Here are the four boxes before I started to build the apparatus. They are all different sizes and shapes.
I first embedded and taped box #2 inside one corner of the base box. I set box #3 on the vertical and decided to partially embed it inside the base box. Why did I embed this box only partially into the base? My thought was to leave a little more space on top of box #1 to offer more area for the children to play on that level. I left the portion of box #3 facing out completely open so the children could have easy access into that box.
I next set the channel box (#4) on top of the base box. Since it was as long as the base box, I cut a hole in box #3 so it ran through box #3. For structural integrity, I added a cardboard wall (#5) inside the opening of the base box underneath the channel box. I also left the flap on the base box so I could tape it down to the table for stability.
Here is a view from the other side of the apparatus. I cut all the flaps but the bottom flap to give the children a big space in which to operate inside that box. I left the bottom flap on because, like on the other side, I wanted to tape it down to the table for stability. Because box #3 is on the vertical, it is higher than the base box. And since the channel box(#4) rests on the base box on the outside, it almost seems like it hangs in the air on the inside of box #3.
One of the more interesting spaces for the children in this apparatus was the square-top box embedded in the corner of the base box. Because children like to fill containers, this box offered them a container that they could fill.
This apparatus offered children a complex variety of spaces on multiple levels for their operations. Please note that the boxes could have been put together in any number of ways. On the day I built it, this is how it came together.
If you build, I would encourage you to think inside, outside and around the boxes. Creating rich spaces for the children to explore lays the foundation for learning not only in math but in all the domains.