Today your mission is...
Make your own fractal to admire one of the most common multiplication models encountered in nature, and the incredible exponential growth.
Ready, Set, Go
Sketch an object or shape that you or your children like. Let’s call this shape the base. Mark the points that stick out, such as the tips of cats’ ears (two), vertices of a triangle (three), or ends of a star (five). Draw smaller versions of the base at each of the marked points.
Mark the same points on each of the smaller versions, and draw even smaller versions of the base at each of these points. Repeat the process as many times as you want. You justmade several levels of a fractal! Fractals of this type are called tree fractals.
Respond to today's task
How to help your child to get started
Get some paper and colored pencils and find a spot where your child can observe you drawing. Talk about what you are doing: “I drew my favorite flower, a daisy. Now I am marking little dots at the top of each petal of the daisy. And now I am drawing smaller daisies growing out of each of these dots.” Invite your child to draw this way too.
Younger children might want to start with something really complicated. We had a four-year old who insisted on drawing an excavator. Go for it! Draw it, but talk about how many “points that stick out” you will need to mark.
Your child might not want to draw, but instead might prefer to observe you do it. Enlist your child’s help with other tasks - choosing colors for each new level, marking points that stick out, and making sure that you do not miss objects.
Toddlers
Small number of branchings (two or three) are easier. Help your toddler remember to draw every part of the picture. Use colored dots, removable stickers or even raisins to mark the places where the next level of pictures goes, or invite your child to do it. Math and art variety keeps kids engaged and invites their own experimentation. For each following level in your tree fractal, try to change the color, scale the shape to be bigger or smaller, rotate it, or reflect it upside down.
Young kids
Offer your child to use software, such as http://www.geom-e-tree.com/ (iOS, has a free version) or http://www.visnos.com/demos/fractal (computer browser). You can use the software to play, or to plan large-scale artistic projects.
Older kids
Play with predictions and estimations. Which tree is easier to draw, with two or with three branchings? How many pictures will we need to draw at the next level? What level has 8 pictures? When will the tree “branches” overlap? What happens to the shape of the tree if we scale pictures up from level to level?
Shift questions towards actions instead of words. For example, if you are using stickers, leaves or building blocks to make a fractal, ask your child to prepare enough objects for the next level. Another idea is to ask your child to point to the level where a certain number of objects would fit.
How is this multiplication?
Everyone does it! Ancient Babylonians did it in base 60. Ancient Mayans did it in base 20. We do it in base 10, unless you are a computer programmer, who does it in base 2. Our number system groups quantities by powers (repeated multiplication by 10s), like levels in the tree fractal with ten branchings. This repeated, recursive multiplication is an incredibly powerful (pun!) idea with profound effects on technology and history, from Egyptian pyramids to modern computers.
But our modern number system has a major drawback: it is very abstract. It’s been developed by adults, for adults. Fractals to the rescue! Making a fractal gives us an opportunity to touch and feel the abstraction, to feel every aspect of modern number systems - the base, the recursion of multiplication, and the sequential arrangement of powers.
Inspired by calculus
Fractals give kids a practical, hands-on recipe: how to make an infinity. The infinity kids make with tree fractals is easy to imagine and to understand, because it’s easy to make and to see. But this easy infinity comes with a more complex structure than, for example, just stairs that go on and on and on. It has built-in ideas of exponential growth, scale, and orders of magnitude. The stair (linear) structure is artificial, but fractal, recursive, nonlinear structures are everywhere in nature.
Algebra = patterns of arithmetic; calculus = patterns of algebra. Let’s look at the example of a doubling fractal tree, called binary tree.
How many pictures are at the next level of the binary tree, if this level has 4?
2*4=8
Algebra
What function gets you to the next level of the binary tree?
f(x)=2*x
What is the speed of growth of that function?
f’(x)=2
When you draw tree fractals, you mostly act at the calculus level, because your main decision is how to branch the tree.
Frequently Asked Question
Ok, so now my child can make tree fractals. But how does it help my child get better at actually multiplying anything?
There are two direct benefits of fractals for calculations. First, they give kids the hands-on, embodied access to the structure of our number system, as we explain above. The second benefit is more subtle: fractals give a big boost to children’s ANS, Approximate Number System, which is one of the cornerstones of successful calculations. Such visual, well-organized patterns help kids to picture the quantities (say, at each level of the fractal), which helps the skill of estimation. Here is a recent study about ANS, explaining why 5-year-olds can (and should) work with algebraic patterns. Like fractals!
Then there are the soft skills of math. Building even the simplest tree fractal is challenging for young children, because they have to be able to keep a pattern going, and because there is a lot of work. The mistakes are easy to notice, though. This way, kids develop the mathematical values of precision, rigor, and perseverance.
Words
Fractal, scale, power, exponent, binary, recursion
Scavenger hunt
Start with the art above, and talk with your children about trees as the lungs of the Earth. Trees and lungs and corals have the same branching structure of tree fractals, and for the same reason! They are maximizing the surface area within a given volume for super-efficient gas exchange.
Watch this slightly spooky video of a fractal hand:
Can you find other examples of fractals in nature, architecture, technology, crafts and art?
Course links
Answer by cleabz · Apr 14, 2014 at 05:14 PM
My daughter's cats and my 6-sided star. I started with something more complex and then realized I needed software to do it. Then I did a 5-pointed star and my daughter said, "That would make a neat snowflake!" So I switched to a six.
The arithmetic makes sense to me but I can't remember the algebra or calculus enough to wrap my brain around it. But I definitely see the limitations of the arithmetic, and where I would have to go into algebra to figure out how many stars there are.
Doing this activity reminded me of when I was about 4 in Montessori school and there was a picture of an image inside an image, and how it really tickled my ideas of infinity. Wondering what my kids will remember of this activity, and if it tickles them in the same way.
Answer by cedamala · Apr 14, 2014 at 06:08 PM
This is a wonderful topic. We noticed that fractals are mentioned in the movie Frozen when Elsa is creating her ice castle. It is mentioned in the lyrics of the song and visually in the creation of the castle. We just watched it again on YouTube to notice the fractals.
Answer by James · Apr 15, 2014 at 05:09 AM
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?
What a story! I like the part about the mass of the universe. The Forktal reminded me of this:
Answer by yileinei · Apr 14, 2014 at 08:22 PM
I did this today with my 4yo daughter and she wasn't very interested at the beginning, she just wanted to paint other things. After a few attempts she got engaged and she understood the point.
This is the first attempt:
This is when she wanted to draw the rest of the picture:
And this is when she understood what we were doing:
I found that just shapes wasn't working for her, with the cats she understood it better.
She didn't think about maths at all and she really didn't ask anything, but at the end she showed she knew how to do it, so I guess it worked.
I can see how this help us think on infinity concept and on regular and exponential growth. We can find this in nature and typical objects are romanesco broccoli, or the seeds in a sunflower.
Answer by champalto · Apr 15, 2014 at 03:07 PM
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 have not seen this technique before. So dramatic! I bet it would look gorgeous on glass, maybe a window.
What a great idea! I'm going to let her do it on a window. It's Crayola Model Magic - that would work easily. And the icicles would match some of the ones we have outside! :)
Answer by mrs123 · Apr 18, 2014 at 03:04 AM
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.
Answer by LeistCatalano · Apr 14, 2014 at 07:38 PM
We just had a few minutes to work on this topic before my boys headed off to school, but I could tell it's something they find fascinating and so do I. I had to really push my oldest to go to school and wait to work more with the website later. (http://www.visnos.com/demos/fractal). Here is one of my creations.
This is also a great activity for hand-drawing. I play to use it often.
Answer by sherylmorris · Apr 14, 2014 at 09:18 PM
I wanted to experiment making tree fractals with Montessori beads. I had difficulties with "crashing" and then materials ran out.
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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.
I'm hanging my tree fractal this time. When I work with these, I'm impressed with how quickly multiples increase. ty
Answer by KaleSprouts · Apr 14, 2014 at 11:24 PM
I did several activities with my 5 yo twins today. We drew fractal trees but it was hard for them to decide how long the lines should be so that they would have plenty of space to keep going, which was frustrating a bit.
We tried the geom-e-tree app but there was too much that I needed to read/understand/do before anything exciting happened, so we aborted before actually making anything. We'll play with that again after I've had a chance to look at it a few times. While on the iPad we looked at YouTube fractal videos a bit and we each voted whether we thought each one was pretty and why. They tended to like the images that resembled items in nature, or items where there was a clear and easy to recognize repetition. The kids also recognized fractals that used the tree structure and found that exciting.
We also got out some blocks and modeled the numbers at each level of a tree fractal that I drew and talked about how quickly the numbers could grow (so we saw exponential growth). Then we took off for the park and saw a tree that followed the tree fractal pattern:
And also a flower where there were 2 white petals to each purple petal:
Answer by Hascoorats · Apr 14, 2014 at 11:25 PM
Our group was just my kiddos today with Dad... Though Dad was VERY interested in the drawing, fractal app and looking for fractals, the kids were less than enthusiastic :( I think it had to do with the time if the day (4pm) & low blood sugar -obviously it couldn't be their presenter ;) They did take notice how quickly they ran out of room in their drawings and how the exponents start "mushing" into each other. They did do some wondering if trees and snowflakes were fractals because the "trees" they built in the Geometwee App reminded them of these things. I'll have our math circle in the morning next time with 3 of their friends and I'm certain they will be more open to new ideas :) A great start over all!
Answer by Caroline_Prochazka · Apr 15, 2014 at 02:54 PM
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.
Caroline, thank you for sharing your process and how everyone felt - it will help other families and mixed-age groups to organize their activities!
Base 60 sounds exotic, but we still use it a lot - directly from the ancient Babylonians to our modern clocks!
Answer by rachelsnowden · Apr 21, 2014 at 01:43 PM
We live in Nepal, so we decided to use the Nepali flag to make a tree fractal. My son was really proud of it and was really excited to show it to people. I actually printed the flags for the base and first two iterations, and we cut them out and glued them, but then my son actually begged to do another level, so we worked together to draw those in.
Answer by MerrilySpinning · Apr 22, 2014 at 10:04 PM
Yesterday gave each of my children (and my husband, too, since he was home!) a large sheet of paper and had them watch while I drew a triangle tree fractal. They've seen me do something like this before, so they thought they knew what I was up to. But that time, each iteration used the same size triangle and rapidly made the pattern larger, so they were surprised when I made each iteration smaller and the size of of the overall pattern increased slowly. As they were working, I asked whether they would ever have to stop drawing the pattern. After some talk, they concluded that as long as they didn't get tired of it and had a microscope powerful enough and a pencil small enough they could pretty much go on forever. My youngest son had chosen a four-sided pattern and as he was drawing his third level, I asked him how many he would wind up drawing. He said "Sixteen," and when I asked him how he knew that, he said that he would be doing 4 twice, and then doing it again -- 4 x 2 is 8, and 4 x 2 again is another 8, and two 8s are 16. My older daughter pointed out that it was easier to say 4 x 4 is 16, so it gave us an opportunity to discuss how there might be more than one way to get to the correct answer and whether one method was better than another.
Answer by Eogruen · Apr 15, 2014 at 02:01 AM
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!
Today we made fractal trees with beads and pipe cleaners. I made mine two branches up to 32. My daughters pooped out on three branch trees at nine branches, but then my 7 year old made a tree of pipe cleaner dolls!
Answer by perbui · Apr 15, 2014 at 06:02 AM
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.)
Yes, this series adds up to infinity - the term for that is divergent series.
What a beautiful minimalist design, and color-coding! A lot of the times, kids choose simple design when they are focusing on the abstract, pure math aspects of the situation. Little engineers, artists, mathematicians, and philosophers approach the same task differently!
The number of lines diverges, but not necessarily the length of the lines, depending on how much shorter you make the lines at each iteration. If the lines in each iteration were a fourth as long as the previous, the total length would be 1+2/4+4/16+8/64+...=1+1/2+1/4+1/8+... which might look familiar. So you could have an infinite number of lines but only need one crayon, one piece of paper, a whole lot of time on your hands, and the ability to relax the laws of physics that only allow things to get so small.
If successive iterations only got smaller by half, you'd get 1+1+1+1+..., which would go up to infinity. If you were to try to draw all the iterations, I think you'd get lines overlapping, and the wax of the crayons would get thicker and thicker — you'd eventually need a ladder to draw the next iterations, then you'd have to take the roof off your house, then you'd need a helicopter, then you'd need a spaceship, then you'd probably get bored of drawing tiny iterations and go exploring with your spaceship instead.
Answer by Lamhita · Apr 15, 2014 at 12:36 PM
we use this mathematic sticks, unfornately we don't have enough mumbers for each colours to do a level all with one colour
this is a draw from my 8ys old, she did it without my help watching the cat of @cleabz
Answer by nikkilinn · Apr 15, 2014 at 01:17 PM
We talked about fractals, using the song "Let it Go" from Frozen as a jumping off point. Then, I drew a fractal with flowers, and another with triangles. The kids (5 & 3) drew there own unrelated pictures beside me. Later, though, looking through their drawings, I found my 5 year old had drawn her own triangle fractal.
It takes younger kids a few times to relate to what we do. But looking at their pictures, sometimes I find bridges I missed during the activities. For example, I've seen kids borrow the shapes I used - triangles, spirals, or trees - and using them in new ways. You can do a pretty detailed analysis of how your work inspires children's drawing if you look at the elements of art such as shape and line, or principles of art such as composition.
I looked up that Frozen song - interesting! It uses fractal as a metaphor to describe a character trait. Wow!
Here I stand And here I'll stay Let the storm rage on My power flurries through the air into the ground My soul is spiraling in frozen fractals all around And one thought crystallizes like an icy blast I’m never going back, The past is in the past
Fractals are used a lot in animation, for example, to build a beautiful forest or a mountain range.
Answer by monika · Apr 15, 2014 at 10:14 PM
We had a lot of fun with this. My daughter colored each "level" of circles and then we counted them together so we could see how quickly the number of circles grew. Once they were too small to count we saw a pattern in the formula so we continued to see how high they would go and my daughter wanted to know how many levels it would take to get over a million.
we worked on this again yesterday and my daughter asked me: "shouldn't there be a larger one too? doesn't infinity always go bigger and smaller?" I told her I didn't think so because I didn't see where a larger shape would fit into the fractal, but then I realized I don't really know the answer. With this type of fractal is there always one largest shape as a start point?
I think there could be a bigger one. You'd just have to lop off one of the 4 main arms of this one, make three copies, then join with a bigger circle. Some fractals, like the Mandelbrot set, I can see as having a largest shape (maybe because they're generated via an algebraic function instead of a geometric pattern?).
Monika, what a perceptive question! The way you drew the fractal, I don't see where the bigger shape would fit, either. But do invite your daughter to play around - maybe she'll have an idea. Speaking in terms of calculus, some functions or sequences can extend in both directions, and some have a "hard beginning" or a "hard end." The reasons for that can be mathematical, such as division by zero being frowned upon, or physical, such as the reluctance to deal with a fractional number of people.
With fractal trees that look like stereotypical trees with trunks and branches, you can imagine zooming out and seeing an even bigger tree. You can modify your fractal to make this happen, if you only keep branching in three. Observe that biggest red bead has four. Here is my sketch, based on your piece:
My kid suggested another possibility. Imagine the boundary of your original fractal as the event horizon, so to speak. You can start another fractal universe beyond it, and then again... tessellating. Like this:
wow, thank you for these replies. I love the multiverse idea! And I'm sure my daughter will as well, I'll show these to her tomorrow :)
Answer by mirandamiranda · Apr 16, 2014 at 04:51 AM
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.
Sorry the link for the Vi Hart video is https://www.youtube.com/watch?v=e4MSN6IImpI
Ahh, you copied your bunnies for kids to color? Interesting! This way, you can experiment with color as a separate variable.
Your images came through, and I made them visible. If you want them to show without people clicking the link, use the Insert Image button that looks like a photo of some mountains:
That pictures with many layers down is a type of fractal too! After the first level, that fractal tree only has one branching.
Fraction, fractal, fracture - kids see similarities over differences. All these words do come from the same root! If you and the kids like words, you can go on an etymology scavenger hunt.
Thanks Maria for the image tips - I will try this next time! And I wondered about the 'fract-' root - definitely a good one to discuss.
Answer by AGray · Apr 16, 2014 at 02:21 PM
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.
Week 2 Task 2: Substitution fractals 24 Answers
Week 2 Task 3: Zoom and powers 22 Answers
Week 2 Task 4: Sequences and series 23 Answers
Week 2 Task 5: Multiplication towers 20 Answers