The Painted Cube

Planning

The first time I saw this problem I had my student hat on. As a participant in in group setting at one of the NYSED/CUNY Teacher Leader of Mathematics Institutes I attended, my fellow group members and I were tasked with solving a version of The Painted Cube.

Note: If this is your first time seeing this problem, you may want to stop here and spend some time working on a solution before continuing.

It was interesting in the way that we individually saw and solved the problem. Some of us looked at the 2-dimensional image on the handout and discerned how to answer the questions. Others needed the support of snap-cubes to be able to visualize the 3-dimensional shape that was represented by the image.  Another one of our group members asked questions that helped us to see a way to expand our thinking of the situation beyond the questions on the handout. This variety of ways we entered the problem and the possible extensions intrigued me and I was inspired to take it into my classroom to try it out. I saw the problem as a way to make connections within and between a variety of math concepts and vocabulary:  spatial structuring, cubes, cubing a number, surface area, faces of a cube, volume, patterns, linear and quadratic functions, and graphing.  Along with that, I love to have my students play with all kinds of manipulatives and this problem begs for the use of snap cubes so I couldn’t resist!

After using the problem with my students and witnessing their level of engagement, I decided to revise the version I had worked on at the math institute so that my students would have additional worksheets to help them see the ways that the problem could be extended to include linear and quadratic functions. I went on to use this new version  with my students, as well as at workshops that I facilitated at the local, regional, and national level.

But something kept coming up. While the problem (as I originally solved it myself and then redesigned it), lent itself to multiple entry points, high-level extensions, and was accessible to students who were functioning at multiple levels in problem solving, I felt it could be improved. Some students had quite a bit of difficulty with the initial phase of the activity, so I wanted to provide more support for those students, while at the same time improve the scaffolding for students who were ready to take the problem to the next level.

To that end, the first change I wanted to make was to provide a launch for the problem. I had been working with introducing problems with I notice/I wonder strategy and the response from students was always more than anything a worksheet could provide. So, I created a first page that has a 2-dimensional representation of a 3x3x3 cube and asks students to write down everything they can about that shape and any questions they have. I reasoned that:

  1. This gives the student opportunity to really examine the shape before he or she needs to start thinking about answering any questions.
  2. This formative assessment gives the teacher valuable information on what students already know.
  3. Students can verbalize what they observe and ask questions before they start to work on problem solving. Perhaps they agree or disagree with another student’s observation. This invests students in following through on further examination of the situation.
  4. Teachers can build on student’s background knowledge and observations in any way that seems appropriate. For example, this would be a time to make sure that students understand what a cube is and what faces on a cube are: essential to understanding the mathematical situation that this problem presents. Some teachers may feel it is appropriate for their group of students to discuss additional vocabulary such as volume and surface area.

I created a Google Slide Presentation that can also be used along with the student pages. This presentation includes color which may be useful for some students.

Next, I wanted to create Push and Support questions, similar to what is in The Paycheck Problem. This was my first attempt at creating this type of resource, so inspired by the ANN Under 10 (Adult Numeracy Network) talk about Crowdsourcing Curriculum, by Eric Appleton, I wrote up some Push and Support questions and reached out to my fellow NYSED Teacher Leaders and asked for some feedback on how to make the resource better. The outstanding feedback I received resulted in page two of the activity which eliminates all of the original questions and presents the problem/situation with a bit of a twist.

This revision benefits students at all levels. For the students who might feel overwhelmed with so many questions on one page, it enables them to  start thinking about the situation first before they are tasked with answering questions that require them to solve for specific values. Other students would, I believed, intuitively begin to seek the answers to all those original questions with only the one prompt:  Write down as much as you can about how many faces are painted on each cube.  This requires more sophisticated problem solving skills because the questions are being formed in the student’s mind: the math is coming from the student rather than the teacher.

As additional support, the Google Slide Presentation (referred to above) presents the same sequence and includes a link to an NRICH animated version of the The Painted Cube problem. Since the NRICH animation shows a red cube being dipped in yellow paint, I reworded the problem to reflect those same color combinations. I also purposefully did not include color on the handouts that I gave to students. This gives them the opportunity to color in the black & white image, if desired, to help them think through the situation.

Supporting Productive Struggle with Push and Support Cards

Knowing that some students would (1) need help being able to visualize a 3-dimensional cube when having only a 2-dimensional image to look and (2) other students would have some trouble knowing where to start, I took the some of the questions that were on the original version of the problem, rephrased some of them, and then made up a few questions of my own to get stuck students moving:

Support Questions:

  • Can you make a model of the cube?
  • Where would  1-inch cubes with no faces painted yellow be?
  • How many faces are painted yellow on each of the 1-inch cubes that are in the corners of the 3x3x3 cube?
  • Sherri thinks that there are twelve 1-inch cubes with 2 faces painted yellow and Connie thinks that there are eight 1-inch  cubes with 2 faces painted yellow on the 3x3x3 cube. What do you see?
  • How many of the 1-inch cubes have more than 3 faces painted yellow?
  • How many total 1-inch cubes are in the 3x3x3 cube?
  • How many of the 1-inch cubes have exactly one face painted yellow?

Then, to extend the thinking of students who finished the initial phase of the problem, I created the following push questions to get them thinking about patterns, functions, and graphing:

Push Questions:

  • How many cubes would have 3 faces painted if the cube was 2x2x2?
  • How many cubes would have no faces painted if the cube was 4x4x4?
  • How many cubes would have one face painted if the cube was 4x4x4?
  • How many cubes would have two faces painted if the cube was 4x4x4?

Along with the above push questions are the following prompts to fill in two input/output tables, which are then followed by prompts to graph the functions:

After making these edits, I was ready to try out the new version in my classroom. I planned to have snap cubes, dice, graph paper, dot paper, markers, and colored pencils available for students who wanted to draw or create a visual model. The graph paper would, of course, also be used by the students who were able to bridge their understanding of the problem to functions.

Analyzing Student Work

Zachary’s Approach

For the initial phase of the activity, Zachary built a model of the 3x3x3 cube with snap cubes to use as a reference, and then he organized his findings using the notice/wonder page of the activity. I did not give him any of the support questions in this phase as he was able to discern the number of faces painted on each cube without needing that support.

He clearly demonstrates his understanding of the total surface area on the bottom left of his work. He also makes clear his understanding that there are 8 cubes that have 3 faces painted, for a total of 24 faces painted on the 1-inch cubes. He does the same with the remaining cubes: 12 x 2 (faces painted) and 6 x 1 (face painted). He adds them up to get 54, which equals the total surface area that he had computed. Although I can’t be sure which he computed first, the total surface area or the individual cubes, his work definitely indicates that he checked his work by comparing the total surface area with the total faces painted.

I knew that I would have to be ready with extension activities for Zachary to keep him involved. He enjoys solving challenging problems when he is tasked to think beyond something he has already encountered, and as I expected, I was handing him push questions fairly soon: one right after the other. Using the input/output table to explore how many cubes would have 2 faces painted if the cube was bigger or smaller than 3x3x3, he was able to discern the pattern, rate of change, and then the explicit rule. He then worked on graphing. At first, his graph did not include the 4th Quadrant of the Coordinate plane but ever the perfectionist, he added another piece of graph paper to the bottom of the graph that he had been working on so that he could include negative numbers to reflect the function he had discovered.

This proved to be challenging for Zachary, as I’d hoped it would be, keeping him moving forward and interested in the problem. Most students did not get to this point, as I’d expected.

Alyssa’s Approach

Alyssa needed some of the support questions in order to make sense of the situation. I gave her the support card that suggests making a model of the situation, so she used the snap cubes to do this. Using the 3x3x3 cube that she created, she was able to discern the number of faces painted on each 1-inch cube, but as she turned the connected cubes around and upside-down to get a final count, she kept coming up with different totals. At this point I suggested that maybe making a model of the 3x3x3 cube with the dice might work for her. I was thinking that if the cubes weren’t connected, like the snap cubes, it might be easier for her to decompose the cube to decipher the number of faces painted on each cube.  What she did with that suggestion was something I hadn’t thought of doing, and I was totally blown away with her strategy pictured below.

Let me explain. She created a 3x3x3 cube and used the numbers on the dice to keep track of how many faces were painted on each individual cube. So, as she built her model, she kept in mind how many faces would be painted on each 1-inch cube and she purposefully faced that number towards her. If a cube had 3 faces painted, she faced the side of the dice with three dots towards her. If a cube had 1 face painted, she faced the side of the dice with one dot towards her, and so on.

Then, she decomposed the completed 3x3x3 cube, putting in piles all the dice with 3 dots, with 2 dots, and with one dot. As for the one die that was in the middle, she knew that it had no faces painted but since the die didn’t have a blank side, she just put it to the side. This strategy was what enabled her to move on to the push questions and by the time class ended, she had progressed up to the point where she was able to graph the input/output tables.

Nannette’s Approach

Nannette needed the support questions, one-by-one, to keep working productively. She started with making a model of the cube with snap cubes and then used the model to answer the remaining support questions. You’ll notice that, like Zachary, she had computed the total surface area of the 3x3x3 cube but she did not use that information to check her work. Instead, she uses the total number of cubes in the 3x3x3 cube (27 = volume) to check to see if her calculations were correct.

Now she was ready for some push questions. She moved through the first few of them but got stuck on the question that asks: How many cubes would have no faces painted if the cube was 4x4x4? To answer that question she built a model of the bigger cube:

This was an awesome aha moment for Nannette! Her model is a perfect representation of the situation and I’m glad that Nannette was able to make it this far into the activity before the end of class time.

Reflections

I learned so much about student thinking and the support that students need as I worked with my students on all three versions of this problem.

(1) The original problem without the extensions is a solid open-ended problem that can stand on its own. Most students really needed visual models to work through this problem.

(2) The  version of the problem that I had bundled, with all the extensions, provided students who were functioning at higher levels the prompts they needed to keep moving forward to higher level thinking, but proved overwhelming for a significant number of other students. Additionally, this version took several class periods if I wanted to try and get all students to the point where they could understand the patterns and functions that came of making the cube smaller or bigger than 3x3x3 cube. I thought it was important that all students made this connection, but realized after hearing some students groan as we revisited the problem on day three, that this was a mistake. I missed the point of what a Low-Threshold High-Ceiling task is: a mathematical activity where everyone in the group can begin and then work at their own level, yet the task also offers lots of possibilities for learners to do much more challenging mathematics too. – NRICH. This realization, that every student does not need to get to the same place, caused me to revise The Painted Cube, yet again.

(3) This new version, with the launch and support and push questions, is much more effective in creating a classroom environment where students at all functioning levels can be supported in struggling productively at their own pace. A big plus to this version (accessible at the link above) is that it can go from launch to finish in one class period. When I speak of a class period, I’m thinking that I spent about 90 minutes in each of my classes on this new version. When I say launch to finish, I mean each student took it as far as he or she could. Students were able to feel comfortable with where they were with the problem and build on their pre-existing understanding of math concepts and vocabulary in some way that was meaningful to each individual student.

How to Use Support and Push Questions

To use these questions, float about the room, with Support and Push questions in hand, and assess students who may need support and students who may be ready to take it to the next level. Simply drop the appropriate question or suggestion in front of the student and move away. This is a non-threatening way to support students. It gives them a chance to think about the support suggestion or question without having the teacher looking over their shoulder. They have the ability to self-pace as they think through this new information. Revisit the student after they’ve had a chance to work with this new information to assess whether they need additional support or push.

Final Thoughts

This version worked extremely well in my classes. Both of my classes are open-enrollment, meaning that it’s possible that a new student may enter my class on any given day of the year. The dynamics of my classrooms are constantly changing. Additionally, I teach in a one-room schoolhouse. Meaning, I have students whose pretests indicate that their NRS benchmark functioning level may be anywhere on the spectrum. Thus, I really like the way the problem is structured so that it is student directed, self-paced, and provides scaffolding for students to take it as far as they want to. I’m really excited to share this version and hope to hear from you when you use it in your classroom. Please share your comments below.

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