I have this written on the board:
“What shape is next?”
I use this structural sequence to add a new perspective on the square numbers. I want them to see how you can build on a pattern by adding to shape and dimension. Essentially, this is a sequence of 3D figurate numbers. Many students attempt to build the next shape in the pattern without counting the exact number of blocks. This is something I discuss with them. They agree that you don’t need to count the exact number of cubes of every step in this sequence. Instead, they can take ownership of this sequence by seeing the shape of the pattern and building the general structure. Once they list out the number of cubes in each step, they recognize that they are dealing with the square numbers. I push them on this, making sure they are able to discuss how they recognized this pattern. Students will rearrange the cubes into squares and discuss how this relates to the day before. The wonderful this is that students are essentially exploring how to transform a flat figurate object from a 2D plane onto a 3D plane. I finish by asking students to think about how they will make structures that use the square numbers (in some way and at any starting point) to build their own 3D square shapes.
To make this work, I separate and color code centimeter cubes, inch cubes, foam chips, pennies and other resources I have so that students can begin to construct a 3D square pattern. I remind them that we will share these at the end and think about how the pattern can help other students solve for the square numbers. For example, my 3D structure can be shifted, separated and rotated to form a square. I give students 30 minutes because this is always a challenging activity. They need to make the connection that these structures are probably best built if each new step has the previous step incorporated into it. For example, in my structure, the third square number’s top two rows are made from the previous square number and the second square number has the single unit cube on top of it.
This type of nested reasoning can really help students generate wonderful square patterns in 3D.
This summary is essentially a show and tell. Students have carefully constructed a 3D square number sequence and we do a gallery walk and visit another tables sculptures. The idea is to review their work, figure out how the pattern is growing (not just in total cubes but in its structural layout), think about how the shape lends itself to an algorithm and then share their thoughts with the class. This is difficult because everyone already “knows” the answer. Our goal is to focus on how the pattern will encourage others to find an answer and how the pattern might lead us to new insight. What makes even trickier is that you can create these patterns and only have a sense of the structural growth. Seldom will students design a pattern thinking, “if I place the cubes this way, that will help students come up with a specific algorithm that is different from anything we have tried so far.” The class conversation focuses on mathematics as a deep and meaningful activity, not just something that has a specific right or wrong answer. Its an attempt to bring more meaning to the process and focus of mathematics.