# Simple Folds that are Hard

Folding a square diagonally is not in most people’s skill set.

There are so many applications of diagonals, not just in paper folding but also in math, that it’s a worthwhile concept to chase down and cozy up to, but it needs to be done carefully.

Having this extra time home these days, I find myself wanting to write about these big/small details that teaching has taught me, that feel important to share.

If I ask a student to fold a piece of paper in half, they fold the paper in half. Generally I ask students to refine their intentionality with the paper so that their paper is folded evenly. The older they are the more they love this lesson. The whole experience of folding paper is half is a pretty happy place. Until it isn’t.

What shape is the paper in when you fold a square in half? Think about it before you raise your hand, and don’t change your answer because of what someone else says, and I’m going to call on everybody who has their hand up.

I see every hand go up. Rapid fire around the room, rectangle, rectangle, rectangle. Then someone say triangle. Just one person. That person is so brave. I ask that student and another student to come to the front and prove their response to me and to the class. They are both correct. We celebrate.

The fact that making a diagonal fold fries brain circuits was driven home to me one day when I asked a class of third graders to make three folds with a square piece of paper. I didn’t anticipate any problem as I already had already done quite a few folds with this group.

The lesson for this day was to do one diagonal fold, unfold it, flip it over, fold vertically, open it, and then fold horizontally.

I could not have been clearer in my instructions. I had drawn out the directions, which were projected on to the classroom smartboard. The papers I handed out were printed so that they could compare their progress easily with my sample.

I have rarely so quickly and completely experienced the chaos that followed. The students were totally confused; there were renegade folds on nearly every paper. Students struggled with paper orientation. They fearfully charged ahead in complete darkness. My perfect simple lesson was simply a disaster.

This was a class of 25 students. I would be facing four more classes who needed to learn this folding sequence.

The problem wasn’t the diagonal fold. The problem was the sequence of folding. Did I lose you for a moment when you read this:

“The lesson for this day was to do one diagonal fold, unfold it, fold vertically, open it, and then fold horizontally.”

Turns out that making the diagonal fold in the first step of this lesson disoriented the class in a way from which they could not recover. Immediately, students were felt unsafe, afraid of doing something wrong, and I could not gather them back to me.

The fix turned out to made perfect sense. I showed the diagonal fold last instead of first.

Students folded papers in half (the “regular” way), opened the paper, folded it in half in the other direction, opened it, flipped it over and made one diagonal fold. Before any confusion took hold of their brains, their folding was done.

I want to write more about the diagonal, how to make it safe, and why it’s important, but that’s another story.

Today’s story is simply about how something that is simple can go haywire.

# Fraction & more Fractions

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.

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.

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.

#### 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.

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

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

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.

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

There was some deeper thinking going on than I expected.

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,

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.

# 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.

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…)

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.

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 http://mathhombre.blogspot.com/2009/08/money-games.html

Simply awesome.

Addendum #2: here’s the link to the final post of this project https://bookzoompa.wordpress.com/2017/04/30/fancy-plane-shapes/

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.

# Re-Framing a Lesson

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.

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.