## The Three Acts Of A (Lousy) Mathematical Story

This is one of the most tragic math problems I’ve ever seen. (Click for larger.) Not because it’s awful, though it is, but because the awfulness conceals something amazing. I mean, how great is it that we can drop a rock in a well and the sound of the splash tells us how deep the well is. That’s wizardry!

I find it completely amazing we get to offer that power to our students. If my goal were to conceal that amazingness, though, to ensure my students would be less interested in mathematical wizardry thanks to my efforts, I’m not sure I could do any better than this problem.

Problems

1. The student experiences act one and act two at the same time. Act one is supposed to hit you in the gut; act two in the head. The only reason your textbook tries to do both at the same time is because printing the same problem on two different pages is logistically impossible. Luckily, you aren’t bound by the same constraints.
2. The problem starts in the second act. And what a second act. Your students have no idea why they’re wading through that long, thickety paragraph outlining the tools, information, and resources (act two) they’ll need to solve the hook (act one) which shows up long after they’ve stopped caring.
3. And what a hook. Seriously, could someone please explain to me which interest group or political constituency is served by slurring what should have been concise, obscuring what should have been clear, and jargoning what should have been conversational. Seriously, how would a human phrase that hook? Would a human need twenty-six words?
4. The act one visual is cheap. Again, we’re dealing with cheap clip art here only because of the constraints on an industry that’s taking on water. Don’t go down with that ship. Can you think of a better visual, one that would make students wonder, “Wow. How deep is that?” without you lifting a finger?
5. The act three payoff is weak. Imagine all the intensity of the final assault on the Death Star in Star Wars. A planet’s survival hinges on an unimaginably long shot. Luke takes that shot as the clock winds down, a shot right at the guts of the Death Star. What if at that moment we cut to some Rebellion functionary announcing in a slow monotone, “The Rebels were successful. They destroyed the Death Star.” That’s what it’s like to read the answer to a visually compelling problem in the back of the book. Show that thing explode.

Solutions

It turns out that Hollywood occasionally makes math problems for us. Click through and have a look.

With Brendan Fraser, you get a fun check on your own answer and an explanation of why his team even cares how deep the cave is. With the Descent team, you get a much deeper cave and a stronger separation between the first and second acts. Both represent massive improvements over our status quo.

To be clear, I’m not saying you can just play act one and two and your students will trot merrily to an answer in act three, deriving that thorny equation for projectile motion all on their own while stopping periodically to smell the constructivism flowers. I’m not saying that. This problem is tricky and will likely require lots of help on your part. What I’m saying for sure is that it makes no sense to offer that mountainous paragraph of helpful text without your students knowing (to say nothing of caring) why you’re offering it.

Related:

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

1. #### Shari

July 18, 2011 - 7:12 am -

You’ve put together some excellent resources with the video clips. I would love to see students chew on these problems!

2. #### Christine Guest

July 18, 2011 - 7:13 am -

This may be tangential, but it’s still fun. When my Dad was a boy, he and his buddy crawled into an abandoned mine shaft in Scranton PA. When the felt the edge with their finger tips, they stopped crawling, and dropped in a pebble, counting elephants. When they didn’t hear a splash, they crawled to daylight in panic.

3. #### Josh Gates

July 18, 2011 - 7:20 am -

…Ocean’s Eleven. Glow sticks and elevator shaft. Damn, now I have to go find it!

4. #### Christopher Danielson

July 18, 2011 - 8:37 am -

Not to pile on here, but the wonder of the problem is further obscured by the fact that it’s number 101 in the problem set.

I’m just sayin’.

5. #### Ben Fulton

July 18, 2011 - 11:05 am -

Drums, drums in the deep. This scenario also comes up in Lord of the Rings, dropping a stone into a well in Moria. But I can’t remember if the movie version elucidates the problem in a usable way.

6. #### luke hodge

July 18, 2011 - 1:38 pm -

I took a shot at something similar to this with my algebra class last year. The video is too long and the sequencing needs to be reworked a bit. You have one video of an object falling from a high bridge and another of a can dropped from a house, in slow motion, with a timer and ruler superimposed. The slow motion really helps to show that the speed is not constant.

To my surprise, a very good model for the motion can be nicely derived using snapshots from the video (without cheating in the editing process). I provided my students with a paper copy of 4 well chosen snapshots of the falling object (2:20 in the video) with a blank table and blank coordinate plane on back. Should have allowed for more time to work or given more direction – ended up needing a lot of very leading questions to develop the model with little time to look at the pattern in the table.

I think it might have been better to just have the object falling from a taller house/building, with the timer & ruler superimposed, then stop the video when the object is half way down or so. Obvious question being when will it hit the ground. The rock from the bridge could be a follow up question on another day. Maybe a little too much going on for one day.

7. #### Timon

July 18, 2011 - 8:29 pm -

Love it! I totally thought of the LOTR scene as well. It doesn’t give the answer unfortunately. A fun extension might be to discuss the speed of sound (I remember counting time between lightning stiles and thunder to know how close it was) and I feel like a fun extra concept might be to ask how long the shaft has to be for the speed of sound to factor in.

8. #### Timon

July 18, 2011 - 8:45 pm -

HAHA! Whoops! The question does talk about the speed of sound as well. Der! I guess that show that the question didn’t even hold my attention, even though I knew that it was a part of it. Hmmm I now have much more to ponder.

9. #### Bowen Kerins

July 18, 2011 - 9:56 pm -

“The only reason your textbook tries to do both at the same time is because printing the same problem on two different pages is logistically impossible.”

Bah. The same problem can and should be printed on two different pages, many times. And it should be okay to write the problem down in a spot where the best we expect from students is “I don’t know but I could try this this and this”… then write the problem down again in a spot where the same students have the mechanism to succeed. Sometimes it’s a day, a week, or a month between the original problem and the final payoff.

Logistically, this is fine. Our books are published by a Very Large Publishing House where they not only let us do that, they understood why we wanted to do that and supported the process. And they even put together some nice custom artwork that isn’t from Getty Images.

The choice made in the book cited here isn’t a question of logistics, it’s a question of purpose. Why was this problem written? I think it was written to make students say “Wow, math *is* used in the real world!” while really teaching graphing calculator skills. The target of graphing calculator fluency was chosen over having the students actually do any math in the problem other than understanding what it means to intersect two graphs.

A second, mildly sinister, purpose is that it helps in a “flip test” to make the book appear more application-friendly, when the reality is… not so much.

I can’t get over the total lack of math in the solution process. Graph these on the same axes given this specific viewing window? Good lord, man. At least have students graph the two pieces as functions then let them decide that graphing f+g would be a good idea.

They won’t even let students use the original variables t and s! It’s x, Y1, and Y2! Not all math is done on a TI-84, or so I hear.

AAAARRRRGGGGHHHH

But I agree with Daniel: surely the previous 100 exercises in this lesson must all be totally awesome.

10. #### Numbat

July 19, 2011 - 1:17 am -

Hi Dan and others,

I’m very interested in what level of Maths the students have before you introduce this and how you would expect them to solve it.

When I first saw the problem on the other site, I thought the expectation was to use the acceleration due to the force of gravity etc. I immediately thought of terminal velocity and all the other variables involved. I tried googling for terminal velocity and so forth and got into a huge mess, and was going to ask you about this. I’m so glad you’ve put this here where I can ask you and others how they’re going to approach this.

Perhaps it’s because I’ve not taught this level of Maths but I’m really at a loss how I might approach this in a classroom. Are you expecting your students to find (or remember) those formulae? Are they something you’d give to younger students?

Cheers,
Chris.

11. #### Dan Meyer

July 19, 2011 - 3:57 am -

Bowen: Logistically, this is fine. Our books are published by a Very Large Publishing House where they not only let us do that, they understood why we wanted to do that and supported the process.

By coincidence, I have four of your textbooks on my desk. Can you give me an example?

Bowen: I can’t get over the total lack of math in the solution process.

Truth.

Numbat: Are you expecting your students to find (or remember) those formulae?

12. #### Jamie

July 19, 2011 - 5:30 am -

Amazing how Hollywood has done the work for us here. What other math concepts does Hollywood highlight for us?

13. #### DMT

July 19, 2011 - 5:53 am -

Luke: can I get a copy of your video? I really liked the rock drop off the bridge and can use that part. I thought I would edit out the rest.

I especially like the clear delay between seeing it hit and hearing it hit.

The advantage of that video for me is it is a simpler problem when you can see the object hit the ground (as opposed to having to include calcuations for the speed of sound). At the level I teach that would be far to difficult.

14. #### DMT

July 19, 2011 - 9:16 am -

Okay, I found a firefox extension to do it for me. I have the video now, so you don’t need to do anything else.

Dan, if you do put up some instructions, pointing at all the automated firefox add-ons might be a good place to start since they make it really simple.

15. #### Bowen Kerins

July 19, 2011 - 9:52 am -

Time to play Pimp My Book, I guess… (CME Project: http://cmeproject.edc.org)

I only have the Algebra 2 book with me here in Park City, so I’ll go with the first problem set in the book (p. 5-6). In Lesson 1.1, students are given input-output tables, and asked to find functions that agrees with each table. Some are linear, some are quadratic, some are exponential, some aren’t any of those. The exercises then appear again later… sometimes much later:

– Exercise 6 (Table H) matches a linear rule (f(0) = 3, f(1) = 8, f(2) = 13, etc). It’s used in Lesson 1.2 to introduce or remind students about function notation and the ability to create a recursive definition for a function. It’s used again in Lesson 1.3 to introduce or remind students about difference tables. In both of these lessons other tables from 1.1 appear in the exercises, and students are asked to revisit them with their new knowledge.

– Exercises 13-15 (Tables O-Q) reappear in the homework exercises for Lesson 1.4. Table O is a simple quadratic (output = 5 * input^2), Table P is a simple linear (output = 4 * input + 1), but Table Q is a mess — turns out O + P = Q. This previews the concept of adding functions that is picked up in Chapter 2. In general I expect students NOT to get a right answer for Table Q in the initial activity, which hopefully makes them curious enough to want to know how to do it later.

– Exercise 2 (“Richard notices there is more than one rule for Table A”) is inspired by a mathematician’s comment that any table can be fit with more than one rule. This idea is introduced immediately, and we try to say “Find -a- function that fits this table” and not “Find -the- function”. The idea is formalized in Lesson 2.7 — 130+ pages later — as students learn they can fit a function to any desired “next” number in a sequence.

There’s more, even in that first lesson, but this is already almost unreadably boring. Many more times, a problem will appear before it is formalized as a worked-out example or discussion — that way, if students figured it out the first time, you skip the discussion, and if they didn’t, you can use their ideas to generate discussion rather than bringing up some new topic from scratch. My overall opinion, shared by the lead author, is that every problem needs to serve a purpose, otherwise it isn’t doing much good and taking up space that could be occupied by a better problem.

As for clip art, it’s still there, but there are also hand-drawn cool things like the people on page 34 constructing a difference table with a crane, or the dude on page 94 trying to find ways to make x^8 from the product of three terms. There’s a lot less clip art than I was expecting, although I can do without the picture of the guy breakdancing on the cover. Algebra 2: Electric Boogaloo!

16. #### luke hodge

July 19, 2011 - 11:25 am -

DMT: Right click on the link about half way down on the right side of the page to download the video. You can also find the original rock drop on youtube by googling big rock from bridge or something like that. Zamzar is a free and easy to use online application you can then use to download the youtube video.

17. #### Deacon

July 19, 2011 - 12:02 pm -

Great stuff people. Keep up the great ideas and dialog.

As I sit here and really begin to get into my curriculum writing for next school year, I made a list of skills that work with the state standards. Some skills are going to have to be re-taught to “refresh” thats a given. Do you want to use these media clips for every new skill/standard/concept or do you just pick and choose for the new skills that they will encounter? I teach pre algebra and linear algebra to 7th and 8th graders. The stuff I see my 7th graders doing is a lot of the same things I saw them do as 5th graders a few years ago. There is some new concepts, but there is still a lot of the basic fundamental math the first half of the school year. With the 8th grade linear algebra, much of it is new to them but its the stuff like combining like terms or graphing inequalities which, maybe I’m not looking hard enough, you don’t see in movies or everyday life. What do we do then? I want to have those kick ass hooks but what about the concepts that are abstract? Dan or anybody thoughts?

18. #### DMT

July 19, 2011 - 2:29 pm -

So now I’ve more seriously started playing around with making my own movie for my class, but need some help on something basic – how do I overlay a timer?

I only have Windows Movie Maker, so maybe I just don’t have the software to do anything interesting. Is it possible with WMM? Or is there something else free out there? Or am I doomed?

19. #### luke hodge

July 20, 2011 - 6:15 am -

DMT: I don’t know much at all about video editors, but AVS was very easy to use, free, and didn’t muck up my computer. The free version leaves a water mark on your video. You can remove the water mark at a later date by purchasing the software (\$40 or \$50 I think).

20. #### Per

July 20, 2011 - 12:57 pm -

I have a bit of an issue with this problem since I am a physics teacher as well as a math teacher. If your model need the precision of taking the time for the sound to travel into account then you also need to calculate with air resistance since it will be bigger than the extra time from the sound to travel.

Here is the simple model ( s = a * t^2 /2 ) is hopefully good enough and if it isn’t then we need to look into differential equations to sort out air resistance before we care about travel time for the sound.

21. #### Daniel Peters

July 21, 2011 - 9:57 am -

This isn’t even math.

You could summarize “Problem 101” by simply telling the student,

“Find a certain numerical value by using your graphing calculator to find the point of intersection of Y1 =… and Y2 = 4. Use XMIN=0, XMAX=300, YMIN=0, and YMAX=5. Refer to the manual for details. Avoid reflecting a possible real-life background at all costs.”

22. #### Garth

July 25, 2011 - 12:05 pm -

This is a computer applications problem, not a math problem. Except for the verbosity it would be a good problem to see if a kid can find the intersection of two graphs on their calculator. It is a bit weak in the math but good to test typing on an itty-bitty keyboard.

23. #### Phil Smith (@liketeaching)

August 26, 2011 - 7:32 am -

Just had the sudden urge to come back to this one to answer the burning question you asked of the previous 3acts (from my point of view).

For me, I don’t think your outline of this problem has quite enough detail about act 2. Not that I wouldn’t be able to use it at all in a class, but I could easily have missed the idea of factoring in the time taken for the sound to get back to you (something I guess a lot of kids wouldn’t think about either). In the act 2 spot I would love to have a bit more guidance on the maths that could be involved.

Come to think of it, you could also put links there to other ideas that could guide the students in the right direction (such as a video of someone listening to thunder to work out its distance)

24. #### Steve Dickie

November 1, 2011 - 12:42 am -

Just a quick “how to” question. What program are you using for video editing? Specifically, how did you get the timer on your falling rocks video?

Thanks,
Steve Dickie

25. #### Ben K

February 18, 2012 - 1:58 pm -

I’m a pre-med student whose taken physics and calculus and is preparing for the MCAT. When I saw the scene where Brandon Frasier dropped the light stick, I immediately thought about a simple kinematics (translational motion) equation (discounting air resistance and the time it takes for the sound to rebound).

Daniel Peters mentioned this above:

x=V0t+a(t^2)/2

Initial velocity is zero and they say it takes approximately 3 seconds to reach the bottom.

Why wouldn’t x=9.8(3^2)/2=45meters or ~148ft

They say it was 200ft. Is my model too simplistic, or are they wrong about the distance of the fall?

26. #### Johanna

May 31, 2012 - 2:03 pm -

Hey Ben,
I was thinking of doing this with my class and coming across some of the same difficulties. I timed it, and it actually falls for 6 seconds. But that’s a problem because you’d end up with 576 feet. Now I’m searching everywhere for the terminal velocity of a glowstick to figure out if that is the problem, or what he is counting.

27. #### James Key

June 1, 2012 - 11:02 am -

@Dan: “To be clear, I’m not saying you can just play act one and two and your students will trot merrily to an answer in act three, deriving that thorny equation for projectile motion all on their own while stopping periodically to smell the constructivism flowers.”

Dude, that is what happens in my class every day! ;-)