Building arithmetic and number sense by talking about geometric series

I’ve been a little busy both at home and at work this week and as of this morning hadn’t given any thought to our Family Math projects for this weekend. More or less on a whim I decided to return to an old favorite topic for today’s project – infinite series, and specifically geometric series.

Two of our prior talks on infinite series are linked below, you can find others on the blog under the tag “infinite series”:

Just for Fun: Some Infinite sums

Talking about Infinite Series

The first part of the talk today was introducing the concept of a geometric series. The main idea I’m trying to get at today is showing how we can extend a common way of showing why 0.9999… = 1 to the problem of summing a geometric series. We talk through some of the basic ideas using the series 1 + 1/2 + 1/4 + 1/8 + . . . as our example.


The next thing that we looked at was the series 1 + 1/3 + 1/9 + 1/27 + . . . . My older son initially believes that this series will also sum to 2 because it goes on forever. My younger son’s initial guess is that it will sum to 1.5. His reason is that (except for the first term) the terms are smaller than the series 1 + 1/2 + 1/4 + 1/8 + . . . .

One theme that shows up here that will continue for the rest of today’s project is that subtracting two infinite series is a little confusing to the boys. I should have found a better way, or at least an alternate way, to explain this idea to them.


In the next talk I wanted to have the boys pick their own series to sum. Unfortunately, I wasn’t clear with them that I wanted to look only at series where the terms went to zero. That lack of clarity caused a small problem at the start of this part of the project.

Once we got on the right path, we worked through the series 1 + 1/5 + 1/25 + 1/125 + . . . without too much difficulty. But the next series caused a little bit of trouble:

1 – 1/3 + 1/9 – 1/27 + . . . .

The subtraction and the negative signs were big stumbling blocks here. I really needed to provide a better way to help them see what was going on when we subtract one series from another.


In the next part of our talk we moved on to talking about an general geometric series. This discussion is a big step up in abstraction. I think this abstraction was not as difficult for my older son as it was for my younger son, which isn’t a huge surprise. Subtracting the individual terms in each series still presented a little bit of difficulty. We did manage to get to find a fairly simple formula for our sum, though. Even with the difficulty we had, I think the discussion here are a nice example of how you can take an idea from a specific setting and use it in a slightly more abstract setting.


The last part of today’s project involved using the formula we found in the last video in the situations that we’d already considered. A few examples showed that our formula seemed to match the prior results. Yay!

We then wrapped up by looking at a few situation where the terms in the series do not go to 0. Here the formula produces some results that seem strange. For now I’m leaving these odd results as fun little paradoxes for the boys to ponder.


Watching these talks as I put this blog together makes me wish I’d done a better job with this project. I think that the important mathematical ideas here can be made accessible to kids if you present them in the right way. The results are neat and some seem strange (you’ll hear my son reference Numberphile’s video about the result 1 + 2 + 3 + . . . = -1/12 in the last video – so these strange results can really make kids think).

Hopefully the next time we return to this topic I’ll remember the lessons from this one and present some ideas in a slightly different (and hopefully slightly better!) way.

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Comments

3 Comments so far. Leave a comment below.
  1. I think you are being too hard on yourself. These are challenging topics and the kids will benefit from pondering them. Feeling like there is something they haven’t quite understood could be a good source of motivation to do so.

    I’m curious: when do you know if you had a good session? How do you assess your overall process and progress with the boys?

    three suggestions:
    (1) you could use this result to look at the repeating decimals again, for example, the old stand-by 0.99999999999….

    (2) would be good to have a place to keep earlier results, perhaps add a whiteboard to the other wall? Having the opportunity to look over what you’ve done earlier in the discussion might help reinforce the concepts and may also lead them to identify patterns which might have been missed.

    (3) it would be lovely to insert the clause “if it exists” in advance of many of these manipulations. On the one hand, it is an important reminder that, sometimes, what we are studying might not exist and this is a key to many trick questions. On the other hand, it is also a valuable tool for investigation: assume this thing exists, what properties would it have? It even works outside math as one habit of creative thinking.

    • Whether rightly or wrongly, I’ve changed my approach to infinite series after reading Ellenberg’s “How not to be Wrong.” Also probably influenced by the Numberphile -1/12 video and subsequently reading Hardy’s “Divergent Series.”

      I want the kids to see that the formulas work and sometimes they don’t work. I think Ellenberg’s line about the 1 -1 + 1 – 1 . . . series was that we can say that the limit doesn’t exist, or we can define it to be 1/2 because all of the theories about convergence say one of those two things.

      I don’t mean to suggest that I’ve thought this approach through carefully at all, I just don’t want to slam the door shut on the divergent series and I want the kids to be puzzled about the times the formulas produce add results.

      As for when things went well and when they didn’t – that’s just more of a gut feel when I’m going through the videos. The value I get out of all of these videos is the opportunity to learn from what feels like it went well and what feels like it could have been better. Today was more of the latter than the former.

  2. Your approach is fair and probably doesn’t do any harm, but my idea is that it would be valuable to help them see what is going on in the different cases and the development:
    (1) we have an easy case we understand well: finite series. We know what the equals sign means here.
    (2) we have a desire to extend to infinite series and find we can do that (cauchy convergence) in a way that lines up with our intuition and matches the results for finite series.
    (3) we have a desire to extend further and find, surprisingly, that we can coherently do it in a way where the earlier cases still are treated the same, we get a bunch of additional results for formerly non-compliant series, but we have to give up our earlier intuition for those rogues.

    Just my feeling that this is one of the coolest parts of modern mathematics. However, I haven’t read JSE’s or Hardy’s books, so maybe I should defer.

    We did a baby version of extending definitions the other night, prompted by some intentional silliness from my middle son: adding all odd integers up to 10

    Also, playing with some partial sums might help them sort out subtracting the series.

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