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
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...
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.
We built a binary fractal tree with paddle pop sticks. We'll revisit it another time with differing numbers of branches.
Look what she came home with from her doodle time today!
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.
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.
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!
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!
I attempted to introduce this lesson by drawing my own fractal picture using bees. This resulted in a fun little lesson in bees, but didn't really get the fractal concept across. Both my four and five-year-olds tried to draw chains as their fractal image.
I showed them all the other images that had been previously uploaded, and they got excited about building fractal sculptures. Thank you @Ria! It's a little hard to see, partially because physics was not in favor of this sculpture, but this is a binary fractal of birds built on each wing of the previous birds.
This is actually a fractal as well, however he wasn't interested in symmetry of length or direction, so it doesn't look very "fractal".
The Multiplcation comes through very nicely with the many groups of equivalent sizes. The Calculus effect as well as the number of groups increases each time. I am contemplating how I could ask future physics students to represent physics concepts such as kinematics in fractals.
In the future, I think I will introduce these exercises more simply so that my children are able to focus on the concept of fractals. They were excited about it once I was able to re-direct them. It seems that fractals are a concept that should be revisited as the child grows and is able to understand more complicated patterns, and has the patience to repeat more iterations.
@ChristyM - nice complexity there! By the way, chains or nested circles are types of fractals, too. A chain is a fractal tree with one branching.
About simple shapes vs. animal or other more complex shapes... The choice of one vs. the other is a strong personal preference of each kid. There are at least two known differences in learning styles that play a role in this choice.
First, sequential thinkers prefer step-by-step instructions, going from simple to harder and harder structures. Nonsequential thinkers jump into complexity with ease.
Second, kids who are into storytelling and pretend-play often (but not always) play with toys and objects that represent people, trees, animals, and other concrete things and characters. Other kids play with abstract art, design, puzzles, and engineering, and thus need non-representational, abstract objects for their play.
Parents usually know ahead of time what their kids prefer, but sometimes a new context surprises!
We watch a couple of videos and then I explained what a fractal is then we drew rabbits and spiders.
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.
This is what I got out of my son today. After this we discussed a tree in our yard and the way in which it branched off. If there is anything that I want to impress upon my soon is the immersion of patterns in the world around us. Perhaps this will help in that area.
Nice variety! That means understanding. With immersion in the world, if you keep up your own interest, and keep noticing things for yourself, it usually spills out into kids' interests too!
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.
I did a tree style fractal in minecraft. Took me a few hours but it was very fun. Sadly my laptop wouldn't screen shot it so I can't share.
Glad the fractals were hours of fun! Have you tried some of these shortcuts for Minecraft screenshots? http://minecraft.gamepedia.com/Screenshots
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 :)
I'm going to work on my tree fractal tomorrow but I found this link useful and thought I would share:https://www.youtube.com/watch?v=e4MSN6IImpI
Here my 9 yr old
son made the fractal useful and visually pleasing by using the rectangle fractals and making it into a flag info graphic.
What a thoughtful, metaphoric social statement!
I attempted making a tree fractal with mega blocks as my 16 month old toddled. He decided to walk on it so I relocated to an air mattress; not the most stable, but my efforts of constantly reconnecting pieces seemed to peak his interest as he got on top of the mattress and struck some yoga poses. I asked him what it looked like and he got really close to the base blocks to look but no answer. I said I thought it looked like tree tops swaying in the wind and then I made the shwee shwaa wind sounds as I waved my arms in the air. His excitement spilled because he loves to make the sounds of everything. He asked to help and immediately sat down to build and attach his creation. He really liked working the part of tree that began to branch off the mattress. He called for dada and reenacted wind through the tree tops and pointed enthusiastically to the blocks. Next time I will use the word fractal. I got so excited along with him that I forgot to introduce the word.
This was a fun experience. Building blocks across the floor/mattress was different for us. We typically build vertical and vehicles. In the future if he ever says he doesn't have enough blocks to build fractals, I will really understand because I was surprised how fast the blocks ran out.
Week 2 Task 1
I discussed the idea with my small group of math students and we ended up having a very interesting discussion. John drew the following:
After examining it for a while, Jane said that John’s fractal was not correct because if the center was a square and one square was coming out of each corner, then in the third stage (second stage is the one with one square in the center and four squares on the vertices,) each square had two empty corners that needed to have squares coming out of them. Then she drew the correct form and realized that it would look like a chess board! They were all excited about this discovery. Then, John explained that the starting shape of his fractal was not a square, but the five squares in the center. He drew the next stage on the board to defend his fractal design:
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