Intended Audience: Math teachers, prospective teachers, and parents (public, private, homeschool).
In this video, 6-year-old Autumn shows how to find the sum
This problem actually has a rich history involving one of the greatest mathematicians who ever lived: Carl Friedrich Gauss. Please share Gauss’ fascinating story with your children or students. The story goes like this: When Gauss was seven, his elementary instructor gave what the teacher thought was an extra-pointless exercise of adding up the first (say) 25 numbers just to keep his students busy for an hour. Almost immediately, young Gauss threw down his slate and declared, “There it lies.” When the hour was up, the teacher inspected Gauss’ answer and found it to be correct. An interesting article investigating the full story can be found here.
Autumn’s solution to this question was probably very similar to how Gauss solved it. In the video, I mention that Autumn ran up stairs and came down 10 minutes later with the solution. Here is her actual work (click Autumns_Work for a .pdf version):
In this picture you can see her thinking: Pairing up 1+25, 2+24, 3+23, etc. and keeping track of which terms she summed on the line below (you can also see the left over 13 at the end). I think what happened next is that she recognized (in her mind) that
because she writes “same as 13 x 25.” Regardless, you can see her calculating the
just as she did in the video, and you can see me checking her work underneath (and actually showing her another mental math technique–look!).
For Teachers who teach Eureka Math: This sum is called a finite series because it is a sum of a finite sequence of numbers. We explore this particular series in many different places in the high school curriculum, but a particularly interesting discussion around this series occurs in Lesson 8 of Module 3 of Grade 9 (Algebra I). In that discussion we show how the sum can be visualized as “triangles:”
This visualization quickly leads (through pictures!) to the general formula of the sum of the first n positive integers:
Plugging 25 in for n shows that , which is what Autumn derived. If you are teaching Lesson 8 (or any of the other lessons where this series shows up), think about sharing this video of Autumn with your class as a way to stimulate a discussion with your students.
As always, comments are welcomed! In particular, we are trying out different places to do math. The math in this video was done on the hood of a Porsche 914. We have friends with all kinds of cool sports cars who may be willing to let us borrow them for an afternoon, so look for more “Math on the hood of sports cars” soon (especially if we get positive feedback to do more videos like this).
CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636