The manner in which tables were stacked up in our classroom that day , it suddenly caught my attention.. I counted them 4+5+6= 15 and Aha ! It's a triangular number.. I was somehow amused by the fact that how come 4+5+6=15? because the triangular expression for 15 is 1+2+3+4+5

I usually include my students too in such investigations when they are around.. and hence this seemingly trivial question was posed to them as well...

They started staring at this structure ... And soon, one of them - Vaishnavi responded:

"Yes, it's easy," she said, " I can visualize this.."

I asked her to explain and this is how she had restructured the given structure in her mind...

As you can see , she had formed the triangle arrangement of 1+2+3 in the top 3 layers.. and then shifted one and two blocks from the second last layer to its upper and lower layers respectively to form 4 and 5 respectively, thus getting the common expression for 15 = 1+2+3+4+5

Her spatial flexibility was worth appreciating.. but what absolutely delighted me was the visualization of my other student - Kanchan.. This is the way she had imagined:

They started staring at this structure ... And soon, one of them - Vaishnavi responded:

"Yes, it's easy," she said, " I can visualize this.."

I asked her to explain and this is how she had restructured the given structure in her mind...

As you can see , she had formed the triangle arrangement of 1+2+3 in the top 3 layers.. and then shifted one and two blocks from the second last layer to its upper and lower layers respectively to form 4 and 5 respectively, thus getting the common expression for 15 = 1+2+3+4+5

Her spatial flexibility was worth appreciating.. but what absolutely delighted me was the visualization of my other student - Kanchan.. This is the way she had imagined:

She noticed that the rightmost column had 6 tables and it's previous two columns had 4 and 5 tables... So she rearranged 6 tables mentally as 1+2+3 before the stacks of 4 and 5 tables , so as to form the triangular equation of 15= 1+2+3+4+5

I asked her how did she know that the last column of 6 could be arranged as 1+2+3, then she immediately replied that 6 is a Triangular number...!

So I asked her with curiousity that will we able to do this restructuring for every 3 consecutive numbers, say for 5+6+7 also?

She thought for a while and said: No, it's only possible when the last number is Triangle number...

"Why do you say so?"

"Because only then we will be able to rearrange the last column (number) in terms of triangular form before it's previous columns.."

"Okay... So can you tell me what would be the next possible case of consecutive numbers?"

It didn't take much time for her to work out that the expression would be 5,6,7,8,9,10..

"Can you explain how?"

"Sir, 10 is the triangle number and hence can be expressed as 1,2,3,4 before 5,6,7,8,9...."

"Hmm... Good thought.... Can we generalize this then?"

She looked at me for more clarity...

"Means... If the number in the last column is Nth triangular number, then what should be the sequence of consecutive numbers , how many numbers, and what triangle number will be eventually formed?"

I knew she would have understood my query... After about half a minute, she said this...

"If the last Nth Triangle number is K, then the sequence should start from N+1 and go on till K.. This will add up to (K-1)th Triangle number..."

And this was just awesome for me....

Do let me know your views about this exploration which was triggered by a casual observation of stack of tables :)

Thank you...