Scaling and Transforming for Design with 7 year-olds

Cut-paper Scalin
Cut-paper Scaling

I’ve been reading on-line conversations lately about scaling and dilation (which are two terms for the same thing, I think). Repeatedly I’ve seeing questions from various people on scaling activities. This confused me at first, not the concept itself, but why did there seem to be such an interest in methods of teaching dilation? Turns out that this idea of proportional change is a key concept to grasp in math, and the nuances of it are tricky to teach. After thinking about it I thought about how artists and designers use scaling all the time, and I mean all the time. So, this week, when I was working with first grade students, I decided that I would present the decorative part of their project as an a math lesson, as an exercise in scaling.

The transformations of two 1.5
The transformations of two 1.5″ x 3″ strips of paper

Each student picked six strips of paper, which were 1.5″ x 3″ each. I told them that these strips are the shape of two squares on top of each other, and I knew that because they are twice as high as they are wide. I don’t think they understood that part, but that was fine. I just wanted them to hear the language, as I think that it conveys a sense of importance. I then showed them how if they lined up the papers so that they made an “L” they could see a square, which they could then cut, as in the photo above. From there we talked about how they could use the square for decoration. Since a plain square is,well, not too exciting I tilted the square for a more interesting look. Next I showed them what happened when the square was cut in half, corner to corner -one cut, two triangles! We then looked at what happened when the triangle was cut in half again, and then again.

Scaling down squares
Scaling down squares

I also showed the students that to make a scaled down square that they had to cut the original square twice, once in both directions. There is something about making “baby squares” that particularly captured their interest.

Scaling down triangles
Scaling down triangles

Baby triangles were a big hit too.

Scaling handsThis is an activity that I’ve taught many times bu it seems to me that, judging from the results, this was the most successful day I’ve had of teaching this kind of embellishment.

Sacling 2

Their project, by the way, was making a book for their animal reports.

Scaling with elephantStudents have done their research, now they’ve made their books, next they will be adding writing to the books. This elephant will eventually be flanked by facts about his habitat and other interesting facts.

Scaling with birdThese students will be able to continue with these embellishments as they complete the rest of the books, I won’t be there for that part, but I think they’ve got the idea and these books well become even more wonderful.

Flip Book

Flip-Book by Committee?

Flip books in progress, holding things together with binder's clips
Flip books in progress, holding things together with binder’s clips

This first set of flip-books, showing some transformations of the graph of a line, have been keeping me occupied. I have a small circle of math-friends to send these to, for comments, but I would like to expand this circle. It has taken me by surprise how many revisions this book has gone through already.



Flip Book: Cutting the pages down to size
Cutting the pages down to size

There’s been revisions in sequence, color, line weight and paper weight. It feels like this tweaking can go on endlessly. First, though, I would like to get this in the hands of some more teachers who are willing to put a copy of this flip book into student’s hands, and then tell me how it goes. I am aiming at something like a template to use as base for his book as well other books like it.

Flip=Book cover pieces
Flip-Book cover pieces

Is there anyone that I can interest in playing with me and my books? If you teach the equation of a line, and would like to have one of these books in exchange for giving me some feedback and suggestions, please let me know. The contents of what’s in the book is similar to the GIF in my last post. As I said there, the GIF can show the concept, but it seems to me that  it ‘s more valuable if students hold the book in their hands, as this then allows them to slow the animation down, so that they can work out for themselves the secret of what’s going on between the equation and the picture.

Three Flip-Books and gold envelopes
Three Flip-Books sliding into gold envelopes

So who wants to play? Leave a comment below, or send an email to me in the address that’s listed under my About tab. Later this week I am hoping that three teachers will receive a little book in a gold envelope.

Art and Math · Math and Book Arts

The Flux Capacity of an Artful Number Line

Envelope Number Line Book, Paula Beardell Krieg
Number Line Book, closed

I like the number line.

The number line is all about relationships: I can look at the number line and actually see and measure the chasm between two quantities, even when, as in the case of negatives, those quantities don’t even exist.

As an adult I’ve realized that I had some misconceptions about the number line, and I have discovered subtleties about it that surprise me.

envelope Number Line Book, Paula Beardell Krieg
Unpacking the Number LIne

I’ve been toying with number lines for quite a while. In my opinion the number line needs to be toyed with. The images that I see of it are not captivating. I’m wanting to rigorously play with this arrangement of symbols in way that captures some of its nuances. I intend to try to investigate numerous bookish solutions which means that I suspect that this topic will keep coming up.  I hope this will be an ongoing bookmaking/discovery journey. I’m not sure exactly where I will be going with this.

But I do know that a few nights ago , after a disappointing evening of cutting and folding, a way of proceeding finally presented itself, but I was too tired to grab hold of it. The inspiration teased me all night, and before 7 am the next  morning I was tending the coffee pot while working out my construction. I’m very pleased with how this particular structure worked out. It was so unexpected and delightful that I am excited to be sharing it.

Number Line Book, expanded, Paula Beardell Krieg
Number Line Book, expanded

It’s built from envelopes, the kind we think of as regular envelopes, though, technically, they are called “No. 6 3/4” envelopes.

Number Line,with negative numbers, Paula Beardell Krieg
Number Line, showing negative numbers too

Here’s are some of the things I like about this piece:

  • it’s a zig-zag
  • it has pockets
  • it scales
  • the structure suggests infinity since it can keep going in either direction
  • it can fold up into a polite accordion-like book.

The pockets are the most distinguishing feature of this number line. These pockets hold cards, which are printed with different sets, or sequences of numbers. This means that the labeling, or the scaling, of the line is always in flux, subject to the whims of whichever algorithm that’s called for.

Number LIne by 10's
Number LIne by 10’s

That’s the crux of it: the flux.

As students proceed through their grasp of numbers, the labeling of the number line constantly changes in scale as needed. Eventually the number gets integrated into the coordinate plane, and becomes the x-axis. I remember seeing the little graphs in math books, and I thought that when I got to grown-up math that the lines would get longer. It never occurred to me that it would be the scale that changed, not the size of the line.

Number LIne in Flux
Number LIne in Flux

You can see that there’s intermediate markings between the numbers. These can be interpreted differently depending on which scale is being used. For instance, when counting by tens, the small lines can be counted as ones, when the number line is increasing by one’s, the intermediate lines become tenths. In my mind, the point of doing this is to drive home the concept that the very same line can morph into whatever one needs it to be for the visuals of the relationship at hand. The maker becomes the master of the line.

Folded Number Line
Folded Number Line

Then the maker gets to fold up the number line into this accordion-like square. Just my style.

Over the next few days I will be working on designing a set of instructions on how to put this line together. It’s likely, however, that if you picked up some envelopes you ‘d  figure this out for yourself.

 Addendum Here’s the link to the tutorial: https://bookzoompa.wordpress.com/2014/11/03/the-envelope-number-line-tutorial/

Some more number line posts:







Flexagons · math · Paper Toy

Hexagon-Flexagons: Post 2, Fractions

Except for the last picture, all of the images in this post exist on one hexagon-flexagon. In my last post I showed hexagon-flexagons that were about making designs with pencil, paper, and gouache. When I was making the design for this post I was thinking more about math.

Hesxagon-Flexagon divided into three rhombuses

Okay, now here I go,showcasing my hexagon-flexagon as way to illustrate fractions, rather than to be a kaleidoscopic toy.
The hexagon-flexagon above has been illustrated to be understood as thirds,showing that three thirds create a whole.
Now, this concept might be better understood if the parts were actually labeled.

Flexing the structure to reveal another design

I worried that labeling the parts could cause a problem, because, after flexing the structure, the design changes. I did a mock up to see what would happen, and this is what I got:

Fractioned parts, labeled

Not bad. I like this way of working with the hexagon-flexagon.

Hexagon-Flexagon, two trapezoid showing dividing the structure in half

One thing is really clear to me, and that is that I would love to work with a graphic designer who would add words and numbers that looks snazzier than my handwriting. Oh, and another thing that is clear is that labeling the hexagon as two halves doesn’t work well after it’s been flexed. (I’m not providing a picture here of how the broken up halves looks).

Six equilateral triangles dividing the hexagon-flexagon into sixths

This side of the hexagon-flexagon shows that six parts equal a whole. Ideally, I would label each of the blue triangles with their own “1/6′ fraction, then write equations all around the outside edges (such as 1/6 + 1/6 = 2/6 =1/3)

Flexed version of the previous image

I think that this is going to be a long-term project, perfecting these images with better text graphics. I like how the geometric patterns work out here. It’s just the labeling that I can’t get right.

So, everyone has heard students question why they have to learn math, particularly algebra and trig, as they don’t foresee ever using it. I have at least one good answer to that age-old question. The reason to master math is that it keeps a person’s options open. I recently spoke to the someone who was helping her 30ish-year single parent daughter with Algebra because it is required for a nursing degree. Today I spoke a woman who has gone back to school to get a degree is Public Health. She is struggling with her required economics course. My son needs a to complete two semesters of Calculus towards his Biology degree, which will qualify him to go on to Chiropractic studies. So, there you have it: learning math a keeps options open for the future.

Now here’s another way of decorating a hexagon-flexagon, and hey, it’s even seasonally correct, as it can easily pass for a six-sided snowflake.

Hexagon-Flexagon by Michele Gannon

Math and art together: always a great idea.