Week 2 Task 1: Tree fractals
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
- What multiplication ideas do you see in this topic? How about ideas inspired by calculus?
- Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?
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.
- Arithmetic
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How many pictures are at the next level of the binary tree, if this level has 4?
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2*4=8
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Algebra
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What function gets you to the next level of the binary tree?
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f(x)=2*x
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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
- All course tasks
- Multiplication Lounge: our open forum for questions, comments, and ideas
- Course participants survey
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!
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.
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.
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
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:
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.
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.
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.
