The Flux Capacity of an Artful Number Line

October 23, 2014

Envelope Number Line Book, Paula Beardell Krieg

Number Line Book, closed

I like the number line.

The number line is all about relationships: I can look at the number line and actually see and measure the chasm between two quantities, even when, as in the case of negatives, those quantities don’t even exist.

As an adult I’ve realized that I had some misconceptions about the number line, and I have discovered subtleties about it that surprise me.

envelope Number Line Book, Paula Beardell Krieg

Unpacking the Number LIne

I’ve been toying with number lines for quite a while. In my opinion the number line needs to be toyed with. The images that I see of it are not captivating. I’m wanting to rigorously play with this arrangement of symbols in way that captures some of its nuances. I intend to try to investigate numerous bookish solutions which means that I suspect that this topic will keep coming up.  I hope this will be an ongoing bookmaking/discovery journey. I’m not sure exactly where I will be going with this.

But I do know that a few nights ago , after a disappointing evening of cutting and folding, a way of proceeding finally presented itself, but I was too tired to grab hold of it. The inspiration teased me all night, and before 7 am the next  morning I was tending the coffee pot while working out my construction. I’m very pleased with how this particular structure worked out. It was so unexpected and delightful that I am excited to be sharing it.

Number Line Book, expanded, Paula Beardell Krieg

Number Line Book, expanded

It’s built from envelopes, the kind we think of as regular envelopes, though, technically, they are called “No. 6 3/4” envelopes.

Number Line,with negative numbers, Paula Beardell Krieg

Number Line, showing negative numbers too

Here’s are some of the things I like about this piece:

  • it’s a zig-zag
  • it has pockets
  • it scales
  • the structure suggests infinity since it can keep going in either direction
  • it can fold up into a polite accordion-like book.

The pockets are the most distinguishing feature of this number line. These pockets hold cards, which are printed with different sets, or sequences of numbers. This means that the labeling, or the scaling, of the line is always in flux, subject to the whims of whichever algorithm that’s called for.

Number LIne by 10's

Number LIne by 10’s

That’s the crux of it: the flux.

As students proceed through their grasp of numbers, the labeling of the number line constantly changes in scale as needed. Eventually the number gets integrated into the coordinate plane, and becomes the x-axis. I remember seeing the little graphs in math books, and I thought that when I got to grown-up math that the lines would get longer. It never occurred to me that it would be the scale that changed, not the size of the line.

Number LIne in Flux

Number LIne in Flux

You can see that there’s intermediate markings between the numbers. These can be interpreted differently depending on which scale is being used. For instance, when counting by tens, the small lines can be counted as ones, when the number line is increasing by one’s, the intermediate lines become tenths. In my mind, the point of doing this is to drive home the concept that the very same line can morph into whatever one needs it to be for the visuals of the relationship at hand. The maker becomes the master of the line.

Folded Number Line

Folded Number Line

Then the maker gets to fold up the number line into this accordion-like square. Just my style.

Over the next few days I will be working on designing a set of instructions on how to put this line together. It’s likely, however, that if you picked up some envelopes you ‘d  figure this out for yourself.

 Addendum Here’s the link to the tutorial: https://bookzoompa.wordpress.com/2014/11/03/the-envelope-number-line-tutorial/

6 Responses to “The Flux Capacity of an Artful Number Line”


  1. […] so that real learning can take place. Paula Beardell Krieg shows several uses of it in her post, the-flux-capacity-of-an-artful-number-line, and promises to give directions on how to make one […]

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  2. […] see this structure as a way to make a number line, but it can be many different things. My last post showed pictures of how the pockets facilitate exchangeable content. I can imagine it with all […]

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  3. I save them Paula, thanks!

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  4. Simon Gregg Says:

    These are beautiful! I want to make them with my class!

    Liked by 1 person


    • I’ve been wanting to do these with a class, but the opportunity hasn’t presented itself yet. My thought is if I was working with kindergarteners that I would make the envelope construction myself and have them make the sets of colorful numbers.

      If I was working with students who are learning fractions I would might still construct the pockets myself but I would want to make this a number line between one and zero. I would a make several sets of numbers: one for fractions, one for percentages, one for decimals.

      I hope you make one of these. Let me know how it goes…just don’t try to fold up the line with the cards inside… After you empty it, twist it,, it practically folds itself up like magic.

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  5. “The maker becomes the master of the line.” This is inspirational. Yet some people seem stuck there. Even remixing other people’s projects is too much. It has to be… by numbers. As in, paint by numbers. How do you get people who are stuck from paint-by-numbers to trying their own “what if?” experiments in making?

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