Cubic Units and Sonobe Cube Construction

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Over the last few months, we have been exploring distance (one dimension), square numbers and triangular numbers (area or two dimensions), and then we have built on those concepts with square and triangular pyramids (volume or three dimensions). To continue our understanding of 3-dimensional objects we need to understand how volume is measured: in cubic units.

There is no better way for children to understand cubic units then to create them and feel their size. First, I showed them different dimensions of Rubik’s cubes from 2x2x2, 3x3x3 (the standard Rubik), all the way up to 12x12x12. The attached document called Cubes.pdf has pictures of each of these cubes and a worksheet for them to learn the number of cubic units that make up each. The answers are on the fourth page. They should try to understand the first five cubic numbers (1, 8, 27, 64, and 125). The younger students will use addition to explore cubic dimension but the older students will see that multiplication is the fastest way to get there. For example, exploring 5x5x5 cubes they will see that the first level of cubes is a 5x5 square of 25 cubic units and there are five levels or stories (in the context of a building) of 25 each. Some will multiply 25x5 and some will simply count by 25s (25, 50, 75, 100, and finally, 125). 

We are starting to work on strategies for building 7x7x7 cubes. They already know that the first 7x7 layer is 49 cubic units. Then they are faced with multiplying 49 x 7 or counting by 49s. Instead, I recommend counting 7 50s (50, 100, 150, 200, 250, 300, and finally, 350). Then we say, “350 represents 50 7s but we only need 49 7s, so what do we subtract from 350?” Most of them can see that one 7 must be subtracted to obtain 343. We will build on these strategies over the year.

I taught the students a different construction of Sonobe Cube as in previous years. This origami construction consists of creating a square piece of paper and then following a series of 22 directions attached on the document Sonobe Cube Design.pdf. Six pieces of paper are required to make each cube (one for each face). 

Through the construction of these pieces, we were able to explore rectangles with 2:1 ratios, right trapezoids, parallelograms (and why this one is not a rhombus and that a “diamond” is not a mathematical term), right triangles, and angles of acute and obtuse measure. Most importantly, we stressed the distinction between parallel (touchdown or field goal signal) and perpendicular (unsportsman-like conduct signal). These pieces will not go together unless they are perpendicular to each other. Do not worry if they cannot put the pieces together. I will do this at the beginning of the next class.

For children who have done this before, I challenged them to create cubes of greater than 1x1x1 dimension. The 2x2x2 cube requires four squares per side (a total of 24 pieces). Here they do not fold both right triangles (step 13 of the Sonobe Cube Design.pdf). They only fold those tabs that will connect 90 degrees around a side. I had a 2x2x2 cube in class for them to use as a model. Finally, the largest cube ever created by a Mathlete was a 4x4x4 cube utilizing 96 squares of paper.  Of course, I do not expect this but it is a great goal to have.

I encourage family involvement and these cubes can make nice storage boxes (if the 6^th face is omitted) as well as great holiday ornaments.

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