Endless Accordion

I’m once again revisiting accordions and number lines, because they are both  infinity fun. What I’ve attempted to do here is to create a classroom friendly accordion book whose pages are pockets which can contain changing content, in this case a variety of number lines.

What makes this project classroom friendly is that it is designed to be used with a ubiquitous material: standard sized, standard weight copy paper. It requires a few simple folds, and very few materials. I’ve made templates that can be printed out, but lacking the resource of a copy machine, this can all be easily constructed without my templates.

Endless accordion with pockets.

The accordion is made from units of full-size sheets of paper, folded, then attached together. For the basic number line I recommend using 6 papers, which will result in 12 pockets. Since zero through 10 needs 11 pockets, the extra pocket at the end conveniently implies “dot dot dot …. on and on …to be continued. ”

The shows where the fold lines occur.

A full sheet of 8.5″ x 11″ (or A4) is folded so it ends up looking like the picture below:

Two pockets from one sheet of paper

The tabs at the side are there to create an attachment surface for other the next pockets.

Attaching pockets together

The tabs of adjoining papers can be attached with glue, tape, sewing, paper fasteners, staples or paper clips. I ch0ose paper clips.

One piece of paper, folded, has room for four numbers

The cards with the numbers are also made from sheets of uncut, folded paper. They are folded so that they are just a bit narrower than the pockets.  Once they’ve been folded they can be glued (or taped etc) shut but I don’t bother doing this, as they seem to stay together just fine without gluing.

Counting by 10’s

One set of numbers can make four different number lines.

Counting by ones

I’m providing links to PDF’s. There’s a PDF for the pocket, which I recommend that you make 6 copies of. This template is in black and white only. I hand colored in the dividing lines.

As for the numbers, I have one full color PDF here, and one that has the black and white outlines of the numbers if you prefer to let have your students color in the numbers themselves. At the moment I only have files for paper measured in inches, but in the next day or two I will update with A4 versions as well.

template for pocket

PDF 8.5 x 11 for accordion pockets lines

Numbers in color

PDF 8.5 x 11 accordion number line, colored numbers

Number to color in yourselves

PDF  8.5 x 1 blank numbers number line for accordion pockets

If you’re interested I’ve posted something about my interest in the number line on my Google+ page https://goo.gl/ScI0nZ

I would love to hear from anyone who constructs this project with a class!

Jacob’s Ladder Details

Jacob’s Ladder in transition

Time to finish up this Jacob’s Ladder. I had it in mind to develop this way of working with the Jacob’s Ladder so that it could be replicated by others, but it’s just way too specific and involved for me to actually encourage someone it do what I’ve done here. Rather than being a project to share, this has been more like designing a product to produce. But, still, there’s a few things here that I want to write about.

(If you’re not sure what a Jacob’s Ladder is take a look at  http://toymakingdad.blogspot.com/2010/09/making-jacobs-ladder.html or https://bookzoompa.wordpress.com/2014/11/16/a-jacobs-ladder-number-line-in-progress/.)

Jacob’s Ladder, in the early stages

I’ve been working with the Jacob’s Ladder in the same way that I would work with a flexagon: in other words, I’ve been interested in making complete transformations happen when the Ladder is flexed. This resulted is having four completely separate sides to discover. Here they are:

Side # 1

Side #2, (Flip side of Side #1)

Side #3

Side #4 (flip side of Side #3)

I had such a good time watching this come together!

The first thing I want to point out is the first and last panels of Side#1 and Side #4: they are both zero’s and both title pages. These two pages are the only two sides that don’t really change. I had to work these panels into the sequence, and I had to keep in mind that, in fact, they do change, but that change is only that they flip upside down. So not only did I have to make them work as part of sharing a sequence, but I had to figure out a way to have them make sense upside-down and right-side up in each of the sequences. And if that sentence just made your head hurt you can empathize with me. (Thank you.)

At first I thought I would begin with a one, since that can go upside-down, but finally decided that a zero was better. It took me awhile to realize that I couldn’t make the other end a number, I just couldn’t make it work. Making a cover page, though, seemed like a good solution.

Cover/ Title Page

I think that this turned out to be equally legible from any vantage point.

If you decide to try to make one of these I do have one major tip for you to consider. I think that most of the sets of directions that are out there have this one fatal flaw in common: they direct you to assemble the different sections in a line, growing it so that it resembles the completely unfolded Ladder. I have followed different renditions of this same theme, and, although the directions have been flawless in terms of accuracy they are confusing to follow. Following directions for sewing book sections together have can also have this same maddeningly confusing accuracy about them, too, so I’ve learned to look for patterns in the sequences, so I can just discard the directions. It’s a great way of working.

For the Jacob’s Ladder, assembling it is much easier if you stack the sections, weaving the ribbon around the blocks in the only logical way to wrap them. The trick is to start out right. Here’s how to start out right:

Step 1

Make sure your three long strips of ribbon are a bit longer than  your finished Jacob’s Ladder will be (some set of directions have you cut lots of small pieces and attach them in certain patterns, but I find this, at best, cumbersome).  In the photo above I’ve been overly generous in the overhanging piece, but you should get the idea that the upper and lower long ends extend to the left and the middle long end extends to the right.

Step 2,

For the second step, lay the second board on top of the first, and wrap the ribbons around the board in the only way that makes sense.( If this makes no sense to you now, you’ll see what I mean if you try it.)

Step 3

Next, lay down the third board, and, again wrap it. Continue doing this until you’ve run out of ribbon or boards. You’ll have a neat little stack that you should hold together well until you are able to attach (with nails or staples) the ribbons to the edges of the board.

Jacob’s Ladder, stacked

The ribbons make this checkerboard-like pattern. Quite distinctive.

I had meant to take some time in this post to write more about why I keep doing these number lines, but I think I will save that for a different day. It’s time for me to sign off for the night. I’ll be counting sheep soon…

The Flux Capacity of an Artful Number Line

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.

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

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, 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

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

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

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/

Some more number line posts:

https://bookzoompa.wordpress.com/2014/11/24/jacobs-ladder-details/

https://bookzoompa.wordpress.com/2016/08/28/two-beautiful/

https://bookzoompa.wordpress.com/2016/08/04/the-great-big-number-line/

https://bookzoompa.wordpress.com/2015/10/14/an-out-of-pocket-experience/

https://bookzoompa.wordpress.com/2015/03/28/a-fractions-number-line-project-for-fourth-graders/