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. Mike Lawler

    May 29, 2014 - 9:56 am -

    I’m not a big fan of this task in general, but there are so many different and super fun directions that you can go with cubes.

    Here’s an overview of some of the fun ways I’ve used Rubik’s cubes to introduce basic math concepts to my younger son. I think holding the cubes in his hand / having them in front of him is a big improvement over the drawings on paper:


    We also recently used smaller cubes as building blocks as a way to discuss / understand a 4-dimensional object. I think this task with snap cubes is a fun way to get at the pattern recognition the painted cube task is trying to get at:


    I’d also point to Patrick Honner’s “visual proof” post here as a fun math exercise with squares and rectangles, though possibly a little more advanced than what the painted cube problem is aiming for:


  2. I’d probably start with discussing how to solve a Rubik’s Cube. Plenty of motivating videos to kick off this context.

    The different permutations available in that context are way more interesting than painted cubes.

  3. I like the way George Lenchner extended it in Creative Problem Solving in School Mathematics. Instead of solving the problem, he presented the basic 3x3x3 painted cube and then challenged readers to create problems of their own.

    Of course, the classic problems get posed, because they are the easiest extensions to come up with, but there is a lot more that one can do with the cubes if you take the time to brainstorm. It’s a fun exercise, and it’s cool to compare the puzzles and see how many different ideas your friends/classmates come up with.

    I wrote about the problems I created in response to this prompt in a blog post here (mostly probability questions) and another one there (different ways to paint the cubes).

  4. This would take a lot of previous work, but… maybe they could get a bunch of little cubes. Some with three painted faces, some with two, some with one, many with none. Then asked them to build the largest possible cube so that its whole exterior is painted.

    They would then start building cubes, probably some group would get the largest one faster, or maybe another group built a smaller one. How many of each type did they use? How many would they need to build a larger one? how many for a humongous 1,000-cube-sided cube?

  5. Initial thoughts:

    I wonder whether you could start by asking a question that forces the organizational questions. If at all possible, I want students to come up with structure and organization on their own.

    I’ve got three small children at home and the puzzles never stay “finished” for very long. In fact, the puzzle pieces like to mingle with each other. Sometimes they get lost. What if I created two of these puzzles in my house: a 4-sided and a 5-sided puzzle? If my kids got to them, how would I know how to separate all the pieces and whether I even had all the pieces?

    This could lead into discussions of painted sides and how many of each type I need to be sure I have before bringing them to school for math class.

  6. I was totally on board when I thought that the puzzle was to make sure that the unpainted faces were supposed to be on the outside. I’ll tell you this, if he gave me the puzzle of 1,000 cubes, I would probably get the outline of the base constructed and lose interest.

    IF I were using this task with my students, I would be emphasizing that we are doing this task because of the math they would learn – not the puzzle. In that case I would break the instructions into a couple of half-sheets and insert a class discussion between each section.

    I would be slightly interested in figuring out which two different sized cubes I would need to make a larger cube that appears fully painted.

  7. What are we specifically trying to redesign this for? If it is about relationships and writing equations for them, I don’t use this at all. I would much rather look at polygons and polyhedrons: angle measures, vertices, faces and edges. There are plenty of relationships there to observe and write formulas for. It also builds a better knowledge base for the students for that HS Geometry class.

    If we are specifically trying to bump this up a bit, I am on board with the Rubik’s cube type of idea. I used CMP and I actually made some sample cubes for this. My students struggled with initial spatial visualization of what was happening so I found it helpful for them to have a concrete version. I never started with a completed cube, but rather gave them a bag of cubes and see what they decided to do with them. Most of the time I got weird looks, but then they started construction pretty quickly. One thing to stay away from with this is to keep the cube smaller, a 4x4x4 cube is more than enough to turn students off. I also had them investigate how big of a cube we would need to turn it inside out: can they take the cube and construct one of the same size with no painted sides showing? We then talked about strategy and if there were multiple ways to create this type of cube from the same blocks. IF students really got into this, we would talk about larger cube building, if not- they had more than enough experience in what the task was trying to present.

    I would also have students contemplate Curmudgeon’s MathArguments180 Post http://matharguments180.blogspot.com/2014/05/day-118-cubes.html, I like this conversation more than painted sides.

  8. I have used this task with primary students. If I did it again I think I’d maybe do 5 minutes unrelated paint-dipping with factoryballs http://www.mathplayground.com/logic_factoryballs.html just to get in the dipping mood, and because I know they’ll go home and want to do more.

    Then the nrich animation: http://nrich.maths.org/2322
    so that everyone is really clear about what we’re doing. Then get going on the 3 X 3 case. Work in pairs, use whiteboards, scribble, then make clearer drawings and notes to share with the class.

    Afterwards a look a the 4×4 case.

    Looking back with the class after each stage, I might use this interactive:

  9. I’m thinking of the cubes as pieces of a larger building. Faces which are against the ground or on top are bases and ceilings, exposed sides are windows.

    Given a set number of cubes (say, 20), students are asked to design a building which provides the most window exposure. If the building must have 2 stories, how does that change the design?

    If the building must not take more than 9 squares of base space, how can we design the building?

    Can we then start to find relationship between number of modular pieces, stories and proportion of window space.

    Also send a few stat-related variants on twitter.

  10. This works pretty well when you have actual Rubik’s Cubes in the room. It is much easier for students to understand the setup, plus actual Rubik’s Cubes exist at ridiculous sizes (I don’t own that one, but the Internet has videos).

    With actual Cubes you can give the problem straight. I would throw the Connected Mathematics version in the trash bin. That is a ridiculous story. I got tired just reading the second sentence and started skimming.

  11. Bob, I like this lots. Fawn Nguyen describes a building lesson similar to this: http://fawnnguyen.com/2012/05/22/20120521.aspx

    I like the idea of continuing the lesson in this direction, on subsequent days.

    I would get out the multilink cubes or interlocking cm cubes and get the children designing

    I usually get my students to draw on dotty isometric paper at some point, along with using an online way too such as
    or, more like Minecraft:

    So it would be a great opportunity for learning how to represent these 3D shapes as well as thinking about how to maximise window area given the constraints.

  12. I don’t know that I have anything terribly insightful to add, but this seems like a fun conversation.

    I don’t really see too much that is wrong with the problem/puzzle itself, which (to me) is something like:

    I have this cube (show picture/tangible) made up of smaller cubes. If I dipped the whole thing in paint, how many of the smaller cubes would have paint on them? Is there a rule or shortcut we can create that would allow us to answer that question for ANY sized cube?

    To me, the issue seems to be that the version we see in your blog post attempts to steer the direction of student thinking and leaves little room for play and divergent thinking/approaches. It “scaffolds” away all of the rich mathematical thinking and play in an attempt to cover standards. In particular, the unspoken assumption in the way it has been printed is that writing and graphing linear, quadratic, and exponential functions is the real “Math” in the task (things we can easily point to as belonging to the discipline/standards).

    But, at it’s core and without all the mechanical scaffolding (as re-posed here), the question allows room for many mathematical strengths and habits of mind to be valued and sends different messages about what the real “math” might be: taking things apart and putting them back together, creating systems of organization, assigning variables, making generalizations, posing extension questions, etc. In addition, because it doesn’t dictate how to proceed, it encourages students to trust their own thinking and allows them to “see themselves” in the work that develops. The work of the teacher becomes to follow the student, looking for mathematically ripe opportunities in their work and thinking.

  13. Paul Jorgens

    May 29, 2014 - 3:00 pm -

    Could this be done in 3Act form?

    Start with a video and 4 buckets each with a different color multilink cube

    Build 2x2x2 – All Red

    Build 3x3x3 – Use Red corners, Blue edge non-corners, Yellow face non-edge, slide a face and drop a white one in middle

    Show 4x4x4 construction

    Stop video at beginning of 5x5x5 with cubes left in bucket.

    What questions would students ask? Would they care?

  14. 1. I think this has a ton of value purely as a puzzle, and that value is increased the more visual the question gets.

    2. The rich math behind this that I would want kids to get at is thinking proportionally about dimensions. There are always 8 corners, but the pieces with two painted faces represent length and increase linearly, while the pieces with one painted face represent area and increase with the square of the side length. There’s some more math that connects these with the volume of the cube in interesting ways. That said, I don’t know how best to motivate all of that.

    3. I think there’s potential to use an office building to make this a bit more concrete. It doesn’t give much extra meaning to the corners, but I think those are the least mathematically interesting anyway. But edge offices are worth the most, and any window is better than being in the middle. There’s a ton of room to stretch the task with skyscrapers of different shapes (I’m especially curious about the new World Trade Center), or an algebraic or optimization problem about cost. There’s a great visual floor, especially for rectangular buildings, and a great moment of abstraction as students have to make inferences about parts of the building they can’t see.

    I wonder if the Pentagon, with it’s symmetry in five floors, five sides, and five rings is worth considering? May be too complicated, but it’s pretty visually compelling.

  15. I’ve never done a three-act, but I can imagine something like this. It’s not the original puzzle, but it’s one of my favorite extensions—less algebra, more spatial reasoning:

    (1) Set the scene: 16 wooden blocks, two colors of paint. Show the blocks being all painted one color, arranged into a cube, and then the outside of that cube painted another color.

    (2) Take fully painted cube of blocks apart, sort them according to paint scheme. Then rearrange with the original color on the outside. Then show a new scene: 27 wooden blocks, three colors of paint.

    (3) Show the final blocks being arranged and rearranged so that they form a cube of each color in turn.

  16. Personally, I can’t get over how small Leon is. Poor Leon seems to be about 35 cm tall.

  17. I like the direction of comment #12 by Bryan. Figure out what the heart of the question is, and put it out there in 1-2 sentences. Everything else is a distraction.

    Get some Rubik’s Cubes (as others have suggested). Put students in groups and pass them out. Then ask: how many stickers? Are there any cubes with 2 stickers? 1? none? more than 2? How many of each?

    Extensions: what if someone built a 4x4x4 Rubik’s cube. Now how many of each? What if it were an nxnxn cube? Can you write down some formulas that take the edge length and return the number of 0-, 1-, 2-, and 3-sticker cubes?

    As Dan has pointed out, paper is part of the problem here. The book authors want to get all their questions out at once, and this is surely not best for the classroom. Putting those questions in a teacher’s guide would be fine, but students should see a single, unencumbered question.

  18. Everyone seems to focusing on paint or puzzles, so this idea might seem a bit out of left field. I decided to take another route that I think could make a great 3 Act lesson.

    In Japan (and probably in other places if you look hard enough), grocery stores carry “square” watermelons. Sometimes the watermelons are rectangular prisms, but other melons are almost perfect cubes.

    Act 1: Start with a picture of a watermelon cube. After answering questions from students about where you can buy the cubes and what other shapes are possible (there’s a lot if you do a simple Google search), show a video of cutting the watermelon cube into cubes.

    Here’s where the driving question arises: Depending on how many cubes I want, how many cuts will I have to make?

    Act 2: Have students work with a few specific cases. I think some valuable discussion could develop as students question if they need to consider the cubes with rind on 1, 2, or 3 faces, if the cubes need to be a certain size, or how we define a cut (does trimming the rind off of some cubes add to the total number of cuts?). Adding onto the complexity, students could develop a model that will predict how many cubes of each type I’ll get as I change the number of cuts in each direction.

    Act 3: There might be multiple solutions to the problem depending on how students approached the situation. Maybe I would have a video of how I cut the melon to get a certain number of cubes, then compare my solution with student work.

    The Sequel: Use other possible shapes or compare the watermelon cubes to a regular ellipsoidal melon.

    If I had a blog, I would create the task and post it. Does this seem reasonable?

  19. I am a HUGE fan of doing this task as a square pyramid instead of a cube. Many of the pieces that give students difficulty with the cube are brushed aside with a new visualization.

  20. I can’t take any credit for this painted cube task, (listed on this CIC Workshop task list) but it was redesigned with Complex Instruction in mind. Last year our math department learned the fundamentals of creating groupworthy tasks that allow for multiple entry points and are open ended.

  21. Could the pieces be re-arranged to create a “painted solid” that is enclosed in an unpainted solid? What is the “largest” such painted solid? What are some ratios involving the original cube and the enclosed painted solid? Is it possible to enclose a painted cube inside an unpainted cube?

    The context for the original problem is a painted cube. But, that context is quickly de-emphasized. Once the table is completed, the questions would seem to steer the student towards simply noticing patterns in a table, finding an equation, etc. The original context has become little more than column titles on a chart.

    This also seems like quite the convoluted approach to helping poor Leon solve his puzzle. Make a big table & graphs & equations — a solution only a math teacher could love.

  22. Ahhh! I knew it was too good to be true!

    I made up this lesson, or thought I did, with Rubik’s Cubes:

    My favorite part comes at the end when I connected the functions to the geometric objects that they represented (vertices, edges, faces, etc).

    I see that a lot of other folks have discussed the Rubik’s Cube idea, but I promise the extension where you write the functions is pretty fun for the kids.

  23. Okay, a lot of people have offered drastically different tasks. Fair enough. It’s like asking “how can we improve checkers?” and hearing someone say, “basketball is more fun!” I don’t disagree. But for argument’s sake, what if we stuck with this task?

    Use actual cubes to establish context? Check. (Jason Dyer, Bryan Anderson, et al.)

    Use imagery to establish context? Check. (Denise Gaskins, Tom Hall, et al.)

    My primary question about the CMP version is “how much help should we offer and when?” CMP has answered that question with “lots and right away!”

    Bryan Meyer’s comment speaks my mind pretty well. Teachers have a luxury that textbooks do not: we can titrate our help more precisely. We can suggest tables and strategies and skills and information as students need that help and not a moment sooner. (I’m willing to bet Bryan and I would disagree on how much help the teacher should give, even though we agree on when the teacher should give it.) Textbooks have to print all that help right away.

    James Key reforms these ideas helpfully and pins the problem where it belongs:

    … paper is part of the problem here. The book authors want to get all their questions out at once, and this is surely not best for the classroom. Putting those questions in a teacher’s guide would be fine, but students should see a single, unencumbered question.

  24. Brian MacNevin

    May 31, 2014 - 3:49 pm -

    CMP2 reminds me of highly scaffolded inquiry in a science setting. You elicit preconceptions (and guesses and estimates), then you engage in a highly structured activity that is meant to lay out some data to help the student see or discern a pattern that they can then apply to other settings or to larger, more unwieldy, numbers.

    I think it’s a major step up from other forms of textbook tasks. But I think I see what you’re getting at: the kids aren’t asked to engage in the initial modeling. In science it’s like an engineering task where you tell the kids: use 1 meter of tape, 1 meter of string, and 20 spaghetti noodles to support a marshmallow as high above the table as you can. And ultimately, you’re left with a “so what”.

    Leon (the kid in the task) is off-task. What he WANTS to do is maximize profits from his game. What he WANTS to do is find the most efficient ways to make the blasted things, or to paint them, or to ship them. Instead, he gets sidetracked by a mathematical relationship he suspects.

    I think I like Leon’s sense of curiosity (I do the same kind of thing); but I do have that feeling of a contrived story meant to steer a student in the right direction. Hmmm.

  25. In this presentation the real math has been done for the student. This should be an exploration, with no initial guidance. Just a “What can you say about this?” question.
    Polya’s advice to simplify, reduce, look at special cases is what should drive the investigation.

    I also wonder whether the students have looked at first, second and third differences, or ratios of successive values. If not then it will be a real trial and error approach.

  26. Paul Jorgens

    May 31, 2014 - 7:27 pm -

    I found a blog that does something similar to my thoughts in an earlier post. In my lesson, the students come in while I am in the midst of building the 4.

    This blog has a pic through 5.


    I stop at 4 in class and let them make observations and ask questions. They wonder about the next stage. Which color is used the most? Will I have enough cubes to make the 5 or 6? Which color will I run out of first?…

  27. Brian MacNevin:

    I think it’s a major step up from other forms of textbook tasks. But I think I see what you’re getting at: the kids aren’t asked to engage in the initial modeling.

    Right! I imagine “modeling” as a sequence of tasks from A to Z. The CMP task (and most textbook modeling) does tasks A through M for the students, has students do N through P, then ignores Q through Z.

    Howard Phillips:

    In this presentation the real math has been done for the student. This should be an exploration, with no initial guidance. Just a “What can you say about this?” question.

    I appreciate your critique of initial guidance, which is different than critiquing guidance altogether.

    Paul Jorgens:

    I found a blog that does something similar to my thoughts in an earlier post. In my lesson, the students come in while I am in the midst of building the 4.

    Great find, Paul.

  28. So, here is another idea that sticks to the original script.

    Provide several collections of cm cubes with various sides painted. Some collections should “work” in that a painted cube can be formed from the pieces and some shouldn’t “work” in that it is impossible to form a painted cube from the pieces. These could be physical cubes or images or both. The student’s job is to turn each collection into a cube or explain why it is not possible. Part two would be for the students to make their own puzzle by describing a collection of cubes. Part three would be to make the tables & graphs.

    I am still not really sure what the original “task” is. Is it to help Leon put the pieces back together? Wouldn’t one just use add hoc reasoning while visualizing what a 10 x 10 x 10 cube looks like?

  29. Like I had stated earlier, I taught using CMP2 and I really liked the direction of the textbooks overall. This is one thing that did bug me, it made me think of those horrid multiple choice tests I used to take in college where if you did not know an answer, skip it and it would be answered for you later by another question. All of interest is taken away because you are given a rigid outline of what to do. I always tell my students there are multiple ways to solve problems, students need to find their path.

    Dan Meyer~
    *Right! I imagine “modeling” as a sequence of tasks from A to Z. The CMP task (and most textbook modeling) does tasks A through M for the students, has students do N through P, then ignores Q through Z.*

    This was one problem I identified early, and one that I circumvented. I only use a classroom set of the books, I use my SMARTBoard to display images that I need. That way, students are not exposed to the mathematics early. After we did these type of investigations I would have students get the textbook and work on the extension problems. This type of system worked really well. I guess I overlooked that when you posted this because it was something I already implemented.

    CMP2 is great for changing how students approach a Mathematics class, and I really like to watch how students grow during the year (they only get exposed to the curriculum my year). As with any curriculum, you need to examine what works for your students and classroom and make those changes.

  30. As someone with intimate knowledge of CMP, I want to add something to the conversation here.

    Dan writes:

    My primary question about the CMP version is “how much help should we offer and when?” CMP has answered that question with “lots and right away!”

    Yes and no.

    It is absolutely the case that the student pages we see here answer that question in that way. Dan is not wrong at all here.

    The teacher’s edition talks about this problem in a somewhat different way, though.

    You can use a Rubik’s Cube to launch the problem…Also having small unit cubes or sugar cubes around so students can build some of the smaller cubes of length 2, 3, or 4 will be very helpful…

    [Ask students] Suppose we paint a cube with edge length of 10 cm and then separate it into 1,000 small centimeter cubes. How many of the small cubes will have paint on three faces [etc.]? (Let students make some conjectures. Some might notice that the cube on a corner will have three faces painted.)

    If I were given a stab at writing that TE text today, I would definitely include some language that explicitly asks teachers not to over-organize things for kids, that explicitly encourages teachers to launch the problem without the student edition in front of them.

    One of the important contributions that CMP has made in US math curriculum is a set of teacher materials that go beyond answer keys and teaching tips. It is a curriculum in which the student edition really isn’t enough to get the whole picture.

    The ongoing tension in every bit of CMP work I have been involved in has been between open and structured. How much structure do teachers need in order to get kids into a place where they notice and work on the important mathematics? Too little structure creates a problem (viz. the basketball game breaking out on the checkerboard in the present discussion). Too much structure creates a problem (this is the premise of the critique here).

    I think two questions are in play here:

    (1) How much structure/direction do we want the lesson to have in an ideal implementation?


    (2) How much structure/direction do student materials need to have in order to get as many classrooms as possible as close as possible to this ideal?

    These are really difficult questions. They are asked regularly among CMP authors, writers, teachers and professional development folks. I am glad that you are asking them, too, Dan.

    That student page deserves the critique you give it. And yet the curriculum’s goals are more nuanced than the student page suggests.

    I am sad to report that I am 40 minutes away from my office where my CMP1 books are housed (the image here is from CMP2). Thus I cannot produce for you an image of the CMP1 version of this problem, and I cannot recall whether it had the same structure. It is absolutely the case that publisher and classroom feedback pushed the writing team to make many problems more structured in CMP2 than they had been in CMP1.

  31. Thanks for the behind-the-scenes perspective, Christopher.

    One of the important contributions that CMP has made in US math curriculum is a set of teacher materials that go beyond answer keys and teaching tips. It is a curriculum in which the student edition really isn’t enough to get the whole picture.

    Do you have a sense of how much teachers took their cues from the student edition vs. the teacher edition?

  32. @Dan&Chris: One thing to consider as well with the topic of teacher cues would be training in the curriculum. Even though we purchased CMP and paid for “training”, it was training of “hey, here are the cool features of CMP”, mostly digital resources. There was no instructional planning. This is the downfall for many curriculums I have dealt with. They need training on lesson design, instructional specific support (not just digital bells and whistles) and an actual lesson observation. If you don’t supply this, it doesn’t matter how good of a curriculum you produce- implementation will be inconsistent to the point you can’t even be sure it’s presented to students as intended.

  33. In response to Bryan Meyer’s comment on offering “less scaffolding” for students: I fully agree, but I think the problem needs to offer scaffolding for teachers. Unfortunately, there is not a lesson plan supplied to the teacher on taking this one question which was originally intended as a passing question within a set of “homework exercises” and developing it into a rich and robust task.

    As a math coach, I frequently field candid requests for training on the things that you and I take for granted in this activity (class discussion, group work, student conjecture, manipulatives, varying responses, improvisation, etc). Most of us reading this post would take Nicole Paris’ photo and run with it. The vast majority of teachers cannot.

    With that said, this post and Bryan’s comment have me thinking how to offer what is best for both students and teachers.

  34. Chris Shore makes a valid point about teachers that can “take [the] image and run with it” and the teachers who need a lesson on the lesson. I am in a co-taught setting and have found that when I use a rich task, I have to walk the other teacher through the task, the things to say (facilitation) and the things not to say (there is a tendency for many teachers to prompt or give answers). As I learn more and more about teaching with these tasks, there is more structure than first appears, to ensure that learning and not parroting is taking place. Each class is different, but the “oh, I get it now!” is so worth the process.
    I would like to see more training for teachers in basic facilitation of these tasks.
    Dan- could you make more videos like the one with the penny pyramid? That showed so clearly HOW to present the task and facilitate it. I’ve shown and shared that when I need to communicate the process of tasks.

  35. Marcia Weinhold

    June 11, 2014 - 11:41 am -

    I served for awhile as “outreach coordinator” for teachers beginning to implement the Core Plus high school curriculum, and one of the things we noted was that teachers really felt the need to organize things for their students. The larger the class size, the greater the need to do this, so that you can be sure everyone is “making progress” on the problem. If every student or group is writing their own organization, the teacher has to stay at each group longer to figure out what is going on, and to catch the mathematical moments. When classroom management is an issue (let’s face it), teachers will organize the math in order to facilitate the management. What sometimes happens, however, is that the students have a somewhat false sense of “understanding” the math, when what they have actually done is filled in someone else’s tables. So the preparation they have for the future is to follow specific directions, not to formulate and solve problems.