Flip Books, nearly done

April 22, 2015

Equation of the Line Flip Books

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.


A bounty of books

Eight Humongous Rubber Band Books, showing off their colorful spines

The sixth grade English teacher in this school likes the idea of each of her students making a book that they can use as (her words) a memory catcher. Writing, pictures, and ephemera will go into these books. The design challenge is that I can’t count on having more than 40 minutes to work with the students. I want them to end up with something large, sturdy, and I want them to enjoy making it.

Setting out the Papers

Setting out the Papers

On my day with these sixth graders, they walked in the library, saw the colorful papers and were immediately delighted. “Do we get to do this today?!” They were all so happy! My papers here are tabloid size, 11″ x 17″ 67lb papers (which, by the way, are getting more expensive and harder to source every time I look).

Choosing papers

Choosing papers

Each student chooses eight papers. We have plenty of space to work. It’s interesting to notice how each student chooses to arrange their stash.

setting out papers

Getting ready to work

Some students choose to work alone and spread their papers out all out in front of them

Getting ready to work

Getting ready to work

Other students work two, three or four to a table and have to stack their papers.

Next step is to fold the papers then nest them together in groups of two.


I’ve worked with these students many times before, and they are all have expert paper-folding skills.


The trick to accurate paper-folding is to hold the paper with one hand, then slide the other hand towards the curl.

foldingThese students have been using my bone folders just about every year they’ve been in school. If I forget to hand them out they will ask for them. In schools I refer to them “folding tools” to avoid  vegetarian discussions. If the fact that they are made of bone comes up, I advise vegetarians not to eat them.

These papers will  get nested together in groups of two.

These papers will get nested together in groups of two.

The students end up with four groups of two folded papers. This grouping is completely non-intuitive: students want to nest them all together, one inside of the other, and wrap one rubber band around the spine and be done. In fact, the book would work just fine that way, but I’m here to show them something different, and, arguably, better. By asking them to make four groups of paper they will end up with a thicker, and much cooler looking book spine, one which shows off some of the colors in the book.

Snips off the top and bottom of the spine edge of the paper

Snips off the top and bottom of the spine edge of the paper

Once the pages are grouped together, there’s one more step before the assembly starts. The corners of the tops and bottoms of the folds are snipped off. These snips create valleys that the rubber bands will settle into.

Attaching nested groups of paper

Attaching nested groups of paper

Two groupings of papers are set next to each other side-by-side, opened in the middle. The rubber band slides over the four adjacent pages, binding the page groupings together.  I use  Quill Brand Rubber Bands, 7Lx1/8″W which are humongous in just the right way. Smaller rubber bands will actually work for this, but the tighter the rubber band stretches, the sooner it will rot and break. I want these books to stay together for a good long time.

Attaching signatures together

Binding together two groupings of papers, with the rubber band

On goes the rubber band! This is done until all four sections are linked, in sequence, one group right next to each other. This book can be made to be just about any number of pages long.

Book made with humongous rubber bands

Book made with humongous rubber bands

It’s a good idea to decorate the cover of this book right away, as the flexible nature of the spine can make it tricky to figure out which page is the front once it’s been opened and looked through. Students make pockets to go on the front and back covers, to store items that will be eventually attached into the books. I’ve been making these books with this school’s sixth graders for a number of years, but I don’t get to see them finished. Students, however, will joyfully tell me about them, and they will also tell me, oh I remember when my brother made these! From what I understand, they hold a plethora of memories.

Pieces of Paper


This story begins in a teachers’ lunchroom, a couple of years ago, in Upstate NY. I was sitting with some teachers when another member of the staff started talking to a first grade teacher, Mrs. K, about a new math mandate. It was something about using manipulatives to create a variety of shapes. I’m a bit foggy on this part but it seems to me that they were required to use rhombuses (or rhombi, both are correct) for their shape building.

A Rhombus

A Rhombus

Upon being told that she would have to incorporate these manipulatives into her math unit Mrs. K asked if there was any money in the budget for manipulatives. The answer was no.

After school I sought out Mrs. K and showed her some paper-folding and shape transformations that referenced rhombuses. This teacher seemed delighted with what I was showing her. I volunteered to send her something that I thought she might find useful, then went home and created these images for her, which are equilateral triangles that become a rhombus.

screen shot rhombi

I never asked Mrs.K if she used what I sent her. I recognize that what I sent was, unfortunately, not a project. Instead, it was just the bones, the beginning of a project that needed to be developed.  Every so often I’ve revisited these images, wondering what I could do with them. Then a few nights ago Malke, from Indiana, asked me about projects for a family night.

It was late, and we decided to resume the conversation the next day. The next morning, before Malke and I reconnected, I saw this post from Simon Gregg, in France:

I had an Aha! moment. It suddenly came together. I sent off this note to Simon:

Malke, who I included in the conversation, responded with a reference to a beautiful manipulative that I wasn’t familiar with, but which also showed that she immediately recognized what I was getting at with my DIY paper version of manipulatives.

Since Malke seemed to know exactly what I was thinking about I got to work creating the pieces for this activity. I’m pretty happy with how this has developed. It requires triangle paper, and matching paper shapes that can be printed on colorful papers. My thought is that simple, bold shapes can be created in sort of a free form way…

Big Shapes

Big Shapes

…or more challenging shapes can be drawn on to the paper…

Drawn Shapes

Drawn Shapes

…and filled in, while trying to make as few cuts as possible and being mindful about cutting along the lines defined by the triangles.

Filling in drawn shapes

Filling in drawn shapes

So, where can you get these papers to do a do-it-yourself shape building set? Right here. I’ve created a couple of PDF’s to get you started:

Triangle paper and Rhombuses

Make beautiful shapes. Send photos. Thank you.

Addendum: Take a look at Malke’s post on hands-on math: she collected and organized many interesting perspectives. It’s a fabulous piece of writing.  http://mathinyourfeet.blogspot.com/2015/04/some-thoughts-on-hands-on-math-learning.html

A Fractions Book

I’ve finished up this fractions/number-line project that I’ve been thinking about. I worked with a class of fourth graders who were just starting their fractions unit. My plan from the start was to try to present a project that was dynamic enough to capture their interest. The center piece of the project was to make a “magic wallet,” which is a shortened variation of a Jacob’s Ladder. I’ve been using the name “Li’l Jacob” instead of “magic wallet” because, originally, I couldn’t remember the magic-wallet name, and Li’l Jacob seems properly descriptive.  This structure opens in two different ways, to reveal two different visuals. It’s tricky, and seems magical. I am happy to report that these students were over-the-moon happy to learn how make this.


The students told me that the first step of arrangement of papers reminded them of a person

After showing the students what the finished book project would look like we dove right into making the L’il Jacobs. Making this requires a completely non-intuitive sequence of precise folding and gluing. The students have to keep track of where they are in the sequence in order to get the folding to work. I was nervous about how I could get them to see for themselves what was going on. A great surprise was that they offered me the best description I could hope for: they saw the arrangement of papers and immediately recognized it a human figure, legs and torso. Perfect! Now I was completely convinced that I would continue calling this a Little Jacob.

???????????????????????????????The students decided that this next step was best described by saying that they were covering Jacob’s innards.


As the sequence of folding continues, the Jacob becomes smaller. (Notice the paper that student is using  to protect the desk from getting mucked up with glue.)

Jacob squared

Li”l Jacob squared

The last fold reduces the paper into a square.

Each student made four Li’l Jacobs. Each of these had a set of equivalent fractions written on and in it. But we didn’t even start with the fraction labeling until the third class. Our second class was about making the book that was going to hold our fraction cards.

We folded a 33″ x 4.5″ paper int halves, then quarters, then eighths, to make an eight page accordion. I know that most people don’t have access to paper this size, but with a some thought this can be created by combining smaller sheets of paper.


Origami Pockets on every other page. Student chose their own colors.


Students then made origami pockets out of 5.5″ squares of paper. Starting with the second page, these were glued on to every other page of the book.

zero page

Creating a cover with a fold-in, to extend the number-line to inside

Next came the cover. The book needed an extension  so that the number line could start at zero. To accomplish this we attached an extra long cover piece then folded it over. I know I’ve explained that badly, so I hope the pictures above are adequate explanation.

fractions-Zero-to-OneFinally we were ready to label the Little Jacobs with equivalent fractions. I talked to students about how fractions could be a way of counting to one: one-fourth, two-fourths, three-fourths, four-fourths(one). I showed them my animated zero-to-one gif and some static images of equivalent fractions  but they seems to like the image above the best. We circled the columns of fractions equivalent to eighths. The really seemed to get the concept, and kept referring to it as they did their labeling.


The picture above is my sample that shows the labeling, with different ways to write equivalent fractions, as well as a simple addition problem, using the fractions.


Here are some images of the students finished books.


The even pages hold the fractions in the pockets.

odd pages

On the odd pages, students wrote out the fractions. These fractions had no equivalents on our chart. Forth graders don’t do fractions beyond the twelfths, so 1/8, 3/8. 5/8, and 7/8 stand alone.

K's book Some students were more inclined than other to do decorations on the books.


One thing that was wonderful about this class was that the students were incredibly helpful to each other. I could have never gotten this far with this project if I had to problem shoot with each child individually. The students who grasped each step were enthusiastic about working with a classmate that didn’t quite get a step.

big line


We lined up the fractions so the eighths showed, thus showing the 1, 2, 3, 4, 5, 6, 7 and 8 in the numerator. By the end of my third meeting with these students, just about everyone had finished with their books. This is one class that I can say is excited about equivalent fractions.





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