math · Math and Book Arts

Fraction & more Fractions

Many Parts of a circle
Many Parts of a circle

I’ve been working with 9 different grade levels, nine different projects, this month, which is kind of wild, and even more wild because of all the snow days and other unexpected shifts in schedules. Most of the projects we’re doing are things I’ve written about enough on these pages, but I have managed to slide in a couple of new things with the fourth graders.

I had some extra time with some of the students  because they chose to stay after school for some extra time with me. Am still racing to finish prep for tomorrow, but want to quickly post about these two extra projects.

Dividing up a circle project
Dividing up a circle project

I brought in circles and sheets of regular shapes. Student cut up the shapes, and rotated them around a center point. The circles were marked with 12 evenly spaces dots around the circumference. We talked about other cyclic things that are divided up into 12 parts (clock, months) and talked about how 12 has so many divisors.

Rotating Shaper around a circle
Rotating Shape around a circle

I printed the shapes on heavy paper. I hadn’t done this with kids before so I didn’t know if they’d have trouble with this. It was no problem for them at all. They were excited, worked creatively, asked questions and were totally engaged.

Student rotations
Student rotations

Here’s the PDFs that I created for this project.

Circles with 12 dots

shapes to rotate in circles

I casually mentioned that ANY shape can be rotated. Well, they didn’t have to hear me say that twice before they were making new shapes.

Crazy Shape rotation
Crazy Shape rotation

The trick is to retain points that can still line up with the center and with a point on the edge of the circle.

Another Crazy Shape Rotation

During class time, I worked with students on a fractions/ bookmaking project that I’ve written about previously on my Books Are Fractions  post.

Fractions book
Fractions book

I knew some students would finish up early, so I showed them some images I had printed up some twitter posts. (If you want to see many more images like this, type in the words Fraction Museum in the twitter search bar and you will be well rewarded)

The kids were enthusiastic about creating fraction museum pieces, which I then photographed.

Fraction Museum hearts
Fraction Museum hearts

The idea is to collect items, see them as part of a whole, then write fractions that describe the collection.

Fraction Museum books
Fraction Museum books

There was some deeper thinking going on than I expected.

Mixed Fraction Museum
Mixed Fraction Museum

I’ve assembled all their images on to 2 large sheets of papers, and will present them to the kids tomorrow….but only if I stop this blogging and get back to work,


Addendum March 26 2018

During my fractions conversations with these kids (who, by the way, had a good grasp of fractions before I ever showed up) I talked about the confusion that can happen when trying to understand why, when the denominator is a bigger number, the unit fraction is smaller. I showed them a piece of paper folded into four sections, then said if I had to fold the same paper into eight sections (which we did) that the number of units had to be smaller to accommodate the larger number of sections. Then I asked “Imagine if we had to divide this paper into 100 sections, how small would those sections have to be? 


Well, that was it. They begged to see a page divided into 100 sections. Each time they saw me, they reminded me. Finally, today, I brought in TWO papers, and asked which one of them had fraction units that were each 1/100. Led by an particularly independent thinker, they figured it out. And figured out why, even though the divisions looked different, that they were all 1/100s.  It was a great conversation. Here’ the PDFs of you can ask kids this question yourself: hundreths

So much fun.

Art and Math · Arts in Education · Making Books with children · Making books with elementary students · Math and Book Arts

Making Books with Money


Oh my gosh, working with second grade students is so rich.

They have skills, they are enthusiastic and uninhibited, and tapping into their learning curve is delightful.

Windshield, with George in the driver's seat
Windshield, with George in the driver’s seat

I’m working with three sections with about 22 students per class, so I’m getting to see about 66 different ways that students are making sense of the 100 cents project that I described in my last post. (oh, there’s an unintended pun in that last sentence…)

Abstract design
Abstract design

Short recap: students were given images of coins, which add up to $3.00, from which they chose $1.00, or 100 cents, worth of coins to create a design.

These students hadn’t started studying money yet, which was fine. Most students seemed to understand how much coins were worth, though certainly a few students had no idea about the value of coins.

It was fun, when adding up the value of nickels, to say, Now you know why it comes in handy to count by fives.


Making the wallet-book to house the 100 cent images, then making the images was what we got done on the first day. Separating out 100 cents was certainly the most challenging part of the project. The designs flowed freely.


Day 2 was a bit more challenging, but I think that the toughest part was just communicating to them what I was looking for, which was for the students to make matching arrays of the coins that they used in their designs, then providing the equation which showed that the value of the coins equal 100.


Turns out that this array-making uncovered a few mistakes. For instance the airplane pictures above was five cents short, so he added a nickel on to the bottom and all was well.

Person in landscape
Person in landscape

There was a wide range of simplicity to complication of images.


If students didn’t have enough coins of a certain value left from their original 300 cent to making the matching array, they would exchange change with another student, at least that was the plan, which worked fairly well. I did bring lots of extra coins, for moments when it seemed better just to hand students what they needed.

Still, everyone should have had 100 cents left over. These coins got glued on to a pocket of their wallet book, along with a statement of the value of these coins.  That little black folder that contains the 100 cent image now has an enlarged section of a colorful buck glued on to the front. After all that figuring and adding, it was great to end yesterday’s class with some playful coloring in.

Okay, one more day with these students. The next piece that goes into the wallet-book has to do with combining shapes to make other shapes, much in the same way that we combined values of coins to make other values.

The most joyful moments during these days is having this opportunity to be a part of these early moments of learning about addition. When students say that they can’t get their numbers to add up to 100, though they know that they do, I can sit with them and help them sort out what’s going on. It’s so illuminating for me hear them tell me what they’ve done, and then to help them see another way of interacting with the numbers.

Addendum: as soon as this post went up the generous and brilliant connector-of-all -things-math offered me this link to some other coin projects

Simply awesome.

Addendum #2: here’s the link to the final post of this project

Addendum #3, April 2019

AFter teaching this many times, one of the biggest changes I’ve made to this project is to hand out only $1.50 in coins at first, from which the students pick and choose and count to make their $1.00 designs. They each get 10 pennies, 6 dimes, 6 nickels and two quarters. Then I give out the appropriate number of coins in arrays for the second page.

BIG TIP: Before teaching this class I encourage the teachers to get the kids practice  doing their skip counting starting at a place other than zero. Also, I think it would be helpful for kids to skip count by 25’s before starting their money unit.

Drawings · Making Books with children · Making books with elementary students

Re-Framing a Lesson

Bookmaking with First Graders
Bookmaking with First Graders

OMG Have I got a teaching tip for anyone who has ever pulled their hair out trying encourage students to make their drawings bigger, to fill up the page. It’s only taken me like 25 years of working with students to figure this out. This is big.

Filling up a page with a drawing
Filling up a page with a drawing

There’s this variations of a bookmaking project that I do with mostly first and second graders that includes a drawing. The bigger and bolder the drawing is, the better it looks in the book. Needless to say, it’s such a struggle for this age of student to make their drawings big enough.


Usually I give the students the paper that their drawing goes on and do everything but beg them to draw bigger. Well, sometimes I beg. Then, yesterday  (Friday)  Carter, a 7 year-old in my first class of the day, suggested that, before they start their drawing, I  lay the paper inside the frame that will surround it. It had never occurred to me to do this, so I tried it out in my next class of the day.


Unbelievable. In my next class, after sliding the paper behind the frame before the drawing began, every single student filled up the paper with large bold drawings to go along with their stories.

Never has this happened before.

Maybe it was just a fluke, maybe this class had been bribed enough times to fill up the page that they now did it instinctively. I had one more class to go.


Next class, same thing happened. They filled up the space with big drawings.


Some students lifted the frame away after the first part of the drawing was done so that they could make their drawings even bigger. OMG I was so happy. My conclusion: if you want students to make a drawing to fill up a space, FRAME THE SPACE with a dark frame! I don’t know why it works, but far be it from me to ever think I can fathom what goes on in the mind of a 7 year-old.

Now here’s the part that gives me chills…I have to ask myself, why did Carter put forth his suggestion? I give credit to this: recently I was impressed by reading Malke Rosenfeld’s book about engaging students in whole body learning. While I teach different subject matter than Malke, I am deeply impressed by how she gives her students permission to explore the learning space before she begins her lessons. I took this to heart, and this week, for the first time, within certain boundaries, I encouraged students to fold and unfold, then explore and examine the materials that we were using together. In some way I think this sense of engagement with the materials led Carter to making a suggestion that was based on what would have worked better for him. I already know that my best teaching tips come from the single digit crowd, I just don’t always know how to tap into them.

So thank you Malke, thank you Carter, and OMG I am so happy.


Number Line · origami · Paper Toy · summer art/math

Origami Boats and Meandering Number Line with 4-year olds

How to Make an Origami Toy Boat
How to Make an Origami Toy Boat

Full disclosure: I did not try out teaching 4-year olds how to make origami boats. It’s not that I didn’t want to, or that I chose not to, it was just that there were other things I wanted to do more, and my time with these little ones was limited. Much to my delight, though, after we used the boats in  our activity, the children asked me about how they were made. I did a demonstration, with hope that this may encourage an interest in  paper-folding.

Toy Boats
origami Boats, stacked

I chose to use these paper boats because they stack. Just for the record, I was curious to see if they floated. Turns out ;Yes! Until the paper absorbs too much water, these vessels are sea worthy. What was more useful for me, though, was that they can stand on their own, so that we could use them as playing pieces for a board game.

During my workshops with these children I noticed that even the most accomplished child in the group could not coordinate counting items with the movement of his hands. In other words, if there was a  pile of 8 stones, these children would end up counting inaccurately because their fingers would move out of sync with the numbers that they were reciting. I was really interested to see this, partially because  I’ve read that there is something about learning to play the piano that  helps children be better at math: this now makes sense to me, as playing notes would help train a person to coordinate fingers with intention.

Toy Boats and Board
Toy Boats and Board, drawn on large paper

Wanting to try out a simple, and, yes, frugal, made-from-paper activity to encourage accurate counting skills, I worked out a sweet game that the kids seemed to like . What we did mimics  classic board games where a die is thrown, and the player advances a certain number of spaces along a line. It was, however,  important to me that I didn’t want to create winners or losers.  This is how it went: the playing pieces were these paper boats, and when two boats land on the same space, they become a team, and stacked together. The point of the game is to get all the boats stacked together as a team before any boat reaches the end of the meandering number line, which, just, for no particular reason other than I ran out of space on my paper, was 42 units long.

Beginning at 0
Beginning at zero

Unfortunately, I didn’t get a photo of the kids playing this game, but, they played in groups of three and four, and they seemed to enjoy watching others play as well as playing the game themselves. Counting spaces, counting the dots on the dice, and (especially!) anticipating what throw of the die would yield the desired outcome were all challenging but doable for these kids.

Each of the five sessions that I worked with these students, one-third of my lesson plan was to focused my interpretation of relationship thinking, such as creating patterns from shaped paper,  developing finger sense, estimating, discussing what was the same and different about shapes and flowers, and this unit counting game.

Other parts of my time of my time with these kids was artful numbers, which I what my next and last post about my time with these students will be about.

I had to learn how to make these origami boats for this project. I looked many different models, but this one that I’ve shown  I found most enchanting. I put together a video of it, that is worth watching because there’s some pointers included that I just can’t fit onto a tutorial page.

Happy boating.

Addendum 5/6/2017 Just came across a tutorial that shows an alternative way of making this boat. An interesting way of altering the folding method by Gregor Müller