The Animated Equation Book

Flip Book Array

My Do-It-Yourself Equation of the Line Flip Books are ready to share.

I’ve been writing about these, on and off for months, but my work on them has been steady. My goal has been to create PDF pages that I can distribute on-line which a class of students can assemble in about 10 to 15 minutes,  and which show how the changing variables in the equation of the line, y = mx + b, changes the look of the graph. I am doing this because it seems to me that the understanding of this particular equation is either a gateway or an obstacle into the continuing study of mathematics. This is my way of contributing to the conversation of how to nurture a more math literate culture.

My thought is that if students can hold the equation in their hands, that it will give them the opportunity make sense of it for themselves.

I’ve made four sets of PDF’s. I  find that heavier weight papers makes the best flip book. I use Hammermill Color Copy Digital Cover paper, 60lb, but that said, if all you have is standard weight copy paper, use it.

Here are the PDF’s!

Changes in b                                                                                                                                    Click to watch a  brisk animation                                                                              Slower animation

The PDF for flip book of changes of b in y= mx +b

Changes in m between negative one and positive one.
Brisk animation
Slower animation

The PDF for flip book of changes of m between negative one and positive one

in y = mx + b

changes in m, greater than positive one , less than negative one.
Brisk animation
Slower animation

The PDF for Flip Book to show the changes in m greater than positive one, less than negative on in y = mx + b

Changes-in-x                   Brisk Animation                                             Slower Animation

 The PDF for Flip Book to show changes in x on the graph of y = mx + b

There, now that you’ve downloaded all the PDF’s you’ll notice that, oddly, they look like this:

Sample of the PDF pages

What may or may not be obvious is that there are six pages on each panel, and half of these pages are upside-down. So now it’s time to give you some hints on how to make these pages into flip books, and tell you what’s up with the upside-down.

Each piece of 8 1/2″ x 11″ copy paper will contain six pages of the flip book. (To my A4 friends, I will be making an A4 version of these, but they are just not ready yet.) The pages need be cut out on the solid gray line. Let students do this with scissors! The pages are numbered, so it should be easy enough to get them in the right order. The important part, when assembling these books, is that the FLIPPING EDGES are even!!! That’s why half the pages are upside-down on the page, so that the flipping edge always fall on the edge of the paper so it doesn’t have to be cut.

You can seen the top surface of the book on the left smooth. This is the flipping edge, which must be smooth. The spine of the book on the right is not smooth, which is fine.

Now, how to bind these small books…

Flip Book Bindings

Bookbinders will figure out their own ways to bind these books. These simple binding solutions are meant for the classroom or home school venue.

Before you start, note that the front and back covers aren’t numbered.  You can figure out which is which. Just be sure to flip over the back page so the graphic shows otherwise your back cover will look boringly blank.

The easiest way to put these pages together is to make sure that the flipping edge is even, then wrap a rubber band tightly around the spine. This simple solution works surprising well, though every so often you might have to remove the rubber band and realign the edges. The thinner the paper, the thinner your rubber band should be. Experiment. You’ll know what works and what doesn’t.

My favorite simple solution is to use strong clips, like on the upper right in the photo above. I just found out that these are called binders’ clips. Excellent name. Use the smallest size that works. The small ones, 3/4″ wide, work just fine.

If you have access to a drill, then doing a simple sewing is swell. Drill three holes evenly spaced, about 1/4″ from the spine and follow this sequence: go through the middle hole, leaving a tail of thread behind, sew through the top hole, travel down to and go through the bottom hole, then go back up through the middle hole and tie your ends together.

If you’re a bit more adventuresome and want to look up Chicago screw posts, these make excellent bindings for flip books.

Books books books

That’s it. Make lots of books. Let me know how it goes.

Flip Books, nearly done

Equation of the Line Flip Books

I didn’t think these would take so long to design and create.

I thought I was starting with something easy. My endgame plan is to make flip books that show hypotrochoids, aka spirograph shapes. I thought that starting  with something as straight forward as a straight line wouldn’t take long. These books show variations of the equation of a line (y=mx+b). Four books, one to show changes in b, one to show changes in x, and two to show changes in m. This process has been anything but fast and easy.

I didn’t count on there being so many decisions to make. I didn’t count on having to make so many revisions. I had to learn a whole lot more about the graphics program that I’m using.

I am ready to finish up this project.

I have loved every second of working on these books.

This is a short post because I hope to finish up these books in two days, and to post PDF’s for Do-It-Yourself flip books, so that anyone can make them.

 

Flip Books as a Bodacious Learning Tool

Flip Books and random pages. The Birthrite flip-book by Ruth Hayes, published by The Real Comet Press, 1988, was a gift from Lewanne Jones and Jim Fleming when my son was born.

After all these years of teaching how to make books I have now become smitten with using books as a teaching tool. Yes, I know, using books for instruction is anything but a new idea. This said, flip books have captured my interest because they are fun and dynamic. Among other things, I’m pairing them up with mathematical equations to illustrate how the picture of an equation changes when there are changes in a variable. So far I’ve shown some of these flip books to a couple of my smart but not particularly mathy friends and what happened next is so worth writing about.

y equals mx plus zero ( this should play through 2 times. Click on it into play again)

The first thing that happens, of course, is that my friends flipped the pages. Everyone loves a show. The gif in the box above is a pretty good representation of what they saw in one of the books. A gif is fun, too, but it’s not as effective as a flip book in that it doesn’t allow the viewer to slow down and examine what’s going on. I saw this happening so clearly: my friends were drawn into the equation by the action of the flipping, then they slowed down, looked at the images more slowly and tried to understand what was going on. They had great questions.

For instance, Sarah thought I had made a mistake in labeling these pages. She hadn’t sorted out how the graph of y=50x could be so similar to y= negative 50, after all 50 is arguably a large number while negative 50 is indisputably a very small number. It was easy enough to explain how this works, and she absolutely understood it. What I understood was that, without the flip book, she would have never been interested in having had this conversation.

Comparing the slopes of a line

John also had some questions. He couldn’t fathom why I showed lines with slopes equaling 1 to 8 in sequence, then started skipping to 12, 20 and 50. When I pointed out how the lines were becoming increasingly indistinguishable from each other as the slope becomes larger, he better understood my choices of which slope values to use.

It’s taken me quite a bit of time to get to the point where I’m ready to show this first equation-of-a-line book to anyone. There will be two or three more books to go with this one: one that shows only “b” changing; one that shows the graph when the changes in “m” are between positive one and negative one; and a book that shows b and m changing at the same time.

I’m hoping to get some feedback on these from classroom teachers. Here’s my plan: after I have this y=mx Flip-Book finished I will make a few extra copies and send some out to teachers who are willing to point out flaws in my presentation. In my next post, when I’m ready (hopefully tomorrow) let me know if you want to be one of my collaborators. Even though I’ve already worked through at least a half-dozen different variations of this one book, there are still things that I am not sure about. More about that when I continue…

Flip-Book, Take #2

Waxed Thread for Sewing Books

Am still trying to hone in on a way of making flip-books. This means creating templates in my graphics program as well as figuring out how I want to bind the books. There were too many things that weren’t right about the Japanese stab-sewn binding that I’ve already written about, so I thought I would take a look through the internet to see how other people bound their flip books. I found some slam-dunk awesome flip books, but not much inspiration for the treatment of the spine.

Some solutions for Flip-Book Bindings that I found on the Internet

One site recommended putting a tight rubber band around the pages. I actually love this solution, as it’s so cheap and easy, but it wasn’t the way I wanted to go for the set of books that I’m planning. Another site suggested using a pad of Post-It notes, which actually seems like sort of an expensive way to go, and doesn’t leave much room for error. Also, the use of Post-its invites the distinct possibility that someone will re-appropriate your animation for use as (gasp) Post-it notes.

I tried out using a lighter weight paper, too, which my daughter immediately nixed. She said that the pages moved too fast, and that the book didn’t have the satisfying clicking sound that she likes. Someone on the internet suggested using filing cards for the pages, which I think would work well.  I think I saw at least one flip-book that was bound with those heavy-duty paper clips, which I just discovered are (appropriately) called binder’s clips. I think these would work well, though that re-appropriation problem would again apply. It looked to me like some people used pre-bound books or pads, and some people just held the pages together with one hand while they flipped with the other. Oh,  and a truly heavy-duty staple gun can do the trick too.  All good solutions, but none that enthralled me.

Gathering materials for bookbinding

I decided to sew again, but to sew a simple 3-hole pamphlet stitch on to a flap. I made the flap out of book cloth. It’s hard to explain the flap in words, but I think the pictures show it well enough.

The book cloth has one accordion fold in it, and the rest wraps around the spine.

Book Sewn to Spine Piece

There, that was simple enough. I’ve actually done something like this with first grade students, but instead of sewing and using cloth, the spine that these first graders used was made of paper,to which we applied a two-hole punch, and used paper fasteners to attach everything together.

Big Bad Dragon storybook

Here it is!

Storybook

That was such a great project. But I digress..

Done

Here’s the finished version of my little book. I’m mostly happy with it. I really like the look of the spine. The book works well, makes a good sound, and feels good in my hands. The last page has some writing, explaining what CMYK is. The text reads as follows  (please feel free to suggest edits to this text if I have made any errors, as I still consider this a work in progress. Thank you):

CMYK is like a secret code that printers use to define and mix color.

C: Cyan, bluish values
M: Magenta, reddish values
Y: Yellow, values of yellow
K: Key, values of Black

The range of numbers that follows each letter defines the concentration of CMYK pigment that the printer uses: 100 is fully concentrated brightness, zero indicates the complete subtraction of color.

This flip-book shows 48 of the possible 104,050,401 permutations of CMYK values 0 through 100.

In my last post I included an unlabeled and incomplete animated of a version of the contents of my book. If you’ve seen that post already and still have an itch to see a flip-book in action, take a look at this thoroughly gratifying brilliant creation of Matthew Shilian:

If you are in need of instructions for how to do the pamphlet stitch that I referred to, take a look at http://www.booklyn.org/education/ispamphlet.pdf

More flip books to come…