Building A Better Taco Cart

And by “taco cart” I mean “digital math curriculum.”

I made Taco Cart out of videos and photos. I’m comfortable making math curricula out of videos and photos but I’d rather build them out of code.

Here’s the Taco Cart I wish I had made. Implicitly, here, I’m admitting I’m in over my head. I need a new set of skills or a new set of collaborators.

Currently, I’m asking students to guess where Ben and I should enter the roadway to get to the taco cart as fast as possible. But how do they register that guess? Do they point to it? Do they make a mark on a printout of the scene?

Let’s give them tablet computers, instead, and let them slide their fingers down the road until they’re happy with their guess.

Then they see all their classmates’ guesses. Ideally those guesses are all attached to faces or names somewhere so you can see how your best friend or the girl you have a crush on guessed. This ratchets up their perplexity. Who guessed closest?

Then we ask them what information would be useful. This is abstraction. We’re giving the students a chance to extract the essential features of the context.

We ask them to discard the inessential features of the context.

The tablet summarizes the class’ responses. The teacher can use this information to seed a brief discussion.

What happens next is violent. We’re going to vaporize the world. We’re going to strip away the sand. We’re going to destroy the buildings. We’re going to wipe Dan and Ben and the taco cart off the map and replace them with points and lines. If you’ve studied math at the university level, it’s possible you’ve lost touch with the violence inherent in mathematical abstraction.

So we scaffold that process briefly. We prepare the student. We say, “We’re going to get rid of a bunch of stuff you said was inessential and represent the rest as neatly as possible.”

Now this is interesting. Each student is given her own task, a task that she, herself, picked. “You guessed that this would be the fastest path,” we say. “Go ahead and figure out how long your path would take.”

This is more fun than evaluating the duration of a generic path and it’s easier than differentiating the generic path and solving for its minimum. It isn’t all that much more difficult for the teacher to check either because everyone is performing the same calculation, just on a different value.

Everybody enters their results. The tablet checks them for correctness and then displays them.

Remember that everybody is doing the same calculation on a different value? [BTW: Christopher Danielson is right that I overstepped myself here.] That means it’s abstraction time again.

We pull out three student pages (never mind that these are all from the same person) and we ask the students to notice what changes and what doesn’t.

We turn the thing that changes into a variable. But why? Because math teachers need the work? Because math teachers amuse themselves with this notation? No. Because it lets us try any value we want really quickly. What are the highest and lowest values we should try?

The student slides her finger along the graph and the path adjusts with it. The tablet snaps to points of interest like minima and maxima and displays the value of the graph at those points.

From here we’d play the video that shows that answer. The tablet would find out which student guessed the closest initially and throw some love on her.

A Few Closing Notes Before I Ask Your Opinion About This Kind Of Curriculum

  • On the upside, this task attempts to clarify abstraction for students and make that process participatory. It involves the entire ladder. Students pick their own math problem. Students are guessing. They’re deciding what information is useful and useless. ¶ Look at the original task and imagine how many more students are included in this re-imagining. ¶ The task is also social in a way that’s difficult to achieve without 1:1 technology. The tablet collects and represents the entire class’ guesses in real-time. A teacher can’t do that.
  • On the downside, I’m not sure what the teacher does in this sketch. At different times, I wobble between having the textbook function as the teacher and having the textbook simply maintain a steady course through the problem. If I taught this problem, I know I’d handle a lot of the exposition (ie. “Here’s why we use variables.”) myself, in conversation with students. But what should the textbook do? ¶ Also, we didn’t differentiate the function and solve for the minimum. We formulated the model and solved it graphically. Does that still count as math?

Now you go.

2012 Oct 11. Dave Major went out on spec and put a lot of this into code. It’s exciting.

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. Your (brilliant) example uses real-time guesses/work from “your classmates”. Could there possibly be an 80/20 benefit of making this exact interactive textbook, but minus the real-time? Or rather, from hard-coding in real guesses from, say, earlier tests of these exercises done in real classrooms? In other words, the student working in this new textbook is in a virtual classroom, but one faithfully re-created from a real one?

    Maybe I’m asking how crucial it is to the program that this be the actual students’ REAL classmates, in real time. I don’t mean just contriving some imaginary virtual class and classmates, but perhaps introducing the student at the beginning of the book to some real group of students somewhere (who did the prototype), complete with names and faces…

  2. Theron J Hitchman

    October 9, 2012 - 8:40 am -

    I like this a lot. Be sure to leave room for two more things
    1. Student “error” and revision
    2. Reflection.
    Those can be handled in person, but are very difficult in a textbook format.

    what if a student hasn’t listed all of the things they will need to solve a problem? Should your interactive book let them stew for a while, realize the trouble and revise their wish list?

    It would be good to have the student reflect on the process. Better at several points, but definitely at the end.

  3. You don’t need new skills or new collaborators; you need better tools. So far, the technically skilled community has failed to provide (math) teachers with software simple yet powerful enough to do what you want. The ideal program would require only this storyboard as input. You’d specify a mathematical model, indicate where key parts of that model intersect the background image, and voila. Then what you could do (not necessarily what you *should* do) is make the tool available to everyone, get a library of problems, and have teachers publish their own curated subsequences from the library.

    But, as you say, this lesson presupposes every student as an iPad or other tablet, and they’re all networked together. Apple’s puts tight restrictions on what you want to do, as far as sharing “code” and so forth. Android is an underdog and highly fragmented. The first obstacle to creating the tool isn’t the code, it’s finding an environment to write the code in, given the companies, products, languages, and so on we have to deal with in the real world. (If every tablet was identical and open, we’d have a much easier time.)

    If you want to go with an extant tool, your best bet if probably Codea:

  4. @Max: Ok, theoretically, someone could code this ubertool for you. But please realize you’re asking for an awful lot from developers for a tool which I have no idea how you’d monetize it (unless, I dunno, Pearson funds it or something).

    I wonder how much of this you could do with a web interface using HTML5 stuff. Hmmmmmm.

    – an intrigued ex-programmer-now-teacher

  5. For that matter, what exists out there as far as classroom-targeted apps that share data with the teacher from students’ iPads / whatever?

    I keep seeing that as the biggest technical hurdle here, but I have no idea if such things exist already. If they do I’d be curious to see how they manage the data and whether it’s worked in practice.

  6. Hi Dan,

    I really enjoyed reading this post. It’s a great problem and it was interesting to read your thought processes throughout the tablet sequence.

    A few minor points that got me thinking/I wasn’t sure about:

    1) Would the students be abstracting this problem (i.e. drawing a diagram) or would the tablet be doing it for them? It would be interesting to see how many of the students would go ahead and draw a diagram, like the one in tablet 7, with all of the necessary information. To what extent does the act of drawing a diagram help a student to identify and organise information to go ahead and solve a problem?

    2) Does the tablet sequence automatically take students in a certain direction? Would students who would naturally solve this problem using calculus be steered into a graphical solution? In this case I don’t think so but there may be other cases in which this should be taken into account so as to keep the strategy as natural as possible.

    3) I may be wrong but I get the sense during this tablet sequence that the whole class are doing the same thing at the same time. Would some students be waiting for other students to finish after they’d calculated their own time?

    Again, these are all small points when you take the entire thing into account. It would be great to see a lesson working in this way.

  7. The step of recording and displaying results is very delicate. I think it’s unlikely students will all present it in a way that looks “the same”. An intermediate step that helps a lot is taking one student’s guess and evaluating it, as a full class, before sending students off to evaluate their own guess. With simple expressions this might not be necessary, but considering the complexity of this expression I think students’ results will come back very differently.

    In my opinion, the graphical solution is definitely still math! A table with zoom-in capability could also work and is a nice potential feature of a tech-based curriculum.

    One could even present this task to precalculus students, then present it again to calculus students when the exact solution (via differentiation) could be found. At a precalculus level, some talk about how this function is more complicated than a parabola would be interesting and would help students understand why they need more machinery to do the work directly.

  8. Going low tech on this as we don’t have 1:1 currently, but I see giving each kid a thin strip of paper to represent the 562.6 ft of road. She can make a tick mark on it for her guess, of course the strip shows which end is taco cart. Then we can stack the strips on board for all to see, or I can transpose all marks onto one class strip.

    Maybe it’s not so insulting to kids any more if we use the textbook AFTER they’ve gone through the 3 Acts. The text can offer follow-up and extension questions like the ones you’d posed in the sequel. If not, then it can stay in time-out.

    The main reason I’m sold on 3-Act lessons is because it is conducive to teacher questioning, lots of it. Normally when a kid asks me a question, it’s not uncommon that my reply is tossing two questions back at him. When we as teachers do our homework before giving kids a task — by setting goals, anticipating student strategies/solutions (Smith’s 5 Practices) — we ask better questions that will nurture a meaningful and productive discourse.

    It’s ironic that there’s been mention of the teacher becoming extinct due to advancing technology, including digital curricula, because all I envision and experience is the opposite — digital media (if it’s any good) only solicit more questioning, exposition, dialogue between the teacher and his students.

    I want to believe that there is no downside, Dan, to great activities like this, whether it’s a 1:1 or paper-pencil setting. Good teachers will make this come alive one way or another, the hard work was done in Act 1 — just give it to the kids. We’ve been hungry for taco carts.

  9. I like this but am slightly worried that the ‘greatness’ of using gestures is a bit overrated (as in more tabletspeak). Sure, using your finger is nice but actually using a stylus or mouse works better for precision. Some of these things -not perfect- could then be done already, but alas, java, so not on a tablet. This is what I see as the biggest problem here, a bit like one of the comments: we sacrifice functionality and authoring abilities for form, dictated by market and company policies. New iOS? Doesn’t work any more. Interactive eBooks with ePub3? Sorry, we needed one more gadget so we made a new format? HTML5? Sorry, the business model doesn’t work as well as offering Apps. Even if one, probably costly project, would lead to the tool you want, it would probably serve some types of questions and tasks and not offer the flexibility we need. Maybe it’s because funding is often based on competition as well. Many more thoughts but maybe later.

  10. Dan,

    I may be able to collaborate with you on something like this. I’m already working on a number of similar ideas over at

    Don’t get too turned around by the puzzle concept. At the end of the day the process the students go through is very similar in the work you’ve described and what I am working on. I’m also a developer who is already building these apps.

    So if you’re interested in working with a developer in SF who is interested in working more closely with teachers around these types of projects, let me know. jared at

  11. No developer/teacher needs to create “the software.” Just let the students create app. I’d start with Fawn’s comment above and then help the students convert that into a digital format.

  12. To comment on your downside, I definitely imagine teachers facilitating discussion throughout the lesson you proposed, and I’d be curious to see how different the discussions would look from teacher to teacher.

    I have felt tension related to this while designing math lessons at my company. On one hand, you want to give the teacher enough support/structure to succeed implementing the task you’ve designed, and on the other hand you don’t want to stifle the teacher by structuring the activity and discussion too much.

    For example, we’ve had completely contradictory opinions of the teacher support materials we write. At first the materials for each lesson were multiple pages with lots of explanations for the teacher and lots of suggestions of questions she could ask students. Who doesn’t like specific suggestions and insight from the people who designed the lesson? Plenty of teachers, apparently.

    So, we listened and shortened our teacher materials to provide all the teacher talk on no more than two pages. Now you have something you can skim quickly and still get the gist of the lesson. That’s the right amount, right? Wrong. Now some teachers (presumably different from before) are saying we’re not providing enough guidance.

    Planning for the teacher and anticipating her needs and desires is a challenging task because there is so much variability among teachers. All I can say is you’ll end up with some teachers who love whatever amount of freedom and structure you provide, and you’ll end with some who just don’t.

  13. While writing my previous comment, I was reminded of another issue relating to the variability of teachers. The digital lessons my company creates are designed to be part of a blended classroom. The teacher still has plenty of “teaching” to do. We are by no means trying to replace the teacher. Apparently, this is a hard concept to grasp, and even accept.

    One of our instructional coaches was meeting with one of our teachers, talking about how the curriculum should work in the classroom. At the end, the teacher said, “Oh, I’m still teaching and working with my kids. The technology is just supporting our work, right?”

    The instructional coach, thrilled that the teacher was finally getting it said, “Yes! That’s it!”

    The teacher’s reply, “Why would I want that? I just want to push a button and let the computer start teaching.”

    Other than being an example of a teacher who’s ready to be replaced by computers, it speaks to the view some (many?) teachers still have with regards to technology in education. Their limited viewpoint is that using computers in class must mean students working with adaptive software programs independently while the teacher sits back and reads a magazine.

    No matter how well designed your curriculum may be, you face an uphill battle conveying what your curriculum is and what it is not, just by virtue of it being a digital product. Don’t let that deter you though. I’d love to see this developed at some point.

  14. 2 things:

    1) I love the idea of putting out there what you would like to have. Isn’t the idea of tech startups to bring together people with ideas and people with tech skills? My impression of edtech right now is “the customer is always right”, meaning what’s on the market is what the market demands. If some niche market (we who think a bit differently) demand something different, someone might just make it.

    2) In the spirit of putting out there what I would like to have: Why would students assume that there is a fastest way to get there? Could the question be: “Does our taco cart angle of attack matter?” Also, I’m confused about when and how the formulas come into play. Do students know these? Are they given a formula if they know to ask “is there a formula for this”? Could they pick their path and see the distance and time, then talk about the relationship of the two? Maybe all that happens, and I’m just missing it from the description :(

  15. Dan Meyer

    Also, we didn’t differentiate the function and solve for the minimum. We formulated the model and solved it graphically. Does that still count as math?

    I think the modeling and representing the task in different ways (i.e. graphically) is what the standards of math practice of the CCSSM is all about. I’ll be highly disappointed when we start assessing this type of task by telling students they have to “differentiate the function and solve for the minimum,” as if this was the only method. That skill is important for students to investigate, but shouldn’t be considered the “only way.”

  16. Getting away from the “how would you build this” and back to the design, re the ‘downside’ listed:

    The key points in your example here for teacher involvement on a whole-class level are when student data gets pooled together. eg. The graphic of everyone’s guesses, the list of useful/useless information from the whole room, and some of the final graphs.

    What if those data displays only showed up on the teacher’s screen to project at the front of the room? That way the book is still pulling those things together, but it’s also drawing the focus (and the responsibility) back onto the teacher to get students discussing what this stuff means.

    The general idea anyway is, think about what stuff should be only in the “teacher’s edition” so as to enable the teacher to do their job well rather than try to supplant it.

  17. I teach science at a cyber school and also dream about this kind of instructional tool. I don’t know if it will help but you may want to watch this guy: I came across him on TED. I’m guessing that Josh has it right- unless you can find a way to make money off it you probably won’t find anyone to make it. Can we all pitch in a few bucks to hire a programmer?

  18. This whole thing can be done with Geometers Sketchpad. I admit that is not a nifty iPad app but Sketchpad still does a lot of cool stuff. Starange to think of Sketchpad as “old school”.

  19. I think there are three key points here:

    1) Participation. The students are asked to *do something* at every stage. Make an initial guess. Do a calculation. Make a drawing. Evaluate your initial guess and revise.

    2) Timing. Controlling the flow of information. Imagine these digital textbooks are “locked,” and only the teacher can unlock the program to show the next screen. Picture this interaction: “You guys did a great job deciding what’s important and what’s not. Now I want you to make a math sketch for this problem using points and lines.” Screens go interactive, and each student (using a stylus?) makes their drawing. 20 secs later, teacher projects student #29’s work for all to see, then remarks, “Many of you made a drawing that looks similar to this.” Then he/she hits the UNLOCK button that advances everyone’s table to the next digital page, where there is a pre-fab drawing. Students could not access this drawing until the teacher unlocked it. Now we go the next stage of the problem…

    3) Presenting a composite view of student responses. Technology handles this in ways a human teacher can’t, as Dan said.

    To the commenter who suggested the responses of a “pilot class” be used: with all due respect, I can’t see this producing the same level of buy-in from the students. “Someone somewhere at some time thought xyz” is not nearly as compelling as “the dude next to me just typed x = 5.3.”

  20. @James Key
    “someone somewhere at some time thought XYZ is not nearly as compelling as ‘the dude next to me typed…”

    Absolutely agree. (I was the one who suggested the ‘pilot class’ results as a (granted, much weaker) substitute for real-time.)

    But in the spirit of not-letting-perfect-be-the-enemy-of-good, I think Dan’s overall idea matters deeply, with or without the real-time. A less-compelling NOT JIT version would still be a hell of a lot better than what currently exists, for all the OTHER things he’s doing here. Also, for scenarios where there’s no classroom.

    However, I did not mean “someone somewhere at some time…”, which is why I suggested the textbook participant be introduced to the *real* students from which the other guesses were derived. Not as good as “the dude next to me”, for sure, but still better than pseudo/made-up “guesses” or real but anonymously provided guesses. Why couldn’t a textbook like this feel more like a virtual class? (again, in the absence of a REAL classroom experience, which would be far better).

    In any case, my response was probably off-topic since this IS about a tool specfically for classroom use. One potential beautiful side-effect if such an 80/20 version was developed, however, is that it might be one more way to inspire more teachers to start thinking in this way…

  21. In the meantime, while we still have to teach without 1-1 tablets, you might consider having students use this GeoGebra applet,, then using various non-techy ways to gather students’ predictions etc. Take students to the computer lab so each kid has their own computer for using the applet, and teach from there. The lacking functionality of submitting predictions electronically is not ideal, but this applet certainly is better than nothing.

    You might also like this related applet and game for students to engage with this problem:

    Two Terrains,

    Dirt Dash game,


  22. Thanks for the feedback, everybody.

    General comments:

    One, I’m not particular to an iPad or Android or HTML5 on a desktop. Touch events could register as click events and vice versa. Let’s not get hung up on SDK restrictions or litigate any browser wars. Let’s just dream for a second.

    Two, nothing currently works out of the box like I’m describing. I’ve looked. Several top tier educators in this forum have suggested Geogebra or Sketchpad. It’d be clunky and you’d make a lot of design concessions but you might be able to get me 60% of what I need. You might. But you’re a hacker and the other 90% of educators need something that doesn’t require hacking. They deserve it too. Teaching is hard enough without asking someone to also understand how to rig up some kind of data structure in the cloud to handle student input from Sketchpad.

    Kathy Sierra:

    Could there possibly be an 80/20 benefit of making this exact interactive textbook, but minus the real-time? Or rather, from hard-coding in real guesses from, say, earlier tests of these exercises done in real classrooms? In other words, the student working in this new textbook is in a virtual classroom, but one faithfully re-created from a real one?

    Curricula are sold to classrooms – both online and brick/mortar – but also to individuals. In the case where an individual wants to use the curriculum, I take no exception to seeding it with data from your pilots. In that instance, I don’t see what we’re sacrificing. In the classroom, though, there’d be something pretty strange about seeing the responses of people you’ve never met while being blind to the responses of your classmates around you.


    what if a student hasn’t listed all of the things they will need to solve a problem? Should your interactive book let them stew for a while, realize the trouble and revise their wish list?

    “What should the application assess and what should it just record?” This is a good question and reasonable people can disagree. For most qualitative data, I vote the application just records it, perhaps displays it to teachers and students, but doesn’t assess it. Down that road lies multiple choice items and checkboxes asking students to “pick the information you think would be useful here.” We’d become servants to computing’s constraints.

    Dan Pearcy:

    Would the students be abstracting this problem (i.e. drawing a diagram) or would the tablet be doing it for them?

    Great question. At one point I had a frame where the student would draw the important details on top of the context. I probably should have kept it. Even if the tablet does nothing except record that sketched abstraction, pass it along to the teacher, and then show the student our prefabricated abstraction, that moment cost us nothing but a few extra kilobytes.

    Dan Pearcy:

    I may be wrong but I get the sense during this tablet sequence that the whole class are doing the same thing at the same time. Would some students be waiting for other students to finish after they’d calculated their own time?

    Ideally, there’d be a back button so if you’re working at a faster rate than your classmates, you can move on ahead without seeing your classmates’ guesses (for example) and then head back a little later to see them, without losing any work.

    Bowen Kerins:

    The step of recording and displaying results is very delicate. I think it’s unlikely students will all present it in a way that looks “the same”.

    Christopher Danielson called me out for the same line. I think you’re both right. My point is that the teacher won’t have to do any extra work assessing students who get to work on problems of their own choosing, not that there isn’t any work.

    Belinda Thompson:

    Why would students assume that there is a fastest way to get there? Could the question be: “Does our taco cart angle of attack matter?”

    The original task might provide some useful context. We’re looking at a sequel to an earlier task that should answer these questions.

  23. @ Belinda,
    “Also, I’m confused about when and how the formulas come into play. Do students know these? Are they given a formula if they know to ask “is there a formula for this”?”

    For my students who acted helpless without being explicitly given every formula for every situation, I worked a lot on developing their ability to turn the units that they speak and take for granted into the formula. Their tendency was to randomly multiply or divide, without thinking why, and then asking me if they had done it right. “Do the units work out?” was my frequent reply. They should look at the units and confirm that the unwanted units were canceled out through the procedure, and that the only thing left is what they wanted. I liked this because they could instantly understand the procedural relationship between three variables, whether it be distance, speed, and time, or mass, volume, and density, or number, total price, and price per unit. No direct formula instruction necessary each time.

    The fact that they didn’t learn to look at units in elementary school when they were taught the rote procedure of “divide the 8 slices of pizza among the 4 friends to see how many slices each one gets” is a major missed opportunity for the students that still plagues seniors in high school who are just now learning about this skill. “Why is it 8/4? Why isn’t it 8×4, since I have 8 slices and 4 friends? Oh, if I want ‘slices per person’, I must take slices/people. Ok.”

  24. @joseph, The grade school student has a very good understanding of the units “slice” and “person”, so they realize they need to divide and that the 2 means 2 slices per person.

    Many middle school and high school students do not seem to have good understandings of rates and the corresponding units. That is why the randomly divide or multiply. Using a formula and units and “cancelling” may be a crutch that enables them to get the job done, but I feel this does not address the underlying issue. If they understand what the units mean, they will understand whether to divide or multiply. I would very much prefer the student to simply divide or multiply – no work shown – when solving a d=rt or m = Vd. If they can’t do that, I don’t think they have internalized the meaning of the units.

  25. @mr bombastic

    “If they understand what the units mean, they will understand whether to divide or multiply.”

    Yes, that’s my point exactly. The high school seniors that can’t break down what speed is made out of (“the distance I traveled divided by the time it took…. oh, I get it now”), don’t understand the units, and haven’t figured out a “crutch” that scaffolds the information into something that they can self-check. I would rather they develop a process for understanding unit relationships that works with any new set of units they come to. And I would rather they have a way to check their own answers, instead of relying on an authority for confirmation. I don’t plan on being with them for the rest of their lives, so they need to be able to do this themselves, with whatever life throws at them. Yes, eventually internalize the formulas, but how do you get a formula in the first place? And how do you know you did it correctly?

    If you say that grade school students understand units, but middle/high school students don’t understand more compex units, it’s because nobody taught them the tools to use while they were in the easy-to-follow elementary math. Nobody was preparing them to use math the way it would be helpful for chemistry or physics, just pseudocontextual math class. Teaching about manipulating units is not too difficult for an elementary or middle school math student, but unit analysis isn’t done there. I was not taught to look at units to determine the correct math operation until high school chem, and I ignored the lessons then because I was smart enough to do it my own way. My crutch was to use proportions, which involves NO real understanding of units, except that the top and bottom units have to match.

    Until college, when my way didn’t work for the more complex, higher-level situations, and I had to go back and relearn the unit factoring method that would have helped me the first time around. I was good at math, so I didn’t think I needed an understanding of units, until I did. Most students hit this road-block in high school, and flounder on simple density calculations because suddenly the units are more abstract than splitting a pizza among your friends. Elementary students got those answers correct because they could see it happening and understand it. High school students can’s see molar mass as well, and won’t intuitively know the relationships. If unit skills were explicitly developed in the pizza questions, those skills would be ready for use when they were needed. It’s short-sighted not to prepare students while the concepts are easy.

    After a new relationship has been figured out by the students, then yes, calculate it in your head. I don’t expect them to do all work with written units on paper, but developing that ability early in the math curriculum will prevent future problems of being stuck without a formula handed to you. Is it a crutch or a learning/checking aid? I still think about the units every time I’m in the grocery store comparing the 2-lb bag of pasta to the 1.5-lb bag. That doesn’t mean I haven’t internalized the relationship. It means I like saving money and knowing that I’m doing it correctly.

  26. Joseph: “It’s short-sighted not to prepare students while the concepts are easy.”

    Um, fractions might be easy in retrospect, but for an elementary school student this is big stuff to tackle. That’s not to say elementary teachers should ignore your points, but I just want to address the assumption that elementary math is inherently easy. Believe me, there are students who would disagree with you.

    To add to your points, another area where teachers should be more thoughtful about units is division (since it is generally taught in a way unrelated to fractions). Teachers often ignore the fact that many division situations result in something like “slices per person”. Unfortunately, the questions students are asked tend to lend themselves to answers like “3 slices”. For example, “A pizza is cut into 12 slices. Four friends want to share the pizza equally. How many slices will each friend receive?” The problem is that if you do the division:

    12 slices ÷ 4 people

    Your answer should be 3 slices per person.

    Unfortunately, the way the question is asked, it is appropriate to answer “3 slices” because that is how much each person will get. Is there any evidence that students cannot handle answering questions with an answer like “3 slices per person”? Maybe there is a reason why it is not discussed.

  27. Thanks Brian, and I absolutely concede that I don’t know where in the curriculum this should live. I don’t want to be the secondary teacher whose knee-jerk reaction is to pass blame for all student shortcomings on inept elementary/middle school teachers, when we don’t understand the problems they are dealing with any more than they understand ours. Secondary teachers who complain but don’t communicate with earlier teachers about how to build the foundations for their future needs are not solving any problems. Elementary might be too early for this unit work. But it has to come in somewhere before the ideas are too abstract for the units to be understood. I should have said, “… while the concepts are concrete,” instead of ,”easy.”

    Your explanation of the use of “slice per person” is the kind of simple, intentional, laying-the-groundwork language that builds the foundation of the math relationships made in the future. I had grade school teachers who never said “three point five.” It was “three and five tenths,” because they knew that it would be important in later grades. If unit analysis is too tough for a particular grade, don’t use it. But make sure they understand the difference between 3 slices and 3 slices per person, so that it’s a smaller step up to the next level of complexity, and they are already speaking the correct language to understand it.

  28. Natalie Sterrett

    October 11, 2012 - 9:37 am -

    Dan, I love the way you presented this problem. I didn’t mind that you worked through it graphically. That was actually my favorite part. I guess you expect calculus students to understand what you’re doing without having to visually show them or explain it, but sometimes kids learn “find a min/max: set the derivative equal to zero” and they actually never think (and, unfortunately, never are taught) about why they’re doing those steps. Then when they’re confronted with a problem that requires that understanding, they have no way to approach it. So, anyway, I loved this problem! And it’s not like you have to ignore the calculus aspect of the problem. You can go through the abstraction process and the graphical solution, and then end with the calculus as an introduction to extrema or optimization. I would use it!

  29. @ Fawn I enjoyed reading your post.

    I also reflected on using this problem with a private tutee – brilliant question. Good work Dan

  30. Wow. That implementation is good, Dave. What IDE do you do HTML5 in? That’s slick.

    I think the community is onto something here. KA says, watch my video, answers these problems. This says, be in the problem.

  31. (Very late reply, sorry.) I agree that a teacher could look at a student’s work with a 300 or a 276.3 and judge the correctness of the number — and wouldn’t it be sweet if the technology could eventually make that judgment!

    What I meant was that the generalization step is very hard to see unless the examples all “look the same” (as they do in your samples). This is a hard problem! And it’s one that a single student or group might not encounter, because they’d run their second number the “same way” as the first. But others might have a totally different but consistent answer.

    In class, my tendency is to work out one example as a full class in an attempt to push everyone to the form I’m after. It’s either that, or expect a conversation about why this crazy thing is really the same as that other crazy thing.

    I agree with Chris D. that the tech can add a lot of access, speed, and community to this process, which is terrific. Now when does the Taco Cart iPad app come out!!

  32. Anyone who was getting an error on the results page should be able to see it now (, I had been slightly lazy with the coding and dropped a clanger.

    Shawn: No IDE, just a mixture of Ruby on Rails for the backend and Javascript for the front. It would be sweet if there was some sort of turnkey solution for this kind of stuff, and I’m sure you could get close to this using one but you’d end up sacrificing something from the original idea.

  33. I’ve just finished a geogebra file which helps you to play around with this problem to find the quickest path. It also allows you to input different speeds for each person on the path and in the sand. Spent all of last night trying to make it work – tried 4 different strategies.

  34. Bowen:

    In class, my tendency is to work out one example as a full class in an attempt to push everyone to the form I’m after.

    Not to be that progressive educator, but aren’t those different representations and the interesting discussion about their equivalence nine tenths of the fun of teaching math?

  35. Keep being that progressive educator, and teach me how to italicize in comments!

    It’s definitely a balance. If each student were asked to do three numeric examples, then generalize, I’d have them do the whole thing completely on their own and see what generalizations come. Multiple generalizations are awesome, and it’s really fun to compare them.

    In this case you were having each student do one numeric example then generalize through sharing. In my opinion, when students do one complex numeric example and report back, you get a sea of responses. For the Taco Cart, some kids will resolve the addition and subtraction in the numerators. Others will make a common denominator. A lot of kids will just report back a decimal, the result of hidden calculations. All of these answers are correct.

    Then comes a tough phase, showing that these answers are the “same”. But: the numeric answers are different. I’d rather do this comparison phase with multiple generalizations instead of multiple numeric answers. With multiple generalizations, you can actually show they are the same! Students usually want to chase this question, since they’ll want to know who’s right. (Everybody!). With multiple numeric answers, students may just accept that the answers are different — of course they’re different, we used different numbers!

    So definitely yes: different representations and the discussion about equivalence are terrific and I love them. I’d just rather have that discussion after each student makes their own generalization. If there is a specifically targeted “form” of generalization (like what appears in Step 5), working out one calculation as a group makes it more likely that students will compute their own the same way, allowing the generalization to come from the different students’ work.

    One example that comes to mind for me is the equation of a line, generalized from the slope formula — if students don’t put the points in the same order each time, the generalization won’t come.

    Either way, it’s still a lot better than the “here’s the formula now evaluate it” manner in which such problems are often presented to students, and I feel these overlooked phases are where the real mathematical thinking is done.

  36. It would be interesting to see what work students show for the “students result” page. The most common approach I see in a non-test setting is more of an ad-hoc approach. Use the calculator – jot down a number on the diagram – use the calculator.

    I agree with Bowen, that having different representations that use different numbers is probably too much of a swamp to wade through.

    Related to Bowen’s comments, I might give them a few minutes to decide on a second pick after seeing the scatter plot of all the guesses and corresponding times. I think running the calculations a few times is essesntial for kids that struggle with variables and generalizations.

    It is a tough balance. The hook is the kid gets to pick a number and see if they “win”. The more you ask a kid to do things in a particular way (show work for a method that would only involve hitting enter on the calculator one time), I suspect the less commited they become to the problem.

  37. When asking students to make a second pick, it takes some care to let students know their target is the process and a generalization and not to “home in” on the best possible answer. Especially for kids who have previously done a lot of solving by guess and check, they may not want to slow down to see the common behavior and potential generalization.

    To fool this I sometimes resort to “guessing badly” — in Dan’s work this is the “give me a number you know is too high” stuff. I might say here “Make a second pick you think is worse than your first pick, and find out if it is.” That takes away the implicit goal of finding the best possible answer, and can help students focus on guess-and-check toward a generalization.

  38. I think it’s interesting that the horizontal distance doesn’t affect the point you should aim for to get there fastest (once it reaches the “fastest point” that is, if short than that , then a straight line wins).

    That feels counter-intuitive on the surface and seems like it should be a good source of discussion in the classroom on two levels:

    Explain in general terms why this might be true.

    Show it mathematically – which involves making a lot of simplifications, which I think is an important skill in mathematics. Too many students seem to feel that math has been invented to make easy problems more complicated instead of the other way around.

  39. If anyone wants to see it I’ve now built out the example to encompass the entirety of Dan’s original idea.

    The next step conceptually (from a technical rather than pedagogical standpoint at least) is to make it work inside of say, iBooks Author. The most logical implementation would be to have it pull live results if it has network connectivity and otherwise fallback to predefined ‘example’ classmates.

  40. Love the task, both the original and Dave Major’s interactive version of this problem. Showed it to a few students. A few tiny tweaks I’d like to throw out there (I can only imagine how difficult it is to code these things):

    1) Students found the screen with circles/stars confusing. The students thought they signified spots in the sand instead of time estimates. Can the graph be placed ABOVE the picture? Also, would be cool to be able to toggle incorrect answers on and off.

    2) When determining best path in the final screen, would love to see y-value (seconds) while sliding across the boardwalk in addition to the x-values (which is less interesting to the problem).

    3) The text box seems to be missing for the question “What does this graph mean and how can it help us find the fastest path?” I’m not sure if the teacher-side of this is built, but I presume the idea is that the teacher can see the different responses – very cool feature!

    Teaching at a school that is committed to individualized instruction via technology as much as possible while preserving good pedagogy and math practice standards, I find this adaptation of this problem exactly the tool I need. If you ever want to pilot stuff like this at Summit San Jose just say the word.