November 23, 2011
After posting about Hexagaon-Flexagons on November 10 and November 16, I started working on making a template for teaching this tricky paper invention. Even though this structure is well covered on the net, what I want to add to the mix is something to make the folding easier.
When I’ve taught this structure the part that people have the hardest time with is creating precise folds. I made the instruction sheet above because it provides a way to create score lines so that the folding is easier.
Scoring is PRESSING lines onto paper, so to help facilitate folding. If a score line is firmly pressed into paper, it will fold easily on the score line. A pen that has no ink in it, or a paper clip, both make good scoring tools, as they will make a thin line. Bookbinders use bonefolders for scoring, but for this template I prefer a paper clip because it makes a thinner score line.
Here’s a picture of my daughter’s hand scoring my template. The template is on a firm but not hard surface: the surface ideally should have a bit of ‘give” so that it can be pressed into. (A stack of newspapers or a catalogue works well.) Place the ruler on the indicated line, holding it securing in place, then run the edge of the paper clip along the edge of the ruler. Press firmly, but not too hard, as you don’t want to rip through the paper. If you look closely at the image above you can see the the score lines that have already been pressed into the paper.
That’s about all I have to say for now. Hopefully the instruction sheet above clear enough to follow. If you want to print out template without the instructions, this is the a link to my non-annotated hexagon-flexagon .
And for some more inspiration, here’s a hexagon-flexagon decorated by Michele Gannon:
Now, here it is again, flexing….
….to next reveal the design that was formerly tucked inside:
addendum: Here are a couple of links to another person’s take on the hexagon-flexagon: http://plbrown.blogspot.com/2011/03/amazing-trihexaflexagon.html and http://plbrown.blogspot.com/2011/12/art-math-magic-its-trihexaflexagon.html
Addendum 3/12/2017: just came across another great flexagon resource! http://www.puzzles.com/hexaflexagon/activities.html
November 16, 2011
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.
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.
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:
Not bad. I like this way of working with the hexagon-flexagon.
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).
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)
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.
Math and art together: always a great idea.
June 8, 2011
Last month I posted a group of photographs documenting flexagons made by fifth grade students. These structures are best seen (and held) to be understood, but, basically, they are a paper toy whose folds, when articulated, reveal hidden surfaces. Got that ? If not, check out Forest of Flexagons. To make a larger copy of the directions shown above here’s the PDF for Flexagon Squared
There are other shapes for flexagon. I also make ones that are hexagon-flexagons, which I like like because its structure can be decorated to mimic fractals. But that’s another story.