Cubes, Jackson Origami — First 12 Cubic Numbers — Midpoints

After spending weeks exploring the wonders of squares and square numbers, the natural extension from two dimensional figures is the study of cubes. 

CHARACTERISTICS OF A CUBE: First I placed a cube in front of the children and asked them to describe it mathematically. There responses included or I filled in:

• 3D
• 6 faces (not sides; polygons have sides)
• polyhedron (solids)
• hexahedron (six faces)
• Platonic solid (1 of 5)
• 8 vertices and 12 edges
• one cubic unit
• one cubic inch

The actual cube they were looking at was a cubic inch. 

ESTIMATING ONE INCH WITH PHALANGES: I introduced the children to the part of their finger that is approximately one inch long or 2.5 centimeters (it is the middle phalanges between the PIP joint and the DIP joint [proximal interphalangeal and distal interphalangeal). Of course, the size of one’s middle phalanges changes as they grow. 

VISUALIZING CUBIC NUMBERS: I then asked them how many cubic inches it would take to create a “2 by” cube. Very few said 8. Most said four as they were very intrenched in the study of squares. I showed them that four cubic inches represented the first floor of a 2x2x2 cube, at which time they easily saw that 8 cubic inches was the correct response. Then 3x3x3 was easier as they realized they would need three floors of 9, or 27. We created 4x4x4=64 and even 5x5x5=125 but then I simply added on to the three dimensions to create the frame for an 8x8x8. The challenges was asking them not to figure out 8x8x8 but to estimate how many cubes in my cube set it would take to fill the 8x8x8 as a fraction of the set. I did not tell them how many cubes the set held. There answers were interesting. Most said it would take half or one quarter of the set to fill the 8x8x8. They were amazed when I told them that 8x8x8=512 and then I showed them that the set of cubes held 510. We would need the whole set and we would still be two cubes short. [I cannot believe that the set did not hold a cubic number of cubes; this was mathematically disappointing. There was clearly room in the set for 512; but not room for 729=9x9x9]. Most of the children were even able to comprehend the 10th cubic number: 10x10x10=1000.

NUMBERS BOTH A PERFECT SQUARE AND CUBE: With the older children we discussed which one of my Mathlete Dollars is both a square and a cubic number. Most understood that the number 1 is both square and cubic. I do not have a 64 or 729 Dollar Bill which would have been the second and third numbers that are both square and cubic. Some children suggested 10,000 which is neither square or cubic. But 1,000,000 is 100 cubed and 1,000 squared.

FIRST 12 CUBES ON FINGERS: I had all of the children use their fingers to count by cubic numbers: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. And with the older children we went two further: 1,331 and 1,728. 11 cubed is interesting because it is a palindrome 1331. 12 cubed is important since it represents the number of cubic inches in a cubic foot. 

CUBE CLOCK: The kindergarteners were given a cube clock to record their first twelve cubes. See attached pdf.

What is more fun than building cubes? Creating your own origami cubes. In the past, we have created Sonobe Cubes but this week we explored the much easier Jackson Cube. 

FINDING MIDPOINT: Starting with a square piece of paper, I taught them how to find the midpoint of an edge. First I had them take a pencil and estimate the midpoint and then check it against the actual midpoint. No-one was able to estimate the actual midpoint but some were within a few millimeters. The method to find actual midpoints is to line up two corners (these are the two vertices of the segment) and then simply pinch down. That small folded segment is the actual midpoint.

JACKSON CUBE MODULE: There are six faces to a cube and six modules to make a Jackson Cube. I created squares with pictures of the 1-6 cubes on them. The 1st and 6th are the same color (pink). The 2nd and 5th are the same color (blue) and the 3rd and 4th are the same color (green). This resembles an actual die were the opposing faces add up to 7. The module directions are attached as a pdf. The key is the mark the midpoint and fold to the middle or midline; it is important not to fold at the midpoint. Putting together the Jackson Cube is a big challenge. Each face connects perpendicular to the other. The tab from one face goes in between the two tabs of the other. It is best to focus on starting with three different colors and then adding the opposite face of the same color until only the top of the cube is remaining. The directions are clear but I did not have the children use the directions in class since I wanted to see if they could follow my instructions without the visual directions. I am aware that origami means without tape but it is important to tape these together to preserve their structure.

CHALLENGE DURING THE WEEK: The children were also given 2.5 inch squares to create small Jackson Cubes. However, the best Jackson Cubes are made with post-it notes, preferably three different colors. They are very firm and go together easily. They should make as many cubes as their heart desires.