One student did come up with a way to calculate the result and had justification through manipulation of dot arrangements. The issue was how to get the rest of the class to a point where presenting this student's ideas would make sense.
I had an idea and decided to see where it would lead. I began by writing 1 + 2 + 3 + 4 + 5 + 6 + 7 on the board and asking students how they would add these in their head. Students are naturally inclined to form 10's as these are nice anchor values to use. Many students said they would add values 3 + 7 and 4 + 6 to get 10’s and then add the remaining values.
I then added an 8 into the mix and wrote out 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8. Students again made pairs to form 10’s. The issue is that the process changes in that different values are now being gathered up at the end. I asked the class if there was a consistent process that they could use when adding the values. A consistent process would enable us to find values of triangular numbers without having to think too much about how to do it.
At this point, several students noticed that adding 1 + 6 = 7, 2 + 5 = 7 and 3 + 4 = 7. The students said you could then multiply 3 x 7 and add the final 7 to get an answer. I wondered if this would work consistently and asked students to try it with 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9. Students did and said it worked. I pushed students to consider how this process worked. They realized that you subtract one from the last value and then multiply the last value by half the second to the last value. They basically thought of the process as (8/2) 9 + 9. The class said this was the same process when adding the integers 1-7.
I asked students to use this process to calculate T75. They readily did this without issue. I then wondered if the formula worked if there were an even number of values instead of an odd number of values. Students tried this and found it worked .
Now they had this part down, it was time to push them to finding a formula. I told the class I wanted to write out the series without having to specify what the final value actually is. I wrote out the sum like this: 1 + 2 + 3 + 4 + 5 + 6 + 7 + … + (n - 1) + n, where n is the final value. I asked students to use the process they were just using to calculate the value of Tn. Students were readily able to write out that Tn = n(n-1)/2 + n.
Students saw that there is no magic about where the formula comes from. Using their logic and processes, the formula was simply an extension of their thinking. It also demonstrates how the formula and math help make life easier by providing them a more efficient and effective way to calculate values of triangular numbers.
Students were now in a position to see a justification tied to the triangular number representation through dots. Besides providing a different way to approach developing a formula, the presentation of an alternative way to derive a formula provided an in to discuss the algebraic equivalence of formulas.
I was glad that I pursued this idea as it helped students build confidence and comfort with the math involved in our work with triangular numbers.
