Tag Archives: Assignment

Fractals & Chaos Recap for 12/10

We saw yesterday that a sequence of geometric transformations can illustrate the same “fixed point attractor” behavior that we’ve seen when iterating a mathematical expression. We explained this by illustrating that such geometric transformations are analogous to iterating a linear function in the Complex number system. Specifically, adding two complex numbers is analogous to translating, and multiplying two complex numbers is analogous to a combined dilation/rotation.

The conventional rectangular form of a complex number a + bi tells us the horizontal and vertical components of the translation achieved by adding the 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.

We’ll dig in to this some more tomorrow, but for now please watch this video recapping polar vs rectangular coordinates (which the video refers to as “Cartesian coordinates”) and how to convert between the two.

Fractals & Chaos Recap for 11/22

We continued our discussion of the logistic map from yesterday by iterating the function for various values of the growth parameter a (which we identified as being bound between 0 and 4). We observed that if a is too low (a < 1), the population will die out and the destination of the orbit is zero. Once a passes 1, the population will eventually settle at some proportion of the maximum population; for example at a = 2.6, the orbit of iterations settles on a value of approximately 0.6154 (precisely, this is 8/13 of the possible maximum population). For a = 3.2, we observed a two-cycle of {0.5130,0.7995}, suggesting that the population here will year by year fluctuate between roughly 51% and 80% of its possible maximum population. For a = 3.6, we observed two bands of chaos bound within (0.32,0.6) and (0.79,0.9), suggesting that the population never dies out, but never settles at a stable value (or set of values).

This weekend, please continue to explore values of a and the destinations of orbits within this graph. Use the apps at MathInsight.org to help, as well as this Logistic Function Cobweb Diagram I made in Desmos. Keep track of your observations on the Catalog of Behavior you got in class. By Monday, you should at least have an observation for every tenth value of a between 2.2 and 4.0.

AP Statistics Assignment for 11/21

Your next Personal Progress Check has been assigned. Log into AP Classroom and complete the Unit 3 MCQ Part B PPC by Monday, November 25

On Monday, we will have our Unit 3 Test. We will spend Friday reviewing, and in preparation for that you should take a look at the review problems for Unit 3: pages 331-336, exercises 1, 3, 7, 9, 11, 15, 19, 20, 23, 29, 32, 40, 41. You’ll get answers and supplemental review tomorrow.

AP Statistics Assignment for 11/15

After swapping responses to the AP review problems, see the scoring guidelines here:

For each, read over the sample solution, then look at the scoring guidelines, scoring each section of the response as an E, P, or I. At the end, convert the E/P/I breakdown to a numerical score (out of 4). Give the scored document back to its owner.


If you missed the videos shown in class, you can watch them here and here. The notes sheet we started in class can be found here.

For homework this weekend, in addition to the PPC below, please read pages 305-313 (stop at “Experiments & Samples) and do exercises 1, 2, 3, 9, 13.

Reminder: Your next PPC (personal progress check) has been assigned: Unit 3 MCQ Part A. It will be due by the start of class on Monday, November 18. As with the previous PPC, there is a 25 minute timer. You are not required to work by the timer, and its expiration will not lock you out of the PPC. It is there only as a guide for how long I expect this PPC should take.

AP Statistics Assignment for 11/13 + PPC

From pages 302-303, do exercises 15, 16, 18, 25, 30, 32

Your next PPC (personal progress check) has been assigned: Unit 3 MCQ Part A. It will be due by the start of class on Monday, November 18. As with the previous PPC, there is a 25 minute timer. You are not required to work by the timer, and its expiration will not lock you out of the PPC. It is there only as a guide for how long I expect this PPC should take.

InCA Assignment for 10/31

We did some more work with Complex Fractions today. Work on the additional practice problems tonight; your first Homework Quiz will be in class tomorrow!

Today’s Files

Cumulative IXL Modules

AP Statistics Assignment for 10/31

Read chapter 10 (pages 267-276) for some more examples of running simulations, then from the exercises on page 277, do 11, 13, 17. For running your simulation in question 17, you should do at least 20 trials, using the random numbers found on Appendix page A-81 in your textbook or elsewhere, or use the random number generator in your calculator (refer to pages 273/274 for instructions).

Fractals & Chaos Lesson Recap for 10/30

We talked some more about hypercubes in class today, including drawing some pictures and building our own models out of gumdrops (take care of them!)

If you’re interested in reading more about the 4th dimension, check out the links below. I especially recommend the short story And He Built a Crooked House.

  • The Adventures of Fred, Bob, and Emily – a detailed look at how the lives of a 2D (Fred), 3D (Bob), and 4D (Emily) creature interact with each other. See especially the “World” section, where the author, Garrett Jones, imagines how wheels, water, and war would work in these universes.
  • And He Built a Crooked House – a short story by “Big Three” science fiction writer Robert A. Heinlein about an eccentric architect who designs a house in the shape of an unfolded hypercube. An earthquake hits, and the house folds back up on itself to concerning results (see also this student film version of the story)
  • Some Notes on the Fourth Dimension – some animations and movies showing the geometry of the fourth dimension, including some of those featured in the Flatland dvd bonus features.

Your homework: You were given a sheet of graph paper at the end of class. This is what you should do with it:

  • Fold it in half lengthwise (“hot dog” style).
  • Unfold and put a dot on the left edge of your crease.
  • Flip a coin (or use some other random procedure). If the coin lands heads, draw a diagonal that goes over and up one square. If the coin lands tails, draw a diagonal that goes over and down one square.
  • Continue with this pattern, creating a zig-zag across your paper, until you reach the other side. Bring that in tomorrow.

Intro to College Algebra Assignment for 10/30

We’ve moved on to the next phase of the rational expressions unit: Complex Fractions. Don’t be intimidated by their name! These are expressions that involve “nested fractions,” meaning the numerator and denominator of the expression are both fractions (or a sum/difference of fractions) themselves.

Remember the basic process for simplifying these beastly expressions:

  1. Combine fractions in the numerator (finding a lowest common denominator and so on) so you have one single rational expression
  2. Combine fractions in the denominator (LCD, etc.)
  3. Keep the top fraction, flip the bottom fraction, and change the operation to multiplication (KFC)
  4. Simplify.

Today’s Files

Cumulative IXL Modules

AP Statistics Assignment for 10/30

Reminder: For Thursday, work on multiple choice question 1-15 from the Unit 2 practice exam, found on pages 264-266 of your textbook. We’ll start things off on Thursday with a brief discussion of these answers. Note: Unlike other assignments from your textbook, this assignment is required and will be checked for completeness.

In addition, you received a playing card today and corresponding instructions about producing 200 coin flips. If the card was black, I’m asking you to actually flip a coin 200 times and record the results. If the card was red, I’m asking you to make up the result of 200 coin flips (not simulate them, totally make them up). Bring in those results tomorrow as well and I will attempt to sort your results based on how they were derived (either legitimate flips or made up ones).

AP Statistics Assignment for 10/29

You had you Unit 2 test in class today, and you’ll do the test redo in groups tomorrow.

For Thursday, work on the Unit 2 practice exam, found on pages 264-266 of your textbook, specifically the multiple choice questions 1-15. We’ll start things off on Thursday with a brief discussion of these answers.

Note: Unlike other assignments from your textbook, this assignment is required and will be checked for completeness.

Fractals & Chaos Recap for 10/28

Report your findings for our final box-counting project here.

After talking so much these past few weeks about fractional dimension—dimension values that fill in the gaps between 1, 2, and 3 dimensions, we turn our perspective in the other direction on the number line: towards the 4th dimension. It stands to reason that the trends we’ve observed could be extended past the 3rd dimension, but considering fractals, or even Euclidean shapes, is immensely challenging for us as 3-dimensional creatures.

What helps is to consider the perspective by analogy: we can better understand the fourth dimension by putting ourselves in the mindset of a 2-dimensional creature considering the third dimension. Fortunately, this is territory that has been well-covered.

In 1884, British teacher and theologian Edwin Abbott Abbott published Flatland: A Romance of Many Dimensions. It is told from the perspective of A. Square, a denizen of the titular 2-dimensional world, and starts off explaining aspects of their universe in great detail. The second part describes his first encounter with the 3-dimensional Spaceland, and the changes to his world-view as a result.

Abbott wrote the book partly as an exercise in geometry, but also partly as a satire on the regimented Victorian-era social hierarchy. As a result, there are some rather uncomfortable characterizations of women as lower-class citizens, among other shocking commentary.

You can read the full book here (I have paper copies if you’d prefer that), but for tomorrow please at least read this excerpt.

Intro to College Algebra Assignment for 10/28

Your first quiz of Unit 2, covering all of our work with Operations with Rational Expressions, is tomorrow, October 29.

Today’s Links

Extra Practice Links (IXL Review)

Intro to College Algebra Assignment for 10/24

We have a formal quiz on operations with rational expressions planned for Tuesday, October 29. We did some more review in class today.

Today’s Links

Extra Practice Links (IXL Review)