Week 2 Task 1: Tree fractals

We looked at this kind of fractal as part of a maths circle session on doubling and halving a little while ago. I started by reading the myth of Hercules and the Hydra heads. Even though my circle is all girls, they were immediately excited by the chopping off of heads. They were all competing with each other to calculate how many heads would be on the next iteration. Later on in the session we drew square based fractals on squared paper and triangle based fractals on isometric paper. Using grid paper encouraged a greater level of accuracy, with the necessary calculations, but became a bit overwhelming for the younger or less able children - although they could quite happily devise colouring schemes for other people's fractals.

We finished by watching Vi Hart's binary tree video and were really inspired by the crashing Hydra heads doodle. When faced with a boring half hour (like waiting for me to have my acupuncture session) this is still my daughter's favourite activity. She's had numerous goes at it and has improved in accuracy and consistency each time. Really looking forward to trying some more fractal activities with her.

My 12 year old daughter and I did a few different tree fractals, drawing shapes on paper. What was really neat about it is how easy that made thinking about powers of any base. When we used a five pointed star it easily led into a discussion of repeated multiplication and then powers/exponents of a single base, 5. Such an elegant introduction! We also discussed the idea of infinity and whether we could add up all the areas of our stars.

echer.jpgMy son (9) thought the fractal hand video was a little spooky also. However it got him to think of infinity. We started with the definition of fractal then the Sierpinski Triangle. We did a chalk drawing of the triangle that then led to infinity. We did simple addition and I explained the power of multiplication. We tried a carrot drawing and I explained the importance of single points that branch out to make fractals. I explained it by drawing stars and looking at our tree in the front yard.

We then looked at some of the art by M.C. Esher.

My kids (6,4) also thought of the lyrics to "Let it go" when I told them about fractals. When I showed them how snowflakes were fractals they became so excited that they now understood what Elsa was singing about! They were not that interested in drawing them on paper, but were very interested in the apps and watching "Fractals-Hunting the Hidden Dimension" documentary (only bits and pieces of it because it's targeted more toward adults) showing fractals in nature and how they are being used in animated films, technology and medicine. They were much more enthusiastic about drawing them with chalk on our driveway. They also have been enjoying looking for fractals in nature in our yard/park or even while driving around.

One request: While you have provided great ideas and instructions for the activities, I was wondering if, along with the assignment you could provide a short lesson/paragraph/script or even a REALLY, REALLY SIMPLE way of explaining what fractals/scale/sequences etc. are to younger children. I've been trying to explain it to them using my own words and videos/pictures, but since I don't have a strong background in math, I'm not so confident that I'm explaining the ideas clearly (with words they understand) and accurately.

We started with a scanned picture of one of her favourite Octonauts characters, and made this into a simple fractal, although this was beyond the capabilities of a 5 year old. Then she had a go at the geom-e-tree app on the ipad, which allowed her to play with fractals interactively. I pointed out how the numbers grew at each point, which didn't seem to generate much interest - she seemed most interested in collapsing the fractal structure down into a single branch, then unfolding it, or changing the pictures that the fractal was made from. I didn't push the explanations with her - I thought it better to just let her explore and absorb it visually, which will hopefully help lay the foundations for developing an intuitive feel for multiplication.

I drew a fractal of the Easter bunny - in both directions - tree fractal style outward and substitution, using the eyes as the spot for repeating pattern. I didn't say anything to my children (ages 5 and 7), but just left the drawings mixed in among the other books they were reading. Boy did it stir up excitement! They were laughing, trying to figure it out, shouting "what is this? Where did it come from?" , counting how many faces they saw. I just casually said "oh, they are fractals". Then they started seeing the patterns and relationships and discussing among themselves. When my husband came home from work they grabbed the papers running to him "daddy we have fractals!". I can't believe the excitement this generated and we haven't even begun to discuss them.

I thought a fractal out of Cuisenare rods would be good for a start, but my timing was bad - too late in the day, so nobody except of me was impressed :) Next day my elder son was drawing chalk fractals - I just threw the idea in, he did the rest:

At some point he stopped drawing and started asking himself how many branches he would have at next level. A rare occasion of him asking a math question, so I shut up and listened.

The younger one (5yo) didn't have much interest in what I was showing, so I am yet to find the object he will be interested in multiplying in this manner.

I started out by drawing fractal triangles. My daughter wasn't really interested because she "has done this before." What sparked a little interest was showing her the other kids drawings. She then drew fractal bunnies. She became more interested when she realized the fractal patterns in nature. We then looked at Interactive Math virtual manipulative generate explore fractal tree www.visnos.com and a Mandelbrot video this really caught her attention with her exclaiming "now I am really interested!" We watched both the Deep Mandelbrot Zoom and The Mandelbrot Set the only video you need to watch. She was fascinated and is now outside searching for examples of fractals.

Wow I feel really inspired by all the comments on here! I want to just carry on making fractals with my kids all week... We have lots of forks at the moment for some reason so I might give that one a try. Although thinking about it not sure we have 16+ so maybe not!

I was working with my children, ages 8, 6, 6 and 21 months. We started off with me demonstrating a fractal triangle, then a bunny head. Then the girls had a go at drawing their own. We had a person, a dog with floppy ears, a flower, and circles. I asked how many points a circle had, and they said zero (or infinitely many? I wonder now) but my daughter just decided to add two, then four on another example, because she said that would fit.

I found they were less interested in making perfect fractals with each layer complete than in seeing how many layers down they could get.

We talked about how we could keep drawing for infinitely long, ie fractals can continue infinitely. We also talked about how the numbers of each unit increase at each level, and the size decreases. We did have some trouble with 'mushing', as someone else so eloquently put it!

We looked at the hand video and at Vi Hart's binary trees:We liked the reference in the video to the hydra's heads as one daughter has a Heracles obsession - we talked about how the heads (which grow two in place of each one that is severed) would grow like a fractal tree.

We had a bit of a go with the ipad app, my oldest was interested but not enough to look for more than a little while. My 21 month old however loved it, although not sure how much that was general ipad love as he is not usually allowed on it!

My oldest then found a fractal weed of some kind outside in the garden.

Thanks for the suggestion of allowing kids to colour in our designs, by the way, all my girls enjoyed that:

All in all a great activity, although they did keep saying fraction instead of fractal which could get confusing...!

PS I hope my images come through ok as I have not posted any before! So apologies if not.

My 9 year olds are at a sleepover, so I tried this out with my 14 year old.

I kept it simple and went with a triangle and then a square. The first level was one triangle, the second level was three triangles, and after that it got more interesting: the third level was six triangles, the fourth level was twelve triangles, the fifth (not drawn) would have been 24, etc.

I asked my daughter to describe an equation for the levels using 3 and 1. For 6 (level three) she said "6=3x(1+1)". I proposed an alternate: "6=3x(3-1)". She wanted to know why it made a difference. I tried to guide her to it with 12 (level four) with the same rule to use 3 and 1, and her first answer was "12=3x(3+1)". Again, I proposed "12=3x(3-1)(3-1)", but she still wasn't buying it.

I drew a square fractal: level one was one square, level two was four squares, and again, things got interesting after that. Level three was 12 squares, level four would have been 36 squares, level five 108 squares and so on.

She immediately recognized that 12=4(4-1), and also ended up arriving at 36=4(4-1)(4-1). When I asked her why, she said because the equation was simpler than 4(1+1+1)...and then she couldn't wait to leave.

The equation for a "regular" n-sided polygon-fractal after level 2 seems to be n(n-1)^(level-2). The rate of change or f'(x) would be n-1 for those levels, but n between level 1 and 2. Would love to know how to express that as a calculus equation.

I worked with my 5yo on drawing fractals and that was good:

But what was really awesome was seeing his reaction to a youtube video that continually zooms in on the Mandelbrot set:

https://www.youtube.com/watch?v=ohzJV980PIQ

Totally blew his mind. (Mine too for that matter). After that, we spent ten minutes watching a documentary with Benoit Mandelbrot talking about it (before his attention span wavered) :)

https://www.youtube.com/watch?v=s65DSz78jW4

I'm not sure the multiplication part is coming through (although we talked about that with the triangle fractal), but wow this stuff is good ammo for making math cool and exciting!

I love this assignment! I've always been interested in fractals, but not too sure how they connect to multiplication. I tried drawing a cat fractal with my 5 yr old, but he wasn't too interested. I think the drawing intimidated him. I'm going to try again today.

My 6-year-old had fun creating the"frozen fractals" from the song "Let it Go!" She is also excited with the thought that our body has fractals!

I showed my 7yo daughter fractals on paper using triangles, flowers, stick figures, stars. When it came time for her to do it herself, she took the easy way out and did it with lines. She said, "It's easier this way." However, she did get into the artistic component of the assignment by using colors of different hues and values, starting with dark valued colors for the base and getting lighter as she moved to the edges. She commented on how fast it took to run out of room and have the branches run into each other.

We talked about exponents, exponential growth, sequences and series. We connected it to a project we did last year on infinite series reaching 1 (half of half of half of half...), and determined that tree fractals are infinite series reaching infinity. (Is that right? I don't know enough calculus to wrap my brain around that.)

I wanted to experiment making tree fractals with Montessori beads. I had difficulties with "crashing" and then materials ran out.

------------------------------------------------------------------------------------------

The way leaves (or cones) can represent an entire tree is actually appreciating fractals. Yes?

Unfinished artwork (below) frames in an area where I keep a nature plate. Children who visit are allowed to explore. I listen for comments like, "Look the leaf is a tree." I don't go into lecture mode, I may simply say, "Oh, you noticed that!" I like creating environments where children can learn or realize things on their own. I believe that they are more inclined to "own" it, more so than when an adult tries to "pour" information into their heads.

We made a "forktal" (well, I made it while my daughter watched - she's not much help at 6 months old). We counted the tines on one fork (4), made a big fork out of four forks (with a knife as a handle), and counted the total tines (16), and noted 4*4=16. Then we made an enormous forks out of big forks, and calculated the tines at 4*4*4=64. Next we made a ginormous fork out of enormous forks (we had to break out the plastic forks partway though construction), getting 4*4*4*4=256 tines. I asked her what would happen if we had another 3 tables with similar setups, and we calculated 4*4*4*4*4=1024 tines. Perhaps more interestingly, I noted that our table had 4 iterations, and a normal fork has 4 tines. What if we did the number of iterations in a ginormous fork? $$4^{256}=1.53\times 10^{154}$$ would be a lot of tines! Actually, even if all the mass in the universe was converted to forks (even really small ones!), we wouldn't have enough to build this rather large fork. Imagine that instead of 4 iterations or even 245 iterations, we used the number of tines in this new rather large fork... How many times should we keep upping the number of iterations before we have an unimaginably large number?

We tried just drawing tree fractals with my 2.5 yr old and 4.5 yr old. The favorite was an ice cream sundae fractal where each cherry had a new sundae put on top of it. We also tried cat, car, and truck. The kids seemed to understand that each layer had smaller objects, and more of them. They didn't seem to get the multiplication aspect of it, but overall, they said "it's pretty good".

My 7 year old and I played with fractals a few days ago and Today we drew one with chalk outside and then she played with the app for a good 20 minutes. I think she understands the idea, we will keep playing with it this week. I love the idea of making one of beads. We might try that!

My son enjoyed playing with http://www.visnos.com/demos/fractal. Right away he noticed it was going by multiples of 2. He started doing them 2, 4, 16, 32, 64 (which is minecraft block limit), 128, 256, etc. He pointed out its the same language used in processing and binary code. He also played with writing it out as exponents. We then used today's activity as a challenge to make a fractal in both Scratch & Minecraft. We found someone else's Scratch program http://scratch.mit.edu/projects/1360458/ for making fractals but we also made our own. http://www.fractal-explorer.com/minecraftbasics.html gave us a start for fractals in minecraft. Our son chose to make one with redstone lamps & wool. Fractals are fun according to my son.

We started out with the spooky hands, followed by an easy hook for the ids: YouTube videos (sadly, kids, there is no hits for 'Star Wars Fractals'). There are some nifty computer graphic rendered fractal videos to be found - many are quite hypnotic. While watching we came up with words for what we thought of these fractals - repeating, growing, patterns, infinity, Universe, swirling.... Then we took it to pencil and paper and sketched some models (stars, lightning bolts, squares)...getting simpler and trying to find how to have the fractal fill more space and less space, and overlap itself more or less. I sent my fidgety 5yo to the fridge to hunt for a fractal (cauliflower), which he then observed under a magnifying glass. Simplifying to a tree fractal, we drew models of base 2, base 3, base 10 (looks like a dandelion, they said) to explore the difference in how quickly the numbers grow. I introduced my 9yo to the idea of base 2 (binary) for computer language and base 10 is what we know as the decimal system. His mind was blown when I noted some Ancient systems used base 20 or even base 60 - neither of us could quite wrap our minds around the scale of either of these. When enthusiasm for the pencil/paper activity waned, I got out the iPad with Geom-e-Twee and they each explored the binary and ternary tree fractal modeling in that mode for another while. My 5yo was most interested in stretching and scaling. My 9yo was interested in the effect of swirling finger movements to make a kaleidoscope effect, and had fun identifying all the shapes that came together - squares, 3Dish boxes, snowflakes, conifers, hearts and many more. By then I needed a break. What high paced exploration! I offered them a challenge: head to the playroom and see if you can build some sort of fractal idea from lego bricks. Curious what they will come up with.