Painted Cube

Today my grade 10s invited the grade 10 class from across the hall to come over to do math together. We worked in co-mingled random groups of 3-4 students to tackle the problem of the painted cube.

The question is: imagine a cube, like a Rubik’s cube, 3x3x3. The cube is dunked in a bucket of paint. When dried off, the cube is taken apart to reveal the little cubes that make it up. How many of those little cubes have no paint? How many have 1 face painted? How many have 2 faces painted, and how many have 3 faces painted? Do any have all faces painted?

Students worked together to understand and represent the problem, and to work through their answers. There were excellent drawings and tables created.

we pushed the thinking to a 4x4x4 and a 5x5x5 situation. Some wanted to do a 100x100x100 cube! Many decided to create a general rule or equation for each number of painted faces.

To help model their thinking, some groups used the linking cubes we have in our room. The concrete tools really help us visualize 3 dimensional problems.

After pushing groups to consider a general formula, we asked them to graph them, and continue their model into the hypothetical range of negative sided cubes. The whole cube scenario falls apart, but the models keep going.

some students were interested in restricting the domain, to tell the region where the model works. We need whole number values for x, and they have to be bigger than 2.

It was fun to see the room so full of active math. Groups worked well together. Some students were concerned about having never solved a problem like this before, but took risks, and tried things, and realized by the end that the thinking is the goal. The thinking about the problem, and modelling the problem is far more valuable than knowing the number of painty sides on a cube. The ability to think, communicate, and represent the question in multiple ways is the skill which will be quite useful in future math classes.

hopefully we can have another problem solving day with our neighbouring class in the near future.