Linear Fun #1: Air Travel

We’re deep into linears

We’ve established “rate” as “something per one something” and the y-intercept as our initial condition, all without using the terms “slope” or “y-intercept” or the variables “y” or “x.”… or Graphing Stories, for that matter, which at the time seemed like the quintessential introduction to linears, dammit. I can’t decide if I hate or love this part of my job.

Maybe you frown at that kind of corner cutting but a) you have no idea how gradually you’ve gotta introduce those abstractions at the level I teach, and b) that’s why I don’t read your blog.

So watch as I take my kids through this tech-drenched project-based assignment. Squint through crossed eyes and I might look like someone you’d see at NECC.

  1. Set It Up

    If I fly 300 miles out of San Francisco, is the duration of my flight predictable? Is the cost? Would the graphs look predictable or random?

  2. Give Them Laptops

    We have a mobile lab of 15 MacBooks. That’ll do.

  3. Fly Out Of United Airlines

    Send them out of the same airport. (SFO is our local hub.) Have them pick fifteen one-way, non-stop destinations. Ask them if they have any international YouSpace or FaceTube friends they’d like to visit, thereby cementing your digital street cred.

    Baghdad, as many curious students found out, is somewhat inaccessible.

  4. Record Flight Data In Hard Copy

    Have them record:

    • airport codes,
    • departure times,
    • arrival times,
    • flight durationsHave fun explaining why a flight takes off from San Francisco at 9:00 AM, lands in Honolulu at 11:30 AM, and lists a 5.5 hour flight time.,

    and most crucially:

    • distance in miles,
    • time in minutes, and
    • cost in dollars.
  5. PDFs here! Getcher PDFs right he-ah!

  6. Enter Flight Data Into Excel

    If you want to skip past the hard copy step straight to this one, just know you’ll need a continuous two hours.

  7. Graph!

    Marvel a bit at how well Time v. Distance fits a regression line and how terribly Cost v. Distance does.

  8. Discuss
    1. What does the rate mean for each? The plane flies at .11 minutes per mile, roughing out to 545 miles per hour. Each flight costs 22 cents per mile.
    2. What does the initial condition mean for each? This one’s truly fantastic, as you’d expect the initial condition for time to be zero. (A flight of zero miles oughtta last zero minutes.) Instead, that 36.9 minutes is the time the airline builds in for sitting on the deck, waiting to take off. Elsewhere, that $51.19 initial condition is the charge just for stepping onto the plane.
    3. What does it mean if a dot is above / below the line? The flight is longer / shorter than you’d expect for that distance. The flight is costlier / cheaper than you’d expect for that distance.
    4. Why is cost v. distance such a terrible fit? What does cost depend on if not distance? Why, for instance, does a short flight to podunk li’l Eureka, CA, cost more than a flight to LA at double the distance? TFJ pulled the answer to that one outta thin air, eyes darting back and forth, putting it all together in what has gotta be one of the most satisfying moments of my career, a moment in which I was pretty much wholly uninvolved: “There’re only one or two flights to Eureka.”
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. My father works as the Aircraft PowerPlant Manager (AKA, he is in charge of the engines) at SFO for United so I was tipped to this post. You might want to expand it to San Jose and Oakland and maybe discuss why those flight times and prices also vary compared to SFO. Maybe even have those really energetic students look at Sacramento, which some people have been known to travel out of because they find SJO or OAK too congested.

  2. “and b) that’s why I don’t read your blog.”

    And I guess that’s why I don’t read yours anymore, either.

    Signing off —

  3. Wonderful lesson! I’ve noted this myself, without ever formalizing the thought – cost per time, instead of distance… Then why do non-stops cost more?!

  4. @John, thanks, man, I try to hit the free weights every two, two-and-a-half months, or so.

    @James, thanks for the extensions. Making a note for next year.

    @Alane, I mean, seeing as how your blogroll comprises both JD2718 and Vlorbik (who, I mean, seriously, bro, get a grip … ) I’m surprised you didn’t jump ship awhile ago.

    I’ve got a lot of respect for their work at the high end of things. But I think that if they toted their styles and predilections on down the hall to the classes I teach – tossing up interesting problems involving the eighth root of pi, telling kids to call it a “linear relationship,” not a “line” – they’d crush ’em.

    We teach different crowds, which require different adaptations of mathematical principles each. I can embrace that fact. For survival’s sake, I’ve had to embrace that fact. I have little use, then, for inflexible, proscribed pedagogy – theirs, yours, or anyone’s. It’s like tossing one of my free weights to a drowning man.

    Let’s push things forward.

  5. @another dan, I reckon it’s because they’ve gotta clean up, restock, and fuel two (or more) planes instead of one. That $51.19 figure is garbage, really, given the fit of the line, but you’re paying that every time you step onto a plane.

  6. I wouldn’t take blogrolls too personal. Mine is just stuff I read, and not all of it. I’m thinking about making it a rotating exhibit, so I bump people off add people not because I love or hate them, but just to change the scenery.

    Re: terms. I teach classes on both ends of the ability spectrum; sometimes I’m a stickler, sometimes I play fast and loose to get a point across. I just try to be careful I don’t do anything that’ll hurt ’em later. Certainly I would introduce the word “slope” *sometime*, since it pops up about a billion times later and is slathered all over the pass-to-graduate test.

    No reason you can’t introduce it slow, though, using all the context and scaffolding you can and slipping in sideways.

  7. I just want one of your students to grab their cellphone and post a lesson of yours on FaceTube. I’m sure it would go viral in about 2 days. I love good, silly, fun that includes learning. Doesn’t everybody?

  8. Have fun explaining why a flight takes off from San Francisco at 9:00 AM, lands in Honolulu at 11:30 AM, and lists a 5.5 hour flight time.

    The plane travels at exactly 88 miles per hour. I think that’s it.

  9. Very nice– well– what *is* that wrapped around your upper body in the picture? A handkerchief? Yes. Very nice handkerchief.

  10. What’s with ‘Math Teacher?’ Pissy ’cause you didn’t use the terms ‘slope’ or ‘y-intercept?’ Good grief.

    That’s what I like about you, Dan. Good or bad, you’re willing to put it all out there (literally)…

  11. Dan: Just curious.

    How long did it take you to Photoshop the lettering on that ice cream truck photo to give the impression you actually own a “t-shirt” labeled in such an uber-Prenskian manner?

    I’m guessing by your skill set and the minutes:to:edits ratio, perhaps 5 minutes (tops) if you went uninterrupted.

    Most of the fabric ripples on the real shirt are echoed decently in along the vertical axis of each letter. Good stuff, that. Some, however, seemed to have escaped from the barn when the graphics team went post-production on the photo. Maybe had the “E” really gone Under Armor “native”, it’d have passed the TSA screeners’ attention over there at the quiet end of the SFO terminal.

    Pixel veracity or not, it is one of the great edu-blogger images of 08. Can a Second Life located Flickr-inspired Ning self-help group with Google Earth tags be far behind?

    Now, if only Kansas’ M. Wesch or the Common Craft folks would mash it up along with one of their next “future of everything digitally curious” videos, then we’d really be jumping the shark along with the Fonz. And viral wouldn’t even be scratching the global surface of it. It’d be Hannah Montana big! It’d be “eleven”, IMHO.

    At the very least — and this is a big ol’ favor I’m asking — please consider opening up your own “Digital Native” (et al) shop over there at so that everyone flying down to NECC can have a shirt of their own before the next unconference breaks out. Even I would watch a grainy UStream workshop video if I was able to play the “Who’s Wearing the dy/dan “Digital Native” t-shirt” drinking game if the speaker audiences were properly attired.

    And if you wouldn’t mind, mind doing me one more favor?

    Can you add a “Dirty Diaper 2.0” tuckus-cover option, too, for the toddler set in my world? Promise to daddy blog the results as Beckett lives up to the hype. And link your way so the good folks at Technorati can do their magic.


    BTW, brilliant math lesson.

    Heck, brilliant lesson period.

    Teaching “different crowds” or not, lessons like this would have kept my head above the math class boredom waterline way back in my own high school years.

    Wish I could figure out a way to graph out Othello inspired assumptions in a similar data-drenched visual way after spring break just as a nice mind-break from the normal routine my literature groupies are subjected to. Instead I’m left to ask my kids to write vocabulary quiz paragraphs inspired by TED videos of hip hop body-benders. Such is life in the English Dept.

    Next step with the linearity progression?

    I’d sure love to see a Feltron job done on your kid’s take-away’s. Might be fun to have the kids do a “What they’re NOT telling you down at reservations desk” mock-up marketing guide for customers trying to price out their next MySpace buddy meet-up.

    Or would that require getting some punk English teacher involved since it’d most likely fall outside of the hours:to:content NCLB math spec sheet?

  12. I hope that “frown at that kind of corner cutting” comment wasn’t aimed at this rant about short cuts.

    If anything, this seems like it’s putting the corners in place – making the kids actually learn and understand the ideas, and only adding the language later once the concepts are understood. It’s what I strive for in my lessons. This one looks good enough to teal. Thanks!

    P.S. Do you still have the graphing stories available?

  13. It’s a nice concrete application of linear regressions. Very nice. But don’t you think that they should be pretty darn familiar with linear relationships after 6th, 7th, and 8th grade? I think 9th grade is the time when we have to start expecting & leading them to a certain level of abstraction. I use the concrete stuff (2-ish days) to remind them of what they already spent 3 years on, but then we get on with some algebra.

    Lines and systems of lines, me and the babes knocked those out of the park. Thanks, I know, to much groundwork-laying by their excellent middle school teachers. But this year they have to somehow internalize geometric sequences and exponential functions (in 8 days, says New York, ha!). The best concrete model I found for them to get their hands on, it involved Yachtzee’ing Skittles and adding/removing the “heads” up ones, which I didn’t like because who freaking cares about Skittles in a cup, and then I found out they already did it in Earth Science, except with pennies.

  14. Kate,

    Not speaking for Dan here, but yeah there are plenty of things that I think my students should know by the time they enter my “Algebra II” class. But on the diagnostic at the beginning of the year most students acted like they’d never seen a coordinate plane before. Never mind a linear relationship. Reading this reminds me that I need to make sure I build into things more slowly than I want to. And using the relationships without the vocab for a while, makes sense.

    @ Dan. Thanks for the idea!

  15. @Mr. K, nah, not that.

    @Christian, however I presented myself and my classes in this post, at the end of this two hour exploratory unit, my kids were intellectually exhausted. TFJ nailed the Eureka, CA, question but other kids had already checked out even in that, my strongest remedial math class.

    Maybe we extend this later with the good, raw stuff suggested by you and James above, but that day, that week even, it would’ve been too much.

    Where do you draw the line, right? How long do you draw out the story before it gets stale and the crowd urges you to move on? When do you decide to get in front of the mob and call it a parade? I don’t have answers. Only guesses.

    @Kate, I should clarify that these are below basic, remedial classes, where the difference between what they should know and what they do know is vast and varies wildly from student to student.

    Sounds like you’ve got an awesome eight days ahead of you, though.

    @Chris P, represent.

  16. “But I think that if they toted their styles and predilections on down the hall to the classes I teach – tossing up interesting problems involving the eighth root of pi, telling kids to call it a “linear relationship,” not a “line” – they’d crush ‘em.”

    Certainly I post puzzles to engage teachers. But hard problems or digital slide shows?

    False dichotomy. I’ve taught math to 9th graders who could barely read. I’ve graphed people walking (conveniently across NY-style numbered cross streets) (position vs time) and spent weeks putting speeds and starting points onto graphs, and into equations, and eventually reading them from graphs and equations, and just sort of, sort of, translating to slope and y-intercept.

    I’ve taught exponents without rules, and how to avoid unnecessary minus signs. And adding signed numbers without rules.

    And I still teach all this, but now what took weeks takes minutes (different kids).

    But I’ve always taught with respect for the math. I don’t cheat the math, even if it takes a long time to get there. (maybe with the next teacher) And I’ve always given kids some responsibility for their own learning.

  17. “But hard problems or digital slide shows?”

    Look, if you’re bent on pigeonholing my work there, feel free, but you’re the only person in this lesson – a lesson which explicitly gives students ownership of their learning – to bring up digital slideshows. I’d mount a more vigorous defense of my work but I’m just not sure you get what’s happening here at all.

  18. Ok, so while I came in via Shareski, I’m adding you to my blog roll. I’m looking forward to additional “math-reluctant, kid-friendly” posts. Or at least contacts.
    Way to make math accessible!

  19. I’m doing a variant of this lesson today with my 7th grade Pre-Algebra class. To streamline it, we’re going to be using Mapquest with driving, instead of flying, but otherwise we’ll still be investigating lines of best fit and then coming up with the slope-intercept equation from that. Also, for better or for worse, driving pretty much yields a y-intercept of 0 (technically, so does flying but all of that taxiing and sitting on the tarmac sort of pushes up the left end of the line so that you get something of a positive y-intercept).

    What I really like about your plan is that *most* of the relationships that we use math to model existed before the math did – in other words, we developed our mathematics (in this case, linear equations) to fit with our observation of the physical world’s behavior. And that’s a more authentic way to expose students to this mathy stuff than to ask them to memorize y = mx + b with nothing to show for it. I’m too often guilty of the latter more than the former.

  20. I prefaced our jump from Rates to Rates With This Hazy Understanding Of An Initial Condition with the line, “I’ve taught this four different ways in four years and I don’t think I’ve got it right yet.” I’ve grounded this so heavily in real-world data and a flurry of “guy running away from home with a head start” problems, I’m not sure we’ll all survive the necessary transition to abstraction.

    In other words, I fully expect to deliver the same speech next year.

    PS. I had considered the Mapquest analog but figured the model would be a dirty fit. I reckon you have ’em driving mostly long-distances, mostly outside cities.

  21. Dan – you’re right, we travelled (virtually) around the entire country. The only journey that we couldn’t (virtually) make was from Orlando to someplace in Alaska, because the doggone teacher had made up a beautiful piece of graph paper that didn’t quite go far enough on the x-axis will miles.

    I think that you’re right if we did trips within our own urban area, it would be a much more scattered plot. In reality, I’m starting to suspect that Mapquest just has some very linear function (with a y-intercept of zero) that calculates their estimated travel time. In other words, our best fit lines were mighty good because they really do fit on a straight line! BTW, how do you find the miles of each of your plane trips (I don’t think that the airline web pages show you that, do they?).

  22. Back in the day – like, not two years ago – we’d have to run through a separate website to calculate airmiles but with United Airlines you just mouseover “Flight Details.” Score!

    Re: Mapquest, so, I guess I could just find out for myself, but was the slope, like, 65 the whole way across? Were any routes noticeably above/below the line. eg. a trip that ran through Montana or straight along an interstate where the speed is 70 or 75.

    Sounds more and more like a great precursor to the air travel activity, interpreting an insignificant and then a significant y-intercept.

  23. That’s my thought exactly (your last sentence) — the Mapquest activity is essentially a direct variation exercise w/ a zero y-intercept, and the United Airlines one is the next step.

    Yes, the slope was very close to 65 across the board for interstate trips, so my hopes that somehow Mapquest knows about every traffic slowdown and wreck have been dashed. Hmm, I’m a member of AAA so I’m heading over there to see if their TripTik website does factor in other stuff (only to satisfy my own curiousity, not to integrate that into a lesson).

    And thanks for that very Web 2.0ish hint about mousing over the flight details! I hadn’t even considered that so I just didn’t try….

    I’d love to collect more real world applications of linear relationships that middle school or high school students can handle, like yours.

  24. I love the shirt. i am a “bridge” myself, sometimes a “refugee” — even though I am only 24.