AP Statistics Assignment for Week of 12/10

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You will spend this week working on your projects. You should strive to be done with data collection by Thursday the 13th, and you’ll have up to Monday the 17th to work on your presentations and written report. Refer to the contents of the Mid-year Project Folder for guidelines on what your presentation and report should look like.

We’ll spend the 18th-21st presenting on your findings, and your reports (one per group!) will be due on the 21st.

Fractals & Chaos Lesson Recap for 12/14

We did some more work with Complex Paint today, in the hopes of “solving” Az+B. To make this easier, we derived the “Polar-Linear” form of this formula, where instead of referring to A in rectangular coordinates, we refer to it in a modified form of polar coordinates, where we use R and T, the fraction of a full turn that the angle of polar form refers to. This allowed us to make the following observations:

  • B has no impact on the type of fixed point we get, just its location
  • If R < 1, the fixed point is an attractor
  • If R = 1, the fixed point is neutral
  • If R > 1, the fixed point is a repeller
  • The denominator of T (in lowest terms) tells us how many spokes we get in a pattern.

The only thing we haven’t figured out yet is where spirals come from, and why spirals turn into spokes when we use values of R closer to 1. We’ll explore that on Monday.

Intermediate Algebra Assignment for 12/14

Your Unit 3 test is on Tuesday, December 18. We started our two days of review by making a study guide, which you should finish for homework if you did not.

Today’s Files

Cumulative IXL Modules

Fractals & Chaos Recap for 12/13

We then introduced our next Very Important Program: Complex Paint. This software was created by former F&C students Devon Loehr and Connor Simpson, updating software originally available only on MacOS9, and will be invaluable to us going forward. We’ll discuss the features of the program as they become relevant, but after downloading/unzipping the folder listed in the Google Drive folder linked, read the README file for help on how to use the software.

For now, we are using “Linear” mode, and working off of Complex Paint Worksheet 1. We’ve observed that in the linear complex expression Az+B, the value of A can affect the type of fixed point we get. For example, the angle of A written in polar form seems to govern the number of spokes we get, and the attracting/repelling/neutral nature of the fixed point is tied to the value of r. The value of B seems to just change its location. We’ve also noticed some unusual differences in the patterns we see: sometimes the path of attraction towards a fixed point forms straight-line spokes, but sometimes it forms spirals instead. Odd…

We’ll have to look at this some more going forward!

Intermediate Algebra Assignment for 12/13, plus Test Warning

The last bit of new material for this unit takes us back to the connection between the roots/solutions of a quadratic and the x-intercepts of its graph. This relationship exists even with polynomials of higher degrees, which means we can use that relationship to factor using its roots! We introduced two tasks today:

  • Using the known roots of a polynomial to write the polynomial, and
  • Using the graphing calculator to find the zeroes, and by extension the factors, of the polynomial.

These are the last two new items, in addition to everything else you’ve learned this unit, that will appear on your Unit 3 test on Tuesday, December 18.

Today’s Files

Cumulative IXL Modules

Intermediate Algebra Assignment for 12/12

We reviewed the key features of parabolas today, but this time from an algebraic perspective. We’ve already seen that the roots of the parabola (aka the x-intercepts or the zeroes) are the solutions we get from the quadratic equation. We saw today that the equation for the Axis of Symmetry can be obtained from the standard form of a quadratic equation using the formula x = –b/(2a). And since the vertex is on the axis of symmetry, the x-coordinate of the vertex is that same value (and the y-value is obtained by plugging that x-value into the equation).

Today’s Files

Cumulative IXL Modules

Fractals & Chaos Recap for 12/11

We spent some time reviewing answers to the Complex Numbers Transformations sheet from yesterday, and used the principles to find the precise value of the fixed point you were finding graphically last week with the transformations Dilation x1/2, Rotation 45°, Translation Up 4.

We then introduced our next Very Important Program: Complex Paint. This software was created by former F&C students Devon Loehr and Connor Simpson, and will be invaluable to us going forward. We’ll discuss the features of the program as they become relevant, but after downloading/unzipping the folder listed in the Google Drive folder linked, read the README file for help on how to use the software.

Your homework tonight is to finish the back of the Complex Numbers Transformations sheet.

Fractals & Chaos Lesson Recap for 12/10

We reviewed that the values of a and b of the Rectangular Form tell us the horizontal and vertical components, respectively, of the translation achieved by adding a complex number, but this form is not useful for when the number is being used as a multiplication factor. For that, we need the number’s Polar Form. Once finding it, the value of r tells us the dilation factor and the value of θ tell us the rotation factor.

To better understand this requires practice, and so we spend a considerable amount of time in class today converting between rectangular and polar forms for a single complex number and identifying the transformations they refer to. This work was on the Complex Numbers Transformations sheet, and your homework is to finish the front as well as questions 3, 4, and 5 on the back.

Intermediate Algebra Assignment for 12/10

We took a moment today to pause and look back on the three methods of solving quadratic equations that we have discussed: factoring, the quadratic formula, and the square root method. We discussed when each method might be most efficient, but reinforced that point that the quadratic formula is always a valid solution method.

Today’s Files

Cumulative IXL Modules

Fractals & Chaos Lesson Recap for 12/7

We closed out our work with the Feigenbaum Plot today and took a moment to reflect. With our work in the real numbers, we started with linear functions (mx+b) and quickly discovered nearly everything there was to find. Linear functions are easy and non-chaotic. When we moved to non-linear functions, things started getting interesting. We looked at a few examples of the form x^2 + c, but did a deep dive with the logistic map of ax(1-x). For a single parameter, we could make a cobweb diagram to show the behavior of that specific function, but the really interesting things happened when we made our catalog of behavior for all parameters, creating the Feigenbaum Plot.

We will follow the same path through the forest of Complex Numbers. We will start with linear functions of the form Az+B, and understand what we can find there. We’ll then move on to non-linear functions, specifically of the form z^2 + C, and look at the behavior of single, specific values of C. Eventually, we will move to a catalog view there and see what we find.

Today was our first step towards that goal, with a discussion of how arithmetic on complex numbers can mimic geometric transformations. Complex numbers have two coordinates, a real part and an imaginary part, so their operations provide a convenient way of movie around the coordinate plane. Specifically, adding complex numbers produces translations and multiplying complex numbers produces a dilation/rotation. Exactly how the dilation/rotation is understood requires another way of referring to these complex numbers: Polar Form. We will discuss this on Monday.

Intermediate Algebra Assignment for 12/7

The third method of solving quadratics is to simply use the square root, but there’s a danger here: you still have two solutions. The + is built into the quadratic formula, but it still needs to be added here. And that falls to you, the solver!

Today’s lesson also included a brief revisit to word problems, this time resulting in a quadratic equation.

Today’s Files

Cumulative IXL Modules

Fractals & Chaos Recap for 12/6

You dived in deep to one of the questions we asked in our exploration of the Feigenbaum Plot: how many cycles of each type are there? It turns out that for every cycle born out of chaos, there is an “evil twin” repelling cycle born as well. As a result, there are actually two 3-cycles. Moreover, not all cycles are necessarily bound by real numbers, instead moving to the realm of complex numbers. This results in many more 4-, 5-, 6-cycles and beyond.

Furthermore, these “evil twin” repelling cycles serve a purpose. As the members of the attracting cycle bifurcate, their patterns branch out further and further. When this pattern of branching reaches one of the values of the repelling cycle – indeed it will reach all members of the repelling cycle simultaneously – the branching stops and the system dissolves into chaos once more.

Intermediate Algebra Assignment for 12/6

You had your solving quadratic equations quiz in class today. I want you to do some more for homework. Tonight, work on IXL Module J.9 Solve a quadratic equation using the quadratic formula to a score of 80 or higher. Remember: any score above 80 will count as extra credit!

Cumulative IXL Modules

Fractals & Chaos Recap for 11/5

We started in class today watching a Numberphile video about the Feigenbaum Plot and an interesting number that can be found in it that appears to have some surprising universality. The video does a great job of recapping what we’ve done over the past few days, so watch it if you’ve missed anything. You were also given an article to read about the plot and this constant.

The video also makes a point that the pattern we see in the Feigenbaum Plot is not unique to the Logistic function we’ve been iterating. In fact, any function that creates a bound area with the x-axis can exhibit such behavior. With the rest of the period, play around with Paul Fischer-York’s Bifurcation Diagram. Use the dropdown in the upper-right corner to examine diagrams for other functions. Some other functionality:

  • Use the Darkness slider to make the image darker and easier to see.
  • Left-click and drag a rectangle around any portion of the diagram to zoom in on that portion.
  • Right-click anywhere in the diagram to run a series of iterations at that value of a along the x-axis. The software will identify the magnitude of the cycle you’re looking at (though sometimes it makes a mistake, so look at the list as well!)

Use this map to explore the Sharkovskii Ordering mentioned in the article above. Why does that ordering make sense given this picture?

Intermediate Algebra Assignment for 12/5

You will have a quiz tomorrow on solving quadratic equations. Today you were given a review sheet (see below) with some more practice. Work on this review and the IXL modules linked below in preparation for tomorrow’s quiz!

Today’s Files

Cumulative IXL Modules

Fractals and Chaos Recap for 12/4

We explored cycles a bit more in class today, noting that if {p,q} are the two values of a 2-cycle, then by definition (p) = q and (q) = p. Put another way, ((p)) = p and ((q)) = q. What’s important to note, though, is that the reverse is not true. If ((n)) = n, then n could be a member of a two cycle (n,m,n,m,n,m,…or it could just be a fixed point: (n,n,n,n,n,n,…)

There were two important takeaways from this algebraic definition of a 2-cycle. The first is that the local slope (Instantaneous Rate of Change) to all points of a 2-cycle, or indeed any cycle, is always the same. As a result, all points of a cycle will pass an IRC of -1 simultaneously, meaning all points of a cycle bifurcate simultaneously. This served as our proof of Question 2 posed early in our exploration of this pattern.

The second takeaway is graphing the function y = ((x)) can be a way of finding new cycles. Fixed points on the graph of y((x)) that are not common with the graph of y = (x) will be the parameters of our 2-cycle. By extension, graphing y = f (((x))), or any number of nested iterations, will give us a tool of finding new cycles. More crucially, this also shows us why cycles spontaneously appear from chaos. Explore the graph here. For a = 3.84, the “wiggliness” of the graph of y = f (((x))) is enough for the fingers of the graph to touch the line y = x. But for a < 3.84, it isn’t. The moment that a becomes large enough for those fingers to touch the line y = x is the moment that a 3-cycle is born!

Intermediate Algebra Assignment for 12/5

We did some more work with the quadratic formula today, specifically with simplifying the radical expressions we sometimes get when we apply it.

Today’s Files

Cumulative IXL Modules

AP Statistics Assignment for Week of 12/4

For the remainder of this week, you will be watching the movie Experimenter, an independent biographical film about the life and career of experimental psychologist Dr. Stanley Milgram, who conceived of an infamous Obedience Experiment in the wake of the Nuremberg War Criminal Trials after WWII.

On Monday of next week, you’ll be asked to use the movie and the articles below to inform a conversation about experimental ethics. You will be assigned a side at the start of the movie, defending one of these two statements:

  • For some experiments, it is okay to lie to subjects in the name of proper experimental design and to preserve the integrity of what is being tested.
  • It is never okay to lie to subjects in an experiment, regardless of what is being tested.

The discussion on Monday will be highly structured. Please read over the Ethics Discussion Guidelines for details. Also, by Monday, read over the following three articles for some more examples that may be useful for your stance.

Fractals & Chaos Lesson Recap for 12/3

We finally found our first 3-cycle, at a value of a = 3.83. Oddly, this was after chaos emerged, suggesting that sometimes, inexplicably, chaos can turn into order. This of course raised a slew of new questions:

  1. We also found a 5-cycle at a = 3.74 and a = 3.906. Is there a 3-cycle before a = 3.74? Or does the 3-cycle necessarily come after the 5-cycle?
  2. For that matter, why do some cycles happen more than once? A 6-cycle can be found at a = 3.63 and 3.845. A 4-cycle can be found at a = 3.5 and 3.96. Where are these coming from?
  3. What even is the order of cycles, anyway? It seems that powers of 2 come first, born out of bifurcations from our original pattern of fixed points.
  4. Finally, where do the cycles that emerge from chaos come from? Why do they emerge? How are they “born”?

To make exploring these equations easier, I’ve created a Logistic Orbit Iterator on Google Sheets. Make a copy of it and save it to your drive, and you can edit the value of a and the seeds to get a picture of the destinations of the orbit (or scroll down to the bottom and see how it behaves after 500 iterations).

Tomorrow, we’ll finally see the full picture of what we’ve been discussing.