About Benjamin Kirk

I am a New York State Master Teacher in Mathematics at Ithaca High School. I have been teaching at Ithaca High School since 2007.

Intermediate Algebra Assignment for 1/28

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 Files

Cumulative IXL Modules

    AP Statistics Assignment for 1/18, plus Test Warning

    The applet for today’s activity can be found here. Using this simulator, find:

    • Two very different values of p and n that create a distribution that you would confidently identify as “normal”
    • Two very different values of p and n that create a distribution that you would confidently identify as “not normal”
    • At least one value of p and n that creates a distribution that might be normal, but you’re on the fence about.

    Report your findings here

    For homework this weekend/next week, please finish reading Chapter 16, pages 423-429, then work on exercises 32, 33, and 34 from page 432.

    Out Unit 4 Test will be on Wednesday, January 30. We’ll spend Monday and Tuesday after we get back from midterm exams reviewing the unit. Over next week, I recommend you work on the following practice problems from the Unit 4 review, pages 434-439: 5, 16, 20, 21, 23, 25, 27, 28, 37.

    AP Statistics Assignment for 1/17

    First, some more about a big topic that came up at the end of class: The Difference Between an Assumption and a Condition

    In order for a model (e.g., a normal model or binomial probability model) to be valid for a scenario, we must be able to make certain assumptions about that scenario.  These assumptions can include the results of individual trials being independent from each other, the distribution of results being sufficiently unimodal and symmetric, etc.

    If it is appropriate to make these assumptions, then go ahead and do it. But if it is not appropriate to make these assumptions, you can still proceed provided certain conditions are satisfied.  These conditions mean that the scenario is “close enough” to allow the model to be valid.

    For example, coin flips are independent.  There is no finite population of coin flips that you are “drawing” a sample from, and so each coin flip’s outcome is independent of the next.  Drawing cards from a deck are not independent, as the deck is finite and the probability of a certain outcome changes with each card that is removed. However, if the population is large enough, or more specifically if the sample is small enough in comparison to the population, then that probability change is very small, small enough to be ignored.

    In general, as long as the sample size is less than 10% of the overall population, the probability change isn’t big enough to be worrisome. The reason why the magic number is 10% has to do with something called the Finite Population Correction Factor, and a thorough description of where it comes from and how it affects probabilities can be found here.

    Your homework tonight

    From pages 431-432, please do 19, 21, 26, and 28.

    Intermediate Algebra Assignment for 1/17

    Your Unit 4 test is tomorrow. To prepare, consider the following:

    Fractals & Chaos Recap for 1/16

    We discussed an open problem related to the connectedness of the Mandelbrot Set today. It seems silly, but “connected” is actually a tough idea to define mathematically. We discussed the following ideas:

    • Not disconnected: That is, the set is not divided into pieces. We can’t draw a line “between” pieces of the set without crossing the set. The Mandelbrot set is not disconnected
    • “Simply” connected: Both the set and its complement (everything not a member of the set) are not disconnected. A circle is not simply connected, because it divides the plane into regions inside and outside of the circle, which are disconnected from each other by the circle. A disk — a circle that includes its interior — is simply connected, as anything not in the disk is outside the disk. The Mandelbrot Set is simply connected
    • “Path” connected: Any two points of the set can be joined by a path that stay in the set. We used the Topologist’s Sine Curve to define an example that is not “path” connected. The Mandelbrot Set, though, is path connected.
    • “Locally” connected: Around any point of the set, we can draw a circle small enough such that the set is not connected inside of it. We used the idea of a comb defined by the line segments y = 1/2^x from x: [0,1] as an example of a space not locally connected. Mathematicians are still unsure whether the Mandelbrot is Locally Connected or not, an open problem known as the MLC.

    The last topic, which we will get into tomorrow, is a brief history lesson of the Julia Set that starts in Newton’s Method for Approximation.

    For tomorrow, please read this article from the November, 1991 issue of Science News, published just after an important discovery was made about the border of the Mandelbrot Set, as well as this follow-up of sorts from 2017 in Scientific American

    Intermediate Algebra Assignment for 1/16

    Our test is on Friday, January 18th and today we started reviewing.

    Today’s Files

    Cumulative IXL Modules

    Fractals & Chaos Recap for 1/15

    We wrapped up yesterday’s game of “Find the Julia Set” and reviewed some other interesting properties of the Mandelbrot Set:

    • “Area” Julia Sets come from “areas” in the Mandelbrot Set, and “string” Julia Sets come from “strings” in the Mandelbrot Set
    • There are “veins” through the main cardioid of the Mandelbrot Set where we can watch the orbit of a single fixed point die towards a 2-, 3-, 4-, or any magnitude cycle. Along these veins, the fixed point forms spokes. Offset slightly from these veins and they become spirals
    • Relatedly, there are balls off of the main cardioid for every possible cycle, and every variant of every cycle (1/5-cycles, 2/5-cycles, etc.). Imagine a circle centered on the cusp of the cardioid. Any angle rotation around that circle points straight at a ball on the edge of the cardioid, whose orbit-cycle magnitude exactly matches the fraction used to find that angle. A 120 degree rotation, representing 1/3 of a full rotation, points straight at the 3-cycle ball on the top of the Mandelbrot set, and a 240 degree rotation, 2/3 a full rotation, points straight at the 3-cycle ball on the bottom. A 72-degree rotation, 1/5 of a full, points straight at the first 5-ball. Rotations of 144, 216, or 288 degrees point straight at the second, third, and fourth 5-balls.

    Our last two topics of the course: an open problem related to the connectedness of the Mandelbrot Set, and a discussion of Newton’s Method of Approximation, and its surprising connection to the Mandelbrot Set

    Intermediate Algebra Assignment for 1/15

    We revisited solving quadratic equations today, now with the capability of evaluating negative radicands!

    Today’s Files

    Cumulative IXL Modules

    Fractals & Chaos Recap for 1/14

    The central question I have asked of you is In what way are the shape of the Julia Set, the size of the cycle with it, and its location in the Mandelbrot Set related to each other? Today, we started to answer this question.

    First, we observed that orbits along the horizontal x-axis of the Mandelbrot Set bifurcate in the same way that they do in the Feigenbaum plot, and for good reason. Furthermore, we noticed that elsewhere along the edge of the central cardioid of the Mandelbrot Set, we can find balls that yield cycles of any magnitude, and in general the size of the ball is inversely related to the magnitude of its cycle (bigger ball = smaller cycle).

    When we look at the Julia Sets we get in each of these balls, we realize that each of them have a central area, centered at the origin, and each with an infinity of “pinch points” found along its edge. The number of areas pinched together at this point exactly matches the size of the orbit contained! Given a mystery Julia Set, then, we can determine the size of its contained orbit just by its shape!

    Finally, we observed that every ball connected to the central cardioid has an infinity of balls attached to it, also of decreasing size. The fractal nature of the Mandelbrot Set manifests itself here: the largest sub-ball corresponds to a bifurcation, the next largest to trifurcations, and so on. Thus, the 2-cycle found in the San Marco Julia Set can bifurcate into a 4-cycle, trifurcate into a 6-cycle, or any other multiple of two. The 3-cycle found in the Douady Rabbit can bifurcate to a 6-cycle, trifurcate to a 9-cycle, and so on for any multiple of 3. And this pattern extends to the Julia Sets themselves: the 3-ball off of the 2-ball creates a Julia Set that is a San Marco of Douady Rabbits:

    3 off 2

    San Marco made up of Douady Rabbits — The 2-cycle of the San Marco trifurcates to create a 6-cycle

    And the 2-ball off of the 3-ball creates a Julia Set that is a Douady Rabbit of San Marcos:

    2 off 3

    Douady Rabbit made up of San Marcos — The 3-cycle of the Douady Rabbit bifurcates to create a different 6-cycle

    With this in mind, we can play the game: Find the Julia Set. For each Julia Set in the attached file, use the shape/structure of the Julia Set to predict where in the Mandelbrot Set it can be found, and as a result the magnitude of the cycle found within it. Then verify your predictions using Fractal Zoomer.


    Intermediate Algebra Assignment for 1/14, plus Test Warning

    As we continue to explore the use of i to represent the square root of -1, and the consequences that such a definition involves, we revisited the topic of rationalizing today, and noted that since i is a square root, the rules of Simplest Radical Form require that we not leave i in the denominator of a fraction.

    Our test on Unit 4 will be on Friday, January 18th. Please do not be absent on that day! If you know for some reason that you will be absent on that day, please arrange a time before the 18th to take the test! Otherwise, we will have to schedule a time for you to take the test during exam week.

    Today’s Files

    Cumulative IXL Modules

    Fractals & Chaos Recap for 1/11

    We spent the period playing around with Fractal Zoomer, and have made a few observations:

    • The central “bulb” of the Mandelbrot Set is a cardioid, and each other bulb off of that central body is a perfect circle.
    • Each bulb has its own cycle of orbits. Some bulbs have the same size orbit as others, but with a different “order” (1,2,3,4,5,6 vs 1,3,5,2,4,6 for example)
    • If you tile the plane with Julia Sets of a sufficient density, the collection of Julia Sets make a photo-mosaic of the Mandelbrot Set. This further reinforces that there is something important between the location of C within the Mandelbrot Set and the shape of the corresponding Julia Set.
    • Along the central “spike” on the left side of the Mandelbrot Set, we can find baby Mandelbrot Sets strung along that string. The main body of these baby Mandelbrots show cycles instead of fixed points. Realizing that the horizontal axis on which the Mandelbrot set is centered is the real number axis, this suggests that there may be a meaningful connection between the Mandelbrot Set and Feigenbaum Plot…

    Keep working with Fractal Zoomer and considering the questions posted yesterday. Please also read the section on Mandelbrot Set from pages 74-81 in your copy of Fractals: The Patterns of Chaos (and bring that book back on Monday).

    Intermediate Algebra Assignment for 1/11

    More work with imaginary numbers today, in particular simplifying expressions that involve multiplying them or with higher powers. We saw in the videos we watched yesterday that powers of i don’t increase like other real numbers, but instead cycle through the numbers 1, i, -1, -i, and then back to 1 again.

    Today’s Files

    Cumulative IXL Modules