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:

http://mikesmathpage.wordpress.com/2014/01/09/a-little-fun-and-a-little-math-wtih-rubiks-cubes/

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:

http://mikesmathpage.wordpress.com/2014/05/26/using-snap-cubes-to-talk-about-the-4th-dimension/

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:

http://mrhonner.com/archives/10239

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).

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.

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.

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.

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
http://illuminations.nctm.org/Activity.aspx?id=4182
or, more like Minecraft:
http://www.nowmakethis.com/

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.

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?

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.

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

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?

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.

As some have likely already noticed, there is a rather lovely conclusion that draws together general rules for the numbers of cubes with zero-painted faces, one-painted face, two-painted faces, and three-painted faces at the bottom of Brian Seitz’s page: http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/SeitzBrian/EMT669/painted.cube/painted.html.

Heading in that direction makes a strong opportunity for Mathematical Practice Standard 7.

Also, just a word about what ideally has gone on before with all primary / elementary age children.

There’s a 2D analogue:
https://picasaweb.google.com/lh/photo/DZjapmBJwxDhjZfbA4m_5NMTjNZETYmyPJy0liipFm0?feat=directlink
– the three characters at the bottom (their names are Paloma, Parsang and Perry). How does the series work?

This is the kind of thing I do with 9 and 10 year olds:
http://pinkmathematics.blogspot.fr/2013/06/make-pattern.html

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.

I accidentally got sent to the Stanford site and came across the real math investigation, the “measure of discness”.

Here it is
http://psc.stanford.edu/psc-materials/disc-problem.html

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.

@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.

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.

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.