The Most Interesting Math Problems To Me Right Now

But what are they? What would you call these problems? What are their essential features? Do you know of other problems like them?

Problem #1

Here are some points in the plane:

(4, 1), (17, 27), (1, -5), (8, 9), (13, 19), (-2, -11), (20, 33), (7,7), (-5, -17), (10, 13)

Choose any two of these points. Check with your neighbor to be sure that you didn’t both choose the same pair of points. Now find the rate of change between the first and the second point. Write it on the board. What do you notice?

Problem #2

Find the length of the third side of this triangle:

140104_1

Problem #3

Choose three consecutive whole numbers and add them. Write your sum on the board. What do you notice?

Problem #4

Everybody pick a number.
Multiply it by four.
Add two.
Divide by two.
Subtract one.
Divide by two again.
Now subtract your original number.

On the count of three, everybody say the number you have.

Problem #5

On a graphing calculator with overhead display, graph f(x)=sqrt(x) and show the graph to your class, but with a viewing window that is very small, like 3.9 < x < 4.2 and 1.95 < y < 2.05 (without showing them what function is being graphed for them). Show the class the graph and ask them what function they think they see. They will say that it is a line. You can also trace along the curves and find two points on the graph, if you want them to find an equation for the line that they think they see. The line they get will be approximately the tangent line. Then zoom out and they can see that the function isn't linear. Credit

Problems 1, 2, and 5 are from Scott Farrand, a math professor at Cal State Sacramento, and his student Janelle Currey. Problem 3 & 4 are my own constructions.

50 Comments

  1. Reply

    All of them are getting at linear behavior in some way. Problem 5 is a classic calculus dupe used to introduce the concept of local linearity leading to instantaneous rate of change. I didn’t do a ton of #3 examples in my head, but I think the point is to show that the sum is related to the original numbers by a non-coincidental amount.

    I smell clever trickery designed to get you thinking about slope.

  2. Reply

    See the activity around page 8 of this,
    http://www.vpri.org/pdf/m2007007a_revolution.pdf

    To me they all involve looking at patterns in how things change, looking at “increase by” patterns i.e. differential relationships.

    Since they deal with relative change, they also have a “kinesthetic” quality to them. You can understand the rate of change in a graph the way you feel your body in motion.

    See also this way of understanding of parabolas,

  3. Reply

    Something I noticed is that all 5 of these problems have a twist. But the twist isn’t, I think, that our initial expectations were wrong. Those would feel more jarring than these problems feel.

    Instead, the feeling is more of dawning realization. It’s more like, wait a second, there’s something going on here.

    Maybe these problems can be created by finding utterly unfamiliar contexts (to which we bring few expectations) and then ask students to do something that every student can do that creates a pattern (or some other sort of unexpected meaning). Dunno, that still feels fuzzy, but maybe it’s a start.

  4. Reply

    #1formula for slope
    #2pythagorean formula
    #3counting formula
    #4doesn’t matter what you do, subtracting what you started with gives everyone the same answer.
    #5function that gives the linear, usually a factor of the actual function.
    These are all skills on the way to somewhere else, like the steps needed for DOK 3 type problems.
    They are solvable by algorithms that don’t require understanding, which points me back to the Chinese parable in one of your past posts.
    I have used similar problems as starters, intros to ideas where I want to see what my Ss know (the formulas) or some, like #4, to gain my students’ attention. Nothing like sleight of hand to explain how math isn’t magic. Even though they sometimes feel like it is.
    I think these are good think starters, if used well.

  5. Reply

    I think we want to be cautious about labeling these kinds of problems in a certain way. For example, if I do problem #2 by itself and discover that the answer is a whole number near 12 (eg. 13) I won’t be surprised unless I have happened to have tried a lot of other numbers and noticed that actually these two things don’t happen all that often.

    In other words, whether or not a student will be surprised by any of these problems depends a lot on their background knowledge so I hesitate to call these problems surprising.

    More later, I need to take my kids outside for a bit.

  6. Reply

    What jumps out at me here is the “open-beginningedness” of the problems that allow students to do some (not a lot) of work and will likely spark some curiosity and/or a debate. Offers a ‘hook’ without necessarily offering any sort of relatable context.

  7. Reply

    The problems all seem fairly scripted. With all of them, there is the opportunity to “play” with the numbers. None of them have a real world context. They might not make the cut for our state test on that alone. In some, students might not know what to look for beyond the answers they compute.

  8. Reply

    I agree with David: there’s some danger in using these problems because if a student misses the aha or trick or issue, there’s not much to pull them back to, and they may feel like they missed a joke.

    These problems are tricky business but can pay off BIG when they hit.

    My personal favorite was in Alg2/Trig when kids were learning to graph sine and cosine variants, and I had them graph the evil function y = sin 666x on the TI-83’s standard trig window. If you’ve got a TI-83, try it!

  9. Richard Mark Yanku

    January 4, 2015 - 3:21 pm -
    Reply

    Discovery … self then shared … #4 looks like it’ll raise the roof … #3 is loaded with a-ha potential … Thanks for Sharing

  10. Reply

    You are asking students to look and think first, before strategies are shown. These problems also pose opportunity for developing conceptual understanding of fundamental algebraic principles. Here is a link on how #4 can used to reinforce basic arithmetic and algebraic skills: http://bit.ly/1tBPcVl

  11. Reply

    These are like magic tricks. Show the trick & hope folks are interested in seeing how the trick works. Some variations would be a line with a hole: y = (x^2 – 2x + 1)/(x – 1); Graphing several equations that all turn out to be the same/similar: y = 2x + 1, 2y – 4x = 2, y^3 = x^3 + 1; plotting points along with the product of the point & a rotational matrix; divisibility rules.

    I don’t see how #2 fits in. Strange to assign “credit” to problems that have been around forever.

  12. Reply

    In my opinion, the “Number Tricks” lesson linked to should also include some opportunity for students to work without the overhead of algebraic expressions. Then, the concept of using and evaluating algebraic expressions can derive from the natural question: why did this trick work?

    I agree that it reinforces basic arithmetic and algebraic skills, while also providing some necessary connections in the links between the two: it becomes more clear that n stands for a number, because you just worked it through for several numbers.

    There’s some detail on elementary-school thinking around this type of task here: http://thinkmath.edc.org/resource/early-algebra-little-kids

  13. Reply

    I like these types of problem, but I do agree with many of the other commenters as they have to be dealt with carefully so that the kids don’t see it as mathemagic. Here are some (higher level) problems that I use in class:
    * Pick an amount of $ to put in a bank. This is a bank run by a bunch of fools and they give you 100% interest. How much is in your account by the end of one year? What is the ratio of earnings? What if you go in monthly to ask them to calculate the interest? What is the ratio? Awesome! Go in daily, what is the ratio? Hourly? Every second? Every microsecond? WTH?
    * Graph some functions where the slope at any point is the same as the function value.
    * What is 1 + 1/1! ? Add on 1/2!. Add on 1/3! …
    And after some of that stuff, you throw them something like this:
    * f(x) = -720+1765 x-1624 x^2+735 x^3-175 x^4+21 x^5-x^6 (spoiler alert: its just x-(x-1)(x-2)(x-3)(x-4)(x-5)(x-6) ). Evaluate f(1), f(2), f(3) … f(6), f(7)…

  14. Reply

    I apologize, because this is a kvetch that I’ve offered before, but I’m somewhat concerned that a curriculum or teacher could overuse this technique. If kids came to expect these weird surprises in class, I suspect that we’d see diminishing returns on both the learning and (especially) the engagement potential of this type of problem.

  15. Reply

    I’ve spoken with other teachers who feel that kids are like bacteria in their ability to adapt to novelty. One day of different is good, the second day the new wears off; the third day they are immune.
    These interesting problems must have follow through and be anchored to something they already know. I think that if they are used well, what the students comes to expect is not the next “show” but the next “what are we going to sink our teeth into next.” Then, the ability to see patterns, to notice -to be curious- has a hope of becoming the norm, rather than just an entertaining “trick.”

  16. Reply

    I appreciate the resources and commentary, everybody. I’ll be integrating all of it into a post later.

    But it interests me how many of you suggest teachers should use these surprises sparingly. Each of these problems attempt to provoke disequilibrium in a student. They start with familiar operations in a familiar context through which something unfamiliar happens, which students then have to accommodate.

    This is disequilibrium, which Piaget argued is essential for learning, which many of you argue we should strictly ration.

  17. Reply

    I’m surprised that you all see #2 as simple. Only my quicker witted students know that the answer is “somewhere between 7 and 17”.

  18. Reply

    To be clear here, I’m not saying that these kinds of resources shouldn’t be used, only that whether or not these have the intended effect seems to me to depend a lot on what the students understand and do not yet understand.

  19. Reply

    I wouldn’t say #2 is simple as much as it offers a “snap” answer of 13. Then you show the kid a 5-12-10 (or better yet, someone else in the room thinks of it) and they go …. ooohhhhhhhhhhh

    Disequilibrium! And then the question evolves to “what lengths are possible”, with some kids forgetting about non-integer lengths, how big 16.99999… can be, and a potential debate about whether 7 and 17 form triangles.

  20. Reply

    Right.

    #2, contra those who thought it was sooo easy, is the one that provokes equilibrium through unexpected disorder, rather than the unexpected order of the others.

    Are we sure kids are going to get super bored of these? Are we sure kids won’t develop some extremely useful intellectual dispositions instead?

  21. Reply

    But it interests me how many of you suggest teachers should use these surprises sparingly. Each of these problems attempt to provoke disequilibrium in a student…This is disequilibrium, which Piaget argued is essential for learning, which many of you argue we should strictly ration.

    I don’t think that this is the right way to characterize some of the above objections. There’s more than one way to provoke disequilibrium than with “twist” problems. You can get protein from hamburgers, but you don’t need to get protein from hamburgers.

  22. Reply

    I provoked disequilibrium this past weekend (at least for some of the students) with an activity I did with 8 to 10 year olds.

    I asked them to count out 100 snap cubes and prove that they had 100 cubes.

    One group was counting out the blocks and snapping them together and not paying attention to the total. They just thought they had to get to lots of blocks first. I asked them how many blocks they had. They had NO IDEA. So they decided to count their blocks.

    It turns out that all three of these students thought that in order to count out their 4x5x5 block structure, they just needed to count the outside. First they thought they had 20 + 20 + 25 + 25 blocks. I asked them how many sides the shape they had constructed had (first disequilibrium – I’m offering them information that directly contradicts their 4 side model).

    Once they had established that they had six sides, they now thought they had 20 + 20 + 20 + 20 + 25 + 25 blocks (counting each face). I then pointed at a block on a corner and asked how many times this block would be counted (second disequilibrium – the fact this individual block gets counted three times contradicts their model of counting the outside).

    One of the kids then tried a different way to count and said he was going to slice up the shape. He then counted five 20s and said the number of blocks should be 20 + 20 + 20 + 20 + 20. Another students, upon seeing this method (third disequilibrium – he’s been prompted to use a different method to think about counting) decides that they could also count it as 4 twenty-fives (25 + 25 + 25 + 25).

    In this case, I think that the task itself, without my prompting and the structure with which kids did the task (prove that you have 100 blocks to someone else rather than just count out 100 blocks, work together to construct this proof, have me around to ask pointed questions) is critical to this disequilibrium.

    Magdalane Lampert recently asked me why we were separating our website resources into instructional resources and curricular resources. “Curriculum resources are used with instructional routines. I never know if something is supposed to be curricular or instructional. It’s all the same pile to me.”(paraphrasing).

    It seems to me that asking if any problem will promote disequilibrium depends on the knowledge of the students and the way the task is actually enacted in their context. Counting to 100 isn’t necessarily going to be a task most people would learn from, but because of the way that I framed the task, the way the students acted on the task, the ways in which I responded to how they acted, it ended up being something to learn from.

  23. Reply

    @David, that’s helpful. Your interjections still feel twisty to me, to some degree anyway. Do you see them otherwise?

    It seems to me that asking if any problem will promote disequilibrium depends on the knowledge of the students and the way the task is actually enacted in their context. Counting to 100 isn’t necessarily going to be a task most people would learn from, but because of the way that I framed the task, the way the students acted on the task, the ways in which I responded to how they acted, it ended up being something to learn from.

    Very much agreed, but those particulars don’t make the task of generalizing completely impossible. The fact that chapter seven appears after chapter six in a textbook is the author’s way of saying, “Your students do this only once they know that.” It accounts (imperfectly) for the knowledge of the students. (eg. The warranty on Problem #2 is that “you shouldn’t do this unless your students understand the Pythagorean theorem.”)

    As for the math knowledge for teaching your dialog with students required, sure, that’s difficult to come by in curriculum. But I believe there are still a lot of useful generalizations we can (and have yet to) make.

  24. Reply

    I am still not seeing the purpose of #2. If students have the Pythagorean Theorem, then it is dirty pool to provide a triangle that appears to be a right. You are not providing a surprise so much as setting the audience up to be the dupe – gotcha, it doesn’t have a right angle marker. There is a certain mean spiritedness to this one that I don’t see in the others.

    Is the punchline really that the side could be between 7 & 17?

  25. Reply

    Go on!

    Your interjections still feel twisty to me, to some degree anyway.

    If we’re trying to characterize every problem that leads to math learning, then I doubt that we’ll land on a meaningful design pattern. That class of problems is too large.

    I would describe some of the problems in your post as “distributed pattern” problems. You take a pattern, but distribute each element of the pattern to different students. Of course, there’s an “undistributed” version of this same problem, where you give each student all the elements of the pattern and ask them to make an observation.

    It’s unclear to me how strong your claim is. Which of these feels right to you?

    1. Students will only feel disequilibrium (and only learn math) from the distributed version of the task.
    2. Students will always feel more disequilibrium (and learn more math) from the distributed version of the task.
    3. Students will often feel more disequilibrium from the distributed version of the task, though it depends on the students’ prior knowledge and other relevant context.

  26. Reply

    Maybe it would be useful to compare these to other problems and construct a taxonomy of task type based on examples and non-examples?

    Also, would it be useful to see how other people have constructed taxonomies for this purpose? For example, Bloom’s taxonomy has much more nuance to it (if you actually read the associated book, which I have not yet read, but know about because of another book I am reading) and is the result of many assessment and curriculum people talking about a very similar issue.

    I agree that being able to talk fairly narrowly about mathematical tasks and label them in certain ways is likely useful for those of us who use tasks, share tasks, and create tasks. I’m just wondering if we can build on someone else’s start to this work?

  27. Reply

    Sorry, when I said “Bloom’s taxonomy has much more nuance to it” I meant “than what is typically discussed and shared with teachers.”

    As I recall each of the six top-level taxonomies has sub-taxonomies with further descriptions and examples.

  28. Reply

    I Hodge: I suspect that what I use #2 for is different from what Dan uses it for.

    I use similar questions to stop students making poor assumptions. To try and stop them thinking they know what a question is about before reading it properly.

    A far too common mistake in geometry is to assume any angle that looks right angle must be right angle (and anything that looks isosceles must be isosceles, anything that look parallel must be parallel).

    Over the years I’ve done it, it generally leads into interesting asides as a “practical” introduction to greater than that is not equal to, one we agree that a triangle of no volume is a line. But that’s just an aside for me.

    There’s no duping involved in my case, as I warn them at the start of the topic very strongly not to make assumptions. If someone puts a hand up and asks if it is right angled, I say “we can’t tell”.

    When we do parallelogram areas I also do one with stated outer lengths (perimeter) and ask what we can say about the area. That gives a lower bound of greater than zero, but an upper of greater than or equal to the area of the equivalent rectangle, which is a nice difference from the triangle.

    It helps start them on the process of moving away from every problem has one, exact, answer.

    Areas/volumes is also where I introduce accuracy versus precision, and the effect of measurement errors (although generally only the clever really get a proper hold on that).

  29. Reply

    @Michael, I’m not sure I’m signing onto this “distributed pattern” pattern. I’m not sure the distribution of the pattern (either b/w students or w/i the same student) correlates all that well to disequilibrium.

  30. Reply

    I’m not sure I’m signing onto this “distributed pattern” pattern.

    Of course, and I didn’t mean to put words in your mouth. But I think the “distributed pattern” thing fairly characterizes #1, #3, #4.

    I’m excited to see how you characterize the sorts of problems that are exciting you. I feel like there’s a thesis floating around here about the way surprise relates to learning that isn’t explicit yet.

  31. Reply

    In response to Dan:

    This is disequilibrium, which Piaget argued is essential for learning, which many of you argue we should strictly ration.

    Rather than ration disequilibrium in our mathematics lessons, we should be thinking long and hard about how we can find more opportunities to bring this technique into our classroom. If we choose to ration, in an attempt to somehow reserve the effectiveness of this type of questioning, we are forced to resort to less interesting questions.

    I think a common mindset amongst math teachers is that we have to work hard to “trick” our students. On quizzes, tests and every once in a while during our lessons. Are we concerned that by introducing disequilibrium in our lessons too often that we won’t be able to trick our students as effectively? Should we be trying to trick them in the first place?

    If we use this type of question in our lessons as often as possible, I think that students will begin to look at all problems more critically as if they are searching for a hidden trick somewhere.

    Would it be all that bad if students became such great problem solvers that they manage to foil our “trick” before we anticipate?

  32. Reply

    This is the showmanship, the magic of our craft, the sparking of curiosity, introducing perplexity, that brings the classroom alive. Good math teachers build up their bag of these tricks over the years so that this becomes second nature to them.

    I was inspired by my physical science C&I instructor when I was a student teacher in the late 60’s. He told a tale of when he was teaching in one of the toughest parts of NYC, he would set up material in the back of the classroom so that 5-10 minutes into the period a small fire would start (the lesson was going to be about spontaneous combustion). He would yell “fire” and run to the back of the room and put it out with his fire extinguisher. He definitely had everyone’s attention. He would then start questioning everyone “Who started that?” “Did you do that?” “Someone must have done that!” Eventually, “A fire can’t just start by itself can it?” After a quick clean-up and class discussion his lesson had everyone fully invested in learning about the chemistry of spontaneous combustion.

  33. Reply

    Here is a common example from the first year of algebra (after doing a bit of factoring).
    Demonstrate fantastic mental calculations orally such as 17^2 – 16^2 equals 33, and 23^2 – 22^2 equals 45, an on and on. As students see the pattern they begin to join in.

    In middle school and high school students can then explain why this works. With 5th graders they are tickled to discover the pattern and the room is alive with a math teacher’s favorite sound “ahh!”

    The same with the pattern 19×21 = 399, 41×39 = 1599, 59×61 = 3599, …

    There are a number of calendar patterns that work their “aha” magic on students as well.

    In advanced algebra and precalc I use to begin one lesson by mentioning that “every polynomial function has another function associated with it call its “cousin function.” I would then ask students to call out a polynomial function and I would give then another function that was the cousin to their function (by taking the derivative). They would continue giving me functions and I would give them their cousin. Eventually someone would give the cousin function for me. This would really get the other on the edge of their seat. 15-20 minutes later the majority of students are taking derivatives. A few days later, I’d mention that a particular polynomial function is a cousin function to some function. What is the original function? And off we’d go again…

  34. Reply

    I’ve been working on transformations on the coordinate plane with 5th graders here in SF. We place different pentominoes on the coordinate grid and make a list of the coordinates of the vertices of the pentomino. Then they sketch the image reflected across the x-axis and record the new coordinates of this image. After a number of examples they are able to describe, in their words, what happens to the coordinates when a figure is reflected across the x-axis. We then do reflections across the y-axis. This is followed with 180° rotations. Our 5th graders were able to generalize both cases. Make it a puzzle, or ask them to predict, and their curiosity takes over.

  35. Reply

    Dan, I think it could be useful and clarifying to the discussion if you could offer some important and varied non-examples of the kind of problem that you’re talking about and looking for.

  36. Scott Farrand

    January 7, 2015 - 8:00 pm -
    Reply

    These problems aren’t of value in and of themselves – it is in their careful use that they can provide significant leverage. The fifth problem for example is one that I use as the warm-up exercise for a lesson on linear approximations in a calculus class. Starting with this exercise gives my students the fundamental idea of what is going on in a linear approximation better than any explanation I have come up with.

  37. Reply

    Here’s one I love and have used it many times, especially when teaching combining like terms. It also helps kids generalize and understand that a variable’s value can change depending on the situation. It’s called “Two Sums” and can be found in Mathematics Teaching in the Middle School, volume 16, no. 2, September 2010, pg. 68.

  38. Elayne Bowman

    January 8, 2015 - 6:15 am -
    Reply

    I have noted that my students and I all spend much more valuable time trying to figure how “WHY?” or “WHY NOT?” than simply practicing the examples in the text. For example, I had a student ask me a question I could not answer this week. “If y = 3x +2 is not direct variation, is it indirect variation?” Hmmm. I told her I didn’t know. We talk about direct, inverse, and joint, but I’d never thought of indirect. But she was THINKING and that tickles me to death!

  39. Reply

    When dividing polynomials, I teach them the basics of long and synthetic division and let them choose which one they want to use.

    I then tell them to divide a given polynomial by 2x-1 using either method. I asked students to share their answers given the two methods and ask what happened and why. Comes to an interesting discussion.

  40. Reply

    I have done this type of problem whenever with high school students, preservice teachers, and inservice teachers. I’m not sure that they are imposing disequilibrium, because often they aren’t expecting anything. More often it seems to be a case of “that’s cool/weird”, followed by the ever more important “I wonder why”. I have two similar favourite problems that had this impact.

    The first of these problems arose from a situation in which I was told the night before that parents would be touring through my class in period 4 on the next day and that I had to have the class using the graphing calculators. The parents touring wasn’t an issue, but the graphing calculators were — the students I had in period 4 had never used the graphing calculators (long story), and we were heading into the final unit of the course: rational expressions. On the fly, I decided that we would spend that period graphing rational expressions (despite rational functions being in the next grade), just to see what the students thought (so far they were familiar with linear and quadratic functions). They very quickly figured out how to enter functions into the graphing calculators and were on to the first rational function, y = 1/x. “Something’s wrong with my calculator”, “mine too”, “hey they are all busted”. I immediately said, well then, I’ll do it up on the one hooked up to the projector. Second later “yours is busted too” — the investigation was on — why does the graph look that way? Soon they were pros at identifying non-permissible values (exclusions from the domain or undefined values, if you prefer), but then I threw in a function that had a common factor on the top and bottom, resulting in another non-permissible value which was not undefined, but indeterminant. “What’s going on now”…. We had over 50 parents through that room (not bad for a community of 800 and school of 300), and neither the students nor I remember any of them coming through; however, I never had to remind any of those students to state and consider non-permissible values when working with rational expressions — beautiful!

    The second problem was similar, but happened around trigonometric identities. Before talking about trig identities or proving them, I gave each student a paper with a list of trig functions on it. Their parter’s paper had a list of trig functions that could be matched to the other to create an identity. All of the matching graphs brought great curiousity: “how can those graphs be the same?” I ended by asking them how they would like to show that the graphs were the same on paper, and they created the identities that we went on to prove.

    I think WTF is a good description of this type of problem, but not quite classroom appropriate. I usually think of them as WAM problems — wait a minute….

  41. Reply

    I agree that these WTF teaching moments really engage students and bring a cognitive awareness that is essential for math teaching. Here are some other examples I can think of:

    1. Squaring both sides of a trig equation and then checking an answer graphically to see that the solutions don’t match up.

    2. Solving a quadratic inequality (like x^2 – 5x + 14 > 0) using the Zero Product Property only to find that it doesn’t match up with the graph.

    3. Finding out that there could be two possible triangles using the Law of Sines.

    In all cases I try to build the drama. For example, for #1 I sell it as an afterthought that we are going to check our answer graphically to give them a way to check their work for an upcoming test.

  42. Reply

    If I had to play the old Sesame Street game of “which of these things is not like the others,” I’d choose problem #3. All of the other use inductive reasoning to get at a pattern that would normally be taught as a rule that you then apply.

  43. Reply

    I get A level students to try and solve two simultaneous equations. If you are selective with your coefficients they may not notice that the lines are parallel. This will likely produce nonsense like 0=1, I think. If the equations happen to be the same line itself you will get truths like 0=0, not false but no help!
    Students are surprised by these results if they don’t stop to think about the graphical solution.
    I USED to teach like this of course. When this lesson was observed I was described as negative, because I was saying ‘can’t you do it!’ The students themselves were fine that I was only teasing them and it was part of the ‘performance’.
    Since then things have got worse. Despite what we are told, everyone wants them and they want to be merely spoonfed!

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