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 njbillips · Apr 25, 2014 at 01:30 PM
We went for a few variations - trees, triangles and then moved up to stars. I think that the video above really helped my daughter to see. We had a great discussion about using smaller and smaller pens to keep drawing.
@njbillips It's interesting to me how the use of familiar objects (pens in your example) can be very evocative for good math. It's a storytelling device very different from, say, exotic worlds for pretend-play!
Answer by Elizabeth02 · Apr 25, 2014 at 02:14 PM
We made a propeller fractal with items we had around the house. It got pretty cramped at the end, but the logarithmic growth (obviously powers of 3 not 10) was pretty quick. We got up to 243 and then we ran out of room and items. The picture is hard to make out, but the math was pretty clear as we went. We talked about infinity and he enjoyed that idea, he made the mistake of trying to double instead of multiply by three once, but then corrected himself. He used repeated addition, especially to do the big multiplications (81x3), but all in all, it was a great exercise!
Answer by lisa.koops · Apr 28, 2014 at 01:44 PM
I started a star fractal doodle during the last piece of a very long college band concert my 7-y-o and I attended yesterday. She grabbed the pen and finished it by the time she caught the idea (after a few of the second-level stars). She did a circle pattern on her own, which was interesting, and then I tried to introduce her to the tree, but she got pretty frustrated (and we weren't able to talk freely at the concert). She stuck with it and seemed proud of it in the end. I'm curious to see if she'll do those herself during her typical doodle times at school (work is finished and waiting for next assignment) or home.
Answer by lisa.koops · Apr 28, 2014 at 01:44 PM
I started a star fractal doodle during the last piece of a very long college band concert my 7-y-o and I attended yesterday. She grabbed the pen and finished it by the time she caught the idea (after a few of the second-level stars). She did a circle pattern on her own, which was interesting, and then I tried to introduce her to the tree, but she got pretty frustrated (and we weren't able to talk freely at the concert). She stuck with it and seemed proud of it in the end. I'm curious to see if she'll do those herself during her typical doodle times at school (work is finished and waiting for next assignment) or home.
Answer by lisa.koops · Apr 28, 2014 at 09:03 PM
Look what she came home with from her doodle time today!
Answer by SarahKrieger · May 01, 2014 at 04:30 AM
We built a binary fractal tree with paddle pop sticks. We'll revisit it another time with differing numbers of branches.
Answer by SarahKrieger · May 02, 2014 at 04:37 AM
Actually, we made our tree into an artwork.
I can't work out how to get my pics in the correct orientation - any suggestions?
@SarahKrieger - we have a (ternary) tree at the local art museum - so it's definitely a subgenre of the arts! As for pictures, here is my favorite quick online editor: http://pixlr.com/express/ Browse for your file, click Adjust button and you can crop, rotate, and do other transformations. Many kids like to mess with photos or to see adults transform photos - and it's very mathematical.
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...
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