Arts in Education · Math and Book Arts

Books, Symmetry, and Students

Pamphlets made by seventh graders
Sewn Pamphlets made by seventh graders

I’m in the busy part of my art-in-ed, itinerant artist season. The challenge is to keep what I do relevant to the students, to the curriculum, to the teachers, and to myself. Most of the work that I do in schools is done with teachers I’ve worked with in previous years. Usually I repeat a project each year with the teachers’ new classes, though there are always tweaks that are made. Then, sometimes, it’s time to retire a project that’s been working well for years.

I’ve just finished up quite a few projects in classrooms, many of which were new this year. I’m going to attempt to write a number of posts about these projects before the next set of classes that I teach start up.

This is a detail on one of my own drawings
This is a detail on one of my own drawings that starts with the graph of functions.

There’s been a shift in my approach to what I offer to the schools. Whereas I used to think of my work as a way to motivate and celebrate literacy, now I am more focused on using our bookmaking projects in a way that supporting the teachers’ math goals. I’ve been realizing that the math part of the curriculum is where many teachers most appreciate support. There is so much in the paper and book arts that can support the math that students need to learn that making this shift has been thoroughly enjoyable to me.

Symmetry is a theme that kept emerging in the projects that I presented these last few weeks. This is partly to do with the nature of making books, but I also deliberately focussed on it more than it other years. I’ve realized, just recently, that the symmetry of shapes is the visual equivalent of mathematical expressions. I probably won’t express this well, but here goes. Think about doing any sort of math problem that has an equal sign in it. 5+3 = 8. It’s balanced. If you add a 3 to one side you have to add a 3 to the other side to keep the expression true. Math calculations are all about symmetry and balance. It is, therefore, completely appropriate and desirable, to help kids develop their natural affinity to symmetry.

Starting with a pile of cards
Starting with a pile of cards

One of the projects that I did with kindergarten students had to do with these piles of square cards. Students worked in teams. The first student puts down a card, then the partner puts down a card that is a symmetrical reflection of color and shape. They take turns putting down a card and then reflecting it.

Making Symmetry
Making Symmetry

It tool a bit of doing for these 6 year olds to get the hang of what we were doing, but, still, quickly, patterns emerged.

Reflection symmetry
Reflection symmetry

These cards, by the way, are an element within a larger math activity book that we made.

 Since these pieces are made from paper, I suggested to the teacher, as this becomes easy for the kids, to cut out one of the smaller squares from some of the cards so that mirroring the shape transformations becomes a bit more challenging.


Another symmetry project I tried out for the first time was with the Pre-K crowd. My friend Joan, who has worked with this age group, showed me this activity that she had developed with kids she had worked with. I’ve been excited to try it out.

Popsicle Stick symmetry
Popsicle Stick symmetry

What I did here was define a line of reflection. Then these five-year olds did the same kind of reflection symmetry that I described above, each taking turns putting down a stick, then the partner reflects it with one of their sticks.

Popsicle Stick symmetry
Popsicle Stick symmetry

Again, it was a struggle to get these students started, but it didn’t take long for them to catch on.

Four student symmetries
Four student symmetries

After a short while I combined groups so, instead of working in pairs, there were four people in a group, which led to different kinds of designs. The pattern above was made near the end of the activity. From start (first handing out the sticks) to finish, this activity took a mere twenty-two minutes, which was how long it took them to began to lose interest. At this point I suggested that they just used the sticks to make whatever arrangements that they wanted to make. Surprisingly, many started trying to use them to spell out their names. I heard their teacher remark something about the fact that they struggle to write their names but they seem to be able to construct them just fine. Which gave me an idea, which I will show in my next post. Now, though I want to jump back to the photo at the top of the post, which is the books made by seventh graders.

Folding and tearing  large paper
Folding and tearing large paper

I’ve been doing this project with the seventh grade for many years. I give them a large piece of paper (23″ x 35″), which they fold and tear to make a pamphlet.

Pamphlets in progress
Pamphlets in progress

I don’t explicitly talk about the symmetry of the folding we do, but I will talk about it in the future. The fact that the sequence of fold and tears results in a scaled down version of the original sheet is something I want them to be aware of.

Glueing out the spine piece of the pamphlet
Glueing out the spine piece of the pamphlet

In fact, every aspect of making this book is symmetrical, even the pattern of the thread that sews the pages together is totally symmetrical.
When building just about anything, even a book, symmetry rules.

Some finished books
Some finished books

These kids are so proud of their books.

Ok, enough for now. More tomorrow…..

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? 

Hundreths
Hundreths

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 · Math and Book Arts

Fancy Plane Shapes

Second Graders are learning their shapes. What a great age to be composing, decomposing and recomposing.

Bird on the left, hexagon & rectangle on the right
Bird on the left, hexagon & rectangle on the right

These images are the final part of the Wallet-Book project that I began with about 66 students earlier this week. After making images that had a value of 100 we moved on to getting up-close and intimate with parallelograms and trapezoids.

 rhombus/parallelogram paper
click here for PDF of rhombus/parallelogram paper

Students got strips of these parallelograms, which I had printed on five different colored papers. We looked at the parallelograms and noticed that they were made of two triangles. After separating a triangle, we fit next to a full parallelogram, to see how the two shapes, together, could make a trapezoid.

Triangle plus parallelogram makes a trapezoid
Triangle plus parallelogram makes a trapezoid

The kids seemed delighted by this discovery. While most students illustrated this connection with three different images…

Three shapes on one
Three shapes on one

…there was one student who took an elegant approach, which was to make just one shape, then label the way it could be broken down into its parts.

Hexagon, rectangles and star
Hexagon, rectangles and star

Next we moved on to hexagons, rectangles, then, finally, a shape of their own choosing. The kids loved noticing how three parallelograms not only are the parts of a hexagon, but also, that their hexagon has the appearance of a cube drawn in perspective. The toughest shape for the students to make was the rectangle, as this meant that they had to divide a triangle in half, the rotate and reflect the pieces. Not so easy. Try it.

Designing
Designing

After making the compulsory curriculum shapes the kids went free-form, designing their own creations. My seventy minutes time slot with each class of 22 students just flew by!

Whale on the water, spouting
Whale on the water, spouting

Some students made abstractions with the shapes, others created scenes or something recognizable.

Playing with shapes
Playing with shapes

Since this was the first time I was doing this project in the classroom, and since I had three classes to work with, I kept changing how I presented the project, getting a better feel each time for better ways to get the kids to interact with the shapes within the time frame I had with them, and within the agenda that needed to be addressed. For each class, though, I kept the curriculum piece at the beginning of my time with the students, ending the class the creative part. I think I’d like to see what happens if I flip that order of working, as perhaps it will let students discover on their own things that I actively try to get them to see.

 

Pieces in pockets of Wallet-Book
Pieces in pockets of Wallet-Book

When we finished, any extra parallelograms were stored in the origami pocket we had made during out last session, looped a humongous rubber band into a hole punched into the front flap, created and ID card for the front, and added some bling, because, well, bling.

wallet-books
wallet-books

The Wallet-folders, by the way, are made from heavy weight 11″ x 17″ paper that has a four-inch fold along the bottom edge, which creates the pockets. There are two 7-inch wide pockets within, and a 3-inch fold-over flap. If you are wondering where I got this awesome paper that is tinted with black, blue, and silver, with a light cast of gold above, well, thank you for wondering. I spray painted the papers (outside, of course, with a mask on) so I could have exactly what I wanted. Yeah, a bit crazy, but that’s what I do.

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

Making Books with Money

Flower
Flower

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.

Person
Person

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.

Bug
Bug

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.

Aiirplane
airplane

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.

Flower
Flower

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/