Hexagonal Centered Numbers and Antiprism Pumpkin Linear and Volume Ratios

After last week’s exploration of triangular, and the introduction to square numbers, we continued looking at polygonal numbers in the form of a hexagon. The children noticed that 36 is both a triangular and square number. This week they were able to see that 91 is both a triangular number and a hexagonal centered number. 

We started with a single tennis ball and had them place six balls around the perimeter for the second hexagonal centered number of 7. Then I asked them to make conjectures about the next hexagonal centered number. Many of them said 13 as they simply added 6 like the difference of 1 and 7. When I had them place the additional six balls around the number 7, they immediately saw that they needed an additional 6 balls for 19 as the third hexagonal centered number. Then they disagreed on whether to add 18 or 24 to generate the next hexagonal centered number. When they added the tennis balls, they soon saw that they needed to add 18 to generate 37 as the fourth hexagonal centered number. 

The attached pdf has a worksheet where they can list the first 18 hexagonal centered numbers by adding to 1, the first hexagonal centered number, 6, then 12, then 18, then 24, and so on. If they have time, let them find more than 18 of these numbers. I also gave them hexagonal graph paper so they can draw their own hexagonal centered numbers (use color).

The 3-6th graders noticed that each hexagonal centered number was formed by three oblong rectangular numbers plus one. So, they can find the nth hexagonal centered number by multiplying an oblong rectangle n(n-1) by 3 and then adding 1. So the 10th hexagonal centered number is 3 x 10 x 9 +1 = 271.

Then the fun began. I created a hexagonal antiprism which is the same as a hexagonal prism but the lateral (vertical) faces are triangles instead of rectangles. I decorated the solids to look like pumpkins. I created two of them in a linear ratio of 1:2. This means that I enlarged the original by 200% so the width, length and height of the larger was twice as long as the original. Using one inch cubes, I showed them that any similar solid in a linear ratio of 1:2 has a volume ratio of 1:8. I also created a larger hexagonal antiprism which was 300% of the original so the linear ratios of 1:2:3 (baby, mama, and papa bear) have volume ratios of 1:8:27. We then filled the baby bear with candy. If two of a certain size of candy can fit in the baby bear, then the children made a conjecture that the mama and papa bear would be a able to hold exactly 16 and 54 pieces of candy of the same size. This was 1x2:8x2:27x2 and we proved it. When I was able to fit three one inch cubes in the baby bear, the children were confident that the mama and papa bear antiprisms would hold exactly 24 and 81 cubic inches respectively. And, it worked.

To make the hexagonal anti prisms out of the two nets I gave the children, they have to cut along the perimeter and fold along every straight line. Then they have to glue or tape the tabs to the triangles. Then after halloween they can store their candy in them and know exactly how many they will have. For example, we made a conjecture that 25 M&Ms would fit in the baby bear so 200 and 675 would fit into the mama and papa bears.

To make the papa bear hexagonal antiprisms you will have to take the pdf and enlarge to 300%. I could not figure out how to post this on my site.