My nine year old DD was asked to find a huge triangular no. for hwk-26th, I think!
We quickly realised adding it up (1+2+3 etc) was going to take forever and mistakes would probably be made on the way. Looking for an easier sum, we added the top and bottom rows of the triangle together
line 1 (1 dot) to line 26 (26 dots) = 27 dots
line 2 (2dots) to line 25 (25 dots) =27 dots
line 3 (3dots) to line 24 (24 dots) = 27 dots
continue doing this until you finish with
line 13 (13 dots) to line14 (14 dots) = 27 dots
This leaves you with 13 lines of 27 dots, and you can then quickly do this much more manageable multiplication to get your answer much more easily.
Sounds complex, but isn't if you sketch it on the bit of paper
. DD loved the concept and quickly grasped pairing the largest and smallest lines. Works for odd triangular nos. too but need to add on the extra 'middle' line after doing the multiplication.
Hope this helps - not sure it conforms to any technical method but DD's teacher liked it. And aren't the children supposed to be being taught to play with and enjoy numbers?