Triangular and Tetrahedral Numbers

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We continued to explore polygonal numbers. Last week it was squares and oblong numbers. This week we learned about triangular numbers which are formed by stacking spheres (eg. tennis balls). The first few triangular numbers are 1,3,6,15, 21, 28, 36, 45, 55 and so on ……..

1                                                                                          1

1 + 2 =                                                                                 3

1 + 2 + 3 =                                                                          6

1 + 2 + 3 + 4 =                                                                   ___

1 + 2 + 3 + 4 + 5 =                                                             ___

1 + 2 + 3 + 4 + 5 + 6 =                                                       ___

The attached pdf contains triangle grid paper to help the children identify triangular numbers. They should record the first 20-30 triangular numbers.

We then built a triangle pyramid out of 220 tennis balls with a base of 10x10x10. I challenged the students to build their own pyramid and showed them my attempt using 35 plastic ups.

Another name for a triangular pyramid with all equilateral sides is a “tetrahedron.” So the number of tennis balls it takes to build a pyramid is considered a tetrahedral number.

1                                                                                             1

1 + 3 =                                                                                    4

1 + 3 + 6 =                                                                             10

1 + 3 + 6 + 10 =                                                                    ___

1 + 3 + 6 + 10 + 15 =                                                            ___

1 + 3 + 6 + 10 + 15 + 21 =                                                   ___

For the older classes, I introduce them to the algebraic formula for finding a triangular number. They were able to see the connection from oblong numbers to triangular numbers by observing that they are half of an oblong. If they want to check their work, they should use the formula of finding triangular numbers which is

n(n+1)/2 ,

where n is the smallest dimension in the oblong number.

Homework is (i) to create a triangular pyramid, (ii) find the first 20-30 triangular numbers, and (iii) if you are ambitious, the first 20-30 tetrahedral numbers.

Again, for the older kids, the algebraic formula for tetrahedral numbers is n(n+1)(n+2)/6.

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