## What is the sum 1+2+3+4+…+24+25?

Intended Audience:  Math teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 6-year-old Autumn shows how to find the sum

1+2+3+4+…+24+25.

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

$26\times 12 + 13 = (25\times 12 + 12) +13 = 25\times 12 +25 = 13\times 25,$

because she writes “same as 13 x 25.”  Regardless, you can see her calculating the

$25 \times 10 = 250$

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:”

–Grade 9, Module 3, Lesson 8 of Eureka Math/EngageNY

This visualization quickly leads (through pictures!) to the general formula of the sum of the first n positive integers:

$S(n)=\frac{n(n+1)}{2}.$

Plugging 25 in for n shows that $S(25)=25\times 13$, 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

Distinguished Professor of Mathematics, LSU. Geometric topologist: gauge theory, exotic 4-manifolds, knot theory. Author: Elementary Mathematics for Teachers.
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### 2 Responses to What is the sum 1+2+3+4+…+24+25?

1. Eleanore says:

I shared this article and video with some algebra teachers who work with students who are having difficulty with algebra. It is a great video to watch a child’s thinking and will help student understand how the formula was derived.

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2. Kelly Moore says:

I feel like I’M learning math all over again. This is really going to help my boys. When I saw this was Common Core related, I balked. Because I’ve heard such horror stories about it. But my son, who is seven and in second grade, struggles in math and so I’m willing to try anything to get him to get it and feel confident. It breaks my heart so when I hear him say, “I’m not good at math!” Age seven is far too young to be believing such nonsense. I only wish I had teachers to help me like this when I was in school. Thanks so much!

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