Triangular Numbers and Cannonballing

Whenever I visit Revolutionary or Civil War battlegrounds, I always marvel at how ammunition was stored. Cannonballs were stacked in triangles of spheres and it was referred to as cannonballing. 

This amazing sequence of numbers where we start with 1 and add 2, 3 + 3, 6 + 4, 10 + 5, etc. The first 10 numbers in this sequence is 1, 3, 6, 10, 15, 21, 28, 36, 45, and 55. We see these numbers come up all the time. In the previous week, where I had the students fill out a special intersection multiplication table the second row and column was the list of triangular numbers.

I hot glued the first 8 layers of these numbers with tennis balls to have them feel how these numbers look and grow. We built triangular pyramids and even rhomboidal pyramids. We studied the equilateral triangles and discussed the differences between this pyramid and the great pyramids of Giza which were rectangular bases.

The attached pdf has pictorial representations of these numbers as well as a list up to the 54th triangular number. With the older children (3rd-6th), we discussed how triangular numbers are formed by taking half of an oblong rectangle (these are rectangles of consecutive dimension like 1x2, 2x3, 3x4, and so on). Thus, they could find the nth triangular number by taking that n value and multiply it by the next integer (the area of the oblong) and then divide it by 2. So the 100th triangular number is 5,050.

All of the students were able to connect two consecutive triangular numbers and see that it makes a square number. This sequence is just as amazing as the above connection.

Have fun exploring the pdf this week. Also, I would love for them to create their own triangular numbers. They can use the graph paper I supplied, or use manipulatives such as golf balls, marshmallows, etc. If they are too large to bring to class, take a picture.