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 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 annettehaddad · Apr 18, 2014 at 12:00 AM
I drew a fractal of the Easter bunny - in both directions - tree fractal style outward and substitution, using the eyes as the spot for repeating pattern. I didn't say anything to my children (ages 5 and 7), but just left the drawings mixed in among the other books they were reading. Boy did it stir up excitement! They were laughing, trying to figure it out, shouting "what is this? Where did it come from?" , counting how many faces they saw. I just casually said "oh, they are fractals". Then they started seeing the patterns and relationships and discussing among themselves. When my husband came home from work they grabbed the papers running to him "daddy we have fractals!". I can't believe the excitement this generated and we haven't even begun to discuss them.
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.
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 Sblair · Apr 18, 2014 at 12:58 PM
echer.jpgMy son (9) thought the fractal hand video was a little spooky also. However it got him to think of infinity. We started with the definition of fractal then the Sierpinski Triangle. We did a chalk drawing of the triangle that then led to infinity. We did simple addition and I explained the power of multiplication. We tried a carrot drawing and I explained the importance of single points that branch out to make fractals. I explained it by drawing stars and looking at our tree in the front yard.
We then looked at some of the art by M.C. Esher.
Answer by Joyce · Apr 18, 2014 at 06:25 PM
My 12 year old daughter and I did a few different tree fractals, drawing shapes on paper. What was really neat about it is how easy that made thinking about powers of any base. When we used a five pointed star it easily led into a discussion of repeated multiplication and then powers/exponents of a single base, 5. Such an elegant introduction! We also discussed the idea of infinity and whether we could add up all the areas of our stars.
Answer by juggling_ginny · Apr 18, 2014 at 08:12 PM
We looked at this kind of fractal as part of a maths circle session on doubling and halving a little while ago. I started by reading the myth of Hercules and the Hydra heads. Even though my circle is all girls, they were immediately excited by the chopping off of heads. They were all competing with each other to calculate how many heads would be on the next iteration. Later on in the session we drew square based fractals on squared paper and triangle based fractals on isometric paper. Using grid paper encouraged a greater level of accuracy, with the necessary calculations, but became a bit overwhelming for the younger or less able children - although they could quite happily devise colouring schemes for other people's fractals.
We finished by watching Vi Hart's binary tree video and were really inspired by the crashing Hydra heads doodle. When faced with a boring half hour (like waiting for me to have my acupuncture session) this is still my daughter's favourite activity. She's had numerous goes at it and has improved in accuracy and consistency each time. Really looking forward to trying some more fractal activities with her.
Answer by pkouch · Apr 18, 2014 at 10:45 PM
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:
Answer by Sdhanks · Apr 19, 2014 at 01:30 AM
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!
Answer by Ria · Apr 19, 2014 at 01:30 AM
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.
Answer by Kris · Apr 19, 2014 at 07:52 AM
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
Answer by zzzeee2000 · Apr 20, 2014 at 10:33 PM
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
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 mistermarty · Apr 21, 2014 at 10:11 PM
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!
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 classicalmama · Apr 22, 2014 at 10:14 PM
We watch a couple of videos and then I explained what a fractal is then we drew rabbits and spiders.
Answer by ChristyM · Apr 24, 2014 at 03:49 AM
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!
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