Art and Math · Arts in Education

Designing Outside My Envelope

I’m continuing to work on coming up with designs to use with some of my paper folding projects. This time around curvy lines are what I’m interested in. Valentine’s day is around the corner so of course I’m thinking about curvy lines.

After seeing some images I posted on twitter ,my friend Kathy H @kathyhen_asked me to blog about how I make these so her students might have some fun with curves. My initial reaction to her inquiry was negative, as I rely heavily on Adobe Illustrator, which isn’t very accessible. It took me a day or two to realize there are other options available for a student/person-with-computer, so here goes.

I start out in a free on-line graphing program called Desmos. To plot curvy lines we need to direct the graphing calculator to plot something that is cyclical. Think of a pulsing wave that goes up and down. The easiest way to tell the graphing calculator to make a wave is to reference a sine function. This is as easy as typing in a few letters. Here, take a look! https://www.desmos.com/calculator/welqj0gbm2 Be sure to play around with changing the numbers on this graph, so you can see how simply changing the numbers changes the curve.

Decorated Boxes
Decorated Origami Boxes

The next thing I do is try to make the curves more interesting. One the ways I do this is to direct the graphing calculator to multiply two cyclical functions together. To see what this looks like, go here https://www.desmos.com/calculator/deea2nnuzb. BTW one of the advantages of going to these graphing links is that you can use and modify these if that is more comfortable for you. The only thing to keep in mind is that if you want to save your own changes you have to make your own account, Which I recommend.

Origami Masu box
Origami Masu box

Last thing I play around with is making curves which relate to the curves that I have, but are different. These secondary curves are derived from the first curves, but follow different rules, Look at this link https://www.desmos.com/calculator/iedzflkoot Be sure to read the notes.

Here’s the graph that I made, after much playing around, to use on the box in the photo above:

Design to apply to origami box
Design to apply to origami box https://www.desmos.com/calculator/2h1bwetltb

What I need to do next is to make the graph into a an image that I can color. For me, that means taking it into Adobe Illustrator and trace it using the pen tool, then color it in with the Live Paint Bucket tool. There are other options.

The simplest option would be to hit the print button in Desmos, then simply trace the pattern you’ve made and color it in. Make copies of this if you have access to a printer. Doing these by hand has a charm that no computer can match.

Another option is to use a different graphing tool called Geogebra that can output a file that can be opened in a free online vector program called Inkscape. https://inkscape.org/ What you have to do is, from the dropdown menu on the upper left hand corner of Geogebra, is choose Download As, then choose SVG. Then, open this file in Inkscape.

Personally, I can’t do everything I want in Geogebra simply because I am not familiar enough with Geogebra. Today I wanted to use the workflow I’m describing here, but I couldn’t figure out how to tell the graphing program what I wanted to do. What I did next was ask for help. Jen Silverman @jensilvermath came to my rescue and inputted my curves. https://www.geogebra.org/m/vtstwgwx Asking for help is a completely reasonable workflow. These programs are so user friendly that, after not-too-long, we won’t need to ask for help. But ask for help for as long as it takes to learn how to do this on your own.

Used Inkscape for the first time today. I didn’t know how to do anything. Googling questions about Inkscape was easy. Again, this is a program that is designed to be user friendly.

Heart, with help from Jen
Heart, with help from Jen

This is a post that makes sense to me, but am not sure if I’ve been clear enough.

Having access to this technology that creates these magnificent curves can be so enjoyable. Be patient, though as it takes a good bit of playing around to get a really satisfying image. Then don’t forget to hit the save button! Happy almost-Valentines day.

Addendum, later today.  Read all about it!

This may make for an easier workflow.

My friend John Golden got me to try out coloring a Desmos file in Paint, which, I think, is standard program on most computers? Well, as least my computers always seemed to come with Paint, so it must be easy to get. It’s a raster program, so images won’t be super smooth, but, for classroom work, it’s looks great.

The workflow would be to save the Desmos file as a PNG by clicking the Share icon that’s in the upper right corner, then choose export. But before you do this, go into the settings by clicking the wrench icon on the near the top right and make sure everything is UNCHECKED! Your png will look like this:

Polar Rose, Desmos
Polar Rose, Desmos https://www.desmos.com/calculator/dgvpoxgyk6

Next open the image in Paint then use the paint bucket to fill in the blanks.

polar rose raster
polar rose raster

It’s true that the edges won’t be perfect. Raster images don’t do curves well.

polar rose raster detail
polar rose raster detail

Even though, close up, the edges are rough, still, this prints up quite nicely.  This is definitely a way of getting the job done!

Here’s what John just did, using this workflow

Arts in Education · Beads on Books · math · Working with Paper

Loose Ends

 

 

A handmade paper book cover
A handmade paper book cover

Last week I worked on organizing my desks and workspace.

This week I’ve been trying to clear ideas out of my thoughts by getting out the things I’ve been thinking about. Writing posts and making videos are like my pensieve in the Harry Potter movies.

So today I did a video dump. Three videos on three different topics. Getting these out will let me move on in a more focused way.

The first one is a video about working with paper. I was tearing some paper for this book cover, and took the opportunity to make a video to show how and why I tear. Here’s the link:

Next video is to accompany a post that I wrote a couple of weeks ago, about a project that I did with 5 year-olds about counting beads to create groups that add up to 10 beads. The kids enjoyed this project so much that I revisit the project with a video.

Bead counting book
Bead counting book

Here’s the video that describes the project. The cover photo of this video makes it look like it might be upside down, but everything is there right as it should be.

Next up is a video about a math problem that I saw on Mike Lawler’s blog  

Calculus problem that is more than just calculus
Calculus problem that is more than just calculus

I was swept away by this problem because it illustrates how a problem with really easy calculus in it (really, I could teach the calculus part in like five minutes) is scaffolded on top of math from geometry, algebra, and pre-calc. I love how skills from many parts of math have to be used together here. In many ways this is like the bookmaking that I do, in which I use many different skills to create one thing.

Thank you WordPress for giving me a place to clear my brain.

In about 10 days I will be teaching a two-day Chinese Thread Book workshop. Between now and the, it what I mostly hope to be thinking about. I will be seriously over preparing! Looking forward to it.

Making designs on papers
Making designs on papers for Chinese Thread books

 

 

 

Arts in Education

Recalculating a Failed Workflow

Each year, as part of a bookmaking project with sixth graders, I bring in my impressive assortment of paper punches, and let the students decorate their handmade books with a colorful array of  items which include stars, crescents, hearts and creatures. It can be a wild free-for-all, with some students slapping on their paper-punched creations willy-nilly, and others making carefully thought out arrangements. It’s generally a messy, high-energy class period, with bits of paper and glue being  put down with excitement and delight.

Reflection, translation, glide-reflection
Reflection, translation, glide-reflection

Last year it occurred to me to rein in some of this excitement and introduce the students to different kinds of symmetry that would enhance their designs, Things seemed to go pretty well last year, so I tried it out again this year.

Paper Punch Symmetry
Paper Punch Symmetry

While some good things happened, it was clear to me that my own  ideas about teaching something new came at a cost: students didn’t do nearly as much decorating as in previous years, and much of the excitement had been sucked out of the project.

Naturally, what I want is the excitement AND to be able to teach something new and valuable.

My experience with the sixth graders was weighing on my mind this past week when I was working with second graders on cut-paper project.

Rhombi as the ready
Rhombi as the ready

What I think had been my biggest misstep with the sixth graders was that I dampened their paper-punching enthusiasm before they ever got a chance to indulge in the novel activity I was laying out for them. I depend on the  use of unusual tools and colorful materials to engage students,  but I hadn’t  given these kids a chance to experience their excitement before laying out the conditions which seemed to challenge and dampen their spirits.

In hindsight, I think things would have gone much better if I had laid out the materials, let the students create their collections of paper punched bats, balloons, dragongflies etc. and then, only after they had gathered together these personalized treasures, I could then proceed with the references to symmetries.

Cutting out Rhombi
Cutting out Rhombi

 

Now, this week, working with second graders, I tried to learn from what happened during my time with the sixth graders.  I had an agenda, which is to use rhombuses to construct plane shapes, such as trapezoids, hexagons, and triangles. This is supposed to be an exciting activity, full of experimentation and discovery. I didn’t want to do anything suck the joy out of playing with the rainbow of colors in search of wisdom.

The students needed to cut out rhombuses from a packet of colorful  printed strips. I decided not to tell them exactly what we’d be doing with the rhombuses. Instead, after the rhombuses were cut, I encouraged to students to slide them around on their desk, and organize them into piles or shapes that appealed to them.

Rhombuses on the run
Rhombuses on the run

This exploration time took only a few minutes, and, as different students finished their cutting at different rates, it kept everyone busy.

Stars
Stars

I traveled around the room, showing some kids how to use these shapes to make all sorts of arrangements.

Hexagon, Cube, Star
Hexagon, Cube, Star

It’s worth mentioning that they were notably impressed that three rhombuses could look like a hexagon, or like a cube in perspective, with color choice playing a significant role in creating the illusion that these same shapes were different.

playing with Rhombi
playing with Rhombi

It was only after this time of playing around that we got down to the business. I tried to keep them in discovery mode by asking them to take just a few pieces in their hands (which at this point included some rhombuses that had been cut in half to form two equilateral triangles) and to try to figure out how to make a trapezoid or a triangle or a hexagon, or a scaled up rhombus.

Plane Shapes
Plane Shapes

It all worked out.

More Plane Shapes
More Plane Shapes

At no point did I feel like my agenda had sucked the air out of the room. Whew.

My note for next time is to remember to let the kids feel the excitement and let them create their own relationships with the materials before I overlay my lessons into the moment.  Whereas I had hijacked their enthusiasm before, I think that this different approach enriches their enjoyment, and hence their learning.

 

 

 

 

 

 

Art and Math · Arts in Education

100-Book for Kindergarten, Trying it Out

Rubber Bands books to use for Hundred Book
Rubber Band book for 100-Book

About a month ago Simon Gregg posted a string of images about 100-books that were assembled by his class of 5 and 6 year olds.  On each of page there was a group of items that added up to 10.  That’s 10 pages of with 10 items, so it’s a 100-book. I want to do this project with 5 and 6 years olds! I ‘ve been in kindergarten classrooms all this week, doing a literacy based book project, but I noticed that there was one 40 minute block of unscheduled time in my schedule so I asked one of the teachers if I could try out this project with her class.

cover of 100-Book
cover of 100-Book

40 minutes was just enough time to get this book started, to get a feel for it. I assembled books for the whole class, just to expedite getting to the content. Although I didn’t realize about how this project would be received, the bottom line is that the kids enjoyed it, were enthusiastic to continue working on it, and seemed to be making some new connections. I debriefed with the teacher the next day and we agreed it’s a project worth developing. She is continuing the book without me: her kids are demanding to finish!

 

five plus five equals ten
five plus five equals ten

I wanted this to be a tactile book, and one that combined objects and finger counting. I brought in lots of foam, sticky-backed items. Students put a number of items on the page, hopefully in an arrangement, then traced the number of fingers they would need to make the total of items and fingers equal ten.

Eight plus two equals 10
Eight plus two equals 10

When there were less than five fingers to trace, the finger image looked rather odd, but the kids didn’t seem to mind.

Our first page, which I don’t have a photo of, was two tracings of hands, so that was five plus five equals ten. I noticed, when looking through the books afterwards, that a few of the kids wrote  5 + 5 = 10 on every page: these kids were not connecting the items on the page to the number sentences.  When I talked to their teacher about this she said that having them assemble the items, make them equal ten, and writing the number sentence was probably too much to do so quickly.

If I get a chance to properly do this project, these are the things I’d change or keep in mind:

  • I’d say we should do groupings of items on all of the pages as being one step (rather than finishing each page completely before moving on)
  • Next step would be going back to figure out how many fingers it would take to make ten and then creating the tracing of the right number of fingers.  Even though the fingers looked funky, I really like including them in this book.
  • Last step would be to revisit each page for a third time, this time to write the number sentence that describes the page.
  • I’d slow down and make sure kids were putting their items in groupings that could be recognizable to them.
Six and four make 10
Six and four make 10

Given enough time, I think that I would have the students make each page on single sheets, then bind them together with a simple pipe-cleaner binding, kind of like this one :  https://bookzoompa.wordpress.com/2011/12/18/pipe-cleaner-bound-scrapbook

I’d probably have the kids make a big origami pocket to store all their pages in before they are bound together.

The three traced fingers
The three traced fingers

I like the idea of using items in the book that are slightly 3D, like these foam sticky-back pieces, but I’d think it would also be great to have other things, like cotton balls or popsicle sticks, that the kids glue in. I like having them work with glue.

The kids that Simon Gregg worked with included some playing cards in their books. This is something I’d like to do. My local thrift store sometimes has used sets of playing cards that I can pick up cheaply.

Hopefully I’ll be writing more about this project as I get more of a chance to do it with more kids.

Do take a look at this twitter thread that got me thinking of this project. The videos towards the end of the thread are precious.