Fractal cookies by Evil Mad Scientist

**Today your mission is…**

Create your own substitution fractal and use it to explore multiplication.

**Ready, Set, Go**

Find a picture you like, with prominent parts inside its outline. The object can contain the parts, such as an owl’s head with two big eyes, or an open book with large illustrations. Or it can be made of the parts, for example, a word out of letters, or a window out of panels. You can also start with an abstract shape, like a triangle split into triangles.

Replace each part with a miniature copy of the whole shape. Repeat several times. You just drew several levels of a substitution fractal!

**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?**

**Help your child to get started**

Some children can’t resist large-scale projects and making a substitution fractal is perfect for them. Get the largest piece of paper you can find and start big, with the base object taking up the entire sheet. Or take this activity outside where you can draw with chalk.

Draw a fractal based on a shape your child loves - a toy, a setting, an idea. Tell a story of what’s happening and encourage your child to add details to the story as you draw new levels of the fractal.

**Toddlers**

If your toddler likes stickers and collages, cut shapes out of different-color paper for different levels. Fold paper to cut several shapes at once. Or you can make fractal cookies out of real or play dough.

**Young kids**

Children realize early on that words are made out of letters. How about letters made out of words? Your kids can use this software to make math birthday cards for friends and relatives. http://www.edboas.com/science/ftext/index.html

**Older kids**

The study of fractals picked up in the late 1960s, with the use of computers. Tech literacy is essential for studying fractals beyond the first introduction to the idea. Make extreme and outlandish art with Droste Effect software: http://www.photospiralysis.com/

**How is this multiplication?**

Like tree fractals, substitution fractals support the foundations of our number system, but in a different way. Tree fractals show groups of groups of groups, for example, tens of tens of tens making a thousand. Substitution fractals can show that too, if you zoom out through their levels. But substitution fractals also show parts of parts of parts, such as one-tenth of one-tenth of one-tenth of a meter making a millimeter. This is one of the most systematic (and gorgeous) ways to explore multiples and fractions, positive and negative powers, multiplication and division within a single unifying pattern.

**Inspired by calculus**

Substitution fractals focus inward: you zoom in, almost like a microscope. They help children imagine not only infinitely large, but infinitely small, infinitesimal objects. It’s interesting to note that in a group, girls often prefer the subtlety and inward focus of substitution fractals, and boys often prefer the big expansion of tree fractals.

**Frequently Asked Question**

*What if I can’t draw at all?!!*

Babies don’t care if their parents finished their opera training before singing lullabies! However poorly you draw, kids will appreciate you drawing for them. But if you are anxious, start from simple geometric shapes, such as squares and triangles. You can also borrow someone’s math goggles, so to speak, to see how more complex objects come together from simple shapes. If all else fails, a circle with two triangles always makes a good cat portrait!

**Words**

Fractal, scale, power, exponent, recursion, zoom, reverse

**Scavenger hunt**

Matryoshka, by Radiology Art

Seek substitution fractals and nested pictures in art. They appear in abstract designs, such as nested dolls, but also in philosophical or symbolic art pieces, such as Bruegel’s “Big fish eat small fish.” Nested structures also appear in traditional architecture (check out these fractal villages in Africa), rhymes, and stories, such as “This is the house that Jack built.”

*Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpet, bricks, and much else besides. Never again will your interpretation of these things be quite the same.*

- Michael Barnsley

**Course links**

- All course tasks
- Multiplication Lounge: our open forum for questions, comments, and ideas
- Course participants survey

Comment

**Answer** by James
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Apr 15, 2014 at 08:07 PM

My daughter was in a fussy mood this morning, so we went with a music fractal, as music often makes her happier. We chose one of the songs from her music class, Frère Jacques. It's 16 measures long, so we pulled out the middle 8, sped up the tune to double time, and inserted that in instead. The double time tune also has a middle (4 bars now), so we put in the original sped up 4 times. Finally, we put in two bars of 8x in the middle of that. We did try even faster, but it was too hard for me to spit out words that fast. Since Frère Jacques is a round, all the iterations can be played at the same time without getting too much clashing of harmonies. Here's the sheet music for "Fractal Jacques", followed by a video me singing the fourth iteration to my daughter along with a midi rendering.

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Wow, just wow. Fractal art of all sorts - visual, stories, now music!

Is it okay to share your video? You have it unlisted.

Sure, share away - I just didn't want to pollute YouTube with my singing! If anyone would like to try singing along with their kids, here's the sound file: http://wardlawbailey.com/FractalJacques.mp3.

Thank you for posting this. We did the fractal activities last week, and my kids (4th grade group of 4 girls) had asked to integrate music. I told them that someone had made a fractal of the song Frere Jacque, and asked them how they think you did it. Then it turned into a discovery experience for them, in which they figured out what you did, then we watched your video, then played the music on the piano.

**Answer** by Rodi.Steinig
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Apr 16, 2014 at 12:18 PM

ATTENDANCE: J (9), S (10), A (10), T (10) – all fourth grade homeschooled girls

SYNOPSIS OF ACTIVITIES

__My tree fractal demonstration__ – I showed them a turtle pin that is a special piece of jewelry, and told them I’d like to do some math art in honor of it, and that I’d show them how and that I’d like their help. Then I started drawing. Several times, I asked them to pick up pencils and help but no one did until the fourth iteration, and then it was only S doing one turtle. As I worked, they brainstormed their own ideas.

As I was working on the 4^{th} iteration, I started asking about/discussing the math behind this – why is this math, etc? We worked together on a chart of number patterns, and did some predicting. It was hard to describe which iteration we were talking about at first, since these kids had never worked with that concept (except possibly in science experiments a smidge). The kids suggested that we use real life vocabulary so we called the first turtle in the center “mom” then the next group “children” etc etc and this helped a lot and generated smiles.

Then the kids started working on their own. They attempted turtles, birds, suns, hearts, triangles, and infinity signs. As they worked challenges and questions arose, so we made running lists of “problems” and “questions” that we added to throughout class. (I’ll put the list at the end of the report.)

__Photo__ I took a group photo now as the girls drew so that I would have it ready to use later on the computer.

__Tree fractal software __The kids played on the visnos.com site, and had a great time. They took turns at the computer, with lots of oohs and aahs and how did you do that’s. One student stayed late afterwards to do more. This was a fantastic way to introduce scale factor from a numerical perspective.

__Fractal hand video__ They loved it, although found it creepy. We watched it 2-3 times. Used it to discuss substitution fractals (since the fingernails in this got replaced by smaller hands). I mentioned that I had seen one somewhere with an eye substituted for a mouth but can’t remember where. The kids want me to email that to them when I encounter it again (help?).

__Substitution fractal__ I drew a quick example of this (a traffic light), and told the kids they could do this type too. I tried to get then to come up with the idea first by asking a bunch of questions about their frustrations with the tree fractal and large scale factors. I don’t think anyone ended up making their own substitution fractal, but some had ideas.

__Letter fractals__ We played with letter fractals on the edboas site. That was fun. I encouraged kids to make their own by hand, but no one did. They were too taken by tree fractals. They loved the idea of using this site for birthday cards.

__Photospiralysis__ They had fun with this using the photo of the 4 of them I took. I think everyone is looking forward to using these various sites at home.

__Fractal scavenger hunt__ – did not do

__Ted Talk on African Designs__ – did not show – I didn’t think that it would hold people’s interest b/c it didn’t hold mine (and I’ve tried to watch this several times over the past year or so). I think the topic is interesting but it needs to be presented in a different way for kids to appreciate it. (My guess is that you intended this for the adults only, but I wasn’t sure.) My issue, though, might also be that I’m overscheduled (like most working homeschooling parents) and it would take something even more spectacular to get me to watch a video for 16 minutes.

PROBLEMS (student-generated list)

- Keeping the shape consistent (with my turtle, sometimes the tail pointed right and sometimes left)
- Keeping size of shape consistent
- Running out of room for additional iterations (which generated nice discussion of adjusting scale factor)
- Having any room at all when doing a tree fractal with a scale factor bigger than one for a shape with width
- With shrinking iterations, drawing tiny things is hard

QUESTIONS (group collaboration; this list was still being added to even as people were walking out the door at the end)

- Will the points eventually connect?
- What’s the difference btwn a fractal and a tessellation?
- How many iterations/shapes could you draw if you had unlimited time, desire, and ability (robot? Computer? Magic?)
- What is the minimum number of points required on your shape to generate a fractal that’s not just a line?
- How do you know you’re finished?
- Is a hybrid tessellation/fractal possible? Has it been done? (This would have a scale factor, but no negative space.)
- If you connect each of your shapes of the second iteration to the first in a different way (i.e. connect one by its feet, another by its tail, etc), is it still a fractal? Is it still multiplication?

POST-CLASS REFLECTIONS

At one point when the kids were struggling with their tree fractals (how many points? curved lines or straight? is the scale factor sufficient for non-overlapping shapes?) I showed them my failed attempts from the night before. Seeing my failures made a huge difference in their confidence.

The above took about 1 hr 45 minutes. The kids then snacked and played with photospiralysis a bit in the dining room while two of the parents joined me in the living room for a bit of conversation. The parents were talking about how their kids do not feel confident in math (especially one who had spent two years in traditional public school). Their kids also do not like “school math” but enjoy “non-school math.” With this in mind, I called the kids back in to reflect on the session with the parents.

First I asked the girls what they would have named this course or session. (They knew I’d be giving feedback to you. Their proposals:

- The Family Tree Gathering (they had named the 4-generation turtle fractal “Our Family Tree”)
- Math Made Fun
- A Get Together Circle
- Math Get Together
- Fractals

Then I told them that you had actually already named this course, and that it was called “Multiplication.” Jaws dropped and eyes opened wide in shock. We had a nice discussion about how we had just been sitting around doing multiplication for a few hours.

Then I asked for specific comments about the session – what they liked, what they didn’t, and what they would change. They liked everything, but here are the things I jotted down as we discussed:

- Liked computer things
- More food (we scheduled this 9:30-11:30 so people were fading with hunger for lunch – I’ll ask the parents to bring more snacks next time)
- Play time afterwards (everyone had different activities and errands they needed to get to)
- Having dog here, but not interfering (my dog is laid back, calm, and quiet but likes to lie in the center of things. The kids liked petting her, but S did not like her trying to sneak the snacks. One parent stayed for the whole session and helped to manage the dog. We live in a tiny house, so dog management is a challenge. I like the kids to have the snacks on the floor while we do math to maximize the time we have to do the math, instead of taking a formal break, which causes this dog problem.)
- Having friends (S, the student who had been in public school – the social aspect was very important to her. Her mom told me that at her school, they had posted everyone’s test/quiz grades on the board to motivate kids – this was in kingergarten and first grade. So math for her was competitive.)
- Real life applications (one student, T, requested more of this)
- Science math (another recommendation from T)
- Integrating art (S stressed that having the art and having the dog really helped her to focus)
- More activities (things petered out after an hour and 45, but kids were hungry and parents were coming. I could have done the scavenger hunt, but didn’t.) I think they really really would have liked something else to draw by hand that was more different from the tree fractal task than was the substitution fractal.
- Recc. From J (with the others agreeing): integrate music and movement and acting and dance
- Recc from S (with others agreeing): a non-competitive game (brainstorming already began for this, so I promised them 10 minutes next time to brainstorm together – maybe they can have a 10 minute eat at the table break next time to do this away from the dog)
- A group fractal where each person did a different shape and put theirs in as a different iteration (T suggested this at the beginning, but everyone else wanted to do their own. By the time she suggested this again at the very end, the others were ready to collaborate but we were out of time)

Finally I asked them if there was anything that they didn’t like. J said that the first activity where I explained to them how to make the turtle fractal was not fun for her. She has grown up in exploratory math circles, so she really noticed that I was basically giving them an algorithm and asking them to execute it. That really cramped her style. Particularly when she had put her fractals aside to draw her favorite thing: interior design sketches. I asked her for today to try to stick with fractals since we had a goal here. (Later, she mentioned privately that it hurt her feelings when I asked her to do this.) The group did agree with her that they all would have preferred an activity that had them doing more thinking as opposed to following directions. I asked them, “Then how could you have known how to make a tree fractal?” They didn’t know. And of course 24 hours later with my 20/20 hindsight I can think of many ways I could have introduced this topic differently that would have involved some discovery on their part.

Rodi Steinig

PS Sorry this is so long. Maria, I tried to attach it as you suggested, but for some reason this file does not appear as a selection option when I click on the paperclip icon through this site. Weird.

**Answer** by yileinei
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Apr 15, 2014 at 07:11 PM

We did this one and my 4yo daughter understood it much better than the one yesterday! She got the point really quickly and she said: "we are doing hearts inside hearts and we are getting tinny little ones! We can't go further because we don't hace a thinner pen!" So she really new we could go on and on... I love this activities!! I asked her how many hearts would she draw in the next turn and she counted the gaps before drawing them so she really understood it :)

I see how she understood dimension and space sharing, which appears to be divide by 2 in this case, isn't it?

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Sweet, she got both the scale and the position of the next heart - all aspects of this, really! Including the idea that infinity lives in our imaginations, even if our pens are too thick to draw it.

We'll never know why substitution fractals made more sense, between the two. Maybe just because it's the second fractal activity in the family?

**Answer** by Eogruen
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Apr 17, 2014 at 02:07 AM

this morning we had a great time drawing chalk fractals in the driveway. My four year old drew the nested tree pattern above the large tree fractal. She was very proud of herself.

My seven year old did a five times five flower design.

I did a four petaled flower substitution fractal, shown here with some cute interference who was more interested in attempting to eat wood chips than learn math.

We wrote on big letters-- tree and pattern fractals, try some at home! Just in case someone walking by wanted to be inspired.

**Answer** by cpw
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Apr 15, 2014 at 08:35 PM

We have really been enjoying fractals. Yesterday we did tree fractals and Sieroinski triangles. We discussed the concept of exponents with these fractals and then went outside looking for examples in nature. Today both kids really enjoyed the substitution fractals, our five year old really loved drawing out the owl eye fractal. Our son drew his won tree fractals today and wrote out the exponents. We discussed the difference between 2+2+2+2=2x4 and 2x2x2x2. It was really fun.

**Answer** by sherylmorris
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Apr 15, 2014 at 09:34 PM

Finding new ways to look and think about Maria Montessori's Pink Tower! Thank you!

1x1x1=1 2x2x2=8 3x3x3=27 4x4x4=64 5x5x5=125

6x6x6=216 7x7x7=343 8x8x8=512 9x9x9=729 10x10x10=1,000

**Answer** by CynthiaDadmun
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Apr 16, 2014 at 05:41 PM

This morning my 3yo and 5yo worked on the Sierpinski fractal (http://fractalfoundation.org/resources/fractivities/sierpinski-triangle/) (Thanks for that link, babyhclimber!) That was just about their speed. Here are their triangles. I had to help my 3yo with some of the lines, and she also had a little trouble cutting! 5yo got it.

We also talked about how the drawn triangles can't really go on forever, since the marker is too thick, and even the printed one is limited by the size of the little globs of ink. But a zoom-in movie on the computer CAN go on forever (http://math.bu.edu/DYSYS/animations/sierp-zoom.html) because the computer is using the mathematical rules to figure it out over and over. Hoping that plants the seed of the idea that the mathematical IDEA is bigger than the representation of it on paper (ie, the printout is just a model of the real Sierpinski triangle, which is a concept). And... I think the ST printout lends itself to a fun little puzzle.

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**Answer** by pkouch
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Apr 18, 2014 at 11:30 PM

My international student from Korea made a lotus in a couple of minutes after I introduced the class to substitution fractals. I didn't have my cell phone with me to take a picture, but I found this similar image online.

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**Answer** by babyhclimber
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Apr 15, 2014 at 03:31 PM

OMG, my son is loving all of these fractals. He did his name on http://www.revisionrads.com/ftext/output/o18531.html and was blown away by how many times his name was written in each letter of his name. Very cool! On http://www.photospiralysis.com/WebAppGamma/RunApp.html (and with google search to help explain) we learned about the droste effect by playing with mirror, stretching, & zenith. He learned that daddy can do this with real pictures inside photoshop which lead to some funny business! We looked at images of Sierpinski Triangle Fractals (http://fractalfoundation.org/resources/fractivities/sierpinski-triangle/) and discussed how doodling like this is combining art and math.

**Answer** by Caroline_Prochazka
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Apr 16, 2014 at 06:26 PM

We only had a little time for this, this week, and so I took the opportunity of a car ride between errands for a discussion about substitution fractals. My 9yo had seem the picture of the owls already at that point and so he caught on quite quickly. We discovered that this was a fractal exercise that would lend itself to Star Wars images - with an example being Darth Vader's helmet copied into the eyes of the helmet much like the owl eyes. My 5yo pointed out that the shape of the mouth grille on Darth Vader's mask is more like the shape of the helmet itself and that he thought this was a better place to zoom in. We talked about how using the eyes is similar to the binary tree fractal where the number doubles each time. Then we wondered if the 'tree fractal' mode of Vader's helmet copied on the mouth grille would be a single line connected to a single line, connected to a single line and so on. Hmmm.

I watched the TED talk on African Village fractals with my 9yo yesterday and he was truly interested. Visual media is always a big sell with him. I think if I had more time, I would seek out larger aerial photos of such villages and have him trace the fractal geometry with different colour markers.

The Breugel painting interested both kids, but we had difficulty finding more such work - does Breugel have more in this vein? Springing from this, I started doodling with my 9yo a string of pearls, where each 'pearl' then became a finer string of pearls, and so on. A string with 13 pearls was simple, 13 smaller 'pearls' in each of those 13 would be 169, already a big enough number. The next iteration got us to 2197 which hit home - "whoa! now that's a LOT of beans".

Only after I we were moving on with our day, did the 5yo show some interest in these doodles. He was equally impressed, I think. Just coming at it as a new discovery - "look! this loop is made up of smaller loops, and there are more loops in that".

So they seem to understand the geometry of the topic, and the elder understands how this is tied to multiplication, but is not really interested in expressing much of it in numerals on paper. I see another participant has noted that the thrill in this is that it makes math 'fun', and we have been craving more of just this type of thing.

Spatial understanding of multiplication is valuable in its own right, even without numerical connections.

I think fish-within-fish was a one-off idea for Bruegel, but check out the Droste Effect group on Flickr. Here's my Old Empire contribution:

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**Answer** by champalto
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Apr 16, 2014 at 09:37 PM

The great thing about this whole experience is my math-skeptical 8-year-old is getting into it! He is actually quite good at math - just unsure why he should care about it. I am very excited! Both he and his sister are Minecraft fans, so here are their Minecraft-inspired substitution fractals. (I am told one is a ghast and one is a creeper.)

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**Answer** by dnamkrane
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Apr 16, 2014 at 11:32 PM

Not a skilled visual artist, so I went into my Paint program and started with a base shape of an oval with a diamond circumscribed within it. I copied and reduced that shape so that it would fit into the base shape and repeated that two more times. I was surprised that what looked something like an eye when I started out ending up looking more like a rose petal, as someone below found as well.

Out of curiosity, I reversed the shapes and created a new fractal. What started out looking more like a ring now looks something like the Superman logo :-)

More seriously, I was able to more levels of fractals into the first than the second. The diamond inscribed more easily into the oval than vice versa. There's something there about width versus height that I can't quite put my finger on.

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**Answer** by mirandamiranda
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Apr 17, 2014 at 05:42 AM

We started off with me showing the girls (ages 6, 6, 8) the owl fractal which they had seen before and talking about substitution fractals and how they worked. I suggested some other things we could try - houses repeating in their windows, people repeating in their limbs...

Daughter P drew her own version of the owl.

Daughter M tried various smiley faces, a single nesting and a triple one:

We talked about how much harder it was with more points of substitution!

Daughter G at first was anxious and felt she could not contribute. So I got out a book I had been meaning to show them, The Mouse and His Child by Russell Hoban. A key element of the story is the 'Last Visible Dog', an image on a dog food can that features a dog holding a smaller version of the can, which has a smaller dog on it, also with a yet smaller can... Another character describes looking at this image as contemplating infinity: "an endlessness of little dogs, receding though progressive diminution to a revelation of the ultimate truth ... Beyond the last visible dog".

This was more inspiring to us all and we attempting a co-authored rendering (although not strictly adhering to the book's description):

Not super clear, sorry... getting out the story book definitely eased the tension G had been feeling, even though she continued to fret about her dog-drawing abilities.

Then we had a go at the Sierpinski triangle which was a big hit with my most methodical child, P. She asked me to draw the triangles in (I had printed out the worksheet) and she coloured them. She talked about going even further than this but she also insisted on counting all the triangles each time and was exhausted by the time we got to 243!

This was for me the most enlightening in terms of the multiplication involved. I really started to see how with each iteration we were trebling the triangles by drawing 3 triangles in each, and we talked about this as well as how we could multiply instead of counting them all.

We also looked at the word fractals which were fun. Later at dinner G spontaneously suggested making letter fractals, tree ones from each point of A, say, and substitution ones with letters like O. And then she noticed a flower fractal in a film we were watching. So it is definitely sinking in!

One thing I am thinking is that my children don't ask loads of questions. I don't know how much this is habit, or a sign of their interest level. I am going to try and make more of an effort to elicit them in future tasks. I also find that the arithmetical side of the fractals is not something that really interests them, similarly to what another poster mentioned.

I am also finding these activities really great for giving me an insight into how my children learn, what interests and engages them and what does not. I feel like I have learnt more about them from this than from all the workbooks and manipulatives and maths story books and so on we have used so far!

Arithmetic just isn't all that interesting to most people, as branches of mathematics go.

There are several games for brainstorming questions in small groups. You can try them together with math. But some kids generate one or two bigger questions per activity, rather than rapid-shooting tons of little questions all time... And some kids interact non-verbally, in gestures and constructions and building ideas!

**Answer** by perbui
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Apr 17, 2014 at 09:42 AM

I showed my 7yo daughter examples of substitution fractals, and she really thought the ones that had anything to do with animals really cute. When it came time for her to draw, she went for abstract shapes. Instead of multiplying the original design by 2 or more for each subsequent level, she only copied it once, which isn't really multiplication, is it? It is the Droste effect, but I don't really think it's multiplication. We did relate to the examples we found in other places to exponential growth, just like tree fractals, so that was good.

On another note, as a literature tie-in, we read *A Grain of Rice** *by Helena Clare Pittman as an illustration of exponential growth. It was a great way to give the kids a physical representation of these abstract concepts.

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**Answer** by Valerie
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Apr 19, 2014 at 08:43 AM

This generated much more interest than tree fractals. After a quick demonstration of hand drawing a series of leaves inside leaves she immediately understood the concept, and produced a face with substitution fractal eyes and nose. Then while we were watching a universe zoom out video as part of the next exercise she drew this as a circular substitution fractal, describing it as a cyclone (something she's very interested in). She was also mesmerized by a musical representation of substitution fractals with zooming - the White Stripes' Seven Nation Army video

**Answer** by AGray
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Apr 21, 2014 at 01:56 PM

I love the connection to stories, like "The House that Jack Built". I am going to see if my kids notice that connection! I am going to introduce substitution fractals today. Last week was busy with the Easter holiday, but we did spend some time drawing cat face tree fractals with sidewalk chalk.

**Answer** by HappyHEmum
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Apr 22, 2014 at 09:48 AM

My 2 loved this activity, we are a little behind schedule but catching up! Very fun way to introduce substitution fractals

**Answer** by mistermarty
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Apr 22, 2014 at 03:29 PM

I spoke to my son about doing the reverse of yesterday's tree fractals asking him about making something smaller. He did something like this a few weeks ago by folding paper into smaller and smaller dimensions. Today he decided to do some cutting and this is what he came up with. Upon seeing this it dawned on him that the scale model of the U.S.S. Yorktown that he received for Christmas and that we built together uses the same idea as this little project. I'd say it was a good exercise.

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**Answer** by mrs123
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Apr 23, 2014 at 02:52 AM

Substitution fractals were a lot easier for my younger children to understand. I'm not sure if it's the concept or if it's because they had already learned about tree fractals and fractals in general the day before. They are really starting to get the idea of infinity through these lessons. Here's my daughter's drawing of a frowning girl:

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Nice spatial reasoning there - using the frown as hair. Very clever!

**Answer** by ChristyM
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Apr 28, 2014 at 09:49 PM

To me, this activity suggests a more complex function than straightforward multiplication or division. I definitely see the calculus feature as the function is repeated.

Both of my children really enjoyed this activity. One did a self portrait and repeated until we reached the limit due to pixels.

The other cut pieces of paper smaller and smaller. She then cut a hole in the big sheet and put the smaller piece inside, then put the next smaller piece inside and so on.

By the way, we actually did these activities day by day in order, not as I posted the answers.

I am hoping to post pictures as my children really love seeing their work next to everyone else's, but need to get some time on the actual computer to reformat the pictures.

Week 2 Task 1: Tree fractals 49 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

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