Halloween Hexagonal Coffin Prism x1, x2, and x3 -- Linear, Area, and Volume Ratios

We had an exploration this week comparing solids with a linear ratio of 1:2:3, an area ratio of 1:4:9, and a volume ratio of 1:8:27. We were able to prove that the small prism held 10 pieces of candy while the medium prism held exactly 80 pieces and the large prism held exactly 270 pieces. 

We spent the class focusing on the discussion of dimensional analysis (points or location has zero dimension; length or distance has one dimension; surface area or area has two dimensions; and  solid or volume has three dimensions). 

I gave them a NET to create the small hexagonal prism in black and white but promised that I would send you a file with the color version.  This is attached. Also, here is a link to the larger formats x2 and x3. 

https://www.dropbox.com/s/5mqvrcu600wmwyh/11x17%20coffin.pdf?dl=0

 I recommend printing these out on harder card stock. Also, the x2 and x3 were printed by me on 12x18 paper at a copy store (not inexpensive so I do not recommend that—you may be able to print at your office or configure the large format on multiple normal pieces of paper). Students are not required to create these but it may be a fun Thanksgiving project.

With the K/1st grade class, I gave them a color version on hard stock and we put together the base in class. They still have to complete the top.

All of the Mathletes were encouraged to master the square and cubic numbers. Also, the other challenge was to use the volume ratio of 1:8:27 to find the number in the middle and large if the small had 2, 3, 4, 5 … The key was to see that if the small was 2, the medium would be 2 x 8 or 16, and the large 2 x 27 or 54. Most students noticed that the medium was counting by 8 and the large by 27s. One strategy is to see 8 as add 10 and take away 2. Also adding 27 is just adding 30 and taking away 3.