Using Mike’s suggestion of starting from the bottom would be more like you described.

## Jim Hays

February 4, 2014 - 1:29 pm“It’s unclear to me what we do if a student just scribbles away on the grapher.”

True enough. I shifted from suggesting “standing in line” to “scoops of ice cream” because the second is a quantity that can be thought of as either continuous or discrete. However this doesn’t solve the problem of multiple values. Same goes for graphing velocity of a ride.

Here’s my revised attempt: I’ve been to amusement parks where the price of some of the specialty rides is adjusted based on attendance. So early in the morning going up the bungee swing might be $10, then later it is $15 or $20, and say $35 in the middle of the day when there are crowds and long lines. If the price were displayed via a vertical number line with a big arrow on it to show the price then multiple arrows could pop up at a time to represent multiple prices.

A few more ideas for which I can’t immediately think of a context I like: Inequalities? Relations that aren’t functions? Graphs containing single points? Something polar?

## Chris Lusto

February 4, 2014 - 2:58 pm@Dan: Yes to h vs. t. I’m thinking that working some epicycles into the mix would be interesting.

## Eli

February 4, 2014 - 3:31 pmI had to google a bunch of these rides. I guess I haven’t been to enough carnivals. Also, don’t search for these rides on youtube — you’ll never want to go to a carnival again. I’ve got a bunch of questions back, specifically about three clusters of suggestions.

(1) Velocity vs time graphs (@Jason, @Jim, @Chuck, @John).

Velocity v. time graphs bring up some extra fun/challenges on our side for interpreting what the student drew. Multiple values — do we split the path of the person? If there’s a gap, does the little guy pick up where he left off? What does it mean to travel with an undefined velocity for 1 second? Also, for the speed folks, are you picturing the graph is “speed” (magnitude of velocity), or something like vertical-component of velocity? In particular, do we want to allow for a negative “speed” on speed graphs?

(2) Graphs with discontinuities (@Jim)

Cannon Man would have a jump discontinuity at the moment the parachute pops if we’re graphing velocity — would that fit the bill?

For discontinuous graphs in general, how do you think we should treat the connected version (with vertical lines) vs the discontinuous, disconnected version? In the case of cannon-man popping a chute, I’m not even sure which would be technically “correct.” What do you think the distinction is between connected step graphs and disconnected ones / and what type of question could we ask to try to tease that distinction out?

(3) inception graphs seem potentially “frightening” — maybe in a good way if there’s an intuitive place to start… (two motions super-imposed) (@Lusto, @Jonathan, @Megan, @Liz, @Dan).

Are you picturing these as parametric graphs? If not (or if so), what grade level / subject matter do you think this would be targeting? Can you guys elaborate on an effective, low-floor way to get into that kind of task? Seems like it could be frighteningly.

@Michael — what did you mean by double-ferris wheel? Separate x-y coordinates graphs?

@Jim — just wanted to quickly second @Dan: the rationales in your list are great. I find it so much easier to think about / tweak an interaction on the foundation of a specific pedagogical purpose.

## Jim Hays

February 4, 2014 - 3:46 pm@Eli That’d do it if you can figure out a way to deal with all the complications that come along with a velocity graph.

As for connecting them or not, I’d say disconnected is probably more accurate to a simple mathematical model, but a graph that is connected is going to be closer to the physical reality. I don’t have strong feelings about which one I think is better.

## Jacob Klein

February 4, 2014 - 5:53 pmGraph height, speed, velocity, and distance travelled of a baseball during, say, this play: http://www.youtube.com/watch?v=EdepjcbE3bU. The ball is pitched (horizontally), hit (way up), caught, held, and thrown back to the pitcher (leisurely). I think it’s important (whether using this example or another) to realize one scenario has multiple variables to graph, answering multiple questions, and that a graph is not a picture.

An extension: allow students to control the Cannon Man or a ragdoll character inside the left window, creating a new loop:

1. Create your own animation puzzle.

2. Can you graph the height over time? The speed? The velocity?

3. See how well you tracked your own animation puzzle.

4. Challenge your classmates…

5. Class discussion: whose puzzle was the toughest? Why?

## Jonathan Claydon

February 4, 2014 - 5:59 pmEli – I agree and I have no idea. I would need to get pencil and paper and see what the output even looks like. I’m so terrible at parametric equations I’m not sure. Rotation within rotation is definitely something I see as the upper levels of this activity.

It could make a fun calculus discussion I think.

## Michael P

February 5, 2014 - 5:01 am@Eil

I mean something like this.

## Chris Lusto

February 5, 2014 - 6:49 amHere is the sort of thing I had in mind. But make it cuter and everything, and maybe even have the cart have non-constant rotational speed to be closer to the way one of these things actually behaves.

## pam rawson

February 5, 2014 - 10:39 am@Jason @Mike I like the ideas of the drop tower and the pendulum rides. I had also thought of these.

At our local amusement park, the drop tower, called Dragon’s Descent, starts near the ground, travels up to the top (220 feet) in a few seconds, then pauses at the top before dropping. It includes a couple of bounces before slowly lowering the riders safely to the starting level. A velocity vs time graph would be pretty interesting. Constant upward velocity at the beginning – zero velocity at the top – downward velocity for the drop followed by some brief oscillation and then constant downward velocity as the carriage is lowered gently toward the ground.

The pendulum ride, called the Sea Dragon, uses a motorized wheel to get it up to speed, to maintain full speed, and then again to slow it down. (It doesn’t do a 360.) It would be interesting to graph both the height vs time and velocity vs time graphs. I wonder if students would realize that the maximum velocity happens at the bottom of the ride.

By the way, our amusement park has Physics Fun Days in May. I’ll be sharing this application with my colleagues. It would be great to have our students think about these rides and motions before we go to the park.

Thanks for continuing to do this great work, Dan (& friends)

## Adam Poetzel

February 6, 2014 - 9:56 pmI think one common misconception that could be addressed would be graphing height over time for a roller coaster that does a loop. The animation could just show a roller coaster that is traveling down hill, does a loop, and then leaves the screen. Many students will want to include a loop in their graph. But we know that the graph can’t go “back in time” and will never form a loop.

## Dan Meyer

February 7, 2014 - 5:00 am@

Adam, I like it. Simple, yet drives straight at a prevalent misconception around function graphing.## Darren Burris

February 7, 2014 - 5:43 amAny of the carnival toss or throwing games would be of interest . . . projectile motion of differing kinds depending on the game.

The large slide that has several “hills” or “bumps” as you slide down on a burlap sack.

Acrobatics under the big top (more circus I guess) … trapeze, juggling, etc.

## Adam Poetzel

February 7, 2014 - 10:10 amLooking at the same “story” through different “lenses” (i.e. height from ground, distance traveled, speed) is always a rich learning experience for students. I was thinking about the use of “speed” as the dependent variable in some of your stories. It is harder to match exactly unless you have units because it’s hard to say how fast or slow something is moving on a graph with no unit labels. For example with your bumper car, “How high should I start the speed on the y-axis?”. One thought on addressing this would be to use a simple icon (like a star) on the y-axis to let the students know where the speed graph will begin. With your bumper cars, they could draw a horizontal line indicating constant speed as the car curved along the path, but then quickly drop to the x-axis and ride it at 0 after the crash. For some speed graphs, there will still be some subjectivity in how much to go up or down, but you could still get a pretty good match of the overall shape. Just a thought.

## Jacob Klein

February 8, 2014 - 2:23 amFor the roller coaster: you know Line Rider? http://www.linerider1.net/

With a similar interaction students could draw their own roller coaster tracks (lines and loops), see the resulting movie, and then model the corresponding graphs, or send the challenge to a classmate.

## Kevin Hall

February 10, 2014 - 10:57 amI agree with Adam–this is a great opportunity to show why functions are different from relations. You kind of see that with the bumper cars, and the animation does nicely show multiple cars appearing at once, but it’s less clear to a novice what’s going on.

## Eric Berger

February 13, 2014 - 8:15 pmAll of us at Desmos really appreciate ideas and discussions like this, so we’re building some new scenarios based on your feedback and would like to keep exploring these ideas together.

A lot of your suggestions were about graphing velocity, so we built velocity sketching on top of the existing rides. Check them out and let us know what you think.

https://labs.desmos.com/carnival/scenario/cannonvelocity

https://labs.desmos.com/carnival/scenario/bumpervelocity

As Jim pointed out above, the conversion to velocity is particularly interesting because it introduces a jump discontinuities in both graphs (when the parachute opens, and when the car crashes).

This also opens the door to what Jacob and Adam discussed, creating multiple different graphs from the same scenario by graphing different variables.

## Jacob Klein

February 13, 2014 - 8:32 pmEric – I like! Very fun to play with different speed curves and see what happens. Also, it’s impressive that your team was able to add this new feature so quickly.

## Josh Bobbitt

February 14, 2014 - 8:57 amWhat kinds of rides might students want to graph?

(And what misconceptions might they have?)

Spoiler alert: I have work samples of students generated graphs and rides from a teacher that I would be willing to share (not publicly) that might help answer that.

## Eric Berger

February 14, 2014 - 3:26 pmAs promised, we’ve built a couple more scenarios based on your ideas. We built Adam’s recommendation of a roller-coaster with a loop in it to demonstrate that although the track has a loop, the graph of height vs time doesn’t:

https://labs.desmos.com/carnival/scenario/rollercoaster#rollercoaster

We also followed Chuck and John’s advice to try graphing velocity vs time on a roller-coaster to provide a more challenging scenario, and to show that the graph isn’t shaped like the track:

https://labs.desmos.com/carnival/scenario/rollercoastervelocity

Do these look like what you were envisioning? Do they inspire any more ideas?

## Eric Berger

February 14, 2014 - 5:39 pmAnd here’s one more scenario for feedback. Here’s an implementation of the zipper (Thanks, Chris!), which we thought was a good example of all the cycle-in-a-cycle ideas that Chris, Jonathan, and Megan were proposing.

https://labs.desmos.com/carnival/scenario/zipper

## Jim Hays

February 15, 2014 - 10:10 amA couple of pieces of feedback:

1) Awesome! Thanks for putting this together! I think you worked out a good way of doing velocity.

2) On the zipper, it goes up too high and too low… The correct height goes off the graph when I try to draw it. Not sure what’s going on with that.

3) I presume y’all will come up with more tailored “misconception” graphs for the second slide (for example, have students respond to feedback about a student who draws a graph with a loop for the roller coaster.

4) What about both position and velocity graphs at the same time? Perhaps one is editable and the other is just shown? Or perhaps both are editable? Obviously there are multiple ways to present the position graph based on the velocity, but something akin to whatever formula you currently use to display the roller coaster car could be used.

## Kevin Hall

February 19, 2014 - 10:45 amThe bumper car activity uses “distance traveled” in a confusing sense. If I travel 3 ft. forward, and then reverse 3 ft., I’ve actually traveled 6 ft. total. You really mean “displacement”, but that’s confusing because students aren’t familiar with the term, and 2-dimensional displacement can be complicated.

In addition, the bumper car activity has strange behavior when I draw a graph that’s not a function. I think the activity used to show multiple cars at the same time, but now it doesn’t. So if I draw a graph that’s a loop (or Jenny’s graph from the writing prompt), it gives me the wrong simulation of what that would mean.

## Eli

February 19, 2014 - 10:50 am@Kevin — bumper car “distance travelled” isn’t showing multiple values? Or the bumper “velocity along path”?

Latter one’s here:

https://labs.desmos.com/carnival/scenario/bumpervelocity#bumpercars

## Kevin Hall

February 19, 2014 - 11:05 am@Eli, hmm, well it is now, but it didn’t seem to 10 minutes ago. Either you fixed something or I was looking at the velocity activity by accident.

## Neil Marshall

February 19, 2014 - 1:09 pmThere is some great stuff around the winter Olympics at the moment…..ski jumping, slopestyle snowboarding as they zip up those curves and do acrobatics…..distance and speed is interesting.

We were discussing this earlier….it seems easier to ask them to graph the horizontal speed and vertical speed….velocity is a vector and asking for speed is hard as it is the length of this vector…

## Dan Meyer

February 19, 2014 - 5:41 pmYou guys should head to this link, check out the new rides, and offer some feedback. Which ones should we include in the carnival? Which should we abandon?