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. Numbers represent items in the grocery basket.

    Ask pupils to devise a formula for the time it will take a queue to be processed, including suitable variables and constants.

    Which queue should you join?

  2. Who will you line up behind, and why?

    Which combination of initial per-customer time (perhaps 30 seconds) and additional per-item time (a second each?) will result in each line taking the same amount of time? Which combination makes the first line take longer, or the second?

  3. It seems to me there are two major questions, or at least two major categories of question.

    The first is about reality: How long will each line take? Which one will go faster?

    The second is about perception: If you walked up right now, which line would you choose? Or, which line *seems* like it would go faster? Why is it not necessarily the one that actually will go faster? It’s hard to tell exactly how many items someone has, plus not all items take an equal time to scan. Are there visual cues that indicate potential slowdowns, for example, a distracted mom whose kids are bothering her, making it hard for her to find her credit card quickly.

    I guess the first question might be more mathematical in some sense, but both are worthy classroom topics, I would say.

  4. Maybe I need some coffee, but I had a little trouble interpreting what the things in the photo were (red thing, black and white thing). The grocery baskets from top view looked like cough drops or something. Once I read the comments, I got it though.

    Nice concept, there’s a formula I would use to estimate how long each line would take, and this might motivate the students to derive it themselves. Of course, there’s some extra variability (are there baggers at each counter, is each customer paying with the new tap-n-go credit cards or are they counting out change) but the basic idea is nice, since this is a common decision people make everyday.

  5. Looks to me like a good one for 2 equations in 2 variables. First I’d get the students to give me their gut reaction. Then we discuss. They’ll be arguing with per item times and per person (getting their money) times.

    So we get a few on the board and pick something in the middle, so they can all agree. We figure the time on each line.

    Then we ask (better if it’s one of them, but if not, I can) either:
    How many people in the left line would make them equal?
    How many items in the right line would make them equal?

  6. P.S. I got the photo, but it took a moment. The red things are hand baskets, I think? Maybe cart-shaped would work better. Maybe a view from an angle, instead of directly overhead.

  7. Linear equations, check.
    Systems of equations, check.
    Long debates about noisy variables, check.

    There’s a lot here. And the opening question is simple, “Which line would you choose?” Any student can access that question, regardless of mathematical ability. It taps into her intuition, which is a nice way to start a lesson.

    A few questions for any interested participants:

    1. Claire rightly calls out two parameters, initial per-customer time and scan per-item time. Let’s see some estimates on both of those.

    2. List five variables that will complicate this model, in descending order of greatest weight.

    3. Rank from fastest form of payment to slowest: cash, check, card.

  8. I really like where Dan is going with this.

    I know this isn’t necessarily the point of WCYDWT, but there are tremendous opportunities for extension here. There is a whole field of mathematics devoted to waiting line models. The calculations are deceptively simple, but there is a tie-in to probability with arrival and service times involved. You can start to look at when it makes more sense to open additional lines from a management perspective vs. when it makes more sense to add baggers to existing lines etc.

    I learned all of this stuff in a graduate math course in applied math, but have had incredible success with showing it to AP students post-exam. They eat up the mathematics where the application to real-life is blatantly evident.

    The book I have is “An Introduction to Management Science: Quantitative Approaches to Decision Making”, although I’m sure 99% of the info is available online.

  9. I thought it was a piece of an alien cell phone. I feel dumb visually.

    Here is my content contribution, though: apparently, under most circumstances, forming a single line that branches toward two (multiple?) cashiers is faster overall than multiple lines to individual cashiers. This can take the problem to another level. Line theory is Serious Business (TM) and there are references to be had there.

    I would approach this with modeling/programming, then students would objectify (thus notice) variables as they tried to program the situation.

  10. Considering I worked as a cashier for a grocery store for 3 years and am a math teacher now, I could probably write a book on this topic, but I won’t. I’d estimate time per item scan to be about 4 seconds (Most items would be less, but items with warped bar codes, produce and bulky items bring it up). Average time per customer would probably be around 40 seconds. Bagging opens up a whole new can of worms though. It’s usually trivial for small orders (everything in one bag), but increases quickly with larger orders (What do I want to keep together or separate, how heavy should I make each bag, uh oh I’m running out of room in the cart…). Assuming a good bagger and the person/cashier help at appropriate times, however, it’s probably irrelevant for the order sizes above.

    As for payment, I’d say it would go card/cash/check, but I’d actually say the difference is probably pretty small unless the person is able to pay with their card while the cashier is still checking.

  11. I realize the visualization is a little iffy here but apparently I overestimated the extent to which the title of the post (“groceries”) would point people in this direction.

    Aaron, I’d like to know more about this kind of quant work. Can you point me to some online resources? Or, better, start a blog and write some of your classroom stuff up yourself.

    Mr. Sweeney, off a few hours of observation and off my interviews with checkers and the manager of the store, the difference between card, cash, and check isn’t close, neither is that the right order. Which is interesting to me.

  12. Hmm, interesting. I suppose I’m relying on just personal and dated information. Checks were usually a pain at my store cause you needed to check id, make sure they were able to use checks at the store, run the check through the register to print on it, so it was definitely the “winner.” I know I’ve seen cashiers at stores unsure of what to do with a check before, so I’m still sticking with that taking the longest. They were also somewhat rare to get. I guess cash would be faster than cards then? (cash only lines) I was trying to take into account that some places allow cards to be put in before the bill is totaled. I guess in general it could be a lot of reading, button presses and waiting. What did you come up with? How were my estimates on TPC/TPI?

  13. @Sean, yeah, cash before credit. No contest, really, especially with machines that count out the coinage. Cash customers know what they’re about, have a general idea of how much money their transaction will require, and typically have it ready.

    Assuming average time per customer means time not spent scanning items (rather than average total time of the entire transaction, which is less useful here) your numbers are pretty good. The actual scan time per item, though, is lower than nearly every estimate I have seen either from you or from others during last night’s conversation on Twitter.

  14. Uf. Those are baskets, not carts. And they’re exactly what baskets look like overhead. I know this because they come from a photo I took of a basket directly overhead.

    Now accepting applications for a better-looking visualization.

  15. That’s OK. What was misleading for me (after I knew what it was) was that I’ve never seen more than one person holding a basket on the same line. I guess I just don’t get out enough.

  16. This really allows a chance for some students to provide some information of their own. I worked for Target for 6 years before I began teaching, and while I wasn’t primarily a cashier, I occasionally had to help out when there were long lines. Something Target does (and I assume most major retail outlets do) is rate their cashiers. Every item gets a certain amount of time to process (divided into “softlines” and “hardlines” — essentially “clothes” and “non-clothes”), and each type of payment gets a certain amount of time to process (if I remember correctly, checks were twice as long as any other method of payment — and they usually still went over the allotted time). Then the cashier is rated with a percentage score, and yada yada yada. I don’t remember all the times, but I know they were there.

    Any student who works in a store with a system like this will either know or will be able to get the information needed (time per item/method of payment). Of course, we could get that information, too, but it might carry more weight for the class coming from a student.

  17. I was a cashier for Longs Drugs and Barnes and Noble during high school and college. We didn’t have any form of rating system, like what is described above for Target.

    I’d say either 1 or 2 seconds to scan an item. With a lot of stuff it would be less than a second. Books were especially quick since the ISBN was in the same place for any book.

    There is a lot of variability in the payment method time. Since we’re only going to care about the time that is additional to “scanning time”, I’ll talk about that: the time spent after a cashier presses [Total].

    Check: customer writes amount (or entire check, which seemed to be the case a lot) and takes out ID for cashier to look at. 10-40 seconds.
    Check must be read and endorsed by computer/printer: +15 seconds.

    Cash: customer hands bills modestly over the amount: 20 seconds for cashier to make change.
    OR customer tries to count exact change: 30-45 seconds
    OR customer needs $80+ worth of change (rare) 1-5 minutes for a manager to arrive with key.

    Credit: sometimes this can overlap almost entirely with the scanning. If the customer is ready: 5-10 seconds to accept charge and sign.
    OR if the customer hands card to cashier, 10-20 seconds.

    Other line-hazards:
    1) pricechecks (dealbreaker)
    2) people returning items (dealbreaker)

    Not a big deal at either Longs or Barnes and Noble. At grocery stores, sometimes there are people who are there specifically to bag. If that’s the case, they are pretty much done before the transaction is finished.

    As for a different visual: what about a 3/4 shot of the queues at eyelevel (or perhaps 8 ft.) Something that would simulate how they might arrive at the lines in a real store.

  18. As for a different visual: what about a 3/4 shot of the queues at eyelevel (or perhaps 8 ft.) Something that would simulate how they might arrive at the lines in a real store.

    Great, let’s see it.

  19. @Aaron: I had a physics professor who would always go off on tangents from the material we were studying, usually relating to classes that several students would later take (for example, “Here, we’re assuming a perfectly rigid body. Of course, there’s no such thing and so to really describe this accurately, you need to deal with stress and strain. If you’re a mechanical engineer, you’ll learn about that next semester in [whatever course]” and so suddenly you’re not just learning “first semester mechanics” — you’re getting the first steps toward something that’s ahead of you, and your current class begins to become part of a bigger picture.

    On the other hand, teachers going off on tangents annoys some students. I think there’s definitely a right way and a wrong way to do it. But anyway, even if it’s beyond the scope of the particular class, it can’t hurt to mention it, and guide students in the right direction if they’re interested to learn more.

  20. Okay, silence broken. Love this one, but I adore it’s LA possibilities.

    I stared at these ‘cough drops’ and immediately had visions of graphic organizers dancing around.

    The question (what line do you choose?) is a great writing warm-up for persuasive writing and argument support. Because more often than not, persuasive writing assignments tend to take root in heavy literature once a semester gets rolling.

    What people above call ‘variables’, I’d call ‘reasons followed by examples’. Pure paragraph development heaven.

  21. As a cashier at a grocery store I can’t resist posting SOMETHING here. Here it is:

    Being the first customer in an empty checkout lane does not ensure the fastest possible checkout time. It all depends on the customer. The reason is that the cashier’s scanning speed is limited by the number of items on the belt. If the customer unloads the cart one item at a time and the cashier begins scanning immediately, then the customer’s unloading speed equals the maximum cashier scanning speed. The fastest customers, as far as ringing up is concerned, are the ones who have already placed their groceries on the belt.

  22. Robert E. Harris

    November 8, 2009 - 8:42 pm -

    Have me select my check out line, then you go to another one. You will get out before me. Or have my wife pick it for you, that will get you out fast.

    Who is checking, and who is in front of you? White women of upper middle class appearance tend to write checks. Men and nonwhites do not. Men usually carry their credit cards and cash in more accessible places than women. Certain sorts of checkers tend to be slow (political correctness prevents specification of sex, race, and size.) Store managers tend to be slow. In some stores the “10 or less” line is operated by the least experienced checker.