We continued our re-introduction to Parabolas and Quadratic Equations today by playing a game similar to the classic game Guess Who, where you had to ask a series of questions to guess which of 25 parabolas your randomly matched partner selected. This taught us the value of having clearly defined vocabulary to describe the key features of such shapes, as several questions that you asked each other led to confusion!
For tonight’s homework, please complete the Parabola Polygraph Debrief reflection. You will also need to bring in your Chromebook once more tomorrow for the third and final introductory activity to this unit.
Today in class, we did a (hopefully fun!) activity introducing our next unit on Parabolas and Quadratic Equations. Your homework tonight is to reflect on two questions:
- Describe one thing that you learned from the Will It Hit the Hoop Desmos activity from class.
- What else do you think you could use a parabola to model the shape/path of?
Write your reflection on a piece of paper or record them in Socrative (room code G102KIRK)
Bring your Chromebook in tomorrow again for Part 2 of this introduction!
With Mr. Drix, you observed some patterns that emerge in cobweb diagrams for certain types of fixed points. For an attractor, the pattern of steps in the cobweb diagram will be drawn towards the fixed point; for a repeller they move away. Furthermore, if the fixed point is direct, the pattern of steps will look like actual steps as they move towards or away from the fixed point, trapped between the two lines defined by the function we are iterating and y = x. If the fixed point is alternating, on the other hand, the pattern will spiral around the fixed point with each step, producing a picture that most clearly gives the cobweb diagram its name.
From there, we took our first look at non-linear functions, realizing quickly that in addition to having more than one fixed point, such functions can have different types of fixed points. One could be a direct repeller, while another an alternating attractor. The cobweb diagram remains our best way to observe these differences, and we used the Geogebra-based applications found here to create them.
We looked at 4a (y = x^2) together, classifying the fixed point at 1 as a direct repeller and the fixed point at 0 as a direct attractor. We furthermore observed that -1 is a “pre-image” to the fixed point at 1, and therefore we have different behavior for different seeds. Your homework tonight is to continue and look at 4b-4f, using the cobweb diagram app linked above and the Iterated Functions Supplement to keep track of your results.
We discussed the reading from Fractals: The Patterns of Chaos and looked at some references to the “Butterfly Effect” in popular culture. We spent most of the rest of the period playing with the Solar System simulator online.
For Monday, please read pages 49-54 from your book, the section on “The Fractals and Chaos of Outer Space.”
We discussed some key terms that we observed in the first part of the Nova documentary (specifically: fixed point, attractor vs repeller, and strange attractor) and we modeled some of these terms in action with some magnet pendulums. At the close of the period, we also looked briefly at a Solar System simulator found here.
For tomorrow, please read the three sections mentioned on yesterday’s post from Fractals: The Patterns of Chaos
We put up the random zig-zags that you created last night and noticed that while the pattern tendered to wander away from the central line, it would eventually trend back as well. The pattern of central line crosses came in “bursts,” and the frequency of these bursts demonstrated fractal-like, self-similar qualities.
We returned to the classroom to examine another version of this, a Chaos Game app designed by Fractals & Chaos alum Istvan Burbank. We saw the impact that different constraints on random movement had on the pattern that movement created, including one very surprising result. We closed out the period investigating and playing with this app some more.
For Monday, read the Discover Magazine article Chaos Hits Wall Street. While this is an old article (1993!) it addresses a lot of the topics we’ll be discussing in the second part of the course. We’ll see some updated takes on the theories presented in the next articles.