Fractals & Chaos Recap for 10/18

If you have a compass (the circle-drawing kind) please bring it with you to class over the next few days.

We discussed Mandelbrot’s Article, then used it to segue into a further discussion of dimension. As we are aware, we still have problems with the Hausdorff dimension formula for calculating dimension of fractals. It can’t handle fractals with stems (i.e., non-iterating segments that never disappear) and with fractals that are not exactly self-symmetric.

Today, we considered a football field, a circle, and a Koch Curve, and looked at how the size of the measuring stick we use to measure the length or perimeter of such things has an impact on the total amount of length we actually calculate. For a football field, the size of the stick makes no difference. We’ll be obtaining 100 yards worth of length even if we use a foot (S = 3) or an inch (S = 36) as our step size.

For a circle, this isn’t the case. Use a measuring stick the length of the diameter, and we can only make two steps before we end where we started. Use a stick the size of the radius (S = 2) and we can make 6 such steps (resulting in a measure of three diameters). Use a half-radius (S = 4), and we wind up with a total length of slightly more than 3 diameters. There is a limit to this, of course: pi*d, which is precisely the formula for the circumference of a circle.

For the Koch Curve, the story is very different. Use a step size the length of the original baseline, and we can make one step. Use a step size of 1/3 the baseline (S = 3), we can make 4 steps, giving a length of 4/3 the base. Use a step size of 1/9 the baseline (S = 9), and we can make 16 steps, for a total length of 16/9 the base. As we shrink the length of the ruler we use, the number of steps increases more quickly, and so the total length increases without bound.

We’ve seen suggestions at this idea before. In the second article we read (The Diversity of Life), we saw that reducing the scale of our perspective dramatically increases the amount of living space we can find. This idea is also found at the center of the coastline paradox, hinted at in the Ants in Labyrinths article (see also this blog post from UK Urban Planner Alasdair Rae)

We will be expanding on this in class tomorrow, including a discussion on what all this has to do with the dimension of what we’re measuring.

Intro to College Algebra Assignment for 10/18

For Period 3

We worked with adding and subtracting rational expressions today, modifying the usual “lowest common denominator” process to accommodate monomial denominators. We’ll do some more practice with this on Monday.

Period 3 Links


For Period 8

We spent today practicing adding, subtracting, multiplying, and dividing with an IXL assignment. Your homework is one of two:


Extra Practice Links (IXL Review)

Fractals and Chaos Lesson Recap for 10/17

We had our official Fractal Art Show Gallery Opening today!

For tomorrow, please read Mandelbrot’s revolutionary paper that sparked the recognition of fractals and fractal geometry How Long is the Coast of Britain? You might also want to read this version, where Mandelbrot himself explains how he originally wrote this paper as a “Trojan Horse” to introduce his vision of fractal dimension into the scientific community conversation.

Intro to College Algebra Assignment for 10/17

For Period 3

In class today, we debriefed Tuesday‘s Jigsaw about Multiplying/Dividing Rational Expressions. Your homework is to do a few additional problems.

Period 3 Links


For Period 8

We worked with adding and subtracting rational expressions today, modifying the usual “lowest common denominator” process to accommodate monomial denominators. We’ll do some more practice with this on Monday.

Period 8 Links


Extra Practice Links (IXL Review)

Intro to College Algebra Assignment for 10/16

In class today, we debriefed yesterday‘s Jigsaw about Multiplying/Dividing Rational Expressions. Your homework is to do a few additional problems.

Today’s Links

Extra Practice Links (IXL Review)

Fractals & Chaos Recap for 10/16

One more chance to sign up for snacks for tomorrow’s Fractal Art Show!

In class today, we finished up the dimension classwork sheet and practiced how the segments of each fractal design that you measure to calculate its dimension can be used to recreate the fractal in FractaSketch (aka “stealing the template”). Neat!

We also have an important reading assignment: By Friday, October 18, please read Mandelbrot’s revolutionary paper that sparked the recognition of fractals and fractal geometry How Long is the Coast of Britain? You might also want to read this version, where Mandelbrot himself explains how he originally wrote this paper as a “Trojan Horse” to introduce his vision of fractal dimension into the scientific community conversation.

Intro to College Algebra Assignment for 10/15

We started a jigsaw activity in class today, exploring the multiplication and division of rational expressions. You started out in groups examining three carefully worked out simplification, multiplication, and division problems. In step 2, you were given some additional examples of that same type of problem to complete in your groups. Step 3 will be to share what you’ve observed with some new people.

We didn’t quite get to finish Step 2 in class, so your homework is to do that on your own. Finish whatever problems you were working on in class. If you weren’t in class, take your pick of which type of problem to analyze and come to class tomorrow prepared to share.

Today’s Links

Extra Practice Links (IXL Review)

 

Fractals & Chaos Recap for 10/15

Please sign up for snacks for this week’s Fractal Art Show on Thursday, October 17.

In class today, we worked on our fourth and final dimension classwork sheet (while a handful of stragglers turned in their Fractal Art Show designs) and looked at how the analysis we do to find the dimension of the shapes can be used to recreate them in FractaSketch. Neat!

We also have an important reading assignment: By Friday, October 18, please read Mandelbrot’s revolutionary paper that sparked the recognition of fractals and fractal geometry How Long is the Coast of Britain? You might also want to read this version, where Mandelbrot himself explains how he originally wrote this paper as a “Trojan Horse” to introduce his vision of fractal dimension into the scientific community conversation.

InCA Assignment for 10/11

Today in class, we started with a Desmos activity exploring the similarity of reducing numeric fractions and simplifying rational expressions. They both involve factoring, then reducing the common factors away.

Today’s Links

Extra Practice Links (IXL Review)

Fractals & Chaos Recap for 10/11

Please sign up for snacks for next week’s Fractal Art Show on Thursday, October 17

Your FractaSketch designs were due today! You should be turning in one entry in three of the following categories:

  • Fern
  • Tree (or shrubs, bushes, weeds, etc.)
  • Spiral
  • Realistic (other natural phenomena)
  • Artistic (patterns, designs, etc.)

On the back of each of your submissions, you should write your name, the category, a notation on which way is up, a title (for the realistic and artistic categories) and a note about whether it is an “official” work, i.e., one of the three expected from all students, or an “additional” work that you would like to be considered in an “Additional Works” category.

In addition, for one of your official submissions, I also need one template. You’ll need to take a screenshot of the template, including its arrows, and print that as a separate page. Use the “snipping tool” in Windows, and leave the image “actual size” (don’t blow it up). My goal is to fit all templates on a single piece of paper for a matching game as a part of the art contest.

Do not print the .png images directly! Instead, I recommend importing your images into a single Google Docs file, one image per page, and printing that document. This allows you to resize your images more convenient and generally makes the resolution of the images less grainy.

Fractals & Chaos Recap for 10/10

Please sign up for snacks for next week’s Fractal Art Show on Thursday, October 17

Your FractaSketch designs are due tomorrow, Friday, October 11. Remember, I expect from each of you one entry in three of the following categories:

  • Fern
  • Tree (or shrubs, bushes, weeds, etc.)
  • Spiral
  • Realistic (other natural phenomena)
  • Artistic (patterns, designs, etc.)

On the back of each of your submissions, you should write your name, the category, a notation on which way is up, a title (for the realistic and artistic categories) and a note about whether it is an “official” work, i.e., one of the three expected from all students, or an “additional” work that you would like to be considered in an “Additional Works” category.

In addition, for one of your official submissions, I also need one template. You’ll need to take a screenshot of the template, including its arrows, and print that as a separate page. Use the “snipping tool” in Windows, and leave the image “actual size” (don’t blow it up). My goal is to fit all templates on a single piece of paper for a matching game as a part of the art contest.

Do not print the .png images directly! Instead, I recommend importing your images into a single Google Docs file, one image per page, and printing that document. This allows you to resize your images more convenient and generally makes the resolution of the images less grainy.

AP Statistics Assignment for 10/10

We will be referring again to the Tale of Two Cities activity from last week. Period 1 can find your data here, and period 7 students can find your data here.

Finish reading Chapter 7 (Pages 192-197). Also, watch this video from APStatsGuy, which gives a great explanation for how r^2, the Coefficient of Determination, is interpreted.

From pages 208-209, do 24a, 33, 34, and 48

Also: read over the Chapter 7 Investigative Task. As with previous tasks, you’ll have the opportunity to ask questions about this in class tomorrow and will have the weekend to complete your response in a Google Document, to be shared with me once you are done. A heads up: your response will be due by 10:00am on Monday, October 14 (not Tuesday when we get back!)

Fractals & Chaos Recap for 10/9

We spent some more time in class working on practicing our Generalized Hausdorf Dimension formula on the second batch of fractals you got on Monday, then started in on a third batch. We’ll spend tomorrow working on our FractaSketch designs (remember that they are due on Friday!)

I would also like you to read Pollack’s Fractals, an article from Discover Magazine about the math underlying Jackson Pollack’s famous paintings.  For some interesting follow-up reading, check out this article from the New York Times about the use of fractal analysis to examine the authenticity of supposed Pollack paintings and this article from the Science Daily blog suggesting that such an analysis is not scientifically valid.