I tried the beginning of my Pairing Up With Gauss lesson with my middle school group, which contains 10- to 13-year-olds at approximately 6th to 8th grade math levels. This is a group of 9 kids, 3 of whom were in this group last year, and 6 who are new to it (but who had been in the approximately 3rd to 5th grade level group I’d worked with last year). I’ve showed this trick in the past to both groups, whenever a need to add a sequence of numbers arose, but I’ve never done an explicit lesson on it.

Since it was my first time meeting with them this year, we first created a list of problem solving strategies (maybe I’ll put those in a separate blog post), to put at the beginning of their problem solving notebook, to refer back to when they are stuck on a problem and don’t know where to start. I also had them add a for-now-blank page for definitions, since experience has taught me that vocabulary is a big part of math problem solving.

Then I asked them to try adding up the numbers from 1 to 100. I told them they’d have 5 minutes, and I wasn’t so worried about whether they got the answer or not, but I wanted them to tell me how they tried to do it. One remembered the trick and got the correct answer. Two got double the correct answer, and realized why while we were discussing everyone’s approaches. Those were my returning students. One started with a “solve a smaller problem” strategy and decided to add up all the numbers from 1 to 50 (the hard way) but she wasn’t sure if just doubling that result would give her the correct answer or not. Another, I think, tried something with averaging the numbers in different sub-ranges, but was unable to describe what he did clearly, and he ended up with a way-too-high result.

The whole group was able to understand and appreciate the Gauss trick when I showed it to them. I handed out the practice problems, and they were able to apply it easily to sequences of consecutive numbers starting from 1. Sequences with odd numbers of terms were not a problem. A sequence of even numbers starting from 2 was not a problem. The problems started with the sequences of consecutive numbers that did not start with 1. Even though we had done an example on the board and talked about getting tricked when figuring out how many numbers were in the list, most of the students were off-by-one on the number of items in the list. (For example, thinking there were 39 terms in the sum: 11 + 12 + 13 + … + 48 + 49 + 50 or 60 in the sum: 5 + 6 + 7 + … + 63 + 64 + 65) Sadly, no one realized right away that the fractional answer they obtained when using 39 terms for the first example should have tipped them off that something was wrong. They so busy doing the multiplication 19.5 x 61 that they forgot that what they were really doing was supposed to be a shortcut for adding a bunch of whole numbers. They all had trouble figuring out how many terms were in the counting by threes example as well. We ran out of time and didn’t have time to really process all of this as well as I’d have liked, so we will re-visit it. I elected not to send the “finding the rules” part of the lesson home as homework since (a) they were still struggling with some of the numeric examples, (b) most of them (except the 3 returning students) have very little experience working with variables, and (c) September scheduling anomalies meant that I wasn’t going to see them again for 2 weeks — usually I give homework to be discussed during the next lesson.

So it will be 2 weeks until I get a chance to follow-up on this and move forward. But overall, I thought it was a good start, and we will be able to build on it the next time we meet.

1. September 18, 2007 5:43 am

Interesting to hear how it turned out.

Maybe you’re better off just omitting the case where the sequence doesn’t start from 1, but instead pursue into variables with the case where it does start with 1.

Then, after that is mastered, one could go back to the case where the sequence does not start with 1. Start with 0 perhaps as the first step because that clearly shows how there’s 1 number more in the sequence than what it looks like…
0, 1, 2, 3, 4, 5, 6 has 7 numbers, 1 more than the difference of the last and the first.

2. September 18, 2007 10:52 am

Hi Maria. Thanks for the comment!

The way we talked about figuring out how many numbers were in a sequence that didn’t start with 1 was this… suppose we are talking about the sequence 11, 12, …, 24, 25. I asked, “How many numbers would there be if it started at 1?” (25) “And how many are missing from the beginning?” (10) “So, how many are there really there?” (15). I showed them how if they subtracted, they’d get the wrong number.

Yet, when they went to do it on their own, they subtracted and got the wrong number. Oh well… At least they were quick to pick up on how they’d been tricked by the very trick I’d warned them about, and I think they do “get it” now.