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 AGray · May 07, 2014 at 01:33 AM
I love seeing everyone's pictures. My kids had fun with tree fractals, here are my 9 yr old's cat fractal and my 5 yr old's self fractal.
@AGray - it's interesting to note how individual pictures can have lots of variety (cats) or no variety at all (copies of a photo). Noticing similarities and differences is a big overarching principle in math...
Answer by babyhclimber · Apr 15, 2014 at 03:12 PM
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.
Would your attach your programs here, using the Insert File tool that looks like a paperclip? Scratch also has an easy uploader you could use. Or screenshots. I'd love to see!
We are having massive problems with our online Scratch program. We got it to run the program earlier just fine but now that we've gone back to re-run it and try to share it only to have the sprite disappeared and not draw. The only thing working is the exponent counter/multiplier. So you see the numbers getting exponentially larger but no drawing. My son thinks its my computer so he is trying to fix it on his and we'll see if we can get it to work properly. We are trying to figure out if its an issue with our computers or the scratch program. Playing with the Scratch program in the link I posted early is very cool. My son's was more of an exponentially increasing drawing of lines than a fractal tree. Disappointed that we can't get the program to repeat... On the old scratch (not web based) which we can't share he can't get the sprite to show but does get the drawing. I admit to not being a scratch expert so hoping daddy can help fix it when he gets home.
Answer by TraceySeier · Apr 15, 2014 at 07:44 PM
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".
Answer by CynthiaDadmun · Apr 16, 2014 at 05:58 PM
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!
Answer by dnamkrane · Apr 16, 2014 at 10:55 PM
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.
Answer by Kristin · Apr 17, 2014 at 05:19 PM
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.
Answer by oxanavashina · Apr 17, 2014 at 08:12 PM
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.
Answer by Valerie · Apr 18, 2014 at 02:12 AM
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.
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