No, this is not a post about the song “Jungle Love” by Morris Day and The Time. (Kudos, though, if your mind free-associates in the same way mine does).
Rather, this was a student’s argument today in my algebra class for why the sum of five consecutive odd numbers is never a multiple of ten. We’ve been investigating sums of consecutive numbers and using numerical and algebraic arguments to justify our findings. This is all a part of exploring algebraic expressions and their uses before digging into equation solving.
Here’s the worksheet in question. The methods we’d been employing to figure out such always/sometimes/never problems have been to plug in some numbers and observe behavior (3+5+7+9+11=35, 7+9+11+13+15=55) and to model the situation with algebraic expressions (2x+1+2x+3+2x+5+2x+7+2x+9=10x+15). The former gives some intuition for and the latter gives a compelling reason why five consecutive odd integers will never sum to a multiple of ten.
But one of my students had another way of seeing this that also used algebraic thinking, albeit of a different kind. Disregarding the fact that the odd numbers in question are consecutive, she said just to add five odd numbers together (o+o+o+o+o), and using sum rules about evens and odds, we can see that the sum will be odd—and thus not a multiple of ten.