Between Simple And Easy

My favorite problems are simple but not easy. The difference hasn’t always been apparent. I’m talking about clear, minimal constraints which require complicated, comprehensive thought. These problems are rare, but some lucky days they arise from a single image, like the one up there, like the one today.

The Question

If that table tennis ball is the Earth:

  1. how big is the Sun?
  2. how far away is the Sun?

Follow Through

You take bets. Is the sun a tennis ball? A beach ball? (A: something closer to a weather balloon.) If you miniaturized the solar system, what solar body would focus the Earth’s orbit? (A: the taqueria down the road.) You pick their pockets with these bets, getting them to buy into the problem unwittingly.

Maybe you put them into groups and wait until they requisition data. (eg. the radius of the Earth, the tennis ball, and the Sun; the mean distance from the Earth to the Sun.) Maybe you give them all laptops and let them scour the ‘tubes for the same data.

And I Wonder Constantly:

  1. do these simple-but-not-easy questions exist for every math standard on the books?
  2. who has them?
  3. are these people easily extorted?
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. Yeah, who has these questions?

    I teach physics in addition to math and I saw a great presentation by a Reed College professor called, “The Physics of Rainbows.” It was filled with interesting questions and great “content” about light, reflection, refraction, etc. And I often wonder if I could teach an entire year’s worth of physics “content” just by asking a series of “How does X work?”

  2. To answer ‘wonder A’, I think these questions exist all over. Teacher around the planet ask these questions.

    The problem is, they probably don’t recognize the genius in asking; instead, they’re exhaling b/c they just came up with something to do in the last minute.

  3. Are there particular standards you are stumped by? You’re more likely to get mileage out of “give me something related specifically to this” than “give me everything you have”.

    In the last two days my students played Let’s Make a Deal (Is it better to switch? I don’t have large doors in the classroom so we did it with envelopes) and put bets on the Birthday Paradox (What are the odds two of us here have the same birthday?).

    But probability is easy. Throw down something hard.

  4. @Ken (and also @Jason, relatedly): I have to note here that these very very constructivistic problems don’t work so hot unless preceded by small measures of very very crucial direct instruction.

    So while I’m curious if you (Jason) have an awesome, concise question for Law of Sines / Law of Cosines (or maybe Ken also, who knows) I realize the question, itself — however spontaneous-seeming — will represent the end of something very deliberate.

  5. I’ll attempt a counterexample: factoring polynomials. Does there exist a simple interesting motivational problem related to factoring polynomials?

    Geometry and probability seem to lend themselves to real-world situations, but I expect there’s some algebraic concepts that are a bit harder to motivate with simple setups.

  6. Dan, is there a lake (or large ditch) nearby?

    Here’s the question: How wide is it across the lake?

    Unfortunately, we don’t have anything that exciting nearby, so we made our field trip to the back of our football bleachers and pretended that was our “lake”. The main thing is you can’t just measure it directly.

  7. I’ve been thinking about exactly these wonders / questions myself as our new education reform here in Quebec is built on learning and evaluation through real-life contextual problems. I would really love to teach my whole course through problems like these, but as others have mentioned here, get stuck on examples for algebraic topics such as factoring, etc.
    Wouldn’t it be wonderful if there was a place for everyone to post their simple, but challenging contextual problems and have them organized by concept?

  8. Dan,

    Regarding Law or Sines (or Cosines), you could try to do something with a treasure map or surveying (similar to the lake comment by Jason). Also estimating tree height with angle and distance, though that is more easily done with a tangent than law of sines.

    I think surveying is a good motivator for lots of trig concepts, thought the law of sines is a bit algebraic. I submit that algebraic concepts are harder to find simple example for.

  9. sweet jesus i’d pay at least $100 for that book. an extra hundred if it had a scenario where a kid needed to graph a parabola. like REALLY NEEDED to, not some b.s. throwing a baseball to first base contrivance.

    and i’m totally stealing your ping pong ball for proportions tomorrow.

  10. Steven:

    This is a decent start —
    Suppose you bought 50 square tiles. If you want them in a rectangle, how many ways can you decorate?

    Segue that into the perimeters (specifically, half the perimeters) and you’ll get factoring x^2+15x+50 and related equations.

    re: abstract problems in general. Yes, it’s hard. I try to use puzzles, like this one for introducing completing the square (4 solutions I know of, and all of them algebraically equivalent to completing the square).


    How high does the rocket go in the last launch? (at 1:10)

    [Note: You’ll have to ignore horizontal movement in this case, so it’ll only be approximate. It’ll be a time vs. vertical height equation.]

  11. I never ‘got’ calculus in calculus classes. It wasn’t until I started to use it in engineering classes that the bulbs began to glow.

    Having said that, I’m not sure that the question of theory vs. application or proof vs. problem solving is easily solved. It’s a bit of a chicken or egg first puzzle isn’t it? You really can’t afford to go after one without the other. Maybe it depends upon what side of the brain you favor or something. I always needed the anchor/example of constrained domains found in engineering and found the 5-n dimensions of theoretical math daunting.


    Here’s a challenging and fascinating real world problem. Try corealis effect. It’s got N/S hemisphere challenges, latitude challenges (function of cos of latitude), scale challenges (doesn’t work small scale), it’s a four dimensional challenge, and it touches us all as an engine of storm circulations. Have kids do it for different planets.

    Here’s the question to kick off the month… Why do high pressure systems circulate clockwise (N Hemisphere)?

  12. For an application type problem of sine law and cosine law, I get the class to measure the height of the church across the street…the one catch…they have to stay on this side of the street to do it.

  13. One of the challenges of public school teaching is five periods per day with 180 days of compressed time– it’s a challenge for learners and teachers. I’m not sure how many of these type of questions exist, but certainly if there are best ways (questions) for key learning concepts, value is added by sharing those best practices. That seems to be a big part of what I’m seeing here–an effort to share how to support students in realizing important learning outcomes.

    I’d wonder if having this type of learning as a part of many units, among other learning strategies wouldn’t make the most sense–an occasional activity, rather than daily fare. It’s a naturally interesting and engaging question which I’d think everyone could have an opinion, an involvement — by a long shot, not all the real world questions are.

  14. Too many worthwhile comments to hit one at a time so briefly, then, to Tom, with a few other drive-by remarks:

    A five-minute scan of Core Plus indicates a curriculum built along the same lines as IMP and CPM, lines which cut straight through constructivism and problem-solving (individual units in each curriculum build toward an overall project or theme). The link to sample exercises is dead, though, so it’s tough to tell if it dodges my two largest concerns. Namely:

    1. We need to practice skills without application. ie. The scale solar system problem wouldn’t have worked had we not run through some scale triangles — just triangles, no word problems — first. IMP sneered at those problem sets and the local PTA ran it out of town at the end of a pitchfork the year before I arrived.
    2. The application problems must prioritize a kid’s senses and perception of the world over a content standard.

    #2 is why I have little hope finding these excellent problems outside an informal bull session like this one here.

    For instance, my Geometry textbook, which is pretty great, writes up a variant of Jason’s lake problem as:

    A surveyor at point A needs to calculate the distance to an island’s dock, point C. He walks 150 meters up the shoreline to point B such that AB is perpendicular to AC. Angle ABC measures 58°. What is the distance between A and C?

    And it’s like, did anyone still care after that first sentence? After the first five words?

    I need problems — one per week for 40 weeks will suffice — which distill these murky, adoption-committee-placating problems, heavy-laden with set-up and jargon, into several compelling sentences or one visceral image or any combination thereof, but which definitely messes with a student’s self-assured sense that she knows exactly how the world works.

    Here’s tomorrow’s opener. One sentence, one image. This question calls for a hammer and, though all my students have a hammer in their toolboxes, their first reaction will be, wtf. This is great. We’ll discuss, what can we possibly measure here?, what can we do with those measurements?, what shapes can we fit around the lake?, etc.

    I am collecting these problems. I find them only rarely in published curriculum but, damn, people, do they abound in here.

  15. Same topic (Law of Sines / Cosines).

    How fast do you have to be driving to outrun a speed camera?

    Start with perpendicular for ease of calculations, and then work your way to pointing towards the road at an angle.

    Also fun: you can declare the angle of your digital camera to be identical to that of whatever speed camera we’re talking about, and the students will have to figure out how to set up an experiment to calculate that.

  16. Also, I really do like your ‘secondary openers’. (Your letter one on this slide.) I’m thinking of trying them next year. How much mileage do you get out of them? Have they been worth the time?

  17. Dan, fastest you can get is 1/1000 sec.

    There’s also 1/500 sec, 1/250 sec, etc. You might want to express the problem with a depiction of an actual camera so you can reference that camera’s specs.

  18. OK, I see what you mean, quick openers, not an extended word problem a day.

    What we really need, imho, is a strong free content curriculum that can be the skeleton for framing and distributing lesson-scale innovation. Or perhaps we need a national curriculum to provide that framework. Unfortunately, since pitchfork-wielding partisans would demand a hand in shaping it, I’m not very hopeful that such a thing could be written in the US in 2008.

    Also, this kind of innovation is what Japanese Lesson Study practices are designed to foster, and why they drink our milkshake, educationally.

  19. @Jason, and the aperture angle on a speed camera? Actually is there a site that lists these specs? A dealer? Good problem. These visceral one-liners are all the better if they involve lawbreaking, of course.

    Also, the total time cost on that miscellaneous final question is around a minute and pays off huge in classroom culture and classroom management.

    @Tom, this isn’t the first time you’ve tossed out the Japanese model. Am I right that — in addition to the professional isolationism of the American teacher — funding is a limiting factor here? Like, if we can’t get cash for a weekly hour with our departments, what’s the over/under on detailed observation time?

  20. Dan,

    I keep throwing it out because it seems like such an exact fit for your point of view. But yes, it takes time, and time takes money. This is why I get so frustrated with arguments that funding is not a central problem for school reform. Even if you aren’t just throwing money at a problem, just about everything meaningful costs some money, and in any other kind of enterprise, that’s understood.

  21. Dan, there’s a good table here.

    @Tom, I’m not sure why we need some sort of national initative to have curriculum the way we want. Can’t we just write it?

  22. Jason,

    In theory, yes, the whole thing can be written from whole cloth.

    In practice, I’m dubious, because a good multi-term math curriculum has to first be built on a pretty serious theoretical foundation. For example, step 1 is “What is mathematics?” That’s the kind of thing which is much easier to do within the structure of, say, a graduate school, than a mailing list and/or a wiki.

    Put another way, open source software projects that start with “Hey, anyone want to help me build an X?” usually don’t get past the “What should an X do?” and “What is the ideal programming language/database/platform for X?” phase. Ones that start with “Hey, I wrote an X, you might find it useful” do much better.

  23. This is kind of funny, These comments have run the gamut of my insights and frustrations from the last 3 years of teaching math.

    – basing lessons on an interesting, engaging problem is where it’s at
    – where do I find these types of problems
    – they have them in Japan, as a result of their lesson-study PD format
    – I don’t read Japanese
    – nothing published in English is what I’m looking for
    – I don’t have the power to completely restructure PD for my district. Conversations with administrators have resulted in uncomfortable silence.
    – Japanese style lesson study would take convincing me and several of my friends to voluteer many unpaid hours to PD. This is where this train of thought runs off the rails.

    I ordered and read the Singapore math books, the problem being their concept development doesn’t line up with my curriculum, like not even close. Just today I ordered an examination copy of Core Plus, so we’ll see. My guess is I will probably find a few good problems I can incorporate into lessons where I haven’t found a good problem yet, but it’s not going to be the holy grail.

    I have found that with wikis and listserves, lots of people want to sponge off them, and hardly anyone will contribute. Observe this fizzled effort: in which dozens of people said they wanted to participate, and only 4 teachers ended up contributing.

    It all leaves me wishing our educational “leadership” would get its head out of its ass.

    @Jason: I need to motivate finding axis of symmetry, plotting 5-9 points, and drawing a smooth curve on 1/4″ graph paper. Rockets aren’t going to work for me. The best I do is a rectangle perimeter vs. area thing, but it’s the most boring day of the entire year.

  24. Kate, the axis of symmetry is exactly the moment when the rocket is highest.

    But if you’re meaning justifying the actual act of drawing out a parabola by hand (admittedly the rocket thing can be done just with algebra on the quadratic) you need your students to make a parabolic hot dog cooker.

    They’ll have to lay out actual graph paper and sketch out the parabola first before they bend their plastic to match.

  25. This topic is morphing into ‘why isn’t this stuff possible in our curriculum’ and at the risk of a shameless plug, let me say, I’m up for that.

    I’m ready to open my Kimono now at When Galaxies Collide which is about just that question.

    What is it that prevents so many smart, dedicated, energetic people from effecting change in our profession and what might we do about it?

  26. Here’s a geometry puzzler I ran across a few years ago…

    Why is it that however you fold a piece of paper, the fold is a perfectly straight line?

  27. @Alex, what I mean is that my students have the tools to solve the opener, they just don’t yet see the application.

  28. Hm…I guess I missed that the axis of symmetry referenced in comment 31 was actually related to the parabola in the rocket problem.

    Speaking of parabolas, I think the best in-class demonstration I’ve ever seen was in a class on Experimental Design and Measurements in undergrad. The setup was a catapult that had several variable settings, such as using 1 or 3 rubber bands, pulling the arm back 90 or 135 degrees, and something else, I don’t remember exactly.

    Anyway, suppose he just varies those two settings between two values, giving 4 different settings to test. He puts down a tape measure in class then launches the ball with those 4 settings, maybe multiple times to get an average. After launching the ball and recording the results in this magical spreadsheet, he says, ok now let’s try something different. He puts on two rubber bands, chooses a different angle that we didn’t test and punches that into the spreadsheet, which linearly interpolates an estimated distance for the ball to travel. We put a bucket at the predicted distance, and this ball lands in the bucket, based on some crazy spreadsheet.

    Since I can’t remember the exact contents of the spreadsheet, I must not have been scaffolded properly or something, but it blew me away. I still talk about it to this day, though in the comments section of my friend’s blog.

  29. Just a comment or two.

    Steven, as far as a motivation for factoring polynomials, there may not be one good opener question that would spark interest. But what about motivation for doing bicep curls? I coach basketball , so I have some cred when it comes to bringing athletics into the classroom. My analogy is you will never be asked to flex your bicep, or do a bicep curl during a (insert athletic event here). However, the training aspect (that may on it’s own appear tedious and unrelated) allows one to perform better at other more elaborate athletic tasks. Learn how to do the exercise well, practice it, and know when it is necessary, and it then becomes useful, part of the bigger picture, even if on its surface it may appear irrelevant.

    That usually is a response to the generic “when are we ever going to use this” comment, and of course it is lost on some students…

    With regards to the general tone of what “style” of questioning/problem solving/curriculum choices is best, I am sure that if there was a one-size-fits-all program that worked for everyone, some Prentice Hall/Wiley/Key Curriculum/Disney/GE/Microsoft conglomerate would be marketing it to all. One of the most frustrating, yet I find interesting things about teaching is the mixing together all of these differing techniques, philosophies, activities, styles and curriculum design all come together to form a classroom environment. The ability to adapt is I feel one of the strongest traits teachers have. If I ever find that holy grail of teaching that works for all students, in all disciplines, on all of the standardized tests, hitting all of the various learning styles, accomodating all student needs, I think I would get bored.