How I used to teach finding the sum of a series:

I used to teach this concept very procedurally. I began by telling a story about Carl Frederick Gauss that I heard when I was in school:

As an elementary student, Gauss and his class were tasked by their teacher, J.G. Büttner, to sum the numbers from 1 to 100. Much to Büttner’s surprise, Gauss replied with the correct answer of 5050 in a vastly faster time than expected.^[18] Gauss had realised that the sum could be rearranged as 50 pairs of 101 (1+100=101, 2+99=101, etc.). Thus, he simply multiplied 50 by 101.^[19] Other accounts state that he computed the sum as 100 sets of 101 and divided by 2.^[20].

https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

Then I would show my students how he did it, and ask them to use the formula to find the sum of a different series.

What happened:

Students were not as interested in the story as I had hoped. The math was boring because I was doing most of it, and they didn’t see the relevance.

How I teach it now:

After I began my teaching transformation, I changed the story to a problem:

As an elementary student, Carl, and his class were tasked by their teacher to find the sum of the numbers from 1 to 100. Carl replied with the correct answer within a few minutes.

Discussion questions:

• What do you think he didn’t do? (He didn’t do 1 + 2 + 3+ … + 100)
• How could he have done this? Brainstorm some ideas with your group.

Use the number tiles or a number line to find the sum of the numbers from 1-20.

Materials:

Every group had access to 20 tiles numbered 1-20. They also had the option of drawing and using a number line.

What happened:

Students collaborated (almost everyone used the tiles), and found multiple ways to find the sum of a series. Generally, students were smiling – even students who claimed they didn’t like math, engaged in working toward a solution. When we did the closing, several groups shared their work and then we worked to find the connections and looked for a way to generalize the solutions. Here’s the full blog post on this lesson.

Some student solutions:

All groups got the correct sum of 210. We had to find a way to generalize these and make a “rule” that would help us find the sum of any arithmetic sequence.

The arithmetic was easy, and made sense to students because they reasoned about it to solve the problem, so we used that to build the algebraic expressions for the arithmetic series.

Possible questions to help students build the general expression for an arithmetic series from their computations:

• What was your common sum?
• How many of that sum did you have?
• What did you do with these two numbers? Why?
• What if we found the sum of the numbers 1-30? What do you think your common sum would be? How many of that common sum do you think you would have?
• What if we found the sum of the numbers 1-50? What do you think your common sum would be? How many of that common sum do you think you would have?
• What patterns do you notice?
• What if it was the sum of the numbers 1 through n? How many of that sum do you think you would have?

The tables below are for teacher information. Students should build these expressions from their understanding of the arithmetic they used to solve the problem.

Note: n=20 (because we found the sum of the numbers 1-20)

When students share their expressions, they notice that they’re all very similar or can be rewritten to be the same. So, it doesn’t really matter which common sum they used, the reason it works for multiple common sums is because of the algebra.

Full disclosure: I have not been able to help students who made sums of 30 generalize to a similar expression, even if we combined the 20s and 10s to make 7 sets of 30. If anyone has an ideas, please share.