All posts by Mr. Kirk

I have been Mathematics Teacher at Ithaca High School since 2007. In that time, I have taught Algebra 1, Intermediate Algebra, Honors Algebra 2, Honors Pre Calculus, Advanced Placement Statistics, Math AIS, and a one-semester elective course in Fractal Geometry and Chaos Theory.

Fractals & Chaos Recap for 1/13

Download Fractal Zoomer.exe from here. Use the navigation guidelines posted yesterday to help you use the program.

Play around with the program, but do so with intent. I have a few things I want you to explore:

  • Take unusual values of C from the Complex Paint worksheet and verify the connected/disconnectness of the corresponding Julia Set
  • Explore: Where in the Mandelbrot Set can we find values of C that correspond to area versus string Julia Sets?
  • Explore: What is the relationship between the location of C in the Mandelbrot Set and the orbit pattern within the corresponding Julia Set?
  • Challenge: We saw earlier that orbits of quadratics can contain up to Nine 6-cycles. All nine can be found as attracting cycles in the Mandelbrot Set. Can you find all of them?

 

InCA Assignment for 1/13, plus Quiz Warning

My two periods are still not quite in sync with each other, so check for your period below. Both periods will have a quiz on the unit so far on Thursday, January 16

3rd Period

We finished the lesson started Friday about simplifying radical expressions

Today’s Links

Cumulative IXL Modules


8th Period

We spent some more time with “nth” roots (the general term for any radical: square root, cube root, fourth root, etc. We observed that evaluating an expression with a radical is similar to but subtly different from solving an equation with a power. See the notes for more details.

Today’s Files

Cumulative IXL Modules

Fractals & Chaos Recap for 1/10

We finished proving some of the observations made yesterday, and also discussed an argument that all Julia sets are either completely connected (any two points can be connected by a path along the Julia Set) or completely disconnected (every point is disconnected from every other point). This difference gives us a convenient way to categorize Julia Sets, allowing us to create a catalog view of them, much like we did for the Feigenbaum Plot in the real numbers.

What we need, then, is a convenient way of identifying whether or not a Julia Set is connected without having to actually draw it. Fortunately, iterations of points contained within the Julia Set give us a way to do that: if we can find seeds that do not diverge to infinity, then the Julia Set is connected. More specifically, we have argued that the origin, z = 0, is a convenient starting seed to iterate. If the orbit of z = 0 eventually tends to infinity, we know the Julia Set is disconnected. But how do we know the path of an orbit will actually tend to infinity and never return?

Fortunately, there’s a radius of no return. We further proved in class that if the path of an orbit of the origin exceeds r = 2the orbit will never come back. This means that any value of |c| > 2 results in a Julia Set that is automatically disconnected (since if z_0 = 0, then z_1 = 0^2 + c = c, and we’re already past 2), and for any value of |c| < 2, we just need to iterate until the orbit exceeds 2.

If we imagine the complex plane as a computer image, where every pixel corresponds to a single value of c, then we can colorize those pixels based on whether or not their orbit has “escaped”. As we increase the number of iterations, more and more pixels will become colorized (we can use the same colors for pixels that escape at the same number of iterations). Eventually, we will see a shape form. That shape is the Mandelbrot Set.

Download and investigate the Mandelbrot Set using the program Fractal Zoomer. Make sure you download Fractal Zoomer.exe (unfortunately, this software only functions on Windows-based computers). There are three modes in the basic view screen: Zoom Mode, Julia Mode, and Orbit Mode

  • In Zoom Mode:
    • Left Click = Zoom in
    • Right click = Zoom out
    • Ctrl+F3 = Set new center
  • Press J to activate Julia Mode
    • Left Click anywhere in the Mandelbrot Set to create the Julia Set at that value of C
    • Ctrl+F3 will allow you to spawn a Julia Set of a specific value of C (use this to investigate Julia Sets from the Complex Paint Worksheet)
  • Press O to activate Orbit Mode
    • Left click anywhere in the Mandelbrot Set and it will superimpose the orbit of seeds within the Julia Set that uses that value of C
    • Ctrl+F3 will allow you to specify the orbit of a particular value of C (or if you are in a Julia Set, of a particular seed).

InCA Assignment for 1/10

All InCA students will have a Homework Quiz on Monday of the following assignments (these assignments came out in different orders for different periods, but the numbering is the same):

  • HW 4.2 – Evaluating Rational Powers
  • HW 4.3 – Multiplying/Dividing Radicals
  • HW 4.4 – Simplifying Radical Expressions with Rational Roots

Review the homework keys and videos for the past few days to prepare!

3rd and 8th periods did different things today (you’ll be aligned again come Monday)

3rd Period

We returned to our work with fractional exponents and used them to simplify radical expressions

Today’s Links

Cumulative IXL Modules


8th Period

Rational exponents are equivalent to radicals, therefore the shortcuts that we can use working with exponents extend to radicals as well. Today we discussed multiplying and dividing radical expressions.

Today’s Links

Cumulative IXL Modules

Fractals & Chaos Recap for 1/9

We spent some time reviewing the results from Complex Paint Worksheet 2. We have observed a few things, the most obvious of which are that all Julia Sets have 180-degree rotational symmetry and the only Julia Sets that have x- and y-axis symmetry have a parameter that is a real number. It turns out that both of these observations are provable facts about Julia Sets, which we proceeded to start, and will finish tomorrow.

InCA Assignment for 1/9

3rd and 8th periods did different things today:

3rd period students

We spent some time noticing some things about the powers reference list you’ve created. Many powers evaluate to the same thing (2^6, 4^3, and 8^2 for example, as do 3^4 and 9^2) and you were challenged to come up with why. After that, we played a competitive game where you tried to find as many different ways of expressing the number 256 using multiplication and exponents.

Your homework tonight is to repeat the game with the number 32. A reminder of the rules:

  • The only operations allowed are exponents and multiplication
  • Each term of your expression must have an exponent
  • You may have at most one exponent = 1

8th period students

We returned to our work with fractional exponents and used them to simplify radical expressions

Today’s Links

Cumulative IXL Modules

InCA Assignment for 1/8

Rational exponents are equivalent to radicals, therefore the shortcuts that we can use working with exponents extend to radicals as well. Today we discussed multiplying and dividing radical expressions.

A reminder: because I want you to be able to work through this chapter without relying on the calculator, we will have a daily Powers Quiz. I will choose 10 powers from the reference table you made and ask you to evaluate them without using the calculator in a brief, timed quiz. We will do this every day through the unit, and only  your highest grade on any one quiz will be the one used in your Unit 4 grade.

Today’s Links

Cumulative IXL Modules

Fractals & Chaos Recap for 1/7

We discussed the results from iterating yesterday’s seeds in the function z², finding that some seeds will attract to zero, some will spiral off to infinity, but others seem trapped in a unit circle around the origin, falling into a cycle or landing on (1,0). This unit circle forms a boundary between seeds that diverge and seeds that do not, and that boundary is called a Julia Set, named after French mathematician Gaston Julia, and developed by Julia and Pierre Fatou (Fatou names the complement of the Julia Set; in the case of C = 0, the Fatou set is the entirety of the rest of the complex plane except for the unit circle).

Not all Julia Sets (in fact pretty much none of them) look as simple as this, and Complex Paint is a great tool to help us see them. Watch this video for instructions on the derivation of Julia Sets, plus instructions on how to use Complex Paint to create and understand this new class of mathematical object. Feel free to explore whatever parameter values you would like, but in particular we will be using the ones found on Complex Paint Worksheet 2. We started this today and will finish it tomorrow. For each, note the type of Julia Set you get (area, string, or dust), any symmetry you observe, and the destination of orbits inside any area Julia Sets.

InCA Assignment for 1/7

More work with rational exponents today, including evaluating expressions that include them.

A reminder: because I want you to be able to work through this chapter without relying on the calculator, we will have a daily Powers Quiz. I will choose 10 powers from the reference table you made and ask you to evaluate them without using the calculator in a brief, timed quiz. We will do this every day through the unit, and only  your highest grade on any one quiz will be the one used in your Unit 4 grade.

Today’s Links

Cumulative IXL Modules

Fractals & Chaos Recap for 1/6

We started today with a brief recap of the work we’ve done with complex linear functions using the back of the Complex Paint worksheet we’ve been working off of.

From there, we started work with complex quadratic functions. We will be replicating the process we did with real numbers: analyzing a family of quadratics where we only adjust a single parameter value and investigate the behavior of iterations for specific values of the parameter. Eventually, we will find a way to categorize and catalog these behaviors.

The family of functions we will be analyzing is  + C, and the first parameter we will look at is C = 0. We found that squaring a complex number in polar form resulted simply in squaring the value of R and doubling the value of θ, so an initial seed like [2,10°] becomes [4,20°], then [16,40°], then [256,80°] and so on, for larger and larger values of R, suggesting that the seed [2,10°] diverges to infinity under iterations of z².

If you weren’t in class, pick two of the seeds below to iterate. Iterate until you’re convinced what the long-term destination might be (diverging? converging? limit cycle?). Note also that θ should never exceed 360°. If the pattern for [2,10°] from above were continued, we would get 160°, then 320°, then 540°. But 540° is larger than 360°, and is equivalent to 540-360=280°. And that’s the angle we would record.

  • [1,45°]
  • [1/3,60°]
  • [1,30°]
  • [5,180°]
  • [1,7°]
  • [1,120°]
  • [1/2,36°]
  • [2,90°]
  • [1,10°]

InCA Assignment for 1/6

We’ve started a new unit, this one on rules of exponents. Today was a review of pre-existing shortcuts, along with an introduction of some new ones.

Finally, because I want you to be able to work through this chapter without relying on the calculator, we will have a daily Powers Quiz. I will choose 10 powers from the powers reference sheet you make in today’s homework. I will ask you to evaluate them without using the calculator in a brief, timed quiz. We will do this every day through the rest of the unit, and only your highest grade on any one quiz will be the one used in your Unit 5 grade.

Today’s Links

Cumulative IXL Modules

What’s Going On in This Graph? (January 8th)

Visit this page on the New York Times and look at the graph showing the reading level and time (in minutes) required to read internet privacy policies for 150 popular websites and apps plus a few books. The graph was taken from elsewhere on the New York Times’s website.

As you look at the graph, consider the following three questions

  • What do you notice? If you make a claim, tell us what you noticed that supports your claim.

  • What do you wonder? What are you curious about that comes from what you notice in the graph?

  • What’s going on in this graph? Write a catchy headline that captures the graph’s main idea.

Write your thoughts as a comment on the New York Times’s page. Respond to others’ comments if you wish. Visit again on Wednesday for a moderated discussion by representatives from the American Statistical Association, who will reply to as many comments as they can.

By Friday, the NYT will link to the article the graph is from and provide additional information and context. Check how your headline compared!